research article mathematical identification of...

16
Hindawi Publishing Corporation e Scientific World Journal Volume 2013, Article ID 268673, 15 pages http://dx.doi.org/10.1155/2013/268673 Research Article Mathematical Identification of Influential Parameters on the Elastic Buckling of Variable Geometry Plate Mirko Djelosevic, 1 Jovan Tepic, 2 Ilija Tanackov, 2 and Milan Kostelac 3 1 Faculty of Mechanical and Civil Engineering Kraljevo, University of Kragujevac, 36000 Kraljevo, Serbia 2 Faculty of Technical Sciences Novi Sad, University of Novi Sad, 21000 Novi Sad, Serbia 3 Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Mirko Djelosevic; [email protected] Received 7 August 2013; Accepted 16 September 2013 Academic Editors: O. D. Makinde and J. Meseguer Copyright © 2013 Mirko Djelosevic et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e problem of elastic stability of plates with square, rectangular, and circular holes as well as slotted holes was discussed. e existence of the hole reduces the deformation energy of the plate and it affects the redistribution of stress flow in comparison to a uniform plate which causes a change of the external operation of compressive forces. e distribution of compressive force is defined as the approximate model of plane state of stress. e significant parameters of elastic stability compared to the uniform plate, including the dominant role of the shape, size, and orientation of the hole were identified. Comparative analysis of the shape of the hole was carried out on the data from the literature, which are based on different approaches and methods. Qualitative and quantitative accordance of the results has been found out and it verifies exposed methodology as applicable in the study of the phenomenon of elastic stability. Sensitivity factor is defined that is proportional to the reciprocal value of the buckling coefficient and it is a measure of sensitivity of plate to the existence of the hole. Mechanism of loss of stability is interpreted through the absorption of the external operation, induced by the shape of the hole. 1. Introduction in-walled plates are the basic components of a large num- ber of modern supporting structures of open and closed type or profiled and box girder that are used in the construction of bridges, airplanes, boats, and other authorized facilities. Plate shaped elements of no uniform geometry are oſten incorporated in these structures which implies a variation in thickness and existence of various shapes holes. Require- ments for using the plates with perforated holes and stepwise variable thickness result from functional, assembly, service, control, and other conditions, and oſten they are the result of weight minimization in order to achieve economic effect in the process of optimum design of structures. Mathematical identification of phenomenon of plate stability and general stability of girder and columns as their components is based on the application of the methods of Kantorovich and Ritz. e first method is based on solving partial differential equa- tions of the fourth order (1), and it is derived from the equi- librium conditions of thin plate of constant geometry. e second method is based on the principle of minimum total potential energy, which provides a stable state of plate bal- ance. eoretical basis for both methods were systematically presented in [13]. e issue of stability of thin-walled ele- ments occupies an important place in the process of support- ing structures design [4]. For plates of constant thickness affecting parameters include geometric sizes, supporting conditions, and load conditions [57], while the stability of the element with a hole in addition to these characteristics is affected by the shape, size, and position of the hole [8, 9]. Most researches in the area of stability refer to the ideal- ized case of a simply supported plate. In contrast to this is the plate with restraint edges along which the displacements and

Upload: buinhu

Post on 28-Jul-2018

225 views

Category:

Documents


0 download

TRANSCRIPT

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 268673 15 pageshttpdxdoiorg1011552013268673

Research ArticleMathematical Identification of Influential Parameters onthe Elastic Buckling of Variable Geometry Plate

Mirko Djelosevic1 Jovan Tepic2 Ilija Tanackov2 and Milan Kostelac3

1 Faculty of Mechanical and Civil Engineering Kraljevo University of Kragujevac 36000 Kraljevo Serbia2 Faculty of Technical Sciences Novi Sad University of Novi Sad 21000 Novi Sad Serbia3 Faculty of Mechanical Engineering and Naval Architecture University of Zagreb 10000 Zagreb Croatia

Correspondence should be addressed to Mirko Djelosevic prostructuresgmailcom

Received 7 August 2013 Accepted 16 September 2013

Academic Editors O D Makinde and J Meseguer

Copyright copy 2013 Mirko Djelosevic et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The problem of elastic stability of plates with square rectangular and circular holes as well as slotted holes was discussed Theexistence of the hole reduces the deformation energy of the plate and it affects the redistribution of stress flow in comparison toa uniform plate which causes a change of the external operation of compressive forces The distribution of compressive force isdefined as the approximate model of plane state of stress The significant parameters of elastic stability compared to the uniformplate including the dominant role of the shape size and orientation of the hole were identified Comparative analysis of the shapeof the hole was carried out on the data from the literature which are based on different approaches and methods Qualitative andquantitative accordance of the results has been found out and it verifies exposed methodology as applicable in the study of thephenomenon of elastic stability Sensitivity factor is defined that is proportional to the reciprocal value of the buckling coefficientand it is a measure of sensitivity of plate to the existence of the hole Mechanism of loss of stability is interpreted through theabsorption of the external operation induced by the shape of the hole

1 Introduction

Thin-walled plates are the basic components of a large num-ber of modern supporting structures of open and closed typeor profiled and box girder that are used in the constructionof bridges airplanes boats and other authorized facilitiesPlate shaped elements of no uniform geometry are oftenincorporated in these structures which implies a variationin thickness and existence of various shapes holes Require-ments for using the plates with perforated holes and stepwisevariable thickness result from functional assembly servicecontrol and other conditions and often they are the result ofweight minimization in order to achieve economic effect inthe process of optimum design of structures Mathematicalidentification of phenomenon of plate stability and generalstability of girder and columns as their components is basedon the application of the methods of Kantorovich and Ritz

The first method is based on solving partial differential equa-tions of the fourth order (1) and it is derived from the equi-librium conditions of thin plate of constant geometry Thesecond method is based on the principle of minimum totalpotential energy which provides a stable state of plate bal-ance Theoretical basis for both methods were systematicallypresented in [1ndash3] The issue of stability of thin-walled ele-ments occupies an important place in the process of support-ing structures design [4]

For plates of constant thickness affecting parametersinclude geometric sizes supporting conditions and loadconditions [5ndash7] while the stability of the element with a holein addition to these characteristics is affected by the shapesize and position of the hole [8 9]

Most researches in the area of stability refer to the ideal-ized case of a simply supported plate In contrast to this is theplate with restraint edges along which the displacements and

2 The Scientific World Journal

rotations are disabled These two extreme and diametricallyopposite cases practically very rarely occur in pure formbut they are most often occur in combination with otherbasic supporting conditions In the case of real girders thereis a case of elastic supported plate meaning that along theindividual edges are beams of certain bending and torsionalrigidity Analysis of constant geometry plate shows thatthe supporting conditions greatly affect the stability [7] Inaddition implementation of staying is the way to increasestability [10ndash13]

Researches of plates with holes andor stepwise variablethickness as components of girder are of special interests[14ndash24] It is important to emphasize that very importantmethods are those having universal application in the studyof the stability of plates and shells It refers to the elementsof the support that may have more holes usually arrangedequidistant (perforation) aswell as suddenor gradual changesin thickness

Rational design of structures requires the use of thesestructures which includes the identification of parametersinfluencing stability In the last ten years development ofcomputer techniques based on the use of finite elementsmethod (FEM) has enabled researchers to make a significantcontribution to this phenomenon [25] Researches [18 22]show that the existence of the hole leads to a reduction ofbuckling coefficient for a certain value which depends onlyon the shape size and position of the hole To this the factthat the distance between the holes in perforated plates is alsoa parameter which in interaction with others affects the valueof the critical buckling stress should be added Due to thecomplexity of the mathematical model between the bucklingcoefficients that is minimum value of the critical bucklingstress and characteristic parameters of the hole researchershave mostly kept on semianalytic approximate and numeri-cal methods and expressions obtained experimentally

2 Analysis and Problem Formulation

A significant number of researchers have examined thephenomenon of linear or elastic buckling of plates indicatingthe importance of such an analysis in the design of structuresStudies of plate elements of girder or columnswith perforatedholes are of particular interest These studies are realizedusing the finite element method finite strip method [26ndash28]and experimental tests Due to the geometric complexity andstress complexity the study of such problems mathematicallyis aimed to the partial analysis of individual components

Analysis of the elastic buckling of plates a number ofresearchers has relied on the application of von Karmanrsquostheory based on the fourth-order partial differential equation[1 2 29]

nabla4119908 = minus

1

119863(119873119909

1205972119908

1205971199092+ 119873119910

1205972119908

1205971199102+ 2119873119909119910

1205972119908

120597119909120597119910) (1)

The function of the deflection depends on the type ofload and boundary conditions and it is presented as a linearcombination of trigonometric and hyperbolic functionsAs analyzed plates have ideally planar geometry without

transversal pressure and initial curvature integration con-stants that define the contour conditions remain indefinite

In this case the deflection function with indefinite coeffi-cients can be used to determine the coefficient of bucklingUnder the influence of compression load plate is in theposition of stable equilibrium until a critical intensity whenit suddenly loses stability and fracture occurs This situationis a consequence of the idealization of the initial state of plategeometry The application of this method is limited to a plateof constant thickness

Numerous studies of linear buckling plates especiallythose that are based on mathematical models refer only tothe problem of uniform plates Study subject are plates ofconstant or variable thickness in one or both directions underthe influence of combined plate loads Analytical solutionfor pressure plate of variable thickness composed of twosegments of the same length is given in [5] The energymethod based on the Rayleigh-Ritz principle is applied toanalyze the plate [29 30] More complex cases are where theplates of gradual variable thickness in two normal directionsare considered by the finite difference method [6]

Thin-walled elements are potentially critical places of thestructure due to the interactive effect between planar stateof stress and elastic buckling phenomenon The existence ofholes in such elements followed by a stress concentration gen-erally reduces their strength to buckling Elastic buckling isthe first step in the analysis of stability of structuresMeasurestaken to increase the critical stress of elastic buckling effect onfavorable on behavior in the zone of inelastic buckling and onmechanism of boundary conditions

The subject of this paper simply supported uniaxialloaded rectangular plate with square rectangular and circu-lar holes as well as slotted holes Elastic buckling of plateswith so-defined geometry and load conditions is modeledin terms of energy method (detailed information is given inSection 3)Themathematical model is based on the principleof minimum total energy of plate that provides a state ofstable equilibriumModel of the elastic buckling of plate witha hole is tightly coupled with the model of a planar state ofthe considered elementThe aimof the authors is to verify andcomplement the existing results regarding the elastic bucklingof plates with various shapes holes The analysis based on thephysical andmathematicalmodel enables exact identificationof mechanism of elastic stability

