Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 230120 18 pageshttpdxdoiorg1011552013230120

Research ArticleDynamic Characters of Stiffened Composite Conoidal ShellRoofs with Cutouts Design Aids and Selection Guidelines

Sarmila Sahoo

Department of Civil Engineering Heritage Institute of Technology Kolkata 700107 India

Correspondence should be addressed to Sarmila Sahoo sarmila juyahoocom

Received 11 December 2012 Revised 3 May 2013 Accepted 14 May 2013

Academic Editor Jun Li

Copyright copy 2013 Sarmila SahooThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Dynamic characteristics of stiffened composite conoidal shells with cutout are analyzed in terms of the natural frequency andmodeshapes A finite element code is developed for the purpose by combining an eight-noded curved shell element with a three-nodedcurved beam element The code is validated by solving benchmark problems available in the literature and comparing the resultsThe size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints of cross-ply andangle-ply laminated composite conoidsThe effects of these parametric variations on the fundamental frequencies andmode shapesare considered in details The results furnished here may be readily used by practicing engineers dealing with stiffened compositeconoids with cutouts central or eccentric

1 Introduction

Laminated composite structures are gainingwide importancein various fields of aerospace and civil engineering Shell roofstructures can be conveniently built with compositematerialsthat have many attributes besides high specific strength andstiffness Among the different shell panels which are com-monly used as roofing units in civil engineering practice theconoidal shell has a special position due to a number ofadvantages it offers Conoidal shells are often used to coverlarge column-free areas Being ruled surfaces they provideease of casting and also allow north light in Hence this shellis preferred inmany places particularly inmedical chemicaland food processing industries where entry of north lightis desirable Application of conoids in these industries oftennecessitates cutouts for the passage of light service lines andalso sometimes for alteration of resonant frequency Inpractice the margin of the cutouts must be stiffened to takeaccount of stress concentration effects An in-depth studyincluding bending buckling vibration and impact isrequired to exploit the possibilities of these curved formsThe present investigation is however restricted only to thefree vibration behaviour A generalized formulation for thedoubly curved laminated composite shell has been presentedusing the eight-noded curved quadratic isoparametric finite

element including three radii of curvature Some of theimportant contributions on the investigation of conoidalshells are briefly reviewed here

The research on conoidal shell started about four decadesago In 1964 Hadid [1] analysed static characteristics ofconoidal shells using the variational method The researchwas carried forward and improved by researchers like Brebbiaand Hadid [2] Choi [3] Ghosh and Bandyopadhyay [4 5]Dey et al [6] and Das and Bandyopadhyay [7] Dey et al[6] provided a significant contribution on static analysis ofconoidal shell Chakravorty et al [8] applied the finite ele-ment technique to explore the free vibration characteristics ofshallow isotropic conoids and also observed the effects ofexcluding some of the inertia terms from the mass matrixon the first four natural frequencies Chakravorty et al [9ndash11]published a series of papers where they reported on free andforced vibration characteristics of graphite-epoxy compositeconoidal shells with regular boundary conditions LaterNayak and Bandyopadhyay [12ndash15] reported free vibrationof stiffened isotropic and composite conoidal shells Das andChakravorty [16 17] considered bending and free vibrationcharacteristics of unpunctured and unstiffened compositeconoids Hota and Chakravorty [18] studied isotropic punc-tured conoidal shells with complicated boundary conditionsalong the four edges but no such study about composite

2 Journal of Engineering

conoidal shells is available in the literature Also they did notfurnish any information on vibration mode shapes It is alsoseen from the recent reviews [19 20] that dynamic character-istics of stiffened conoidal shells with cutout are still missingin the literature The present study thus focuses on the freevibrations of graphite-epoxy laminated composite stiffenedconoids with cutout both in terms of the natural frequenciesandmode shapesThe results so obtainedmay be readily usedby practicing engineers dealing with stiffened compositeconoids with cutouts The novelty of the present study lies inthe consideration of vibration mode shapes of stiffened com-posite conoids in presence of cutouts

2 Mathematical Formulation

A laminated composite conoidal shell of uniform thicknessℎ (Figure 1) and radius of curvature 119877119910 and radius of crosscurvature 119877119909119910 is considered Keeping the total thicknessthe same the thickness may consist of any number of thinlaminae each of which may be arbitrarily oriented at an angle120579 with reference to the 119909-axis of the coordinate system Theconstitutive equations for the shell are given by (a list of nota-tions is separately given)

119865 = [119864] 120576 (1)where

119865 = 119873119909 119873119910 119873119909119910119872119909119872119910119872119909119910 119876119909 119876119910119879

[119864] = [

[

[119860] [119861] [0]

[119861] [119863] [0]

[0] [0] [119878]

]

]

120576 = 1205760

119909 1205760

119910 1205740

119909119910 119896119909 119896119909119910 120574

0

119909119911 1205740

119910119911119879

(2)

The force and moment resultants are expressed as

119873119909 119873119910 119873119909119910119872119909119872119910119872119909119910 119876119909 119876119910119879

=int

ℎ2

minusℎ2

120590119909 120590119910 120591119909119910 120590119911 sdot 119911 120590119910 sdot 119911 120591119909119910 sdot 119911 120591119909119911 120591119910119911119879

119889119911

(3)The submatrices [119860] [119861] [119863] and [119878] of the elasticitymatrix[119864] are functions of Youngrsquos moduli shear moduli and Pois-sonrsquos ratio of the laminates They also depend on the anglewhich the individual lamina of a laminate makes with theglobal 119909-axis The detailed expressions of the elements of theelasticity matrix are available in several references includingVasiliev et al [21] and Qatu [22]

The strain-displacement relations on the basis of im-proved first-order approximation theory for thin shell (Deyet al [6]) are established as

120576119909 120576119910 120574119909119910 120574119909119911 120574119910119911119879

= 1205760

119909 1205760

119910 1205740

119909119910 1205740

119909119911 1205740

119910119911119879

+ 119911119896119909 119896119910 119896119909119910 119896119909119911 119896119910119911119879

(4)

a

b998400

b

a998400

ws

hs

hh

hl

X

Y Zh

Figure 1 Conoidal shell with a concentric cutout stiffened along themargins

where the first vector is the midsurface strain for a conoidalshell and the second vector is the curvature

3 Finite Element Formulation

31 Finite Element Formulation for Shell An eight-nodedcurved quadratic isoparametric finite element is used forconoidal shell analysisThe five degrees of freedom taken intoconsideration at each node are 119906 V 119908 120572 120573 The followingexpressions establish the relations between the displacementat any point with respect to the coordinates 120585 and 120578 and thenodal degrees of freedom

119906 =

8

sum

119894=1

119873119894119906119894 V =8

sum

119894=1

119873119894V119894 119908 =

8

sum

119894=1

119873119894119908119894

120572 =

8

sum

119894=1

119873119894120572119894 120573 =

8

sum

119894=1

119873119894120573119894

(5)

where the shape functions derived from a cubic interpolationpolynomial [6] are

119873119894 =(1 + 120585120585119894) (1 + 120578120578119894) (120585120585119894 + 120578120578119894 minus 1)

4 for 119894 = 1 2 3 4

119873119894 =(1 + 120585120585119894) (1 minus 120578

2)

2 for 119894 = 5 7

119873119894 =(1 + 120578120578119894) (1 minus 120585

2)

2 for 119894 = 6 8

(6)

The generalized displacement vector of an element is ex-pressed in terms of the shape functions and nodal degrees offreedom as

[119906] = [119873] 119889119890 (7)

that is

119906 =

119906

V119908

120572

120573

=

8

sum

119894=1

[[[[[

[

119873119894119873119894

119873119894119873119894

119873119894

]]]]]

]

119906119894V119894119908119894120572119894120573119894

(8)

Journal of Engineering 3

311 Element Stiffness Matrix The strain-displacement rela-tion is given by

120576 = [119861] 119889119890 (9)

where

[119861] =

8

sum

119894=1

[[[[[[[[[[[[[[[[[

[

119873119894119909 0 0 0 0

0 119873119894119910 minus119873119894

119877119910

0 0

119873119894119910 119873119894119909 minus2119873119894

119877119909119910

0 0

0 0 0 119873119894119909 0

0 0 0 0 119873119894119910

0 0 0 119873119894119910 119873119894119909

0 0 119873119894119909 119873119894 0

0 0 119873119894119910 0 119873119894

]]]]]]]]]]]]]]]]]

]

(10)

The element stiffness matrix is

[119870119890] = ∬[119861]119879[119864] [119861] 119889119909 119889119910 (11)

312 Element Mass Matrix The element mass matrix isobtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (12)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(13)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (14)

32 Finite Element Formulation for Stiffener of the ShellThree-noded curved isoparametric beam element (Figure 2)are used to model the stiffeners which are taken to run onlyalong the boundaries of the shell elements In the stiffenerelement each node has four degrees of freedom that is 119906119904119909119908119904119909 120572119904119909 and 120573119904119909 for 119883-stiffener and V119904119910 119908119904119910 120572119904119910 and 120573119904119910for119884-stiffenerThe generalized force-displacement relation ofstiffeners can be expressed as

119883-stiffener 119865119904119909 = [119863119904119909] 120576119904119909 = [119863119904119909] [119861119904119909] 120575119904119909119894

119884-stiffener 119865119904119910 = [119863119904119910] 120576119904119910 = [119863119904119910] [119861119904119910] 120575119904119910119894

(15)

1

23

4

5

6

7

8

120578

120585

(a)

(i)

(ii)

1 2 3

1

2

3120578

120585

(b)

Figure 2 (a) Eight-noded shell element with isoparametric coor-dinates (b) Three-noded stiffener element (i) 119883-stiffener (ii) 119884-stiffener

where

119865119904119909 = [119873119904119909119909 119872119904119909119909 119879119904119909119909 119876119904119909119909119911]119879

120576119904119909 = [119906119904119909sdot119909 120572119904119909sdot119909 120573119904119909sdot119909 (120572119904119909 + 119908119904119909sdot119909)]119879

119865119904119910 = [119873119904119910119910 119872119904119910119910 119879119904119910119910 119876119904119910119910119911]119879

120576119904119910 = [V119904119910sdot119910 120573119904119910sdot119910 120572119904119910sdot119910 (120573119904119910 + 119908119904119910sdot119910)]119879

(16)

The generalized displacements of the119884-stiffener and the shellare related by the transformation matrix 120575119904119910119894 = [119879]120575where

[119879] =

[[[[

[

1 +119890

119877119910

symmetric

0 1

0 0 1

0 0 0 1

]]]]

]

(17)

This transformation is required due to curvature of 119884-stif-fener and 120575 is the appropriate portion of the displacementvector of the shell excluding the displacement componentalong the 119909-axis

Elasticity matrices are as follows

[119863119904119909]=

[[[[[[

[

11986011119887119904119909 1198611015840

11119887119904119909 119861

1015840

12119887119904119909 0

1198611015840

11119887119904119909 119863

1015840

11119887119904119909 119863

1015840

12119887119904119909 0

1198611015840

12119887119904119909 119863

1015840

12119887119904119909

1

6(11987644 + 11987666) 119889119904119909119887

3

1199041199090

0 0 0 11988711990411990911987811

]]]]]]

]

[119863119904119910]=

[[[[[[

[

11986022119887119904119910 1198611015840

22119887119904119910 119861

1015840

12119887119904119910 0

1198611015840

22119887119904119910

1

6(11987644 + 11987666) 119887119904119910 119863

1015840

12119887119904119910 0

1198611015840

12119887119904119910 119863

1015840

12119887119904119910 119863

1015840

111198891199041199101198873

1199041199100

0 0 0 11988711990411991011987822

]]]]]]

]

(18)

where

1198631015840

119894119895= 119863119894119895 + 2119890119861119894119895 + 119890

2119860 119894119895

1198611015840

119894119895= 119861119894119895 + 119890119860 119894119895

(19)

4 Journal of Engineering

and 119860 119894119895 119861119894119895 119863119894119895 and 119878119894119895 are explained in an earlier paper bySahoo and Chakravorty [23]

