Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 844-868A9626886, NAG1-1162, F49620-93-1-0184, AIAA Paper 96-1636

Thermomechanical buckling and postbuckling responses of composite panelswith skewed stiffeners

Ahmed K. NoorNASA, Langley Research Center, Hampton, VA

James H. Starnes, Jr.NASA, Langley Research Center, Hampton, VA

Jeanne M. PetersNASA, Langley Research Center, Hampton, VA

IN:AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and MaterialsConference and Exhibit, 37th, Salt Lake City, UT, Apr. 15-17, 1996, Technical Papers. Pt. 2(A96-26801 06-39), Reston, VA, American Institute of Aeronautics and Astronautics, 1996,

p. 844-868

The results of a detailed study of the buckling and postbuckling responses of composite panels with skewed stiffenersare presented. The panels are subjected to applied edge displacements and temperature changes. A first-ordershear-deformation geometrically nonlinear shallow-shell theory that includes the effects of laminated anisotropicmaterial behavior is used to model each section of the stiffeners and the skin. A mixed formulation is used in theanalysis with the fundamental unknowns consisting of the generalized displacements and the stress resultants of thepanel. The nonlinear displacements, strain energy, transverse shear stresses, transverse shear strain energy density, andtheir hierarchical sensitivity coefficients are evaluated. The hierarchical sensitivity coefficients measure the sensitivityof the buckling and postbuckling responses to variations in the panel stiffnesses, the effective material properties of theindividual layers, and the constituent material parameters. Numerical results are presented which show the effects ofvariations in the material properties of the skin and the stiffener on the buckling and postbuckling responses of thepanel as well as on the sensitivity coefficients. (Author)

Page 1

A96-26886 AIAA-96-1636-CP

THERMOMECHANICAL BUCKLING AND POSTBUCKLING RESPONSES OFCOMPOSITE PANELS WITH SKEWED STIFFENERS

Ahmed K. Noor,* James H. Starnes, Jr.** and Jeanne M. Peters***Center for Computational Structures Technology

University of VirginiaNASA Langley Research Center

Hampton, VA 23681

Abstract

The results of a detailed study of thebuckling and postbuckling responses ofcomposite panels with skewed stiffeners arepresented. The panels are subjected to appliededge displacements and temperature changes.A first-order shear-deformation geometricallynonlinear shallow-shell theory that includes theeffects of laminated anisotropic materialbehavior is used to model each section of thestiffeners and the skin. A mixed formulation isused in the analysis with the fundamentalunknowns consisting of the generalizeddisplacements and the stress resultants of thepanel. The nonlinear displacements, strainenergy, transverse shear stresses, transverseshear strain energy density, and theirhierarchical sensitivity coefficients areevaluated. The hierarchical sensitivitycoefficients measure the sensitivity of thebuckling and postbuckling responses tovariations in three sets of interrelatedparameters; namely, the panel stiffnesses; theeffective material properties of the individuallayers; and the constituent material parameters(fibers, matrix, interface and interphase).Numerical results are presented for rectangularpanels with open section I-stiffeners, subjectedto edge shortening and uniform temperaturechange.

*Professor of Aerospace Structures andApplied Mechanics, and Director, Center forComputational Structures Technology,University of Virginia, Fellow AIAA.**Head, Structural Mechanics Branch, FellowAIAA.***Senior Programmer Analyst, Center forComputational Structures Technology,University of Virginia

The results show the effects of variationsin the material properties of the skin andthestiffener on the buckling and postbucklingresponses of the panel, as well as on thesensitivity coefficients.

Introduction

The potential of using the directionaldependence of composite properties indesigning tailored structures to improvestructural performance has received increasingattention in recent years (see, for example,Refs. 1 and 2). Structural tailoring can beachieved by using the bending-torsionalstiffness coupling of composite structures, ashas been demonstrated in the X-29 wing. Itcan also be achieved in metallic stiffened panelsby skewing the stiffeners with respect to theaxes of the panel, thereby introducingextensional-shear stiffness coupling. A recentnumerical and experimental study has beenreported on the combined effects of tailoringboth skin laminate anisotropy and stiffenerorientation on the buckling and postbuckling ofcomposite panels, subjected to edgedisplacements3. The present paper is anextension of the study reported in the citedreference. The extensions include: a) study ofthe effect of uniform temperature change; andb) development of hierarchical sensitivitycoefficients of the buckling and postbucklingresponses.

