Engineering Structures 38 (2012) 78–88

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Engineering Structures

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

An analytical model for the buckling of plates under mixed boundary conditions

Eugenio Ruocco a,⇑, Massimiliano Fraldi b

a Department of Civil Engineering, Second University of Naples, Aversa, Italyb Department of Structural Engineering, University of Naples ‘‘Federico II’’, Naples, Italy

a r t i c l e i n f o

Article history:Received 22 August 2011Revised 11 December 2011Accepted 22 December 2011Available online 17 February 2012

Keywords:Buckling analysisAnalytical solutionsBiaxial loadsDiscontinuous constraints

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.engstruct.2011.12.049

⇑ Corresponding author.E-mail address: eugenio.ruocco@unina2.it (E. Ruoc

a b s t r a c t

An analytical approach for the buckling analysis of rectangular plates under mixed boundary conditionsis presented. In order to solve the partial differential equation governing the problem at hand a method ofseparation of variables is here adopted, by introducing the displacement field as a result of the scalarproduct of two vectors which combine prescribed and unknown scalar functions. By following this strat-egy, exact buckling solutions for a wide class of problems, in which mixed boundary conditions can beassigned relaxing some usual constraints, are determined, and buckling load of plates, where biaxial ten-sile and compressive loads are applied in presence of piecewise clamped and partially supported sides,obtained analytically. Several cases of engineering interest are finally analyzed, and comparisons of thetheoretical outcomes with literature data and Finite Element-based numerical results are also shown,in order to highlight the effectiveness of the proposed strategy.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Motivated by the great interest in numerous engineering appli-cations, buckling of two-dimensional systems has been extensivelyinvestigated in literature. In particular, with reference to the clas-sical Kirchhoff theory, elastic stability of rectangular plates hasbeen largely studied and – due to the difficulties encountered toanalytically solve the governing equilibrium equation – severalsemi-analytical approaches and numerical strategies have beenproposed in the last decades, such as finite element, boundary ele-ment, finite difference, finite strip as well as power series methods,all widely summarized in many texts and handbooks [1–3].

In this framework, however, Finite Element Method (FEM) rep-resents the main way to analyze the buckling behavior of structuralmembers, because of its versatility and capability to determineeigenvalues and eigenvectors, as well as stress fields, in platesunder arbitrary boundary conditions. Nevertheless, to provideaccurate predictions of the buckling response, some problemsrequire the use of FE models characterized by a very large numberof elements and, as a consequence of non-linearities, unforeseeablecomputational times have to be taken into account: this is thesituation one could for example encounter when expected closebuckling modes produce numerical instabilities [4]. In order toovercome such difficulties some alternative numerical strategieshave been proposed in literature: for instance, in [5–7] Aliabadiand co-workers show as the boundary element method (BEM) canbe a viable way for analysing buckling problems. Furthermore,

ll rights reserved.

co).

the Finite Strip Method (FSM), based on the discretization of thestructure along the sole transverse direction, can be efficientlyemployed in buckling analyses, being in these cases both computa-tional times and numerical instabilities drastically reduced [8].However, these advantages are lost when no prismatic elementsor systems constrained differently from simply supported edgeshave to be analyzed.

When the geometry of the structure is sufficiently regular andthe boundary conditions do not exhibit discontinuities in termsof loads and constraints, ad-hoc semi-analytical techniques canbe instead successfully adopted. Most of models based on suchtechniques assume the out-of-plane displacement, w(x,y), in thefollowing uncoupled form

wðx; yÞ ¼ w1ðxÞw2ðyÞ ð1Þ

where w1 and w2 are two functions of the separated in-plane vari-ables, {x,y}. Also, the assumption (1) is chosen in order to reducethe complexity of the partial differential equation governing theequilibrium of the system, so finally obtaining a set of uncoupledordinary one-dimensional differential equations from which, underspecific boundary conditions, analytical solutions can be searched.Numerical models based on closed form solutions are not affectedby the adopted discretization, and reliable results can be obtainedwith the minimum number of elements required for a completedescription of the geometry. In Fig. 1 a comparison between standardFSM, FEM and semi-analytical methods is shown with reference to atypical stiffened plate, highlighting the needed equivalent discretiza-tion for having the same accuracy in terms of results.

Starting from some classical solutions proposed by Timoshenkoand Gere in [1], further efforts have recently been made on analytical

Fig. 1. Typical stiffened structure (A), and pictorial comparison between analytical (B), FSM (C) and FEM (D) required discretization.

Fig. 2. Full 2D structures which cannot be analyzed adopting an analyticalapproach.

