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American Institute of Aeronautics and Astronautics1

AIAA-2002-1517

Finite Element Modeling of the Buckling Response of Sandwich Panels

Cheryl A. Rose1

NASA Langley Research CenterHampton, VA 23681

David F. Moore2

Lockheed-Martin Engineering and Sciences CompanyHampton, VA 23681

Norman F. Knight, Jr.3

Veridian Systems DivisionChantilly, VA 20151

Charles C. Rankin4

Lockheed-Martin Advanced Technology CenterPalo Alto, CA 94304

1 Aerospace Engineer, Mechanics and Durability Branch. Member AIAA.2 Aerospace Engineer. Member ASME.3 Staff Scientist. Associate Fellow AIAA. Member ASME. Corresponding author.4 Staff Scientist. Associate Fellow AIAA.Copyright 2002 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U. S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes.All other rights are reserved by the copyright owner.

AbstractA comparative study of different modeling approachesfor predicting sandwich panel buckling response isdescribed. The study considers sandwich panels withanisotropic face sheets and a very thick core. Resultsfrom conventional analytical solutions for sandwichpanel overall buckling and face-sheet-wrinkling typemodes are compared with solutions obtained usingdifferent finite element modeling approaches. Finiteelement solutions are obtained using layered shellelement models, with and without transverse shearflexibility, layered shell/solid element models, with shellelements for the face sheets and solid elements for thecore, and sandwich models using a recently developedspecialty sandwich element. Convergence characteristicsof the shell/solid and sandwich element modelingapproaches with respect to in-plane and through-the-thickness discretization, are demonstrated. Results ofthe study indicate that the specialty sandwich elementprovides an accurate and effective modeling approach

for predicting both overall and localized sandwich panelbuckling response. Furthermore, results indicate thatanisotropy of the face sheets, along with the ratio ofprinciple elastic moduli, affect the buckling responseand these effects may not be represented accurately byanalytical solutions. Modeling recommendations arealso provided.

IntroductionThe increased performance requirements of futureaeronautical and aerospace vehicles, and the projectedincreased demand for air travel, suggest that moreefficient aeronautical and aerospace flight vehicleconcepts are needed. An example of a revolutionaryconcept for an efficient, large transport aircraft is ablended-wing-body type (BWB) of aircraft, whichblends the wings and fuselage into a single liftingsurface. Due to the shape of the BWB airplane, thepressurized centerbody region, which includes both thepassenger area and the cargo area, is non-circular. The


non-circular centerbody region is challenging from thestandpoint of structural design since the basic coverpanel structure carries both internal transverse pressureload and normal wing bending and torsion loads. Inorder to satisfy the performance and weightrequirements for the BWB aircraft, and other advancedconcepts that are subjected to bending or pressureloadings, advanced sandwich-type constructions withcomposite material face sheets and relatively thick cores(see Figure 1) offer a potential design advantage overconventional metallic materials and stiffened skinconstruction [1].

Most sandwich structures are defined using a three-layertype of construction, as illustrated in Figure 1. Theouter layers are thin, stiff, high-strength material; whilethe middle layer is a thick, weak, low-density material.Initial analytical work on sandwich structures treated thethree-dimensional sandwich structure as a pair ofmembrane face sheets held apart by a core material witha relatively large transverse shear stiffness. The bendingstiffness of the face sheets is ignored and the core isassumed to be inextensional in the transverse directionand has negligible stiffness in the in-plane directions.This type of model, called a sandwich of the first kind,has been applied successfully in many applications.However, a more robust formulation with additionalfidelity is needed to model complex nonlinear structuralbehavior including local failures such as face-sheetbuckling and face-sheet disbond. A sandwich of thesecond kind accounts for the out-of-plane response ofthe face sheets and the full three-dimensional behaviorof the core material.

Early analysis work for metallic sandwich structuresincludes Plantema [2], and design guidelines forsandwich structures are given in Ref. [3]. Generalinstability and face-sheet wrinkling are described byBenson and Mayers [4]. One aspect of their work wasrelated to defining a stability boundary between face-sheet wrinkling and general instability. Researchershave also studied the analysis of sandwich structureswith emphasis on the use of composite material facesheets and foam cores. Finite element formulations forsandwich panels are reviewed by Ha [5, 6].Displacement-based formulations and hybridformulations are considered as well as different through-the-thickness kinematic models. No numerical studiesare presented. Frostig [7] investigated sandwich panelbuckling using a higher-order theory which accounts fordifferent boundary conditions on the upper and lowerface sheets. Using a closed-form solution, he studied theinfluence on buckling of different boundary conditionsfor various panel aspect ratios and for both soft and stiffcores. Tessler et al. [8] present a {1,2}-order theoryaccounting for transverse shear and normal stresses and

strains. They present linear stress solutions forthermally loaded, simply supported sandwich plateswith laminated face sheets using an analytic, closed-form solution. Vonanch and Rammerstorfer [9]presented a Rayleigh-Ritz solution for face-sheetwrinkling of general unsymmetric sandwich panels withorthotropic face sheets. Comparison of their analyticalresults with unit-cell three-dimensional finite elementresults is used to verify their approach.

