first ply failure study of composite conoidal shells used as roofing units in civil engineering

TECHNICAL ARTICLE—PEER-REVIEWED

First Ply Failure Study of Composite Conoidal ShellsUsed as Roofing Units in Civil Engineering

Kaustav Bakshi • Dipankar Chakravorty

Submitted: 26 February 2013 / in revised form: 3 July 2013

� ASM International 2013

Abstract In practical civil engineering, the necessity of

covering large column free open areas with shell surfaces is

often an issue. Such areas in medicinal plants and auto-

mobile industries prefer entry of north light through the

roofing units. Doubly curved singly ruled conoidal shells

are stiff and easy to fabricate as surfaces and fit excellently

to the above-mentioned industrial requirements. Today, the

engineers intend to use laminated composites to fabricate

these shell forms. Engineers are also concerned with the

performance evaluation of different stacking sequences to

maximize the stiffness for a given quantity of material

consumption. First ply failure load analysis of composite

plates appears abundantly in the literature, but on com-

posite shells, only a few papers are found (though not on

conoidal shells). This paper addresses an important issue

with which the practical engineers are often concerned

regarding performance evaluation of different laminations

(including antisymmetric and symmetric cross and angle

plies) in terms of first ply failure load of composite

conoids. The paper uses the finite element method as the

mathematical tool and concludes logically to a set of

inferences of practical engineering significance.

Keywords Composite materials � Conoidal shells �Finite element method � Failure investigations �Failure loads � Failure modes

Symbols

A Area of the shell

{d} Displacements of the shell

{de} Element displacements

E11, E22, E33 Elastic moduli

1, 2 and 3 Local coordinates of a lamina

G12, G23, G13 Shear moduli

ne Number of elements

Ryy Radius of curvature of the

conoidal shell along the ‘‘y’’ axis

Rxy Radius of cross curvature

of the conoidal shell

T Shear strength of lamina

Te Allowable shear strain of lamina

v/ Volume of the shell

XT, XC Normal strengths of lamina in tension

and compression, respectively

XeT, XeC Allowable normal strains of lamina

in tension and compression, respectively

�y y/b

YT, YC Normal strengths of matrix in

tension and compression,

respectively

YeT, YeC Allowable normal strains of matrix

in tension and compression,

respectively

mij Poisson’s ratio

r1, r2 Normal stresses acting along

1 and 2 axes of a lamina,

respectively

r6 Shear stress acting on 1–2 surface

of a lamina

sxy, sxz, syz Shear stresses of the shell

kx, ky, kxy Curvature changes of the shell

due to loading

K. Bakshi (&) � D. Chakravorty

Department of Civil Engineering, Jadavpur University,

Kolkata 700 032, India

e-mail: [email protected]

D. Chakravorty

e-mail: [email protected]

123

J Fail. Anal. and Preven.

DOI 10.1007/s11668-013-9725-y

Introduction

In civil engineering, it is advantageous to use thin shells

instead of flat plates to cover large column free open spaces

as one sees in airports, parking lots, hangers, and the like.

Doubly curved shell forms are more esthetically appealing

and rigid structures compared to the singly curved ones and

they enjoy a preference in the industry. Among the dif-

ferent doubly curved shell forms used in the civil

engineering, the conoidal shells are highly preferred by the

practicing engineers as these shells have ruled surfaces and

hence are easy to cast and fabricate. These shells are

esthetically appealing too. The civil engineers started to

use the laminated composites to fabricate the conoidal

shells from the second half of the last century as the high

specific stiffness and strength properties of these materials

result in less gravity forces and mass-induced forces

(seismic force) on the laminated shells compared to their

isotropic counterparts. All these taken together reduce the

foundation costs to a great extent.

Realizing the importance of laminated conoids in the

industry, different aspects of these shells are being reported

by different researchers even today. A number of

researchers like Nayak and Bandyopadhyay [1–3], Das and

Chakravorty [4–6], Kumari and Chakravorty [7, 8], and

Pradyumna and Bandyopadhyay [9, 10] are still working

on static bending, free and forced vibration responses of

conoidal shells. Free and forced vibration responses of the

conoidal shells were reported by Nayak and Bandyopad-

hyay [1–3]. Das and Chakravorty [4–6] used the finite

element method to study bending and free vibration char-

acteristics of graphite-epoxy conoidal shells with and

without stiffeners, while bending response of the delami-

nated graphite-epoxy conoidal shells was reported by

Kumari and Chakravorty [7, 8]. Pradyumna and Bandyo-

padhyay [9] carried out static bending and free vibration

investigations on laminated composite conoidal shells

using the higher-order shear deformation theory. The

authors [10] further worked on dynamic instability

behavior of laminated conoids.

