Exact stress functions implementation in stability analysisof plates with different boundary conditions underuniaxial and biaxial compression

Olga Mijušković n, Branislav Ćorić, Biljana ŠćepanovićUniversity of Montenegro, Faculty of Civil Engineering, Cetinjski put bb, 81000 Podgorica, Montenegro

a r t i c l e i n f o

Article history:Received 8 November 2013Received in revised form8 March 2014Accepted 12 March 2014Available online 15 April 2014

Keywords:Elastic stability of platesExact stress functionsRitz energy methodMixed boundary conditions

a b s t r a c t

Analytical approach used for critical load determination is based on Ritz energy technique in which twofactors are crucial for the accuracy of results. First factor is deflection function. Herein, double Fourierseries are used to represent buckled shape of the plates under arbitrary external loads. The second andthe most important factor is adoption of adequate realistic stress distribution within plate prior tobuckling. Based on the Baker&Pavlović&Tahan and Pavlović&Liu investigations, exact stress distributionswere introduced herein. In that way, with the adequate deflection functions and precise stressdistribution, Ritz energy method can produce highly accurate results.

Through several examples for the plates with different boundary conditions, for which availableliterature offers very few analytical solutions, accuracy of the presented analytical approach is proved.Results obtained by analytical approach are reaffirmed by numerical finite-element runs.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Previous studies on stability of rectangular plates under influ-ence of variable loads were mostly based on assumptions ofsimplified stress distribution, which led to questions as the accuracyof results thus obtained. This paper shows how the stability ofplates with mixed boundary conditions under arbitrary in-planeload distributions, analyzed up to now almost exclusively bynumerical methods, can be tackled analytically in an accurate,practically exact manner. The proposed method uses an approachbased on Ritz energy technique. Introduction of the exact two-dimensional elasticity solutions for in-plane stress distributions iscrucial contribution and main condition for the wide applicability ofthe presented analytical method. By adopting exact stresses withina plate under any type of external loads and by using the doubleFourier series to represent any possible buckled profile for plateswith different boundary conditions, the critical loads can beobtained in very accurate way. The results produced, for the firsttime, by the analytical approach proposed in this article, arecompared to those from numerical finite element method.

It is interesting that the analytical procedure for determiningthe exact stress distribution within a rectangular plate, loaded by

in-plane arbitrary external load is based on the solution datedfrom XIX century. In fact, in 1890 Mathieu [1] was the first todetermine the stress function for the particular case of a rectan-gular plate loaded along the edges by an arbitrary compressivestress through superposition of two (DEA and DEB), so called, basictypes of load (Fig. 1).

In the series of papers by Baker and Pavlović [2–5], one of thesetypes of load (DEA), was applied during the stability analysis ofsimply supported plates, loaded by the locally distributed com-pressive stress. Several years later, in 1993, more than one centuryafter the Mathieu's original paper had appeared, the same authorsreturned to the basic problem [6]. Their aim was to obtain thesolution of the exact stress distribution for the general case of therectangular plate loaded by the arbitrary external in-plain load.The fact is that any in-plane load (normal and/or shear) which actsalong the edges of the plate, can be described by the chosenfunctions (even and/or odd in relation to the coordinate axes), sothe total solution is obtained by the adequate combination of 8/16basic load cases (eight for each of two orthogonal directions)(Fig. 2).

To enable a detailed description of the analytical procedure forcritical buckling load determination for the plates with differentboundary conditions, the basic load case DEA is used in this paper.So far, according to the literature, this kind of analysis has beendone almost exclusively for simply supported plates [2–5,7–9]with very few exceptions [10–12].

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.03.0060263-8231/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ382 69 458 099; fax: þ382 20 241 903.E-mail addresses: olja_64@yahoo.com, olgam@ac.me (O. Mijušković).

Thin-Walled Structures 80 (2014) 192–206

2. Basic outline

Before proceeding with solution, it is necessary to summarizethe main governing expressions of two-dimensional elasticity, as,in common with much of XIX century work, Mathieu's notationand approach depart from current conventions.

In his paper [1], Mathieu (1890) expressed the known equili-brium equations, without the presence of body forces, in terms ofdisplacements

∂sx

∂xþ∂τxy

∂y¼ 0

∂τyx∂x

þ∂sy

∂y¼ 0

Mathieu ) Δu¼ �1ε

dνdx

and Δv¼ �1ε

dνdy

ð1Þ

where ν is volumetric dilatation given by

ν¼ ∂u∂x

þ∂v∂y

U ð2Þ

Using relatively simple mathematical operations, the equationsfrom the system (Eq. (1)) can be written as follows:

Δν¼ 0: ð3Þ

Mathieu's approach to the 2D elasticity problem starts with thecareful selection of two ordinary Fourier series for ν (Eq. (3)) withinfinite unknown coefficients, taking into account the symmetryor anti-symmetry of the stresses with respect to the x and ydirections

ν¼ ν1þν2: ð4ÞThe next step presents the introduction of the function F,

(F¼F1þF2), from the conditions that the following equation isfulfilled

ΔF ¼ �1εν ) ΔF1 ¼ �1

εν1 and ΔF2 ¼ �1

εν2: ð5Þ

Finally, when displacements u and v are determined as follows

u¼ dFdx

þα

Zν1dx and v¼ dF

dyþα

Zν2dy; ð6Þ

direct stresses N1 and N2 along axes x and y, as well as the in-planeshear stress T3, are defined by Eqs.(7a) and (7b).

N1 ¼ λνþ2μαν1þ2μd2Fdx2

; N2 ¼ λνþ2μαν2þ2μd2Fdy2

ð7aÞ

Notation

a plate length (measured along the x-axis)Ao, An, Am Fourier-series coefficients for arbitrary external

loadings f(y), f(x)b plate width (measured along the y-axis)Bn, B0 unknown group of coefficients defined by Eqs. (13)

and (14)D¼Et3/12(1�ν2) flexural rigidity of the platee(x)¼sinh (x) abbreviationE(x)¼cosh (x) abbreviationE Young's modulusf(x), f(y) functions defining the applied load on the edges

y¼7b/2 and x¼7a/2F, (F ¼F1þF2) function introduced to satisfy Eq. (6)Gm unknown group of coefficients defined by Eq. (12b)Hn unknown group of coefficients defined by Eq. (12a)l1, l2 load length on the edges x¼7a/2 and y¼7b/2

respectivelyK buckling coefficient for the case of distributed loadKT buckling coefficient for the case of concentrated forceN1, N2 analytical solutions for direct stresses along x and y

axis respectivelyNmin minimal number of terms in one direction for deflec-

tion functionpx, py load intensity in x and y direction respectivelyP intensity of the concentrated force

t thickness of the webT3 analytical solution for shear stress in the plane x-yu, v displacements along the x and y directions

respectivelyU strain energy due to bendingV potential energy associated with the work done by

external loadsw out-of-plane deflection functionWmn coefficients for deflection functionα¼(λþ2μ)⧸μ constant defined in terms of Lame's parametersβm, β0 unknown group of coefficients defined by Eqs. (13)

and (14)γx ¼ l1/b patch-load ratio on the edges x ¼7a/2γy ¼ l2/a patch-load ratio on the edges y ¼7b/2Δ Laplase's operatorε¼μ⧸(λþμ) constant defined in terms of Lame's parametersλ¼νΕ/(1þν)(1�2ν) Lame's parameterΛi functions in non-dimensional form defined by Eq. (15)μ¼Ε⧸2(1þν) Lame's parameterν,(ν¼ν1þν2) volumetric dilatationν Poisson's ratioΠ total potential energy of the systemsx, sy direct stressesτxy shear stress in the plane x–yτ(x)¼E(x)�x/e(x) abbreviationϕ¼a/b aspect ratio of the plateΨ(x)¼e(x)/τ(x) abbreviation

Fig. 1. Decomposition of loads (cases DEA and DEB).