E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88 79

approaches in stability of plates. In this framework, the work byShan and Qiao [9] presents an exact solution able to find bucklingand vibration responses of isotropic rectangular plates having nosimply supported edges, and Iuspa and Ruocco [10] adopt aclosed-form solution in a topological optimization procedure foranalysing stability of composite stiffened plates. Also, Hosseini-Has-hemi et al. [11] define a closed form solution able to analyze the elas-tic buckling of transversely isotropic Mindlin-Reissner plates withtwo simply supported parallel edges and a variety of constraint con-ditions applied on the complementary part of the object boundary,while Chen et al. [12], by invoking both the classical Kirchhoff theoryof plates and the Ritz method, derive a concise formula for determin-ing critical buckling stresses of a simply supported plate under biax-ial compression and shear loads. Composite panels elasticallyrestraint by torsion bars are studied in [13], where the authors deter-mine an explicit solution starting from the analysis of simply sup-ported and clamped cases, even if, because of its derivation, thisformulation cannot be properly claimed to be a closed form solution,as stated in the work by Qiao et al. [14]. Additionally, Wang and Red-dy [15] present a relationship between the elastic buckling loads andthird-order shear deformation polygonal plates with simply sup-ported edges. Leissa and Kang, by using the classical power seriesmethod to obtain the exact solutions for free vibration and bucklingof simply supported-free [16] and simply supported-clamped thinplates [17], analyze the effects of linearly varying forces and mo-ments on the solution. Furthermore, Reddy and Pan [18] obtain ex-act buckling loads and natural frequencies of simply supportedrectangular plates by using a higher-order shear deformationtheory.

However, to the authors knowledge, with reference to the buck-ling of rectangular plates only a limited amount of boundary con-ditions can be treated analytically. In fact, analogously to the FSM,most of the closed form solutions and analytical approaches avail-able in literature start from the hypothesis (1), by assumingw1(x) = sinx, and so requiring simply supported boundary condi-tions on the y-sides of the plate. Additionally, the limit about thepossibility of considering different boundary conditions does notconstitute the sole restriction for these models. Due to the intrinsicone-dimensional nature of the displacement field, which assumesone scalar function in x-direction, a fully description of the buck-ling response of two-dimensional systems is in fact inhibited, pre-cluding the possibility of applying constraints on a limited regionalong the x-direction. As a consequence of these considerations,

two-dimensional structures as those illustrated in Fig. 2 cannotbe generally analyzed by exploiting closed form solutions, at leastin the form recalled above, and therefore most of the models whichadopt a sinusoidal displacement function in x-direction shouldactually be considered one-dimensional models, coherently withthe input required in a numerical procedure (Fig. 3). Finally, the‘‘forced’’ sinusoidal behavior in x-direction excludes, ab-initio,eventual critical loads associated to more complex or mixedmodes. Motivated by these arguments, the present paper is aimedto explore the possibility of finding an enhanced form of the dis-placement field for buckled plates, in order to remove some ofthe above highlighted restrictions on the boundary conditions tobe applied. The model enriches a previous one proposed in [19],in which the restriction on the boundary conditions, at least interms of constraints, is removed, by constructing a model able todescribe the critical behavior of structures subjected to compres-sive loads in x direction.

Fig. 3. Analytical model, input required and corresponding 2D structure.

x

y

Nx

O

Nxy

Nxy

Ny

l

h

Nxy

NxyNy

Nx

Fig. 4. Analytical model: tensile and shear loads along the four sides of a thin plate.

80 E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88

In what follows, the adoption of a model able to represent awide class of kinematics of two-dimensional plates under a varietyof loads and constraints is presented. Examples are reported at theend and the theoretical forecasts are illustrated in comparison withresults obtained by making use of FEM-based commercial codes(ANSYS�) as well as deduced by available literature analyticalsolutions.

2. Problem formulation

Consider, in a Cartesian coordinates system (O,x,y), a thin, rect-angular, linearly elastic, homogeneous and isotropic plate ofdimensions h � l and uniform thickness t, as shown in Fig. 4. Then,with reference to the Kirchhoff kinematical model and under theVon Karman hypothesis, if Nx = rxt, Ny = ryt, Nxy = sxyt representthe generalized normal and in-plane shear stresses produced byuniform loads acting on the plate edges, the equilibrium differen-tial equation governing the buckling can be written as

Dr4w� Nxw;xx � 2Nxyw;xy � Nyw;yy ¼ 0 ð2Þ

where w is the transverse displacement, r4 is the bi-harmonic dif-ferential operator, comma denotes differentiation and D representsthe flexural rigidity of the plate as usual defined as