As a wide spectrum of advantages is offered by the

composites, one needs to know and explore the material

characteristics comprehensively and failure study of these

materials is a part of that overall study. Failure of lami-

nated composites is progressive in nature (as it was

observed by Singh and Kumar [11], Akhras and Li [12],

and Ganesan and Liu [13]) initiating with the first ply

failure and finally culminating in the ultimate ply failure.

The aforesaid authors also reported that the ultimate failure

load of laminated composites is much higher than the

corresponding first ply failure load. Therefore, assigning

the engineering factor of safety to the first ply failure load

underestimates the ultimate material capacity and leads to a

highly conservative design. However, it is important to

know the load value at which failure initiates, otherwise the

latent damages progress gradually within the laminate and

lead to a sudden total ply failure under service conditions.

In a review of the literature to follow, the focus is on papers

published on first ply failure of composite plates and shells

and an attempt is made to identify an area where failure

study may be extended.

Pandey and Reddy [14] worked on first ply failure of

simply supported graphite-epoxy plates using the linear

strain displacement relationship and four-noded and nine-

noded Lagrangian elements. Linear and nonlinear first ply

failure loads of thin and thick plates subjected to transverse

and in-plane loads were reported by Reddy and Reddy [15].

Experimental first ply failure loads of centrally loaded

square plates were reported by Kam and Jan [16], Kam and

Sher [17], and Kam et al. [18]. The authors also carried out

theoretical failure investigations of those plates using lay-

erwise linear displacement theory [16], the Ritz method

[17], and finite element method [18]. It was also found that

the finite element method yields fairly close results to the

experimental values. Turvey and Osman [19] worked on

nonlinear failure of carbon-epoxy unsymmetrical cross-ply

laminated strips (i.e., large aspect ratio rectangular plates)

subjected to uniform lateral pressure. Linear and nonlinear

first ply failure loads of graphite-epoxy thick and thin

plates subjected to uniformly distributed load were repor-

ted by Sciuva et al. [20]. Cho and Yoon [21] used the

postprocess method to study first ply failure loads of cross-

ply laminated plates subjected to uniformly distributed and

sinusoidal loads. The proposed method was validated by

comparing the results with the values obtained from exact

elastic analysis and first-order shear deformation theory.

First and ultimate ply failure loads along with the buckling

loads of symmetrically laminated plates under positive and

negative in-plane shear were reported by Singh and Kumar

[11]. The buckling loads of the plates were found to be

considerably small when compared to the failure loads of

those plates. The first ply failure loads of stiffened plates

were studied by Ray and Satsangi [22] and Kumar and

Srivastava [23]. The authors used geometrically linear

finite element formulations in their studies. Chang and

Chiang [24] carried out experimental and theoretical first

ply failure investigations of laminated composite plates,

and a detailed review on damage modeling and finite ele-

ment analysis for composite laminates was reported by

Zheng and Liu [25]. Recently, Lal et al. [26] worked on

stochastic nonlinear failure analysis of laminated compos-

ite plates under compressive transverse loading. First ply

failure loads of cylindrical and spherical shell panels were

obtained by Prusty et al. [27]. The authors used linear finite

element formulation to evaluate first ply failure loads of

both bare and stiffened shells.


123

The perusal of the literature reveals that research reports

on first ply failure loads of laminated composite plates are

available abundantly, but similar work on shells is much

less in general and absolutely missing for conoidal shells.

Although different behavioral aspects like static bending,

free vibration, and instability behaviors of laminated co-

noids are reported recently, a confident application of such

shells needs detailed knowledge about the load which the

shell is capable to withstand before failure. Hence, the

present study intends to report the first ply failure of uni-

formly loaded clamped composite conoidal shells using

established failure criteria. Moreover, the failure modes

and the failure tendencies in the case of interactive failure

and failure locations are studied critically to grossly

understand the failure behavior of the composite conoids.

