Buckling Behavior of Long Symmetrically Laminated Plates ...mln/ltrs-pdfs/NASA-97-tp3659.pdf · A parametric study of the buckling behavior of infinitely long symmetrically lam-inated

National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-2199

NASA Technical Paper 3659

Buckling Behavior of Long SymmetricallyLaminated Plates Subjected to Shear andLinearly Varying Axial Edge LoadsMichael P. NemethLangley Research Center • Hampton, Virginia

July 1997

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Abstract

A parametric study of the buckling behavior of infinitely long symmetrically lam-inated anisotropic plates that are subjected to linearly varying edge loads, uniformshear loads, or combinations of these loads is presented. The study focuses on theeffects of the shape of linearly varying edge load distribution, plate orthotropy, andplate flexural anisotropy on plate buckling behavior. In addition, the study examinesthe interaction of linearly varying edge loads and uniform shear loads with plateflexural anisotropy and orthotropy. Results obtained by using a special purpose non-dimensional analysis that is well suited for parametric studies of clamped and simplysupported plates are presented for[±θ]s thin graphite-epoxy laminates that are repre-sentative of spacecraft structural components. Also, numerous generic buckling-design charts are presented for a wide range of nondimensional parameters that areapplicable to a broad class of laminate constructions. These charts show explicitly theeffects of flexural orthotropy and flexural anisotropy on plate buckling behavior forlinearly varying edge loads, uniform shear loads, or combinations of these loads. Themost important finding of the present study is that specially orthotropic and flexurallyanisotropic plates that are subjected to an axial edge load distribution that is tensiondominated can support shear loads that are larger in magnitude than the shear buck-ling load.

Introduction

Buckling behavior of laminated plates that are sub-jected to combined loads is an important consideration inthe preliminary design of aircraft and launch vehicles.The sizing of many structural subcomponents of thesevehicles is often determined by stability constraints. Onesubcomponent that is of practical importance in struc-tural design is the long rectangular plate. These platescommonly appear as subcomponents of stiffened panelsused for wing structures and as semimonocoque shellsegments used for fuselage and launch vehicle structures.Buckling results for infinitely long plates are importantbecause they often provide a useful conservative estimateof the behavior of finite-length rectangular plates, andthey provide information that is useful in explaining thebehavior of these finite-length plates. Moreover, knowl-edge of the behavior of infinitely long plates can provideinsight into the buckling behavior of more complexstructures such as stiffened panels.

An important type of long plate that appears as asubcomponent of advanced composite structures is thesymmetrically laminated plate. In the present paper, theterm “symmetrically laminated” refers to plates in whichevery lamina above the plate midplane has a correspond-ing lamina located at the same distance below the platemidplane, with the same thickness, material properties,and fiber orientation. Symmetrically laminated platesremain flat during the manufacturing process and exhibitflat prebuckling deformation states. These characteristicsand the amenability of these plates to structural tailoringprovide symmetrically laminated plates with a significantpotential for reducing structural weight of aircraft andlaunch vehicles. Thus, understanding the buckling

behavior of symmetrically laminated plates is an impor-tant part of the search for ways to exploit plate orthotropyand anisotropy to reduce structural weight.

In many practical cases, symmetrically laminatedplates exhibit specially orthotropic behavior. However,in some cases, such as [±45]s laminates, these platesexhibit anisotropy in the form of material-induced cou-pling between pure bending and twisting deformations.This coupling is referred to herein as flexural anisotropy,and it generally yields buckling modes that are skewed inappearance. The effects of flexural orthotropy and flex-ural anisotropy on the buckling behavior of long rectan-gular plates that are subjected to single and combinedloading conditions are becoming better understood. Forexample, recent in-depth parametric studies that showthe effects of anisotropy on the buckling behavior of longplates that are subjected to compression, shear, pure in-plane bending, and various combinations of these loadshave been presented in references 1 through 5. Theresults presented in these references indicate that theimportance of flexural anisotropy on the buckling resis-tance of long plates varies with the magnitude and typeof the combined loading condition. However, none ofthese studies supply results for plates loaded by uniformshear and a general linear distribution of axial load acrossthe plate width. Both the uniform axial compression andthe pure in-plane bending loads are special cases of thegeneral linear distribution of axial edge loads. Results forthis class of loadings are useful in the design of aircraftspar webs and panels that are located off the neutral axisof a fuselage or launch vehicle that is subjected to overallbending and torsion loads. Moreover, the importance ofneglecting flexural anisotropy in a buckling analysis ispractically unknown for this class of loadings.

2

One objective of the present paper is to present buck-ling results for specially orthotropic plates that are sub-jected to uniform shear, a general linear distribution ofaxial load across the plate width, and combinations ofthese loads in terms of useful nondimensional designparameters. Other objectives are to identify the effects offlexural anisotropy on the buckling behavior of longsymmetrically laminated plates that are subjected to thesame loading conditions and to present some previouslyunknown results that show some unusual behavior.Results are presented for plates with the two long edgesclamped or simply supported and that are free to move intheir plane. Several generic buckling-design curves thatare applicable to a wide range of laminate constructionsare also presented in terms of the nondimensional param-eters described in references 1, 2, 5, and 6.

Symbols

Am, Bm displacement amplitudes (seeeq. (22)), in.

b plate width (see fig. 1), in.

D11, D12, D22, D66 orthotropic plate-bendingstiffnesses, in-lb

D16, D26 anisotropic plate-bendingstiffnesses, in-lb

E1, E2, G12 lamina moduli, psi

nondimensional buckling coeffi-cient associated with criticalvalue of an eccentric in-planebending load (see eq. (21) andfig. 1(a))

nondimensional buckling coeffi-cient associated with criticalvalue of a uniform shear load(see eq. (20) and fig. 1(a))

shear and in-plane bendingbuckling coefficients, defined byequations (20) and (21), respec-tively, in which anisotropy isneglected in the analysis

nondimensional buckling coeffi-cient associated with criticalvalue of a uniform axial com-pression load (see eq. (18) andfig. 1(a))

nondimensional buckling coeffi-cient associated with criticalvalue of a uniform transversecompression load (see eq. (19)and fig. 1(a))

L1, L2, L3, L4 nondimensional load factorsdefined by equations (14)through (17), respectively

nondimensional membranestress resultants of system ofdestabilizing loads defined byequations (10) through (13),respectivelynondimensional membranestress resultants of system ofsubcritical loads defined byequations (10) through (13),respectively

N number of terms in seriesrepresentation of out-of-planedisplacement field at buckling(see eq. (22))

Nb intensity of eccentric in-planebending load distributiondefined by equation (5), lb/in.

Nxc intensity of constant-valuedtension or compression loaddistribution defined byequation (5), lb/in.

Nx, Ny, Nxy longitudinal, transverse, andshear membrane stress result-ants, respectively (see eqs. (5),(7), and (8)), lb/in.

membrane stress resultants ofsystem of destabilizing loads(see eqs. (6) through (9)), lb/in.

membrane stress resultants ofsystem of subcritical loads (seeeqs. (6) through (9)), lb/in.

nondimensional loading parame-ter (see eqs. (14) through (17))and corresponding value atbuckling (see eqs. (18) through(21)), respectively

out-of-plane displacementfield at buckling defined byequation (22), in.

x, y plate rectangular coordinatesystem (see fig. 1), in.

nondimensional parametersdefined by equations (1), (2),(3), and (4), respectively

in-plane bending load distribu-tion parameters (see fig. 1 andeq. (5))

Kb nb1( )cr≡

Ks nxy1( )cr≡

Ks γ =δ=0Kb γ =δ=0

,

Kx nx1c( )cr≡

Ky ny1( )cr≡

nx1c

ny1 nxy1 nb1, , ,

nx2c

ny2 nxy2 nb2, , ,

Nx1c

Ny1 Nxy1 Nb1, , ,

Nx2c

Ny2 Nxy2 Nb2, , ,

p̃ p̃cr,

wN ξ η,( )

α∞ β γ δ, , ,

ε0 ε1,

3

nondimensional platecoordinates

θ fiber angle of a lamina (seefig. 1), deg

λ half-wavelength of bucklingmode (see fig. 1), in.

