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European Journal of Mechanics A/Solids 39 (2013) 120e133

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European Journal of Mechanics A/Solids

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Nonlinear buckling analysis of laminated composite curved panels constrained byWinkler tensionless foundation

Danial Panahandeh-Shahraki, Hamid Reza Mirdamadi*, Ali Reza ShahidiDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e i n f o

Article history:Received 29 July 2012Received in revised form24 September 2012Accepted 18 October 2012Available online 8 November 2012

Keywords:Uni-lateral bucklingTensionless foundationLaminated composite curved panel

* Corresponding author. Tel.: þ98 311 391 5248; faE-mail addresses: [email protected]

[email protected], [email protected]@cc.iut.ac.ir (A.R. Shahidi).

0997-7538/$ e see front matter � 2012 Elsevier Mashttp://dx.doi.org/10.1016/j.euromechsol.2012.10.010

a b s t r a c t

This article addresses structural analysis of laminated composite cylindrical panels resting on tensionlessfoundation under axial compression. The problem is inherently and highly nonlinear. The governingequations are derived based on classical shell theory and principle of minimum total potential energy.Major contributions of this paper consider the effects of curvature and composite material properties inderiving energy-based governing equations. The numerical results show that ignoring initial curvature ofreinforcing panels for modeling columns having cross sections of curved boundaries would cause thatthe buckling load would be estimated less than that of actual value. Further, the influence of tensionlessfoundation on the uni-lateral buckling behavior of panels is severely dependent on effective parameterssuch as central angle, aspect ratio, thickness, and the degree of foundation modulus. In addition,increasing the number of panel layers, keeping the thickness constant, and choosing an appropriate plyangle for fibers, might increase impressively the influence of tensionless foundation on the buckling load.Moreover, the effects of parameters like aspect ratio, thickness, central angle, the number and angle ofplies, lamination scheme, material orthotropy and foundation modulus on buckling load are investigated.The results are compared with case studies, whenever available in the literature.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

Load-carrying structures under investigation as a systemmay bein contact with other structural parts or members having differentstiffness. In many engineering applications, foundation has nocapability of inserting tensile as well as compressive forces to theinterface between foundation and structure. This bi-modulusbehavior of foundation causes the contact area would be indeter-minate. Therefore, the problem is inherently and highly nonlinear.This kind of foundation is known as uni-lateral, one-sided, ortensionless. If the buckling phenomenon takes place, because of theexistence of a foundation, the structure is distorted only in one ofthe directions perpendicular to the contact area. This kind ofbuckling is known as uni-lateral buckling. In the literature, uni-lateral buckling is categorized as contact problems. In the sequel,some of important studies conducting on tensionless foundationare reviewed.

x: þ98 311 391 2628.(D. Panahandeh-Shahraki),t.ac.ir (H.R. Mirdamadi),

son SAS. All rights reserved.

Of the first researchers working on the tensionless foundations,we may point out to Schindler (1959), Tasi and Westmann (1967),and Weitsman (1970). With introduction of one parameter,Schindler (1959) divided the contact area between the structureand foundation as in-contact and no-contact areas, and derived theequation of equilibrium for each area, separately. Then, regarding tothis way, studied lateral deflection of a beam on the tensionlessfoundation under two separate loadings of concentrated point loadand moment. Tasi and Westmann (1967) introduced an iterativescheme for detecting contact area. Then they extended theirscheme for transverse deformations of a beam on the tensionlessfoundation. Weitsman (1970) extended the tensionless foundationmodel based on the two models of Winkler and Reissner. Then heinvestigated the degree of area in-contact and transverse defor-mation analysis of beams and flat plates subjected to concentratedlateral force including weight.

So far, extensive studies have been exercised on the analysis ofplates resting on tensionless foundations. Do (1976, 1977) wasamong the first researchers that investigated buckling analysis ofrectangular thin plates on tensionless foundations. He derivedgoverning equations based on von Karman’s and bifurcation theo-ries and using variational inequality formulation. Hobbs (1989)investigated buckling of a rectangular plate having weight ona rigid (tensionless) foundation. This plate was under biaxial in-

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Nomenclature

a, b, h, R, a side, width, thickness, radius and central angle ofpanel

a/b aspect ratiou, v, w displacement field components in x, y and z global

coordinate systemu0, v0, w0mid-plane displacement field components in the x, y

and z directions{ε} total strain vector{ε(01)} linear part of membrane strain vector{ε(02)} nonlinear part of strain vector, von Karman’s strain

vector{ε(1)} bending strain vectorsij, εij, i, j¼ x,y stress and strain vector components.{N} total force resultant vector of in-plane stresses{M} total moment resultant vector of out-of-plane stresses[A], [B] and [D] extensional, bending-extensional coupling, and

bending stiffness matricesfg(x,h) gth term of polynomial for out-of-plane displacement

fieldnp number of two-dimensional polynomial terms for out-

of-plane displacement fieldfb(x,h) out-of-plane boundary conditions polynomialx, h dimensionless coordinates (natural coordinates)x, y Cartesian coordinates{U} total displacement field vectorfbUg unknown coefficient vector[N]R displacement field matrix[CM] matrix resulting from membrane strain vector[CB] matrix resulting from bending strain vectorHiðxÞ i ¼ 3; :::;10HiðhÞ i ¼ 3; :::; 10

Hierarchical functions in naturecoordinates

HiðxÞ i ¼ 1;2HiðhÞ i ¼ 1;2

first order Lagrange functions in naturecoordinates

dUe potential energy variation resulting by deformation ofsprings

dV potential energy variation of panel resulting from in-plane loads

dU strain energy variations of paneldP total potential energy variation[Ke] total stiffness matrix resulting from elastic foundation½Kww

e � stiffness matrix resulting from lateral displacementX contact functionkf spring stiffnessb foundation modulus(is logarithmic of spring stiffness

based on ten)[KT] total stiffness matrix[Kmm] stiffness matrix resulting from membrane loads[Kmb] stiffness matrix resulting from coupling loads[Kbb] stiffness matrix resulting from bending loads[KG] geometric stiffness matrix[H] matrix resulting from von Karman’s strain[N0] matrix resulting from in-plane forcesl, Ncr bucking critical loadNcr non-dimensional buckling critical loadm, n number of half waves in longitudinal and

circumferential directionsinc the percentage of the difference between uni-lateral