The panels considered are rectangular andflat with a single centrally located I-shapedstiffener. Both the skin and the stiffenerconsist of a number of perfectly bonded layers.The individual layers are assumed to behomogeneous and anisotropic. A plane ofthermoelastic symmetry exists, at each point ofthe skin and the stiffener sections, parallel tothe middle plane of the section. The loading

844

consists of an applied edge displacement and auniform temperature change. The skin andeach section of the stiffener are modeled asplate elements.

Mathematical Formulation

The analytical formulation is based on afirst-order shear-deformation shallow-shelltheory, with the effects of large displacementsand laminated anisotropic material behaviorincluded. A linear Duhamel-Neumann typeconstitutive model is used and the materialproperties are assumed to be independent oftemperature. The thermoelastic constitutiverelations used in the present study are given inAppendix I. The panel is discretized by usingtwo-field mixed finite element models. Thefundamental unknowns consist of the nodaldisplacements and the stress resultantparameters. The stress resultants are allowedto be discontinuous at interelement boundariesin the model. The sign convention for thegeneralized displacements is shown in Fig. 1.The external loading consists of an appliededge displacement qe and a uniformtemperature change T (independent of thecoordinates XL X2 and xa).

Governing Finite Element Equations

The governing finite element equationsdescribing the postbuckling response of thepanel can be written in the following compactform:

{f(Z)}=[K]{Z}+{G(Z)}-qi{QO)}-q2{Q(2)}=0(1)

where [K] is the global linear structure matrixwhich includes the flexibility and the linearstrain-displacement matrices; {Z} is theresponse vector which includes both unknown(free) nodal displacements and stress-resultantparameters; (G(Z)} is the vector of nonlinearterms; qi and q2 are edge displacement andthermal strain parameters, respectively; {QW}is the vector of normalized mechanical strains;and {Q(2)| is the vector of normalized thermalstrains. The form of the arrays [K], (G(Z)},{Q(0} and {Q(2>} is given in Ref. 4.

The procedures for determining thestability boundary, and the postbifurcationequilibrium configurations, corresponding tospecified values of the parameters qi and q2are described in Ref. 4. If an incremental-iterative technique such as the Newton-Raphson method is used, the recursion formulafor the rth iteration can be written in thefollowing form:

and

(2)

(3)

where {AZJW is the change in the responsevector during the rth iteration cycle; and therange of I,J is the total number of componentsof the response vector {Z}.

Sensitivity of the Postbuckling Response

The derivatives of the postbucklingresponse with respect to the laminate, layer andmicromechanical parameters are obtained bydifferentiating Eqs. 1. The resulting linearalgebraic equations have the following form:

(4)

Note that the matrix on the left-hand side ofEqs. 4 is identical to that used in the Newton-Raphson iterative process (see Eq. 2).Therefore, if the Newton-Raphson technique isused in generating the postbuckling response,the evaluation of each sensitivity coefficientrequires the generation of the right-hand side ofEqs. 4, and a forward-reduction/back-substitution operation only (no decompositionof the left-hand side matrix is required). Theexplicit form of the arrays {DQ^VaX,} and

is given in Ref. 4.

845

Multiple-parameter reduction methodshave been developed for substantially reducingthe number of degrees of freedom used in theinitial discretization in order to reduce the costof generating the stability boundary, thepostbifurcation equilibrium configurations andthe sensitivity coefficients5'6. The methods arebased on successive applications of the finiteelement method and the classical Rayleigh-Ritztechnique. The finite element method is used togenerate a few global approximation vectors (ormodes) for approximating each of the nonlinearequations, Eqs. 1; and the equations for thesensitivity coefficients, Eqs. 4. The Rayleigh-Ritz technique is then used to generate thereduced sets of equations in terms of theamplitudes of these modes.

An effective set of modes forapproximating the nonlinear equations wasfound to be the path derivatives of the responsequantities with respect to the parameters qi andq2- The modes used in approximating theequations for the sensitivity coefficients includeboth the path derivatives and their derivativeswith respect to A,. The equations used inevaluating the path derivatives, and theirderivatives with respect to A, are obtained bysuccessive differentiation of the originalnonlinear equations, Eqs. 1, with respect to qi,<\2 and A,. The left-hand side matrix in theseequations is the same as that of Eqs. 2. Thedetails of applying reduction methods to thegeneration of the stability boundary, and thepostbifurcation equilibrium path are given inRefs. 5 and 6, and their application to theevaluation of the sensitivity coefficients isdescribed in Refs. 7 and 8.