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 193

T3 ¼ μ 2d2Fdxdy

þα

Zdν1dy

dxþα

Zdν2dx

dy

" #ð7bÞ

Mathieu's approach is best illustrated by reference to hisoutline for the problem DEA. This will now be given in somedetail to illustrate the procedural strategy to be followed in all8/16 cases.

3. Exact stress and displacement functions for thefundamental problem DEA

Solution approach begins by expressing any arbitrary externalloading f(y) (Fig. 3) in terms of half-range (even) Fourier expansionover the whole edges y¼7 b/2.

f ðyÞ ¼ A0þ∑nAn cos ny ð8Þ

The choice of functions for the component dilatations ν1 and ν2is stipulated by the presence of double symmetry.

ν1 ¼ B0þ∑nBn coshðnxÞ cos ny n¼ 2qπ

bq¼ 1;2;3;… ð9aÞ

ν2 ¼ β0þ∑mβm coshðmyÞ cos mx m¼ 2pπ

ap¼ 1;2;3; ::: ð9bÞ

There is unique choice of the series for the dilatations (ν1, ν2)which will satisfy the appropriate boundary conditions and sym-metry requirements of displacement and stress for each of the 8/16cases. It is clear that the choice of the functional form for dilatationspredetermines the subsequent form of displacement and stress.

In the same way, in which the volumetric dilatation is pre-sented by superposition of two functions, F is divided into

Fig. 3. Fundamental case DEA.

Fig. 2. Eight/sixteen fundamental load cases.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206194

components F1 and F2 which represents the solution of Eq. (5).

F1 ¼ � 12ε

B0x2�12ε

∑n

1nBnxeðnxÞ cos nyþ∑

nHnEðnxÞ cos ny ð10aÞ

F2 ¼ � 12ε

β0y2� 1

2ε∑m

1m

βm ye ðmyÞ cos mx þ ∑mGm EðmyÞ cos mx

ð10bÞFinally, by introducing the functions F1 and F2 into expressions

for stresses (Eqs. (7a) and (7b)), the solution of the problem isreduced to determination of four unknown groups of coefficientsBn (B0), βm (β0), Hn and Gm from available boundary conditions. Ofcourse, for the case of rectangular plate, there are eight boundaryconditions, (two at each edge), but because of characters ofadopted volumetric dilatations, they are reduced to necessaryfour. In the example of DEA, they are

N1 ¼ f ðyÞ x¼ 7a2; ð11aÞ

N2 ¼ 0 y¼ 7b2; ð11bÞ

T3 ¼ 0 x¼ 7a2; y¼ 7

b2: ð11cÞ

The solution technique involves the choosing of two boundaryconditions, in order to obtain explicit expressions for one group ofunknown coefficients, while the remaining two are used to reducethe problem to the infinite system of linear equations whichcontain parameters (An, A0) of external load.

The original Mathieu's procedure is applied during the solvingof this issue. Conditions that the values of the shear stresses atcontours equal to zero (Eq. (11c)) make possible the definition ofthe coefficient groups Hn and Gm.

Hn ¼ Bn � 12n2þ

a4nε

Eð1=2naÞeð1=2naÞ

� �ð12aÞ

Gm ¼ βm � 12m2þ

b4mε

Eð1=2mbÞeð1=2mbÞ

� �ð12bÞ

The second group of the boundary conditions, which refers tothe distribution of the normal stresses (Eqs. (11a) and (11b)), isused to produce infinite system. Finally, with expressions forconstants Bn (B0), and βm (β0) (Eqs. (13a), (13b) and (13c)), whichare functions of external force coefficients An (A0), it is possible todetermine displacements, as well as exact stress distribution forthe plate under non-uniform compression.

B0 ¼ðλþ2μÞA0

4μðλþμÞ ; β0 ¼ � λA0

4μðλþμÞ ð13aÞ

Bn ¼An

ðλþμÞτð1=2naÞ�8n2 cos 1=2nb

bτð1=2naÞ ∑mβm

með1=2mbÞ cos 1=2ma

ðm2þn2Þ2ð13bÞ

βm ¼ �8m2 cos 1=2maaτð1=2mbÞ ∑

nBn

ne ð1=2 naÞ cos 1=2nbðm2þn2Þ2

ð13cÞ

By recursive iteration, Mathieu proposed a way to solve thesystem by so-called “reduction” method in which infinite sets ofinfinite series are recursively “looped” into one another to solve“exactly” infinite system of equations. Of course, in practice, only afinite number of terms are used to obtain the solutions. Obviously,the number of terms should be carefully chosen from the con-vergence and accuracy point of view.

Bq ¼Aq

ðλþμÞτðqπϕÞþ16ϕ4q2ð�1ÞqðλþμÞπ2τðqπϕÞ∑q0

q0Aq0 ð�1Þq0Ψ ðq0πϕÞ

�fΛ1ðq; q0Þþð16ϕ4=π2ÞΛ3ðq; q0Þþð16ϕ4=π2Þ2Λ5ðq; q0Þþ…gð14aÞ

βp ¼ � 4ϕp2ð�1ÞpðλþμÞπτðpπ=ϕÞ∑q

qAqð�1ÞqΨ ðqπϕÞ

�fΛ0ðp; qÞþð16ϕ4=π2ÞΛ2ðp; qÞþð16ϕ4=π2Þ2Λ4ðp; qÞþ…g ð14bÞΛi functions are in non-dimensional form:

Λ0ðp; qÞ ¼1

ðp2þϕ2q2Þ2; ð15aÞ

Λ1ðq; q0Þ ¼∑p

p3Ψ ðpπ=ϕÞðp2þϕ2q02Þ2

Λ0ðp; qÞ; ð15bÞ

Λ2ðp; qÞ ¼∑q0

q03Ψ ðq0πϕÞðp2þϕ2q02Þ2

Λ1ðq; q0Þ; ð15cÞ

Λ3ðq; q0Þ ¼∑p

p3Ψ ðpπ=ϕÞðp2þϕ2q02Þ2

Λ2ðp; qÞ; ð15dÞ

Λ4ðp; qÞ ¼∑q0

q03Ψ ðq0πϕÞðp2þϕ2q02Þ2

Λ3ðq; q0Þ; ð15eÞ

Through the use of Eq. (6) it is possible to obtain the followingequations for the displacements u and v:

u¼ B0xþ∑nBn

α

2nþ a4ε

Eð1=2naÞeð1=2naÞ

� �eðnxÞ�xEðnxÞ

2ε

� �cos ny

þ∑mβm

12m

� b4ε

Eð1=2mbÞeð1=2mbÞ

� �EðmyÞþyeðmyÞ

2ε

� �sin mx; ð16aÞ

v¼ β0yþ∑nBn

12n

� a4ε

Eð1=2naÞeð1=2naÞ

� �EðnxÞþxeðnxÞ

2ε

� �sin ny

þ∑mβm

α

2mþ b4ε

Eð1=2mbÞeð1=2mbÞ

� �eðmyÞ�yEðmyÞ

2ε

� �cos mx: ð16bÞ

Finally, according to Eqs. (7a) and (7b), expressions for theexact stress functions for the plate under load case DEA, are

N1 ¼ A0þðλþμÞ∑nBn 1þna

2Eð1=2naÞeð1=2naÞ

� �EðnxÞ�nx eðnxÞ

� �cos ny

þðλþμÞ∑mβm 1�mb

2Eð1=2mbÞeð1=2mbÞ

� �EðmyÞþmy eðmyÞ

� �cos mx;

ð17aÞ

N2 ¼ ðλþμÞ∑nBn 1�na

2Eð1=2naÞeð1=2naÞ

� �EðnxÞþnxe ðnxÞ

� �cos ny

þðλþμÞ∑mβm 1þmb

2Eð1=2mbÞeð1=2mbÞ

� �EðmyÞ�my eðmyÞ

� �cos mx;

ð17bÞ

T3 ¼ ðλþμÞ∑nBn �na

2Eð1=2naÞeð1=2naÞeðnxÞþnxEðnxÞ

� �sin ny

þðλþμÞ∑mβm �mb

2Eð1=2mbÞeð1=2mbÞeðmyÞþmyEðmyÞ

� �sin mx:

ð17cÞIt is very easy to show that all the solutions (Eqs. (16a), (16b)

and (17a), (17b), (17c)) indeed satisfy the specified stress boundaryconditions, as well as the obvious displacement constraints.