Bert [10] summarized different theories for sandwichplates with laminated composite face sheets that accountfor both transverse shear and transverse normal effects.Noor et al. [11] presented an exhaustive reference list(over 1300 citations) of analytical and computationalprocedures for sandwich structures. Librescu and Hause[12] presented a further survey and extended theformulation to include buckling and postbucklingresponse of flat and curved sandwich structuressubjected to mechanical and thermal loads. A Rayleigh-Ritz procedure for simply supported sandwich plateswas developed by Rao [13] where the bending stiffnessof the face sheets was ignored. Kim and Hong [14]extended Rao’s work to account for the face-sheetbending stiffness. Hadi and Matthews [15] presented aRayleigh-Ritz procedure based on a zigzag theory andaccounts for shear deformation in the face sheets.Comparisons with other Rayleigh-Ritz solutions werepresented for sandwich panels with thin face sheets.Results are reported for different face-sheet stackingsequences. Dawe and Yuan [16, 17] presented a finitestrip formulation using B-splines for sandwich panelswith anisotropic face sheets. However, no comparativenumerical studies of different finite element modelingstrategies for predicting the buckling response ofsandwich panels have been identified in the literature.

The present paper describes the basic buckling behaviorand response of sandwich panels loaded in axialcompression and compares buckling predictions forvarious levels of finite element modeling fidelity.Buckling results obtained from approximate analyticalexpressions are also compared with buckling resultsobtained from the finite element analyses. Differentfinite element models of the sandwich panel areconsidered including layered shell models, specialtysandwich element models, and layered shell/solidmodels. Numerical results obtained using the STAGS(STructural Analysis of General Shells) nonlinear finiteelement code [18] for the three different finite elementmodeling approaches are presented for selectedsandwich panel design parameters. Parameters varied inthe study include the face-sheet thickness and the corethickness as well as modeling fidelity. Particularattention is given to examining the buckling behavior fora specific panel aspect ratio, as the core thickness


becomes large. The results presented demonstrate theinterplay between finite element models with differentlevels of fidelity and the modeling requirements foraccurate prediction of sandwich panel bucklingresponse. General modeling guidelines are provided.

Sandwich Panel BucklingSandwich panel buckling involves several possiblemodes. Three buckling modes are considered in thisstudy. One mode is an overall panel buckling mode orgeneral instability mode where the face sheets and thecore buckle together into long wavelength buckles - longin the sense that the length of a half-wave is equal to oneof the planar dimensions of the sandwich panel.Another possible buckling mode consists of shortwavelength buckles where the face sheets and the coredo not exhibit any transverse extension through thethickness of the sandwich. This mode is a shortwavelength panel-buckling mode, and is commonlyreferred to as an asymmetrical wrinkling mode. Anothershort wavelength buckling mode is one where the corematerial is either stretched or compressed. This mode isa short wavelength face-sheet-wrinkling mode, and isreferred to as a symmetrical wrinkling mode. Othersandwich panel failure modes, such as face-sheetdimpling, core crushing, and face-sheet disbond, are notincluded in the present study.

Analytical predictions of overall sandwich panelbuckling often ignore the bending stiffness of the coreand assume that the core serves primarily to move theface sheets away from the midsurface of the panel. Asimple approach for predicting overall panel buckling ofa sandwich panel with homogeneous, isotropic facesheet and core materials loaded by in-plane compressionand with simply supported conditions on all edges ispresented by Brush and Almroth [19]. Vinson [20]developed an analytical model that includes the effectsof shear deformation for predicting overall panelbuckling of composite sandwich panels with orthotropicface sheets. This model, incorporated into PANDA2[21], is used herein as the analytical model for generalinstability and local face-sheet wrinkling predictions.

Buckling loads for general and face-sheet-wrinklinginstabilities are computed for a given sandwichconfiguration to determine the minimum buckling loadand the critical buckling mode shape for the sandwichpanel. Parameters having a significant influence on thebuckling response are the face-sheet thickness, tf, thecore thickness, hc, and the core shear stiffness, Gc aswell as the panel planar dimensions a and b.Researchers, such as Allen and Feng [22], have derivednon-dimensional parameters to characterize thestructural response of the sandwich panel. These

parameters can be used to guide an analyst in selecting amodeling fidelity for sandwich panel buckling analyses.

Finite Element ModelingSandwich panel finite element modeling has generallytaken one of three approaches depending on panelgeometry and constituent materials. The bucklingbehavior of a sandwich panel is dependent on thesandwich panel geometry and properties and mayinvolve a general instability mode, a local instabilitymode, or an interaction between the two. Accurateprediction of general or overall instability modesrequires adequate representation of the sandwichstiffnesses whereas prediction of local instabilitiesrequires detailed through-the-thickness modeling. Thefirst modeling approach exploits standard shell finiteelements. These models are referred to herein as layeredshell models, and for thin sandwich panels may providea first approximation of the global behavior. The secondapproach uses standard shell finite elements for the facesheets and solid three-dimensional finite elements forthe core. These models are referred to as layeredshell/solid models and provide a modeling approach forboth general and local response predictions. Theaccuracy of this approach, as well as its computationalcost, is related to the through-the-thickness modeling ofthe core material. The third approach is a full three-dimensional finite element model, in which solid three-dimensional elements are used to model both the facesheets and the core. These models are referred to asthree-dimensional solid models and they are typicallyreserved for detailed local modeling because of theircomputational cost. Recently, an additional modelingapproach has been implemented in the STAGS nonlinearfinite element code. This modeling approach, referred toherein as sandwich element models [18, 23], uses aspecialty element developed specifically for the analysisof sandwich structures. Sandwich element modelsembody the kinematics and stiffness of the sandwichstructure for less computational cost than the layeredshell/solid models. Sandwich element models provide acost-effective analysis approach for capturing sandwichbehavior for large-scale sandwich structure simulationsas well as for detailed local analyses. Details for thesemodeling approaches are described in the next sections.