3 The Mathematical Model

The approach in this paper based on an analytical approachrequires forming a mathematical model according to definedphysical conditions of supporting and load Beside the modelof elastic stability it is necessary to form a model of planestate of stress of plate with rectangular hole In this paper theresearch is directed to analyze plates with holes of differentshapes in the domain of elastic material behavior Supportingstructures are formed by different geometry plates intercon-nected into a functional entirely

The model formed on the basis of the plates system pro-vides a real picture of the behavior of the whole cross sectionApproach to solving was given in [31] where the problem

The Scientific World Journal 3

of the static load capacity of box girder partially exposedto horizontal loads is analyzed Partial analysis of stabilityof girder includes independent examination of certain plates[32] which greatly simplifies the mathematical model whichwill be presented in this chapter

31 Core of the Buckling Problem The mathematical modelis based on the application of energy method using totalpotential energy of plate The most general application of theenergy method refers to the case of transversely loaded platesandor plates with initial curvature simultaneously exposedto the effects of plane forces Transversal and plane forcesrepresent known external load whose effect is manifestedthrough external work 119864

119882 which leads to a deformation

of plate As a result of external load we have occurrenceof deformation energy 119864

119868 manifested through the process

of bending and torsion which leads to the induction of anadditional plane forces due to the existence of connections ofplate supports if any Transversal loading affects the bendingof plate Plane forces affect increase or reducing of the platedeflection in dependence on whether they are pressed ortightening The existence of initial curvature is equivalent tothe existence of a fictive transversal load [1]

Themathematicalmodel is formedbased on the followingassumptions

(i) considered plates have ideal geometry (without sig-nificant warping)

(ii) behavior of isotropic plates is in the area of elasticstability

(iii) the effect of supporting to inducing forces in a platersquosplane is ignored

(iv) the impact of the external transversal loading iseliminated

Generally the energy balance of plate which is located inthe state of stable equilibrium is

119864119882= 119864119868 (2)

Work of external forces (119864119882) is defined by the following

formula

119864119882=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(3)

The internal deformation energy (119864119868) is

119864119868=119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

(4)

A necessary and sufficient condition for plate in order tobe in stable equilibrium position is that the total energy 119864has a minimum Total energy of plate whose components aregiven by (3) and (4) and in pursuance of [1 2] is

119864 = 119864119882+ 119864119868

=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

+119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592)

times [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(5)

First term refers to the deformation energy or to workperformed by forces acting in the plane of the plate Theinduction of these forces is a consequence of boundaryconditions or restrictions of plate their effect has passivecharacter and it manifests itself by reducing deflection andthus reducing the deformation energy plate too An exampleis limited supporting plate that is plate supported on fixedsupports to prevent extension caused by the deflection ofplate The occurrence of passive plane forces is always con-nected to the phenomenon of deformation or internal energyand it is characteristics of geometry material and boundaryconditions of plate Conversely if the plate is exposed toexternal planar load then their effect is active because initiatestrain and the corresponding energy is classified as externalwork

32 Approximate Model of Planar State of Stress of Plate withRectangular Hole To determine the work carried out byplane compression forces (3) it is necessary to know theirdistributions across the whole plate surface If a plate is ofconstant thickness with no holes then the distribution ofthese forces is uniform and this case of load is the easiestcase for studying stability In case where there is a hole onthe plate andor stepwise variable thickness disregardingthe edges of plates exposed to constant planar load thereis a redistribution of these forces in the highly nonuniformfunctions This phenomenon is explained by the existenceof discontinuous changes through which the load cannot betransferred but it is diverted to other areas of plates thatare ldquobottlenecksrdquo in the transfer of load These are the mostusually areas above and below the hole in whose immediatevicinity there is a concentration of the load while in front ofand back of the plate density of load stream lines decreases(Figure 1)

4 The Scientific World Journal

Nx

Unload zone

Nx

Figure 1 Redistribution of stress flow of plate with a hole

Model of planar state of plate with rectangular hole(Figure 2) is formed under the following conditions

(i) distribution of load at place where elements of plateare separated is linearized

(ii) the effect of the transversal forces along the connect-ing line among the elements of plate is ignored

Justification for these assumptions is reflected in the factthat linearized force function119873

119909has the same mean value as

real distribution that the difference of the external work is asmall value in comparison to the work of real load Externalshearing forces do not affect the plate and the induceddeformation energy due to the fact that shear stress is smallvalue in comparison to the normal stress

The components of the reaction force 11987310158401(119910) and 119873

1015840

2(119910)

defined by (6) and (7) are determined from the equilibriumcondition of plate element ldquo1rdquo plates

1198731015840

1(119910) =

119887

1198871

119887 minus 1198872

2119887 minus 1198871minus 1198872

119873 = 1198911119873 0 le 119910 le 119887

1 (6)

1198731015840

2(119910) =

119887

1198872

119887 minus 1198871

2119887 minus 1198871minus 1198872

119873 = 1198912119873 (119887 minus 119887

2) le 119910 le 119887 (7)

Elasticity of plate affects the induction of additional reac-tions of element ldquo1rdquo whichmanifest asmoments (8) and linearvariable force (9)

1198721=11990421198902+ 11990441198901

11990421199043minus 11990411199044

119873 = 1198921119873

1198722= (

1198901

1199042

+1199041

1199042

11990421198902+ 11990441198901

11990421199043minus 11990411199044

)119873 = 1198922119873

1198723= [

1199051

1199033

minus1199032

1199033

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199032

1199033

+1199031

1199033

)]119873 = 1198923119873

1198724= [

1199051

1199036

minus1199038

1199036

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199038

1199036

+1199032

1199036

)]119873 = 1198924119873

(8)

The coefficients 11990318

11990414

11989012

and 11990512

are defined inAppendix A

Distribution of linear variable forces caused by the influ-ence of the moments of elastic clamp is given by

11987310158401015840

1(119910) = (

2119910

1198871

minus 1)21198721

1198871

= 1198911119873 0 le 119909 le 119887

1

11987310158401015840

1(119910) = (

2 (119887 minus 119910)

1198872

minus 1)21198722

1198872

(119887 minus 1198872) le 119909 le 119887

(9)

The distribution of reaction forces (6) (7) and (9) along119909 = 119886

1is stepwise variable which means that at part 119887

1lt

119910 lt (1198871+ ℎ) does not act load Developing into a single

trigonometric series and by their adding coming up are tounique function (12) which determines the conditions at theedge 119909 = 119886

1of segment ldquo1rdquo of plate is obtained

1198731(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198911119873

119899120587sin 1198991205871198871

2119887sin 1198991205871198871

2119887minus

81198921119873

119899212058721198872

1

times cos 1198991205872(119899120587 cos 1198991205871198871

2119887

minus 2119887

1198871

sin 11989912058711988712119887

)]

times sin119899120587119910

119887

= 119873

infin

sum

119899=1

119875119899sin

119899120587119910

119887

(10)

1198732(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198912119873

119899120587sin

119899120587 (2119887 minus 1198872)

2119887sin 1198991205871198872

2119887

+81198922119873

119899212058721198872

2

cos119899120587 (2119887 minus 119887

2)

2119887

times (119899120587 cos 11989912058711988722119887

minus 2119887

1198872

sin 11989912058711988722119887

)]

times sin119899120587119910

119887= 119873

infin

sum

119899=1

119876119899sin

119899120587119910

119887

(11)

119873(119909 = 1198861 119910) = 119873

1(119909 = 119886

1 119910) + 119873

2(119909 = 119886

1 119910)

= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887

(12)

Defining a planar state of stress of plate Airy stressfunction is applied This function has the following form

1205974119880

1205971199094+

1205974119880

12059711990921205971199102+1205974119880

1205971199104= 0 (13)

The Scientific World Journal 5

N = const

y

1

b

a1

b2

h

b1

N2(y) = N9984002 + N998400998400

2

M2

N

c

N1(y) =998400 + 998400998400N1 N1

M1 M3

M4

EI1 EI3

EI4

EI2

h

N

N = const

b

y

1

a1

h

3

4

2

c a2

b2

b1

x

NN2(y) N4(y)

N3(y)N1(y)

oplus

oplus

oplus

oplus

oplus

oplusoplus

⊖⊖

oplusoplus

oplus

⊖ ⊖

oplus oplus

Figure 2 Model of planar state of stress of plate with rectangular hole

A function that satisfies (13) is presented by the trigono-metric series

119880(119909 119910) =

infin

sum

119899=1

[119860119899sinh 119899120587119909

119887+ 119861119899(119899120587119909

119887) sinh 119899120587119909

119887

+ 119862119899cosh 119899120587119909

119887+ 119863119899(119899120587119909

119887) cosh 119899120587119909

119887]

times sin119899120587119910

119887

(14)

Planar forces 119873119909and 119873

119909119910are defined through the com-

ponent stresses 120590119909and 120590

119909119910formulated by

119873119909=120590119909

120575=1

120575

1205972119880

1205971199102

=1205872

1205751198872

infin

sum

119899=1

1198992[119860119899sinh 119899120587119909

119887

+ 119861119899(2 cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)

+ 119862119899cosh 119899120587119909

119887

+ 119863119899(2 sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)]

times sin119899120587119910

119887

119873119909119910=

120590119909119910

120575= minus

1

120575

1205972119880

120597119909120597119910

= minus1205872

1205751198872

infin

sum

119899=1

1198992[119860119899cosh 119899120587119909

119887

+ 119861119899(sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)

+ 119862119899sinh 119899120587119909

119887

+ 119863119899(cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)]

times cos119899120587119910

119887

(15)

Integration constants 119860119899 119861119899 119862119899 and 119863

119899are determined

from the boundary conditions as follows

(1) 119873119909= 119873 =

2119873

120587

infin

sum

119899=1

1

119899(1 minus cos 119899120587) sin

119899120587119910

119887= const

119873119909119910= 0 119909 = 0

(2) 119873119909= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887 119873119909119910= 0 119909 = 119886

1

(16)

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

2 The Scientific World Journal

rotations are disabled These two extreme and diametricallyopposite cases practically very rarely occur in pure formbut they are most often occur in combination with otherbasic supporting conditions In the case of real girders thereis a case of elastic supported plate meaning that along theindividual edges are beams of certain bending and torsionalrigidity Analysis of constant geometry plate shows thatthe supporting conditions greatly affect the stability [7] Inaddition implementation of staying is the way to increasestability [10ndash13]

