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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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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=
ℎ
61198641198681
1199033=
119888
61198641198682
1199034=
119888
31198641198682
+ℎ
31198641198683
1199035=
ℎ
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
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
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
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
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