Layered Shell ModelsLayered shell models exploit the existing plate and shellfinite element analysis features available in most finiteelement codes. The sandwich panel is modeled usingtwo-dimensional shell elements with at least threegroups of layers. The first group of layers correspondsto the laminate of one face sheet. The next group oflayers corresponds to the core material, and the lastgroup of layers represents the laminate of the other face


sheet. This approach gives an equivalent single-layerresult using classical lamination theory to compute thesandwich stiffness coefficients. In this model all layershave a common, unique rotation through the crosssection of the sandwich. Because the core materialgenerally only offers shear stiffness, shear-flexible C0

shell elements are typically used in layered shell models.Shell elements based on the classical Kirchhoff-Lovetheory (C1 shell elements) can be used; however, suchan approach ignores the transverse shear flexibilityoffered by the sandwich structure. Within STAGS, the4-node C1 shell element is called the 410 element (see[24]) and the 9-node ANS C0 shell element is called the480 element (see [25, 26]). Full integration is used forboth shell elements (i.e., 2×2 and 3×3, respectively).Finite element formulations for sandwich panels basedon high-order theories, such as Frostig [7] and Tessler etal. [8], are not available within STAGS and hence notincluded in this study.

Layered Shell/Solid ModelsThe next level of modeling uses shell elements to modeleach individual face sheet and solid elements to modelthe core material. These models are referred to herein aslayered shell/solid models. Each face sheet may be amulti-layer laminate and is modeled using shellelements, while the core is modeled using solidelements. Multiple solid elements through the corethickness may be required to represent the coredeformation accurately. Displacement-fieldcompatibility between the shell element and the solidelement must be considered during the modelingprocess. For example, the use of 4-node C1 shellelements for the face sheets with an 8-node solidelement leads to an incompatibility for the normaldisplacements. Lagrangian shell elements based on a C0

formulation, used with standard solid elements of thesame order, give displacement-field compatibility for thetranslational degrees of freedom. Combinations of the4-node, 8-node or 9-node C0 shell element with the 8-node, 20-node or 27-node solid element, respectively,give translational displacement compatibility. WithinSTAGS, the compatible set of shell and solid elementsare the 9-node ANS C0 shell element (480 element) andthe 27-node ANS C0 solid element (883 element). Fullintegration is used for both the shell and solid elements(i.e., 3×3 and 3×3×3, respectively). For the layeredshell/solid models, the shell element reference surface isdefined to coincide with the bounding surface of theadjacent solid element. In addition, the rotationaldegrees of freedom of grid points associated with thecore and not connected to the shell elements of the facesheets are constrained to zero.

Sandwich Element ModelsA specialty finite element for sandwich panel analysishas been formulated by Riks and Rankin [23] and isdescribed in the STAGS manual [18]. This specialtyelement exploits the existing shell finite elementtechnology available in the finite element code itself.The deformation of the face sheets is modeled by usingindividual shell elements for each face sheet. Couplingbetween the two shell elements is carried out byapplying an appropriate penalty function, consistent withthe material behavior of the core, to enforce thekinematics of the core as a function of the kinematics ofthe face sheets. The core is assumed to have generallyanisotropic three-dimensional elastic properties whosedeformation is defined by the change in distancebetween adjacent face sheets. For this special sandwichelement, the face sheets are intrinsically modeled usingthe standard STAGS 410 quadrilateral shell elements.Multiple sandwich elements may be stacked through thethickness of the core to provide a refined through-the-thickness discretization of the core material wherein theintermediate face sheets are treated as “phantom” facesheets with zero thickness. Between adjacent facesheets, the transverse shear strain varies linearly with thethickness coordinate and the transverse normal strain isconstant. The number of integration points through thecore thickness of each sandwich element can be chosento be either one or two depending on whether a spuriousmode is triggered by the boundary conditions in the facesheets. Stiffer results are generally obtained when twointegration points are used. Within STAGS, thisspecialty sandwich element is called the 840 element[18, 23]. This modeling approach provides acomputationally attractive approach for sandwich panelanalysis for large-scale structures and also provides acapability to assess detailed response characteristicsusing a local strip model.

Numerical Results and DiscussionThe basic geometry of the sandwich panels analyzed inthis study is defined in Figure 2. The upper and lowerface sheets are identical. Analytical predictions aremade following the approach of Vinson [20] asimplemented in PANDA2 [21]. Finite element analyses,using the STAGS nonlinear finite element analysis code[18], are conducted using layered shell models, layeredshell/solid models, and sandwich element models. Thelayered shell models use the 4-node C1 410 shellelement and the 9-node ANS C0 480 shell element. Thefinite element approximation for the out-of-planedisplacement is cubic for the 4-node element andquadratic for the 9-node element. For the layeredshell/solid models, the face sheets are modeled using the9-node ANS 480 shell element and the core is modeled


using the 27-node ANS 883 solid brick elementsthrough its thickness. This approach provides acompatible displacement field between the finiteelements in the face sheets and those in the core. For thesandwich element models, the 840 sandwich elements[18, 23] are used through the core thickness.

Full panel models are defined as finite element modelsof the entire sandwich panel. Spatial discretization of thefull panel is related to the anticipated buckling mode.Thus, some knowledge of the panel buckling response isneeded to develop these full panel analysis models. Fora rectangular panel, the number of half-waves, m, alongthe panel length in a general-instability panel-bucklingmode is a function of the panel aspect ratio a/b ,assuming a single half-wave across the width of thepanel. Generally, five to six grid points per half-waveare required for accurate buckling predictions. Tocapture a face-sheet-wrinkling mode, the finite elementmesh needs to be sufficiently refined to represent short-wavelength buckles.