and bi-lateral buckling load relative to bi-lateralbuckling load

Ei i¼ 1, 2 Young’s modulusy Poisson’s ratioU0 surface of panel

D. Panahandeh-Shahraki et al. / European Journal of Mechanics A/Solids 39 (2013) 120e133 121

plane compressive load. Shahwan and Waas (1991, 1994) startedbuckling analysis of a rectangular orthotropic thin plate of infiniteand finite dimensions, on tensionless foundation. They concludedthat uni-lateral constraint was an appropriate mathematical modelfor studying the behavior of delamination buckling in laminatedplates. Thereafter, using analytical and experimental methods,Shahwan and Waas (1998) investigated buckling behavior ofa laminated rectangular plate, constrained uni-laterally to a foun-dation, more precisely. They used an iterative method for obtainingthe response of structure to uni-lateral buckling phenomena. Honget al. (1999) engaged in static response analysis of plates and shellsof revolution on tensionless foundations. They assumed that platesand shells of revolution were subject to transverse point load andextracted governing equations, considering large deflections, andbased on finite element method. Smith et al. (1999a,b) studied thestability of a homogeneous rectangular plate on tensionless foun-dation under shearing force and based on RayleigheRitz method. Inaddition, Smith et al. (1999c), and Bradford et al. (2000) continuedsimilar studies for a rectangular plate under simultaneous loadingsof axial compression, bending moment and shear forces. In anotherstudy, Smith et al. (1999d) compared and validated their previousresults by their data resulting from experimental techniques. Theyimplemented an iterative method for obtaining uni-lateralresponse in all their studies. Hedayati et al. (2007) investigatedlocal buckling of a rectangular plate bolted to a concrete foundationand used for reinforcing that concrete foundation. They solved theproblem by RayleigheRitz method together with Lagrange’smultipliers scheme and an iterative method for finding contact

area. The plate under study was subjected to axial compressive andshear forces. Ma et al. (2007) initiated contact buckling analysis ofa laminated composite plate in which the layers of the plate underinvestigation had been separated in some regions. In this work,delamination phenomenon had been modeled as a contactproblem. Further, Ma et al. (2008) investigated buckling analysis ofprofiled steel sheets forming the skin of concrete-filled compositewall panels on a foundation. They modeled that profile by aninfinite orthotropic plate resting on a tensionless foundation. Theplate was under separate loadings of axial compression andshearing forces. The difference between this work and theirprevious study was the presence of a foundation in their later studywhile in the former, there was no foundation and the buckling oflaminated composite plate, suffering from delamination, had beenstudies as a contact problem. Again, Ma et al. (2011) were involvedin the study of buckling of an infinite thin plate on a tensionlessfoundation under shear loading. The infinite plate had beenmodeled as a 1Dmechanical system. In all of their threeworks, theyobtained contact area by using lateral buckling mode function andreached two nonlinear equations for two regions of in-contact andnon-contact areas.

In this paper, in particular, we pay attention to uni-lateralbuckling analysis of laminated composite panels of curved profilewhich may be bolted, riveted, or welded to other structuralmembers or parts, all around. These curved panels are used forstiffening or reinforcing steel or concrete beams, columns, ormoment-resisting frame, mechanical connections and joints, aswell as storage vessels and industrial pipeline and lifeline systems.

D. Panahandeh-Shahraki et al. / European Journal of Mechanics A/Solids 39 (2013) 120e133122

The contact interface between these two classes of structuralmembers with dissimilar stiffness may be modeled as a tensionlessfoundation. Almost all the research conducted on uni-lateralbuckling is limited to beams and flat plates constrained toa tensionless foundation of rigid/elastic type.

In the present study, we have investigated buckling character-istics of laminated composite curved panels on tensionless foun-dations, subject to compressive axial force, based on shell classictheory and using RayleigheRitz method of solution. The purpose ofthis study is investigating the influence of tensionless foundationon the buckling behavior of laminated composite curved panelsubjected to compressive force. For validation purposes, aftersimplifying the composite curved model to an isotropic/homoge-neous flat model, we have compared the results obtained from thismodel with what had been accomplished earlier in the literature.Extensive investigations have been conducted for identifying theeffects of parameters, such as panel aspect ratio, panel thickness,curved profile central angle, lamination scheme, the number oflayers, material orthotropy, and foundation stiffness, on the uni-lateral buckling load.

2. Formulation

A curved panel resting on a foundation is shown in Fig. 1. Thegeometry includes side a, with central angle a, constant curvatureR, width b¼ Ra, and thickness h. Regarding Fig. 1 and themechanical behavior of thin panels, according to classical shelltheory, the displacement field is assumed as follows (Reddy, 2004):

uðx; y; zÞ ¼ u0ðx; yÞ � zvw0

vx

vðx; y; zÞ ¼ v0ðx; yÞ � zvw0

vy

wðx; y; zÞ ¼ w0ðx; yÞ

(1)

where u, v, and w are displacement field components for anygeneric point (x, y, z) in the x, y, z directions, respectively, andu0, v0,and w0 are the corresponding components on the mid-surface.

Fig. 1. Illustration of a laminated composite cylindrical panel resting on a foundationin a global (x, y, z) coordinate system.

Therefore, the strain field (considering moderate rotations initi-ated by von Karman) is derived as (Reddy, 2004):

fεg ¼nεð01Þ

oþnεð02Þ

oþ z

nεð1Þ

o(2)

where {ε(01)} is the membrane (or in-plane) strain vector, {ε(02)},von Karman’s strain vector, and {ε(1)}, bending (or out-of-plane)strain vector:

nεð01Þ

o¼

8>><>>:εð01Þxx

εð01Þyy

gð01Þxy

9>>=>>; ¼

8>>>>>>>><>>>>>>>>:

vu0vx

vv0vy

þw0

Rvv0vx

þ vu0vy

9>>>>>>>>=>>>>>>>>;(3)

nεð02Þ

o¼

8>><>>:εð02Þxx

εð02Þyy

gð02Þxy

9>>=>>; ¼

8>>>>>>>>>><>>>>>>>>>>:

12

�vw0

vx

�2

12

�vw0

vy

�2

vw0

vxvw0

vy

9>>>>>>>>>>=>>>>>>>>>>;(4)

nεð1Þ

o¼

8>><>>:εð1Þxx

εð1Þyy

gð1Þxy

9>>=>>; ¼

8>>>>>>>>>>><>>>>>>>>>>>:

�v2w0

vx2

�v2w0

vy2

�2v2w0

vxvy

9>>>>>>>>>>>=>>>>>>>>>>>;(5)

Reduced plane stressestrain relation in 1e2 material coordinatesystem for the kth orthotropic layer of the laminated compositecurved panel, with a total thickness h, and N layers, as shown inFig. 1, is expressed as follows:8<:

s1s2s6

9=;k

¼24Q11 Q12 0Q21 Q22 00 0 Q66

35k 8<: ε1ε2ε6

9=;k

(6)

where si and εi are the stress and strain components in the materialcoordinate system, respectively. Reduced elastic plane stress coef-ficients for the kth layer, Qk

ij , are as follows:

Q11 ¼ E11� y12y21

; Q22 ¼ E21� y12y21

;

Q12 ¼ Q21 ¼ y12E21� y12y21

; Q66 ¼ G12

(7)

The plane stressestrain relation after transforming to the globalcoordinates system is written as (Reddy, 2004):8<:

sxxsyysxy

9=;k

¼

264Q11 Q12 Q16

Q21 Q22 Q26

Q16 Q26 Q66

375k 8><>:

εxx

εyy

gxy

9>=>;k

(8)

Therefore, the stress resultant vector of in-plane stresses, {N}¼{NxNyNxy}T and the stress resultant vector of out-of-plane stresses,{M}¼ {MxMyMxy}T for a circular cylindrical panel (without consid-ering von Karman strains) is obtained (Reddy, 2004):

� fNgfMg

�¼

� ½A� ½B�½B� ½D�

�(�εð01Þ�εð1Þ

)(9)


In the above equation, [A] is the extensional stiffness matrix, [D],the bending stiffness matrix, and [B], the bending-extensionalcoupling stiffness matrix. Because of choosing a symmetricscheme for layering the panel, the stiffness matrix [B] woulddiminish. In this case, the in-plane behavior will be independent ofthe out-of-plane behavior.