Researches of plates with holes andor stepwise variablethickness as components of girder are of special interests[14ndash24] It is important to emphasize that very importantmethods are those having universal application in the studyof the stability of plates and shells It refers to the elementsof the support that may have more holes usually arrangedequidistant (perforation) aswell as suddenor gradual changesin thickness

Rational design of structures requires the use of thesestructures which includes the identification of parametersinfluencing stability In the last ten years development ofcomputer techniques based on the use of finite elementsmethod (FEM) has enabled researchers to make a significantcontribution to this phenomenon [25] Researches [18 22]show that the existence of the hole leads to a reduction ofbuckling coefficient for a certain value which depends onlyon the shape size and position of the hole To this the factthat the distance between the holes in perforated plates is alsoa parameter which in interaction with others affects the valueof the critical buckling stress should be added Due to thecomplexity of the mathematical model between the bucklingcoefficients that is minimum value of the critical bucklingstress and characteristic parameters of the hole researchershave mostly kept on semianalytic approximate and numeri-cal methods and expressions obtained experimentally

2 Analysis and Problem Formulation

A significant number of researchers have examined thephenomenon of linear or elastic buckling of plates indicatingthe importance of such an analysis in the design of structuresStudies of plate elements of girder or columnswith perforatedholes are of particular interest These studies are realizedusing the finite element method finite strip method [26ndash28]and experimental tests Due to the geometric complexity andstress complexity the study of such problems mathematicallyis aimed to the partial analysis of individual components

Analysis of the elastic buckling of plates a number ofresearchers has relied on the application of von Karmanrsquostheory based on the fourth-order partial differential equation[1 2 29]

nabla4119908 = minus

1

119863(119873119909

1205972119908

1205971199092+ 119873119910

1205972119908

1205971199102+ 2119873119909119910

1205972119908

120597119909120597119910) (1)

The function of the deflection depends on the type ofload and boundary conditions and it is presented as a linearcombination of trigonometric and hyperbolic functionsAs analyzed plates have ideally planar geometry without

transversal pressure and initial curvature integration con-stants that define the contour conditions remain indefinite

In this case the deflection function with indefinite coeffi-cients can be used to determine the coefficient of bucklingUnder the influence of compression load plate is in theposition of stable equilibrium until a critical intensity whenit suddenly loses stability and fracture occurs This situationis a consequence of the idealization of the initial state of plategeometry The application of this method is limited to a plateof constant thickness

Numerous studies of linear buckling plates especiallythose that are based on mathematical models refer only tothe problem of uniform plates Study subject are plates ofconstant or variable thickness in one or both directions underthe influence of combined plate loads Analytical solutionfor pressure plate of variable thickness composed of twosegments of the same length is given in [5] The energymethod based on the Rayleigh-Ritz principle is applied toanalyze the plate [29 30] More complex cases are where theplates of gradual variable thickness in two normal directionsare considered by the finite difference method [6]

Thin-walled elements are potentially critical places of thestructure due to the interactive effect between planar stateof stress and elastic buckling phenomenon The existence ofholes in such elements followed by a stress concentration gen-erally reduces their strength to buckling Elastic buckling isthe first step in the analysis of stability of structuresMeasurestaken to increase the critical stress of elastic buckling effect onfavorable on behavior in the zone of inelastic buckling and onmechanism of boundary conditions

The subject of this paper simply supported uniaxialloaded rectangular plate with square rectangular and circu-lar holes as well as slotted holes Elastic buckling of plateswith so-defined geometry and load conditions is modeledin terms of energy method (detailed information is given inSection 3)Themathematical model is based on the principleof minimum total energy of plate that provides a state ofstable equilibriumModel of the elastic buckling of plate witha hole is tightly coupled with the model of a planar state ofthe considered elementThe aimof the authors is to verify andcomplement the existing results regarding the elastic bucklingof plates with various shapes holes The analysis based on thephysical andmathematicalmodel enables exact identificationof mechanism of elastic stability

3 The Mathematical Model

The approach in this paper based on an analytical approachrequires forming a mathematical model according to definedphysical conditions of supporting and load Beside the modelof elastic stability it is necessary to form a model of planestate of stress of plate with rectangular hole In this paper theresearch is directed to analyze plates with holes of differentshapes in the domain of elastic material behavior Supportingstructures are formed by different geometry plates intercon-nected into a functional entirely

The model formed on the basis of the plates system pro-vides a real picture of the behavior of the whole cross sectionApproach to solving was given in [31] where the problem

The Scientific World Journal 3

of the static load capacity of box girder partially exposedto horizontal loads is analyzed Partial analysis of stabilityof girder includes independent examination of certain plates[32] which greatly simplifies the mathematical model whichwill be presented in this chapter

31 Core of the Buckling Problem The mathematical modelis based on the application of energy method using totalpotential energy of plate The most general application of theenergy method refers to the case of transversely loaded platesandor plates with initial curvature simultaneously exposedto the effects of plane forces Transversal and plane forcesrepresent known external load whose effect is manifestedthrough external work 119864

119882 which leads to a deformation

of plate As a result of external load we have occurrenceof deformation energy 119864

119868 manifested through the process

of bending and torsion which leads to the induction of anadditional plane forces due to the existence of connections ofplate supports if any Transversal loading affects the bendingof plate Plane forces affect increase or reducing of the platedeflection in dependence on whether they are pressed ortightening The existence of initial curvature is equivalent tothe existence of a fictive transversal load [1]

Themathematicalmodel is formedbased on the followingassumptions

(i) considered plates have ideal geometry (without sig-nificant warping)

(ii) behavior of isotropic plates is in the area of elasticstability

(iii) the effect of supporting to inducing forces in a platersquosplane is ignored

(iv) the impact of the external transversal loading iseliminated

Generally the energy balance of plate which is located inthe state of stable equilibrium is

119864119882= 119864119868 (2)

Work of external forces (119864119882) is defined by the following

formula

119864119882=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(3)

The internal deformation energy (119864119868) is

119864119868=119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

(4)

A necessary and sufficient condition for plate in order tobe in stable equilibrium position is that the total energy 119864has a minimum Total energy of plate whose components aregiven by (3) and (4) and in pursuance of [1 2] is

119864 = 119864119882+ 119864119868

=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

+119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592)

times [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(5)

First term refers to the deformation energy or to workperformed by forces acting in the plane of the plate Theinduction of these forces is a consequence of boundaryconditions or restrictions of plate their effect has passivecharacter and it manifests itself by reducing deflection andthus reducing the deformation energy plate too An exampleis limited supporting plate that is plate supported on fixedsupports to prevent extension caused by the deflection ofplate The occurrence of passive plane forces is always con-nected to the phenomenon of deformation or internal energyand it is characteristics of geometry material and boundaryconditions of plate Conversely if the plate is exposed toexternal planar load then their effect is active because initiatestrain and the corresponding energy is classified as externalwork

32 Approximate Model of Planar State of Stress of Plate withRectangular Hole To determine the work carried out byplane compression forces (3) it is necessary to know theirdistributions across the whole plate surface If a plate is ofconstant thickness with no holes then the distribution ofthese forces is uniform and this case of load is the easiestcase for studying stability In case where there is a hole onthe plate andor stepwise variable thickness disregardingthe edges of plates exposed to constant planar load thereis a redistribution of these forces in the highly nonuniformfunctions This phenomenon is explained by the existenceof discontinuous changes through which the load cannot betransferred but it is diverted to other areas of plates thatare ldquobottlenecksrdquo in the transfer of load These are the mostusually areas above and below the hole in whose immediatevicinity there is a concentration of the load while in front ofand back of the plate density of load stream lines decreases(Figure 1)

4 The Scientific World Journal

Nx

Unload zone

Nx

Figure 1 Redistribution of stress flow of plate with a hole

Model of planar state of plate with rectangular hole(Figure 2) is formed under the following conditions

(i) distribution of load at place where elements of plateare separated is linearized

(ii) the effect of the transversal forces along the connect-ing line among the elements of plate is ignored

Justification for these assumptions is reflected in the factthat linearized force function119873

119909has the same mean value as

real distribution that the difference of the external work is asmall value in comparison to the work of real load Externalshearing forces do not affect the plate and the induceddeformation energy due to the fact that shear stress is smallvalue in comparison to the normal stress

The components of the reaction force 11987310158401(119910) and 119873

1015840

2(119910)

defined by (6) and (7) are determined from the equilibriumcondition of plate element ldquo1rdquo plates

1198731015840

1(119910) =

119887

1198871

119887 minus 1198872

2119887 minus 1198871minus 1198872

119873 = 1198911119873 0 le 119910 le 119887

1 (6)

1198731015840

2(119910) =

119887

1198872

119887 minus 1198871

2119887 minus 1198871minus 1198872

119873 = 1198912119873 (119887 minus 119887

2) le 119910 le 119887 (7)

Elasticity of plate affects the induction of additional reac-tions of element ldquo1rdquo whichmanifest asmoments (8) and linearvariable force (9)

1198721=11990421198902+ 11990441198901

11990421199043minus 11990411199044

119873 = 1198921119873

1198722= (

1198901

1199042

+1199041

1199042

11990421198902+ 11990441198901

11990421199043minus 11990411199044

)119873 = 1198922119873

1198723= [

1199051

1199033

minus1199032

1199033

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199032

1199033

+1199031

1199033

)]119873 = 1198923119873

1198724= [

1199051

1199036

minus1199038

1199036

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199038

1199036

+1199032

1199036

)]119873 = 1198924119873

(8)

The coefficients 11990318

11990414

11989012

and 11990512

are defined inAppendix A

Distribution of linear variable forces caused by the influ-ence of the moments of elastic clamp is given by

11987310158401015840

1(119910) = (

2119910

1198871

minus 1)21198721

1198871

= 1198911119873 0 le 119909 le 119887

1

11987310158401015840

1(119910) = (

2 (119887 minus 119910)

1198872

minus 1)21198722

1198872

(119887 minus 1198872) le 119909 le 119887

(9)

The distribution of reaction forces (6) (7) and (9) along119909 = 119886

1is stepwise variable which means that at part 119887

1lt

119910 lt (1198871+ ℎ) does not act load Developing into a single

trigonometric series and by their adding coming up are tounique function (12) which determines the conditions at theedge 119909 = 119886