As it may be seen, analytical procedure for exact stress anddisplacement function determination is very complex anddemanding for herein analyzed DEA case as well as for other basicload cases. In order to enable stability analysis of plates underarbitrary external load, with possibility to vary plate dimensionsand boundary conditions, building of program (symbolic program-ming in the Mathematica) was started with intention to imple-ment all basic load types (8/16 cases, Fig. 2). Up to now, program

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 195

contains general solutions for stress and displacement functionsfor 6/12 basic load cases (four/eight in self-equilibrium DEA(ROT),DOA(ROT), SOA(ROT) and SOB(ROT), and two/four that involve rigid-body translation DEB(ROT) and SEB(ROT)).

Of course, the basis of accuracy of each method is dataconvergence control. That is why it has been programmed thatfor each new load case implemented into program control ofnumber of terms in Eqs. (14a) and (14b) necessary to stabilize

values of coefficients (Bn, βm) should be done. Detailed analysisproved that coefficients of stress and displacement functions (Bn,βm) for DEA case converge rather fast. They might be consideredconstant (i.e. the difference between two consequent terms is lessthan 10-4) with as much as 20 terms. However, in cases with shearstresses (particularly SEB case), adequate accuracy is obtained with60 terms. That is why this number is implemented in program asdefault value for calculation of stress and displacement functions

Table 1aStresses distributions within plate obtained by analytical procedure and by software (ANSYS) based on the finite element method (FEM).

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206196

coefficients for load cases analyzed up to now. Of course, it ispossible to recommend minimal necessary number of terms foreach load case separately. Since adopted number (60, at themoment) does not affect the speed and performances of theprogram, it has been considered as most convenient to have thesame number for all analyzed cases. In the moment of implemen-tation of the rest load cases (two/four that involve rigid-bodyrotation SEA(ROT) and DOB(ROT)), complete convergence study willbe repeated and, if necessary, the number of terms will becorrected.

After determination of coefficients (Bn, βm), stress functions(Eqs. (17a) – (17c)) and in-plane displacements functions (Eqs.(16a) – (16b)) are uniquely defined. In line with the analyticalprocedure applied in the stability analysis of plates, their numberof terms needs to correspond with the number of terms ofdeflection function. More about type and character of deflectionfunction, as well as about analysis of necessary number of itsterms, will be said in next chapters. In this moment, it is importantto point out that stresses and in-plane displacements diagrams(Tables 1a and 1b) are obtained with 20 terms for each of twoorthogonal directions, what is compatible with the deflectionfunction implemented in the stability analysis procedure.

At the end, when it comes to the accuracy of analytical solutionfor in-plane stress distribution, it may be concluded that analyticalsolution has potential to reach exact values, by appropriate choiceand implementation of higher number of series terms. Therefore itmight be said that, due to necessary limitation of series termsnumber in program, suggested analytical solution has a tendencyto produce values slightly above the exact ones. However, it shouldbe pointed out that during convergence analysis of coefficients (Bn,βm) it was noticed that values of first five terms (B1-5, β1–5) aredominant for stress definition (in DEA case they make 97% of finalvalue). Hence, the important enlargement of terms number willnot significantly affect the solution quality in this case. One moreadvantage of analytical solution is possibility of calculation of

stresses with the same accuracy level in all plate points, contraryto FEM comparative results, which have been interpolated amongthe chosen mesh points.

All in all, analysis of diagrams in Tables 1a and 1b shows thatshapes and values of analytical and numerical solutions have highlevel of match.

In Tables 1a and 1b stress and displacement distributions forthe case of biaxial loading are presented. These functions areobtained, on the one side, by analytical approach and, on the other,by software ANSYS based on the finite element method.

Apart from the stress analysis, for the same example (ϕ¼1,px¼py, γx¼0.5, γy¼0.1) control of displacements in x–y plane hasbeen done. Aiming simplier comparative analysis, components uand v of displacements in x and y directions, respectively,determined by analytical expressions (Eqs. (16a) and (16b)) andby ANSYS are shown in Table 1b by means of contour lines withidentical values. Overlap of diagrams is shown in the last column,in order to emphasize good match of results.

4. Analytical approach to plate buckling

The problem of the elastic stability of rectangular plates withdifferent boundary conditions is investigated using the Ritz energytechnique. The strain energy due to bending of the plate is definedin the traditional way. On the other hand, the exact stressdistribution of Mathieu's theory of elasticity is introduced throughthe potential energy of the plate associated with the work done byexternal loads. By adopting the exact stresses within a plate underany type of external loads and using the double Fourier series torepresent any possible buckled profile, the buckling loads can beobtained in a very accurate way. Analytical approach to platebuckling is reaffirmed through several examples of the plates withdifferent boundary conditions (SSSS, CSCS and CCCC) and differentaspect ratios under uni and biaxial concentrated, sinusoidal and

Table 1bIn-plane displacements within plate obtained by analytical procedure and by software (ANSYS) based on the finite element method (FEM).

x

y

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 197

patch compressions. In order to verify the results from analyticalmethod, the finite-element method (ANSYS) is used to producebuckling coefficients for the problem under consideration.

4.1. The adopted deflection series

In order to guarantee the accuracy, the double Fourier series areused to represent buckled profile of the simply supported plate(SSSS), plate with two edges simply supported and other twoclamped (CSCS) and clamped plate (CCCC) (Figs. 4–6). These series(Eqs. (18)–(20)) satisfy all boundary conditions, term by term, andare capable of representing any possible buckled profiles for verywide range of aspect ratios and load cases. It is necessary toemphasize that presently available literature has very few recordson analytical solutions dealing with plates which are not simplysupported.

w¼ ∑s

m ¼ 1∑s

n ¼ 1Wmn sin

mπ

axþa

2

� �sin

nπb

yþb2

� �ð18Þ

w¼ ∑s

m ¼ 1∑s

n ¼ 1Wmn cos

m�1ð Þ πa

xþa2

� ��

� cosmþ1ð Þ π

axþa

2

� ��sin

nπb

yþb2

� �ð19Þ

w¼ ∑s

m ¼ 1∑s

n ¼ 1Wmn cos

ðm�1Þπa

xþa2

� �� cos

ðmþ1Þπa

xþa2

� �� �

� cosðn�1Þπ

byþb

2

� �� cos

ðnþ1Þπb

yþb2

� �� �ð20Þ

As well as in case of stresses and displacements functions,detailed analysis of necessary terms number in deflection func-tions (Eqs. (18)–(20)) is demanded in the procedure of platestability analysis.

The knowledge of this number is desirable for two reasons. Atfirst, the number of terms retained should be large enough so thatthe results for the buckling coefficients should converge within agiven error tolerance. Secondly, the number of terms should alsobe sufficiently large in order that all possible buckles within adeflected profile can be accurately depicted.

The number of terms required to lead to converged results ofthe buckling coefficients depends, of course, on the set errortolerance. In this study, this convergence is investigated for valuescomputed to four significant figures. The differences betweenbuckling coefficients of consecutive iteration steps with respectto the number of terms in the deflection series were recorded andused as a criterion for convergence.

The convergence analysis and condition are taken from Ref. [7],where they have been applied only in case of simply supported

Case 1Plate SSSS

edges x = a/2 simply supported (S)edges y = b/2 simply supported (S)

±±

Fig. 4. Case 1 Plate SSSS.

Case 2Plate CSCS

edges x = a/2 clamped (C)edges y = b/2 simply supported (S)±

±

Fig. 5. Case 2, Plate CSCS.

Case 3Plate CCCC

edges x = a/2 clamped (C)

edges y = b/2 clamped (C)

±

±

Fig. 6. Case 3 Plate CCCC.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206198

plate (SSSS) and the following approximating rule is suggested

Nmin ¼ ½12þ0:8ϕ�: ð21Þ

The bracket means rounding its content to the nearest, largerinteger.