3. Discretization

For a series discretization using Rayleigh-Ritz method (RRM),the displacement field is approximated by some predefined trialfunctions with independent unknown coefficients, or the so called,generalized coordinates. These functions are selected so that theycan, at least, satisfy geometrical or kinematical boundary condi-tions. However, satisfying static or natural boundary conditions isnot necessary. Therefore, out-of-plane displacement field w0(x,h) isobtained by using Pascal polynomials and by applying someauxiliary functions that could satisfy the geometric boundaryconditions, we have (Smith et al., 1999c):

w0ðx; hÞ ¼ fbðx; hÞXpq¼0

Xqr¼0

bwgfgðx; hÞ (10)

In matrix notation, we obtain:

w0ðx; hÞ ¼ Nwðx; hÞ

�� bw(11)

where p is a polynomial function order, f bwg, RayleigheRitzunknown coefficients vector, called the generalized coordinates,and hNwi, a vector resulting from the polynomial functions, satis-fying the geometric boundary conditions. The function fg(x,h) is thegth term of the polynomial and is obtained as follows:

fgðx; hÞ ¼ xrhq�r (12)

With the following definition for the subscript g:

g ¼ ðqþ 1Þ ðqþ 2Þ2

� r (13)

The number of polynomial terms is calculated as follows:

np ¼ ðpþ 1Þ ðpþ 2Þ2

(14)

In Eq. (10), the function fb(x,h) is used (see Fig. 1) for satisfyingdifferent types of out-of-plane boundary conditions on the paneledges. Its values for different types of boundary conditions aretabulated in Table 1. Regarding the panel edges shown in Fig. 1, S, C,and F denote, respectively, simple, clamped and free boundaryconditions. The in-plane displacement field could be defined byEqs. (15) and (16):

u0ðx; hÞ ¼ hNuðx;hÞi�bu (15)

Table 1fb(x,h)function for satisfying free (F), simple (S) and clamped (C) boundaryconditions.

Edgesa Boundary condition

F S C

1 (x� 1)0 (x� 1)1 (x� 1)2

2 (h� 1)0 (h� 1)1 (h� 1)2

3 (xþ 1)0 (xþ 1)1 (xþ 1)2

4 (hþ 1)0 (hþ 1)1 (hþ 1)2

a See Fig. 1 for numbering scheme.

v0ðx; hÞ ¼ hNvðx;hÞi�bv (16)

where fbug and fbvg are RayleigheRitz coefficient vector, hNui andhNvi are the vectors resulting from polynomial functions, which areobtained by multiplying the first 10 terms of the hierarchicalfunctions (Zienkiewicz and Taylor, 1989) by x and h coordinates asdefined in the Appendix. This set of hierarchical functions consti-tutes 100 shape functions in the entire domain.

In the above equations, the first two functions are the first-orderLagrange interpolation functions, one of themmay be eliminated indifferent in-plane boundary conditions. Regarding (11), (15) and(16), the panel displacement field can be written as:

8<: u0v0w0

9=; ¼24 hNui 0 0

0 hNvi 00 0 hNwi

358><>:

�bu�bv� bw9>=>;0fUg ¼ ½N�R

nbUo(17)

In all of the above equations, x¼ (2x/a)� 1 and h¼ (2y/b)� 1.Thus, we can obtain strain vector, defined in (3)e(5), interpolatedby the shape functions used in the definition of the totaldisplacement field in RRM:

nεð01Þ

o¼ ½CM�

nbUo(18)

nεð1Þ

o¼ ½CB�

nbUo(19)

In the above relations, fbUg is a vector representing the totalnumber of unknown coefficients (or the total number of generalizedcoordinates), and [CM] and [CB] are the transformation matrices,operating on total discretized domain, expressed as follows:

½CM� ¼

26666666664

vNu

vx0 0

0vNv

vyNw

RvNu

vyvNv

vx0

37777777775(20)

½CB� ¼ �

266666666664

0 0v2Nw

vx2

0 0v2Nw

vy2

0 0 2v2Nw

vxvy

377777777775(21)

4. Stability analysis

The influence of a tensionless elastic foundation is governed bya rheological model, which is dependent on the displacementdirection, as illustrated in Fig. 1. As the cylindrical panel leaves thefoundation, the force resulting from the foundation reaction is zero,however, as the opposite occurs, the linear reaction of the foun-dation is proportional to the panel lateral displacement. If weassume the elastic foundation as a group of linear springs attachedto the panel, the variation in elastic potential energy resulting froma virtual axial deformation of the springs are expressed as follows:


dUe ¼ dnbUoT ½Ke�

nbUo(22)

where ½Kwwe �, the stiffness matrix resulting from the presence of the

elastic foundation in lateral direction, and [Ke], the total stiffnessmatrix, are written as follows:

�Kwwe

¼Xri¼1

Xsj¼1

kf ½Nw�T ½Nw�0½Ke� ¼

26403�3 03�3 03�3

03�3 03�3 03�3

03�3 03�3�Kwwe

375

(23)

In the above equation, r and s are the number of springs in thelongitudinal and circumferential panel directions, kf, the axialstiffness of each spring (showing a discretized version of founda-tion modulus), and 03�3, is null matrix of size 3� 3. In order toeliminate the spring forces in a tension case, a switching on-offcontact function X is defined, i.e., it is equal to zero as the panelleaves the foundation, and equal to 1, when the panel tends topenetrate into the foundation:�X ¼ 0 ðseparationÞX ¼ 1 ðcontactÞ (24)

The panel strain energy variation is as follows:

dU ¼ZZ

U0

�dnεð01Þ

oTfNg þ dnεð1Þ

oTfMg�dxdy (25)

By substituting Eqs. (9), (18) and (19) for RRM discretization intoEq. (25), we have:

dU ¼ dnbUoT ½KT�

nbUo½KT� ¼ ½Kmm� þ ½Kmb� þ ½Kbm� þ ½Kbb�

(26)

where:

½Kmm� ¼ZZ

U0

�½CM�T ½A� ½CM�

�dxdy (27)

½Kmb� ¼ ½Kbm�T ¼ZZ

U0

�½CM�T ½B� ½CB�

�dxdy (28)

½Kbb� ¼ZZ

U0

�½CB�T ½D� ½CB�

�dxdy (29)