D ¼ Et3

12ð1� m2Þ ð3Þ

where E is the Young’s modulus and m is the Poisson’s ratio. Byadopting the dimensionless coordinates n = mpx/l and g = npy/h,where (m,n) indicate respectively the number of half-waves in xand y directions in the buckling modes, Eq. (2) can be reduced tothe following form

w;nnnn þ 2k2w;nngg þ k4w;gggg � nxw;nn � 2nxyw;ng � nyw;gg ¼ 0 ð4Þ

where k ¼ lm

nh is assumed as a measure of the plate aspect ratio and

nx ¼Nxl2

p2m2D; ny ¼ k2 Nyl2

p2m2D; nxy ¼ k

Nxyl2

p2m2Dð5Þ

are dimensionless stresses. Now, let the displacement w be as-sumed in the following multiplicative form

wðn;gÞ ¼ nðnÞ �wðgÞ ð6Þ

where n(n) is a vector collecting the prescribed functions

nðnÞ ¼

sin n

cos n

2n

1

2666437775 ð7Þ

and w(g) the vector containing the four unknown functions

wðgÞ ¼

w1ðgÞw2ðgÞw3ðgÞw4ðgÞ

2666437775 ð8Þ

whose explicit expressions have to be determined by imposing theequilibrium Eq. (4).

By replacing Eq. (6) into Eq. (4) and then collecting the terms inn it is possible to reduce the equilibrium differential equation to

sin n½k4w1;gggg � ðny þ 2k2Þw1;gg þ ðnx þ 1Þw1 þ 2nxyw2;g�þ cos n½k4w2;gggg � ðny þ 2k2Þw2;gg þ ð1þ nxÞw2 � 2nxyw1;g�þ n½k4w3;gggg � nyw3;gg� þ ½k4w4;gggg � nyw4;gg � 2nxyw3;g� ¼ 0 ð9Þ

E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88 81

which, by invoking the polynomial identity law, becomes vanishingfor any n if and only if the single terms contained in the squarebrackets are zero. As a consequence, two uncoupled systems consti-tuted by two differential equations in the sole variable g can be ob-tained. The first system involves w1(g) and w2(g)

w1;gggg � ðgy þ 2j2Þw1;gg þ ðgx þ j4Þw1 þ 2gxyw2;g ¼ 0

w2;gggg � ðgy þ 2j2Þw2;gg þ ðgx þ j4Þw2 � 2gxyw1;g ¼ 0ð10Þ

and the second system, only containing w3(g) and w4 (g), is given by

w3;gggg � gyw3;gg ¼ 0

w4;gggg � gyw4;gg � 2gxyw3;g ¼ 0ð11Þ

In Eq. (10) j, gx, gy, gxy are the dimensionless quantities writtendown

j ¼ k�1 ¼ ml

hn; gx ¼ j2 Nxh2

p2n2D; gy ¼

Nyh2

p2n2D;

gxy ¼ jNxyh2

p2n2D

The systems (10) and (11) admit respectively the following closedform solutions:

w1ðgÞ ¼ d1h1e�ffiffiffis1p

g þ d2effiffiffis1p

g þ d3h2e�ffiffiffis2p

g þ d4effiffiffis2p

g

þ d5h3e�ffiffiffis3p

g þ d6effiffiffis3p

g þ d7h4e�ffiffiffis4p

g þ d8effiffiffis4p

g

w2ðgÞ ¼ d1e�ffiffiffis1p

g þ d2h1effiffiffis1p

g þ d3e�ffiffiffis2p

g þ d4h2effiffiffis2p

g

þ d5e�ffiffiffis3p

g þ d6h3effiffiffis3p

g þ d7e�ffiffiffis4p

g þ d8h4effiffiffis4p

g

ð13Þ

and

w3ðgÞ ¼1gy

d9effiffiffiffigyp

g þ d10e�ffiffiffiffigyp

g� �

þ d11gþ d12

w4ðgÞ ¼gxy

2g5=2y

2gffiffiffiffiffigy

p� 5Þd9e

ffiffiffiffigyp

g þ d10ð2gffiffiffiffiffigy

pþ 5

� �e�

ffiffiffiffigyp

g� �

� d11gxy

gyg2 þ 1

gyd13e

ffiffiffiffigyp

g þ d14e�ffiffiffiffigyp

g� �

þ d15gþ d16 ð14Þ

in Eqs. (13) and (14) {d1,d2, . . . ,d16} represent 16 unknown coeffi-cients to be determined by imposing the boundary conditions atn = ±mp/2 and g = ±np/2, whereas

hi ¼2gxy

ffiffiffiffisip

ðgx þ j4Þ � ðgy þ 2j2Þsi þ s2i

; i � f1;2;3;4g ð15Þ

in Eq. (15) si(j,gx,gy,gxy) correspond to the four roots of the follow-ing algebraic equation:

ðgx þ j4Þ2 þ 4g2xy � 2ðgx þ j4Þðgy þ 2j2Þ

� �s

þ ð2ðgx þ j4Þ þ ðgy þ 2j2Þ2Þs2 � 2ðgy þ 2j2Þs3 þ s4 ¼ 0 ð16Þ

The Eqs. (15) and (16) depend parametrically, through j, gx, gy, gxy,on the applied loads Nx, Ny, Nxy, the flexural rigidity D and the char-acteristic dimensions (l,h) of the plate. Therefore, the solutions tothe differential Eq. (2) given above pave the way for solving inclosed-form the buckling problem. In fact, by substituting Eqs.(13) and (14) into (8) it is thus possible to rewrite the displacement(6) in the following more compact form

wðx; yÞ ¼ nðnÞ �X1ðgÞ 0

0 X2ðgÞ

� ��

d1

d2

� �ð17Þ

where n(n) indicates the vector already defined in Eq. (7),d1 = [d1 � � �d8], d2 = [d9 � � �d16] are vectors containing the unknowncoefficients and the sub-matrices appearing in (17) have the formwritten down.