Analogous, very rigorous convergence control has been donefor plates with different boundary conditions (CSCS, CCCC). It hasbeen proved that in certain cases even 10 terms can give very goodresults and satisfy demanded convergence condition. However,more often the presence of higher number of terms is desirable,depending on plate aspect ratio, boundary conditions and types ofthe load.

For the particular DEA load case, deflection function with only12 terms (for plates with aspect ratio ϕ¼0.5–3) produces resultswhich differ no more than 0.3% from final values presented in thispaper. However, DEA case is only one of 8/16 basic load typesunder consideration. Parallel accuracy and convergence analysishas been conducted for several basic load types in process ofbuilding general program for buckling load determination forrectangular plates with different boundary conditions under anyarbitrary external load. After very detailed analysis conducted forchosen plates samples not only for this (DEA case) but also for 5/10other basic load cases (DEB, SEB, SOA, DOA and SOB, Fig. 2) solvedand implemented into general program to this point, deflectionfunctions with 20 terms in each of two orthogonal directions haveproved to be optimal from the convergence and accuracy point ofview, as well as in line with recommended condition for SSSSplates (Eq. (21)).

It is necessary to point out that it is difficult to check allpossible load combinations in the whole range of plate dimen-sions. Therefore the adopted number of 20 terms will continuouslybe subject of control for each new analysis.

4.2. Strain energy due to bending and work done by external loads

During the evaluation of the total potential energy of the plate,the first step is defining the strain energy due plate bending in thetraditional way

U ¼ 12DZ a=2

�a=2

Z b=2

�b=2

∂2w∂x2

þ∂2w∂y2

� �2

�2ð1�νÞ ∂2w∂x2

∂2w∂y2

� ∂2w∂x∂y

� �2" #" #dxdy:

ð22Þ

The part of the potential energy of the plate associated with thework done by external loads is presented by Eq. (23). In thisexpression, the stresses within the plate N1, N2 and T3 are given byEqs. (17a) – (17c) that represent solutions of the Mathieu's exact

approach.

V ¼ � t2

Z a=2

�a=2

Z b=2

�b=2N1

∂w∂x

� �2

þN2∂w∂y

� �2

þ2T3∂w∂x

∂w∂y

" #dxdy ð23Þ

Introducing the exact stress functions for arbitrary type ofexternal load makes the expression for the work done by externalforces (Eq. (23)) more complex. It presents the basic difference inrelation to the previous analyses of the stability of plates which arenot simply supported along all edges.

Concerning that one of the chosen numerical examples repre-sents the plate with mixed boundary conditions under non-uniform biaxial loading, it is also necessary to define the expres-sions for the stress distributions within the plate for the rotatedconfiguration (DEAROT, Fig. 7).

Namely, during the all previous studies of the plate bucklingproblems with stress solutions of the Mathieu's exact approach,only simply supported plates were considered [2–5,7–9]. In thosecases, it is possible to determine the stress values for the rotatedconfigurations using original expressions and matrix of transfor-mation, because all types of integrals, which figure in the expres-sion for the work done by external forces (Eq. (23)), are identicalfor two considered load positions. On the other hand, in the case ofplate with mixed boundary conditions (CSCS), it is a big differencebetween load acting on clamped and on simply supported edgessince quite different groups of integrals have to be solved for theoriginal and rotated configurations.

Hence, in order to enable general analysis of plates with mixedboundary conditions (CSCS etc.) it is necessary to have derivedsolutions for stress functions for arbitrary load in both orthogonaldirections (LOAD-CASE, LOAD-CASEROT).

Surely, the procedure of definition of the exact stress distribu-tion within the plate for the rotated configuration (DEAROT) isanalogous to the basic case of load (DEA). That is why its detailedpresentation was considered unnecessarily.

4.3. Formulation of eigenvalue problem

Finally, after definition of the strain energy of plate bending U,and of the work done by external forces V, the total potentialenergy of the system can be written as

Π¼UþV : ð24Þ

From the minimum potential energy principle, the condition (Eq. (24))is given by

∂Π∂Wmn

¼ ∂U∂Wmn

þ ∂V∂Wmn

¼ 0; ð25Þ

Fig. 7. Original and rotated configurations.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 199

which basically represents linear system of m �n homogenous equa-tions per unknown coefficients Wmn. The existence of nontrivialsolution, expressed through condition that the determinant of thesystem is equal to zero, leads to the solution of the classical eigenvalueproblem. In its scope, the lowest value has, for us, the only practicalimportance, presenting the requested critical load.

After derivation and programming solutions for stress func-tions, the second part of program has been built in order to enablefast and precise analysis of stability problem of rectangular plateswith different boundary conditions, under arbitrary load. Byimplementation of presented deflection functions (Eqs. (18) –

(20)) and exact stress solutions (Eqs. (17a) – (17c)) in standardRitz energy procedure, the program analyzing fast and efficientlyone very complex and demanding problem as plate stability hasbeen completed. Its basic purpose is to enable variation of severalparameters – from load type and character, over plate dimensions,to boundary conditions – and to analyze their influence on criticalload value as well as on buckling shapes. For sure it would bemuch more difficult and time demanding to do such a complexparametric analyses without similar program.

5. Examples and results

Examples proposed in this article are used not only as a way toprove precision of presented analytical approach but also to

control behavior of the basic load type DEA in the cases of plateswith mixed boundary conditions. As it was already said, DEA loadcase is only one of the 8/16 fundamental cases (Fig. 2), but in theprocess of external load simulation it is essential. Namely, most ofthe real-life construction loads can be exactly described byappropriate superposition of several basic solutions, but patch

Table 2Buckling load coefficient for four types of plate under uniaxial half-cosines distributed compressive load, example 1.

Results (K) ϕ¼0.5 ϕ¼0.6 ϕ¼0.75 ϕ¼0.8 ϕ¼1.0 ϕ¼1.5 ϕ¼2.0 ϕ¼3.0

Plate SSSSAnalytical solution 7.4522 6.2387 5.4646 5.3657 5.4184 5.9335 5.7364 5.8493FEM (ANSYS) 7.4441 6.2319 5.4582 5.3590 5.4110 5.9278 5.7307 5.8430Wang [13,14] 7.452 6.239 5.465 5.366 5.419 5.933 5.737 5.849Deverakonda [12] 5.14 5.43 5.74

Plate CSCSAnalytical solution 21.5097 16.2465 12.1247 11.2964 9.3829 8.5382 7.4409 6.8762FEM (ANSYS) 21.5200 16.2540 12.1290 11.2990 9.3819 8.5353 7.4383 6.8726Wang [13,14] 21.51 16.25 12.12 11.30 9.383 8.538 7.441 6.876Deverakonda [12] 8.88 6.98 6.61

Plate SCSCAnalytical solution 8.8784 8.2950 8.7061 9.0776 9.4938 9.3126 9.3616 9.6038FEM (ANSYS) 8.8402 8.2711 8.7286 9.1234 9.4578 9.3995 9.4088 9.6375Wang [13,14]Deverakonda [12] 9.33 9.54 10.42

Plate CCCCAnalytical solution 22.2644 17.5772 14.5317 14.1179 14.0701 12.0869 11.7381 11.4304FEM (ANSYS) 22.2760 17.5870 14.5390 14.1250 14.0740 12.0900 11.7410 11.4320Wang [13,14] 22.26 14.07 11.43Deverakonda [12] 13.92 11.49 11.52

Fig. 9. Geometry, boundary conditions and loading of the plate – example 2.

Fig. 10. Buckling coefficients for uniaxial concentrated forces in x direction,example 2.Fig. 8. Geometry, boundary conditions and loading of the plate – example 1.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206200

compression or concentrated force (DEA) is almost inevitable partof the combination.