Here the shear correction factor is taken as 56 Thesectional parameters are calculated with respect to the mid-surface of the shell by which the effect of eccentricities of stif-feners is automatically included The element stiffness matri-ces are of the following forms

for 119883-stiffener [119870119909119890] = int [119861119904119909]119879[119863119904119909] [119861119904119909] 119889119909

for 119884-stiffener [119870119910119890] = int [119861119904119910]119879

[119863119904119910] [119861119904119910] 119889119910

(20)

The integrals are converted to isoparametric coordinates andare carried out by 2-point Gauss quadrature Finally theelement stiffness matrix of the stiffened shell is obtained byappropriate matching of the nodes of the stiffener and shellelements through the connectivity matrix and is given as

[119870119890] = [119870she] + [119870119909119890] + [119870119910119890] (21)

The element stiffness matrices are assembled to get the globalmatrices

321 Element Mass Matrix The element mass matrix for shellis obtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (22)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(23)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (24)

Element mass matrix for stiffener element

[119872119904119909] = ∬[119873]119879[119875] [119873] 119889119909 for119883-stiffener

[119872119904119910] = ∬[119873]119879[119875] [119873] 119889119910 for 119884-stiffener

(25)

Here [119873] is a 3 times 3 diagonal matrix

Consider

[119875] =

3

sum

119894=1

[[[[

[

120588 sdot 119887119904119909119889119904119909 0 0 0

0 120588 sdot 119887119904119909119889119904119909 0 0

0 0120588 sdot 119887119904119909119889

2

119904119909

120

0 0 0120588 (119887119904119909 sdot 119889

3

119904119909+ 1198873

119904119909sdot 119889119904119909)

12

]]]]

]

for 119883-stiffener

[119875] =

3

sum

119894=1

[[[[[

[

120588 sdot 119887119904119910119889119904119910 0 0 0

0 120588 sdot 119887119904119910119889119904119910 0 0

0 0

120588 sdot 1198871199041199101198892

119904119910

120

0 0 0

120588 (119887119904119910 sdot 1198893

119904119910+ 1198873

119904119910sdot 119889119904119910)

12

]]]]]

]

for 119884-stiffener(26)

The mass matrix of the stiffened shell element is the sum ofthe matrices of the shell and the stiffeners matched at theappropriate nodes

[119872119890] = [119872she] + [119872119909119890] + [119872119910119890] (27)

The element mass matrices are assembled to get the globalmatrices

33 Modeling the Cutout The code developed can take theposition and size of cutout as inputThe program is capable ofgenerating nonuniform finite element mesh all over the shellsurface So the element size is gradually decreased near thecutoutmargins One such typicalmesh arrangement is shownin Figure 3 Such finite element mesh is redefined in stepsand a particular grid is chosen to obtain the fundamentalfrequencywhen the result does not improve bymore than onepercent on further refining Convergence of results is ensuredin all the problems taken up here

34 Solution Procedure for Free Vibration Analysis The freevibration analysis involves determination of natural frequen-cies from the condition

10038161003816100381610038161003816[119870] minus 120596

2[119872]

10038161003816100381610038161003816= 0 (28)

This is a generalized eigen value problem and is solved bythe subspace iteration algorithm

4 Numerical Examples

The validity of the present approach is checked through solu-tion of benchmark problemsThe first problem free vibrationof stiffened clamped conoid was solved earlier by Nayakand Bandyopadhyay [13] The second is the free vibration ofcomposite conoid with cutouts solved by Chakravorty et al[11] The results obtained by the present method along withthe published results are presented in Tables 1 and 2 res-pectively

Additional problems for conoids with cutouts are solvedvarying the size and position of cutout along both of the plan

Journal of Engineering 5

Table 1 Fundamental frequencies (radsec) of clamped conoidal shell with central stiffeners

Stiffenerposition

Stiffener along 119909-direction Stiffener along 119910-direction Stiffener along both 119909- and 119910-directionsNayak and

Bandyopadhyay[13]

Present model Nayak andBandyopadhyay

[13]

Present model Nayak andBandyopadhyay

[13]

Present model8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12

Concentric 1728 1750 1737 1731 2083 2115 2085 2076 2090 2124 2101 2084Eccentric attop 1783 1792 1778 1773 2157 2176 2153 2145 2239 2286 2251 2232

Eccentric atbottom 1755 1780 1761 1752 2231 2266 2240 2224 2292 2325 2300 2287

119886 = 50m 119887 = 50m ℎ = 02m ℎℎ = 10m ℎ119897 = 25m 119864 = 254910 times 109 ] = 015 120588 = 2500 kgm3 119908119904 = 03m and ℎ119904 = 1m

Table 2 Nondimensional fundamental frequencies (120596) for laminated composite conoidal shell with cutout

1198861015840119886

Corner point supported Simply supported ClampedChakravorty et al [11] Present model Chakravorty et al [11] Present model Chakravorty et al [11] Present model

00 23863 23494 75450 74892 124736 12330601 23554 23872 75098 75278 123811 12398702 23746 23485 73668 73324 122074 12058803 23510 23768 69979 69763 120515 11910104 23205 23101 61824 61524 116924 115924119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 25 and ℎ119897ℎℎ = 025

y

x

Figure 3 Typical 10 times 10 nonuniformmesh arrangements drawn toscale

directions of the shell for different practical boundary con-ditions In order to study the effect of cutout size on the freevibration response results for unpunctured conoids are alsoincluded in the study

5 Results and Discussion

It is found from Table 1 that the fundamental frequencies ofstiffened conoids obtained by the present method agree wellwith those reported byNayak and Bandyopadhyay [13] Heremonotonic convergence is noted as themesh ismade progres-sively finerThus the correctness of the stiffened shell elementused here is established It is evident from Table 2 that thepresent results agree with those of Chakravorty et al [11] andthe fact that the cutouts are properly modeled in the presentformulation is thus established

51 FreeVibration Behaviour of Shells withConcentric CutoutsTables 3 and 4 show the results for the non-dimensional fun-damental frequency 120596 of composite cross-ply and angle-plystiffened conoidal shells for different cutout sizes and variouscombinations of boundary conditions along the four edgesThe shells considered are of square planform (119886 = 119887) and thecutouts are also taken to be square in plan (119886

1015840= 1198871015840) The

cutout sizes (ie 1198861015840119886) are varied from 0 to 04 and boundaryconditions are varied along the four edges Cutouts are con-centric on shell surface The stiffeners are placed along thecutout periphery and extended up to the edge of the shellTheboundary conditions are designated by describing the sup-port clamped or simply supported as C or S taken in ananticlockwise order from the edge 119909 = 0 This means thata shell with CSCS boundary is clamped along 119909 = 0 simplysupported along 119910 = 0 clamped along 119909 = 119886 and simplysupported along 119910 = 119887 The material and geometricproperties of shells and cutouts are mentioned along with thefigures

511 Effect of Cutout Size on Fundamental Frequency FromTables 3 and 4 it is seen that when a cutout is introducedto a stiffened shell the fundamental frequencies increaseThis increasing trend continues up to 119886

1015840119886 = 04 for both

cross- and angle ply shells except some angle ply shellswith 119886

1015840119886 gt 02 The initial increase in frequency may be

explained by the fact that when a cutout is introduced toan unpunctured surface the number of stiffeners increasesfrom two to four in the present study When the cutout sizeis further increased the number and dimensions of the stif-feners do not change but the shell surface undergoes loss ofboth mass and stiffness As the cutout grows in size the lossof mass is more significant than that of stiffness and hence

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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2 Journal of Engineering

conoidal shells is available in the literature Also they did notfurnish any information on vibration mode shapes It is alsoseen from the recent reviews [19 20] that dynamic character-istics of stiffened conoidal shells with cutout are still missingin the literature The present study thus focuses on the freevibrations of graphite-epoxy laminated composite stiffenedconoids with cutout both in terms of the natural frequenciesandmode shapesThe results so obtainedmay be readily usedby practicing engineers dealing with stiffened compositeconoids with cutouts The novelty of the present study lies inthe consideration of vibration mode shapes of stiffened com-posite conoids in presence of cutouts

2 Mathematical Formulation

A laminated composite conoidal shell of uniform thicknessℎ (Figure 1) and radius of curvature 119877119910 and radius of crosscurvature 119877119909119910 is considered Keeping the total thicknessthe same the thickness may consist of any number of thinlaminae each of which may be arbitrarily oriented at an angle120579 with reference to the 119909-axis of the coordinate system Theconstitutive equations for the shell are given by (a list of nota-tions is separately given)

119865 = [119864] 120576 (1)where

119865 = 119873119909 119873119910 119873119909119910119872119909119872119910119872119909119910 119876119909 119876119910119879

[119864] = [

[

[119860] [119861] [0]

[119861] [119863] [0]

[0] [0] [119878]

]

]

120576 = 1205760

119909 1205760

119910 1205740

119909119910 119896119909 119896119909119910 120574

0

119909119911 1205740

119910119911119879

(2)

The force and moment resultants are expressed as

119873119909 119873119910 119873119909119910119872119909119872119910119872119909119910 119876119909 119876119910119879

=int

ℎ2

minusℎ2

120590119909 120590119910 120591119909119910 120590119911 sdot 119911 120590119910 sdot 119911 120591119909119910 sdot 119911 120591119909119911 120591119910119911119879

119889119911

(3)The submatrices [119860] [119861] [119863] and [119878] of the elasticitymatrix[119864] are functions of Youngrsquos moduli shear moduli and Pois-sonrsquos ratio of the laminates They also depend on the anglewhich the individual lamina of a laminate makes with theglobal 119909-axis The detailed expressions of the elements of theelasticity matrix are available in several references includingVasiliev et al [21] and Qatu [22]

The strain-displacement relations on the basis of im-proved first-order approximation theory for thin shell (Deyet al [6]) are established as

120576119909 120576119910 120574119909119910 120574119909119911 120574119910119911119879

= 1205760

119909 1205760

119910 1205740

119909119910 1205740

119909119911 1205740

119910119911119879

+ 119911119896119909 119896119910 119896119909119910 119896119909119911 119896119910119911119879

(4)

a

b998400

b

a998400

ws

hs

hh

hl

X

Y Zh

Figure 1 Conoidal shell with a concentric cutout stiffened along themargins

where the first vector is the midsurface strain for a conoidalshell and the second vector is the curvature

3 Finite Element Formulation

31 Finite Element Formulation for Shell An eight-nodedcurved quadratic isoparametric finite element is used forconoidal shell analysisThe five degrees of freedom taken intoconsideration at each node are 119906 V 119908 120572 120573 The followingexpressions establish the relations between the displacementat any point with respect to the coordinates 120585 and 120578 and thenodal degrees of freedom

119906 =

8

sum

119894=1

119873119894119906119894 V =8

sum

119894=1

119873119894V119894 119908 =

8

sum

119894=1

119873119894119908119894

120572 =

8

sum

119894=1

119873119894120572119894 120573 =

8

sum

119894=1

119873119894120573119894

(5)

where the shape functions derived from a cubic interpolationpolynomial [6] are

119873119894 =(1 + 120585120585119894) (1 + 120578120578119894) (120585120585119894 + 120578120578119894 minus 1)

4 for 119894 = 1 2 3 4

119873119894 =(1 + 120585120585119894) (1 minus 120578

2)

2 for 119894 = 5 7

119873119894 =(1 + 120578120578119894) (1 minus 120585

2)

2 for 119894 = 6 8

(6)

The generalized displacement vector of an element is ex-pressed in terms of the shape functions and nodal degrees offreedom as

[119906] = [119873] 119889119890 (7)

that is

119906 =

119906

V119908

120572

120573

=

8

sum

119894=1

[[[[[

[

119873119894119873119894

119873119894119873119894

119873119894

]]]]]

]

119906119894V119894119908119894120572119894120573119894

(8)