A number of antisymmetric and symmetric cross and

angle-ply laminates are analyzed to find the lamination that

would best suit a practical situation with a given quantity of

material consumption.

Mathematical Formulation

A doubly curved laminated composite conoidal shell

(Fig. 1) of uniform thickness h and radii of curvatures Ryy

and Rxy is considered in the present study. The thickness of

the conoidal shell (h) consists of any numbers of thin

laminae which are bonded together to form the laminate.

Fibers in each lamina of the composite shell are oriented

along the local axis of the lamina and at an angle h with

reference to the global x-axis (Fig. 1). The x and y axes are

taken at the mid-surface of the shell which has sides a and

b, respectively.

Strain–Displacement and Constitutive Relationship

of the Shell

First-order shear deformation theory is used in the present

study. The strain–displacement [B/] and constitutive rela-

tionship [D/] matrices of the shell adopted in the present

study are those used by Kumari and Chakravorty [8]. While

obtaining the shear stress resultants, proper shear correc-

tion factors (as used by Kumari and Chakravorty [8]) are

applied. Though the authors [8] worked on delaminated

conoids, the matrices ([B/], [D/], and [N]) adopted here are

for the undelaminated ones.

Finite Element Formulation

A displacement-based isoparametric finite element formu-

lation is developed in the present study using eight-noded

curved quadratic elements with C0 continuity as depicted in

Fig. 2. There are five degrees of freedom considered at

each node of the element including three displacements

Fig. 1 A uniformly loaded

laminated composite conoidal

shell


123

(u, v, and w) and two rotations (a and b). The displacement u

is considered to act along the x-axis, v tangential to the mid-

surface, and w normal to it. The rotations a and b are acting

about y and x axes of the shell, respectively. The element

degrees of freedom {u} are interpolated from their nodal

values {de} with the help of the interpolation functions [N].

uf g ¼ N½ � def g

where uf g ¼ u v w a bf gTand ui vi wi ai bif g

i ¼ 1 to 8

The interpolation functions [N] used in the present for-

mulation are adopted from Kumari and Chakravorty [8].

Governing Equations

For an elastic continuum undergoing small deformations,

the governing differential equation is derived based on

minimization of the total potential energy. The total

potential energy p is expressed as sum of strain energy U

and work done due to external load W.

p ¼ U þW

Strain energy of the shell is expressed by the volume

integral,

U ¼ 1

2

Z

v=

egf T rgf dv ðEq 1Þ

By using the laminate constitutive relationship matrix

[D/], Eq. (1) can be written as,

U ¼ 1

2

ZZA

ef gTD=h i

ef gdxdy ðEq 2Þ

By using the strain–displacement matrix [B/], Eq. (2)

can be written as,

U ¼ 1

2

Z Z

A

B=h iT

df gTD=h i

B=h i

df g dxdy ðEq 3Þ

Work done by external load (W) is expressed as an area

integral,

W ¼ �Z Z

A

df gTqf g dA ðEq 4Þ

External load on the shell can be expressed as

qf g ¼ 0 0 qz 0 0f gT, where qz represents

transverse load intensity on the shell.

To minimize the total potential energy of the shell with

respect to its deformations, the shell has to satisfy the

following condition:

opofdg ¼ 0 ðEq 5Þ

Consider the laminated composite shell discretized into

ne elements. The strain energy and work done of the shell

are expressed, respectively, as

U ¼Xne

i¼1

Ui

¼ 1

2

Z Z

A

B=h iT

D=h i

B=h i

dxdy

24

35

i

def gTdef g

ðEq 6Þ

W ¼Xne

i¼1

Wi ¼ �Z Z

A

N½ �T q½ � dx dy

24

35

i

def gT ðEq 7Þ

By applying Eqs. (6) and (7), Eq. (5) can be written as,

Xne

i¼1

Z Z

A

B=h iT

D=h i

B=h i

dx dy

0@

1A

24

35

i

def g

�ZZ

A

½N�T ½q��

dxdy

� �i

¼ 0

Xne

i¼1

Ki½ � def g ¼ Qif g ðEq 8Þ

where Ki½ � ¼ ½R R

A

B=� �T

D=� �

B=� �

dx dy�i and Qif g ¼ ½R R

AN½ �T ½q�dx dy�i.The element stiffness matrix [Ki] and load vector {Qi}

are transformed to isoparametric coordinates n and g,

respectively, for numerical integration by 2 9 2 Gauss

quadrature rule. Global stiffness matrix and load vector are

obtained by assembling the element matrices with proper

transformations due to the curved geometry of the shell and

they are expressed as,

K½ � df g ¼ Qf g ðEq 9Þ

where K½ � ¼Pne

i¼1

Ki½ � and Q½ � ¼Pne

i¼1

Qi½ �.