1of segment ldquo1rdquo of plate is obtained

1198731(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198911119873

119899120587sin 1198991205871198871

2119887sin 1198991205871198871

2119887minus

81198921119873

119899212058721198872

1

times cos 1198991205872(119899120587 cos 1198991205871198871

2119887

minus 2119887

1198871

sin 11989912058711988712119887

)]

times sin119899120587119910

119887

= 119873

infin

sum

119899=1

119875119899sin

119899120587119910

119887

(10)

1198732(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198912119873

119899120587sin

119899120587 (2119887 minus 1198872)

2119887sin 1198991205871198872

2119887

+81198922119873

119899212058721198872

2

cos119899120587 (2119887 minus 119887

2)

2119887

times (119899120587 cos 11989912058711988722119887

minus 2119887

1198872

sin 11989912058711988722119887

)]

times sin119899120587119910

119887= 119873

infin

sum

119899=1

119876119899sin

119899120587119910

119887

(11)

119873(119909 = 1198861 119910) = 119873

1(119909 = 119886

1 119910) + 119873

2(119909 = 119886

1 119910)

= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887

(12)

Defining a planar state of stress of plate Airy stressfunction is applied This function has the following form

1205974119880

1205971199094+

1205974119880

12059711990921205971199102+1205974119880

1205971199104= 0 (13)

The Scientific World Journal 5

N = const

y

1

b

a1

b2

h

b1

N2(y) = N9984002 + N998400998400

2

M2

N

c

N1(y) =998400 + 998400998400N1 N1

M1 M3

M4

EI1 EI3

EI4

EI2

h

N

N = const

b

y

1

a1

h

3

4

2

c a2

b2

b1

x

NN2(y) N4(y)

N3(y)N1(y)

oplus

oplus

oplus

oplus

oplus

oplusoplus

⊖⊖

oplusoplus

oplus

⊖ ⊖

oplus oplus

Figure 2 Model of planar state of stress of plate with rectangular hole

A function that satisfies (13) is presented by the trigono-metric series

119880(119909 119910) =

infin

sum

119899=1

[119860119899sinh 119899120587119909

119887+ 119861119899(119899120587119909

119887) sinh 119899120587119909

119887

+ 119862119899cosh 119899120587119909

119887+ 119863119899(119899120587119909

119887) cosh 119899120587119909

119887]

times sin119899120587119910

119887

(14)

Planar forces 119873119909and 119873

119909119910are defined through the com-

ponent stresses 120590119909and 120590

119909119910formulated by

119873119909=120590119909

120575=1

120575

1205972119880

1205971199102

=1205872

1205751198872

infin

sum

119899=1

1198992[119860119899sinh 119899120587119909

119887

+ 119861119899(2 cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)

+ 119862119899cosh 119899120587119909

119887

+ 119863119899(2 sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)]

times sin119899120587119910

119887

119873119909119910=

120590119909119910

120575= minus

1

120575

1205972119880

120597119909120597119910

= minus1205872

1205751198872

infin

sum

119899=1

1198992[119860119899cosh 119899120587119909

119887

+ 119861119899(sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)

+ 119862119899sinh 119899120587119909

119887

+ 119863119899(cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)]

times cos119899120587119910

119887

(15)

Integration constants 119860119899 119861119899 119862119899 and 119863

119899are determined

from the boundary conditions as follows

(1) 119873119909= 119873 =

2119873

120587

infin

sum

119899=1

1

119899(1 minus cos 119899120587) sin

119899120587119910

119887= const

119873119909119910= 0 119909 = 0

(2) 119873119909= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887 119873119909119910= 0 119909 = 119886

1

(16)

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 3

of the static load capacity of box girder partially exposedto horizontal loads is analyzed Partial analysis of stabilityof girder includes independent examination of certain plates[32] which greatly simplifies the mathematical model whichwill be presented in this chapter

31 Core of the Buckling Problem The mathematical modelis based on the application of energy method using totalpotential energy of plate The most general application of theenergy method refers to the case of transversely loaded platesandor plates with initial curvature simultaneously exposedto the effects of plane forces Transversal and plane forcesrepresent known external load whose effect is manifestedthrough external work 119864

119882 which leads to a deformation

of plate As a result of external load we have occurrenceof deformation energy 119864

119868 manifested through the process

of bending and torsion which leads to the induction of anadditional plane forces due to the existence of connections ofplate supports if any Transversal loading affects the bendingof plate Plane forces affect increase or reducing of the platedeflection in dependence on whether they are pressed ortightening The existence of initial curvature is equivalent tothe existence of a fictive transversal load [1]

Themathematicalmodel is formedbased on the followingassumptions

(i) considered plates have ideal geometry (without sig-nificant warping)

(ii) behavior of isotropic plates is in the area of elasticstability

(iii) the effect of supporting to inducing forces in a platersquosplane is ignored

(iv) the impact of the external transversal loading iseliminated

Generally the energy balance of plate which is located inthe state of stable equilibrium is

119864119882= 119864119868 (2)

Work of external forces (119864119882) is defined by the following

formula

119864119882=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(3)

The internal deformation energy (119864119868) is

119864119868=119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

(4)

A necessary and sufficient condition for plate in order tobe in stable equilibrium position is that the total energy 119864has a minimum Total energy of plate whose components aregiven by (3) and (4) and in pursuance of [1 2] is

119864 = 119864119882+ 119864119868

=1

2int

119886

0

int

119887

0

[119873119909(120597119908

120597119909)

2

+ 119873119909(120597119908

120597119909)

2

+ 2119873119909119910

120597119908

120597119909

120597119908

120597119910]119889119909119889119910

+119863

2int

119886

0

int

119887

0

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus 2 (1 minus 120592)

times [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119909119889119910

minus int

119886

0

int

119887

0

119908119902119889119909119889119910

(5)

First term refers to the deformation energy or to workperformed by forces acting in the plane of the plate Theinduction of these forces is a consequence of boundaryconditions or restrictions of plate their effect has passivecharacter and it manifests itself by reducing deflection andthus reducing the deformation energy plate too An exampleis limited supporting plate that is plate supported on fixedsupports to prevent extension caused by the deflection ofplate The occurrence of passive plane forces is always con-nected to the phenomenon of deformation or internal energyand it is characteristics of geometry material and boundaryconditions of plate Conversely if the plate is exposed toexternal planar load then their effect is active because initiatestrain and the corresponding energy is classified as externalwork

32 Approximate Model of Planar State of Stress of Plate withRectangular Hole To determine the work carried out byplane compression forces (3) it is necessary to know theirdistributions across the whole plate surface If a plate is ofconstant thickness with no holes then the distribution ofthese forces is uniform and this case of load is the easiestcase for studying stability In case where there is a hole onthe plate andor stepwise variable thickness disregardingthe edges of plates exposed to constant planar load thereis a redistribution of these forces in the highly nonuniformfunctions This phenomenon is explained by the existenceof discontinuous changes through which the load cannot betransferred but it is diverted to other areas of plates thatare ldquobottlenecksrdquo in the transfer of load These are the mostusually areas above and below the hole in whose immediatevicinity there is a concentration of the load while in front ofand back of the plate density of load stream lines decreases(Figure 1)

4 The Scientific World Journal

Nx

Unload zone

Nx

Figure 1 Redistribution of stress flow of plate with a hole

Model of planar state of plate with rectangular hole(Figure 2) is formed under the following conditions

(i) distribution of load at place where elements of plateare separated is linearized

(ii) the effect of the transversal forces along the connect-ing line among the elements of plate is ignored

Justification for these assumptions is reflected in the factthat linearized force function119873

119909has the same mean value as

real distribution that the difference of the external work is asmall value in comparison to the work of real load Externalshearing forces do not affect the plate and the induceddeformation energy due to the fact that shear stress is smallvalue in comparison to the normal stress

The components of the reaction force 11987310158401(119910) and 119873

1015840

2(119910)

defined by (6) and (7) are determined from the equilibriumcondition of plate element ldquo1rdquo plates

1198731015840

1(119910) =

119887

1198871

119887 minus 1198872

2119887 minus 1198871minus 1198872

119873 = 1198911119873 0 le 119910 le 119887

1 (6)

1198731015840

2(119910) =

119887

1198872

119887 minus 1198871

2119887 minus 1198871minus 1198872

119873 = 1198912119873 (119887 minus 119887

2) le 119910 le 119887 (7)

Elasticity of plate affects the induction of additional reac-tions of element ldquo1rdquo whichmanifest asmoments (8) and linearvariable force (9)

1198721=11990421198902+ 11990441198901

11990421199043minus 11990411199044

119873 = 1198921119873

1198722= (

1198901

1199042

+1199041

1199042

11990421198902+ 11990441198901

11990421199043minus 11990411199044

)119873 = 1198922119873

1198723= [

1199051

1199033

minus1199032

1199033

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199032

1199033

+1199031

1199033

)]119873 = 1198923119873

1198724= [

1199051

1199036

minus1199038

1199036

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199038

1199036

+1199032

1199036

)]119873 = 1198924119873

(8)

The coefficients 11990318

11990414

11989012

and 11990512

are defined inAppendix A

Distribution of linear variable forces caused by the influ-ence of the moments of elastic clamp is given by

11987310158401015840

1(119910) = (

2119910

1198871

minus 1)21198721

1198871

= 1198911119873 0 le 119909 le 119887

1

11987310158401015840

1(119910) = (

2 (119887 minus 119910)

1198872

minus 1)21198722

1198872

(119887 minus 1198872) le 119909 le 119887

(9)

The distribution of reaction forces (6) (7) and (9) along119909 = 119886

1is stepwise variable which means that at part 119887

1lt

119910 lt (1198871+ ℎ) does not act load Developing into a single

trigonometric series and by their adding coming up are tounique function (12) which determines the conditions at theedge 119909 = 119886

1of segment ldquo1rdquo of plate is obtained

1198731(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198911119873

119899120587sin 1198991205871198871

2119887sin 1198991205871198871

2119887minus

81198921119873

119899212058721198872

1

times cos 1198991205872(119899120587 cos 1198991205871198871

2119887

minus 2119887

1198871

sin 11989912058711988712119887

)]

times sin119899120587119910

119887

= 119873

infin

sum

119899=1

119875119899sin

119899120587119910

119887

(10)

1198732(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198912119873

119899120587sin

119899120587 (2119887 minus 1198872)