The potential energy variation of the panel, resulting from in-plane forces, {N}¼ {NxNyNxy}T, displaced through the mid-planevirtual movement, suffering from GreeneLagrange strains,including bucking theory of moderate rotations, are rewritten asfollows:

dV ¼ �ZZ

U0

�dnεð02Þ

oTfNg�dx dy ¼ �d

nbUoT ½KG�nbUo

(30)

In the above equation, [KG], the geometric stiffness matrix isexpressed as follows:

½KG� ¼ZZ

U0

�½H�T ½N0� ½H�

�dxdy (31)

where [N0] is amatrix resulting from the in-plane forces, as follows:

½N0� ¼�Nx NxyNxy Ny

�(32)

The [H] matrix for a panel discretized by RRM is written asfollows:

½H� ¼

266640 0

vNw

vx

0 0vNw

vy

37775 (33)

Regarding the principle of minimum total potential energy:

dP ¼ dU þ dUe þ dV ¼ 0 (34)

By substituting Eqs. (22), (26) and (30) in Eq. (34), we have:

ð½KT� þ ½Ke� � l½KG�ÞnbUo

¼ 0 (35)

The system of Eq. (35) leads to an eigenvalue system of equa-tions. After applying the boundary conditions, the minimumeigenvalue l can be obtained. This value is the buckling critical loadcoefficient and the corresponding eigenvector is the first mode ofbuckling. The solution procedure is complicated as the final contactregion between the curved panel and the foundation surface is notknown a priori. The bi-modulus nature of the foundation intro-duces a high non-linearity to the system behavior and an iterativeprocedure, as set out in the flowchart of Fig. 2, is therefore neces-sary to calculate the contact region and solve the non-linear contactproblem. The problem is solved numerically using a developedalgorithm. Its results bi-lateral (without any foundation involved)and uni-lateral (with tensionless foundation) deformation, areshown in Fig. 3, as input mode and output mode, respectively.

5. Results and discussion

In this section, in the beginning, the buckling analysis results forthe special case of no foundation (bi-lateral buckling of laminatedcomposite flat and curved panels) are compared with those avail-able in the literature for the same case, starting the simplest form ofa validation process. Then, the results for the second case of uni-lateral buckling of isotropic and homogeneous flat panel restingon tensionless foundation are compared with earlier works againavailable in the literature in order to continue the validationprocess.

In the sequel, a number of examples are offered for demon-strating the effect of uni-lateral constraint on the buckling behaviorof laminated composite cylindrical panels. Eventually, for bothcases of bi-lateral and uni-lateral buckling, the influence ofparameters such as panel aspect ratio, panel thickness, curved edgecentral angle, the number of laminates, lamination scheme, angle ofplies, and foundation modulus are investigated on the bucklingload. It could be worth mentioning that the bi-lateral bucklinganalysis might be obtained by assigning a zero value to the foun-dation modulus.

By virtue of validating the accuracy of the present method andinvestigating the algorithm results for calculating buckling load forboth cases, a cylindrical panel having the geometry shown in Fig. 1is designated. The simply supported boundary conditions (SSSS) areassumed as follows:

v0 ¼ w0 ¼ 0 at x ¼ 0 and a

u0 ¼ w0 ¼ 0 at y ¼ 0 and bð¼ R� aÞ

The material and geometric properties, whenever notmentioned specifically, are as follows.

Fig. 2. A flowchart for nonlinear solution of uni-lateral bucking algorithm.


E1=E2 ¼ 25; G12=E2 ¼ 0:5; y12 ¼ 0:25h ¼ 10�2ðmÞ; b ¼ 1

In addition, it is assumed the plies have the same thickness.

5.1. Validation for bi-lateral analysis

In Table 2, the dimensionless buckling loads of orthotropic flatpanels in the case of no foundation included are compared with the

Fig. 3. (a) Bi-lateral deformations for the case of no foundation involved as inputmodes, and (b) uni-lateral deformations for the case of a tensionless foundation asoutput modes.

work of Reddy and Phan (1985). In this table, the effects of panelaspect ratio, and Young’s moduli ratio are investigated on thebuckling load of a symmetric cross-ply laminated panel withlamination scheme of [0/90/90/0], subject to uniform compressiveaxial load along x-direction. As it can be seen from Table 2, theresults obtained in this work, based on RRM, are in good agreementwith the work of Reddy and Phan (1985).

In Table 3, the bi-lateral buckling loads of a symmetric cross-plycurved panel, having two types of [0/90] and [0/90/90/0] as stack-ing sequences, with aspect ratio of a/b¼ 1, width-to-thicknessratios of b/h¼ 10,100, and radius-to-thickness ratios of R/b¼ 1,5,under a uniform compressive force along y-direction, in the case ofno foundation involved, are contrasted with the work of Singh et al.(2001). The corresponding panel buckling modes are designated by(m,n) describing m as the number of half-waves along longitudinaland n as the number of half-waves along circumferential directions.Another comparison is performed in Table 4, for bi-lateral bucklingloads of a symmetric cross-ply curved panel with stacking sequenceof [0/90/0] and panel aspect ratio of a/b¼ 1, under uniformcompressive axial load along x-direction, in the case of no foun-dation included, with the work of Carrera (1991). As it may beobserved from Tables 2e4, the results obtained from the presentsolution method of RRM could be considered to have enough andsatisfactory performance and accuracy.

5.2. Uni-lateral buckling analysis

As mentioned in the beginning of this section, due to beingnonlinear the system of equations, the numerical solution of uni-lateral buckling load is an iterative scheme; see Fig. 1. For startingthe solution process, we initialize the algorithm by calculatingpanel bi-lateral buckling load as an initial guess for possibledeformations of uni-lateral buckling mode, i.e., using the resultsof a no foundation buckling. Paying attention to the response of bi-lateral buckling, the contact area between the panel and foundationcan be determined. With due attention to this area, we can obtainthe corresponding mode arising from a uni-lateral buckling defor-mation. Then, the bi-lateral and uni-lateral modes are comparedwith one another. This process is repeated until a convergencecriterion is satisfied. In this away, we have calculated the uni-lateralbuckling load and mode of deformation.

5.2.1. Validation uni-lateral analysisAs explained earlier, the foundation is discretized by a 2D

network of discrete springs. The stiffness of springs, kf in [N/m], asa variable parameter, would change from 0 (no foundation) to 1010

[N/m] (rigid foundation). With the attention to the convergenceinvestigations for reaching on appropriate model for this type offoundation, the intensity of spring distributions in both durationsof longitudinal and circumferential would be a minimum numberof 40 springs per unit length (1 m), that would be compatible withconvergence studies, conducted by Smith et al. (1999c) for a flatplate. If would be emphasized that the degree of spring stiffness(the foundation modulus) could determine the type of foundation,i.e., elastic, rigid or something in-between. For a convenience indesignating the foundation modulus, we have implemented thelogarithm of kf as a parameter of foundation, i.e., b¼ log10kf.