Equation (9) is solved by the Gauss Elimination Method.

Fig. 2 The shell element


123

Lamina Stress Calculation

The displacements of the laminated shell obtained from the

solution of Eq. (9) are used to obtain the strain vector. The

strains acting on the surface of a lamina situated at a dis-

tance z from the laminate mid-surface are evaluated in

global axes as

ex ¼ e0x þ zkx; ey ¼ e0

y þ zky and cxy ¼ c0xy þ zkxy ðEq 10Þ

where ex, ey are the strains acting along x and y axes of the

shell and cxy is the shear strain working on x–y surface. ex0,

ey0, and cxy

0 denote corresponding strains at the laminate

mid-surface.

Lamina strains are transformed from the global axes of

the shell to the local axes of the lamina using the trans-

formation matrix

e1

e2e6

2

9=;

8<: ¼

sin2 h cos2 h 2sinhcoshcos2 h sin2 h �2sinhcosh

� sinhcosh sinhcosh sin2 h� cos2 h

24

35 ex

eycxy

2

9=;

8<:ðEq 11Þ

where e1, e2 are the strains of lamina acting along the 1 and

2 axes. e6 is the shear strain working on 1–2 surfaces of the

lamina. Lamina stresses are obtained using the constitutive

relationship of the lamina.

r1

r2

r6

9=;

8<: ¼

Q11 Q12 0

Q12 Q22 0

0 0 Q66

24

35 e1

e2

e6

9=;

8<: ðEq 12Þ

where Q12 ¼ 1� m12m21ð Þ�1E11m21; Q11 ¼ 1� m12m21ð Þ�1

E11; Q22 ¼ 1� m12m21ð Þ�1E22, and Q66 = G12.

Well-accepted failure theories like the maximum stress,

maximum strain, Hoffman, Tsai-Hill, and Tsai-Wu failure

criterion are used to obtain the failure loads of the com-

posite conoid. The failure theories are adopted from Reddy

and Reddy [15].

Failure Modes

Failure modes of a composite lamina are dependent on the

applied load. If a lamina is subjected to tensile stresses,

then its failure modes could be fiber breakage, transverse

matrix cracking in the surface of the lamina, or shear

failure of the matrix. In the case of compressive stresses,

the failure modes could be fiber buckling which dominates

the failure of the lamina along the fiber direction, and

matrix crushing leads to the failure of the composite

matrix. All these failure modes can be identified through

maximum stress and maximum strain failure theories fol-

lowing the guidelines furnished in Tables 1 and 2.

It is important to note here that in interactive failure

theories, individual lamina stresses are not utilized up to

their full strength. On the contrary, interaction of the

lamina stresses leads to failure. In the case of such failure,

the individual stress values developed may be compared to

their corresponding permissible values to investigate which

stress component contributing to the interactive criteria

plays the most significant role in the failure. The stress

component for which the ratio of developed to permissible

stress is nearest to unity may be identified as the most

significant component contributing to the failure following

the guidelines furnished in Tables 1 and 2.

Numerical Problems

The finite element formulation discussed here is used to

solve a number of numerical problems. The formulation is

first used to study the static deflections of a uniformly

loaded isotropic conoidal shell. The spatial variation of

static displacements along x-axis of the shell is reported in

Fig. 3 with which the material and geometric properties of

the shell are also furnished. The present results are com-

pared with the values published by Hadid [28]. The

material properties are carefully adjusted in the current

computer code developed for composite shells to model the

isotropic material as a special one.