2119887sin 1198991205871198872

2119887

+81198922119873

119899212058721198872

2

cos119899120587 (2119887 minus 119887

2)

2119887

times (119899120587 cos 11989912058711988722119887

minus 2119887

1198872

sin 11989912058711988722119887

)]

times sin119899120587119910

119887= 119873

infin

sum

119899=1

119876119899sin

119899120587119910

119887

(11)

119873(119909 = 1198861 119910) = 119873

1(119909 = 119886

1 119910) + 119873

2(119909 = 119886

1 119910)

= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887

(12)

Defining a planar state of stress of plate Airy stressfunction is applied This function has the following form

1205974119880

1205971199094+

1205974119880

12059711990921205971199102+1205974119880

1205971199104= 0 (13)

The Scientific World Journal 5

N = const

y

1

b

a1

b2

h

b1

N2(y) = N9984002 + N998400998400

2

M2

N

c

N1(y) =998400 + 998400998400N1 N1

M1 M3

M4

EI1 EI3

EI4

EI2

h

N

N = const

b

y

1

a1

h

3

4

2

c a2

b2

b1

x

NN2(y) N4(y)

N3(y)N1(y)

oplus

oplus

oplus

oplus

oplus

oplusoplus

⊖⊖

oplusoplus

oplus

⊖ ⊖

oplus oplus

Figure 2 Model of planar state of stress of plate with rectangular hole

A function that satisfies (13) is presented by the trigono-metric series

119880(119909 119910) =

infin

sum

119899=1

[119860119899sinh 119899120587119909

119887+ 119861119899(119899120587119909

119887) sinh 119899120587119909

119887

+ 119862119899cosh 119899120587119909

119887+ 119863119899(119899120587119909

119887) cosh 119899120587119909

119887]

times sin119899120587119910

119887

(14)

Planar forces 119873119909and 119873

119909119910are defined through the com-

ponent stresses 120590119909and 120590

119909119910formulated by

119873119909=120590119909

120575=1

120575

1205972119880

1205971199102

=1205872

1205751198872

infin

sum

119899=1

1198992[119860119899sinh 119899120587119909

119887

+ 119861119899(2 cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)

+ 119862119899cosh 119899120587119909

119887

+ 119863119899(2 sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)]

times sin119899120587119910

119887

119873119909119910=

120590119909119910

120575= minus

1

120575

1205972119880

120597119909120597119910

= minus1205872

1205751198872

infin

sum

119899=1

1198992[119860119899cosh 119899120587119909

119887

+ 119861119899(sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)

+ 119862119899sinh 119899120587119909

119887

+ 119863119899(cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)]

times cos119899120587119910

119887

(15)

Integration constants 119860119899 119861119899 119862119899 and 119863

119899are determined

from the boundary conditions as follows

(1) 119873119909= 119873 =

2119873

120587

infin

sum

119899=1

1

119899(1 minus cos 119899120587) sin

119899120587119910

119887= const

119873119909119910= 0 119909 = 0

(2) 119873119909= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887 119873119909119910= 0 119909 = 119886

1

(16)

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

4 The Scientific World Journal

Nx

Unload zone

Nx

Figure 1 Redistribution of stress flow of plate with a hole

Model of planar state of plate with rectangular hole(Figure 2) is formed under the following conditions

(i) distribution of load at place where elements of plateare separated is linearized

(ii) the effect of the transversal forces along the connect-ing line among the elements of plate is ignored

Justification for these assumptions is reflected in the factthat linearized force function119873

119909has the same mean value as

real distribution that the difference of the external work is asmall value in comparison to the work of real load Externalshearing forces do not affect the plate and the induceddeformation energy due to the fact that shear stress is smallvalue in comparison to the normal stress

The components of the reaction force 11987310158401(119910) and 119873

1015840

2(119910)

defined by (6) and (7) are determined from the equilibriumcondition of plate element ldquo1rdquo plates

1198731015840

1(119910) =

119887

1198871

119887 minus 1198872

2119887 minus 1198871minus 1198872

119873 = 1198911119873 0 le 119910 le 119887

1 (6)

1198731015840

2(119910) =

119887

1198872

119887 minus 1198871

2119887 minus 1198871minus 1198872

119873 = 1198912119873 (119887 minus 119887

2) le 119910 le 119887 (7)

Elasticity of plate affects the induction of additional reac-tions of element ldquo1rdquo whichmanifest asmoments (8) and linearvariable force (9)

1198721=11990421198902+ 11990441198901

11990421199043minus 11990411199044

119873 = 1198921119873

1198722= (

1198901

1199042

+1199041

1199042

11990421198902+ 11990441198901

11990421199043minus 11990411199044

)119873 = 1198922119873

1198723= [

1199051

1199033

minus1199032

1199033

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199032

1199033

+1199031

1199033

)]119873 = 1198923119873

1198724= [

1199051

1199036

minus1199038

1199036

1198901

1199042

minus11990421198902+ 11990441198901

11990421199043minus 11990411199044

(1199041

1199042

1199038

1199036

+1199032

1199036

)]119873 = 1198924119873

(8)

The coefficients 11990318

11990414

11989012

and 11990512

are defined inAppendix A

Distribution of linear variable forces caused by the influ-ence of the moments of elastic clamp is given by

11987310158401015840

1(119910) = (

2119910

1198871

minus 1)21198721

1198871

= 1198911119873 0 le 119909 le 119887

1

11987310158401015840

1(119910) = (

2 (119887 minus 119910)

1198872

minus 1)21198722

1198872

(119887 minus 1198872) le 119909 le 119887

(9)

The distribution of reaction forces (6) (7) and (9) along119909 = 119886

1is stepwise variable which means that at part 119887

1lt

119910 lt (1198871+ ℎ) does not act load Developing into a single

trigonometric series and by their adding coming up are tounique function (12) which determines the conditions at theedge 119909 = 119886

1of segment ldquo1rdquo of plate is obtained

1198731(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198911119873

119899120587sin 1198991205871198871

2119887sin 1198991205871198871

2119887minus

81198921119873

119899212058721198872

1

times cos 1198991205872(119899120587 cos 1198991205871198871

2119887

minus 2119887

1198871

sin 11989912058711988712119887

)]

times sin119899120587119910

119887

= 119873

infin

sum

119899=1

119875119899sin

119899120587119910

119887

(10)

1198732(119909 = 119886

1 119910)

=

infin

sum

119899=1

[41198912119873

119899120587sin

119899120587 (2119887 minus 1198872)

2119887sin 1198991205871198872

2119887

+81198922119873

119899212058721198872

2

cos119899120587 (2119887 minus 119887

2)

2119887

times (119899120587 cos 11989912058711988722119887

minus 2119887

1198872

sin 11989912058711988722119887

)]

times sin119899120587119910

119887= 119873

infin

sum

119899=1

119876119899sin

119899120587119910

119887

(11)

119873(119909 = 1198861 119910) = 119873

1(119909 = 119886

1 119910) + 119873

2(119909 = 119886

1 119910)

= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887

(12)

Defining a planar state of stress of plate Airy stressfunction is applied This function has the following form

1205974119880

1205971199094+

1205974119880

12059711990921205971199102+1205974119880

1205971199104= 0 (13)

The Scientific World Journal 5

N = const

y

1

b

a1

b2

h

b1

N2(y) = N9984002 + N998400998400

2

M2

N

c

N1(y) =998400 + 998400998400N1 N1

M1 M3

M4

EI1 EI3

EI4

EI2

h

N

N = const

b

y

1

a1

h

3

4

2

c a2

b2

b1

x

NN2(y) N4(y)

N3(y)N1(y)

oplus

oplus

oplus

oplus

oplus

oplusoplus

⊖⊖

oplusoplus

oplus

⊖ ⊖

oplus oplus

Figure 2 Model of planar state of stress of plate with rectangular hole

A function that satisfies (13) is presented by the trigono-metric series

119880(119909 119910) =

infin

sum

119899=1

[119860119899sinh 119899120587119909

119887+ 119861119899(119899120587119909

119887) sinh 119899120587119909

119887

+ 119862119899cosh 119899120587119909

119887+ 119863119899(119899120587119909

119887) cosh 119899120587119909

119887]

times sin119899120587119910

119887

(14)

Planar forces 119873119909and 119873

119909119910are defined through the com-

ponent stresses 120590119909and 120590

119909119910formulated by

119873119909=120590119909

120575=1

120575

1205972119880

1205971199102

=1205872

1205751198872

infin

sum

119899=1

1198992[119860119899sinh 119899120587119909

119887

+ 119861119899(2 cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)

+ 119862119899cosh 119899120587119909

119887

+ 119863119899(2 sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)]

times sin119899120587119910

119887

119873119909119910=

120590119909119910

120575= minus

1

120575

1205972119880

120597119909120597119910

= minus1205872

1205751198872

infin

sum

119899=1

1198992[119860119899cosh 119899120587119909

119887

+ 119861119899(sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)

+ 119862119899sinh 119899120587119909

119887

+ 119863119899(cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)]

times cos119899120587119910

119887

(15)

Integration constants 119860119899 119861119899 119862119899 and 119863

119899are determined

from the boundary conditions as follows

(1) 119873119909= 119873 =

2119873

120587

infin

sum

119899=1

1

119899(1 minus cos 119899120587) sin

119899120587119910

119887= const

119873119909119910= 0 119909 = 0

(2) 119873119909= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887 119873119909119910= 0 119909 = 119886

1

(16)

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 5

N = const

y

1

b

a1

b2

h

b1

N2(y) = N9984002 + N998400998400

2

M2

N

c

N1(y) =998400 + 998400998400N1 N1

M1 M3

M4

EI1 EI3

EI4

EI2

h

N

N = const

b

y

1

a1

h

3

4

2

c a2

b2

b1

x

NN2(y) N4(y)

N3(y)N1(y)

oplus

oplus

oplus

oplus

oplus

oplusoplus

⊖⊖

oplusoplus

oplus

⊖ ⊖

oplus oplus

Figure 2 Model of planar state of stress of plate with rectangular hole

A function that satisfies (13) is presented by the trigono-metric series

119880(119909 119910) =

infin

sum

119899=1

[119860119899sinh 119899120587119909

119887+ 119861119899(119899120587119909

119887) sinh 119899120587119909

119887

+ 119862119899cosh 119899120587119909

119887+ 119863119899(119899120587119909

119887) cosh 119899120587119909

119887]

times sin119899120587119910

119887

(14)