Numerical Studies

Extensive numerical studies wereperformed to determine the effects of thestiffener skew angle and fiber orientation of theskin and the stiffener sections on the bucklingand postbuckling responses and theirsensitivity coefficients. The loading on thepanels consisted of an applied edge shorteningqe and a uniform temperature change T.Hierarchical sensitivity coefficients wereevaluated for each problem. The sensitivitycoefficients are the derivatives of the different

response quantities with respect to: a) skin andstiffener stiffnesses, and b) material propertiesand fiber angles of the individual layers. Thematerial properties and geometric characteristicsfor the panels considered in the present studyare given in Fig. 1. The material properties,the fiber orientations and the stacking sequenceselected are those typical of composite panelsconsidered for modern aircraft. The number oflayers and fiber orientation in the differentsections of the panel are given in Table 1. Theskin and each section of the stiffener aremodeled as plate elements. The middle planesof each of the top flange, web and skin aretaken as their respective reference planes. Forthe bottom flange, the middle plane of the skinis taken as its reference plane.

Two parameters were varied in thepresent study, namely, the skew angle of thestiffener, a, and the fiber reference angle of theskin, P. The stiffener skew angle and thelaminate reference angle for the five panelsconsidered in the present study are given inTable 2. Mixed finite element models wereused for the discretization of each section of thestiffener and the skin. A typical finite elementmodel used in the analysis is shown in Fig. 2.Biquadratic shape functions were used forapproximating each of the generalizeddisplacements, and bilinear shape functionswere used for approximating each of the stressresultants. The characteristics of the finiteelement model are given in Ref. 9. The modelhad a total of 638 finite elements (12,229nonzero displacement degrees of freedom).For each panel, the multiple parameterreduction methods outlined in Refs. 5 and 8were used for generating the buckling andpostbuckling responses, and evaluating thesensitivity coefficients. Comparisons weremade with available experimental results3-10.Typical results are presented in Table 3, inFigs. 3-9 for the buckling response and inFigs. 10-16 for the postbuckling response, andare described subsequently.

Bifurcation Buckling

The critical values of qe and T obtainedby the present finite element model are given inTable 3. The critical values of qe are comparedwith both the experimental values reported in

846

Ref. 3, and the values obtained by the STAGSfinite element program10.

As can be seen from Table 3, rotating thestiffener has a more pronounced effect on thecritical values of qe and T than changing thereference angle of the skin. Panel 1 has thelargest values of qcr and Tcr; panel 3 has thesmallest value for qcr and panel 4 has thesmallest value of Tcr. .

Contour plots of the transversedisplacement w, and surface plots depicting themode shapes associated with the critical valuesof qe and T for the five panels, are shown inFigs. 3 and 4. Note that all the mode shapessatisfy the following inversion symmetry (orantisymmetry) conditions:

f(x,,x2)=±f(-x,,-x2)

where f refers to any of the generalizeddisplacement components.

For all the panels considered, the lowestcritical values of qe and T are associated withtwo half waves in the X2-direction. For the qe-case, when cc=0°, the lowest buckling load isassociated with three half waves in the axialdirection. This result is to be contrasted withtwo half waves for the case a=20°. For panelswith P=0°, the buckling modes associated withqcr and Tcr are different (exhibiting a differentnumber of half waves in the axial direction).

The normal and tangential edge forcesassociated with the critical values of qe (withT=0, 100 and 200°F), and T (qe=0) are shownin Fig. 5 for the five panels. For the qe case,an increase in the temperature from 0 to 200°Freduces the total normal force Nt in panels 1through 4. For panel 4, an increase in T resultsin a decrease in the magnitude of the tangentialforce Ns. An opposite trend is observed inpanel 5 for which an increase in T increases themagnitudes of both Nt and Ns.

For the T case, increasing the magnitudeof either a or p or both results in decreasing

the total axial force Nt. The tangential forcesfor positive nonzero a and P are of oppositesigns. Therefore, the values of Ns for panels4 and 5 are less than, and greater than,respectively, those for panels 2 and 3.