λ/b buckle aspect ratio (see fig. 1)

ν12 lamina major Poisson’s ratio

basis functions used to representbuckling mode (see eq. (22))

Analysis Description

In preparing generic design charts for buckling of asingle flat plate, a special purpose analysis is often pre-ferred over a general purpose analysis code, such as afinite element code, because of the cost and effort usuallyinvolved in generating a large number of results with ageneral purpose code. The results presented herein wereobtained by using such a special purpose analysis. Theanalysis details are lengthy; hence, only a brief descrip-tion of the analysis is presented.

Symmetrically laminated plates can have many dif-ferent constructions because of the wide variety of mate-rial systems, fiber orientations, and stacking sequencesthat can be used to construct a laminate. A way of copingwith the vast diversity of laminate constructions is to useconvenient nondimensional parameters. The bucklinganalysis used in the present paper is based on classicalplate theory and the classical Rayleigh-Ritz method andwas derived explicitly in terms of the nondimensionalparameters defined in references 1, 2, 5, and 6. Thisapproach was motivated by the need for generic (inde-pendent of laminate construction) parametric results forcomposite plate buckling behavior that are expressed interms of the minimum number of independent parame-ters needed to fully characterize the behavior and thatindicate the overall trends and sensitivity of the results tochanges in the parameters. The nondimensional parame-ters used in the present paper are given by

(1)

(2)

(3)

(4)

whereb is the plate width andλ is the half-wavelength ofthe buckle pattern of an infinitely long plate (see fig. 1).The subscriptedD-terms are the bending stiffnesses ofclassical laminated plate theory. The parameters andβ characterize the flexural orthotropy, and the parametersγ andδ characterize the flexural anisotropy.

The loading combinations included in the analysisare uniform transverse tension or compression, uniformshear, and a general linear distribution of axial loadacross the plate width, as depicted in figure 1. The longi-tudinal stress resultant is partitioned in the analysisinto a uniform tension or compression part and a linearlyvarying part corresponding to eccentric in-plane bendingloads. This partitioning is given by

(5)

where denotes the intensity of the constant-valuedtension or compression part of the load, and the termcontaining defines the intensity of the eccentric in-plane bending load distribution. The symbols anddefine the distribution of the in-plane bending load, andthe symbol is the nondimensional coordinate given by

(see fig. 1).

The analysis is based on a general formulation thatincludes combined destabilizing loads that are propor-tional to a positive-valued loading parameter that isincreased until buckling occurs and independent subcriti-cal combined loads that remain fixed at a specified loadlevel below the value of the buckling load. Herein, theterm “subcritical load” is defined as any load that doesnot cause buckling to occur. In practice, the subcriticalloads are applied to a plate prior to the destabilizingloads with an intensity below that which will cause theplate to buckle. Then, with the subcritical loads fixed, thedestabilizing loads are applied by increasing the magni-tude of the loading parameter until buckling occurs. Thisapproach permits certain types of combined load interac-tion to be investigated in a direct and convenient manner.

The distinction between the destabilizing and sub-critical loading systems is implemented in the bucklinganalysis by partitioning the prebuckling stress resultantsas follows:

(6)

(7)

(8)

η y/b, ξ x/λ==

Φm η( )

α∞bλ---

D11D22---------

1/4

=

βD12 2D66+

D11D22( )1/2

------------------------------=

γD16

D113

D22( )1/4

------------------------------=

δD26

D11D223( )

1/4------------------------------=

α∞

Nx

Nx Nxc Nb ε0 ε1 ε0–( )η+[ ]–=

Nxc

Nbε0 ε1

ηη y/b=

p̃

Nxc Nx1c

– Nx2c

+=

Ny Ny1– Ny2+=

Nxy Nxy1 Nxy2+=

4

(9)

where the stress resultants with the subscript 1 are thedestabilizing loads, and those with the subscript 2 are thesubcritical loads. The sign convention used herein forpositive values of these stress resultants is shown infigure 1. In particular, positive values of the general lin-ear edge stress distribution parametersand correspond to compression loading. Negativevalues of and or negative values of eitheror yield linearly varying stress distributions thatinclude tension. The two normal stress resultants of thesystem of destabilizing loads, and are definedas positive valued for compression loads. This conven-tion results in positive eigenvalues being used to indicateinstability caused by uniform compression loads.

The buckling analysis includes several nondimen-sional stress resultants associated with equations (6)through (9). These dimensionless stress resultants aregiven by

(10)

(11)

(12)

(13)

where the subscriptj takes on the values of 1 and 2. Inaddition, the destabilizing loads are expressed in terms ofthe loading parameter in the analysis by

(14)

(15)

(16)

(17)

whereL1 throughL4 are load factors that determine thespecific form (relative magnitude of the load compo-nents) of a given system of destabilizing loads. Typi-cally, the dominant load factor is assigned a value of 1,and all others are given as positive or negative fractions.

Nondimensional buckling coefficients used hereinare given by the values of the dimensionless stress result-ants of the system of destabilizing loads at the onset ofbuckling; that is,

(18)

(19)

(20)

(21)

where is the magnitude of the loading parameterat buckling. Positive values of the coefficients and

correspond to uniform compression loads, and thecoefficient corresponds to uniform positive shear.The direction of a positive shear stress resultant acting ona plate is shown in figure 1. The coefficient corre-sponds to the specific in-plane bending load distributiondefined by the selected values of the parameters and

The mathematical expression used in the variationalanalysis to represent the general off-center and skewedbuckle pattern is given by

(22)

where and are nondimensional coor-dinates, is the out-of-plane displacement field, and

and are the unknown displacement amplitudes.In accordance with the Rayleigh-Ritz method, the basisfunctions are required to satisfy the kinematicboundary conditions on the plate edges at and 1.For the simply supported plates, the basis functions usedin the analysis are given by

(23)

for values ofm = 1, 2, 3, ..., N. Similarly, for the clampedplates, the basis functions are given by

(24)

Nb Nb1 Nb2+=

Nb1, Nb2, ε0,ε1

Nb1 Nb2, ε0ε1

Nx1c

Ny1,

nxjc Nxj

cb

2

π2 D11D22( )1/2

------------------------------------=

nyjNyjb

2

π2D22---------------=

nxyjNxyjb

2

π2 D11D223( )

1/4------------------------------------=

nbjNbjb

2

π2 D11D22( )1/2

------------------------------------=

p̃

nx1c

L1 p̃=

ny1 L2 p̃=

nxy1 L3 p̃=

nb1 L4 p̃=

Kx nx1c( )cr≡

Nx1c( )crb

2

π2 D11D22( )1/2

------------------------------------ L1 p̃cr= =

Ky ny1( )cr≡Ny1( )crb

2

π2D22------------------------ L2 p̃cr= =

Ks nxy1( )cr≡Nxy1( )crb

2

π2 D11D223( )

1/4------------------------------------ L3 p̃cr= =

Kb nb1( )cr≡Nb1( )crb

2

π2 D11D22( )1/2

------------------------------------ L4 p̃cr= =

p̃crKx

KyKs

Kb

ε0ε1.

wN ξ η,( ) Am sin πξ Bm cosπξ+( )Φm η( )m=1

N

∑=

ξ x/λ= η y/b=wN

Am Bm

Φm η( )η 0=

Φm η( ) sin mπη[ ]=

Φm η( ) cos m 1–( )πη[ ] cos m 1+( )πη[ ]–=

5

For both boundary conditions, the two long edges of aplate are free to move in thex-y plane.

Algebraic equations governing the buckling behav-ior of infinitely long plates are obtained by substitutingthe series expansion for the buckling mode given byequation (22) into the second variation of the total poten-tial energy and then by computing the integrals appearingin the second variation in closed form. The resultingequations constitute a generalized eigenvalue problemthat depends on the aspect ratio of the buckle pattern(see fig. 1) and the nondimensional parameters and non-dimensional stress resultants defined herein. The small-est eigenvalue of the problem corresponds to bucklingand is found by specifying a value of and by solvingthe corresponding generalized eigenvalue problem for itssmallest eigenvalue. This process is repeated for succes-sive values of until the overall smallest eigenvalueis found.

Results obtained by using the analysis describedherein for uniform compression, uniform shear, pure in-plane bending (given byε0 = −1 andε1 = 1), and variouscombinations of these loads have been compared withother results for isotropic, orthotropic, and anisotropicplates obtained by using other analysis methods. Thesecomparisons are discussed in references 1 and 2, and inevery case the results described herein were found to bein good agreement with those obtained from otheranalyses. Results obtained for isotropic and speciallyorthotropic plates that are subjected to a general lineardistribution of axial load across the plate width werealso compared with results presented in references 7through 13. In every case, the agreement was good.