Planar forces 119873119909and 119873

119909119910are defined through the com-

ponent stresses 120590119909and 120590

119909119910formulated by

119873119909=120590119909

120575=1

120575

1205972119880

1205971199102

=1205872

1205751198872

infin

sum

119899=1

1198992[119860119899sinh 119899120587119909

119887

+ 119861119899(2 cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)

+ 119862119899cosh 119899120587119909

119887

+ 119863119899(2 sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)]

times sin119899120587119910

119887

119873119909119910=

120590119909119910

120575= minus

1

120575

1205972119880

120597119909120597119910

= minus1205872

1205751198872

infin

sum

119899=1

1198992[119860119899cosh 119899120587119909

119887

+ 119861119899(sinh 119899120587119909

119887+119899120587119909

119887cosh 119899120587119909

119887)

+ 119862119899sinh 119899120587119909

119887

+ 119863119899(cosh 119899120587119909

119887+119899120587119909

119887sinh 119899120587119909

119887)]

times cos119899120587119910

119887

(15)

Integration constants 119860119899 119861119899 119862119899 and 119863

119899are determined

from the boundary conditions as follows

(1) 119873119909= 119873 =

2119873

120587

infin

sum

119899=1

1

119899(1 minus cos 119899120587) sin

119899120587119910

119887= const

119873119909119910= 0 119909 = 0

(2) 119873119909= 119873

infin

sum

119899=1

(119875119899+ 119876119899) sin

119899120587119910

119887 119873119909119910= 0 119909 = 119886

1

(16)

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

6 The Scientific World Journal

By substitution of (16) in (15) required coefficients definedby the coefficients (17) are obtained

119860119899= 1198601015840

119899119873

119861119899= 1198611015840

119899119873

119862119899= 1198621015840

119899119873

119863119899= 1198631015840

119899119873

(17)

where

120572119899=1198991205871198861

119887 (18)

Coefficients 1198601015840119899 1198611015840119899 1198621015840119899 and1198631015840

119899are given in Appendix B

The coefficients given by (17)-(18) are derived for elementldquo1rdquo and for applying them to element ldquo2rdquo is necessary tocorrect factor 120572

119899by ratio 119886

11198862 and then its value is 120572

119899=

1198991205871198862119887 If the hole is not centric along the 119909-axis that is if

the center of hole is nearer to one of the sides 119909 = 0 or 119909 = 119887then the elements of plates ldquo1rdquo and ldquo2rdquo are different whichis manifested through coefficient 120572

119899 Especially if the hole is

centric then we have 120572119899= 1198991205871198862119887

33 Determining the Buckling Coefficient and Critical StressAnalysis of elastic buckling of perforated plate by energymethod is presented on the example of simply supported plateof ideal planar geometry and without effect of transversalload noting that this does not reduce the generality of themethodology Deflection is assumed by trigonometric series(7) which satisfies the boundary conditions of a simplysupported plate

119908 (119909 119910) =

infin

sum

119898=1

infin

sum

119899=1

119860119898119899

sin 119898120587119909119886

sin119899120587119910

119887 (19)

The work carried out by uniaxial variable load (119873119909)

affecting the plate with a rectangular hole is

119864119882=1

2int119860

119873119909(120597119908

120597119909)

2

119889119860

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

119886

1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

1198871

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

+1

2int

1198862

1198861

int

119887

1198872

119873119909(120597119908

120597119909)

2

119889119909 119889119910

(20)

The total external work is determined by the componentsthat correspond to elements of plates ldquo1ndash4rdquo according toFigure 2

1198641198821

=1

2int

1198861

0

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119860119899119868119898119899

+ 119861119899119869119898119899

+ 119862119899119870119898119899

+ 119863119899119871119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198601015840

119899119868119898119899

+ 1198611015840

119899119869119898119899

+ 1198621015840

119899119870119898119899

+ 1198631015840

119899119871119898119899)119873

= 1205761198981198991198602

119898119899119873

1198641198822

=1

2int

119886

119886minus1198862

int

119887

0

119873119909(120597119908

120597119909)

2

119889119909 119889119910

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (119864119899119881119898119899

+ 119865119899119882119898119899

+ 119866119899119877119898119899

+ 119867119899119879119898119899)

=1

2

120587411989821198992

119886211988721205751198602

119898119899

times (1198641015840

119899119881119898119899

+ 1198651015840

119899119882119898119899

+ 1198661015840

119899119877119898119899

+ 1198671015840

119899119879119898119899)119873

= 1205881198981198991198602

119898119899119873

(21)

Integral forms given by coefficients 119868119898119899 119869119898119899 119870119898119899 and

119871119898119899

which correspond to element ldquo1rdquo or by coefficients 119881119898119899

119882119898119899 119877119898119899 and 119879

119898119899belonging to element ldquo2rdquo are formulated

in Appendix C

1198641198823

=1

2int

1198862

1198861

int

119887

119887minus1198872

1198731(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198761198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3119899120587 minus cos

119899120587 (119887 minus 1198872)

119887)

minus (cos 119899120587 minus cos119899120587 (119887 minus 119887

2)

119887)]

= 1205931198981198991198602

119898119899119873

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 7

1198641198824

=1

2int

1198862

1198861

int

1198871

0

1198732(119910) (

120597119908

120597119909)

2

119889119909 119889119910

=1

41198602

1198981198991198751198991198731198982119887120587

1198991198862(119888 +

119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times [1

3(cos3 1198991205871198871

119887minus 1) minus (cos 1198991205871198871

119887minus 1)]

= 1205961198981198991198602

119898119899119873

(22)

Adequate internal or deformation energy of plate con-sisted of bending and torsion of the components is given by

119864119868=119863

2int119860

(1205972119908

1205971199092+1205972119908

1205971199102)

2

minus2 (1 minus 120592) [1205972119908

1205971199092

1205972119908

1205971199102minus (

1205972119908

120597119909120597119910)

2

]119889119860

(23)

Flexion component of the internal energy is

119864119868119891

=1

81198631198602

1198981198991205874(1198984

1198864+1198994

1198874+ 2120592

11989821198992

11988621198872)

times [119886119887 minus (119888 minus119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ minus119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205821198981198991198602

119898119899

(24)

Torsional component of the internal energy is

119864119868119905=1

4119863 (1 minus 120592)119860

2

119898119899120587411989821198992

11988621198872

times [119886119887 minus (119888 +119886

119898120587cos

2119898120587119901

119886sin 119898120587119888

119886)

times (ℎ +119887

119899120587cos

2119899120587119902

119887sin 119899120587ℎ

119887)]

= 1205831198981198991198602

119898119899

(25)

The plate is in a state of stable equilibrium if the followingcondition is satisfied

120597119864119879

120597119860119898119899

= 0 resp120597 (119864119868119891

+ 119864119868119905+ 119864119882)

120597119860119898119899

= 0 (26)

where the total potential energy of the plate is given by

119864119879= 119864119868119891

+ 119864119868119905+ 119864119882 (27)

Substitution of (20) (24) and (25) in (26) leads to (28) whichdefines the critical stress of elastic buckling

120590cr =119873cr120575

=120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

(28)

The critical buckling force can be written in compactform

120590cr = 1198961205872119863

1198872120575 (29)

where 119896 is the coefficient of elastic buckling and it can bepresented as

119896hole =120582119898119899

+ 120583119898119899

120576119898119899

+ 120588119898119899

+ 120593119898119899

+ 120596119898119899

times1198872120575

1205872119863

(30)

Ratio of buckling coefficient of uniform plate and platewith a hole is defined by sensitivity factor 119905which is ameasureof plate sensitivity to discontinuous changes in geometryThis factor is proportional to the reciprocal value of thecoefficient of elastic buckling 119896hole and it is important becauseit represents the ability of plate with hole and a particularform and orientation to keep its behavior in the domain ofelastic stability

119905 =119896unhole119896hole

=4

119896hole (31)

Characteristic value of correction factor 119905 is 1 corre-sponding to a uniform plate (Figure 3(a)) If the plate iswith hole this value is adjusted by coefficient (1119896hole) andsensitivity factor 119905 can be larger smaller or possibly equal to1 depending on the values of influential parameters

4 Comparative Analysis andVerification of Results

By analyzing the mathematical model of elastic bucklingit can be concluded that the work of external load has acrucial role in the stability of plate This conclusion comes tothe fore when the plate with elongated holes is consideredSpecifically then the change of deformation energy of plate isminor while the existence of the hole initiate redistributionof stresses and significant change in the external energyThe consequence of this phenomenon is the correction ofbuckling coefficient or sensitivity factor reducing or increas-ing their value In this paper an analysis of the influentialparameters for some of the characteristic forms will beconsidered

41 The Case of Rectangular Hole Figure 4 shows the com-parative analysis of the buckling coefficient by presentedmethod with data for the square plate whose dimensionsare 119886119909119886 with centric rectangular hole dimensions of 119888119909ℎwhere 119888 = 025119886 represents width of the hole Commonlythe buckling coefficient 119896 is expressed as a function ofdimensionless value (ℎ119886) where ℎ is the height of thehole (Figure 3(b)) By analyzing the diagram (Figure 4) canbe concluded that the data [9 16 22 24] have the samedistribution trend while the values differ

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

8 The Scientific World Journal

Nx

b y

a

120575

x

Nx

(a)

Nx

b y

a

120575

x

h

Nxc

(b)

Figure 3 Uniaxial stressed rectangular plate (a) uniform geometry (b) with a rectangular hole

20

30

40

50

60

70

80

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

k(ha)-distribution of a buckling coefficient

Present methodMVM [24]Brown [16]

Azhari [9]ANSYS [22]

Figure 4 Comparative analysis of buckling coefficient of squareplate with a rectangular hole

The main difference between presented procedure andliterature is reflected in the shape of the distribution untilthe values 119910119886 = 025 where the function 119896(ℎ119886) has aconcave shape According to [22] this form is linear whileother researches show that the function is convex in thewholedomain On the domain ℎ119886 gt 025 function of bucklingcoefficient asymptotic tends to the distribution according to[16] while its values are the closest to the data correspondingto [22] The research [18] by analyzing the vertical slottedhole which is closest in shape to a rectangular hole showsthe existence of the concave side in the field to ℎ119886 = 025 aspresented in the discussion on the slotted hole

A special case of the previous considerations is shownin Figure 5 where the square plate with a centric squarehole by dimensions of 119889119909119889 is analyzed Comparative analysisof this procedure is given in studies [14 22] By increasingdimensions of the hole in comparison with plate dimensionsthe buckling coefficient decreases permanently The function

k(ha)-distribution of a buckling coefficient

Present methodSabir [14]ANSYS [22]