In Table 5, the dimensionless buckling load coefficients of a flatpanel for both cases of no foundation involved ðNBil

cr Þ, andtensionless foundation ðNUni

cr Þ, under uniform compressive loadalong x-direction, boundary conditions of SSSS-type, are comparedwith available results (Smith et al., 1999c), for a flat plate. In thisinvestigation, the foundation is b¼ 5. Furthermore, (d) is thebending stiffness as defined by(d¼ Eh3/(12(1� y2))). It is observed

Table 2The comparison of non-dimensional bi-lateral buckling load, Ncr ¼ Ncrb2=p2D22, for a cross-ply laminated flat panel, under compressive in-plane load along x direction, withmaterial properties E1/E2¼ varied, G12/E2¼ 0.5, y12¼ 0.25, and geometric characteristic b/h¼ 1000, with the work of Reddy and Phan (1985).

a/b Four-layer symmetric laminate [0/90/90/0]

E1/E2¼ 5 E1/E2¼ 10 E1/E2¼ 20 E1/E2¼ 25 E1/E2¼ 40

RRM Reddy RRM Reddy RRM Reddy RRM Reddy RRM Reddy

0.5 13.90 13.90 18.1265 18.126 21.877 21.87 22.8738 22.874 24.589 24.591 5.650 5.650 6.3471 6.347 6.9611 6.961 7.1238 7.124 7.4037 7.4041.5 5.233 5.233 5.2768 5.277 5.3099 5.310 5.3182 5.318 5.3322 5.332

Table 3The comparison of non-dimensional bi-lateral buckling load, N

ycr ¼ Ny

crb2=E2h3, fora cross-ply laminated curved panel, under compressive in-plane load along ydirection, with material properties E1/E2¼ 40, G12/E2¼ 0.6, y12¼ 0.25, and geometriccharacteristic a/b¼ 1, with the work of Singh et al. (2001).

R/b b/h Four-layer symmetriclaminate [0/90/90/0]

Two-layer anti-symmetriclaminate [0/90]

RRM Singh RRM Singh

1 10 e e 18.70471 (1,1) 18.705 10 28.99114 (1,2) 28.99127 13.18720 (1,1) 13.187265 100 30.33037 (1,2) 30.33047 26.46004 (1,2) 26.46009


from Table 5 that the results of present solution have enoughaccuracy for further development.

5.2.2. Parametric studies5.2.2.1. The effects of aspect ratio. Tables 6e8 illustrate the influ-ence of increasing foundation modulus, on the buckling loads andtheir variations with panel aspect ratio and foundation modulus. Tothis end, laminated symmetric cross-ply panels with stackingsequence of [0/90/90/0], central angles of curved edge in the rangeof 15

�, 30

�, and 45

�, and a width-to-thickness ratio of b/h¼ 100, are

assumed. The panel aspect ratios are varied from a/b¼ 1 to a/b¼ 4.Regarding Table 6, for the investigated aspect ratios a/b up to 2, theuni-lateral foundation has no effect on the buckling load. Becausein this interval of a/b, the panel bi-lateral buckling mode is(m, n)¼ (1, 1); i.e., one half-wave along each longitudinal andcircumferential direction. Since the panel does not have anytendency to penetrate into the stiff foundation, the percentage ofincrease in the uni-lateral buckling load with respect to bi-lateralone is zero, i.e., inc¼ 0. As far as the panel aspect ratio gets

Table 4The comparison of non-dimensional bi-lateral buckling load, Ncr ¼ Ncrb2=p2D22,for a cross-ply laminated curved panel, under compressive in-plane load along xdirection, with material properties E1/E2¼ 25, G12/E2¼ 0.5, y12¼ 0.25, and geometriccharacteristic a/b¼ 1, with the work of Carrera (1991).

a/R R/h Three-layer symmetric laminate [0/90/0]

Carrera

RRM Love’s CLT Donnell’s CLT Flugge’s CLT

b/h¼ 101 10 28.1721 (1,1) 27.69 (1,1) 28.17 (1,1) 27.67 (1,1)0.5 20 24.6642 (1,1) 24.54 (1,1) 24.66 (1,1) 24.54 (1,1)0.1 100 23.5417 (1,1) 23.54 (1,1) 23.54 (1,1) 23.54 (1,1)b/h¼ 201 20 42.2034 (1,1) 41.72 (1,1) 42.20 (1,1) 41.72 (1,1)0.4 50 26.4883 (1,1) 26.41 (1,1) 26.49 (1,1) 26.41 (1,1)0.2 100 24.2433 (1,1) 24.22 (1,1) 24.22 (1,1) 24.22 (1,1)b/h¼ 505 10 399.728 (2,4) 353.5 (2,4) 387.3 (2,4) 351.4 (2,4)1 50 81.0577 (1,2) 79.62 (1,2) 81.06 (1,1) 79.61 (1,2)0.5 100 52.7269 (1,1) 52.62 (1,1) 52.74 (1,1) 52.62 (1,1)b/h¼ 1001 100 165.1461(1,2) 163.7 (1,2) 165.2 (1,2) 163.7 (1,2)

greater than 2, specifically a/b¼ 3, the resulting mode from bi-lateral buckling becomes the mode corresponding to (2,1) half-waves. Therefore, the panel tends to penetrate into the founda-tion. However, the foundationwould withstand against those partsof the panel with the tendency of penetration into the foundation.Thus, the foundation would make stiffer those parts of the panel.This added stiffness could enhance the buckling load. As it isobserved from Table 6, for a/b¼ 3, with an increase in the foun-dation modulus, uni-lateral buckling load would increase. Animportant conclusion drawn from Tables 6e8 could be that whenthe modulus of foundation would be less than 1 (b� 1), the foun-dation would have no much influence on the panel deformationsand dimensionless buckling load. Thus, the panel would deformfreely subject to uniform compressive axial load and would pene-trate into the foundation easily. With an increase in b, the foun-dation would get stiffer. Accordingly, the foundation wouldwithstand more than before, against the panel deformations andwould not allow the panel to penetrate into it easily. Accordingly,the foundation would make the panel to buckle in a higher mode.As it might be observed in Tables 6e8, with the presence of thefoundation, dimensionless buckling load would increase, and withincrease in b up to a specific value of 8, more increase in thepercentage of buckling load increment could be observed.However, from this specific value of 8 on, the foundation could nothave noticeable influence on the buckling load and correspondingmode anymore. In other words, the foundation stiffness relative tothe panel stiffness would have increased so much that moreincrease could not have any more influence on the panel bucklingload. Accordingly, for the laminated composite curved panel withconstituent material under study, a rigid foundation would bemodeled with a modulus greater than 8 and an elastic foundationwith a modulus less than 8.