Journal of Engineering 3

311 Element Stiffness Matrix The strain-displacement rela-tion is given by

120576 = [119861] 119889119890 (9)

where

[119861] =

8

sum

119894=1

[[[[[[[[[[[[[[[[[

[

119873119894119909 0 0 0 0

0 119873119894119910 minus119873119894

119877119910

0 0

119873119894119910 119873119894119909 minus2119873119894

119877119909119910

0 0

0 0 0 119873119894119909 0

0 0 0 0 119873119894119910

0 0 0 119873119894119910 119873119894119909

0 0 119873119894119909 119873119894 0

0 0 119873119894119910 0 119873119894

]]]]]]]]]]]]]]]]]

]

(10)

The element stiffness matrix is

[119870119890] = ∬[119861]119879[119864] [119861] 119889119909 119889119910 (11)

312 Element Mass Matrix The element mass matrix isobtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (12)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(13)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (14)

32 Finite Element Formulation for Stiffener of the ShellThree-noded curved isoparametric beam element (Figure 2)are used to model the stiffeners which are taken to run onlyalong the boundaries of the shell elements In the stiffenerelement each node has four degrees of freedom that is 119906119904119909119908119904119909 120572119904119909 and 120573119904119909 for 119883-stiffener and V119904119910 119908119904119910 120572119904119910 and 120573119904119910for119884-stiffenerThe generalized force-displacement relation ofstiffeners can be expressed as

119883-stiffener 119865119904119909 = [119863119904119909] 120576119904119909 = [119863119904119909] [119861119904119909] 120575119904119909119894

119884-stiffener 119865119904119910 = [119863119904119910] 120576119904119910 = [119863119904119910] [119861119904119910] 120575119904119910119894

(15)

1

23

4

5

6

7

8

120578

120585

(a)

(i)

(ii)

1 2 3

1

2

3120578

120585

(b)

Figure 2 (a) Eight-noded shell element with isoparametric coor-dinates (b) Three-noded stiffener element (i) 119883-stiffener (ii) 119884-stiffener

where

119865119904119909 = [119873119904119909119909 119872119904119909119909 119879119904119909119909 119876119904119909119909119911]119879

120576119904119909 = [119906119904119909sdot119909 120572119904119909sdot119909 120573119904119909sdot119909 (120572119904119909 + 119908119904119909sdot119909)]119879

119865119904119910 = [119873119904119910119910 119872119904119910119910 119879119904119910119910 119876119904119910119910119911]119879

120576119904119910 = [V119904119910sdot119910 120573119904119910sdot119910 120572119904119910sdot119910 (120573119904119910 + 119908119904119910sdot119910)]119879

(16)

The generalized displacements of the119884-stiffener and the shellare related by the transformation matrix 120575119904119910119894 = [119879]120575where

[119879] =

[[[[

[

1 +119890

119877119910

symmetric

0 1

0 0 1

0 0 0 1

]]]]

]

(17)

This transformation is required due to curvature of 119884-stif-fener and 120575 is the appropriate portion of the displacementvector of the shell excluding the displacement componentalong the 119909-axis

Elasticity matrices are as follows

[119863119904119909]=

[[[[[[

[

11986011119887119904119909 1198611015840

11119887119904119909 119861

1015840

12119887119904119909 0

1198611015840

11119887119904119909 119863

1015840

11119887119904119909 119863

1015840

12119887119904119909 0

1198611015840

12119887119904119909 119863

1015840

12119887119904119909

1

6(11987644 + 11987666) 119889119904119909119887

3

1199041199090

0 0 0 11988711990411990911987811

]]]]]]

]

[119863119904119910]=

[[[[[[

[

11986022119887119904119910 1198611015840

22119887119904119910 119861

1015840

12119887119904119910 0

1198611015840

22119887119904119910

1

6(11987644 + 11987666) 119887119904119910 119863

1015840

12119887119904119910 0

1198611015840

12119887119904119910 119863

1015840

12119887119904119910 119863

1015840

111198891199041199101198873

1199041199100

0 0 0 11988711990411991011987822

]]]]]]

]

(18)

where

1198631015840

119894119895= 119863119894119895 + 2119890119861119894119895 + 119890

2119860 119894119895

1198611015840

119894119895= 119861119894119895 + 119890119860 119894119895

(19)

4 Journal of Engineering

and 119860 119894119895 119861119894119895 119863119894119895 and 119878119894119895 are explained in an earlier paper bySahoo and Chakravorty [23]

Here the shear correction factor is taken as 56 Thesectional parameters are calculated with respect to the mid-surface of the shell by which the effect of eccentricities of stif-feners is automatically included The element stiffness matri-ces are of the following forms

for 119883-stiffener [119870119909119890] = int [119861119904119909]119879[119863119904119909] [119861119904119909] 119889119909

for 119884-stiffener [119870119910119890] = int [119861119904119910]119879

[119863119904119910] [119861119904119910] 119889119910

(20)

The integrals are converted to isoparametric coordinates andare carried out by 2-point Gauss quadrature Finally theelement stiffness matrix of the stiffened shell is obtained byappropriate matching of the nodes of the stiffener and shellelements through the connectivity matrix and is given as

[119870119890] = [119870she] + [119870119909119890] + [119870119910119890] (21)

The element stiffness matrices are assembled to get the globalmatrices

321 Element Mass Matrix The element mass matrix for shellis obtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (22)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(23)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (24)

Element mass matrix for stiffener element

[119872119904119909] = ∬[119873]119879[119875] [119873] 119889119909 for119883-stiffener

[119872119904119910] = ∬[119873]119879[119875] [119873] 119889119910 for 119884-stiffener

(25)

Here [119873] is a 3 times 3 diagonal matrix

Consider

[119875] =

3

sum

119894=1

[[[[

[

120588 sdot 119887119904119909119889119904119909 0 0 0

0 120588 sdot 119887119904119909119889119904119909 0 0

0 0120588 sdot 119887119904119909119889

2

119904119909

120

0 0 0120588 (119887119904119909 sdot 119889

3

119904119909+ 1198873

119904119909sdot 119889119904119909)

12

]]]]

]

for 119883-stiffener

[119875] =

3

sum

119894=1

[[[[[

[

120588 sdot 119887119904119910119889119904119910 0 0 0

0 120588 sdot 119887119904119910119889119904119910 0 0

0 0

120588 sdot 1198871199041199101198892

119904119910

120

0 0 0

120588 (119887119904119910 sdot 1198893

119904119910+ 1198873

119904119910sdot 119889119904119910)

12

]]]]]

]

for 119884-stiffener(26)

The mass matrix of the stiffened shell element is the sum ofthe matrices of the shell and the stiffeners matched at theappropriate nodes

[119872119890] = [119872she] + [119872119909119890] + [119872119910119890] (27)

The element mass matrices are assembled to get the globalmatrices

33 Modeling the Cutout The code developed can take theposition and size of cutout as inputThe program is capable ofgenerating nonuniform finite element mesh all over the shellsurface So the element size is gradually decreased near thecutoutmargins One such typicalmesh arrangement is shownin Figure 3 Such finite element mesh is redefined in stepsand a particular grid is chosen to obtain the fundamentalfrequencywhen the result does not improve bymore than onepercent on further refining Convergence of results is ensuredin all the problems taken up here

34 Solution Procedure for Free Vibration Analysis The freevibration analysis involves determination of natural frequen-cies from the condition

10038161003816100381610038161003816[119870] minus 120596

2[119872]

10038161003816100381610038161003816= 0 (28)

This is a generalized eigen value problem and is solved bythe subspace iteration algorithm

4 Numerical Examples

The validity of the present approach is checked through solu-tion of benchmark problemsThe first problem free vibrationof stiffened clamped conoid was solved earlier by Nayakand Bandyopadhyay [13] The second is the free vibration ofcomposite conoid with cutouts solved by Chakravorty et al[11] The results obtained by the present method along withthe published results are presented in Tables 1 and 2 res-pectively

Additional problems for conoids with cutouts are solvedvarying the size and position of cutout along both of the plan

Journal of Engineering 5

Table 1 Fundamental frequencies (radsec) of clamped conoidal shell with central stiffeners

Stiffenerposition

Stiffener along 119909-direction Stiffener along 119910-direction Stiffener along both 119909- and 119910-directionsNayak and

Bandyopadhyay[13]

Present model Nayak andBandyopadhyay

[13]

Present model Nayak andBandyopadhyay

[13]

Present model8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12

Concentric 1728 1750 1737 1731 2083 2115 2085 2076 2090 2124 2101 2084Eccentric attop 1783 1792 1778 1773 2157 2176 2153 2145 2239 2286 2251 2232

Eccentric atbottom 1755 1780 1761 1752 2231 2266 2240 2224 2292 2325 2300 2287

119886 = 50m 119887 = 50m ℎ = 02m ℎℎ = 10m ℎ119897 = 25m 119864 = 254910 times 109 ] = 015 120588 = 2500 kgm3 119908119904 = 03m and ℎ119904 = 1m

Table 2 Nondimensional fundamental frequencies (120596) for laminated composite conoidal shell with cutout

1198861015840119886

Corner point supported Simply supported ClampedChakravorty et al [11] Present model Chakravorty et al [11] Present model Chakravorty et al [11] Present model

00 23863 23494 75450 74892 124736 12330601 23554 23872 75098 75278 123811 12398702 23746 23485 73668 73324 122074 12058803 23510 23768 69979 69763 120515 11910104 23205 23101 61824 61524 116924 115924119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 25 and ℎ119897ℎℎ = 025

y

x

Figure 3 Typical 10 times 10 nonuniformmesh arrangements drawn toscale

directions of the shell for different practical boundary con-ditions In order to study the effect of cutout size on the freevibration response results for unpunctured conoids are alsoincluded in the study

5 Results and Discussion

It is found from Table 1 that the fundamental frequencies ofstiffened conoids obtained by the present method agree wellwith those reported byNayak and Bandyopadhyay [13] Heremonotonic convergence is noted as themesh ismade progres-sively finerThus the correctness of the stiffened shell elementused here is established It is evident from Table 2 that thepresent results agree with those of Chakravorty et al [11] andthe fact that the cutouts are properly modeled in the presentformulation is thus established

51 FreeVibration Behaviour of Shells withConcentric CutoutsTables 3 and 4 show the results for the non-dimensional fun-damental frequency 120596 of composite cross-ply and angle-plystiffened conoidal shells for different cutout sizes and variouscombinations of boundary conditions along the four edgesThe shells considered are of square planform (119886 = 119887) and thecutouts are also taken to be square in plan (119886

1015840= 1198871015840) The

cutout sizes (ie 1198861015840119886) are varied from 0 to 04 and boundaryconditions are varied along the four edges Cutouts are con-centric on shell surface The stiffeners are placed along thecutout periphery and extended up to the edge of the shellTheboundary conditions are designated by describing the sup-port clamped or simply supported as C or S taken in ananticlockwise order from the edge 119909 = 0 This means thata shell with CSCS boundary is clamped along 119909 = 0 simplysupported along 119910 = 0 clamped along 119909 = 119886 and simplysupported along 119910 = 119887 The material and geometricproperties of shells and cutouts are mentioned along with thefigures

511 Effect of Cutout Size on Fundamental Frequency FromTables 3 and 4 it is seen that when a cutout is introducedto a stiffened shell the fundamental frequencies increaseThis increasing trend continues up to 119886

1015840119886 = 04 for both

cross- and angle ply shells except some angle ply shellswith 119886

1015840119886 gt 02 The initial increase in frequency may be

explained by the fact that when a cutout is introduced toan unpunctured surface the number of stiffeners increasesfrom two to four in the present study When the cutout sizeis further increased the number and dimensions of the stif-feners do not change but the shell surface undergoes loss ofboth mass and stiffness As the cutout grows in size the lossof mass is more significant than that of stiffness and hence

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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International Journal of