20

30

40

50

00 01 02 03 04 05 06 07

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (ha)

Figure 5 Comparative analysis of buckling coefficient of squareplate with a square hole

119896(119889119886) the presented method agrees well with the distri-bution corresponding to [22] both in terms of trend andquantity According to [14] in the area above the 119889119886 gt 04 thebuckling coefficient 119896 first stagnates and then tends to growslowly

In the special case when there is no rectangular hole wehave the case of geometric uniform simply supported anduniaxial stressed plate (Figure 3(a)) with buckling coefficient(32) corresponding to the data from [1 2]

119896 = (119898119887

119886+

119886

119898119887)

2

(32)

The minimum value of the critical stress is obtained for(119898 119899) = 1

120590cr (119898 119899 = 1) = 41205872119863

1198872 (33)

where 119896 = 4 is buckling coefficient of uniform square plate

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 9

Nx

b y

a

120575

x

Nx

120601d

Figure 6 A rectangular plate with a circular hole

42 A Rectangular Plate with a Circular Hole Determinationof the coefficient of elastic buckling of a rectangular plate witha circular hole (Figure 6) by previously presented procedureis not possible in a direct manner The reason for this relatesto the complexity of integrant due to circular contours andinterdependent coordinates119909 and119910Therefore it is necessaryto use approximate solving approach that is based on theresults of the previous analysis Namely the essence of thisprocedure is that the circular hole with diameter 119889 is dividedin 119904 rectangular gaps with dimensions ℎ

119894and 119888

119894whose

positions are defined by 119901119894and 119902119894 The number of rectangular

gaps has to be large enough in order to follow the contour ofcircular hole in a best way

In order to be applicable for a circular hole it is necessaryto make a correction of model of planar state of stress devel-oped for rectangular hole according to Figure 7 Element ldquo1rdquodimensions of 119886

1119909119887 is adjusted to the length of 1198861015840

1to which

the smaller influence of moments corresponds (distributiontends to be uniform form)

Defined geometric values of rectangular gaps (34) areused in the expression for the deformation energy (24) and(25) instead of values 119901 119888 119902 and ℎ

119901119894= 119901 minus

119889

2+2119894 minus 1

2119888 119894 = 1 2 119904

119888119894=119889

119894 119894 = 1 2 119904

119902119894= 119902 = const

ℎ119894= radic1198892 minus [(2119894 minus 1) 119888]

2 119894 = 1 2 119904

(34)

The function of the buckling coefficient 119896(119889119886) has atendency to decline in value over the area of consideration(Figure 8) Progressive decline of value is in the interval119889119886 = (01ndash04) and values over 09 The obtained data arequalitatively and quantitatively consistent with [14] whichjustifies the assumptions in the process of model formationand its implementation to other similar contour shapes

N = consty

b

a1

N2(y)

N1(y)h

b1

b2

45∘

a1998400

y

b

a

y

ci

hi

q

x zp

pi

120575

1

120601d

oplus

oplus

Figure 7 Distribution of the circular hole in rectangular gaps

20

30

40

50

00 01 02 03 04 05 06 07 08 09 10

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodQEM [8]

Figure 8 Comparative analysis of buckling coefficient of squareplate with a circular hole

43 Rectangular Plate with a Slotted Hole Research of platewith rectangular hole has shown that the orientation of thehole to the plate position has a great influence on the stabilityOn the other hand analysis of plate with hole indicates anincrease in work performed by an external force 119873

119909causing

a decrease of the buckling coefficient Slotted hole representsa combination of the previous two cases

Analysis of buckling coefficient 119896 indicates the depen-dence of the hole orientation Figure 9(a) corresponds toa horizontal hole position in relation to the direction of

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

10 The Scientific World Journal

Nx

x

e

d 15d

120575

y

f

b

Nx

a

(a)

x

Nx

120575

e

15d d

f

b y

a

Nx

(b)

Figure 9 A rectangular plate with a slotted hole (a) horizontal (b) vertical orientation

10

20

30

40

50

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 10 Comparative analysis of buckling coefficient of squareplate with a slotted hole (horizontal position)

the force 119873119909 Buckling coefficient according to [18] defined

by FEM method is linearly decreasing The distribution of119896(119889119886) obtained by the presented methodology has the sametrend as the maximum deviation of 5 for 119889119886 = 025

(Figure 10)However if the hole is rotated by an angle of 90∘

(Figure 9(b)) while the other features unchanged the behav-ior of the plate is very different The function 119896(119889119886) can bedivided into two parts in the domain of examination Thefirst part refers to the interval [0 02] where 119896 is a concavefunction and it decreases regressively with increasing 119889119886 Inthe domain [02 05] function has convex form that affectsprimarily the stagnation and then to progressively increaseFunctions of buckling coefficients for the case with slottedhole and rectangular hole are analogous The value of 119896 islarger for the rectangular hole in relation to the slotted holefor the same value of dimensionless argument as a result oflower external work (Figure 11)

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

buc

klin

g co

effici

ent

Normalized hole dimension (da)

k(da)-distribution of a buckling coefficient

Present methodMaiorana [18]

Figure 11 Comparative analysis of buckling coefficient of squareplate with a slotted hole (vertical position)

5 The Influence of the Hole on the ExternalWork and Elastic Stability of Plate

Previously it was concluded that the work of external forcesdominantly influence the elastic stability of plate Defor-mation work of plate exclusively depends on the hole sizewhile the size position and orientation of the hole havesecondary impact Deformation work is a linear functionof plate rigidity Distributions of planar compressive forcescause induction of external work so their identification iscrucial in analysis of stability Model of planar state of stresswas developed for the rectangular hole it was implementedwith correction to circular and slotted form The analysiscarried out this section clearly shows that increasing of sizeof the one-dimensional hole (square and circular) leads toreduction in elastic stability In the case of two-dimensionalholes (rectangular shape and slotted) it should distinguishhorizontal and vertical orientationThe first case is analogousto the previous analysis while the second variant affects

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 11

00

05

10

15

00 01 02 03 04 05

Valu

e of a

coeffi

cien

t ext

erna

l wor

k

Normalized hole dimension (ha)

120582(ha)-distribution of a coefficient external work

Rectangular holeQuadrat holeCircular hole

Slotted hole (horizontal)Slotted hole (vertical)

Figure 12 Change of coefficient 120582 in relation to the parameter (ℎ119886)for the considered hole

the elastic stability It firstly decreases and then increasesdominantly and even exceed the value of 4 which is char-acteristic for the plate without hole Rectangular hole whosegreater dimension is normal to the direction of load action isexposed to considerable stress concentration which is nega-tive in terms of static but it is perfect condition for stabilityConsidering that the work of the external compression forcesdirectly affects the buckling coefficient it is necessary tointroduce the coefficient of external work 120582 which presentsthe absorbed energy caused by the shape of the hole inrelation to the uniform plate

The coefficient of external work is defined as the ratioof external energy of plate with hole and uniform geometrythrough relation

120582 =119864119882hole

119864119882unhole

(35)

Value of coefficient 120582 for slotted hole in the horizontalposition is greater than 1 in the interval from 0 to 05(Figure 12) This value decreases for larger domains Coeffi-cient 120582 for the other shapes of hole has a tendency to fall sincetheir shape affects the reduction ofwork of compressive forcesin relation to the uniform plate The minimum value of 120582 isfor the vertical orientation of slotted hole

There is certain similarity between the diagrams shownin Figures 12 and 13 Slotted hole in horizontal positionhas the most sensitivity This form redistributes the loadabove and below the hole over nominal value of 119873 while infront of and behind the hole circular configuration preventsturbulence of fluid flow and distribution near the value119873119909is formed Because the value of 119864

119908is greater than the

value corresponding to the uniform plate where the pressuredistribution along the plate corresponds to the nominal valueof119873119909 The situation of plate with rectangular hole is opposite

to the above mentioned As a result we have maximumsensitivity for horizontal orientation slotted shape while thesame shape rotated by 90∘ has a minimum Other variationsare between these two cases

00

10

20

30

40

50

60

00 01 02 03 04 05

Valu

e of a

resp

onse

fact

or

Normalized hole dimension (ha)

t(ha)-distribution of response factor

Rectangular holeQuadratCircular

Slotted hole (horizontal)Slotted hole (vertical)

Figure 13 Sensitivity factor 119905 as a function of the (ℎ119886) for the holesof the considered forms

The diagram given in Figure 13 is a basic guideline forselection of the hole shape in terms of the plate stabilityVertical position of slotted hole (Figure 9(b)) in this respectrepresent of optimal shape which is important in the con-struction process of thin-walled supporting structures

6 Conclusions

In this paper the problem of elastic stability of plate with ahole using a mathematical model developed on the basis ofthe energy method has been addressedThe analysis includesconsideration of four types of holes rectangular square cir-cular and slotted (the combination of rectangular and circu-lar)The existence of the hole reduces the deformation energyof plate depending on the form which most commonlyaffects the decrease in external work due to compressiveforces The dominant factor on the buckling coefficient 119896 isthe work of external forcesThat is the reason why coefficient120582 which manifests absorption ability of plate with the holein relation to the uniform plate is defined Through theexposedmethodology analyzedmechanismof elastic stabilityof plates using energy balance in a state of stable equilibriumis analyzed The mathematical identification of influentialparameters on stability which include dimensions and typeof plate material shape size position and orientation of thehole was carried out

Verification of results of the presented method has beenverified by studies [9 14 16 18 22 24] Sensitivity factor119905 defines criteria for selection of the best form of hole interms of stability Results of the application of this study areimportant for the optimization of the structures with specialemphasis on thin-walled erforated elements of high bayware-houses in order to increase the economy of distribution ofsupply chains The developed model (Section 3) is applicableto the methodology [31] and it can be implemented in orderto further studying of the phenomenon of buckling of thin-walled structures