An indication of the sensitivity of thecritical values of qe and T to variations in thematerial properties of the individual layers andthe skin reference angle P is given in Figs. 6and 7, respectively. The sensitivity coefficientsof qe and T are evaluated at T=0°F and qe=0,respectively. For the qe case, panels 2 and 5are more sensitive to variations in EL than theother panels; panel 1 is more sensitive tovariations in GLT than the other panels; panel 5is insensitive to variations in GLT; and as to beexpected, all panels are sensitive to variationsin P. For the T case, panels 1 and 4 are moresensitive to variations in EL than the otherpanels, and panel 5 is more sensitive tovariations in P than the other panels.

An indication of the sensitivity of thecritical values of qe and T to variations in theskin, web and flange stiffnesses is given inFigs. 8 and 9. For the qe case, the panels aremore sensitive to variations in AH of the skinand D22 of the web than to variations in theother extensional and bending stiffnesses of thepanel. By contrast, for the T case, the panelsare more sensitive to variations in the thermalforce NTU of the skin and the thermal momentMx2i of the bottom flange (see Eqs. A. 10,Appendix I).

Postbuckling Response

The postbuckling response of thedifferent panels subjected to combined edgeshortening qe, and uniform temperature changeT=100°F is shown in Fig. 10. Plots of thetotal axial force Nt versus the applied edgeshortening qe and the total strain energy U areshown. Also shown are the ratios of thetangential force to the axial force Ns/N tversus qe for the different panels. The bottomright section of Fig. 10 shows thecorresponding plot of Ns versus qe normalized

847

by dividing Ns and qe by the critical valuesNtcr and qecr for panel 1 (a=0°, p=0°). Thenormalized plots of Ns/N t versus q e /qC c r

are almost linear.

As can be seen from Fig. 10, the Nt

versus qe and Nt versus U plots for thedifferent panels are close to each other. For agiven Nt, both qe and U have their smallestand largest values for panels 1 and 5,respectively. The panels with the rotatedstiffener (panels 3, 4 and 5) have higher shearstiffness than those with the unrotated stiffener(panels 1 and 2). The tangential force Ns forpanel 1 is equal to zero, and Ns for panel 2 issmaller than Ns for other panels. In theadvanced postbuckling stage, the ratio N s /N tfor panel 4 is higher than that for all otherpanels.

An indication of the effect of temperatureon the postbuckling response of the differentpanels is given in Fig. 11. An increase intemperature results in a parallel shift of the Nt

versus qe plots. The same is true for the Ntversus U plots (results not shown).

Normalized contour plots for thetransverse displacement w of the skin and thetotal strain energy density D, at two differentvalues of Nt , namely, Nt / [ETh2J = 0.8 and

Nt/ (ETh2J = 1.6 , are shown in Figs. 12 and13. As can be seen from Fig. 12, a modechange occurs in the advanced postbucklingstage for panel 1. The distributions of the totalstrain energy density are different in differentpanels. For each panel the distributions changein the advanced postbuckling stage. For thetwo panels with cc=0 (panels 1 and 2), thedistributions of Dare similar. The sameapplies to the three panels with cc=2 (panels 3,4 and 5).

The sensitivity coefficients of the totalstrain energy U for the different panels with

respect to the layer properties EL, Ey, GLT>GTT> aL and cc-p; and the skin reference angle Pare shown in Figs. 14 and 15 for differenttemperatures. As can be seen from Figs. 14and 15, the sensitivity of the total strain energyto variations in EL increases rapidly with theincrease in qe. Other sensitivity coefficientsexhibit smaller variation with changes in qe.For T < 100°F, the normalized sensitivitycoefficient EL(9U/3EL) becomes the dominantsensitivity coefficient in the advancedpostbuckling stage. For T > 100°F, the totalstrain energy is very sensitive to variations in(XT and Ej and somewhat sensitive to variationsin EL in the initial postbuckling stage.

Figure 15 shows that for T=0°F, thepanels with P ^ 0° (panels 2, 4 and 5) areconsiderably more sensitive to variations in Pthan the two panels with P=0° (panels 1 and3). In the advanced postbuckling stage, anincrease in temperature results in decreasing3U/3P (for a given value of qe).

Normalized contour plots for thesensitivity coefficient of the total strain energydensity with respect to EL and p, for twodifferent values of N t , namely,N t /(ETh2) = 0.8 and N t / (ETh2) = 1.6,are shown in Fig. 16. Each contour plot isnormalized by dividing by the maximum valueof the sensitivity coefficient. The contour plotsfor the different sensitivity coefficients aredifferent. The contour plots for each of thesensitivity coefficients change with changingN t .