X1¼h1e�

ffiffiffis1p

g effiffiffis1p

g h2e�ffiffiffis2p

g effiffiffis2p

g h3e�ffiffiffis3p

g effiffiffis3p

g h4e�ffiffiffis4p

g effiffiffis4p

g

e�ffiffiffis1p

g h1effiffiffis1p

g e�ffiffiffis2p

g h2effiffiffis2p

g e�ffiffiffis3p

g h3effiffiffis3p

g e�ffiffiffis4p

g h4effiffiffis4p

g

" #

X2¼effiffiffiffiffigxyp

g

gxy

e�ffiffiffiffiffigxyp

g

gxyg 1 0 0 0 0

2effiffiffiffiffigxyp

gffiffiffiffiffigxyp

ðgxy�gyÞ�2e�

ffiffiffiffiffigxyp

gffiffiffiffiffigxyp

ðgxy�gyÞ�gxy

gyg2 0 e

ffiffiffiffigyp

g

gy

e�ffiffiffiffigyp

g

gyg 1

264375

ð18Þ

Moreover, by renaming and collecting all the above quantities,it is also possible to write

XðgÞ ¼X1 00 X2

� �; d ¼

d1

d2

� �ð19Þ

The boundary conditions required to define the vector d can beestablished, in terms of the displacement field, as follows. Atn = ±mp/2

w ¼ 0

My ¼ 0) k2w;gg þ mw;nn ¼ 0ð20Þ

in case of simply supported edges,

w ¼ 0ux ¼ w;g ¼ 0

ð21Þ

for the case where edges are clamped and

Vy ¼ 0) k4w;ggg þ k2ð2� mÞw;nng � nyw;g � nxyw;n ¼ 0

My ¼ 0) k2w;gg þ mw;nn ¼ 0ð22Þ

must hold true if the edges are free. Additionally, at g = ± np/2 it is

w ¼ 0

Mx ¼ 0) w;nn þ k2mw;gg ¼ 0ð23Þ

for simply supported edges,

w ¼ 0uy ¼ 0) w;n ¼ 0

ð24Þ

for clamped edges and

Vx ¼ 0) w;nnn þ ð2� mÞk2w;ngg � nxw;n � nxyw;g ¼ 0

Mx ¼ 0) k2w;gg þ mw;nn ¼ 0ð25Þ

for free edges.In Eqs. (20)–(25) the quantities (Mx,My) and (Vx,Vy) represent

bending moments and transverse shear forces per unit length,applied on the n-side and on the g-side of the plate, respectively.

Adopting for the boundary conditions the displacement fieldsdetermined in Eqs. (13) and (14), in the matrix form (17), and pre-scribing such conditions, on the representative points (A,B,C,D) ofeach side of the structure as illustrated in Fig. 5, the Eqs. (20)–(25)can be rewritten as

n n;nn½ �n¼�mp2� X k2X;gg

0 mX

" #g¼0

� d ¼ 0 ð26Þ

n n½ �n¼�mp2�

X 00 X;g

� �g¼0

� d ¼ 0 ð27Þ

n n;n n;nn½ �n¼�mp2�

k4X;ggg � nyX;g k2X;g

�nxyX 0

ð2� mÞk2X;g mX

264375

g¼0

� d ¼ 0 ð28Þ

x

y

O AB

D

C

A1

l/2

l/4

l/4l/2

l/4l/4

h/2h/4h/4D

C

B

1

1

1

h/4h/4h/2

Fig. 5. Representative points adopted in numerical analysis.

Nα

Nα

Nα

Nα

Nβ

Nβ

Nα

Nα

Nα

Nα

Nβ

Nβ

Clamped Simply Supported

h

h

l l

Fig. 6. SSSS and CFFF examples with boundary and load conditions.