Studying Tables 7 and 8, it could be observed that for foundationmodulus greater than 4 (b> 4), with an increase in the aspect ratiothe value of inc (the percentage of the difference between uni-lateral and bi-lateral buckling load relative to bi-lateral bucklingload) would decrease. Further, for the rigid foundation with

Table 5The comparison of the non-dimensional uni-lateral and bi-lateral buckling loadcoefficients, Ncr ¼ Ncrb2=p2D22, of a flat panel with a plate under uniformlydistributed load along x direction, with the work of Smith et al. (1999c).

a/b NBilcr Buckling

ModeNUnicr Buckling

ModePresentstudy RRM

Smith Presentstudy RRM

Smith

1 4.00 4.00 4.00 4.00

2 4.00 4.00 4.51 4.51

3 4.00 4.00 4.54 4.51

Table 7Effect of a foundation modulus increase on the non-dimensional uni-lateral buckling load and variation of these two parameters with respect to aspect ratio, for a four-layersymmetric laminated panel, [0/90/90/90], with b/h¼ 100, a¼ 30

�.

a/b¼ 1 a/b¼ 1.5 a/b¼ 2 a/b¼ 3

b

Bilateral

Ncr inc % Ncr inc % Ncr inc % Ncr inc %

109.0291 (1,2) 101.6817 (2,2) 104.4582 (3,2) 102.0783 (3,2)

1 109.0312 0.0019 101.6834 0.0017 104.46 0.0017 104.9504 0.00312 109.2311 0.1853 101.8512 0.1667 104.6343 0.1686 105.258 0.29623 110.9828 1.7919 103.2542 1.5465 105.9482 1.4264 107.1386 2.08824 122.771 12.6039 111.0238 9.1876 111.3736 6.6203 112.7572 7.44195 135.9922 24.7302 120.6271 18.6321 117.3875 12.3775 118.0033 12.44076 140.6309 28.9847 123.3221 21.2825 119.5423 14.4403 119.6826 14.04097 141.5362 29.8151 123.9439 21.894 120.1706 15.0418 120.1885 14.52298 e 27.4465 124.1194 22.0666 120.3698 15.2325 120.3663 14.69249 e 27.4729 124.1815 22.1277 120.4351 15.295 120.4479 14.770110 e e 124.2443 22.1894 e e e e

Table 6Effect of a foundation modulus increase on the non-dimensional uni-lateral buckling load and variation of these two parameters with respect to aspect ratio, for a four-layersymmetric laminated panel, [0/90/90/90], with b/h¼ 100, a¼ 15

�.

a/b¼ 1 a/b¼ 1.5 a/b¼ 2 a/b¼ 3

b

Bilateral


55.7862 (1,1) 49.0502 (1,1) 49.6753 (1,1) 50.8794 (2,1)

1 55.7862 0 49.0502 0 49.6753 0 49.0638 0.02772 55.7862 0 49.0502 0 49.6753 0 50.0182 1.97353 55.7862 0 49.0502 0 49.6753 0 51.5667 5.13054 55.7862 0 49.0502 0 49.6753 0 52.0128 6.03995 55.7862 0 49.0502 0 49.6753 0 52.0858 6.18886 55.7862 0 49.0502 0 49.6753 0 52.1013 6.22047 55.7862 0 49.0502 0 49.6753 0 52.1087 6.23548 55.7862 0 49.0502 0 49.6753 0 52.1089 6.23599 55.7862 0 49.0502 0 49.6753 0 52.1131 6.244410 55.7862 0 49.0502 0 49.6753 0 e e


a modulus of 8, the percent of increase in buckling load, in average,for different aspect ratios under study, and for central angles 15

�,

30�, and45

�are respectively, 6.2%, 20%, and 39%. These numerals

would give us an approximate viewwith respect to the influence offoundation on the buckling load of panel with assumed geometricproperties. However, it would be necessary to notice that fora panel with different geometric properties, a different number ofplies, stacking sequence, and different angles of plies, we wouldobserve a different trend in our results. It is necessary to emphasizethat in some examples, for a high modulus, a numerical instabilitymight occur.

Figs. 4e9 would depict the effect of the panel aspect ratio onthe buckling load, for the foundationmodulus of b¼ 4, respectively,

Table 8Effect of a foundation modulus increase on the non-dimensional uni-lateral buckling loadsymmetric laminated panel, [0/90/90/90], with b/h¼ 100, a¼ 45

�.

a/b¼ 1 a/b¼ 1.5

b

Bilateral

Ncr inc % Ncr inc %

146.39 (1,2) 141.0685 (2,2)

1 146.3948 0.0014 141.0702 0.00122 146.5952 0.14 141.2390 0.123 148.3957 1.36 142.7347 1.184 164.3427 12.26 154.5932 9.585 192.0741 31.20 184.5094 30.796 213.4026 45.77 197.4123 39.947 220.2260 50.43 206.7402 46.558 221.5279 51.32 208.0733 47.499 221.8432 51.53 208.5534 47.8310 221.9893 51.63 e e

[0/90/0] and [0/90/90/0] stacking sequences, and for twosymmetric cross-ply laminated panels with curved edges havingcentral angles of 15

�, 30

�and 45

�. The region 1� a/b� 4 would have

been divided into sub-regions that would have the same bucklingmodes. A noticeable point in Figs. 4 and 5 is that in the interval ofmode (1,1), the influence of uni-lateral constraint on buckling loadis zero, because in this interval, the panel has no tendency topenetrate into the foundation. However, with an increase in panelaspect ratio and consequently, a change from mode (1,1) to (2,1),and further, due to the onset of penetration of the panel intofoundation, the effect of uni-lateral would become visible. Anincrease in the uni-lateral buckling load with respect to bi-lateralbuckling load, in average, is about 1%, 2%, 12%, 8%, 17% and 10%,

and variation of these two parameters with respect to aspect ratio, for a four-layer

a/b¼ 2 a/b¼ 3

Ncr inc % Ncr inc %

144.3339 (3,2) 144.2885 (4,2)

144.3357 0.0012 144.2919 0.0024144.5129 0.12 144.6146 0.22146.0380 1.17 147.1217 1.96156.1914 8.21 158.9916 10.19177.2988 22.83 175.1676 21.40187.7528 30.08 180.5452 25.12190.9858 32.32 182.4413 26.44191.9672 33.00 183.0164 26.84192.2874 33.22 e e

e e e e

Fig. 4. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buckling loads of a three-layer symmetric laminated panel, [0/90/0], with (a¼ 15�, b/h¼ 100).

Fig. 5. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buckling loads of a four-layer symmetric laminated panel, [0/90/90/0], with (a¼ 15�, b/h¼ 100).



Fig. 7. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buckling loads of a four- layer symmetric laminated panel, [0/90/90/0], with (a¼ 30�, b/h¼ 100).


Fig. 9. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buckling loads of a four-layer symmetric laminated panel, [0/90/90/90], with (a¼ 45�, b/h¼ 100).