Concluding Remarks

A study is made of the buckling andpostbuckling responses of composite panelswith skewed stiffeners. The panels aresubjected to applied edge displacements andtemperature changes. Each section of thestiffeners and skin is modeled by using a first-order shear-deformation shallow-shell theory,with the effects of large displacements andlaminated anisotropic material behavior

848

included. A linear, Duhamel-Neumann typeconstitutive model is used and the materialproperties are assumed to be independent oftemperature. The different sections of thestiffeners and skin are discretized by usingtwo-field mixed finite element models with thefundamental unknowns consisting of the nodaldisplacements and stress-resultant parameters.The stress resultants are allowed to bediscontinuous at interelement boundaries.

The buckling, postbuckling responsesand hierarchical sensitivity coefficients aregenerated. The hierarchical sensitivitycoefficients measure the sensitivity of thedifferent response quantities to variations inthree sets of interrelated parameters; namely,laminate, layer and constituent (fiber, matrixand interface or interphase) parameters. Anefficient multiple-parameter reduction method isused for generating the buckling andpostbuckling responses, and evaluating thesensitivity coefficients.

Numerical studies are presented whichshow the effects of variations in the stiffenerskew angle and the fiber orientation of the skinon the buckling and postbuckling responses,and the sensitivity coefficients.

The results show that rotating thestiffener has a more pronounced effect on thecritical values of the edge shortening andtemperature than changing the materialreference angle of the skin. Also, rotating thestiffener increases the shear stiffness of thepanels in the postbuckling range.

Acknowledgments

This work was partially supported byNASA Grant NAG-1-1162 and AFOSR GrantF49620-93-1-0184. The numerical studieswere performed on the CRAY C-90 computerat NASA Ames Research Center. The authorsacknowledge the assistance of CatherineRichter of the University of Virginia inpreparing the final manuscript and improvingthe figures.

References

1. Gimmestad, D. W., "An AeroelasticOptimization Procedure for Composite

High Aspect Ratio Wings," Proceedings ofthe 20th ASME/ASCE/AIAA/AHS/ASCStructures, Structural Dynamics andMaterials Conference, pp. 79-86 (AIAAPaper No. 79-0726).

2. Mansfield, E. H., "Some StructuralParameters for Aero-Isoclinic Wings,"Aircraft Engineering, Vol. 24, 1952, p.283.

3. Young, R. D., Starnes, J. H., Jr. andHyer, M. W., "Effects of SkewedStiffeners and Anisotropic Skins on theResponse of Compression-LoadedComposite Panels," Proceedings of theTenth DoD/NASA/FAA Conference onFibrous Composites in Structural Design,Hilton Head Island, SC, Nov. 1-4, 1993,Naval Air Warfare Center Report No.NAWACADWAR-94096-60, Vol. 1, April1994, pp. 11-109 to 11-123.

4. Noor, A. K., "Finite Element Buckling andPostbuckling Analyses," in Buckling andPostbuckling of Composite Plates, G. J.Turvey and I. H. Marshall (eds.),Chapman and Hall, London, 1995, pp. 58-107.

5. Noor, A. K. and Peters, J. M., "Multiple-Parameter Reduced Basis Technique forBifurcation and Postbuckling Analyses ofComposite Plates," International Journalfor Numerical Methods in Engineering,Vol. 19, 1983, pp. 1783-1803.

6. Noor, A. K. and Peters, J. M., "RecentAdvances in Reduction Methods forInstability Analysis of Structures,"Computers and Structures, Vol. 16, Nos.1-4, 1983, pp. 67-80.

7. Noor, A. K. and Peters, J. M., "ReducedBasis Technique for Calculating SensitivityCoefficients of Nonlinear StructuralResponse," AIAA Journal, Vol. 30, No. 7,1992, pp. 1840-1847.

8. Noor, A. K., "Recent Advances in theSensi t ivi ty Analysis for theThermomechanical Postbuckling ofComposite Panels," Journal of EngineeringMechanics , ASCE (to appear).

9. Noor, A. K. and Andersen, C. M., "MixedModels and Reduced/Selective IntegrationDisplacement Models for Nonlinear ShellAnalysis," International Journal forNumerical Methods in Engineering, Vol.18, 1982, pp. 1429-1454.