Journal of Engineering 3

311 Element Stiffness Matrix The strain-displacement rela-tion is given by

120576 = [119861] 119889119890 (9)

where

[119861] =

8

sum

119894=1

[[[[[[[[[[[[[[[[[

[

119873119894119909 0 0 0 0

0 119873119894119910 minus119873119894

119877119910

0 0

119873119894119910 119873119894119909 minus2119873119894

119877119909119910

0 0

0 0 0 119873119894119909 0

0 0 0 0 119873119894119910

0 0 0 119873119894119910 119873119894119909

0 0 119873119894119909 119873119894 0

0 0 119873119894119910 0 119873119894

]]]]]]]]]]]]]]]]]

]

(10)

The element stiffness matrix is

[119870119890] = ∬[119861]119879[119864] [119861] 119889119909 119889119910 (11)

312 Element Mass Matrix The element mass matrix isobtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (12)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(13)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (14)

32 Finite Element Formulation for Stiffener of the ShellThree-noded curved isoparametric beam element (Figure 2)are used to model the stiffeners which are taken to run onlyalong the boundaries of the shell elements In the stiffenerelement each node has four degrees of freedom that is 119906119904119909119908119904119909 120572119904119909 and 120573119904119909 for 119883-stiffener and V119904119910 119908119904119910 120572119904119910 and 120573119904119910for119884-stiffenerThe generalized force-displacement relation ofstiffeners can be expressed as

119883-stiffener 119865119904119909 = [119863119904119909] 120576119904119909 = [119863119904119909] [119861119904119909] 120575119904119909119894

119884-stiffener 119865119904119910 = [119863119904119910] 120576119904119910 = [119863119904119910] [119861119904119910] 120575119904119910119894

(15)

1

23

4

5

6

7

8

120578

120585

(a)

(i)

(ii)

1 2 3

1

2

3120578

120585

(b)

Figure 2 (a) Eight-noded shell element with isoparametric coor-dinates (b) Three-noded stiffener element (i) 119883-stiffener (ii) 119884-stiffener

where

119865119904119909 = [119873119904119909119909 119872119904119909119909 119879119904119909119909 119876119904119909119909119911]119879

120576119904119909 = [119906119904119909sdot119909 120572119904119909sdot119909 120573119904119909sdot119909 (120572119904119909 + 119908119904119909sdot119909)]119879

119865119904119910 = [119873119904119910119910 119872119904119910119910 119879119904119910119910 119876119904119910119910119911]119879

120576119904119910 = [V119904119910sdot119910 120573119904119910sdot119910 120572119904119910sdot119910 (120573119904119910 + 119908119904119910sdot119910)]119879

(16)

The generalized displacements of the119884-stiffener and the shellare related by the transformation matrix 120575119904119910119894 = [119879]120575where

[119879] =

[[[[

[

1 +119890

119877119910

symmetric

0 1

0 0 1

0 0 0 1

]]]]

]

(17)

This transformation is required due to curvature of 119884-stif-fener and 120575 is the appropriate portion of the displacementvector of the shell excluding the displacement componentalong the 119909-axis

Elasticity matrices are as follows

[119863119904119909]=

[[[[[[

[

11986011119887119904119909 1198611015840

11119887119904119909 119861

1015840

12119887119904119909 0

1198611015840

11119887119904119909 119863

1015840

11119887119904119909 119863

1015840

12119887119904119909 0

1198611015840

12119887119904119909 119863

1015840

12119887119904119909

1

6(11987644 + 11987666) 119889119904119909119887

3

1199041199090

0 0 0 11988711990411990911987811

]]]]]]

]

[119863119904119910]=

[[[[[[

[

11986022119887119904119910 1198611015840

22119887119904119910 119861

1015840

12119887119904119910 0

1198611015840

22119887119904119910

1

6(11987644 + 11987666) 119887119904119910 119863

1015840

12119887119904119910 0

1198611015840

12119887119904119910 119863

1015840

12119887119904119910 119863

1015840

111198891199041199101198873

1199041199100

0 0 0 11988711990411991011987822

]]]]]]

]

(18)

where

1198631015840

119894119895= 119863119894119895 + 2119890119861119894119895 + 119890

2119860 119894119895

1198611015840

119894119895= 119861119894119895 + 119890119860 119894119895

(19)

4 Journal of Engineering

and 119860 119894119895 119861119894119895 119863119894119895 and 119878119894119895 are explained in an earlier paper bySahoo and Chakravorty [23]

Here the shear correction factor is taken as 56 Thesectional parameters are calculated with respect to the mid-surface of the shell by which the effect of eccentricities of stif-feners is automatically included The element stiffness matri-ces are of the following forms

for 119883-stiffener [119870119909119890] = int [119861119904119909]119879[119863119904119909] [119861119904119909] 119889119909

for 119884-stiffener [119870119910119890] = int [119861119904119910]119879

[119863119904119910] [119861119904119910] 119889119910

(20)

The integrals are converted to isoparametric coordinates andare carried out by 2-point Gauss quadrature Finally theelement stiffness matrix of the stiffened shell is obtained byappropriate matching of the nodes of the stiffener and shellelements through the connectivity matrix and is given as

[119870119890] = [119870she] + [119870119909119890] + [119870119910119890] (21)

The element stiffness matrices are assembled to get the globalmatrices

321 Element Mass Matrix The element mass matrix for shellis obtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (22)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(23)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (24)

Element mass matrix for stiffener element

[119872119904119909] = ∬[119873]119879[119875] [119873] 119889119909 for119883-stiffener

[119872119904119910] = ∬[119873]119879[119875] [119873] 119889119910 for 119884-stiffener

(25)

Here [119873] is a 3 times 3 diagonal matrix

Consider

[119875] =

3

sum

119894=1

[[[[

[

120588 sdot 119887119904119909119889119904119909 0 0 0

0 120588 sdot 119887119904119909119889119904119909 0 0

0 0120588 sdot 119887119904119909119889

2

119904119909

120

0 0 0120588 (119887119904119909 sdot 119889

3

119904119909+ 1198873

119904119909sdot 119889119904119909)

12

]]]]

]

for 119883-stiffener

[119875] =

3

sum

119894=1

[[[[[

[

120588 sdot 119887119904119910119889119904119910 0 0 0

0 120588 sdot 119887119904119910119889119904119910 0 0

0 0

120588 sdot 1198871199041199101198892

119904119910

120

0 0 0

120588 (119887119904119910 sdot 1198893

119904119910+ 1198873

119904119910sdot 119889119904119910)

12

]]]]]

]

for 119884-stiffener(26)

The mass matrix of the stiffened shell element is the sum ofthe matrices of the shell and the stiffeners matched at theappropriate nodes

[119872119890] = [119872she] + [119872119909119890] + [119872119910119890] (27)

The element mass matrices are assembled to get the globalmatrices

33 Modeling the Cutout The code developed can take theposition and size of cutout as inputThe program is capable ofgenerating nonuniform finite element mesh all over the shellsurface So the element size is gradually decreased near thecutoutmargins One such typicalmesh arrangement is shownin Figure 3 Such finite element mesh is redefined in stepsand a particular grid is chosen to obtain the fundamentalfrequencywhen the result does not improve bymore than onepercent on further refining Convergence of results is ensuredin all the problems taken up here

34 Solution Procedure for Free Vibration Analysis The freevibration analysis involves determination of natural frequen-cies from the condition

10038161003816100381610038161003816[119870] minus 120596

2[119872]

10038161003816100381610038161003816= 0 (28)

This is a generalized eigen value problem and is solved bythe subspace iteration algorithm

4 Numerical Examples

The validity of the present approach is checked through solu-tion of benchmark problemsThe first problem free vibrationof stiffened clamped conoid was solved earlier by Nayakand Bandyopadhyay [13] The second is the free vibration ofcomposite conoid with cutouts solved by Chakravorty et al[11] The results obtained by the present method along withthe published results are presented in Tables 1 and 2 res-pectively

Additional problems for conoids with cutouts are solvedvarying the size and position of cutout along both of the plan

Journal of Engineering 5

Table 1 Fundamental frequencies (radsec) of clamped conoidal shell with central stiffeners

Stiffenerposition

Stiffener along 119909-direction Stiffener along 119910-direction Stiffener along both 119909- and 119910-directionsNayak and

Bandyopadhyay[13]

Present model Nayak andBandyopadhyay

[13]

Present model Nayak andBandyopadhyay

[13]

Present model8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12

Concentric 1728 1750 1737 1731 2083 2115 2085 2076 2090 2124 2101 2084Eccentric attop 1783 1792 1778 1773 2157 2176 2153 2145 2239 2286 2251 2232

Eccentric atbottom 1755 1780 1761 1752 2231 2266 2240 2224 2292 2325 2300 2287

119886 = 50m 119887 = 50m ℎ = 02m ℎℎ = 10m ℎ119897 = 25m 119864 = 254910 times 109 ] = 015 120588 = 2500 kgm3 119908119904 = 03m and ℎ119904 = 1m

Table 2 Nondimensional fundamental frequencies (120596) for laminated composite conoidal shell with cutout

1198861015840119886

Corner point supported Simply supported ClampedChakravorty et al [11] Present model Chakravorty et al [11] Present model Chakravorty et al [11] Present model

00 23863 23494 75450 74892 124736 12330601 23554 23872 75098 75278 123811 12398702 23746 23485 73668 73324 122074 12058803 23510 23768 69979 69763 120515 11910104 23205 23101 61824 61524 116924 115924119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 25 and ℎ119897ℎℎ = 025

y

x

Figure 3 Typical 10 times 10 nonuniformmesh arrangements drawn toscale

directions of the shell for different practical boundary con-ditions In order to study the effect of cutout size on the freevibration response results for unpunctured conoids are alsoincluded in the study

5 Results and Discussion

It is found from Table 1 that the fundamental frequencies ofstiffened conoids obtained by the present method agree wellwith those reported byNayak and Bandyopadhyay [13] Heremonotonic convergence is noted as themesh ismade progres-sively finerThus the correctness of the stiffened shell elementused here is established It is evident from Table 2 that thepresent results agree with those of Chakravorty et al [11] andthe fact that the cutouts are properly modeled in the presentformulation is thus established

51 FreeVibration Behaviour of Shells withConcentric CutoutsTables 3 and 4 show the results for the non-dimensional fun-damental frequency 120596 of composite cross-ply and angle-plystiffened conoidal shells for different cutout sizes and variouscombinations of boundary conditions along the four edgesThe shells considered are of square planform (119886 = 119887) and thecutouts are also taken to be square in plan (119886

1015840= 1198871015840) The

cutout sizes (ie 1198861015840119886) are varied from 0 to 04 and boundaryconditions are varied along the four edges Cutouts are con-centric on shell surface The stiffeners are placed along thecutout periphery and extended up to the edge of the shellTheboundary conditions are designated by describing the sup-port clamped or simply supported as C or S taken in ananticlockwise order from the edge 119909 = 0 This means thata shell with CSCS boundary is clamped along 119909 = 0 simplysupported along 119910 = 0 clamped along 119909 = 119886 and simplysupported along 119910 = 119887 The material and geometricproperties of shells and cutouts are mentioned along with thefigures

511 Effect of Cutout Size on Fundamental Frequency FromTables 3 and 4 it is seen that when a cutout is introducedto a stiffened shell the fundamental frequencies increaseThis increasing trend continues up to 119886

1015840119886 = 04 for both

cross- and angle ply shells except some angle ply shellswith 119886

1015840119886 gt 02 The initial increase in frequency may be

explained by the fact that when a cutout is introduced toan unpunctured surface the number of stiffeners increasesfrom two to four in the present study When the cutout sizeis further increased the number and dimensions of the stif-feners do not change but the shell surface undergoes loss ofboth mass and stiffness As the cutout grows in size the lossof mass is more significant than that of stiffness and hence

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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4 Journal of Engineering

and 119860 119894119895 119861119894119895 119863119894119895 and 119878119894119895 are explained in an earlier paper bySahoo and Chakravorty [23]

Here the shear correction factor is taken as 56 Thesectional parameters are calculated with respect to the mid-surface of the shell by which the effect of eccentricities of stif-feners is automatically included The element stiffness matri-ces are of the following forms

for 119883-stiffener [119870119909119890] = int [119861119904119909]119879[119863119904119909] [119861119904119909] 119889119909

for 119884-stiffener [119870119910119890] = int [119861119904119910]119879

[119863119904119910] [119861119904119910] 119889119910

(20)