All parameters chosen to represent analytical procedure and tocheck its accuracy in critical buckling load determination areobtained under the same conditions in order to provide validconclusions. Coefficients (Bn, βm) of stresses functions N1, N2 and T3are determined with default 60 terms of series (Eqs. (14a) and

(14b)), that guarantees their convergence. The total number ofterms in stress functions is conditioned by derived analyticalprocedure of critical load determination and by chosen deflectionfunctions. Stability analyses for all DEA cases shown herein aredone with stress functions (Eqs. (17a) - (17c)) and deflectionfunctions (Eqs. (18) – (20)) with 20 terms in each of twoorthogonal directions.

Table 3Comparative values of buckling coefficients for the plates SSSS (ϕ¼0.5–3, γx¼γy¼0.1–1, px¼py), example 2.

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 SSSS ϕ¼0.5

/ – 116.8295 40.8298 26.5028 20.6975 17.3323 16.0000 Analytical solution– 116.5734 40.7407 26.4455 20.6543 17.2986 15.9702 FEM (ANSYS)

γx¼0.1 30.4570 27.4864 22.9479 19.5946 16.8781 14.5192 13.4659 Analytical solution30.4250 27.4579 22.9245 19.5754 16.8627 14.5068 13.4530 FEM (ANSYS)

γx¼0.3 11.1184 10.6933 9.9280 9.2473 8.6087 7.9738 7.6553 Analytical solution11.1066 10.6821 9.9176 9.2378 8.6001 7.9661 7.6474 FEM (ANSYS)

γx¼0.5 7.7387 7.5255 7.1319 6.7742 6.4326 6.0824 5.9005 Analytical solution7.7303 7.5173 7.1241 6.7668 6.4258 6.0761 5.8941 FEM (ANSYS)

γx¼0.7 6.6272 6.4650 6.1644 5.8934 5.6379 5.3753 5.2371 Analytical solution6.6196 6.4574 6.1572 5.8865 5.6314 5.3694 5.2311 FEM (ANSYS)

γx¼0.9 6.2858 6.1359 5.8588 5.6116 5.3818 5.1470 5.0228 Analytical solution6.2781 6.1284 5.8516 5.6047 5.3753 5.1409 5.0167 FEM (ANSYS)

γx¼1.0 6.2500 6.1002 5.8238 5.5786 5.3524 5.1220 5.0000 Analytical solution6.2426 6.0930 5.8168 5.5719 5.3460 5.1161 4.9941 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 SSSS ϕ¼1.0

/ – Analytical solution– FEM (ANSYS)

γx¼0.1 26.1936 13.2999 Analytical solution26.1387 13.2723 FEM (ANSYS)

γx¼0.3 9.2096 6.8825 4.6543 Symmetry Analytical solution9.1903 6.8680 4.6446 FEM (ANSYS)

γx¼0.5 6.0748 4.9608 3.6881 3.0537 Analytical solution6.0621 4.9507 3.6803 3.0473 FEM (ANSYS)

γx¼0.7 4.8794 4.1285 3.2051 2.7148 2.4438 Analytical solution4.8692 4.1200 3.1985 2.7092 2.4388 FEM (ANSYS)

γx¼0.9 4.2457 3.6621 2.9152 2.5041 2.2721 2.1232 Analytical solution4.2371 3.6547 2.9094 2.4991 2.2675 2.1189 FEM (ANSYS)

γx¼1.0 4.0000 3.4769 2.7963 2.4161 2.1997 2.0598 2.0000 Analytical solution3.9920 3.4699 2.7907 2.4113 2.1953 2.0556 1.9960 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 SSSS ϕ¼1.5

/ – 11.2677 4.0246 2.7313 2.2905 2.1260 2.0864 Analytical solution– 11.2527 4.1092 2.7275 2.2872 2.1229 2.0834 FEM (ANSYS)

γx¼0.1 29.3081 8.9552 3.6828 2.5663 2.1713 2.0218 1.9854 Analytical solution29.2595 8.9430 3.6778 2.5627 2.1682 2.0188 1.9826 FEM (ANSYS)

γx¼0.3 10.2697 6.3458 3.1576 2.2965 1.9724 1.8468 1.8158 Analytical solution10.2526 6.3367 3.1533 2.2933 1.9696 1.8441 1.8132 FEM (ANSYS)

γx¼0.5 6.7109 5.0017 2.7899 2.0947 1.8208 1.7128 1.6860 Analytical solution6.6997 4.9943 2.7860 2.0918 1.8182 1.7103 1.6835 FEM (ANSYS)

γx¼0.7 5.3242 4.1966 2.5168 1.9373 1.7013 1.6071 1.5836 Analytical solution5.3154 4.1907 2.5134 1.9346 1.6989 1.6047 1.5813 FEM (ANSYS)

γx¼0.9 4.6016 3.5941 2.2831 1.7972 1.5936 1.5115 1.4912 Analytical solution4.5940 3.5894 2.2801 1.7947 1.5914 1.5094 1.4891 FEM (ANSYS)

γx¼1.0 4.3403 3.3246 2.1708 1.7277 1.5396 1.4633 1.4444 Analytical solution4.3333 3.3202 2.1679 1.7253 1.5374 1.4612 1.4424 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 SSSS ϕ¼3.0

/ – 4.9882 1.9531 1.4510 1.2852 1.2357 1.2345 Analytical solution– 4.9853 1.9521 1.4503 1.2846 1.2350 1.2339 FEM (ANSYS)

γx¼0.1 29.9522 4.7215 1.9162 1.4335 1.2724 1.2236 1.2223 Analytical solution29.8921 4.7187 1.9152 1.4328 1.2718 1.2229 1.2217 FEM (ANSYS)

γx¼0.3 10.4755 4.2266 1.8436 1.3988 1.2471 1.1999 1.1985 Analytical solution10.4552 4.2240 1.8425 1.3981 1.2465 1.1993 1.1979 FEM (ANSYS)

γx¼0.5 6.7956 3.7697 1.7708 1.3633 1.2214 1.1765 1.1751 Analytical solution6.7829 3.7671 1.7699 1.3627 1.2209 1.1759 1.1745 FEM (ANSYS)

γx¼0.7 5.2788 3.3557 1.6970 1.3259 1.1945 1.1523 1.1511 Analytical solution5.2695 3.3531 1.6961 1.3253 1.1939 1.1517 1.1506 FEM (ANSYS)

γx¼0.9 4.3653 2.9970 1.6228 1.2863 1.1652 1.1261 1.1251 Analytical solution4.3585 2.9945 1.6219 1.2858 1.1646 1.1255 1.1245 FEM (ANSYS)

γx¼1.0 4.000 2.8388 1.5862 1.2659 1.1496 1.1119 1.1111 Analytical solution3.9941 2.8364 1.5852 1.2653 1.1490 1.1113 1.1105 FEM (ANSYS)

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 201

All numerical results obtained by ANSYS program and used forcomparison are calculated under absolutely identical conditions.Detailed analysis of FEM results convergence has been doneconsidering chosen element and mesh density. It has been shownthat SHELL93 (8 nodes) element and square mesh with minimum40 elements along the shorter edge of plate provide appropriatestabilization of solution for DEA load case. All comparative

numerical values that figure in presented examples, includingstress and displacement analysis in Tables 1a and 1b, are calcu-lated under these conditions.

First example chosen to prove accuracy of presented approachis the case of sinusoidal edge compressive load (Fig. 8). For thisparticular case it is possible to compare obtained results with fewanalytical solutions existing in literature [12], as well as with some

Table 4Comparative values of buckling coefficients for the plates CSCS (ϕ¼0.5–3, γx¼γy¼0.1–1, px¼py), example 2.