Results and Discussion

Results are presented for clamped and simply sup-ported plates loaded by a general linear distribution ofaxial load across the plate width, uniform shear load, andcombinations of these loads. For convenience, platesloaded by a general linear distribution of axial loadacross the plate width are referred to herein as platesloaded by linearly varying edge loads. To obtain the vari-ous edge load distributions used herein,ε1 = 1 was speci-fied andε0 was varied. Sketches that show the linearlyvarying edge loads for several values ofε0 are shown infigure 2. For loading cases that involve shear, a distinc-tion is made between positive and negative shear loadswhenever flexural anisotropy is present. A positive shearload corresponds to the shear load shown in figure 1.Although the analysis presented herein previouslyincludes the means for applying combined loads by usingsubcritical loads, the combined loads considered in thepresent paper were applied as primary destabilizingloads.

Results are presented first for the familiar [±θ]sangle-ply plates that are loaded by linearly varying edgeloads or by combined loads. (Several results for corre-sponding [±θ]s angle-ply plates that are subjected touniform uniaxial compression, uniform shear, or purein-plane bending loads have been presented in refs. 1and 2.) These thin laminates are representative of space-craft structural components and are made of a typicalgraphite-epoxy material with a longitudinal modulusE1 = 127.8 GPa (18.5× 106 psi), a transverse modulusE2 = 11.0 GPa (1.6× 106 psi), an in-plane shear modulusG12 = 5.7 GPa (0.832× 106 psi), a major Poisson’s ratioν12 = 0.35, and a nominal ply thickness of 0.127 mm(0.005 in.). Generic results are presented next, in terms ofthe nondimensional parameters described herein, forranges of parameters that are applicable to a broad classof laminate constructions. The term “generic” is usedherein to emphasize that these buckling results are verygeneral because they are presented in a form that is inde-pendent of the details of laminate construction; i.e.,stacking sequence and ply materials. The ranges of thenondimensional parameters used herein are given by

and The resultspresented in references 1 and 2 indicate thatγ = δ = 0.6corresponds to a highly anisotropic plate. For isotropicplates,β = 1 andγ = δ = 0. Moreover, for symmetricallylaminated plates without flexural anisotropy,γ = δ = 0.(These plates are referred to herein as specially orthotro-pic plates.) Values of these nondimensional parametersthat correspond to several practical laminates (andseveral material systems) are given in references 1 and 2.

To simplify the presentation of the fundamentalgeneric behavioral trends, results are presented hereinonly for plates in whichγ andδ have equal values (e.g.,[±45]s laminates). However, these behavioral trends areexpected to be applicable to laminates with nearly equalvalues ofγ andδ such as a [+35/−15]s laminate made ofthe typical graphite-epoxy material described herein. Forthis laminate,β = 1.95,γ = 0.52, andδ = 0.51. Further-more, results showing the effects of or equivalently

(D11/D22)1/4, on the buckling coefficients are not pre-

sented. References 1 and 2 have shown that variations inthis parameter affect the critical value of the buckleaspect ratioλ/b but not the buckling coefficient (i.e., thebuckling coefficient remains constant) of plates that aresubjected to uniform tension or compression loads, uni-form shear loads, and pure in-plane bending loads. Thistrend was found in the present study to be valid also forthe plates loaded by linearly varying edge loads anduniform shear loads considered herein. For clarity, theshear and in-plane bending buckling coefficients, definedby equations (20) and (21), respectively, are expressed as

and when the generic results are

λ/b

λ/b

λ/b

0.1 β 3.0,≤ ≤ 0 γ 0.6,≤ ≤ 0 δ 0.6.≤ ≤

α∞,


6

described for plates in which flexural anisotropy isneglected in the buckling calculations.

Plates Loaded by Linearly Varying Edge Loadsor Shear

Results are presented in figures 2 and 3 for simplysupported and clamped [±45]s plates, respectively, thatare subjected to linearly varying edge loads that corre-spond to values ofε0 = −2, −1.5,−1, −0.5, 0, 0.5, and 1.In these figures, the minimum value of the loadingparameter found by solving the generalized eigen-value problem for a given value ofλ/b, is shown by thesolid lines for values of (flexural anisotropyis included in the analysis). The overall minimum valueof the loading parameter for each curve is indicated by anunfilled circle, and these minimum values of the loadingparameter correspond to the value of the buckling coeffi-cient for each curve. The corresponding values ofare the critical values of the buckle aspect ratio.

The results presented in figures 2 and 3 show theeffect of the load distribution shape and boundary condi-tions on the buckling coefficient and the correspondingcritical value of As the amount of tension load inthe load distribution increases (ε0 decreases), the buck-ling coefficient increases substantially, and the criticalvalue of decreases. Moreover, the results show thatthe clamped plates exhibit larger buckling coefficientsand smaller critical values of than correspondingsimply supported plates. As the amount of tension load ina linearly varying edge load distribution increases, theamount of the plate width that is in compressiondecreases. As a result, the buckles appearing in the plateare typically confined to the narrower compressionregion. The widthd of this narrower compression regionof a plate is obtained from the equation that definesthe corresponding neutral axis of the in-plane bendingload component (defined by ); that is,

where ε0 < 0. For plates with largenegative values ofε0, the critical value ofλ/b may bemuch less than a value of 1. For these cases, a more accu-rate expression of the critical buckle aspect ratio is givenby λ/d.

Results that indicate the effect of the edge load dis-tribution shape on the buckling coefficients for simplysupported and clamped [±θ]s plates with θ = 0°, 30°, 45°,60°, and 90° (see fig. 1) are shown in figure 4. The solidand dashed lines correspond to buckling coefficients forclamped and simply supported plates, respectively, inwhich the flexural anisotropy is neglected in the analysis.These results show that the clamped plates always havehigher buckling coefficients than the simply supportedplates and that the buckling coefficients for the clampedplates are more sensitive to variations in the load

distribution parameterε0 (indicated by the slope of thecurves). The results also show very large increases inbuckling coefficient as the amount of tension in the edgeload distribution increases (ε0 decreases) for all valuesof θ. For a given value ofε0, the largest buckling coeffi-cient is exhibited by the plates withθ = 45°, followed bythe plates withθ = 30° and 60°, and then by the plateswith θ = 0° and 90°. Moreover, the results indicate thatthe plates withθ = 30° and 60° have the same bucklingcoefficients as do the plates withθ = 0° and 90°.

The importance of neglecting the anisotropy in thecalculation of the buckling coefficients given in figure 4for [±θ]s plates is indicated in figure 5. In figure 5, theratio of the anisotropic-plate buckling coefficientKb tothe corresponding specially orthotropic-plate bucklingcoefficient is given as a function of the load

distribution parameterε0 and the fiber angleθ. The solidand dashed lines shown in the figure correspond toresults for clamped and simply supported plates, respec-tively. The results indicate that the simply supportedplates generally exhibit larger reductions in the bucklingcoefficient ratio because of anisotropy and that the sim-ply supported plates are more sensitive to the load distri-bution parameterε0 than are the clamped plates. Inparticular, the results predict that the simply supportedplates are only slightly sensitive to variations inε0 andthat the clamped plates exhibit practically no sensitivityto variations inε0. The largest reductions in the bucklingcoefficient ratio are predicted for the plates with θ = 45°,followed by the plates withθ = 60° andθ = 30°, respec-tively. The simply supported plates withθ = 45° and withε0 = −2 andε0 = 1 (uniform compression) have values ofapproximately 0.76 and 0.74, respectively, for the buck-ling coefficient ratio.

Generic effects of plate orthotropy.Generic buck-ling results for specially orthotropic (γ = δ = 0) simplysupported and clamped plates that are subjected to lin-early varying edge loads are presented in figures 6through 10. The solid and dashed lines in the figures cor-respond to results for clamped and simply supportedplates, respectively. The results presented in figure 6show the buckling coefficient as a function of the orthot-ropy parameterβ for selected values of the load distribu-tion parameterε0 = −2, −1.5, and−1. Similar results arepresented in figure 7 for values ofε0 = −0.5, 0, 0.5,and 1. The results presented in figures 8 through 10 showthe buckling coefficient as a function of the load distribu-tion parameterε0 for discrete values of the orthotropyparameterβ = 0.5, 1, 1.5, 2, 2.5, and 3. In figures 8through 10, results are presented for−2 ≤ ε0 ≤ −1,−1 ≤ ε0 ≤ 0, and 0≤ ε0 ≤ 1, respectively, because of thelarge variation in the buckling coefficient withε0.

p̃,

0 λ/b 2≤ ≤

λ/b

λ/b.