The integrals are converted to isoparametric coordinates andare carried out by 2-point Gauss quadrature Finally theelement stiffness matrix of the stiffened shell is obtained byappropriate matching of the nodes of the stiffener and shellelements through the connectivity matrix and is given as

[119870119890] = [119870she] + [119870119909119890] + [119870119910119890] (21)

The element stiffness matrices are assembled to get the globalmatrices

321 Element Mass Matrix The element mass matrix for shellis obtained from the integral

[119872119890] = ∬[119873]119879[119875] [119873] 119889119909 119889119910 (22)

where

[119873] =

8

sum

119894=1

[[[[[

[

119873119894 0 0 0 0

0 119873119894 0 0 0

0 0 119873119894 0 0

0 0 0 119873119894 0

0 0 0 0 119873119894

]]]]]

]

[119875] =

8

sum

119894=1

[[[[[

[

119875 0 0 0 0

0 119875 0 0 0

0 0 119875 0 0

0 0 0 119868 0

0 0 0 0 119868

]]]]]

]

(23)

in which

119875 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

120588 119889119911 119868 =

119899119901

sum

119896=1

int

119911119896

119911119896minus1

119911120588 119889119911 (24)

Element mass matrix for stiffener element

[119872119904119909] = ∬[119873]119879[119875] [119873] 119889119909 for119883-stiffener

[119872119904119910] = ∬[119873]119879[119875] [119873] 119889119910 for 119884-stiffener

(25)

Here [119873] is a 3 times 3 diagonal matrix

Consider

[119875] =

3

sum

119894=1

[[[[

[

120588 sdot 119887119904119909119889119904119909 0 0 0

0 120588 sdot 119887119904119909119889119904119909 0 0

0 0120588 sdot 119887119904119909119889

2

119904119909

120

0 0 0120588 (119887119904119909 sdot 119889

3

119904119909+ 1198873

119904119909sdot 119889119904119909)

12

]]]]

]

for 119883-stiffener

[119875] =

3

sum

119894=1

[[[[[

[

120588 sdot 119887119904119910119889119904119910 0 0 0

0 120588 sdot 119887119904119910119889119904119910 0 0

0 0

120588 sdot 1198871199041199101198892

119904119910

120

0 0 0

120588 (119887119904119910 sdot 1198893

119904119910+ 1198873

119904119910sdot 119889119904119910)

12

]]]]]

]

for 119884-stiffener(26)

The mass matrix of the stiffened shell element is the sum ofthe matrices of the shell and the stiffeners matched at theappropriate nodes

[119872119890] = [119872she] + [119872119909119890] + [119872119910119890] (27)

The element mass matrices are assembled to get the globalmatrices

33 Modeling the Cutout The code developed can take theposition and size of cutout as inputThe program is capable ofgenerating nonuniform finite element mesh all over the shellsurface So the element size is gradually decreased near thecutoutmargins One such typicalmesh arrangement is shownin Figure 3 Such finite element mesh is redefined in stepsand a particular grid is chosen to obtain the fundamentalfrequencywhen the result does not improve bymore than onepercent on further refining Convergence of results is ensuredin all the problems taken up here

34 Solution Procedure for Free Vibration Analysis The freevibration analysis involves determination of natural frequen-cies from the condition

10038161003816100381610038161003816[119870] minus 120596

2[119872]

10038161003816100381610038161003816= 0 (28)

This is a generalized eigen value problem and is solved bythe subspace iteration algorithm

4 Numerical Examples

The validity of the present approach is checked through solu-tion of benchmark problemsThe first problem free vibrationof stiffened clamped conoid was solved earlier by Nayakand Bandyopadhyay [13] The second is the free vibration ofcomposite conoid with cutouts solved by Chakravorty et al[11] The results obtained by the present method along withthe published results are presented in Tables 1 and 2 res-pectively

Additional problems for conoids with cutouts are solvedvarying the size and position of cutout along both of the plan

Journal of Engineering 5

Table 1 Fundamental frequencies (radsec) of clamped conoidal shell with central stiffeners

Stiffenerposition

Stiffener along 119909-direction Stiffener along 119910-direction Stiffener along both 119909- and 119910-directionsNayak and

Bandyopadhyay[13]

Present model Nayak andBandyopadhyay

[13]

Present model Nayak andBandyopadhyay

[13]

Present model8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12

Concentric 1728 1750 1737 1731 2083 2115 2085 2076 2090 2124 2101 2084Eccentric attop 1783 1792 1778 1773 2157 2176 2153 2145 2239 2286 2251 2232

Eccentric atbottom 1755 1780 1761 1752 2231 2266 2240 2224 2292 2325 2300 2287

119886 = 50m 119887 = 50m ℎ = 02m ℎℎ = 10m ℎ119897 = 25m 119864 = 254910 times 109 ] = 015 120588 = 2500 kgm3 119908119904 = 03m and ℎ119904 = 1m

Table 2 Nondimensional fundamental frequencies (120596) for laminated composite conoidal shell with cutout

1198861015840119886

Corner point supported Simply supported ClampedChakravorty et al [11] Present model Chakravorty et al [11] Present model Chakravorty et al [11] Present model

00 23863 23494 75450 74892 124736 12330601 23554 23872 75098 75278 123811 12398702 23746 23485 73668 73324 122074 12058803 23510 23768 69979 69763 120515 11910104 23205 23101 61824 61524 116924 115924119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 25 and ℎ119897ℎℎ = 025

y

x

Figure 3 Typical 10 times 10 nonuniformmesh arrangements drawn toscale

directions of the shell for different practical boundary con-ditions In order to study the effect of cutout size on the freevibration response results for unpunctured conoids are alsoincluded in the study

5 Results and Discussion

It is found from Table 1 that the fundamental frequencies ofstiffened conoids obtained by the present method agree wellwith those reported byNayak and Bandyopadhyay [13] Heremonotonic convergence is noted as themesh ismade progres-sively finerThus the correctness of the stiffened shell elementused here is established It is evident from Table 2 that thepresent results agree with those of Chakravorty et al [11] andthe fact that the cutouts are properly modeled in the presentformulation is thus established

51 FreeVibration Behaviour of Shells withConcentric CutoutsTables 3 and 4 show the results for the non-dimensional fun-damental frequency 120596 of composite cross-ply and angle-plystiffened conoidal shells for different cutout sizes and variouscombinations of boundary conditions along the four edgesThe shells considered are of square planform (119886 = 119887) and thecutouts are also taken to be square in plan (119886

1015840= 1198871015840) The

cutout sizes (ie 1198861015840119886) are varied from 0 to 04 and boundaryconditions are varied along the four edges Cutouts are con-centric on shell surface The stiffeners are placed along thecutout periphery and extended up to the edge of the shellTheboundary conditions are designated by describing the sup-port clamped or simply supported as C or S taken in ananticlockwise order from the edge 119909 = 0 This means thata shell with CSCS boundary is clamped along 119909 = 0 simplysupported along 119910 = 0 clamped along 119909 = 119886 and simplysupported along 119910 = 119887 The material and geometricproperties of shells and cutouts are mentioned along with thefigures

511 Effect of Cutout Size on Fundamental Frequency FromTables 3 and 4 it is seen that when a cutout is introducedto a stiffened shell the fundamental frequencies increaseThis increasing trend continues up to 119886

1015840119886 = 04 for both

cross- and angle ply shells except some angle ply shellswith 119886

1015840119886 gt 02 The initial increase in frequency may be

explained by the fact that when a cutout is introduced toan unpunctured surface the number of stiffeners increasesfrom two to four in the present study When the cutout sizeis further increased the number and dimensions of the stif-feners do not change but the shell surface undergoes loss ofboth mass and stiffness As the cutout grows in size the lossof mass is more significant than that of stiffness and hence

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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Journal of Engineering 5

Table 1 Fundamental frequencies (radsec) of clamped conoidal shell with central stiffeners

Stiffenerposition

Stiffener along 119909-direction Stiffener along 119910-direction Stiffener along both 119909- and 119910-directionsNayak and

Bandyopadhyay[13]

Present model Nayak andBandyopadhyay

[13]

Present model Nayak andBandyopadhyay

[13]

Present model8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12 8 times 8 10 times 10 12 times 12

Concentric 1728 1750 1737 1731 2083 2115 2085 2076 2090 2124 2101 2084Eccentric attop 1783 1792 1778 1773 2157 2176 2153 2145 2239 2286 2251 2232

Eccentric atbottom 1755 1780 1761 1752 2231 2266 2240 2224 2292 2325 2300 2287

119886 = 50m 119887 = 50m ℎ = 02m ℎℎ = 10m ℎ119897 = 25m 119864 = 254910 times 109 ] = 015 120588 = 2500 kgm3 119908119904 = 03m and ℎ119904 = 1m

Table 2 Nondimensional fundamental frequencies (120596) for laminated composite conoidal shell with cutout

1198861015840119886

Corner point supported Simply supported ClampedChakravorty et al [11] Present model Chakravorty et al [11] Present model Chakravorty et al [11] Present model

00 23863 23494 75450 74892 124736 12330601 23554 23872 75098 75278 123811 12398702 23746 23485 73668 73324 122074 12058803 23510 23768 69979 69763 120515 11910104 23205 23101 61824 61524 116924 115924119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 25 and ℎ119897ℎℎ = 025

y

x

Figure 3 Typical 10 times 10 nonuniformmesh arrangements drawn toscale

directions of the shell for different practical boundary con-ditions In order to study the effect of cutout size on the freevibration response results for unpunctured conoids are alsoincluded in the study

5 Results and Discussion

It is found from Table 1 that the fundamental frequencies ofstiffened conoids obtained by the present method agree wellwith those reported byNayak and Bandyopadhyay [13] Heremonotonic convergence is noted as themesh ismade progres-sively finerThus the correctness of the stiffened shell elementused here is established It is evident from Table 2 that thepresent results agree with those of Chakravorty et al [11] andthe fact that the cutouts are properly modeled in the presentformulation is thus established

51 FreeVibration Behaviour of Shells withConcentric CutoutsTables 3 and 4 show the results for the non-dimensional fun-damental frequency 120596 of composite cross-ply and angle-plystiffened conoidal shells for different cutout sizes and variouscombinations of boundary conditions along the four edgesThe shells considered are of square planform (119886 = 119887) and thecutouts are also taken to be square in plan (119886

1015840= 1198871015840) The

cutout sizes (ie 1198861015840119886) are varied from 0 to 04 and boundaryconditions are varied along the four edges Cutouts are con-centric on shell surface The stiffeners are placed along thecutout periphery and extended up to the edge of the shellTheboundary conditions are designated by describing the sup-port clamped or simply supported as C or S taken in ananticlockwise order from the edge 119909 = 0 This means thata shell with CSCS boundary is clamped along 119909 = 0 simplysupported along 119910 = 0 clamped along 119909 = 119886 and simplysupported along 119910 = 119887 The material and geometricproperties of shells and cutouts are mentioned along with thefigures

511 Effect of Cutout Size on Fundamental Frequency FromTables 3 and 4 it is seen that when a cutout is introducedto a stiffened shell the fundamental frequencies increaseThis increasing trend continues up to 119886

1015840119886 = 04 for both

cross- and angle ply shells except some angle ply shellswith 119886

1015840119886 gt 02 The initial increase in frequency may be

explained by the fact that when a cutout is introduced toan unpunctured surface the number of stiffeners increasesfrom two to four in the present study When the cutout sizeis further increased the number and dimensions of the stif-feners do not change but the shell surface undergoes loss ofboth mass and stiffness As the cutout grows in size the lossof mass is more significant than that of stiffness and hence

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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International Journal of