In order to verify the capability of the current code in

predicting accurately the first ply failure loads, the linear

failure loads of a laminated composite plate are evaluated

and the values are compared with those originally reported

by Kam et al. [18] for a partially clamped plate. In-plane

degrees of freedom along the plate boundaries are released

Table 1 Identification of failure modes/tendencies using maximum

stress theory

Stress ratio Failure mode

r1

XT[ 1 Fiber breakage

r2

YT[ 1 Matrix cracking

r6j jT

[ 1 Shear failure of the matrixr1

XC[ 1 Fiber buckling

r2

YC[ 1 Matrix crushing

Table 2 Identification of failure modes/tendencies using maximum

strain theory

Strain ratio Failure mode

e1

XeT[ 1 Fiber breakage

e2

YeT[ 1 Matrix cracking

e6j jTe

[ 1 Shear failure of the matrixe1

XeC[ 1 Fiber buckling

e2

YeC[ 1 Matrix crushing


123

to model the partially clamped boundary condition. The

comparative results are reported in Table 3. Dimensions of

the plate are furnished in Table 3 and the material prop-

erties of the graphite-epoxy plate used by Kam et al. [18]

are the same as those adopted here and are reported in

Table 4. The radii of curvatures of the present element are

assigned high values (1030) to model the plate.

Once the accuracy of the present formulation is verified, it

is used to compute first ply failure loads of laminated com-

posite clamped conoidal shells (refer Fig. 4 for boundary

condition). A number of shallow, moderately thin conoidal

shells having varying laminations including symmetric and

antisymmetric stacking sequences of cross and angle-ply

laminates are considered for the additional problems. The

shells are taken to be loaded by uniformly distributed surface

pressure. The first ply failure load is achieved by increasing

the magnitude of the pressure in a stepwise manner and by

continuously checking the developed stresses and strains

against the failure criteria. The first ply failure loads of these

shells are furnished in Table 6 along with the locations from

where the failure initiates. The modes and tendencies of

failure are also furnished. Table 6 must be read in conjunc-

tion with Tables 4 and 5 where the material and geometric

properties, respectively, of the conoidal shells are reported.

Plies are numbered from the top of the laminate downward,

i.e., the topmost ply is numbered one and bottommost ply has

the highest number.

Results and Discussion

The uniformly distributed first ply failure load values of

clamped conoid are furnished in Table 6. The results

obtained from different failure criteria are indicated toge-

ther with the failure modes to which they correspond. In

the case of interactive failure, the failure tendencies are

mentioned which can be understood in a way as is indi-

cated earlier in this paper. It is obvious that for any

particular lamination, the minimum value of the failure

load should be taken up by an engineer for applying

engineering factor of safety to arrive at the working value

of the load.

When two-layered cross-ply laminates are compared, it

is found that the 90�/0� shell is convincingly better than the

0�/90� one and they tend to fail through compressive stress

working along the fibers in a lamina (r1) and transverse

tension (r2), respectively, and at the second and first

lamina, respectively, but at the same location which is the

crown of the lower arch of the shell. Failure occurring at

the support is quiet expected because for the clamped

boundary condition, the points on the support or in the

vicinity are subjected to the worst values of bending

moment and shear force. For 0�/90� shell, the failure mode

is the transverse tensile stress the permissible value of

which is 41.3 MPa and for the 90�/0� shell, the tendency of

failure is through compressive stress along the fibers the

permissible value of which is 2457 MPa, and this is why

the 90�/0� shell has a failure load value which is about 65%

more than its 0�/90� counterpart.

When the three-layered cross-ply shells are examined, it

is found that the 0�/90�/0� and the 90�/0�/90� shells show

comparable performances and both have the same tendency

to fail in transverse tension. These two shells have the 0�and the 90� lamina as the first layer and still they exhibit

almost identical failure load values. This indicates that for

these two laminations, a diagonal tension is the root cause

of failure which has equal components along the beam and

arch directions.