10. Young, R. D., Private Communication.

849

11. Jones, R. M., Mechanics of CompositeMaterials, McGraw-Hill, New York, 1975.

12. Tsai, S. W. and Halm, H. T., Introductionto Composite Materials, TechnomicPublishing Co., Westport, CT, 1980.

13. Padovan, J., "Anisotropic Thermal StressAnalysis," Thermal Stresses I, ed. R. B.Hetnarski, Elsevier Science Publishers,Amsterdam, 1986, pp. 143-262.

14. Bert, C. W., "Analysis of Plates," Vol. 7 -Structural Design and Analysis, Part I, ed.C. C. Chamis, Composite Materials,Academic, New York, 1975, pp. 149-206.

Appendix I - Thermoelastic ConstitutiveRelations for the Laminate

The thermoelastic model used in thepresent study is based on the followingassumptions:

1) The laminates are composed of anumber of perfectly bonded layers.

2) Every point of the laminate is assumedto possess a single plane of thermoelasticsymmetry parallel to the middle plane.

3) The material properties areindependent of temperature.

4) The constitutive relations aredescribed by lamination theory, and can bewritten in the following compact form:

[A][B] 0[B]'[D] 0

0 0 [AJ_

(Al)

where {N}, {M}, {Q} and {e}, {K}, {y} arethe vectors of extensional, bending andtransverse shear stress resultants and straincomponents of the laminate given by:

N

:1 M2 M12l

h Q2]

e2 2E12]

(A2)

(A3)

(A4)

(A5)

K 2K12]

and

'= [2*31 2£3J

(A6)

(A7)

The matrices [A], [B], [D] and [As]contain the extensional, coupling, bending andtransverse shear stiffnesses of the laminatewhich can be expressed in terms of the layerstiffnesses as follows:

[W [B] [D]]

dx3

(A8)

(A9)

where [QJ 'k' and [Q,] (k' are the extensional andtransverse shear stiffnesses of the kth layer; [I]is the identity matrix; h^ and hk_ i are thedistances from the top and bottom surfaces ofthe kth layer to the middle surface; and NL isthe total number of layers in the laminate. Theexpressions for the different coefficients of thematrices Q a n d Q . in terms of thematerial and geometric properties of theconstituents (fiber and matrix) are given inRefs. 11 and 12.

The vectors of thermal effects,{Mr}, are given by:

i\

ic,

and

x3]Tdx3

(A10)where {oc} is the vector of coefficients ofthermal expansion (referred to the coordinatesX j ; X2 and X3 - see, for example, Refs. 13 and14). Note that the skin and the bottom flangestiffnesses and thermal effects are referred tothe middle plane of the skin. The top flangeand web section stiffnesses and thermal effectsare referred to their respective middle plane.

850

Table 1 - Number of layers and fiber orientations in the different sections

Top flange

WebBottom flange

Skin

Section12

• 3456

Numberof Layers

161616161616

Fiber Orientation[+45/0/90/+45/0/90/90/0/+45/90/0/+45][+45/0/90/+45/0/90/90/±45/90/0/+45][±45/0/90]2s[±45/0/90/+45/0/90/90/0/±45/90/0/±45][+45/0/90/+45/0/90/90/0/+45/90/0/+45][±45/+45/03/90]s*

*Note: The laminate reference angle f}° should be added to all fiber angles of the skin.

Table 2 - Panels considered in the present study

Panel

Stiffenerskew angle,

ct°0 0 20 20 20

Laminatereferenceangle, PO

0 20 0 20 20

851

Table 3 - Critical values of qe and T for the different panels

1f)2 QcrVi"skin

Experiment(Ref. 3)

STAGS(Ref. 8)

Present

Tcr, °F (Present)

Panel

1

12.36

14.46

14.19

408.8

2

12.28

13.6

13.38

333.0

3

11.86

10.98

10.95

292.6

4

10.58

11.25

11.25

240.6

5

10.49

10.49

10.44

287.7

852

qe/2 T

X2, U2

Stiffener

I. t_

1.12

•1.20

•1.50-

Laminate propertiesEL = 19.625 Msi

ET = 1.455 Msi

GLT = 0.82 Msi

GTT = 0.49 Msi

VLT = 0.295

Thickness of individual layers = .005 in.L1 = 24 in.

L2=16.5in.