Local strip models are defined as finite element modelsof a localized region with significant through-the-thickness modeling detail so that short wavelengthwrinkling modes can readily be detected. A local stripmodel is used to verify short-wavelength face-sheet-wrinkling behavior predictions obtained with the fullpanel models. Local strip models represent a thinlongitudinal slice of a panel away from edge effects. Inthe present study this strip model has a width equal to2.54 mm and has 25 layers of sandwich elementsthrough the core thickness. One integration point isused through the thickness of each core layer. Thelength of the local strip model depends on thewavelength of the buckling mode. Therefore, severallocal models, with different lengths, are analyzed toensure that the lowest buckling load is determined.

Numerical results for two cases are presented. The firstcase considers a square sandwich panel with single-layerorthotropic face sheets and an isotropic core. This casestudies the influence of different modeling approachesand boundary condition applications on the bucklingresponse predictions for a given sandwich configurationloaded in uniaxial compression. The second caseconsiders a rectangular sandwich panel with isotropicface sheets and an isotropic core. This case examinesthe influence of different modeling approaches on thebuckling response predictions for sandwich panels withdifferent core thickness and face-sheet thickness.

Case 1: Square Sandwich PanelSeveral researchers [13-15, 17] have analyzed thebuckling, under uniaxial compressive stress, of a simplysupported 225-mm square sandwich panel (a/b=1) with

identical single-layer orthotropic face sheets of a givenfiber orientation angle and isotropic core. The facesheets have thickness tf equal to 0.2 mm and the core hasthickness hc equal to 10 mm (h c/a=0.044). Themechanical properties of the face-sheet material are:E1=229.0 GPa, E2=13.35 GPa, G12=5.25 GPa andν12=0.315 (i.e., E2/E1=0.058). Core material data aregiven in Table 1. The fiber orientation of the face-sheetmaterial is varied from zero to ninety degrees where thezero-degree orientation is parallel to the loadingdirection. For fiber orientation angles other than 0- and90-degrees, the face sheets exhibit anisotropic behaviorand the D16 and D 26 bending stiffness terms becomenonzero and large relative to the other bending stiffnessterms. Consequently when the D16 and D26 bendingstiffness terms become nonzero, the buckling modeshapes may become skewed, relative to the loading axiswith non-straight modal node lines. In addition, as theface-sheet fiber-angle orientation increases, the numberof half-waves in the general-instability mode increases.These complexities in the buckling mode shape imposeadditional modeling requirements on the analysismodels over those required for the analysis of a panelwithout anisotropic effects.

The sandwich panel is loaded by a uniform endshortening, uo, applied to the entire loaded edge (Edge 1or x=0 in Figure 2). The in-plane pre-buckling stressstate has uniform stress in the longitudinal direction, andthe other two in-plane stress components are equal tozero. To achieve this pre-buckling stress state, theboundary conditions applied in the finite elementmodels to all grid points along each edge are:u u w x y z= = = = =0 0 and θ θ θ along Edge 1 (x=0);

w x y z= = = =θ θ θ 0 along Edges 2 and 4 (y=b and

y=0, respectively); and 0= ==== zyxw u θθθalong Edge 3 (x=a). In addition, a point at the center ofthe panel has the transverse displacement v set equal tozero to remove rigid-body motion. These boundaryconditions result in a uniform stress state in the facesheets.

Having established a uniform uniaxial stress state,boundary conditions for the buckling analyses aredefined next. All nodes through the thickness along thefour edges of the sandwich panel are simply supportedfor the buckling calculations. These buckling boundaryconditions are: 0 = == xwv θ along Edges 1 and 3

(x=0 and x=a, respectively); and, 0= == yw u θ

along Edges 2 and 4 (y=b and y=0, respectively). Forthe layered shell modeling approach used in the presentstudy, these simply supported boundary conditions arestraightforward to impose on the finite element model.However, for the layered shell/solid modeling approach


and the sandwich element modeling approach,independent approximations may be made for thebending rotations of each face sheet and core. In theseapproaches, the simply supported boundary conditionsare defined so that shear deformations are prevented inthe cross-sectional planes along the panel edges [15].For the sandwich element modeling approach, thesimply supported boundary conditions for buckling maybe imposed using rigid links (multi-point constraints),along the boundaries, that tie the lower face sheet to theupper face sheet – similar to the kinematics relations ofclassical plate theory – when the pre-stress loading is anapplied in-plane force and when constraints areemployed to impose uniform end shortening. For thebuckling calculations, the degrees of freedom associatedwith the end-shortening response are permitted to befree. For models with multiple sandwich elementsthrough the core thickness, boundary conditionsimposed along the panel edges at intermediate or“phantom” face sheets are the same as those imposed onthe bounding face sheets.

Results of a convergence study conducted using thesandwich element modeling approach to predict thebuckling response of a square sandwich panel with a 30-degree face-sheet fiber orientation angle are shown inFigure 3. The open symbols on dashed lines denoteresults obtained using models with one integration point(IP) through the thickness of the sandwich element, andthe filled symbols on solid lines denote results obtainedusing models with two integration points. One, two,four, and eight layers of sandwich elements wereanalyzed for meshes with different levels of in-planediscretization. The predicted buckling loads arenormalized by the solution obtained using the layeredshell/solid approach with four solid elements throughthe core thickness and a 25×25 planar mesh of nodes(i.e., the buckling load used in the normalization is340.0 N/mm). This solution obtained using the highestfidelity model is referred to herein as the referencesolution and is denoted in Figure 3 by the “×” symbol ona solid line. Hadi and Matthews [15] reported a value of467.8 N/mm, and Yuan and Dawe [17] reported a valueof 382.6 N/mm. Hadi and Matthews [15] used aRayleigh-Ritz solution with tr igonometricapproximations for the in-plane variation and a zigzagtheory through the thickness. Their results give abuckling solution stiffer than the present solution due toimplicit boundary conditions from the displacementfield approximations. Yuan and Dawe [17] used a finitestrip solution with B-spline approximations for the in-plane variation and the through-the-thicknessapproximations for the in-plane displacements arequadratic and the out-of-plane displacements are linear.Their results are much improved due to their treatmentof the boundary conditions, yet still stiffer than the

present reference solution that uses a piecewisequadratic approximation through the core thickness.