82 E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88

on the representative points considered on the n-side, and

n n;nn½ �n¼0 �X k2mX;gg

0 X

" #g¼�np

2

� d ¼ 0 ð29Þ

n n;n½ �n¼0 �X 00 X

� �g¼�np

2

� d ¼ 0 ð30Þ

n n;n n;nn n;nnn½ �n¼0 �

�nxyX;g k2X;gg

ð2� mÞk2X;gg � nxX 00 mX

X 0

266664377775

g¼�np2

� d ¼ 0

ð31Þ

on the representative points considered on the g-side.By collecting all the possible prescribed boundary conditions

(26)–(31), a linearly independent eighth-order system can be final-ly obtained in the form

Kðgx;gy;gxyÞ � d ¼ 0 ð32Þ

Given that the components of K are exponential functions of gx,gy, gxy, the system (32) does not admit an algebraic solution andtherefore the associated eigen-problem has to be solved by adopt-ing an iterative numerical procedure. Assuming that the half-wavenumbers n and m vary in the ranges n ¼ 1;2; . . . �n andm ¼ 1;2; . . . �m, �n� �m eigenvalues can be then obtained, the lowestrepresenting the actual critical buckling load.

3. Results and discussion

The analytical method proposed in the previous section is hereimplemented in order to obtain exact solutions for the buckling ofa rectangular thin plates subject to the following in-plane loads

Nx ¼ aN; Ny ¼ bN; Nxy ¼ cN fa;b; cg 2 R ð33Þ

In Eq. (33), {a,b,c} are scalar parameters used to modulate the loadratios.

In a first example two different loading cases are considered,namely: (1) uniaxial in-plane compressive forces in the x-direction(a = �1,b = c = 0); (2) biaxial in-plane compressive loads with

X1 ¼sinh q1g cosh q1g sin q2g cos q2g 0 0

0 0 0 0 sinh q1g cosh q1g s

�X2 ¼

g3 g2 g 1 0 0 0 00 0 0 0 g3 g2 g 1

" #

Nxy = 0 and Ny variable within the prescribed parameters ranges(a = �1,b1 6 b 6 b2,c = 0). Poisson’s ratio v = 0.33 and Young mod-ulus E = 100 GPa are considered for all the performed numericalanalyses.

For convenience of notation it will be indicated with S, F and Crespectively Simply supported, Free and Clamped plate sides and,as a consequence, the overall boundary conditions established foreach case will be associated to an acronym recalling the constraintson the edges, starting from n = �mp/2, n = mp/2, g = �np/2 andg = np/2.

To show the effectiveness of the proposed strategy, two buck-ling analyses are executed for each of the above mentioned loadconditions, by considering different constraints (namely SSSS andCFFF plates). The obtained results are thus compared with theiranalytical or FEM-based counterparts presented in literature. Thesketches of the four structures analyzed are summarized in Fig. 6.

Additionally, being the proposed kinematical model capable todetermine the critical load of plates exhibiting discontinuousboundary conditions, a third example, illustrated in Fig. 7, is also ta-ken into consideration, where the y-sides are partially clamped, theside x = �a1 is simply supported and the side x = a2 is considered free.

3.1. Uniaxial in-plane compressive load (Ny = Nxy = 0)

If in Eq. (33) it is set b = c = 0, a = �1, the solution of Eq. (9) inthe matricial form (19) becomes

0 0inq2g cos q2g

�ð34Þ

A

B

x

y

C

D G

E

F

O

N x

a

b

1 a 2

1

11x

y

O2

2 2

Clamped Simply Supported

Nx

Fig. 7. Example associated to mixed boundary conditions and representative points.

E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88 83

where q1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 þ ffiffiffiffiffigx

ppand q2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigxp � j2

pconstitute, through gx

and j defined in (12), the ‘‘memory’’ of the geometry and of theapplied compressive load. By collecting the boundary conditions(20) and (23) related to a SSSS plate, the matrix K defined in Eq.(32) assumes the following explicit form.

K ¼

0 cos mp2 0 cos mp

2 0 �mpl 0 1

0 cos mp2 0 cos mp

2 0 mpl 0 1

� sinh npq12 cosh npq1

2 � sin npq22 cos npq2

2 0 0 n2p2

4h2 1

sinh npq12 cosh npq1

2 sin npq22 cos npq2

2 0 0 n2p2

4h2 1

0 D mq21k2 � 1

� �cos mp

2 0 �D mq22k2 þ 1

� �cos mp

2 � 2pDnkmh3 0 2Dk2m

h2 0

0 D mq21k2 � 1

� �cos mp

2 0 �D mq22k2 � 1

� �cos mp

22pDnkm

h3 0 2Dk2mh2 0

D m� k2q21

� �sinh q1np

2 D k2q21 � m

� �cosh q1np

2 D k2q21 þ m

� �sin q2np

2 �D k2q21 þ m

� �cos q2np

2 0 0 2Dk2

h2 0

D k2q21 � m

� �sinh q1np

2 D k2q21 � m

� �cosh q1np

2 �D k2q21 þ m

� �sin q2np

2 �D k2q21 þ m

� �cos q2np

2 0 0 2Dk2

h2 0

266666666666666666664

377777777777777777775

With the classical position detðKðgxÞÞ ¼ 0 it is then possible toobtain a parametric expression of the critical value gcr