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CSCS ϕ¼0.5

/ – 166.1140 60.1542 41.0857 33.8398 29.6524 27.8915 Analytical solution– 165.8803 60.0781 41.0384 33.8052 29.6275 27.8709 FEM (ANSYS)

γx¼0.1 82.9067 76.6150 54.9501 38.7773 31.7139 27.4489 25.7224 Analytical solution82.8848 76.5975 54.9097 38.7432 31.6896 27.4319 25.7084 FEM (ANSYS)

γx¼0.3 30.9996 30.2504 28.5754 26.4120 23.8681 21.4396 20.3361 Analytical solution30.9947 30.2458 28.5718 26.4087 23.8644 21.4357 20.3323 FEM (ANSYS)

γx¼0.5 22.0930 21.7650 21.0687 20.2220 19.1254 17.8181 17.1323 Analytical solution22.0893 21.7614 21.0655 20.2194 19.1232 17.8161 17.1302 FEM (ANSYS)

γx¼0.7 19.1541 18.9147 18.4227 17.8726 17.1781 16.2753 15.7604 Analytical solution19.1500 18.9106 18.4186 17.8689 17.1749 16.2727 15.7580 FEM (ANSYS)

γx¼0.9 18.2268 17.9904 17.5079 17.0144 16.4574 15.7340 15.3004 Analytical solution18.2215 17.9850 17.5023 17.0088 16.4524 15.7298 15.2966 FEM (ANSYS)

γx¼1.0 18.1874 17.9318 17.4054 16.8873 16.3558 15.6984 15.2987 Analytical solution18.1834 17.9275 17.4006 16.8821 16.3509 15.6945 15.2953 FEM (ANSYS)

K ¼ scr tb2

π2D/ γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CSCS ϕ¼1.0

/ – 39.9453 14.4903 9.9815 8.4325 7.8278 7.6927 Analytical solution– 39.9038 14.4762 9.9723 8.4248 7.8207 7.6858 FEM (ANSYS)

γx¼0.1 47.6257 25.5627 13.5845 9.8814 8.3683 7.7784 7.3408 Analytical solution47.5857 25.5404 13.5709 9.8724 8.3608 7.7715 7.3353 FEM (ANSYS)

γx¼0.3 16.6288 12.8628 9.0263 7.3883 6.5779 6.0092 5.7388 Analytical solution16.6168 12.8533 9.0190 7.3823 6.5728 6.0048 5.7348 FEM (ANSYS)

γx¼0.5 10.7787 9.0624 6.9943 5.9751 5.4352 5.0389 4.8467 Analytical solution10.7714 9.0561 6.9891 5.9731 5.4313 5.0354 4.8424 FEM (ANSYS)

γx¼0.7 8.4494 7.3570 5.9402 5.1960 4.7827 4.4724 4.3199 Analytical solution8.4438 7.3519 5.9359 5.1922 4.7793 4.4693 4.3169 FEM (ANSYS)

γx¼0.9 7.1955 6.3867 5.2948 4.6993 4.3599 4.0997 3.9708 Analytical solution7.1906 6.3822 5.2909 4.6959 4.3568 4.0969 3.9681 FEM (ANSYS)

γx¼1.0 6.7432 6.0267 5.0457 4.5035 4.1912 3.9499 3.8300 Analytical solution6.7388 6.0226 5.0420 4.5002 4.1882 3.9472 3.8274 FEM (ANSYS)

K ¼ scr tb2

π2D/ γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CSCS ϕ¼1.5

/ – 14.9398 5.4542 3.8310 3.3216 3.1419 3.0991 Analytical solution– 14.9258 5.4492 3.8275 3.3185 3.1391 3.0962 FEM (ANSYS)

γx¼0.1 47.8902 11.7935 4.9727 3.5883 3.1388 2.9787 2.9403 Analytical solution47.8354 11.7835 4.9684 3.5851 3.1361 2.9761 2.9378 FEM (ANSYS)

γx¼0.3 16.4781 8.2876 4.2306 3.1883 2.8307 2.7011 2.6697 Analytical solution16.4639 8.2808 4.2271 3.1857 2.8284 2.6989 2.6676 FEM (ANSYS)

γx¼0.5 10.4280 6.4160 3.6917 2.8761 2.5842 2.4767 2.4505 Analytical solution10.4201 6.4110 3.6888 2.8739 2.5821 2.4747 2.4485 FEM (ANSYS)

γx¼0.7 7.8059 5.2273 3.2721 2.6188 2.3766 2.2860 2.2639 Analytical solution7.7997 5.2233 3.2695 2.6168 2.3747 2.2842 2.2621 FEM (ANSYS)

γx¼0.9 6.0301 4.3636 2.9166 2.3895 2.1878 2.1112 2.0925 Analytical solution6.0256 4.3603 2.9144 2.3876 2.1861 2.1095 2.0909 FEM (ANSYS)

γx¼1.0 5.3749 4.0074 2.7550 2.2813 2.0974 2.0269 2.0098 Analytical solution5.3705 4.0044 2.7528 2.2795 2.0957 2.0253 2.0082 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CSCS ϕ¼3.0

/ – 5.0362 1.9818 1.4907 1.3487 1.3206 1.3244 Analytical solution– 5.0331 1.9807 1.4899 1.3480 1.3198 1.3236 FEM (ANSYS)

γx¼0.1 42.4848 4.7504 1.9412 1.4706 1.3337 1.3066 1.3104 Analytical solution42.3957 4.7475 1.9400 1.4698 1.3330 1.3058 1.3097 FEM (ANSYS)

γx¼0.3 14.2812 4.2313 1.8615 1.4307 1.3038 1.2788 1.2826 Analytical solution14.2600 4.2286 1.8604 1.4299 1.3031 1.2781 1.2818 FEM (ANSYS)

γx¼0.5 8.6689 3.7706 1.7828 1.3900 1.2730 1.2501 1.2539 Analytical solution8.6588 3.7679 1.7818 1.3893 1.2724 1.2495 1.2532 FEM (ANSYS)

γx¼0.7 6.2534 3.3659 1.7044 1.3479 1.2406 1.2198 1.2235 Analytical solution6.2469 3.3634 1.7035 1.3472 1.2400 1.2192 1.2229 FEM (ANSYS)

γx¼0.9 4.8905 3.0173 1.6273 1.3045 1.2062 1.1874 1.1910 Analytical solution4.8856 3.0149 1.6263 1.3038 1.2057 1.1868 1.1904 FEM (ANSYS)

γx¼1.0 4.4065 2.8630 1.5896 1.2825 1.1886 1.1706 1.1741 Analytical solution4.4020 2.8607 1.5886 1.2818 1.1879 1.1700 1.1735 FEM (ANSYS)

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206202

interesting numerical methods [13,14]. As it has already beenpointed out, the main problem of most of the previous analyticalanalysis was assumptions of simplified stress distribution. Hence,implementation of exact stresses solutions (Eqs. (17a)–(17c)) forparticular edge load (DEA) in stability analysis will lead to veryaccurate results for critical load, as can be seen in Table 2.

Second numerical example is plate under concentrated andlocally distributed compression (Fig. 9), which is one of the mostcommon load types in civil engineering practice and very importantcomponent in simulation of more complex external loads. In order toassure accuracy and good behavior of presented solution, three typesof plates with different boundary conditions (SSSS, CSCS and CCCC),

Table 5Comparative values of buckling coefficients for the plates CCCC (ϕ¼0.5–3, γx¼γy¼0.1–1, px¼py), example 2

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CCCC ϕ¼0.5

/ – 259.3330 89.4793 56.5140 42.4837 34.4187 31.4743 Analytical solution– 258.8076 89.4046 56.4878 42.4668 34.4052 31.4634 FEM (ANSYS)

γx¼0.1 83.5580 76.6378 60.7990 46.1419 36.6236 30.4185 28.0477 Analytical solution83.5372 76.6211 60.7861 46.1313 36.6146 30.4104 28.0405 FEM (ANSYS)

γx¼0.3 31.3774 30.5020 28.6134 26.4466 24.0786 21.7298 20.6213 Analytical solution31.3728 30.4981 28.6103 26.4437 24.0760 21.7275 20.6182 FEM (ANSYS)

γx¼0.5 22.5921 22.1957 21.3451 20.3432 19.1516 17.8192 17.1336 Analytical solution22.5888 22.1929 21.3430 20.3414 19.1499 17.8179 17.1313 FEM (ANSYS)