The predicted buckling mode shape is a general-instability mode with a skewed single half-wave in boththe x and y directions. As the number of grid pointsalong each edge of the panel increases, the normalizedbuckling load converges from above, for all levels of thethrough-the-thickness discretizations and planardiscretizations. In addition as the number of sandwichelement layers increases, the convergence trends arewell behaved, consistent, and approach the referencesolution. The buckling load predicted using eight layersof sandwich elements is 5% higher than the referencesolution obtained using the layered shell/solid approach.As the number of sandwich layers used to model thecore thickness increases, the results are less sensitive tothe number of integration points used through thethickness of each sandwich element. Use of oneintegration point gives a more flexible solution than thesolution obtained using two integration points.However, when using one integration point, it ispossible in some situations to trigger a spurious mode.In such cases, two integration points should be used.

Results of the present analysis approach, of Hadi andMatthews [15], and of Yuan and Dawe [17], expressedin terms of the in-plane stress resultant, are shown inFigure 4 as a function of the fiber orientation in the facesheet. The present results were generated using a fullpanel model with 25 grid points in each planar direction.These 625 grid points were used to define a finiteelement mesh of either 576 4-node quadrilateral shellelements or 169 9-node quadrilateral shell elements.Results of the present analysis are presented for layeredshell models with and without shear deformation,layered shell/solid models, and sandwich elementmodels. The results shown in Figure 4 indicate that thelayered shell finite element models are stiffer andpredict higher buckling loads than the layered shell/solidfinite element model (reference solution). The resultsshown in Figure 4 indicate that the layered shell modelwithout shear deformation (410 element models)provides a much stiffer solution (higher buckling loads)than the layered shell model with shear deformation(480 element models) and the buckling load decreases asthe fiber angle increases from zero (results indicated byfilled symbols). Results for the layered shell model withshear deformation (480 element models) indicate a moreflexible solution (open symbol results) than thatobtained using the 410 element. Results obtained for thesandwich element model (840 element models) withrigid links along the boundary edges, denoted by the “×”symbol, indicate a response similar to that predictedusing a layered shell model with shear deformation. Therigid links impose classical plate theory kinematics


along the boundaries while internally the shearflexibility of the core material is included in thesimulation. The buckling load is nearly constant up to afiber angle of 30 degrees and then decreases as the fiberangles increases further.

The results shown in Figure 4 indicate that all finiteelement models of the present study and the analysisapproach of Yuan and Dawe [17] predict a maximumbuckling load when the face-sheet fiber orientation angleequals zero degrees and a progressive decrease inbuckling load occurs as the fiber orientation angleincreases. Hadi and Matthews [15] and additionalearlier analyses [13, 14] using a Rayleigh-Ritz approachwith trigonometric approximations, however, predictthat the buckling load increases with increase in theface-sheet fiber angle from 0-degrees up to 40-degrees.Then the buckling load decreases rapidly to the bucklingload value for the 90-degree case which isapproximately half the buckling load value for the 0-degree case. This trend is similar to the trend obtainedfor a laminate composite plate with an aspect ratiogreater than one (e.g., Ashton and Whitney [27]).However, for a square plate with these materialproperties, the buckling load decreases as the fiber angleincreases. Yuan and Dawe [17] report a decrease inbuckling load with an increase in fiber angle for thesquare sandwich panel under consideration and indicatethat Rayleigh-Ritz solutions based on trigonometricseries overly constrain the panel as material anisotropyincreases. Double sine-series solutions for finite panelswith simply supported boundary conditions and havingsymmetric laminates give rise to artificial derivativeboundary constraints that prevent convergence to thecorrect solution [28, 29]. Stone and Chandler [29]report that buckling loads will be overestimated and maycontribute to misleading trends or conclusions.

All finite element results from the present study indicatethe same buckling mode shape (general instability) forany face-sheet fiber orientation. Initially (0-degreecase) the mode shape involves only one half-wave inboth directions. As the angle increases, the bucklingmode shape becomes skewed and tends to follow thefiber angle. At approximately 60 degrees, the skewedmode shape has two longitudinal half-waves and onetransverse half-wave. This pattern becomes less skewedas the face-sheet fiber angle approaches 90 degrees. InFigure 5, contour plots of the out-of-plane displacementcomponent of the buckling mode shape are shown toillustrate this effect for different face-sheet fiber angles.

The effect of fiber angle on the buckling results obtainedusing the sandwich element modeling approach isshown in Figure 6. Results are shown for all modelsusing one and four sandwich elements through the core

thickness with each sandwich element having one or twointegration points in the thickness direction (open andfilled symbols, respectively). The finite element modelsof the full panel have 25 grid points in each planardirection. The present results bracket the solutionsreported by Yuan and Dawe [17] and appear to beconverging to the reference solution. As the number ofsandwich elements through the core thickness increases,the sensitivity to the number of integration pointsdecreases. These results illustrate the effectiveness ofthe STAGS sandwich element formulation.