6 Journal of Engineering

Boundary 0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

a998400

ararrcondition

darr

yx

y x y xy y

x

y x

y x

y x

yx

yx

y x

y x

y x

y x

y x

yx

yx

y x

x

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

yx

yx

yx

yx

yx

y x

yx

yx

yx

yx

yx

yx

y x

y x

yx

yx

y x

yx

y x

y x

y x

y x

yx

y x

y x

y x

y x

y x

yx

Figure 4 First mode shapes of laminated composite (090090) stiffened conoidal shell for different sizes of the central square cutout andboundary conditions

the frequency increases But for some angle ply shells withfurther increase in the size of the cutout the loss of stiffnessgradually becomes more important than that of mass result-ing in decrease in fundamental frequency This leads to theengineering conclusion that cutouts with stiffened marginsmay always safely be provided on shell surfaces for functionalrequirements

512 Effect of Boundary Conditions on Fundamental Fre-quency The boundary conditions may be divided into sixgroups considering number of boundary constraints Thecombinations in a particular group have equal number ofboundary reactions These groups are

Group I CCCC shellsGroup II CSCC CCSC and SCCC shells

Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

International Journal of

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RotatingMachinery

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Journal of Engineering 7

yx

yx

y x y yx

y x

yx

y x

y x

yx

yx

y x

y x

y x

y x

y x

y x

yx

x

yx

y x

yx

yx

yx

y x

y x

y x

y x

yx

y x

yx

y x

y x

yx

yx

yx

y x

yx

yx

yx

y x

yx

y x

yx

yx

yx

yx

y x

yx

y x

y x

y x

y x

y x

yx

y x

yx

y x

y x

y x

yx

y x

yx

y x

y x

y x

0 01 02 03 04

CCCC

CSCC

CCSC

CCCS

CSSC

CCSS

CSCS

SCSC

CSSS

SSSC

SSCS

SSSS

Point supported

Boundary a998400

ararrcondition

darr

Figure 5 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different sizes of the central square cutoutand boundary conditions

Group III CSSC SSCC CSCS and SCSC shells

Group IV CSSS SSSC and SSCS shells

Group V SSSS shells

Group VI Corner point supported shell

It is seen from Tables 3 and 4 that fundamental frequen-cies of members belonging to the same groups of boundarycombinations may not have close values So the differentboundary conditions may be regrouped according to perfor-mance According to the values of 120596 the following groupsmay be identified

8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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8 Journal of Engineering

x yyxy x y xy x y y xx

xyyxy

xy

xy

xy y xx

xyyxy x y xy

xy y xx

x yyx

yx

yxy x y y xx

x yyx

yx

y xy x y y xx

x yyxyx

y xy xy y xx

x yyx

y x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 6 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCCC boundary condition

x yyxy x y xy x y y xx

xyyxy x

yx

yx y y xx

x yyxy xy

xy x y y xx

x yyxy xy xy x y y xx

x yyxy xy

xy xy y xx

x yyxy xy

xy x y y xx

x yyxy x y xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 7 First mode shapes of laminated composite (090090) stiffened conoidal shell for different positions of the square cutout withCCSC boundary condition

Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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Journal of Engineering 9

xyy

xy xy

xyx

y yxx

xyyxy x y x

y x y y xx

x yyxy x yx

y x y y xx

x yyxy x y x

yx y y xx

xyy

xy x y x

yx y y xx

xyy

xy x yx

y x y y xx

xyy

xy xy

xy x y y xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 8 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCCC boundary condition

x yyx

y x yx

yx y y

xx

x yyxy x yx

yx y y

xx

xyy

xy

x yx

yx

y yxx

x yyxyx y

xy x y y

xx

x yyx

y x y xy x y yxx

xyyxy x y

x

y x y yxx

xyyxy x y

x

yx y y

xx

02 03 04 05 06 07 08

02

03

04

05

06

07

08

x

y

rarrdarr

Figure 9 First mode shapes of laminated composite (+45minus45+45minus45) stiffened conoidal shell for different positions of the square cutoutwith CCSC boundary condition

10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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10 Journal of Engineering

Table 3 Non-dimensional fundamental frequencies (120596) for lam-inated composite (090090) stiffened conoidal shell for differentsizes of the central square cutout and different boundary conditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1057679 1189136 124386 1274786 1244929CSCC 793544 878992 914718 964778 970499CCSC 1040259 1173182 1208486 1225918 119586CCCS 790009 86457 911477 958592 970069CSSC 767919 84417 87445 911546 90359CCSS 764645 83126 871684 906361 903194CSCS 708733 760322 80215 865974 919972SCSC 964607 1061886 1122644 1165409 1148718CSSS 657562 695302 728659 775964 811403SSSC 657398 7072 750802 799808 822204SSCS 623308 652487 693811 747187 804246SSSS 547734 566793 59276 623683 652097Pointsupported 211813 21686 226166 242309 255361

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

Table 4 Non-dimensional fundamental frequencies (120596) for lam-inated composite (+45minus45+45minus45) stiffened conoidal shell fordifferent sizes of the central square cutout and different boundaryconditions

Boundaryconditions

Cutout size (1198861015840119886)0 01 02 03 04

CCCC 1375363 149748 156932 1546134 1470345CSCC 1276366 1399845 1426213 1422069 1337437CCSC 1331847 157494 1659908 1576214 1498912CCCS 1239139 13605 1392721 1409553 1349193CSSC 1193132 1292799 1295345 1198123 1074374CCSS 1141784 1242418 1269562 1194147 1073405CSCS 1191939 1292601 1331178 1369644 1284247SCSC 1176296 1292476 135318 1411146 1437385CSSS 1027485 1069623 1095824 1103696 1031902SSSC 1052055 1110251 113482 1107912 1018345SSCS 1030713 1076771 1118604 1188297 1235415SSSS 898159 916626 943912 89284 833021Pointsupported 264022 268267 273666 28542 301836

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 =0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

For cross ply shells

Group 1 Contains CCCC CCSC and SCSC bound-aries which exhibit relatively high frequencies

Group 2 Contains CSCC CCCS CSSC CCSS CSCSSSSC CSSS and SSCS which exhibit intermediatevalues of frequencies

Group 3 Contains SSSS and corner point supportedboundaries which exhibit relatively low values of fre-quencies

Similarly for angle ply shells

Group 1 Contains CCCC CCSC CSCC CCCSCSSC CCSS SCSC and CSCS boundaries whichexhibit relatively high frequencies

Group 2 Contains SSSC CSSS SSCS and SSSS boun-daries which exhibit intermediate values of frequen-cies

Group 3 Contains corner point supported shellswhich exhibit relatively low values of frequencies

It is evident from the present study that the free vibrationcharacteristics mostly depend on the arrangement of bound-ary constraints rather than their actual number It can be seenfrom the present study that if the higher parabolic edge along119909 = 119886 is released from clamped to simply supported there ishardly any change of frequency for cross ply shell But forangle ply shells if the edge along higher parabolic edge isreleased fundamental frequency even increases more thanthat of a clamped shell For cross ply shells if the edge along119910 = 0 or 119910 = 119887 is released that is along the straight edgesfrequency values undergo marked decrease The results indi-cate that the edge along 119910 = 0 or 119910 = 119887 should preferablybe clamped in order to achieve higher frequency values andif the edge has to be released for functional reason the edgealong 119909 = 0 and 119909 = 119886 of a conoid must be clamped to makeup for the loss of frequency But for angle ply shells if anytwo edges are released the change in fundamental frequencyis not so significant

Tables 5 and 6 show the efficiency of a particular clampingoption in improving the fundamental frequency of a shellwith minimum number of boundary constraints relative tothat of a clamped shell Marks are assigned to each boundarycombination in a scale assigning a value of 0 to the frequencyof a corner point supported shell and 100 to that of a fullyclamped shell These marks are furnished for cutouts with1198861015840119886 = 02 These tables will enable a practicing engineer to

realize at a glance the efficiency of a particular boundary con-dition in improving the frequency of a shell taking that ofclamped shell as the upper limit

513 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 4 and5 for cross-ply and angle ply shells respectively The normal-ized displacements are drawnwith the shell midsurface as thereference for all the support conditions and for all the lamina-tions used here The fundamental mode is clearly a bendingmode for all the boundary conditions for cross-ply andangle-ply shells except for corner point supported shell Forcorner point supported shells the fundamental mode shapesare complicated With the introduction of cutout modeshapes remain almost similar When the size of the cutout isincreased from02 to 04 the fundamentalmodes of vibrationdo not change to an appreciable amount

Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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Journal of Engineering 11

Table 5 Clamping options for 090090 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges

Improvement offrequencies with respectto point supported shells

Marks indicating theefficiencies of number

of restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Good improvement 35

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 46(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 49(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 52

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Marked improvement 57

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkableimprovement 88

(c) Any two edges except for the above option(CSSC CCSS) Marked improvement 63

33 edges including the two parabolic edges (CSCCCCCS) Marked improvement 45

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Remarkableimprovement 96

4 All sides (CCCC) Frequency attains thehighest value 100

Table 6 Clamping options for +45minus45+45minus45 conoidal shells with central cutouts having 1198861015840119886 ratio 02

Number of sidesto be clamped Clamped edges Improvement of frequencies with

respect to point supported shells

Marks indicating theefficiencies of number of

restraints0 Corner point supported mdash 00 Simply supported no edges clamped (SSSS) Marked improvement 52

1(a) Higher parabolic edge along 119909 = 119886 (SSCS) Marked improvement 65(b) Lower parabolic edge along 119909 = 0 (CSSS) Marked improvement 64(c) One straight edge along 119910 = 119887 (SSSC) Marked improvement 66

2

(a) Two alternate edges including the higher andlower parabolic edges 119909 = 0 and 119909 = 119886 (CSCS) Remarkable improvement 81

(b) 2 straight edges along 119910 = 0 and 119910 = 119887 (SCSC) Remarkable improvement 83(c) Any two edges except for the above option(CSSC CCSS) Remarkable improvement 77ndash79

33 edges including the two parabolic edges (CSCCCCCS) Remarkable improvement 86ndash89

3 edges excluding the higher parabolic edge along119909 = 119886 (CCSC)

Frequency attains more than afully clamped shell 107

4 All sides (CCCC) Remarkable improvement 100

52 Effect of Eccentricity of Cutout Position

521 Fundamental Frequency The effect of eccentricity ofcutout positions on fundamental frequencies is studied fromthe results obtained for different locations of a cutout with1198861015840119886 = 02 The non-dimensional coordinates of the cutout

centre (119909 = 119909119886 119910 = 119910119886) were varied from 02 to 08 alongeach direction so that the distance of a cutout margin fromthe shell boundary was not less than one tenth of the plandimension of the shell The margins of cutouts were stiffenedwith four stiffeners The study was carried out for all thethirteen boundary conditions for both cross ply and angle ply

shellsThe fundamental frequency of a shell with an eccentriccutout is expressed as a percentage of fundamental frequencyof a shell with a concentric cutoutThis percentage is denotedby 119903 In Tables 7 and 8 such results are furnished

It can be seen that eccentricity of the cutout along thelength of the shell towards the parabolic edges makes it moreflexible It is also seen that towards the lower parabolic edge119903 value is greater than that of the higher parabolic edge Thismeans that if a designer has to provide an eccentric cutoutalong the length he should preferably place it towards thelower height boundary The exception is there in some casesof cross ply shells For cross ply shells with three edges simply

12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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12 Journal of Engineering

Table 7 Values of ldquo119903rdquo for 090090 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 82951 90621 99503 101707 93088 85111 79437