λ/b

λ/b

Nx 0=d b/ 1 ε0–( )=

Kb γ =δ=0

7

The generic results presented in figures 6 through 10show that the buckling coefficient increases substantiallyas the orthotropy parameterβ increases. In contrast, thebuckling coefficient decreases substantially as the loaddistribution parameterε0 increases and the amount ofcompression in the load distribution increases. In addi-tion, the results presented in figures 6 and 7 indicate thatthe clamped plates are more sensitive to variations in theorthotropy parameterβ (indicated by the slope of thecurves) than the simply supported plates for the fullrange of load distribution parameters considered. More-over, the results presented in figures 8 through 10 indi-cate that for a given value ofβ, the clamped plates aretypically more sensitive to variations in the load distribu-tion parameterε0 than are the simply supported plates.

Some generic buckling results for specially orthotro-pic simply supported and clamped shear-loaded platesthat have been presented in references 1, 2, and 5 are pre-sented in figure 11 for completeness of the present studyand for convenience. The solid and dashed lines in thefigure show the buckling coefficient as a function of theorthotropy parameterβ for clamped and simply sup-ported plates, respectively. The results indicate that theshear buckling coefficient increases substantially as theorthotropy parameterβ increases and that the clampedplates are typically more sensitive to variations inβ thanare the simply supported plates. This trend for the shear-loaded plates is the same as the corresponding trend pre-dicted for the plates that are loaded by linearly varyingedge loads.

Generic effects of plate anisotropy.Results arepresented in figures 12 through 16 for simply supportedand clamped plates that are subjected to linearly varyingedge loads. In figures 12 through 16, the ratio of theanisotropic-plate buckling coefficient to the corre-

sponding specially orthotropic-plate buckling coefficient (see figs. 6 through 10) is given for equal val-

ues of the anisotropy parameters (γ = δ) ranging from 0.1to 0.6. In figures 12 and 13, the generic effects of plateanisotropy on the buckling coefficient ratio are given forsimply supported and clamped plates, respectively, as afunction of the orthotropy parameterβ. For each value ofγ = δ given in figures 12 and 13, two curves are pre-sented. The solid and dashed lines correspond to valuesof the load distribution parameterε0 = 1 (uniform com-pression) andε0 = −2 (the maximum amount of tensionin the load distributions considered herein), respectively.In figures 14 through 16, the buckling coefficient ratio isgiven as a function of the load distribution parameterε0for discrete values of the orthotropy parameterβ = 3, 1.5,and 0.5, respectively. The solid and dashed lines in

figures 14 through 16 correspond to results for clampedand simply supported plates, respectively.

Figures 12 through 16 show that the anisotropic-plate buckling coefficient is always less than the corre-sponding orthotropic-plate buckling coefficient for allvalues of parameters considered. In addition, theseresults predict that the effects of neglecting anisotropyare typically more pronounced for the simply supportedplates than for the clamped plates, but only by a smallamount. Moreover, for the full range of anisotropy con-sidered, figures 12 and 13 show a trend of monotonicincrease in the buckling coefficient ratio as the orthot-ropy parameterβ increases. The results in figures 12through 16 also predict that the effects of neglecting plateflexural anisotropy in the buckling analysis of simplysupported plates become slightly less pronounced as theload distribution parameterε0 decreases and the amountof tension in the edge load distribution increases. More-over, this effect is practically negligible in the corre-sponding clamped plates. This behavioral trend, in whichthe importance of anisotropy is reduced as the amount oftension in the edge load distribution increases, is similarto a behavioral trend given in reference 5. There, theimportance of anisotropy on the buckling load of a platethat is subjected to destabilizing uniform axial compres-sion is shown to be reduced as the amount of subcriticaltransverse tension loadNy2 that is applied to the plate isincreased. However, for this case, the clamped platesexhibit more sensitivity to the transverse tension loadthan do the simply supported plates.

Results that show the importance of flexural anisot-ropy on the buckling behavior of shear-loaded simplysupported and clamped plates have been presented inreference 1 and are presented in figure 17 in a differentform for convenience. In this figure, the ratio of theanisotropic-plate buckling coefficientKs to the corre-sponding specially orthotropic-plate buckling coefficient

(see fig. 11) is given as a function of the

orthotropy parameterβ for equal values of the anisotropyparameters (γ = δ) ranging from 0.2 to 0.6. Two groupsof curves that correspond to positive and negative shearloads are shown in the figure. For each value ofγ = δgiven in figure 17, two curves are presented in eachgroup. The solid and dashed lines correspond to resultsfor clamped and simply supported plates, respectively.

The results presented in figure 17 for simply sup-ported and clamped plates that are subjected to positiveshear loads indicate that the anisotropic-plate bucklingcoefficient is always less than the correspondingorthotropic-plate buckling coefficient. However, thistrend is reversed for negative shear loads. The results infigure 17 also show that the effects of neglecting plate

Kb

Kb γ =δ=0Ks γ =δ=0

8

anisotropy become smaller as the orthotropy parameterβincreases and the anisotropy parametersγ andδ decrease.Furthermore, these results show that the effects ofneglecting plate anisotropy are only slightly more pro-nounced for simply supported plates than for clampedplates. Comparing the results presented in figures 12, 13,and 17 indicates that the reductions in buckling coeffi-cient caused by neglecting anisotropy are, for the mostpart, more pronounced for the shear-loaded plates thanfor the plates that are subjected to linearly varying edgeloads.

Plates Loaded by Shear and Linearly VaryingEdge Loads

Buckling interaction curves obtained by neglectingthe plate anisotropy in the buckling calculations for sim-ply supported [±45]s plates that are subjected to uniformshear and linearly varying edge loads are presented infigure 18. Negative values ofKb indicated on the figurecorrespond to results in which the sign ofNb1 is reversed,and negative values ofKs correspond to negative shearloadings. Several curves that indicate the stability bound-aries corresponding to values of the load distributionparameterε0 = −2, −1.5, −1, −0.5, 0, 0.5, and 1 arepresented in figure 18. Each curve shown in the figureis symmetric about the lineKs = 0, and the curve forε0 = −1 (pure in-plane bending) is also symmetric aboutthe lineKb = 0. In addition, all the curves pass throughthe same two points on the lineKb = 0; i.e., the pointsthat correspond to positive and negative shear buckling.

The curves shown in figure 18 for values ofε0 = 0,0.5, and 1 are open, parabola-like curves that extendindefinitely in the negativeKb-direction. For these com-bined loadings, shear loads that are greater in magnitudethan the shear buckling load can be sustained only whenthe linearly varying edge loads are tensile loads. In con-trast, specially orthotropic plates that are subjected topure in-plane bending (ε0 = –1) can never sustain shearloads that are greater in magnitude than the shear buck-ling load. However, the curves shown in figure 18 forε0 = −2, −1.5, and−0.5 are significantly different fromthe conventional buckling interaction curves found in theliterature and shown in figure 18 for plates that are sub-jected to uniform shear and uniform axial compression(ε0 = 1) or pure in-plane bending loads. Specifically, theresults predict that shear loads that are larger in magni-tude than the shear buckling load can be supported by anunbuckled plate when a tension-dominated linearly vary-ing edge load distribution is applied to the plate first. It isimportant to observe that the loading withε0 = −0.5 istension dominated for negative values ofKb but not forpositive values. In general, when trying to determinewhether a linearly varying edge load is tension domi-nated, both positive and negative values ofKb should be

considered. The ability to carry shear loads greater inmagnitude than the positive and negative shear bucklingloads is attributed to the fact that the stabilizing effect ofthe tension part of the linearly varying edge load isgreater than the destabilizing effect of the compressionpart.

Additional buckling interaction curves that corre-spond to the curves shown in figure 18 are presented infigure 19 for simply supported [±45]s plates that are sub-jected to shear and linearly varying edge loads. Thecurves shown in figure 19 include the effects of flexuralanisotropy which are manifested by skewing and transla-tion of the curves presented in figure 18 in theKb-Ksplane. Thus, the results indicate that for a given value ofKb, the anisotropic plates can carry a negative shear loadthat is much greater in magnitude than the correspondingpositive shear load. Moreover, the results predict thatthe anisotropic plates with a tension-dominated linearlyvarying edge load distribution (i.e.,ε0 = −2, −1.5, and−0.5) can also support shear loads (positive or negative)that are larger in magnitude than the shear buckling loadand that this effect is much more pronounced for platesthat are loaded in negative shear. This greater negative-shear load capacity for the plates is attributed to thegreater shear buckling resistance of these plates undernegative shear loads. That is, flexurally anisotropic platesgenerally exhibit two unequal plate bending stiffnessesalong the directions of the diagonal compression and ten-sion generated by the shear load. For a negative shearload, the higher plate bending stiffness acts in the direc-tion of the diagonal compression generated by the load.