Fig. 10. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buckling loads of a three- layer symmetric laminated panel, [0/90/0], with b/h¼ 100, andcentral angles a¼ 0

�, 15

�.


respectively, for Figs. 4e9. A comparison of these figures wouldshow us that the influence of uni-lateral constraint on the three-layer panel would be greater, as compared to the four-layer panelwith the above-mentioned stacking sequences, with an exceptionof the panel with a central angle 15

�. Furthermore, it is seen from

Figs. 4e9 that increasing the surface of panel caused by increasingthe aspect ratio, it does not necessarily lead to an increase in thecontact region between panel and foundation; and as result ofbuckling critical load. Different patterns in variations of bucklingcritical load have been observed.

5.2.2.2. The effect of central angle. In Figs. 10 and 11, symmetriccross-ply laminated composite panels with stacking sequence of [0/90/0] at different central angles and for two cases of a foundationwith modulus of b¼ 4 and bi-lateral foundation would have beencompared. In Fig. 10, two panels with central angles of a¼ 0

�, and

15�, and in Fig. 11, two panels with central angles of a¼ 15

�, and 30

�

would have been contrasted. As it could be seen from these twofigures, with an increase in the central angle from 0

�to 15

�and then

from 15�to 30

�, dimensionless buckling load would increase

considerably. This would confirm that increase in the curvature of

Fig. 11. Influence of the aspect ratio on the non-dimensional uni-lateral and bi-lateral buccentral angles a¼ 15

�, 30

�.

panel could stiffen the structure against buckling phenomenon anddelay its occurrence. Accordingly, ignoring the curvature of panels,which might rest on other structures (playing the role of founda-tion), might cause an underestimation of buckling load in bothcases of uni-lateral and bi-lateral constraints. Another noticeablepoint would be the degree of increase in buckling load, due to a uni-lateral constraint. In those intervals of the above-mentioned figureswhose modes would not be (1,1), the average of increase in thebuckling loads for central angles of 0

�, 15

�, and 30

�would be

respectively, 20 %, 5 %, and 15 %. In other words, in the beginning,with the increase in the central angle up to 15

�, the effect of uni-

lateral constraint on the buckling load would be a decrease, andthen, after increasing the central angle from 15

�on, its effect would

change to an increase in the buckling load. From a quantitativepoint of view, Table 9 would investigate the effect of central angleon the buckling load for the different foundation moduli. To thisend, a symmetric cross-ply laminated composite panel withstacking sequence [0/90/90/0] and central angles 0

�to 45

�, panel

aspect ratio a/b¼ 2.5, and width-to-thickness ratio b/h¼ 100, fordifferent foundation moduli would have been investigated. As itcould be observed from Table 9, with an increase in the central

kling loads of a three-layer symmetric laminated panel, [0/90/0], with b/h¼ 100 and

Table 9Effect of a foundation modulus increase on the non-dimensional uni-lateral buckling load and variation of these two parameters with central angle, for a four-layer symmetriclaminated panel, [0/90/90/90], with a/b¼ 2.5, b/h¼ 100.

a¼ 0�

a¼ 15�

a¼ 30�

a¼ 45�

b

Bilateral


18.8199 (2,1) 50.8794 (2,1) 102.0783 (3,2)

1 18.8277 0.0414 50.8872 0.0153 102.0818 0.0034 140.8679 0.00252 19.408 3.1249 51.3031 0.8328 102.4153 0.3301 141.212 0.24683 20.6304 9.6201 51.765 1.7406 104.6318 2.5015 144.0218 2.24144 21.2357 12.8364 51.9557 2.1154 111.2109 8.9467 158.7433 12.69235 21.3699 13.5495 52.0122 2.2264 115.9809 13.6195 178.1185 26.44686 21.3935 13.6749 52.0273 2.2561 117.4865 15.0945 185.1519 31.43987 21.3996 13.7073 52.0322 2.2657 117.9612 15.5595 187.3237 32.98168 21.4031 13.7259 52.0382 2.2775 118.1164 15.7116 187.9901 33.45479 21.4248 13.8412 52.1374 2.4725 118.1686 15.7627 e e

10 21.4687 14.0745 52.2859 2.7644 e e e e


angle from 0�to 15

�, the effect of uni-lateral constraint on the

buckling load would decrease and after that angle, the effect wouldhave a considerable increase.

5.2.2.3. The effect of thickness. In Table 10, the influence of thick-ness increment on dimensionless buckling loads and modes forsymmetric cross-ply laminated panels with lamination scheme of[0/90/0], and the central angle of 45

�, and dimensions a¼ b¼ 1, for

different foundation moduli, have been investigated. As it would beseen from Table 10, with an increase in the panel thickness, thedimensionless buckling load would decrease. With a notice to thedimensionless buckling load formula, which would have an inverserelation with the cube of thickness, as a whole, the buckling loaditself would increase. This means that with an increase in the panelthickness, the onset of buckling would be delayed. The importantand noteworthy point is that with an increase in the panel thick-ness, the influence of foundation on the buckling load woulddecrease, i.e., a decrease in the increasing rate of uni-lateral buck-ling load. In other words, it would be claimed that the foundationcould have more influence on the buckling behavior of thinnerpanels.

5.2.2.4. The effect of number of plies and lamination scheme.In Table 11, the effect of number of plies and lamination schemes onthe dimensionless buckling load of symmetric cross-ply laminatedpanels have been investigated. To this end, panels with centralangle of 45

�, aspect ratio of a/b¼ 1, and width-to-thickness ratio of

b/h¼ 100 have been considered. In a comparison of three-layercomposite panels of [0/90/0] and [90/0/90], the dimensionless bi-

Table 10Influence of the thickness increase on the non-dimensional uni-lateral buckling loadand variation of these two parameters with foundation modulus, for a three-layersymmetric laminated panel, [0/90/90], with a¼ b¼ 1, a¼ 45

�.

b

Bilateral

Three-layer symmetric laminate [0/90/0]

h ¼ 0.01 h ¼ 0.02 h ¼ 0.1

Ncr inc % Ncr inc % Ncr inc %

122.188 (1,2) 70.3182 (1,2) 26.3800 (1,1)