γx¼0.7 19.9012 19.6232 19.0289 18.3215 17.4470 16.4135 15.8600 Analytical solution19.8974 19.6201 19.0265 18.3196 17.4453 16.4122 15.8578 FEM (ANSYS)

γx¼0.9 19.2366 18.9894 18.4629 17.8363 17.0504 16.0988 15.5801 Analytical solution19.2314 18.9851 18.4596 17.8337 17.0482 16.0970 15.5774 FEM (ANSYS)

γx¼1.0 19.3406 19.0928 18.5671 17.9453 17.1653 16.2148 15.6940 Analytical solution19.3369 19.0899 18.5652 17.9441 17.1644 16.2140 15.6923 FEM (ANSYS)

K ¼ scr tb2

π2D/ γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CCCC ϕ¼1.0

/ – Analytical solution– FEM (ANSYS)

γx¼0.1 67.7444 37.1264 Analytical solution67.7049 37.1152 FEM (ANSYS)

γx¼0.3 24.0774 19.1035 13.0892 Symmetry Analytical solution24.0691 19.0988 13.0872 FEM (ANSYS)

γx¼0.5 16.1085 13.6335 10.3533 8.6037 Analytical solution16.1036 13.6303 10.3520 8.6028 FEM (ANSYS)

γx¼0.7 12.7037 11.1241 8.8809 7.5817 6.7841 Analytical solution12.6991 11.1215 8.8798 7.5809 6.7834 FEM (ANSYS)

γx¼0.9 10.7885 9.6222 7.9086 6.8674 6.2066 5.7177 Analytical solution10.7848 9.6199 7.9075 6.8667 6.2060 5.7171 FEM (ANSYS)

γx¼1.0 10.0760 9.0424 7.5105 6.5642 5.9563 5.5032 5.3038 Analytical solution10.0729 9.0405 7.5095 6.5635 5.9556 5.5026 5.3032 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CCCC ϕ¼1.5

/ – 29.4092 10.6724 7.3726 6.2699 5.8819 5.8258 Analytical solution– 29.4002 10.6703 7.3712 6.2685 5.8803 5.8246 FEM (ANSYS)

γx¼0.1 64.2124 24.9768 10.0593 7.0838 6.0656 5.7046 5.6518 Analytical solution64.1495 24.9711 10.0579 7.0828 6.0646 5.7033 5.6509 FEM (ANSYS)

γx¼0.3 22.3021 18.2709 8.9531 6.5437 5.6783 5.3666 5.3205 Analytical solution22.2905 18.2665 8.9521 6.5430 5.6776 5.3657 5.3200 FEM (ANSYS)

γx¼0.5 14.2605 13.3959 7.9766 6.0402 5.3089 5.0414 5.0022 Analytical solution14.2553 13.3918 7.9758 6.0397 5.3084 5.0408 5.0019 FEM (ANSYS)

γx¼0.7 10.9178 10.4231 7.0716 5.5389 4.9296 4.7031 4.6707 Analytical solution10.9141 10.4199 7.0709 5.5385 4.9292 4.7026 4.6704 FEM (ANSYS)

γx¼0.9 9.0356 8.7092 6.2211 5.0229 4.5241 4.3357 4.3096 Analytical solution9.0326 8.7065 6.2206 5.0227 4.5238 4.3353 4.3090 FEM (ANSYS)

γx¼1.0 8.3521 8.0788 5.8228 4.7638 4.3155 4.1446 4.1213 Analytical solution8.3496 8.0767 5.2820 4.7633 4.3149 4.1440 4.1209 FEM (ANSYS)

K ¼ scr tb2=π2D / γy¼0.1 γy¼0.3 γy ¼0.5 γy¼0.7 γy¼0.9 γy¼1.0 CCCC ϕ¼3.0

/ – 14.1956 5.9147 4.7057 4.3636 4.2814 4.3053 Analytical solution– 14.1930 5.9142 4.7053 4.3632 4.2806 4.3047 FEM (ANSYS)

γx¼0.1 66.7923 13.2056 5.7770 4.6468 4.3308 4.2563 4.2807 Analytical solution66.5155 13.2035 5.7765 4.6464 4.3304 4.2556 4.2802 FEM (ANSYS)

γx¼0.3 22.9226 11.2239 5.4876 4.5185 4.2578 4.2006 4.2262 Analytical solution22.8904 11.2224 5.4872 4.5181 4.2575 4.2001 4.2258 FEM (ANSYS)

γx¼0.5 14.2520 9.4146 5.1809 4.3720 4.1684 4.1308 4.1576 Analytical solution14.2427 9.4133 5.1805 4.3717 4.1680 4.1304 4.1573 FEM (ANSYS)

γx¼0.7 10.3858 7.9067 4.8645 4.2077 4.0587 4.0412 4.0686 Analytical solution10.3813 7.9054 4.8641 4.2073 4.0584 4.0408 4.0683 FEM (ANSYS)

γx¼0.9 8.1573 6.7084 4.5478 4.0295 3.9296 3.9308 3.9571 Analytical solution8.1544 6.7072 4.5474 4.0292 3.9293 3.9305 3.9568 FEM (ANSYS)

γx¼1.0 7.3610 6.2109 4.3917 3.9364 3.8584 3.8550 3.8606 Analytical solution7.3585 6.2096 4.3912 3.9361 3.8580 3.8544 3.8602 FEM (ANSYS)

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 203

Table 6Comparative values of buckling coefficients for plate CSCSϕ¼ 1.5 (py¼ f px, f¼0.1–5), example 2.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206204

with very wide range of plate aspect ratio (ϕ¼a/b¼0.5, 1, 1.5 and 3)and load aspect ratios (γx and γy in the interval 0.1–1) have beenanalyzed. Besides effects of locally distributed uni and biaxial stresses(DEAþDEAROT), as the ultimate control concentrated forces in one orin both directions are applied and analyzed.

As a special case of the second example, buckling coefficientsfor the plates SSSS, SCSC, CSCS and CCCC, with aspect ratio ϕ¼0.1,0.3, 0.5, 0.7, 0.9, 1, 1.2, 1.5, 2, 3, 4 and 5, under concentrated forceapplied in horizontal direction are presented in chart form(Fig. 10). In order to be able to correctly interpret proposedsolutions, it is necessary to point out that buckling coefficients(KT) for wide strips (ϕo1) are presented in form Pa/π2D, while forthe case of long plates (ϕZ1) the form is Pb/π2D.

All other results for plates with three types of boundaryconditions (SSSS, CSCS and CCCC) under the effects of uniaxialand biaxial (px¼py) compressive stresses (γx and γy in the interval0.1–1), as well as under concentrated forces are presented inTables 3–7. In these tables, there are not only buckling coefficientsobtained by analytical approach, but also, as some kind ofcomparative values, the results of the finite element method(ANSYS).

Also, in case of the plate CSCS with aspect ratio ϕ¼a/b¼1.5,under a different load intensity in two orthogonal directions

(px¼ f � py, f¼0.1–5), behavior of presented analytical solution isanalyzed. Table 6 contains results of analytical as well as finiteelement method approach (ANSYS and FINEL [15]).

6. Final remarks

Buckling analysis of rectangular plates with different boundaryconditions subjected to nonlinear (sinusoidal) in plane loading iscarried out in the first example. The exact in-plane stress solutions(Eqs.(17a) – (17c)) for case DEA, implemented in stability problem(Fig. 8), have given extremely accurate buckling coefficients(Table 2) reaffirmed by numerical methods [13–15]. On the otherhand there is obvious results discrepancy regarding anotheranalytical approach dealing with the same problem [12], especiallyfor plates with mixed boundary conditions (up to 8% for the plateSCSC with aspect ratio ϕ¼ 3).