Among four-layered antisymmetrically laminated cross-

ply shells, the (0�/90�)2 one has a failure load value about

16% more than the (90�/0�)2 one. Both of them fail in the

Hoffman failure criterion in the transverse tensile stressFig. 3 Deflection profile of isotropic conoid under uniformly

distributed load along �y = 0.5

Table 3 Comparison of first

ply failure loads in Newton for a

(02�/90�)s plate

Quarter plate is divided into a

5 9 5 mesh

Length = 100 mm, ply

thickness = 0.155 mm, load

details = central point load

Failure criteria Side/thickness

First ply failure loads

Kam et al. [18]

First ply failure loads

(present formulation)

Maximum stress 105.26 108.26 108.81

Maximum strain 122.86 122.56

Hoffman 106.45 105.86

Tsai-Wu 112.77 112.50

Tsai-Hill 107.06 106.90


123

mode and at the crown of the lower parabolic edge of the

shell. But interestingly, the (0�/90�)2 shell fails at the

second lamina, while the (90�/0�)2 shell fails at its first

lamina. This means that in both the cases, the 90� lamina

lying above the mid-surface fails. This indicates a bending

failure due to flexure along the beam direction where the

bending stress is more as one moves away from the mid-

surface. Naturally, when in case of the (0�/90�)2 shell the

maximum bending tensile stress in the first lamina is

appropriately countered by the 0� fibers, the load-carrying

capacity increases. But, the 90� fibers in the second lamina

being aligned perpendicular to the direction of the bending

stress, the tensile stress directly comes in the matrix which

cracks in transverse tension. Just the reverse phenomenon

is observed in the (90�/0�)2 shell and the matrix cracking

occurs in the first lamina, i.e., in the zone of the highest

bending stress, resulting in a lower value of the failure

load.

Table 6 shows that the behavioral patterns of symmet-

rically laminated four-layered cross-ply shells [(0�/90�)s

and (90�/0�)s] are quite different from each other. They fail

in absolutely two different failure modes—while the (0�/

90�)s shell tends to fail through transverse matrix cracking

when the transverse tensile stress of the matrix approaches

its permissible value, the (90�/0�)s shell fails in transverse

tensile strain criterion. However, both of these shells fail at

the same location and the failure load values are almost

within 10% of each other, and hence the performances are

comparable from an engineering standpoint.

The two-layered antisymmetric angle-ply laminates

[(45�/�45�) and (�45�/45�)] fail in transverse tensile strain

of the matrix and yield a failure load value which is much

less than that of the two-layered cross-ply laminates [(0�/

90�) and (90�/0�)]. It is very interesting to observe that the

failure mode or tendency largely depends on lamination,

and as a natural consequence the load-carrying capacities

vary widely for the two-layered conoids even though the

material consumption is fixed. This means that for most

effective utilization of the material, different stacking

sequences are to be tried to achieve the maximum benefit.

It is further noted that for three-layered shells, the angle-

ply laminates yield lower values of failure load than the

cross-ply ones, but only by a margin of around 10%. The

point of occurrence of failure is also different for angle and

cross-ply shells. For four-layered antisymmetric angle-ply

laminates [(45�/�45�)2 and (�45�/45�)2], the maximum

transverse tensile strain again becomes the criterion like the

two-layered ones [(45�/�45�) and (�45�/45�)] and the

failure load values are much less than what one gets for

four-layered antisymmetric cross-ply laminates. Interest-

ingly, the symmetric four-layered angle-ply laminates

[(45�/�45�)s and (�45�/45�)s] offer failure loads which are

comparable to what are obtained for four-layered sym-

metric cross-ply laminates [(0�/90�)s and (90�/0�)s]. These

observations lead to infer that the cross-ply stacking

sequences are better than the angle-ply ones, but the

Fig. 4 The clamped boundary condition

Table 5 Geometric dimensions of the conoidal shell

Conoidal shell dimensions Values

Length (a) 1000 mm

Width (b) 1000 mm

Thickness (h) 10 mm

Higher height (hh) 200 mm

Lower height (hl) 50 mm

Table 4 Material properties of Q-1115 graphite-epoxy composite

material

Material constants Strengths

E11 142.50 GPa

E22 9.79 GPa

E33 9.79 GPa

G12 = G13 4.72 GPa

G23 1.192 GPa

m12 = m13 0.27

m23 0.25

XT 2193.50 MPa

XC 2457.0 MPa

YT = ZT 41.30 MPa

YC = ZC 206.80 MPa

R 61.28 MPa

S 78.78 MPa

T 78.78 MPa

Xet 0.01539

Xec 0.01724

Yet = Zet 0.00412

Yec = Zec 0.02112

Re 0.05141

Se 0.01669

Te 0.01669


123

Table 6 Uniformly distributed first ply failure loads of clamped conoidal shell

Lamination

(degree) Failure theory

Failure load

(N/mm2)