Due to the nature of this problem and the differingresults obtained by various researchers, two additionalstudies were performed to study the effect of the facesheet moduli ratio and panel aspect ratio on the generalinstability response of a sandwich panel. Results wereobtained using the sandwich element modeling approachwith four elements through the core thickness, and twointegration points through the thickness of each element.Results for the present square panel with E2/E1 = 0.058are shown in Figure 7 along with results for arectangular panel with an aspect ratio, a/b, equal to two,and face-sheet moduli ratio E2/E1 = 0.058 , and for asquare panel with the transverse modulus, E2, doubled(E2/E1 = 0.116). Results for the square panels withE2/E1 = 0.058 and 0.116 are shown by the solid anddashed lines, respectively, and results for the rectangularpanel are shown by the dotted line. As shown in Figure7, doubling the ratio of the face-sheet moduli ratio in thesquare panel causes an overall increase in the panelbuckling load. Increasing the panel aspect ratio to two,while holding the face-sheet moduli ratio constant,results in a variation in the buckling load with face-sheetfiber angle similar to the variation obtained for acomposite plate with similar material properties and anaspect ratio greater than one. For this case, the bucklingload increases with an increase in face-sheet fiber angleup to 25 degrees, and then declines with increase inface-sheet fiber angle.

Case 2: Rectangular Sandwich PanelThe panel considered has a rectangular planform and is508-mm long and 254-mm wide (a/b=2). Identicalaluminum face sheets of thickness tf on an aluminumhoneycomb core of thickness hc define the sandwichpanel. Two face-sheet thicknesses are considered in thisstudy: a thin face sheet with thickness equal to 0.508mm and a thick face sheet with thickness equal to 2.794mm. The core thickness is varied from very thin, equalto the face-sheet thickness, to very thick, approachinghalf the panel width. As the core thickness increases,the buckling response transitions from a generalinstability mode to a short wavelength face-sheet-wrinkling mode. Young’s modulus, E , for thealuminum alloy is equal to 68.95 GPa and Poisson’s


ratio is equal to 0.3. The honeycomb core material istreated as homogeneous isotropic material with elasticmechanical properties given in Table 1. Comparativestudies are performed for a specific panel geometry andmaterial data. Analytical results are computed as thefirst level of analysis following Vinson’s [20] approachas implemented in PANDA2 [21]. Finite elementanalyses are also conducted using STAGS. Finiteelement results are obtained using full panel models andlocal strip models. All modeling approaches are used inthe full panel models, and only the sandwich elementmodeling approach is used in local strip models.

In the pre-stress condition, the sandwich panel is loadedby a uniform end shortening, uo, applied to the sandwichface sheets on the ends of the panel at x=0 and x=a (seeFigure 2). For the buckling calculations, the boundaryconditions applied to all grid points along each edge are:u u w x z= = = =0 0 and θ θ along Edge 1 (x = 0 );

w y= =θ 0 along Edges 2 and 4 (y=b and y=0,

respectively); and u u w x z= − = = =0 0 and θ θ along

Edge 3 (x=a). In addition, a point at the center of thepanel has v=0 to remove rigid body motion.

The finite element mesh for the full panel used in thepresent study has 81 grid points along the panel lengthand 21 grid points across the panel width. This spatialdiscretization (1,701 grid points) is held constant duringthe parametric studies and is adequate to representbuckling modes with up to sixteen half-waves along thepanel length and four half-waves across the panel width.The mesh of 4-node elements (3,200 elements) has fourtimes the number of elements as the mesh of 9-nodeelements (800 elements). A layered shell modelingapproach is attractive when general instability isanticipated because of the simplicity, ease of modelingand low computational cost of this modeling approachas compared to the other modeling approaches.

In the present study, the local strip model with a widthequal to 2.54 mm and 25 sandwich elements through thecore thickness is used to investigate face-sheetwrinkling. One integration point is used through thethickness of each sandwich element. The local stripmodel has a uniform stress state in the face sheets andbuckling boundary conditions associated with asymmetric response. For small values of hc/a, thePANDA2 wrinkling loads correlate well with those fromthe STAGS local strip model. For larger values of hc/a,however, the PANDA2 wrinkling loads are much lowerthan those predicted by the STAGS model. Apparentlythe through-the-thickness assumptions made inPANDA2 are quite conservative. Based on thePANDA2 predictions, the core thickness thatcorresponds to a transition from a general-instability

mode to a face-sheet-wrinkling mode is approximatelyhc/a=0.020 for the thin face-sheet case and hc/a=0.060for the thick face-sheet case.

Linear buckling analysis results obtained from the fullpanel layered shell models indicate overall panelbuckling regardless of the core thickness and the face-sheet thickness. In all cases, the buckling modecorresponds to an overall panel mode with two half-waves along the length and one half-wave across thepanel width. In addition, the buckling results obtainedusing the 410 element are higher than the bucklingresults obtained using the 480 shear-flexible element, asexpected. Good correlation between the layered shellmodel results and the analytical results from PANDA2is obtained for sandwich structures with a very thin core.However, the finite element results based on the layeredshell modeling approach quickly tend to deviate fromthe analytical results as the core thickness increases for agiven panel length (increasing values of hc/a) due tolimitations in the layered shell models for modeling thecore shear behavior and through-the-thicknessflexibility.