03 82557 90021 99158 103034 93743 85175 79190

04 81946 89117 97773 101280 92554 84271 78405

05 81955 88983 97220 100015 91853 83955 78247

06 81946 89115 97771 101280 92555 84271 78405

07 82557 90020 99159 103036 93743 85175 79189

08 82887 90511 99407 101694 93070 85088 79424

CSCC

02 93862 100890 106419 103740 95957 89329 85169

03 96374 104672 110505 108656 101815 94918 89726

04 93858 102085 108275 106920 100353 93653 88508

05 90589 97867 102329 100 93876 88419 84453

06 88150 94257 97223 94591 89294 84754 81462

07 87105 92080 94299 920577 87583 83555 80475

08 87065 91375 93317 91458 87383 83527 80519

CCSC

02 80918 87869 96510 101953 95020 87039 81087

03 80426 87123 95768 102726 95727 87269 81107

04 79615 85977 94270 101072 94674 86601 80675

05 79448 85705 93791 100 93987 86365 80627

06 79612 85977 94271 101073 94674 86601 80675

07 80426 87122 95770 102726 95727 87268 81106

08 80837 87743 96390 101907 95007 87017 81075

CCCS

02 87175 91751 93546 91679 87294 83462 80475

03 87180 92361 94480 92063 87489 83463 80416

04 88309 94562 97427 94587 89193 84660 81409

05 90816 98202 102571 100 93771 88341 84427

06 94105 102438 108564 106999 100349 93673 88565

07 96611 105025 110815 108872 101974 95062 89875

08 93762 101038 106545 103958 96168 89383 85123

CSSC

02 91560 99045 106094 105722 98527 91427 86004

03 91466 99489 107457 108698 103151 96316 90410

04 89314 96955 104290 105667 101025 94902 89407

05 88004 95094 100516 100 95159 90036 85737

06 87238 93625 97473 95882 91217 86821 83288

07 87033 92615 95568 94032 89939 85975 82704

08 87165 92303 94864 93620 89886 86065 82823

CCSS

02 87339 92674 95098 93607 89769 85968 82749

03 87146 92875 95730 94015 89811 85845 82608

04 87407 93903 97657 95859 91084 86689 83196

05 88217 95390 100738 100 95034 89925 85675

06 89541 97258 104563 105762 101012 94895 89435

07 91697 99797 107751 108900 103289 96435 90531

08 91033 98862 106134 105856 98729 91447 85903

CSCS

02 87176 90165 91941 90278 87528 86028 85511

03 90236 93667 95133 93168 90295 88544 87299

04 93274 97392 98931 97165 94295 92109 89886

05 95713 99837 101259 100 97767 96416 93011

06 93271 97392 98930 97154 94291 92103 89883

07 90232 93666 95133 96907 90294 88544 87296

08 87045 90057 91800 90237 87504 86015 85510

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 13

Table 7 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 86114 93707 101983 100984 91907 84864 79915

03 85570 92996 101670 102014 92324 84978 79910

04 84341 91527 100008 100904 91661 84571 79679

05 83782 90945 99224 100 91321 84472 79663

06 84342 91527 100007 100904 91662 84571 79680

07 85570 92996 101673 102014 92325 84979 79911

08 86038 93588 101881 100967 91891 84845 79907

CSSS

02 90957 94019 96237 95319 93007 91535 90574

03 93329 96650 98463 97318 95163 93907 92929

04 94987 98154 99743 99094 97663 96789 95576

05 96157 98431 100035 100 99376 99252 98900

06 94984 98155 99743 99092 97661 96784 95572

07 93325 96649 98463 97317 95162 93907 92926

08 90826 93898 96032 95227 92974 91510 90575

SSSC

02 97801 107461 112236 106669 97842 91027 86339

03 100320 109209 113744 109522 101475 94421 89126

04 99538 107571 110641 105969 98634 92339 87472

05 97804 105127 105975 100 93276 88232 84425

06 96806 103135 102653 96372 90208 85948 82902

07 96657 101965 101055 95197 89518 85604 82855

08 96784 101605 100554 94955 89461 85654 82981

SSCS

02 93160 96470 95571 91662 88790 87812 87727

03 96515 99494 98334 94512 91801 90728 90027

04 99292 102150 101314 97900 95242 93801 92363

05 100559 103324 102905 100 97629 96284 94946

06 99290 102155 101314 97896 95240 93802 92362

07 96514 99481 98332 94512 9180 90726 90024

08 93059 96355 95442 91626 88753 87798 87733

SSSS

02 91510 96363 98295 97493 95587 93691 91976

03 95252 98790 99925 98895 97114 95560 94407

04 97747 100113 100764 99692 98064 96681 95608

05 98751 100497 101006 100 98510 97233 96154

06 97742 100114 100764 99690 98063 96682 95608

07 95251 98786 99922 98897 97113 95560 94405

08 91451 96286 98154 97412 95567 93666 91951

CS

02 137892 128050 114295 104647 100791 101361 109077

03 135205 125573 111227 101093 97006 97834 106312

04 132315 124337 110519 100187 95990 97045 105585

05 131075 123720 110365 100 95813 96912 105499

06 132292 124356 110550 100211 96017 97088 105603

07 135228 125582 111219 101090 96984 97817 106303

08 135480 125754 112435 102528 98801 99577 107915

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

supported when cutout shifts towards the lower parabolicedge (including the lower parabolic edge) the shell becomesstiffer Again for corner point supported shells 119903 valuesincrease towards the parabolic edges and aremaximum alonglower parabolic edge For cross ply shells four out of thirteenboundary conditions yield the maximum value of 119903 along119909 = 05 four yield maximum values of 119903 along 119909 = 04 andothers show maximum values within 119909 = 04 to 05

It is observed from Table 7 that if the eccentricity of acutout is varied along the width the shell becomes stifferwhen the cutout shifts towards clamped edges So for func-tional purposes if a shift of central cutout is required eccen-tricity of a cutout along the width should preferably betowards the clamped straight edge For shells having twostraight edges of identical boundary condition themaximumfundamental frequency occurs along 119910 = 05 For corner

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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Control Scienceand Engineering

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Journal of Engineering

Table 8 Values of ldquo119903rdquo for +45minus45+45minus45 conoidal shells

Edge condition 119910119909

02 03 04 05 06 07 08

CCCC

02 72823 78236 82758 83828 81568 76803 71923

03 74884 80770 86548 88893 86815 81249 75551

04 76701 83522 91226 95703 93788 87111 80040

05 77112 84768 94005 100 97403 89411 81517

06 76805 83652 91325 95680 93763 87160 80108

07 75008 80906 86626 88855 86795 81298 75617

08 72647 77908 82266 83759 81579 76633 71755

CSCC

02 81077 91056 97803 91446 86032 83235 78655

03 83525 93130 102170 99590 94922 89192 82211

04 84074 93824 101696 103807 101498 94596 86412

05 83003 90427 97733 100 99474 96299 87547

06 81276 87968 94605 96334 95817 92081 84895

07 79646 86287 92866 94104 92136 86701 80220

08 78630 85619 92366 92585 89172 83321 77139

CCSC

02 70077 77727 85937 83800 79229 74361 69326

03 71072 78703 87768 88368 83336 77766 72302

04 71984 80017 89895 95290 89759 82905 76264

05 72468 81214 91396 100 93470 85277 77810

06 71937 79893 89673 95311 89871 83044 76355

07 71086 78669 87710 88508 83481 77891 72371

08 70006 77569 85788 83973 79340 74231 69123

CCCS

02 78059 85085 91568 92620 90166 84849 78646

03 78792 85576 91983 93758 92489 87571 81169

04 80502 87433 93909 95951 95851 92337 85557

05 82841 90481 97657 100 99876 96861 88986

06 85401 94488 102330 104306 102177 96765 88819

07 85313 94803 103136 100952 96520 91365 84617

08 81938 91745 98552 93014 87371 84653 80319

CSSC

02 80251 90347 102001 98147 92559 90103 85813

03 81688 90911 103714 105159 101546 97766 88928

04 81340 89735 99850 103865 102503 98783 91054

05 81528 89780 96778 100 99635 97691 92230

06 81914 89289 95251 97877 98401 97867 92317

07 81213 88071 94478 97330 98425 94980 87252

08 80276 87186 94253 97479 97826 90750 83637

CCSS

02 79271 85734 92215 96177 97750 92440 85311

03 80024 86471 92663 96147 97781 95350 88242

04 81065 87896 94083 97107 98178 97531 92578

05 82087 90360 97023 100 100117 98606 94251

06 82521 90825 101803 104549 103368 100351 93755

07 82941 91821 103719 105727 102486 99509 91399

08 80346 90089 100955 99195 93707 91345 87507

CSCS

02 78163 86022 92995 92450 88854 85808 80401

03 79871 86979 93768 95528 94670 90304 83255

04 81380 88475 95483 97758 98510 95335 87720

05 83417 91209 98288 100 101457 99522 90636

06 83285 90593 97462 99337 99875 96908 89038

07 81807 89179 96019 97206 95825 91360 84354

08 79311 87551 94710 93330 88907 85864 80620

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

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DistributedSensor Networks

International Journal of

Journal of Engineering 15

Table 8 Continued

Edge condition 119910119909

02 03 04 05 06 07 08

SCSC

02 85127 94329 95136 88093 85184 82426 78466

03 86016 95544 99993 92933 89052 85588 80885

04 86853 97202 105501 97941 92882 88930 83293

05 87280 98396 107888 100 94353 90351 84256

06 86654 96782 104675 97456 92684 88887 83209

07 85821 95213 99433 92519 89003 85631 80778

08 84798 93899 94867 87932 85323 82385 78165

CSSS

02 84944 92468 99487 102083 102496 102211 95697

03 87001 94415 100612 101921 103359 105519 99083

04 87482 95487 101492 100522 100612 102250 100822

05 87931 97254 102629 100 99420 100620 99680

06 88552 96975 102732 101723 101674 102885 100813

07 88608 96436 102875 103969 105120 106825 99475

08 85793 93995 101843 104209 103551 102657 95378

SSSC

02 90301 101510 103781 96224 91653 89271 87137

03 91100 102460 108098 102693 98673 95766 90177

04 90295 101404 106028 102757 98698 95253 90038

05 89523 99110 102720 100 96670 94017 90034

06 88793 96857 100664 98472 96351 94739 91563

07 88172 95669 99702 97818 96797 96041 90722

08 87395 94930 98931 96571 95997 95427 88365

SSCS

02 88795 97759 99936 94981 92168 90694 87596

03 90150 98974 101928 98665 97216 96201 89404

04 91309 100310 103409 99753 97841 97300 92291

05 92288 101662 104357 100 97487 96678 93643

06 92516 101539 104514 100363 98033 97305 92917

07 91859 100918 103929 99899 98067 97444 90291

08 89927 99404 101683 95907 92707 91701 87789

SSSS

02 88805 93985 97537 99443 9934104 97458 93660

03 90116 95853 98238 99300 9955631 98869 96461

04 89799 95332 98006 97770 96847 96012 94848

05 89665 94925 97577 100 95462 94360 93375

06 90247 95512 98509 98722 97792 96672 95013

07 90943 96728 99725 101290 101508 100518 97141

08 88912 94653 99280 101663 101056 99261 95529

CS

02 124611 120401 110994 104153 102644 103581 108893

03 120487 117591 108489 101849 100596 101659 107324

04 117053 116177 107584 100426 99256 100691 105789

05 115794 115786 107429 100 98788 100425 105330

06 118003 116844 107793 100623 99239 100341 105584

07 122490 118779 109260 102384 100470 100978 106534

08 122076 118091 109558 102712 101196 101869 107386

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

point supported shells themaximum fundamental frequencyalways occurs along the boundary of the shell All these aretrue for cross ply shells only For an angle ply shell suchunified trend is not observed and the boundary conditionsand the fundamental frequency behave in a complex manneras evident from Table 8 But for corner point supported

angle-ply shells also the maximum values of 119903 are along theboundary

Tables 9 and 10 provide themaximum values of 119903 togetherwith the position of the cutout These tables also show therectangular zones within which 119903 is always greater than orequal to 95 and 90 It is to be noted that at some other points

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Journal of Engineering

Table 9 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for 090090 conoidalshells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 103036 119909 = 05 119910 = 07119909 = 06

02 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSCC 110505 119909 = 04 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSC 102726 119909 = 05 119910 = 03

119909 = 05 119910 = 07

119909 = 04 0602 le 119910 le 08

119909 = 0502 le 119910 le 08

CCCS 110815 119909 = 04 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSSC 108698 119909 = 05 119910 = 0303 le 119909 0506 le 119910 le 08