Boundary conditionsAt x1 = ± L.,/2

"1 = ± qe/2u2 = w = ̂ = <|)2 = (|)3 = 0

At x1 = ± .4375 L1 and at x2 = ± .4848 L2

w = 0

X2,U2

Fig. 1 - Panels considered and stiffener geometry.

853

Fig. 2 - Typical finite element model used.

854

Panel 1(a=0°, p=0°)

Panel 2(a=0°, P=20°)

(r

\ i f \x >I'AX;

qe/2 ±

Panel 3(a=20°, £=0°)

Panel 4(a=20°, p=20°)

Panel 5(a=20°, P=-20°)T\

a) uniform edge displacement qe

Panel 1(a=0°, p=0°)

Panel 2(a=0°, P=20°)

(o v,

Panel 3(<x=20°, p=0°)

Panel 4(a=20°, p=20°)

Panel 5(a=20°, p=-20°)

b) uniform temperature change JFig. 3 - Contour plots of transverse displacement w depicting the buckling mode shapes for panelssubjected to uniform edge displacement qe and uniform temperature change T. Spacing of contourlines is 0.2. Dashed lines refer to negative contours. Location of maximum values identified by x.

855

Panel 1(cx=0°, p=0°) Panel 2(a=0°, P=20°)

Panel 3(a=20°, p=0°) Panei •!•(a=20°, PI.20°) Panel 5

(a=20°, p=-20°)

Panel 1(a=0°, p=0°)

a) uniform edge displacement qe

Panel 2(a=0°, p=20°)

Panel 3(a=20°, p=0°) Panel 4

(<x=20°, P=20°)Panel 5

(a=20°, p=-20°)

b) uniform temperature change TFig. 4 - Surface plots depicting the buckling modes for panels subjected to uniform edge displacement

qe and uniform temperature change T.

856

oo01

Nt

ETh2

NSETh2

N

0.8

0.6

0.4

0.2

0

0.2

0.1

0

-0.1

-0.2

0.1

0.05

0

0.05

-0.1

-

-

•

.

a=20°P=-20°Panel 5

T=0°F

T=100°F

T=200°F

qe=0

Fig. 5 - Normal and tangential edge forces associated with the critical values of qe and T for the different panels.

dq

cc=0°p=o°

Panel 1

0-0.01-0.02-0.03-0.04-0.05

cc=20°p=o°

Panel 3

A.-E,

ar

o=20°p=20°

Panel 4

0.730.725

0.720.715

0.710.705

0.7

o=20°P=-20°Panel 5

A, =

°Meax.

0.0440.042

0.040.0380.0360.034

-0.62-0.64-0.66-0.68

-0.7-0.72

'LT 'LT

"Meax

0

-0.01

-0.02

-0.03

-0.04

-0.05-0.1

-0.15-0.2

-0.25

TT

0.02

0.015

0.01

0.005

-0.76-0.78

-0.8-0.82-0.84-0.86

Fig. 6 - Sensitivity of the critical values of qe and T to variations in the material properties ofindividual layers.

858

<x=0°p=0°

Panel 1

cc=0°P=20°

Panel 2

ct=20°p=20°

Panel 4

o=20°p=-20°Panel 5

30

20

10 Illli50

-50

-100

Fig. 7 - Sensitivity of the critical values of qe and T to variations in the skin reference angle p\

859

o=0°p=o°

Panel 1

Io=0°P=20°

Panel 2

o=20°p=o°

Panel 3

ct=20°P=20°

Panel 4

o=20°p=-20°Panel 5

M1 k12

-1.00-1.02-1.04-1.06-1.08-1.10IFP 0.50

0.400.300.200.100.00 Mill

L12 '11

0.20

0.15

0.10

0.05

0.00 [Mill0.500.400.300.200.100.00 • nil

= U11 '66

~

0.500.400.300.200.100.00mil

0.40

0.30

0.20

0.10

0.00 HIM'66

0.400.30

0.20

0.10

0.00Illii 3

0.00

-0.50

-1.00

-1.50urn

Fig. 8 - Sensitivity of the critical values of qe and T to variations in the skin stiffnesses of the panels.860

o=0°p=o°Panel 1

.025

.020

.015

.010

.0050.00

0.050.040.030.020.010.00

.025

.020

.015

.010

.0050.00

e

Ia=0°P=20°

Panel 2

= D22(3)

o=20°p=o°

Panel 3

III...= D22(5)

..IIIA. =

.015

.010

.005

0.00

8T

3T

3T

9T

ct=20°(3=20°

Panel 4

o=20°P=-20°Panel 5

0.050.040.030.020.010.00

0.06

0.04

0.02

0.00

0.03

0.02

0.01

0.00

.020

.015

.010

.005

0.00

D22(3)

- D22(5)

L.lil

Fig. 9 - Sensitivity of the critical values of qe and T to variations in the web and flange stiffnessesof the panels. Superscripts refer to the stiffener section (see Table 1).