The importance of neglecting anisotropy in the cal-culation of buckling interaction curves for simply sup-ported [±θ]s plates that are loaded by shear and atension-dominated linearly varying edge load distribu-tion (ε0 = −2) is indicated in figure 20. Curves are shownin this figure for values ofθ = 30°, 45°, and 60°. Thesolid and dashed lines are buckling interaction curves inwhich the plate anisotropy is neglected (specially ortho-tropic) and included, respectively. Figure 20 indicatesthat the specially orthotropic plates withθ = 45° havelarger (in magnitude) buckling coefficients than the cor-responding plates withθ = 30° and 60°. Moreover, thebuckling interaction curves for the specially orthotropicplates withθ = 30° and 60° are identical. For the aniso-tropic plates, however, these two trends are not valid.That is, the buckling coefficients for the anisotropicplates withθ = 45° are not always larger in magnitudethan the buckling coefficients for the correspondingplates withθ = 30° and 60°. Moreover, the bucklinginteraction curves for the anisotropic plates withθ = 30°and 60° are different. The results also predict the capabil-ity of carrying shear loads that are greater in magnitudethan the shear buckling loads (positive and negative) for

9

all the specially orthotropic and anisotropic platesconsidered.

Generic effects of plate orthotropy.Generic buck-ling interaction curves for specially orthotropic (γ =δ = 0) simply supported plates that have a value ofβ = 3and that are subjected to shear and linearly varying edgeloads are presented in figure 21. In particular, severalcurves that indicate the stability boundaries thatcorrespond to values of the load distribution parameterε0 = −2, −1.5,−1, −0.5, 0, 0.5, and 1 are presented in thefigure. The generic buckling interaction curves shown infigure 21 exhibit the same characteristics as the corre-sponding curves presented in figure 18 for the [±45]splates in which the effects of anisotropy are neglected.That is, each curve shown in the figure is symmetricabout the line given by a zero value of , and the

curve forε0 = −1 (pure in-plane bending) is also symmet-ric about the line given by a zero value of . In

addition, all the curves pass through the same two pointson the line given by a zero value of , i.e., the

points that correspond to positive and negative shearbuckling. Like the results presented in figure 18 for the[±45]s plates, the generic curves shown in figure 21for values ofε0 = 0, 0.5, and 1 are open, parabola-like curves that extend indefinitely in the negative

-direction. Thus, for these three loadings, shear

loads that are greater in magnitude than the shear buck-ling load can be sustained only when the linearly varyingedge loads are tensile loads. However, the plates that aresubjected to pure in-plane bending (ε0 = −1) can neversustain shear loads greater in magnitude than the shearbuckling load. Furthermore, the generic results also pre-dict an ability to carry shear loads that are larger in mag-nitude than the corresponding shear buckling load forplate loadings withε0 = −2, −1.5, and−0.5 (i.e., tension-dominated load distributions). This ability to carry shearloads greater in magnitude than the positive and negativeshear buckling loads is again attributed to the fact thatthe stabilizing effect of the tension part of the linearlyvarying edge load is greater than the destabilizing effectof the compression part.

Generic buckling interaction design curves for spe-cially orthotropic (γ=δ=0) simply supported and clampedplates that are subjected to shear and linearly varyingedge loads are presented in figures 22 through 28 for val-ues ofε0 = −2, −1.5,−1, −0.5, 0, 0.5, and 1, respectively.The solid lines and dashed lines in the figures correspondto results for clamped and simply supported plates,respectively, and curves are given for values of the

orthotropy parameterβ = 0.5, 1, 1.5, 2, 2.5, and 3. More-over, curves are shown for positive shear loading only.Results for negative shear loading are obtained by notingthat buckling interaction curves for specially orthotropicplates that are loaded by shear are symmetric about the

-axis. Points on the curves correspond to

constant values of the stiffness-weighted load ratio

as illustrated in figure 29 by the line

emanating from the origin of the plot. An important char-acteristic of all the buckling interaction curves presentedin figures 22 through 28 is that, for a given stiffness-weighted load ratio, the magnitude of the buckling coef-ficients increases substantially as the orthotropy parame-ter β increases.

The results presented in figure 22 for the plates withε0 = −2 indicate that the maximum value of

occurs within the range and

for the simply supported

and clamped plates, respectively, as the orthotropyparameterβ increases from 0.5 to 3.0. Similarly, theresults presented in figure 23 for the plates withε0 = −1.5indicate that the maximum value of occurs

within the range and


and clamped plates, respectively. Moreover, the resultspresented in figure 25 for the plates withε0 = −0.5indicate that the maximum value of occurs

within the range and


and clamped plates, respectively. Furthermore, theresults presented in figure 24 for the plates withε0 = −1indicate that the maximum value of occurs

when = 0.

The results presented in figures 24, 26, 27, and 28for the plates withε0 = −1, 0, 0.5, and 1, respectively, in-dicate that the buckling interaction curves for the simply

Ks γ =δ=0

Kb γ =δ=0

Kb γ =δ=0

Kb γ =δ=0

Kb γ =δ=0

Nb1Nxy1------------

D22D11---------

1/4

,

Ks γ =δ=0

2.86Nb1Nxy1------------

D22D11---------

1/4

4.44≤ ≤

3.00Nb1Nxy1------------

D22D11---------

1/4

4.13≤ ≤

Ks γ =δ=0

2.64Nb1Nxy1------------

D22D11---------

1/4

3.97≤ ≤

2.58Nb1Nxy1------------

D22D11---------

1/4

3.64≤ ≤

Ks γ =δ=0

8.88–Nb1Nxy1------------

D22D11---------

1/4

5.72–≤ ≤

8.31–Nb1Nxy1------------

D22D11---------

1/4

6.00–≤ ≤


10

supported plates generally have more curvature than thecurves for the corresponding clamped plates. Thus, thesimply supported plates are generally more sensitive tovariations in the stiffness-weighted load ratio.

Generic effects of plate anisotropy.Generic buck-ling interaction curves for simply supported anisotropicplates withβ = 3 andγ = δ = 0.6 that are subjected toshear and linearly varying edge loads are presented infigure 30. More specifically, curves that indicate the sta-bility boundaries forε0 = −2, −1.5,−1, −0.5, 0, 0.5, and 1are presented. The generic buckling interaction curvesshown in figure 30 exhibit the same characteristics as thecurves presented in figure 19 for the corresponding[±45]s anisotropic plates (these plates haveβ = 2.28 andγ = δ = 0.517). That is, the effects of anisotropy are man-ifested by skewing and translation in theKb-Ks plane ofthe curves for the corresponding specially orthotropicplates presented in figure 21. Thus, as was seen for the[±45]s anisotropic plates, the generic results for theanisotropic plates also predict that, for a given value ofKb, negative shear loads can be carried that are muchgreater in magnitude than the corresponding positiveshear load. Moreover, the anisotropic plates with tension-dominated axial-edge load distributions (ε0 = −2, −1.5,and−0.5) can also support shear loads that are larger inmagnitude than the shear buckling load. This effect ismuch more pronounced for plates that are loaded by neg-ative shear. This difference in behavior exhibited byanisotropic and specially orthotropic plates loaded bynegative and positive shear is attributed to directionaldependence of the shear buckling resistance of anisotro-pic plates previously described herein.

Buckling interaction curves that show the genericeffects of plate anisotropy are presented in figure 31 forsimply supported and clamped plates withβ = 3 that aresubjected to shear and a linearly varying edge load corre-sponding toε0 = −2 (tension dominated). Several curvesthat indicate the stability boundaries for equal values ofthe anisotropy parameters (γ = δ) ranging from 0 to 0.6are shown in the figure. The solid and dashed lines in fig-ure 31 correspond to results for clamped and simply sup-ported plates, respectively. These results indicate that thecurve corresponding to a specially orthotropic plate(γ = δ = 0) becomes more skewed and is translated morein the negativeKs-direction as the values of the anisot-ropy parameters increase. These effects of anisotropyappear to be nearly the same for the clamped and simplysupported plates.