Results obtained using the analytical model inPANDA2, the STAGS local strip wrinkling model, andthe STAGS full panel layered shell/solid model arecompared in Figure 8. PANDA2 wrinkling predictionsare shown by the dotted line, the STAGS wrinklingmodel predictions are shown by the dashed line, and theSTAGS full-panel layered shell/solid model predictionsare shown by the symbols. In the layered shell/solidmodels, the core material is discretized using either oneor four 27-node solid brick elements through the corethickness (open symbols and filled symbols,respectively, in Figure 8). PANDA2 overall panelbuckling results are also shown as a solid curve inFigure 8. The results shown in Figure 8 indicate that thetransition from an overall panel buckling mode to awrinkling buckling mode occurs at a small value of thecore thickness to panel length ratio (hc/a). In addition,these results show excellent agreement between thePANDA2 wrinkling results and the wrinkling resultsobtained with the local strip model for moderately thicksandwich panels (for ratio of core thickness to face-sheetthickness less than 50). For panels with a ratio of corethickness to face-sheet thickness greater than 50,PANDA2 wrinkling results appear to be conservative, asa result of an assumed linear displacement responsethrough the thickness of the core. The local strip modelallows a piecewise linear distribution using 25 sandwichelements through the core thickness. In addition, thebuckling results summarized in Figure 8 show that forthe thin face-sheet case results obtained using thelayered shell/solid models correlate well with theanalytical results when just a single solid element


through the core thickness is used. Good correlation isalso shown between the results obtained using the localstrip model and results obtained using the layeredshell/solid model when four solid elements through thecore thickness are used. For the thick face-sheet casewith tf=2.974 mm and hc/a=0.4, the layered shell/solidmodels predict a buckling load that is approximately 8%higher than the buckling load predicted by the local stripmodel. The results for tf=2.974 mm and hc/a=0.4indicate that the buckling load predicted by the layeredshell/solid models is converging to 430.03 kN. Theconvergent behavior of the layered shell/solid modelingapproach is evident even for short wavelength face-sheet-wrinkling behavior with only a 2% changebetween the buckling load predictions obtained with oneelement and four elements through the core thickness.

Similar results for sandwich element models aresummarized in Figure 9. The sandwich element modelshave the same discretization as the finite element modelswith 4-node quadrilaterals. In these models, the corematerial is discretized using either one or four sandwichelements through the core thickness with “phantom”face sheets for the intermediate layers (open symbolsand filled symbols, respectively, in Figure 9). Results inFigure 9 show that for the thin face-sheet case, thesandwich element models with one integration pointcorrelate well with the analytical results and with resultsobtained using the local strip model. Use of twointegration points through the thickness of eachsandwich element results in a stiffer response and higherbuckling loads. Similar trends are observed for the thickface-sheet cases. However, as the core thicknessincreases, this convergent behavior is less evident. Forexample, results obtained using one integration point perlayer for the thick face-sheet case and hc/a=0.4, indicatean increase in buckling load as the number of layersincreases from one to four. Further analyses conductedusing six and eight sandwich elements through thethickness predict buckling loads of 432.99 kN and435.31 kN, respectively, thereby indicating a slowconvergence. Use of two integration points per layerresults in a very stiff solution for a single layer model,and the buckling load is nearly four times the valuepredicted with a four layer model. Additional analyseswere performed using six and eight sandwich elementswith predicted buckling loads of 479.28 kN and 461.93kN, respectively. Again the results appear to beconverging but extremely slowly for the case of twointegration points per layer. These results indicate thatthe sandwich element modeling approach, while quitepowerful for modeling large-scale structures exhibiting ageneral instability, is not effective in a large-scale sensefor predicting face-sheet-wrinkling buckling modesunless local strip models are used.

Finally, a limited review of the computational costassociated with the sandwich element and layeredshell/solid element modeling approaches was performedfor the thin face-sheet case with hc/a=0.1. Thesandwich element approach exploits the 4-node C1 shellelement formulation while the layered shell/solidelement approach is based on the ANS formulation for a9-node C0 shell element and a 27-node solid element.For a fixed finite element discretization (same numberof grid points) in the plane of the sandwich panel, thecritical buckling load for the sandwich elementmodeling approach changes from 52.1 kN to 67.6 kN to76.8 kN as the number of through-the-thickness layersincreases from one to two to four, respectively.Similarly, the layered shell/solid modeling approachgives critical buckling loads that change from 77.7 kN to75.2 kN to 73.7 kN as the number of through-the-thickness layers increases from one to two to four,respectively. Clearly for this wrinkling mode response,the layered shell/solid modeling approach convergesmore quickly than the sandwich element modelingapproach. The advantage of using the sandwich elementmodeling approach from a computational costperspective decreases as the number of sandwichelement layers increases. The computational effortassociated with a sandwich element model is defined interms of the number of through-the-thickness layerswhich in turn increases the size of the global stiffnessmatrix. The number of equations in the global stiffnessmatrix for a model with a single layer of sandwichelements is approximately 60% of that for a layeredshell/solid model with a single solid element in the core.As the number of layers of sandwich elements increases,the number of equations increases proportionally inaddition to the computational effort expended at theelement level. Buckling values obtained using eitherapproach with four layers through the thickness arenearly equal. However, the solutions obtained with asingle layer of sandwich elements require only 60% ofthe computational effort required to obtain a solutionwith a single solid element in the core. Approximatelythe same level of solution accuracy is achieved by usinga single solid element in the core of a layered shell/solidmodel as is obtained by using four layers of thesandwich elements. In this case, the computational costfor the sandwich element modeling approach is overfour times the computational cost of the layeredshell/solid model to achieve equal levels of solutionaccuracy. Extrapolation of this cost differential to large-scale structural panels is not expected to apply becauseof solid element aspect ratio limits. However, use of thesandwich element modeling approach for large-scalesandwich structures provides an effective means todetermine its general-instability buckling response.


ConclusionsA comparative study of different modeling approachesfor predicting sandwich panel buckling response hasbeen presented. The study considered sandwich panelswith anisotropic face sheets and a very thick core.Results from conventional analytical solutions forsandwich panel overall buckling and face-sheet-wrinkling type modes were compared with solutionsobtained using different finite element modelingapproaches. The numerical analyses were conductedusing the STAGS nonlinear finite element code, and theanalytical results were computed using PANDA2.Finite element solutions were obtained using layeredshell element models, with and without transverse shearflexibility, layered shell/solid element models, that useshell elements for the face sheets and solid elements forthe core, and sandwich models using a recentlydeveloped specialty sandwich element. Convergencecharacteristics of the shell/solid and sandwich elementmodeling approaches with respect to in-plane andthrough-the-thickness discretization, were demonstrated.