861

x2, u2

ETh2

2.0 r

1.5

1.0

.5

.15

.10

.05

0.00

-.05

qe/2

25 50 75 100

25 50 75 100

Nt

ETh2

2.0 r

1.5

1.0

.5

Ntor.(0,0)

.3

.2

.1

0.0 \=

Panel12345

<x°00202020

P°0

20020-20

=f=

50 100

UL2/(ETh4)150

Fig. 10 - Postbuckling responses for the different panels when subjected to edge displacement qeand uniform temperature change T=100°F.

862

NtETh2

2.0

1.5

1.0

.5

0.0

Panel 1

NtETh2

j25 50 75 100 125

2.0 r

1.5

1.0

.5

0.0

Panel 2

_j____i____i____|____|25 50 75 100 125

ETh2

NtETh2

1.81.51.2.9.6.30.0

1.81.51.2.9.6.30.0

Panel 3

25 50aqe h2

Panel 5

75

ETh2

100

1.81.51.2.9.6.30.0

Panel 4

25 50 75 100

T,°F

0

100

200

25 50 75 100

Fig. \ \ - Effect of temperature change on the postbuckling response of different panels subjectedto edge displacement qe.

863

Panel 1 Panel 2

^^\^2

x2,u2

Panel 3 Panel 4 Panel 5

a) total edge force Nt/(ETh2) = .8

Panel 1 Panel 2

Panel 3 Panel 4

-*i o

-1Panel 5

b) total edge force Nt/(ETh2) = 1.6Fig. 12 - Normalized contour plots depicting the effects of the stiffener skew angle a and skin

reference angle f$ on the transverse displacement w of the skin. Combined edge displacement qeand uniform temperature change T=100°F. Spacing of contour lines is 0.2. Dashed lines refer to

negative contours. Location of maximum values identified by x.864

Panel 1 Panel 2

x2,u2

qe/2 1

Panel 3 Panel 4 Panel 5

a) total edge force Nt/(ETh2) = .8Panel 1 Panel 2

Panel 3

b) total edge force Nt/(ETh2) = 1.6

Fig. 13 - Normalized contour plots depicting the effects of the stiffener skew angle a and skinreference angle f$ on the total strain energy density in the skin. Combined edge displacement qe

and uniform temperature change T=100°F. Spacing of contour lines is 0.2. Location of maximumvalues identified by x.

865

Panel 2

50 100 150 200 250 300 350

50 100 150 200 250 300 350

T,°F

0100

Fig. 14 - Sensitivity of the postbuckling responses of the different panels to variations in the materialproperties of the individual layers. Combined edge displacement qe and uniform temperature

change T=0° and 100°F.

866

125

100

50

25

125 r

100

75

50

25

-°125

125

100

75

50

25

-°125

-125 -100 -75 -50 -25 0a

25

-100 -75 -50 -25 25

50 75 100

Panel12345

a°00202020

P°0

20020-20=i

50 75 100

T = 200°Fqe/2 1

-100 -75 -50 -25 0

ayap

25 50 75 100

Fig. 15 - Sensitivity of the postbuckling responses of the different panels to variations in the skinreference angle (3. Combined edge displacement qe and uniform temperature change T=0°, 100°

and 200°F.

867

= CL

- CL

::-*:< A"*-• y

Panel 1(a=0°, P=0°)

Panel 2(a=0°, p=20°) Panel 3(a=20°, P=0°)

Panel 4(a=20°, P=20°)

Panel 5(a=20°, p=-20°)

a) total edge force Nt/(ETh2) = .8

b) total edge force Nt/(ETh2) = 1.6Fig. 16 - Normalized contour plots depicting the effects of the stiffener skew angle a and skin reference

angle P on the sensitivity coefficients of the total strain energy density in the skin. Combined edgedisplacement qe and uniform temperature change T=100°F. Spacing of contour lines is 0.2. Dashed

lines refer to negative contours. Location of maximum values identified by x.868