The results presented in figure 31 also show thatfor some values of the stiffness-weighted load ratio

the buckling interaction curves for plates

with γ = δ = 0 and intersect. Thus, for thesevalues of the stiffness-weighted load ratio, neglectinganisotropy has no effect on the critical value of theloading parameter (see eqs. (18) through (21)).

Moreover, the results indicate that as the stiffness-weighted load ratio varies, so does the importance ofthe plate anisotropy. An indication of the importance ofplate anisotropy, with respect to the stiffness-weightedload ratio, is obtained by introducing the angle thatthe line emanating from the origin of the bucklinginteraction curve plot shown in figure 29 makes withthe Ks-axis as an independent variable. This angle isreferred to herein as the stiffness-weighted loadratio angle and is denoted by the symbolΨ. Then, fora given stiffness-weighted load ratio (constant valueof Ψ), the importance of anisotropy on the buckling

coefficients is expressed by the ratio where

Results are presented in figure 32 that show theimportance of plate anisotropy on the buckling behaviorof simply supported and clamped plates withβ = 3 thatare subjected to shear and a linearly varying edge loadcorresponding toε0 = −2. The results for the simply sup-ported plates withγ = δ = 0.6 correspond to those pre-sented in figure 30. Curves that indicate the buckling

coefficient ratio for equal values of the anisot-

ropy parameters (γ = δ) ranging from 0 to 0.6 are shownin figure 32 as a function of the stiffness-weighted loadratio angleΨ. The solid and dashed lines in the figurecorrespond to results for clamped and simply supportedplates, respectively. The results presented in the figureshow a variation in the buckling coefficient ratio ofapproximately±0.5, with the greatest variations occur-ring at values of 0° ≤ Ψ ≤ 70°, 100° ≤ Ψ ≤ 200°, and 290°≤ Ψ ≤ 360°. Moreover, the results indicate that theimportance of plate anisotropy is only slightly more pro-nounced for the simply supported plates than for theclamped plates.

Concluding Remarks

A parametric study of the buckling behavior of infi-nitely long symmetrically laminated anisotropic platesthat are subjected to linearly varying edge loads, uniformshear loads, or combinations of these loads has been pre-sented. A special purpose nondimensional analysis that iswell suited for parametric studies of clamped and simply

Nb1Nxy1------------

D22D11---------

1/4

,

γ δ 0≠=

p̃cr

p̃crp̃cr γ =δ=0

-----------------------

p̃crp̃cr γ =δ=0

-----------------------Kb

Kb γ =δ=0

----------------------Ks

Ks γ =δ=0

----------------------.= =

p̃crp̃cr γ =δ=0

-----------------------

11

supported plates has been described, and its mainfeatures have been discussed. The results presentedherein have focused on the effects of the shape of the lin-early varying edge load distribution, plate flexuralorthotropy, and plate flexural anisotropy on the bucklingbehavior. In addition, results have been presented thatfocus on the interaction of linearly varying edge loadsand uniform shear loads with plate flexural anisotropyand orthotropy. In particular, results have been presentedfor [±θ]s thin graphite-epoxy laminates that are represen-tative of spacecraft structural components. Also, numer-ous generic buckling results have been presented that areapplicable to a broad class of laminate constructions thatshow explicitly the effects of flexural orthotropy andflexural anisotropy on plate buckling behavior underthese combined loads. These generic results can be usedto extend the capability of existing design guides forplate buckling.

An important finding of the present study is thatthe buckling coefficients increase significantly as the

orthotropy parameter increases or

as the load distribution parameterε0 decreases (theloading distribution becomes tension dominated). Incontrast, the buckling coefficients decrease significantly

as the anisotropy parameters and

with equal values (γ = δ) increase, for

all linearly varying edge loads considered and for theplates that are subjected to positive shear loads. For neg-ative shear loads, the trend is generally reversed. Theeffect of linearly varying edge load distribution shape(determined byε0) on the importance of plate anisotropyis generally small, with uniform compression-loadedplates exhibiting the largest reductions in buckling resis-tance. The results presented herein also show that theeffects of plate anisotropy are slightly more pronouncedfor simply supported plates than for clamped plates whenthe plates are subjected to either linearly varying edgeloads, uniform shear loads, or combinations of theseloads. The most important finding of the present study isthat specially orthotropic and flexurally anisotropicplates that are subjected to a tension-dominated axialedge load distribution (e.g.,ε0 = −2, −1.5, and−0.5) cansupport shear loads that are larger in magnitude than theshear buckling load. This ability to carry a shear loadgreater in magnitude than the corresponding shear buck-ling load is attributed to the fact that the stabilizing effectof the tension part of the linearly varying edge load isgreater than the destabilizing effect of the compression

part. Moreover, this unusual behavior is much morepronounced for anisotropic plates than for speciallyorthotropic plates that are loaded in negative shear. Thistrend is reversed for plates that are loaded in positiveshear. This difference in behavior exhibited by anisotro-pic plates and specially orthotropic plates loaded by neg-ative shear and the trend reversal for positive shear loadsis attributed to directional dependence of the shear buck-ling resistance of anisotropic plates.

NASA Langley Research CenterHampton, VA 23681-2199January 27, 1997

References1. Nemeth, M. P.:Buckling Behavior of Long Symmetrically

Laminated Plates Subjected to Combined Loadings. NASATP-3195, 1992.

2. Nemeth, M. P.: Buckling of Symmetrically Laminated PlatesWith Compression, Shear, and In-Plane Bending.AIAA J.,vol. 30, no. 12, Dec. 1992, pp. 2959–2965.

3. Zeggane, Madjid; and Sridharan, Srinivasan: Buckling UnderCombined Loading of Shear Deformable Laminated Anisotro-pic Plates Using ‘Infinite’ Strips.Int. J. Numer. Methods inEng., vol. 31, May 1991, pp. 1319–1331.

4. Zeggane, M.; and Sridharan, S.: Stability Analysis of LongLaminate Composite Plates Using Reissner-Mindlin ‘Infinite’Strips.Comput. & Struct., vol. 40, no. 4, 1991, pp. 1033–1042.

5. Nemeth, Michael P.: Buckling Behavior of Long AnisotropicPlates Subjected to Combined Loads. NASA TP-3568, 1995.

6. Nemeth, Michael P.: Importance of Anisotropy of Buckling ofCompression-Loaded Symmetric Composite Plates.AIAA J.,vol. 24, no. 11, Nov. 1986, pp. 1831–1835.

7. Rockey, K. C.: Shear Buckling of Thin-Walled Sections.Thin-Walled Structures, A. H. Chilver, ed., John Wiley & Sons, Inc.,1967, pp. 261–262.

8. Oyibo, G. A.: The Use of Affine Transformations in the Anal-ysis of Stability and Vibrations of Orthotropic Plates. Ph.D.Thesis, Rensselaer Polytechnic Inst., 1981.

9. Brunnelle, E. J.; and Oyibo, G. A.: Generic Buckling Curvesfor Specially Orthotropic Rectangular Plates.AIAA J., vol. 21,no. 8, Aug. 1983, pp. 1150–1156.

10. Timoshenko, Stephen P.; and Gere, James M.:Theory of Elas-tic Stability, Second ed. McGraw-Hill Book Co., 1961,pp. 373–379.

11. Bulson, P. S.:The Stability of Flat Plates. American ElsevierPubl. Co. Inc., 1969, pp. 63–77.

12. Bleich, Friedrich:Buckling Strength of Metal Structures.McGraw-Hill Co., Inc., 1952, pp. 399–413.

13. Galambos, Theodore V., ed.:Guide to Stability Design Crite-ria for Metal Structures. John Wiley & Sons, Inc., 1988,p. 103.

βD12 2D66+

D11D22( )1/2

------------------------------=

γD16

D113

D22( )1/4

------------------------------=

δD26

D11D223( )

1/4------------------------------=

12

(a) Primary (destabilizing) loading system.

(b) Secondary (subcritical) loading system.

Figure 1. Loading systems and sign convention for positive-valued stress resultants

λ

x

y

ε1Nb1

ε0 Nb1

Ny1

N x1c Nxy1

θ

b

λ

x

y

ε1Nb2

ε0 Nb2

Ny2

N x2c Nxy2

b

(ε0 ε1 0 shown above).> >

13

Figure 2. Buckling results for simply supported [±45]s plates subjected to linearly varying edge loads.