Results of the study indicate that the specialty sandwichelement formulation implemented in STAGS providesan accurate and effective modeling approach forpredicting both overall and localized behavior. Thismodeling approach provides the flexibility for modelingan entire panel or for modeling local detail through thecore thickness. Results of the study indicate thattransition from a general instability mode to a shortwavelength face-sheet-wrinkling mode occurs as thecore thickness is increased. Modeling fidelity to captureboth possible modes must be provided near thetransition region. Furthermore, results indicate thatanisotropy of the face sheets, along with the ratio ofprinciple elastic moduli, significantly affect the bucklingresponse, and these effects may not be accuratelyrepresented by analytical solutions.

AcknowledgementThe work of the second, third and fourth authors wassupported by the NASA Langley Research Center undercontracts NAS1-00135, GS-35F-0038J (Task 1730), andNAS1-99069, respectively.

References

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Table 1. Elastic mechanical propertiesfor core materials

Material Properties Case 1 Case 2Ecx = Ecy, MPa 200 0.6895

Ecz, MPa 200 68.95Gcxy, MPa 146 0.265Gcxz, MPa 146 82.74Gcyz, MPa 90.4 49.64

νcxy 0.3 0.3νcxz = νcyz 0.3 0.01


Figure 1. Photograph of thick sandwich panel segment.

b

hc

x,u

y,v

z,w

tf

tfa

Face sheet

Face sheet

Core

θ

θ

θz

y

θx

Edge 2

Edge 4

Edge 1 Edge 3

x

y

Figure 2. Sandwich panel geometry.


x

x

x x x

1.6

1.3

1.0

0.70 10 20 30

Number of grid points per edge

Nor

mal

ized

buc

klin

g lo

ad, (

Nx)

cr/(

Nx)

cr_3

D

Layered shell/solid - 1, 2, 4 layersOne sandwich layer - 2 IPTwo sandwich layers - 2 IPFour sandwich layers - 2 IPEight sandwich layers - 2 IPOne sandwich layer - 1 IPTwo sandwich layers - 1 IPFour sandwich layers - 1 IPEight sandwich layers - 1 IP y

x

Nx

Nx

θ

Figure 3. Effect of mesh refinement for sandwich element models for the square sandwich panel with a 30-degree fiber angle.


800

600

400

200

00 30 60 90

Ply layup angle, θ, degrees

Buc

klin

g lo

ad, (

Nx)

cr, N

/mm

y

x

Nx

Nx

x x

x

x x

x

xx

x x

θ

Reference solution (layered shell/solid)Hadi & Matthews [15]Yuan & Dawe [ 17]Layered shell model using STAGS 410 elementsLayered shell model using STAGS 480 elementsSingle-layer sandwich element modelwith rigid links

Figure 4. Effect of face-sheet fiber angle on buckling load for layered shell/solid and layered shell modelsof the square sandwich panel.


a) 0-degree case

c) 60-degree case

b) 40-degree case

d) 90-degree case

1.0

-1.0

0

Figure 5. Contour plots of the out-of-plane displacement component of the buckling mode shape fordifferent face-sheet fiber angles.


800

600

400

200

00 30 60 90


Buc

klin

g lo

ad, (

Nx)

cr, N

/mm

y

x

Nx

Nx

θ

Reference solution (layered shell/solid)Hadi & Matthews [15]Yuan & Dawe [ 17]Single-layer sandwich element model - 2 IPFour-layer sandwich element model - 2 IPSingle-layer sandwich element model - 1 IPFour-layer sandwich element model - 1 IP

Figure 6. Effect of face-sheet fiber angle on buckling load for layered shell/solid model and sandwichelement models of the square sandwich panel.


800

600

400

200

00 30 60 90


Buc

klin

g lo

ad, (

Nx)

cr, N

/mm

y

x

Nx

Nx

θ

b

a

E2/E1 = 0.058, a/b = 1E2/El = 0.116, a/b = 1E2/E1 = 0.058, a/b = 2

Figure 7. Effect of face-sheet fiber angle, aspect ratio, and face-sheet elastic modulus ratio (E2/E1) onbuckling load for a sandwich element model of the square sandwich panel.


1000

600

800

400

200

00 .15 .30 .45

Ratio of core thickness to panel length, hc/a

Buc

klin

g lo

ad P

, kN

STAGS solid/shell model - 1 solid - thin face sheetSTAGS solid/shell model - 4 solids - thin face sheetSTAGS solid/shell model - 1 solid - thick face sheetSTAGS solid/shell model - 4 solids - thick face sheetPANDA2 panel bucklingPANDA2 face sheet wrinkling STAGS local strip model

tf = 0.508 mm

tf = 2.794 mm}

}

Figure 8. Buckling load predictions from analytical model, local strip sandwich element model and fullpanel layered shell/solid model.


1000

600

800

400

200

00 .15 .30 .45

Ratio of core thickness to panel length, hc/a

Buc

klin

g lo

ad P

, kN

STAGS sandwich model, 1 layer, 1 IP- thin face sheetSTAGS sandwich model, 4 layers, 1 IP - thin face sheetSTAGS sandwich model, 1 layer, 1 IP- thick face sheetSTAGS sandwich model, 4 layers, 1 IP- thick face sheetPANDA2 panel buckling PANDA2 wrinklingSTAGS local strip model

tf = 0.508 mm

tf = 2.794 mm}

}

Figure 9. Buckling load predictions from analytical model, local strip sandwich element model and fullpanel sandwich element model.

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