x in termsof h/l, the dimensionless geometrical ratio contained in j. InFig. 8 the normalized values gð1Þcr =gmax

cr obtained analytically for

K¼

�sinmp2 cos mp

2

cos mp2 sin mp

2

�D mq22k2þ1

� �sin mp

2 D mq21k2�1

� �cos mp

2

�D ð2�mÞq22k2þ1

� �cos mp

2 D 1�ð2�mÞq21k2

� �sin mp

2

0 D q21k2�m

� �cosh npq1

2 D mþk�

0 D ð2�mÞq1�k2q31

� �sinhnpq1

2 �D ð2�mÞq�

0 D k2q21�m

� �cosh npq1

2 �D k2q22

�0 D �ð2�mÞq1þk2q3

1

� �sinh npq1

2 �D ð2�mÞq�

2666666666666666666664

different ratios h/l in [1], numerical values gð2Þcr =gmaxcr obtained by

means of FEM analyses and gð3Þcr =gmaxcr found through the proposed

formulation are all illustrated, showing the efficacy of the formula-tion and the accuracy of the results.

0 cos mp2 0 �mp

l 0 10 sin mp

2 0 2l 0 0

0 �D mq22k2þ1

� �cos mp

22Dmk2mp

lh2 0 2Dk2mh2 0

0 D 1þð2�mÞq22k2

� �sin mp

24Dk2ð2�mÞ

lh2 0 0 0

2q22

�sin npq2

2 �D mþq22k2

� �cos npq2

2 0 0 2Dk2

h2 0

2þk2q32

�cos npq2

2 �D ð2�mÞq2þk2q31

� �sinhnpq1

2 0 0 0 0

þm�

sinnpq22 �D k2q2

2þm� �

cos npq22 0 0 2Dk2

h2 0

2þk2q32

�cos npq2

2 D ð2�mÞq2þk2q31

� �sin npq2

2 0 0 0 0

3777777777777777777775

The second example is related to a CFFF plate. By imposing theboundary conditions (22) on the n-side and (24) and (25) on theg-side the corresponding matrix K, defined in Eq. (32), takes in thiscase the form:

maxcr

cr

ηη

Analytical solution

Proposed model

FEM analysis

m=1m=2 m=3 m=4

m=5

h l

Fig. 8. SSSS example subject to uniaxial compressive load: dimensionless gcr versus h/l.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

maxcr

cr

ηη

h l

h=125 h=175 h=200h=100

Analytical solution

Proposed model

FEM analysis

Fig. 9. CFFF example subject to uniaxial compressive load: dimensionless gcr versus h/l.

84 E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88

Fig. 9 shows the critical load obtained imposing detðKðgxÞÞ ¼ 0 fordifferent values of h/l, again compared with analytical [17] andnumerical results, these ones obtained by means of FEM analyses.

It is worth noting that, differently from the classical displace-ment-based Finite Element approach, the proposed procedure canbe complementarily interpreted as a static formulation, in the sensethat the chosen displacement functions producing stresses satisfy-ing the equilibrium equations everywhere leave compatibilityconditions verified only on selected representative nodes. Accord-ingly, the eigenvalues representing the critical loads seem weaklydepend on the adopted discretization and a mesh refinement couldbe then required only to improve the accuracy of the eigenvectors,associated to the critical modes. In other words, in the present for-mulation the critical modes strongly depend on both the adoptedmesh and the representative points where boundary conditionsare prescribed, while, if the representative nodes are changed,critical modes accordingly change, without to significantly modifythe critical load, at least for the considered examples. To stress this

concept, in Fig. 10 comparisons of the critical mode associated to aCFFF square plate analyzed by means of FEM (Fig. 10A) with thecorresponding results obtained by following the proposedapproach for two different choices of representative nodes, areillustrated. In particular, Fig. 10B and C show the critical modesobtained choosing to prescribe the boundary conditions on thenodes (A,B,C,D) and (A1,B1,C1,D1), respectively, as reported inFig. 5.

3.2. Biaxial in-plane compressive load (Nxy = 0,Ny = bNx)

If in Eq. (33) is assumed c = 0 and a = �1, the structure is sub-jected to biaxial load, with Nxy = 0 and Ny = bNx, b parametricallyvarying to define the load ratio. As a result, b < 0 refers to biaxialcompressive load (b = �1 representing equi-biaxial load), andb > 0 is associated to the case in which compressive load in x direc-tion and tensile load in y direction are contemporarily exerted onthe plate.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

FEM analysis

Proposed model

maxcr

cr

ηη

h l

h=30

h=40

h=50

Fig. 11. SSSS example subject to equi-biaxial compressive load: dimensionless gcr versus h/l.