Nb1

0

40

20

80

60

100

Minimums

.5 1.0 1.5 2.0Buckle aspect ratio, λ/b

Loadingparameter,

Nb1 b2

π2(D11D22)1/2

p = ~

ε0 Nb1ε0 = –1.5

ε0 = 0

ε0 = 0.5

ε0 = 1

ε0 = –1

ε0 = –0.5

ε0 = –2

λ

b

14

Figure 3. Buckling results for clamped [±45]s plates subjected to linearly varying edge loads.

Nb1

0

40

20

80

60

100

Minimums

.5 1.0 1.5 2.0Buckle aspect ratio, λ/b

Loadingparameter,

Nb1 b2

π2(D11D22)1/2

p = ~

ε0 Nb1

ε0 = –1ε0 = –1.5

ε0 = –2

ε0 = 0.5

ε0 = 1

ε0 = 0

ε0 = –0.5

λ

b

15

Figure 4. Buckling coefficients for simply supported and clamped [±θ]s plates subjected to linearly varying edge loads(anisotropy neglected in the analysis).

Nb1 λ

b

0

60

20

40

120

80

100

140

–1.5–2.0 –.5–1.0 0 1.0.5Load distribution parameter, ε0

ε0 Nb1

Clamped

Simply supported

Bucklingcoefficient,

(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

θ = 45°

30°, 60°

0°, 90°

16

Figure 5. Ratio of anisotropic-to-orthotropic buckling coefficient for simply supported and clamped [±θ]s platessubjected to linearly varying edge loads.

Nb1

.5

.7

.6

.9

.8

1.0

–1.5–2.0 –.5–1.0 0 1.0.5Load distribution parameter, ε0

ε0 Nb1

Clamped

Simply supported

θ = 45°

60° 30°

Bucklingcoefficient

ratio,Kb

Kb γ=δ=0

λ

b

17

Figure 6. Effect of orthotropy parameterβ on buckling coefficients for specially orthotropic plates (γ = δ = 0) subjected to linearlyvarying edge loads (ε0 = −2, −1.5, and−1).

Nb1 λ

b

0

60

20

40

120

80

100

140

.5 1.51.0 2.0 3.02.5

ε0 Nb1

Clamped

Simply supported


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

Orthotropy parameter, β = D12 + 2D66

(D11 D22)1/2

ε0 = –1.5

ε0 = –1

ε0 = –2

18

Figure 7. Effect of orthotropy parameterβ on buckling coefficients for specially orthotropic plates (γ = δ = 0) subjected to linearlyvarying edge loads (ε0 = −0.5, 0, 0.5, and 1).

Nb1 λ

b

0

15

5

10

35

20

30

25

40

.5 1.51.0 2.0 3.02.5

ε0 Nb1

Clamped

Simply supported


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =


(D11 D22)1/2

ε0 = 0.5

ε0 = 0

ε0 = –0.5

ε0 = 1

19

Figure 8. Effect of load distribution parameterε0 on buckling coefficients for specially orthotropic plates (γ = δ = 0)subjected to linearly varying edge loads.

Nb1 λ

b

0

50

100

150

–2.0 –1.5 –1.0

ε0 Nb1

Clamped

Simply supported


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

Load distribution parameter, ε0

β = 3.0

2.5

2.0

1.5 β = 1.0

β = 0.5

20


Nb1 λ

b

0

30

20

10

60

50

40

70

–1.0 –.5 0

ε0 Nb1

Clamped

Simply supported


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =


β = 3.0

2.5

2.0

1.5

β = 1.0

β = 0.5

21


Nb1 λ

b

0

10

20

30

.5 1.0

ε0 Nb1

Clamped

Simply supported


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =


β = 3.0

2.52.0

1.5

β = 1.0

β = 0.5

22

Figure 11. Effect of orthotropy parameterβ on buckling coefficients for specially orthotropic plates (γ = δ = 0) subjected to shearloads.

Nxy1

0

6

2

4

14

8

10

12

16

.5 1.51.0 2.0 3.02.5

Clamped

Simply supported


(Nxy1)cr b2

π2 √D11D32 2

Ks γ=δ=0 =


(D11 D22)1/2

4

λb

23

Figure 12. Effect of orthotropy parameterβ and anisotropy parametersγ andδ on buckling coefficients for simplysupported plates subjected to linearly varying edge loads.


(D11 D22)1/2

0.5

γ = δ = 0.6

0

.2

1.2

.5 1.5 2.5 3.0

Nb1

1.0 2.0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Kb γ=δ=0

Kb

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

0.4

0.3

0.2

0.1

ε0 Nb1

ε0 = 1

ε0 = –2

λ

b

24

Figure 13. Effect of orthotropy parameterβ and anisotropy parametersγ andδ on buckling coefficients for clamped platessubjected to linearly varying edge loads.


(D11 D22)1/2

0.5

γ = δ = 0.6

0

.2

1.2

.5 1.5 2.5 3.0

Nb1

1.0 2.0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Kb γ=δ=0

Kb

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

0.4

0.3

0.2

0.1

ε0 Nb1

ε0 = 1

ε0 = –2

λ

b

25

Figure 14. Effect of load distribution parameterε0 and anisotropy parametersγ andδ on buckling coefficients for platessubjected to linearly varying edge loads (β = 3).

0.5γ = δ = 0.6

γ = δ = 0.1

0

.2

1.2

–1.5–2.0 –.5 .5 1.0

Nb1

–1.0 0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Kb γ=δ=0

Kb

0.4

0.30.2

ε0 Nb1

λ

b

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

β = 3


Clamped

Simply supported

26

Figure 15. Effect of load distribution parameterε0 and anisotropy parametersγ andδ on buckling coefficients for platessubjected to linearly varying edge loads (β = 1.5).

0.5

γ = δ = 0.6

γ = δ = 0.1

0

.2

1.2

–1.5–2.0 –.5 .5 1.0

Nb1

–1.0 0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Kb γ=δ=0

Kb0.4

0.30.2

ε0 Nb1

λ

b

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

β = 1.5


Clamped

Simply supported

27

Figure 16. Effect of load distribution parameterε0 and anisotropy parametersγ andδ on buckling coefficients for platessubjected to linearly varying edge loads (β = 0.5).

γ = δ = 0.1

0

.2

1.2

–1.5–2.0 –.5 .5 1.0

Nb1

–1.0 0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Kb γ=δ=0

Kb

0.30.2

ε0 Nb1

λ

b

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

β = 0.5


Clamped

Simply supported

28

Figure 17. Effect of orthotropy parameterβ and anisotropy parametersγ andδ on buckling coefficients forplates subjected to shear loads.

0.5

0.5

γ = δ = 0.6

γ = δ = 0.6

0

.2

2.0

1.8

1.6

1.4

1.2

.5 1.5 2.5 3.0

Nxy1

1.0 2.0

.4

.6

.8

1.0

Bucklingcoefficient

ratio,

Ks γ=δ=0

Ks

0.4

0.4

0.3

0.3

0.2

0.2

λb

D16

(D31 1 D22)1/4

γ =D26

(D11 D32 2)

1/4δ =

Clamped

Simply supported


(D11 D22)1/2

Negativeshear

Positiveshear

29

Figure 18. Effect of load distribution parameterε0 on buckling interaction curves for [±45]s simply supported plates subjected tolinearly varying edge loads and shear loads (anisotropy neglected).

125

100

75

–25 –20 20–15 15–10 10–5 50

Nxy1

–25

–50

–75

–125

–150

–175

–200

–225

–100

0

50

25

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks =

Nb1

ε0 Nb1

(Nb1)cr b2

π2 √D11D22Kb =

ε0 = –2

ε0 = 0ε0 = –1.5

ε0 = –1

ε0 = –0.5

ε0 = –0.5

ε0 = –2

ε0 = 1ε0 = –1

ε0 = –1.5

ε0 = 1

ε0 = 0.5

ε0 = 0

30

Figure 19. Effect of load distribution parameterε0 on buckling interaction curves for [±45]s simply supported plates subjected tolinearly varying edge loads and shear loads (anisotropy included).