Analysis of results summarized in Table 2 shows that analyticalsolutions for plates with three types of boundary conditions (SSSS,CSCS and CCCC) differ less than 0.14% from ANSYS results.Analytical solution for plate with boundary conditions SCSC isslightly less stable, differing from numerical solutions from –0.43%(ϕ¼0.5) to 0.93% (ϕ¼1.5). Having in mind the well known fact

Table 7Comparative analysis for biaxial concentrated load, example 2.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206 205

that the finite element method produces results below exact onesfor the plate stability problems, it is necessary to comment the factthat in certain cases analytically obtained critical solution is lowerfrom numerical one. For all the other presented results thisrelation remains the intact. Hence, it might be concluded that incase of nonlinear (sinusoidal) load the accuracy of load applicationin both analytical and numerical procedure plays a certain role.Low discrepancy of results is in favor of such conclusion.

In the case of uniaxial and biaxial concentrated force in thesecond example (Fig. 10, Table 7) discrepancy between analyticaland finite element method results for all chosen plate types iswithin 1% (for plates with 0.3rϕr3 it is even less than 0.52%),with exceptions of very wide strips (ϕ¼0.1) where, for the case ofboundary conditions CSCS and force applied in x direction,discrepancy goes up to 1.8%.

It has been said in previous chapters that plates in all presentedexamples have been analyzed under the same conditions in order toenable comparative analysis. Regarding analytical solution, for allplates both stress and displacement functions have 20 series terms.Regarding numerical solution, all plates have the same mesh densityand the same finite element. That is why it is interesting to check thecase of wide strip CSCS (ϕ¼0.1) under concentrated load and toanalyze conditions that will lead to reduction of the difference of 1.8%between analytical and FEM results. Check of FEM procedure provedthat further increase of mesh density does not significantly affectsolution quality (KT¼4.0903). On the other hand, in analyticalsolution increase in number of series terms from 20 to 26 decreasesbuckling coefficient from KT¼4.1649 to KT¼4.1362, which reducesdifference from numerical solution to 1.1%. Hence, even thoughgeneral convergence analysis of analytical solution depending onadopted number of terms in deflection function series has been doneat the beginning, it may not include all variants of plate and loadtypes. Therefore, after examination and analysis of all results for DEAcase, the conclusion concerning necessity of correction of termsnumber in deflection function series will be made at the end of thischapter.

Analysis of diagrams in Fig. 10 leads to interesting observation thatfor the category of wide strips (ϕ¼0.1–0.3) critical load chart lines forCSCS and CCCC plates from one side and for SSSS and SCSC plates fromthe other side start from almost identical values. Increase in values ofratio ϕ causes obvious separation of lines for four characteristic typesof boundary conditions. At the end, for values of ratioϕ43 lines forCCCC and SSSS plates define lower and upper limit of bucklingcoefficient, while lines for SCSC and CSCS plates practically convergeto the same value of critical load.

Besides analysis of uni and biaxial concentrated load, in thesecond numerical example (Fig. 9) detailed analysis of uni andbiaxial locally distributed compression was done, having in mindthat this is rather common load type in civil engineering practice.

After detailed analysis of results (Tables 3–5), very goodbehavior of analytical solution can easily be noted in the completeconsidered range of plate aspect ratio (ϕ¼0.5, 1, 1.5 and 3) andload ratios (γx¼0.1–1, γy¼0.1–1, py¼px).

In case of SSSS and CSCS plates under distributed load(Tables 3 and 4) differences between analytically and numericallyobtained buckling coefficients are less than 0.23%. In case of platewith all four clamped edges (CCCC) discrepancy is below 0.15%,except for plate with ratios ϕ¼3, γx¼0.1, γy¼0, where discrepancyis 0.41%.

In Table 6 very high degree of the results’ concordance (up to0.1%) can be seen in the case of a different load intensity in twoorthogonal directions (pyapx) for one selected type of plate (CSCS,ϕ¼1.5).

Having in mind that buckling coefficients obtained by FEM arebelow exact values, as a result of limited number of terms ininterpolation functions, what leads to certain discontinuity of

slope between adjacent elements, small existing discrepancybetween presented results confirms accuracy of the analyticalapproach.

At the end, after detailed analysis of results, it has beenconcluded that in DEA case it is not rational to burden the programwith increase in terms number above 20 only because of oneisolated case of very wide strip (ϕ¼0.1) with concentrated load,which is not quite typical for civil engineering practice. In favor ofthis conclusion stands the fact that for the plate of the samedimensions with locally distributed load in range γx¼0.1–1 theexisting deflection function (with 20 terms) provides solutionsthat differ from FEM results less than 0.2%.

Hence, it can be concluded that the adopted number of 20terms in the series of deflection and stress functions is sufficientfor the analyzed cases of plates with mixed boundary conditions,subjected to variable uni and biaxial compressive load (DEA) foranalytical approach to produce highly accurate solutions.

Finally, future goals regarding presented analytical solution wouldbe to complete all 8/16 fundamental load types and to be able tobuild more advanced models in order to analyze behavior of plateswith mixed boundary conditions, presented in this paper, under anyreal-life external load. Until now four more programs are completedand implemented in the main computer program (for load cases DEB,SEB, DOA and SOA) while the sixth one is in the trial phase (SOB).Regardless the fact that the solutions for two load cases are not yetimplemented in the main program (DOB, SEA), some very interestingproblems can be easily analyzed by means of five formulatedprograms. One now in the process of consideration and computationis patch-loading (DEAþDEBþSEBROT).

Acknowledgment

Authors’ immense gratitude and fond memories go to ProfessorMilija N. Pavlović.

References

[1] Mathieu E. Theorie de l'elasticite des corps solides, Seconde partie. Paris:GauthierVillars; 1890.

[2] Baker G, Pavlović MN. Elastic stability of simply supported rectangular platesunder locally distributed edge forces. J Appl Mech 1982;104:177–9.

[3] Baker G, Pavlović MN. Rectangular plates compressed by series of in-planeloads: stability and stress distribution. Aeronaut J 1983;87:183–8.

[4] Pavlović MN, Baker G. Buckling of non-uniformly compressed plates,. TheorAppl Mech Trans Yugosl Soc Mech 1983;vol. 9:91–104.

[5] Baker G, Pavlović MN. Stability of wide strips under locally distributed forces. JAppl Mech 1985;107:232–4.

[6] Baker G, Pavlović MN, Tahan N. An exact solution to the two-dimensionalelasticity problem with rectangular boundaries under arbitrary edge forces.Philos Trans R Soc Lond, A 1993;393:307–36.

[7] Y. G. Liu, Buckling of plates under non-uniform stresses, with particularemphasis on shear [Ph.D. thesis]. London: Imperial College; 2006.

[8] Liu YG, Pavlović MN. Elastic stability of flat rectangular plates under patchcompression. Int J Mech Sci 2007;49(8):970–82.

[9] Liu YG, Pavlović MN. A generalized analytical approach to the buckling ofsimply-supported rectangular plates under arbitrary loads. Eng Struct 2008;30(5):1346–59.

[10] O. Mijušković, Stability analysis of rectangular plates with exact in-plane stressfunctions [Ph.D. thesis]. Faculty of Civil Engineering, Beograd; 2008.

[11] Mijušković O, Ćorić B. Analytical approach to critical load determination inplates with mixed boundary conditions, Građevinar. J Croat Assoc Civil Eng2012;64–3:185–94.

[12] Deverakonda KKV, Bert CW. Buckling of rectangular plate with nonlinearlydistributed compressive loading on two opposite sides: comparative analysisand results. Mech Adv Mater Struct 2004;11:433–44.

[13] Wang X, Wang X, Shi X. Differential quadrature buckling analyses of rectan-gular plates subjected to non-uniform distributed in-plane loadings. Thin-Walled Struct 2006;44:837–43.

[14] Wang X, Wang X, Shi X. Accurate buckling loads of thin rectangular platesunder parabolic edge compressions by the differential quadrature method. IntJ Mech Sci 2007;49:447–53.

[15] D. Hitchings, FE77 user manual, Technical report, London: Department ofAeronautical Engineering, Imperial College; 2003.

O. Mijušković et al. / Thin-Walled Structures 80 (2014) 192–206206