1 122.2084 0.016 70.3207 0.003 26.3800 02 122.3905 0.165 70.3435 0.036 26.3800 03 124.1837 1.633 70.5695 0.357 26.3800 04 139.4477 14.12 72.6533 3.320 26.3800 05 179.7819 47.13 81.8013 16.33 26.3800 06 195.8777 60.30 86.4757 22.97 26.3800 07 201.2142 64.67 87.1809 23.98 26.3800 08 201.5262 64.93 87.2962 24.14 26.3800 09 201.8527 65.19 87.318 24.17 26.3800 010 201.8423 65.18 87.3213 24.18 26.3800 0

lateral buckling load of the panel with the latter laminationscheme relative to the panel with the former lamination scheme isgreater. However, the result for the dimensionless uni-lateralbuckling load of the panels would be reversed. Consequently, thepercentage of increase in the dimensionless buckling load, for thepanel with the former lamination is greater relative to the panelwith the latter lamination scheme. This could show us that theinfluence of a uni-lateral constraint for the panel with the formerlamination relative to the latter would be greater. In constant, forthe four-layer composite panels, the dimensionless buckling loadsare greater for both laminations of [0/90/90/0] and [90/0/0/90] andfor both cases of uni-lateral and bi-lateral constraints. Furthermore,for the former lamination, there would be more increase in thepercentage of increase in the buckling load. A comparison of twopanels of [0/90/0] and [0/90/90/0] schemes would show us that forboth cases of uni-lateral and bi-lateral, the dimensionless bucklingload of the latter lamination would be greater, as compared withthe panel having the former lamination. However, the uni-lateralconstraint would have more influence on the panel with theformer lamination relative to the panel with the latter lamination;because there would be more percentage of increase in the buck-ling load of the former lamination as contracted with the latter.However, in a comparison of [90/0/90] and [90/0/0/90] laminationschemes, it would be seen that the uni-lateral constraint wouldhave more influence on the panel with the latter lamination rela-tive to the panel with the former lamination. Accordingly, it couldbe concluded that with an appropriate choice of lamination schemeand the number of plies, it could be possible to channelize, direct,and delay the occurrence of buckling at the favorite of the structuraldesigner.

5.2.2.5. The effect of orthotropy. In Fig.12, the effects of variations ofangle of plies of a symmetric angle-ply laminated panel with two

Table 11Influence of the number of layer and lamination in constant thickness on the non-dimensional uni-lateral buckling load, for a symmetric laminated panel, with a/b¼ 1, b/h¼ 100,a¼ 45

�.

Lamination NBilcr Elastic foundation

(b¼ 4)Rigid foundation(b¼ 8)

NUnicr inc % N

Unicr inc %

[0]3 or [0]4 107.4110 (1,2) 122.0466 13.6258 153.7185 43.1124[90]3 or [90]4 108.6551 (4,2) 109.7989 1.0527 137.1477 26.2230[0/90/0] 122.1882 (1,2) 139.4477 14.1253 201.5262 64.9310[90/0/90] 124.1717 (4,2) 125.3437 0.9439 164.6581 32.6052[0/90/90/0] 146.3928 (1,2) 164.3427 12.2615 221.5279 51.3244[90/0/0/90] 141.1049 (3,2) 143.2263 1.5034 191.8667 35.9745

Fig. 12. Influence of the orientation of fibers on the non-dimensional uni-lateral and bi-lateral buckling loads of a panel with two stacking sequences [q/�q/q] and [q/�q/�q/q], (a/b¼ 1,b/h¼ 100, a¼ 15

�).


different lamination schemes of [q/�q/q]and [q/�q/�q/q], andaspect ratio a/b¼ 1, width-to-thickness ratio of b/h¼ 100, andcentral angle of a¼ 15

�have been investigated on the uni-lateral

and bi-lateral dimensionless buckling loads. Due to the variationsin ply angles and panel curvature in some intervals of the above-mentioned plot, the corresponding buckling mode would not beclearly detectable. As a result, a mode partitioning has not beenaccomplished. However, in some partitions of aspect ratios wherethe uni-lateral and bi-lateral buckling load curves would trace thesame path, we would be sure that the (1,1) mode would haveoccurred and in the other partitions, modes other than (1,1) haveoccurred. Noticing Fig. 12, when the angle of plies for the two typesof lamination schemes, i.e., q would be about 40

�, the uni-lateral

buckling load would be a minimum.

Hh1ðhÞ ¼ 0:5ð1þ hÞ; Hx1ðxÞ ¼ 0:5ð1þ xÞ;Hh2ðhÞ ¼ 0:5ð1� hÞ; Hx2ðxÞ ¼ 0:5ð1� xÞ;Hh3ðhÞ ¼ 1� h2; Hx3ðxÞ ¼ 1� x2;

Hh4ðhÞ ¼ h� h3; Hx4ðxÞ ¼ x� x3;

Hh5ðhÞ ¼ 1� 10h2 þ 9h4; Hx5ðxÞ ¼ 1� 10x2 þ 9x4;Hh6ðhÞ ¼ h� 5h3 þ 4h5; Hx6ðxÞ ¼ x� 5x3 þ 4x5;Hh7ðhÞ ¼ 1� 4h2 þ 9h4 � 6h6; Hx7ðxÞ ¼ 1� 4x2 þ 9x4 � 6x6;Hh8ðhÞ ¼ h� 3h3 þ 7h5 � 5h7; Hx8ðxÞ ¼ x� 3x3 þ 7x5 � 5x7;Hh9ðhÞ ¼ 1� 5h2 þ 6h4 � 9h6 þ 7h8; Hx9ðxÞ ¼ 1� 5x2 þ 6x4 � 9x6 þ 7x8;Hh10ðhÞ ¼ h� 4h3 þ 6h5 � 10h7 þ 7h9: Hx10ðxÞ ¼ x� 4x3 þ 6x5 � 10x7 þ 7x9:

6. Conclusions

In the present article, nonlinear behavior of laminatedcomposite panels resting on tensionless foundation, subjected tocompressive axial forces, using RayleigheRitz method and based onclassical shell theory was investigated. The effects of parameterssuch as aspect ratio, thickness, the central angle, and the founda-tion modulus, on the uni-lateral buckling load were investigated.For the panels with the specific characteristics under study, thefollowing observations were obtained that follow:

(1) A panel would experience uni-lateral buckling that the numberof half-waves, at minimum along one of the two directions oflongitudinal and circumferential, would be more than one. Asa result, the foundation would withstand against penetratingthe panel and this reaction would cause stiffening in someparts of the panel. This added stiffness would make an increasein the buckling load.

(2) The results showed that, increasing the surface of panel causedby increasing the panel aspect ratio, it does not necessarily leadto an increase in the contact region between panel and foun-dation; and as result of buckling critical load. Different patternsin variation of buckling critical load have been observed.

(3) Modeling curved panels with flat plates would cause thatbuckling load would be underestimated. Accordingly, it wouldbe recommended that in deriving the governing equations of

buckling analysis of curved panels, the curvature would not beignored.

(4) With an increase in the central angle from 0�to 15

�, the effect of

uni-lateral constraint on the buckling load would decrease, butfrom 15

�on this angle wouldmake considerable increase in the

buckling load.(5) An increase in the panel thickness would cause a decrease in

the effect of uni-lateral constraint on the buckling load(6) By a knowledgeable select of the number of laminates, stacking

sequence, and angle of plies in a constant-thickness it would bepractical to direct the uni-lateral buckling load in a desired wayand to maximize the load-carrying capacity of the panel.

(7) Generally, the influence of foundation on the uni-lateralbuckling load would be highly dependent on influentialparameters, such as central angle, aspect ratio, thickness, andfoundation stiffness. Because a variation in each geometricparameter would change the corresponding buckling mode;subsequently, a change in mode could have a remarkableinfluence on the degree of penetration of panel into thefoundation.

Appendix

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nonlinear buckling analysis of laminated composite curved panels constrained by winkler tensionless...

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