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

12 The Scientific World Journal

Appendices

A

Consider

1199041=11990321199037

1199036

+11990311199035

1199033

1199042= 1199036minus11990371199038

1199036

minus11990321199035

1199033

1199043= 1199033minus11990321199035

1199036

minus11990311199034

1199033

1199044=11990351199038

1199036

+11990321199034

1199033

1198901= 1199052minus (

1199035

1199033

+1199037

1199036

) 1199051 119890

2= 1199052minus (

1199035

1199036

+1199034

1199033

) 1199051

1199031=

119888

31198641198682

+ℎ

31198641198681

1199032=

61198641198681

1199033=

119888

61198641198682

1199034=

119888

31198641198682

+ℎ

31198641198683

1199035=

61198641198683

1199036=

119888

61198641198684

1199037=

119888

31198641198684

+ℎ

31198641198683

1199038=

119888

31198641198684

+ℎ

31198641198681

1199051=

ℎ3

241198641198681

1199052=

ℎ3

241198641198683

1198681=1205751198863

1

12 119868

2=1205751198873

1

12

1198683=1205751198863

2

12 119868

4=1205751198873

2

12

(A1)

B

Consider

1198601015840

119899= [

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587)

minussinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

119875119899+ 119876119899

1198992]1205751198872

1205872

1198611015840

119899= [

sinh2120572119899minus 120572119899sinh120572

119899(cosh120572

119899+ 1)

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198621015840

119899= [

2120572119899sinh120572

119899(cosh120572

119899+ 1) minus sinh2120572

119899minus 1205722

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) minus

2120572119899sinh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

1198631015840

119899= [minus

120572119899sinh2120572

119899+ sinh120572

119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times2

119899120587(1 minus cos 119899120587) +

sinh120572119899+ 120572119899cosh120572

119899

sinh2120572119899minus 1205722119899

times119875119899+ 119876119899

1198992]1205751198872

1205872

(B1)

C

Consider

119868119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198861

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198861119886

cosh 1198991205871198861119887

minus 1)

+ 2119898119887 sin 21198981205871198861119886

sinh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119869119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 sinh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886)) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 13

119870119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198861

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198861119886

sinh 1198991205871198861119887

+ 2119898119887 sin 21198981205871198861119886

cosh 1198991205871198861119887

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119871119898119899

= int

1198861

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198861119887

((411988721198982+ 11988621198992)2

1205871198861

+ 11988621198992(411988721198982+ 11988621198992) 1205871198861

times cos 21198981205871198861119886

+ 1611988611988741198983 sin 21198981205871198861

119886)

+ 119887 cosh 1198991205871198861119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198861119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198861sin 21198981205871198861

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+ (411988721198982+ 11988621198992)2

) ]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119881119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

sinh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587(cosh 1198991205871198862

119887minus 1) +

119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 (cos 21198981205871198862119886

cosh 1198991205871198862119887

minus 1)

+ 2119898119887 sin 21198981205871198862119886

sinh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119882119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 cosh 119899120587119909119887

+119899120587119909

119887sinh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 cosh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992) 1205871198862

times cos 21198981205871198862119886

+ 1611988611988741198983 sin 21198981205871198862

119886)

+ 119887 sinh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119877119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

cosh 119899120587119909119887

sin3119899120587119910

119887119889119909 119889119910

= [119887

2119899120587sinh 1198991205871198862

119887+119886119887

2120587

1

411988721198982 + 11988621198992

times (119886119899 cos 21198981205871198862119886

sinh 1198991205871198862119887

+ 2119898119887 sin 21198981205871198862119886

cosh 1198991205871198862119887

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

119879119898119899

= int

1198862

0

int

119887

0

cos2119898120587119909119886

(2 sinh 119899120587119909119887

+119899120587119909

119887cosh 119899120587119909

119887)

times sin3119899120587119910

119887119889119909 119889119910

= [119899 sinh 1198991205871198862119887

((411988721198982+ 11988621198992)2

1205871198862

+ 11988621198992(411988721198982+ 11988621198992)

times 1205871198862cos 21198981205871198861

119886

+1611988611988741198983 sin 21198981205871198862

119886)

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

14 The Scientific World Journal

+ 119887 cosh 1198991205871198862119887

(11988621198992(1211988721198982+ 11988621198992)

times cos 21198981205871198862119886

+ (411988721198982+ 11988621198992)

times (411988721198982+ 11988621198992

+ 211988611989811989921205871198862sin 21198981205871198862

119886))

minus 119887 (11988621198992(1211988721198982+ 11988621198992)

+(411988721198982+ 11988621198992)2

)]

times [119887

119899120587(1

3cos3119899120587 minus cos 119899120587 + 2

3)]

(C1)

Acknowledgments

This paper is made as a result of research on project oftechnological development TR 36030 financially supportedby the Ministry of Education Science and TechnologicalDevelopment of Serbia

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill Book Company New York NY USA1959

[2] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill Book Company New York NY USA 1961

[3] H G Allen and P S Bulson Background To Buckling McGraw-Hill London UK 1980

[4] K H Gerstle Basic Structural Design McGraw-Hill BookCompany New York NY USA 1967

[5] V Radosavljevic and M Drazic ldquoExact solution for bucklingof FCFC stepped rectangular platesrdquo Applied MathematicalModelling vol 34 no 12 pp 3841ndash3849 2010

[6] A JohnWilson and S Rajasekaran ldquoElastic stability of all edgessimply supported stepped and stiffened rectangular plate underuniaxial loadingrdquo Applied Mathematical Modelling vol 36 no12 pp 5758ndash5772 2012

[7] J K Paik andAKThayamballi ldquoBuckling strength of steel plat-ing with elastically restrained edgesrdquo Thin-Walled Structuresvol 37 no 1 pp 27ndash55 2000

[8] H Zhong C Pan and H Yu ldquoBuckling analysis of sheardeformable plates using the quadrature element methodrdquoApplied Mathematical Modelling vol 35 no 10 pp 5059ndash50742011

[9] MAzhari A R Shahidi andMM Saadatpour ldquoLocal andpostlocal buckling of stepped and perforated thin platesrdquo AppliedMathematical Modelling vol 29 no 7 pp 633ndash652 2005

[10] J Petrisic F Kosel and B Bremec ldquoBuckling of plates withstrengtheningsrdquoThin-Walled Structures vol 44 no 3 pp 334ndash343 2006

[11] O K Bedair and A N Sherbourne ldquoPlatestiffener assembliesunder nonuniform edge compressionrdquo Journal of StructuralEngineering vol 121 no 11 pp 1603ndash1612 1995

[12] T Mizusawa T Kajita and M Naruoka ldquoVibration and buck-ling analysis of plates of abruptly varying stiffnessrdquo Computersand Structures vol 12 no 5 pp 689ndash693 1980

[13] J P Singh and S S Dey ldquoVariational finite difference approachto buckling of plates of variable stiffnessrdquo Computers andStructures vol 36 no 1 pp 39ndash45 1990

[14] A B Sabir and F Y Chow ldquoElastic buckling of flat panelscontaining circular and square holesrdquo in Proceedings of theInternational Conference on Instability and Plastic Collapseof Steel Structures L J Morris Ed pp 311ndash321 GranadaPublishing London UK 1983

[15] C J Brown and A L Yettram ldquoThe elastic stability of squareperforated plates under combinations of bending shear anddirect loadrdquo Thin-Walled Structures vol 4 no 3 pp 239ndash2461986

[16] C J Brown A L Yettram and M Burnett ldquoStability of plateswith rectangular holesrdquo Journal of Structural Engineering vol113 no 5 pp 1111ndash1116 1987

[17] C J Brown ldquoElastic buckling of perforated plates subjected toconcentrated loadsrdquoComputers and Structures vol 36 no 6 pp1103ndash1109 1990

[18] E Maiorana C Pellegrino and C Modena ldquoElastic stabilityof plates with circular and rectangular holes subjected to axialcompression and bending momentrdquo Thin-Walled Structuresvol 47 no 3 pp 241ndash255 2009

[19] I E Harik and M G Andrade ldquoStability of plates with stepvariation in thicknessrdquo Computers and Structures vol 33 no1 pp 257ndash263 1989

[20] H C Bui ldquoBuckling analysis of thin-walled sections undergeneral loading conditionsrdquoThin-Walled Structures vol 47 no6-7 pp 730ndash739 2009

[21] C J Brown and A L Yettram ldquoFactors influencing the elasticstability of orthotropic plates containing a rectangular cut-outrdquoJournal of Strain Analysis for Engineering Design vol 35 no 6pp 445ndash458 2000

[22] K M El-Sawy and A S Nazmy ldquoEffect of aspect ratio on theelastic buckling of uniaxially loaded plates with eccentric holesrdquoThin-Walled Structures vol 39 no 12 pp 983ndash998 2001

[23] C D Moen and B W Schafer ldquoElastic buckling of thin plateswith holes in compression or bendingrdquoThin-Walled Structuresvol 47 no 12 pp 1597ndash1607 2009

[24] A R Rahai M M Alinia and S Kazemi ldquoBuckling analysisof stepped plates using modified buckling mode shapesrdquo Thin-Walled Structures vol 46 no 5 pp 484ndash493 2008

[25] N E Shanmugam V T Lian and V Thevendran ldquoFiniteelement modelling of plate girders with web openingsrdquo Thin-Walled Structures vol 40 no 5 pp 443ndash464 2002

[26] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers andStructures vol 56 no 1 pp 75ndash83 1995

[27] S C W Lau and G J Hancock ldquoBuckling of thin flat-walled structures by a spline finite strip methodrdquo Thin-WalledStructures vol 4 no 4 pp 269ndash294 1986

[28] Y K Cheung F T K Au and D Y Zheng ldquoFinite strip methodfor the free vibration and buckling analysis of plates with abruptchanges in thickness and complex support conditionsrdquo Thin-Walled Structures vol 36 no 2 pp 89ndash110 2000

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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

The Scientific World Journal 15

[29] C Bisagni and R Vescovini ldquoAnalytical formulation for localbuckling and post-buckling analysis of stiffened laminatedpanelsrdquoThin-Walled Structures vol 47 no 3 pp 318ndash334 2009

[30] D G Stamatelos G N Labeas and K I Tserpes ldquoAnalyticalcalculation of local buckling and post-buckling behavior ofisotropic and orthotropic stiffened panelsrdquo Thin-Walled Struc-tures vol 49 no 3 pp 422ndash430 2011

[31] MDjelosevic VGajic D Petrovic andMBizic ldquoIdentificationof local stress parameters influencing the optimum design ofbox girdersrdquo Engineering Structures vol 40 pp 299ndash316 2012

[32] M Djelosevic I Tanackov M Kostelac V Gajic and J TepicldquoModeling elastic stability of a pressed box girder flangerdquoApplied Mechanics and Materials vol 343 pp 35ndash41 2013

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