Fig. 10. CFFF example, comparison between FEM analysis (case A) and proposed model, adopting (B) ABCD and (C) A1B1C1D1 representative points reported in Fig. 5.

E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88 85

For such a case, the equations in the systems (10) and (11)become

w1;gggg � ð2j2 � bgxÞw1;gg þ ðj4 � gxÞw1 ¼ 0

w2;gggg � ð2j2 � bgxÞw2;gg þ ðj4 � gxÞw2 ¼ 0w3;gggg þ bgxw3;gg ¼ 0w4;gggg þ bgxw4;gg ¼ 0

ð35Þ

whose solutions can be again reduced to the form represented in Eq.(17), providing that matrices X1 and X2, defined in Eq. (18), aresubstituted by the following ones

eX1 ¼ef1g e�f1g ef2g e�f2g 0 0 0 00 0 0 0 ef1g e�f1g ef2g e�f2g

� �eX2 ¼

egyg e�gyg g 1 0 0 0 00 0 0 0 egyg e�gyg g 1

� � ð36Þ

by also setting

f1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

2j2 � bgx �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigxð4þ b2gx � 4bj2Þ

q� �s

f2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

2j2 � bgx þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigxð4þ b2gx � 4bj2Þ

q� �s ð37Þ

Exploiting (36) and imposing the boundary conditions it is final-ly possible to determine the critical loads, whose parametricvariations with h/l are illustrated in Figs. 11 and 12 for SSSS andCFFF plates under equi-biaxial compressive loads. The analyticaloutcomes are there compared with numerical solutions obtainedby performing FEM analyses.

In the case of SSSS plate, the determinant of the matrix K con-taining the boundary conditions assumes the following form

detðKÞ ¼ D4e�hðf1þf2Þð�1þ 2e2hf1 Þð�1þ 2e2hf2 Þm2 f21 � f2

2

2 ð38Þ

Replacing Eq. (37) into (38), setting j = k�1 = m/l � h/n, and impos-ing the buckling condition detðKðgx;h; l;m;bÞÞ ¼ 0, one finallyobtains the following analytical expression of the critical load.

gx ¼ Dl2p4 þ 2h2m2p4 þ h4m4p4

l2

� �h2m2p2ðh2 � b2p2Þ

ð39Þ

In Eq. (39) the number of half-waves related to the minimum criticalload can be achieved imposing gx,m = 0, in this way obtaining thatmin(m) = l/h and then, for square plates, min(m) = 1.

In Fig. 13 the plot of the critical load (39) against the parameterb, obtained by making h/l = 1 and �2 6 b 6 3, is illustrated andcompared once again with the corresponding FEM results. Thegraphic shows how the critical load increases with b, exhibiting

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3β

ηcr

Fig. 13. SSSS example subject to biaxial compressive load: dimensionless gcr versus b.

0

0,2

0,4

0,6

0,8

1

1,2

0 0,5 1 1,5 2 2,5 3 3,5

FEM analysis

Proposed model

maxcr

cr

ηη

h l

h=30h=40

h=50

Fig. 12. CFFF example subject to equi-biaxial compressive load: dimensionless gcr versus h/l.

86 E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88

an asymptote at b = 1, that is in presence of a tensile load iny-direction opposite in sign to the compressive force in x direction.This limit situation can be referred to a state of pure shear in theplate unbuckled configuration and thus the analytical as well asthe FEM results find a theoretical divergent critical load. However,as a further increasing b is considered (i.e. b > 1) the tensile gener-alized stress Ny becomes greater than Nx and instability can occuronly by changing the sign of the applied loads.

3.3. Uniaxial in-plane compressive load with mixed boundaryconditions

As final example, the buckling analysis of the structure repre-sented in Fig. 7 is performed. It is characterized by uniaxialin-plane load Nx and homogeneous boundary conditions on the x-sides, that is – adopting the two local reference system representedin Fig. 7b – simply supported at x1 = �a1/2 and free at x2 = +a2/2. They-sides present inhomogeneous boundary conditions, that isclamped at y1 = b/2 and y2 = �b/2, and free at y1 = �b/2 and y2 = b/

2. Results accuracy requires that, to take properly into accountthe imposed boundary conditions, the numerical model be consti-tuted at least by two elements, with seven corresponding represen-tative points, as reported in Fig. 7B. The displacement field can bethen defined for each element, and collected in the following form:

wI ¼ wðx1; y1Þ ¼ nðn1Þ �X1ðg1Þ 0

0 X2ðg1Þ

" #�

d11

d21

� �

wII ¼ wðx2; y2Þ ¼ nðn2Þ �X1ðg2Þ 0

0 X2ðg2Þ

" #�

d21

d22

� � ð40Þ

where ni = mpxi/ai, gi = npyi/b, (i = 1,2) are the dimensionless coordi-nates related to each element, n; X1; X2 the matrices defined bythe Eqs. (7) and (34) for the displacement field and dij (i = 1,2;j = 1,2) vectors which contains 8 unknown constants obtainableby imposing the boundary conditions (26)–(31) on the representa-tive nodes A–C and E–G, as reported in Fig. 7B. On the common nodeD, assuming continuity in terms of both equilibrium and displace-ment, the following conditions are required:

�n �X 00 �n �X;g1

" #n1¼þmp

2 ;g1¼0

�n �X 00 �n �X;g2

" #n2¼þmp

2 ;g2¼0

k4n �X;g1g1g1k2n �X;g1

ð2� mÞk2n;n1n1 �X;g1mn;n1n1 �X

" #n1¼þmp

2 ;g1¼0

k4n �X;g2g2g2k2n �X;g2

ð2� mÞk2n;n2n2 �X;g2mn;n2n2 �X

" #n2¼�mp

2 ;g2¼0

26666664

37777775 �d11

d12

d21

d22

2666437775 ¼ 0 ð41Þ

E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88 87

Therefore, by collecting the Eq. (41) and the external boundaryconditions, it is possible to carried out a sixteenth-order system oflinearly independent equations, whose eigenvalues and eigenvec-tors allow to completely define the critical behavior of thestructure.

In Fig. 14 the analysis of a plate characterized by geometrical ra-tios (a1 + a2)/b = 10, a1/a2 = 1 and thickness t = 0.1a is performed,following both the proposed method with two elements and aFEM strategy using different mesh sizes. The comparison highlightshow the proposed method converges to the actual result indepen-dently from the number of elements chosen to discretize the prob-lem, while FEM analyses require to adjust the mesh for obtainingacceptable results.

D

Ncr

0

200

400

0 10 20 3

Mesh

Fig. 14. Example with mixed boundary condition: comparison between results ob

a

b

1 a2

b

a2

Nx

a

b

1

a

b

1

a

b

1

Nx

Nx

Nx

Nx

Fig. 15. Example with mixed boundary condition: SFCF plate (a1/a2 = 0), SFFC plate (a1/a2

1, 4).

Moreover, in order to estimate the sensitivity of the model withrespect to variations of element sizes, the same example has beenused to perform a parametric analysis over the geometrical ratioa1/a2. The limit cases a1/a2 = 0 and a1/a2 ?1 represent the SFCFand SFFC plates, respectively (see Fig. 15). All the intermediatecases described by the geometrical ratio variations are also illus-trated in Fig. 15.

Finally, in Fig. 16 are shown the numerical results for limit caseswhere a single element is adopted for the analysis, as well as theresults related to intermediate geometrical ratios, in which theanalyses have been performed choosing two-elements. The out-comes prove that the proposed method is stable and essentiallyinsensitive to the varying geometrical ratios.

0 40 50 60

Analytical solution with 2 elements

FEM solution

size

tained with proposed method and via FEM analysis, for different mesh-size.

a2

a2

a1 = 02a

=14

= 1

= 4

∞

Nx

Nx

Nx

Nx

Nx

a1

2a

a1

2a

a1

2a

a1

2a

?1) and three intermediate boundary conditions applied to the plates (a1/a2 = 1.4,

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1 1.2

crN

crN max

a +aa2

1 2

Analytical solution with 2 elements

FEM solution

Fig. 16. Example with mixed boundary condition: dimensionless critical load versus (a2/(a1 + a2)) ratio.

88 E. Ruocco, M. Fraldi / Engineering Structures 38 (2012) 78–88

4. Conclusion

The paper has presented an analytical approach for determiningexact buckling solutions for plates exhibiting mixed boundary con-ditions, with constraints and load eventually applied piecewisealong the object edges. The solutions have been obtained by intro-ducing an enriched kinematical model where the displacement isconstructed as the result of the scalar product of a vector contain-ing four prescribed functions, responsible for the behavior of theplate in x-direction, and a vector of four unknown functionsdepending on the y-variable. By replacing the displacement withthis mathematical form into the governing equilibrium equation,a family of closed form solutions is finally found. Then, followingthe standard technique, the critical buckling load has been deter-mined by numerically solving the corresponding eigenvalue prob-lem. It has also been demonstrated that the proposed strategy canbe efficiently employed for the buckling analysis of isotropic plates,finding exact closed solutions associated to four main cases,namely monoaxial in-plane compressive load in the x-directionand equi-biaxial in-plane compressive loads in both SSSS and CFFFplates. Additionally, the critical behavior of structures with morecomplex discontinuous mixed boundary conditions has been alsoanalyzed and the critical loads have been obtained for each ofthe cases. At the end, the accuracy and the effectiveness of the pro-posed method are shown through comparisons of the theoreticalforecasts with previously published results obtained following dif-ferent numerical procedures and analytical techniques.

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