125

100

75

–25 –20 20–15 15–10 10–5 50

Nxy1

–25

–50

–75

–125

–150

–175

–200

–225

–100

0

50

25

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks =

Nb1

ε0 Nb1

(Nb1)cr b2

π2 √D11D22Kb =

ε0 = –2

ε0 = 0

ε0 = –1.5

ε0 = –1

ε0 = –0.5

ε0 = –0.5

ε0 = –2

ε0 = –1

ε0 = –1.5

ε0 = 0.5

ε0 = 1

ε0 = 0

31

Figure 20. Importance of anisotropy on buckling interaction curves for [±θ]s simply supported plates subjected to linearlyvarying edge loads (ε0 = −2) and shear loads.

100

80

–25 –20 20–15 15–10 10–5 50

Nxy1

40

20

0

–20

60

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks =

Nb1

–2Nb1

(Nb1)cr b2

π2 √D11D22Kb =

θ = 45°

θ = 45°

30°

30°

30°

45°

45°

60°30°, 60°

60°

Anisotropy included

Anisotropy neglected

32

Figure 21. Effect of load distribution parameterε0 on buckling interaction curves for simply supported specially orthotropic platessubjected to shear and linearly varying edge loads (β = 3).

125

100

75

–25 –20 20–15 15–10 10–5 50

Nxy1

–25

–50

–75

–125

–150

–175

–200

–225

–100

0

50

25

λb

Nb1

ε0 Nb1

(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

ε0 = –2

ε0 = 0 ε0 = –1.5

ε0 = –1

ε0 = –0.5

ε0 = –0.5

ε0 = –2

ε0 = 1ε0 = –1

ε0 = –1.5

ε0 = 1

ε0 = 0.5

ε0 = 0

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks γ=δ=0 =

33

Figure 22. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = −2).

160

140

0 255 201510

Nxy1

80

60

Clamped

Simply supported

40

20

0

–20

100

120

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Buckling coefficient, Ks γ=δ=0 =

Nb1

–2 Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 3 2.5

1.5

0.5

2

1

1

34

Figure 23. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = −1.5).

120

0 205 1510

Nxy1

80

60

Clamped

Simply supported

40

20

0

–20

100

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4

Nb1

–1.5 Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 32.5

1.5

0.5

2

1

1

Buckling coefficient, Ks γ=δ=0 =

35

Figure 24. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = −1).

70

60

155 10

Nxy1

30

Clamped

Simply supported

20

10

0

50

40

λb

(Nxy1)cr b2

π2(D11D32 2)


Nb1

–Nb1

β = D12 + 2D66

(D11 D22)1/2

β = 32.5

1.5

1.5

0.5

0.5

2

1

1


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

36

Figure 25. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = −0.5).

50

0

0 255 201510

Nxy1

Clamped

Simply supported

–300

–50

–100

–150

–200

–250

λb

Nb1

–0.5 Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 3

2.5

1.50.5

2

1

(Nxy1)cr b2

π2(D11D32 2)


37

Figure 26. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = 0).

25

20

0 205 1510

Nxy1

0

–5

Clamped

Simply supported

–10

–15

–20

–25

10

5

15

λb

Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 3

2.5

1.5

0.5

2

2

1

(Nxy1)cr b2

π2(D11D32 2)


38

Figure 27. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = 0.5).

20

15

0 205 1510

Nxy1

0

–5

Clamped

Simply supported

–10

–15

–20

10

5

λb

Nb1

0.5 Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 3

2.5

1.5

0.5

2

1

(Nxy1)cr b2

π2(D11D32 2)


39

Figure 28. Effect of orthotropy parameterβ on buckling interaction curves for specially orthotropic plates (γ = δ = 0) subjectedto shear and linearly varying edge loads (ε0 = 1).

15

10

0 255 201510

Nxy1

0

–5

Clamped

Simply supported

–10

–15

–20

5

λ

b

Nb1

β = D12 + 2D66

(D11 D22)1/2


(Nb1)cr b2

π2 √D11D22Kb γ=δ=0 =

β = 3

2.5

1.5

1.5

0.5

2

1

(Nxy1)cr b2

π2(D11D32 2)


40

Figure 29. Illustration of stiffness-weighted load ratio angleΨ for specially orthotropic and anisotropic platessubjected to shear and linearly varying edge loads.

180

80

160

60

–40 –30 50–20 40–10 300 2010

140

40

120

20

0

–40

–20

100

(Nxy1)cr b2

π2(D11D32 2)

1/4

D22 1/4

D11

Ks =

= constant

(Nb1)cr b2

Nb1Nxy1

π2 √D11D22Kb =

Ks γ=δ=0

Kb γ=δ=0

Anisotropic

Orthotropic

KbΨ

Ks

41

Figure 30. Effect of load distribution parameterε0 on buckling interaction curves for simply supported anisotropic platessubjected to shear and linearly varying edge loads (β = 3;γ = δ = 0.6).

125

100

75

–25 –20 20–15 15–10 10–5 50

Nxy1

–25

–50

–75

–125

–150

–175

–200

–225

–100

0

50

25

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks =

Nb1

ε0 Nb1

(Nb1)cr b2

π2 √D11D22Kb =

ε0 = –2

ε0 = –1.5

ε0 = –1

ε0 = –0.5

ε0 = –0.5

ε0 = –2ε0 = –1

ε0 = –1.5

ε0 = 0.5

ε0 = 1

ε0 = 0

42

Figure 31. Effect of anisotropy parametersγ andδ on buckling interaction curves for plates subjected to shear and linearlyvarying edge loads (β = 3;ε0 = −2).

160

140

–40 –10 30–20–30 10 200

Nxy1

60

40

Clamped

Simply supported

0

–20

–40

–60

20

80

120

100

λb

(Nxy1)cr b2

π2(D11D32 2)

1/4Ks =

Nb1

–2 Nb1γ = δ = 0

γ = δ = .6

.1

.2

0

.1

.2

.3

.4

.5

.5

.4

.3

.6

(Nb1)cr b2

π2 √D11D22Kb =

D16

(D31 1 D22)1/4

γ =

D26

(D11 D32 2)

1/4δ =

β = 3

43

Figure 32. Effect of anisotropy parameters and stiffness-weighted load ratio angleΨ on behavior of plates subjected toshear and linearly varying edge loads (β = 3;ε0 = −2).

.5

γ = δ = .6

γ = δ = .6

0

2.0

1.5

90 180 360270

.5

1.0

Bucklingcoefficient

ratio,

pcr γ=δ=0

pcr

.4.3

.2

.6

.1

.1

.10

λb

D16

(D31 1 D22)1/4

γ =D26

(D11 D32 2)

1/4δ =

Clamped

Simply supported

Stiffness–weighted load ratio angle Ψ, deg

~

~

Nb1Nxy1

–2 Nb1

β = 3

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July 1997 Technical Paper

Buckling Behavior of Long Symmetrically Laminated Plates Subjected toShear and Linearly Varying Axial Edge Loads WU 505-63-50-08

Michael P. Nemeth

L-17605

NASA TP-3659

Paper presented at the 11th DoD/NASA/FAA Conference on Fibrous Composites in Structural Design, Ft. Worth,Texas, August 26–30, 1996.

A parametric study of the buckling behavior of infinitely long symmetrically laminated anisotropic plates that aresubjected to linearly varying edge loads, uniform shear loads, or combinations of these loads is presented. Thestudy focuses on the effects of the shape of linearly varying edge load distribution, plate orthotropy, and plate flex-ural anisotropy on plate buckling behavior. In addition, the study examines the interaction of linearly varying edgeloads and uniform shear loads with plate flexural anisotropy and orthotropy. Results obtained by using a specialpurpose nondimensional analysis that is well suited for parametric studies of clamped and simply supported platesare presented for [±θ]s thin graphite-epoxy laminates that are representative of spacecraft structural components.Also, numerous generic buckling-design charts are presented for a wide range of nondimensional parameters thatare applicable to a broad class of laminate constructions. These charts show explicitly the effects of flexuralorthotropy and flexural anisotropy on plate buckling behavior for linearly varying edge loads, uniform shear loads,or combinations of these loads. The most important finding of the present study is that specially orthotropic andflexurally anisotropic plates that are subjected to an axial edge load distribution that is tension dominated cansupport shear loads that are larger in magnitude than the shear buckling load.

Buckling; Combined loads; Anisotropy; Composite plates 44

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NASA Langley Research CenterHampton, VA 23681-2199

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Buckling Behavior of Long Symmetrically Laminated Plates ...mln/ltrs-pdfs/NASA-97-tp3659.pdf · A parametric study of the buckling behavior of infinitely long symmetrically lam-inated