03 le 119909 0602 le 119910 le 05

CCSS 108900 119909 = 05 119910 = 0703 le 119909 0502 le 119910 le 04

03 le 119909 0605 le 119910 le 08

CSCS 101259 119909 = 04 119910 = 05

03 le 119909 0502 le 119910 le 03

07 le 119910 le 08

03 le 119909 0504 le 119910 le 06

SCSC 102014 119909 = 05 119910 = 07119909 = 03 0502 le 119910 le 08

04 le 119909 0502 le 119910 le 08

CSSS 100035 119909 = 04 119910 = 0502 le 119909 0802 le 119910 le 08

03 le 119909 0603 le 119910 le 07

SSSC 113744 119909 = 04 119910 = 0306 le 119909 0702 le 119910 le 04

02 le 119909 0502 le 119910 le 08

SSCS 103324 119909 = 03 119910 = 0505 le 119909 0803 le 119910 le 07

02 le 119909 0403 le 119910 le 07

SSSS 101006 119909 = 04 119910 = 05119909 = 08

02 le 119910 le 08

02 le 119909 0703 le 119910 le 07

CS 137892 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

119903 values may have similar values but only the zone rectan-gular in plan has been identified This study identifies thespecific zones within which the cutout centre may be movedso that the loss of frequency is less than 5 or 10 respec-tively with respect to a shell with a central cutout This willhelp a practicing engineer to make a decision regarding theeccentricity of the cutout centre that can be allowed

522 Mode Shapes The mode shapes corresponding to thefundamental modes of vibration are plotted in Figures 6 7 8and 9 for cross-ply and angle-ply shells of CCCC and CCSCshells for different eccentric positions of the cutout As CCCCand CCSC shells are most efficient with respect to numberof restraints the mode shapes of these shells are shown astypical results All the mode shapes are bending modes It isfound that for different position of cutout mode shapes aresomewhat similar only the crest and trough positions change

The present study considers the dynamic characteristicsof stiffened composite conoidal shells with square cutout interms of the natural frequency and mode shapes The size ofthe cutouts and their positions with respect to the shell centreare varied for different edge constraints of cross-ply andangle-ply laminated composite conoids The effects of theseparametric variations on the fundamental frequencies and

mode shapes are considered in details However the effectof the shape and orientation of the cutout on the dynamiccharacters of the conoid has not been considered in the pre-sent study Future studies will evaluate these aspects

6 Conclusions

The following conclusions are drawn from the present study

(1) As this approach produces results in close agreementwith those of the benchmark problems the finiteelement code used here is suitable for analyzing freevibration problems of stiffened conoidal roof panelswith cutouts The present study reveals that cutoutswith stiffened margins may always safely be providedon shell surfaces for functional requirements

(2) The arrangement of boundary constraints along thefour edges is far more important than their actualnumber so far the free vibration is concernedThe relative-free vibration performances of shells fordifferent combinations of edge conditions along thefour sides are expected to be very useful in decisionmaking for practicing engineers

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 17

Table 10 Maximum values of 119903 with corresponding coordinates of cutout centre and zones where 119903 ge 90 and 119903 ge 95 for +45minus45+45minus45conoidal shells

Boundarycondition

Maximum valuesof 119903 Co-ordinate of cutout centre Area in which the value of

119903 ge 90

Area in which the value of119903 ge 95

CCCC 100000 119909 = 05 119910 = 05119909 = 04 0604 le 119910 le 06

119909 = 0504 le 119910 le 06

CSCC 103807 119909 = 05 119910 = 04119909 = 03 02 le 119910 le 0506 le 119909 07 03 le 119910 le 06

04 le 119909 le 0503 le 119910 le 05

CCSC 100000 119909 = 05 119910 = 0504 le 119909 le 06

119910 = 05

119909 = 05

04 le 119910 le 06

CCCS 104306 119909 = 05 119910 = 0604 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 06

CSSC 105159 119909 = 05 119910 = 03

04 le 119909 le 0707 le 119910 le 08

119909 = 08 04 le 119910 le 06

04 le 119909 le 0703 le 119910 le 06

CCSS 105727 119909 = 05 119910 = 07

119909 = 05 0702 le 119910 le 04

119909 = 03 0805 le 119910 le 07

05 le 119909 le 0602 le 119910 le 04

04 le 119909 le 0705 le 119910 le 07

CSCS 101457 119909 = 06 119910 = 0504 le 119909 le 07

119910 = 03

04 le 119909 le 0704 le 119910 le 07

SCSC 107888 119909 = 04 119910 = 0505 le 119909 le 0604 le 119910 le 06

03 le 119909 le 0403 le 119910 le 07

CSSS 106825 119909 = 07 119910 = 07119909 = 03

02 le 119910 le 08

04 le 119909 le 0802 le 119910 le 08

SSSC 108098 119909 = 04 119910 = 0307 le 119909 le 0803 le 119910 le 07

03 le 119909 le 0602 le 119910 le 07

SSCS 104514 119909 = 04 119910 = 06119909 = 02 0803 le 119910 le 07

03 le 119909 le 0703 le 119910 le 07

SSSS 101663 119909 = 05 119910 = 08119909 = 03 0802 le 119910 le 08

04 le 119909 le 0702 le 119910 le 08

CS 124611 119909 = 02 119910 = 02 nil 02 le 119909 0802 le 119910 le 08

119886119887 = 1 119886ℎ = 100 11988610158401198871015840 = 1 119886ℎℎ = 5 ℎ119897ℎℎ = 025 1198641111986422 = 25 11986623 = 0211986422 11986613 = 11986612 = 0511986422 and ]12 = ]21 = 025

(3) The information regarding the behaviour of stiffenedconoids with eccentric cutouts for a wide spectrumof eccentricity and boundary conditions for cross plyand angle ply shells may also be used as design aidsfor structural engineers

Notations

119886 119887 Length and width of shell in plan1198861015840 1198871015840 Length and width of cutout in plan

119887st Width of stiffener in general119887119904119909 119887119904119910 Width of119883- and 119884-stiffeners respectively119861119904119909 119861119904119910 Strain-displacement matrix of stiffener

elements119889st Depth of stiffener in general119889119904119909 119889119904119910 Depth of119883- and 119884-stiffeners

respectively119889119890 Element displacement119890 Eccentricities of both 119909- and 119910-direction

stiffeners with respect to shell midsurface11986411 11986422 Elastic moduli

11986612 11986613 11986623 Shear moduli of a lamina with respect to 12 and 3 axes of fibre

ℎ Shell thicknessℎℎ Higher height of conoidℎ119897 Lower height of conoid119872119909 119872119910 Moment resultants119872119909119910 Torsion resultant119899119901 Number of plies in a laminate1198731ndash1198738 Shape functions119873119909 119873119910 Inplane force resultants119873119909119910 Inplane shear resultant119876119909 119876119910 Transverse shear resultant119877119910 119877119909119910 Radii of curvature and cross curvature of

shell respectively119906 V 119908 Translational degrees of freedom119909 119910 119911 Local coordinate axes119883 119884 119885 Global coordinate axes119911119896 Distance of bottom of the 119896th ply from

midsurface of a laminate120572 120573 Rotational degrees of freedom

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

18 Journal of Engineering

120576119909 120576119910 Inplane strain component120601 Angle of twist120574119909119910 120574119909119911 120574119910119911 Shearing strain components]12 ]21 Poissonrsquos ratios120585 120578 120591 Isoparametric coordinates120588 Density of material120590119909 120590119910 Inplane stress components120591119909119910 120591119909119911 120591119910119911 Shearing stress components120596 Natural frequency120596 Nondimensional natural

frequency = 1205961198862(12058811986422ℎ

2)12

References

[1] H A Hadid An analytical and experimental investigation intothe bending theory of elastic conoidal shells [PhD thesis] Uni-versity of Southampton 1964

[2] C Brebbia and H Hadid Analysis of plates and shells usingrectangular curved elements CE571 Civil engineering depart-ment University of Southampton 1971

[3] C K Choi ldquoA conoidal shell analysis bymodified isoparametricelementrdquo Computers and Structures vol 18 no 5 pp 921ndash9241984

[4] B Ghosh and J N Bandyopadhyay ldquoBending analysis of con-oidal shells using curved quadratic isoparametric elementrdquoComputers and Structures vol 33 no 3 pp 717ndash728 1989

[5] BGhosh and JN Bandyopadhyay ldquoApproximate bending anal-ysis of conoidal shells using the galerkin methodrdquo Computersand Structures vol 36 no 5 pp 801ndash805 1990

[6] A Dey J N Bandyopadhyay and P K Sinha ldquoFinite elementanalysis of laminated composite conoidal shell structuresrdquoComputers and Structures vol 43 no 3 pp 469ndash476 1992

[7] A K Das and J N Bandyopadhyay ldquoTheoretical and experi-mental studies on conoidal shellsrdquo Computers and Structuresvol 49 no 3 pp 531ndash536 1993

[8] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFreevibration analysis of point-supported laminated compositedoubly curved shells-Afinite element approachrdquoComputers andStructures vol 54 no 2 pp 191ndash198 1995

[9] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of point supported laminatedcomposite cylindrical shellsrdquo Journal of Sound and Vibrationvol 181 no 1 pp 43ndash52 1995

[10] D Chakravorty J N Bandyopadhyay and P K Sinha ldquoFiniteelement free vibration analysis of doubly curved laminatedcomposite shellsrdquo Journal of Sound and Vibration vol 191 no4 pp 491ndash504 1996

[11] D Chakravorty P K Sinha and J N Bandyopadhyay ldquoApplica-tions of FEM on free and forced vibration of laminated shellsrdquoJournal of Engineering Mechanics vol 124 no 1 pp 1ndash8 1998

[12] A N Nayak and J N Bandyopadhyay ldquoOn the free vibrationof stiffened shallow shellsrdquo Journal of Sound and Vibration vol255 no 2 pp 357ndash382 2003

[13] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisand design aids of stiffened conoidal shellsrdquo Journal of Engineer-ing Mechanics vol 128 no 4 pp 419ndash427 2002

[14] A N Nayak and J N Bandyopadhyay ldquoFree vibration analysisof laminated stiffened shellsrdquo Journal of Engineering Mechanicsvol 131 no 1 pp 100ndash105 2005

[15] ANNayak and JN Bandyopadhyay ldquoDynamic response anal-ysis of stiffened conoidal shellsrdquo Journal of Sound and Vibrationvol 291 no 3ndash5 pp 1288ndash1297 2006

[16] H S Das andD Chakravorty ldquoDesign aids and selection guide-lines for composite conoidal shell roofsmdasha finite element appli-cationrdquo Journal of Reinforced Plastics and Composites vol 26no 17 pp 1793ndash1819 2007

[17] H S Das and D Chakravorty ldquoNatural frequencies and modeshapes of composite conoids with complicated boundary con-ditionsrdquo Journal of Reinforced Plastics and Composites vol 27no 13 pp 1397ndash1415 2008

[18] S S Hota and D Chakravorty ldquoFree vibration of stiffened con-oidal shell roofs with cutoutsrdquo JVCJournal of Vibration andControl vol 13 no 3 pp 221ndash240 2007

[19] M S Qatu E Asadi andWWang ldquoReview of recent literatureon static analyses of composite shells 2000ndash2010rdquoOpen Journalof Composite Materials vol 2 pp 61ndash86 2012

[20] M S Qatu R W Sullivan and W Wang ldquoRecent researchadvances on the dynamic analysis of composite shells 2000ndash2009rdquo Composite Structures vol 93 no 1 pp 14ndash31 2010

[21] V V Vasiliev R M Jones and L L Man Mechanics of Com-posite Structures Taylor and Francis New York NY USA 1993

[22] M S Qatu Vibration of Laminated Shells and Plates ElsevierLondon UK 2004

[23] S Sahoo and D Chakravorty ldquoFinite element bendingbehaviour of composite hyperbolic paraboloidal shells with var-ious edge conditionsrdquo Journal of Strain Analysis for EngineeringDesign vol 39 no 5 pp 499ndash513 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of