Location

(x, y) (m, m) First failed ply

Failure mode/ failure

tendency

0/90 Maximum stress 2.7191 (0, 0.5) 1 Matrix cracking

Maximum strain 2.9378 (0, 0.5) 1 Fiber breakage

Hoffman 1.7807a (0, 0.5) 1 Matrix cracking

Tsai-Hill 1.9999 (0, 0.5) 1 Matrix cracking

Tsai-Wu 2.0374 (0, 0.5) 1 Matrix cracking

90/0 Maximum stress 2.9909 (0, 0.5) 2 Fiber breakage


Hoffman 3.8899 (0, 0.5) 2 Fiber breakage

Tsai-Hill 2.9564a (0, 0.5) 2 Fiber breakage

Tsai-Wu 5.2755 (0, 0.5) 2 Fiber breakage

0/90/0 Maximum stress 2.9719 (0, 0.5) 1 Matrix cracking





90/0/90 Maximum stress 1.9504 (0, 0.5) 1 Matrix cracking

Maximum strain 1.9507 (0, 0.5) 1 Matrix cracking




0/90/0/90 or (0/90)2 Maximum stress 2.2527 (0, 0.5) 2 Matrix cracking





90/0/90/0 or (90/0)2 Maximum stress 1.9449 (0, 0.5) 1 Matrix cracking





0/90/90/0 or (0/90)s Maximum stress 2.7524 (0, 0.5) 2 Matrix cracking





90/0/0/90 or (90/0)s Maximum stress 1.9672 (0, 0.5) 1 Matrix cracking

Maximum strain 1.9602a (0, 0.5) 1 Matrix cracking

Hoffman 1.9640 (0, 0.5) 1 Matrix cracking



45/�45 or �45/45 Maximum stress 0.9587 (0, 0.5) 1 Matrix cracking






123

differences in performance are most noticeable when the

number of layers is two, and with gradual increase in

number of lamina, the performances are somewhat

comparable.

An overall analysis of failure loads for different lami-

nations clearly indicates that the (90�/0�) and (0�/90�)2

shells are very good in terms of stiffness among which

again the (90�/0�) one is the best choice. A two-layered

laminate is easy to fabricate and an engineer should prac-

tically pick up the (90�/0�) lamination to fabricate the

conoidal shell surface if he has a number of options as

given in Table 6. The general behavior of different lami-

nations in terms of their failure load values indicates a

complex interaction between stacking sequence and shell

curvature and it is very difficult to arrive at a unified

conclusion unless the numerical experimentation is done.

Conclusions

The present finite element code is capable of successfully

assessing the first ply failure loads of laminated composite

conoidal shells. The two-layered cross-ply shells fail in the

bending mode and the 90�/0� lamination exhibits a failure

load which is much higher than the corresponding load for

the 0�/90� shell. The three- and four-layered cross-ply

laminates show somewhat comparable performances,

although the root cause of failure is not the same in all the

cases. Among the different laminations considered in the

present study, the cross-ply stacking sequences are con-

vincingly better than the angle-ply ones, and should be

preferred by the practicing engineers, among which again

the 90�/0� shell emerges as a simple solution with a very

high value of first ply failure load.

Acknowledgments The first author gratefully acknowledges the

financial assistance of the Council of Scientific and Industrial

Research (India) through the Senior Research Fellowship vide Grant

No. 09/096 (0686) 2k11-EMR-I.

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Table 6 continued

Lamination

(degree) Failure theory

Failure load

(N/mm2)

Location

(x, y) (m, m) First failed ply

Failure mode/ failure

tendency

45/�45/45 Maximum stress 1.8728 (0, 0.25) 1 Matrix cracking

Maximum strain 1.8314 (0. 13,0.13) 3 Matrix cracking




45/�45/45/�45 Maximum stress 1.5026 (0, 0.5) 1 Matrix cracking





45/�45/�45/45 Maximum stress 1.9333 (0, 0.25) 1 Matrix cracking

Maximum strain 1.8984 (0.13, 0.13) 4 Matrix cracking




a Indicates lowest magnitude of failure load


123

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first ply failure study of composite conoidal shells used as roofing units in civil engineering

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