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VIRGINIA TECH

CENTER FO RCOMPOSITE MATERIALSA N D STRUCTURES

i

Calculation of Skin-Stiffener InterfaceStresses in S t i f f & e M m e I s

David &theMich el Hyer

Fourth &mal Review

11-14 May 1987

CCMS-87-17VPI-E-87-32

7 / 7 3

Virginia PolytechnicInstitute

and

State University

Blacksburg, Virginia24061

U

( bAS A - C R - 111 46 6 2 ) CA L C U L A1 IC I C F Hag- I 78995 K l E - S ' I I P F E L E J i I B ' I E B FA C E S'IGESSES 1AS S i F E f i b E G C C H E G E I IZ FA k E L S 1 C f E r i p e beport( Vi r q i n i a P o l y t r c h i i c l n s t . a & d S t a t e U n i v. ) U n c l a s1 5 3 F C S C L 20K G3/39 G 1 9 C C 6 0

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Col l ege of Eng inee r ingVirg in ia Poly technic Ins t i tu te and Sta te Univers i ty6 lacksb u rg, Virg inia 2406 1

December 1987

CCMS-87-17VPI-E-87-32

Calculation of Skin-Stiffener lnterfaceStresses in Stiffened Composite Panels

Dav id Cohen M i c h a e l W. Hyer2

Depa r tmen t of Eng inee r ing Sc i ence and Mecha n ic s

In t e r im Repor t 68The NASA -Vi rg in i a Tech Com pos i t e s P rog ramNASA Gran t NAG-1-343

Prepared for: Structural Mechanics BranchNat iona l Ae ronau t i c s and Space Admin i s t r a t i onLang ley Resea rch Cen te rHampton, Vi rg in ia 23665

1 Graduate S tudent, Department of Engineering Science and Mechanics,Virginia Polytechnic Institute and State University

2 Professor, Department of Engineering Science and Mechanics,Virginia Polytechnic Institute and State University

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CALCULATION OF SKIN-STIFFENER INTERFACESTRESSES IN STIFFENED COMPO SITE PANELS

(ABSTRACT)

A method for computing the skin-stiffener interface stresses in stiffened composite pa nels

is developed. Both geometrically linear and nonlinear analyses are considered. Particular

attention is given to the flange termination region where stresses are expected to exhibit un-

bounded characteristics. The method is based on a finite-element analysis and an elasticity

solution. The finite-element analysis is standard, while the elasticity solution is based on an

eigenvalue expansion of the stress functions. The eigenvalue expansion is assumed to be

valid in the local flange termination region and is coupled with the finite-element analysis

using collocation of stresses on the local region boundaries. In the first part of the investi-

gation the accuracy and convergence of the local elasticity solution are assessed using a

geom etrically linear ana lysis. It is found that the finite-elemen t/local elasticity solution sche me

produces a very accurate interface stress representation in the local flange termination re-

gion. The use of 10 to 15 eigenvalues, in the eige nvalue ex pansion series, and 100 collocation

points results in a converged local elasticity solution. In the secon d part of the investigation,

the local elasticity solution is extended to include geometric non linearities. Using this analysis

procedure, the influence of geometric nonlinearities on skin-stiffener interface stresses is

evaluated. It is found that in flexible stiffened skin structures, which exhibit out-of-plane de-

formation o n the order o f2 to 4 times the skin thickness, inclusion of geom etrically nonlinear

effects in the calculation of interface stresses is very important. Thus, the use of a geomet-

rically linear analysis, rather than a nonlinear analysis, can lead to considerable err or in the

computation o f the interface stresses. Finally, using the analytical tool de veloped in this in-

vestigation, it is possible to study the influence of stiffener param eters on the stateof interface

stresses.

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Acknowledgements

This study was supported by the NASA-Virginia Tech Comp osites Program under the NA SA

Contract Grant NAG 1-343. We would l ike to acknowledge, in particular, Ors. J. H. Starnes ,

Jr. and M. P. Nemeth for their useful contributions and support of this project.

Acknowledgements iii

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Table of Contents

1. Introduction .......................................................... 1

2.Analytical Method Development ........................................... 62.1 Analysis Overview and Relevant Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Elasticity Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Development of the S tress Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Expressions for the Stresses and Displacements . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Application of the Boundary and Interface Conditions . . . . . . . . . . . . . . . . . . . . . 24

2.3.3.a The Eigenvalue Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3.b The Remaining Part of the So lution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 .4 Collocation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Global Finite-Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Verification of the Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.Application of the Methodology to Stiffened Composite Plates . . . . . . . . . . . . . . . . . . .693.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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3.2 Geom etrically Non linear Elasticity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7273

3.3.1 Structura l Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.3.1.a Details of Finite-Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.1.b Details of Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.2 Sub structura l Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3.3 Ap plication o f the Loca l Elasticity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Structural-Substructural-Local Analysis Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.Results ............................................................ 974.1 The influence of Geometric Nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.1 Stiffener P ara me tric Study Res ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2.2 Stiffener Design Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.Conclusion and Recommendations ...................................... 13 15.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.2.1 Ana lytical Recom mendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2.2 Exp erim ental Recommen dations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.References ......................................................... 137

Appendix A . Mate rial Constitutive Relations ................................. 141A.1 Transforma tion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1

A.2 Integrated Ma terial Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Appendix B. The Eigenvalue Problem ...................................... 1486.1 The 6 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.2 Th e X Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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8.3 Eigenvector Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 .4 Eigenfunction Expansion Represen tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Appendix C . Finite-Element Formulation .................................... 162C.l PE2D Finite-Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

C.2 EAL Finite-Element Prog ram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1

Appendix D. Material Properties and Eigenvalue Data .......................... 176

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List of Illustrations

Figure 1. Skin-Stiffener Separation Initiation at the Flange Termination Region.. . . . . . . 3Figure 2. Skin-Stiffener Cross-Section Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 7

Figure 3. Elasticity Solution Body Geometry. . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . 9

Figure 4. Loading and Finite-Element D iscretizations of Skin-Stiffener90 Flange Termi-nation Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 5. Loading and Finite-Element Discretizationsof Skin-Stiffener45" Flange Termi-nation Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 6. Skin-Stiffener Stress (T, fo r 90 Flange Termination Angle, Finite-Element andElasticity Results Using15 Eigenvalues and 10 0 Collocation points. . . . . . . . 47

Figure 7. Skin-Stiffener Stress cy fo r 90 Flange Termination Angle, Finite-Element andElasticity Results Using15 Eigenvalues and 10 0 Collocation points. . . . . . . . 48

Figure 8. Skin-Stiffener Stress t v fo r 90' Flange Termina tion Angle, Finite-Element andElasticity Results Using15 Eigenvalues and 10 0 Collocation points. . . . . . . . 49

Figure 9. Skin-Stiffener Stress CT, fo r 45O Flange Term inatio n Angle, Finite-Element and. . . . . . . 50Elasticity Results Using15 Eigenvalues and 100 Collocation points.

Figure IO . Skin-Stiffener Stress (T, fo r 45O Flange Termination Angle, Finite-Element andElasticity Results Using15 Eigenvalues and 100 Collocation points. . . . . . . . 51

Figure 11. Skin-Stiffener Stress T~ fo r 45' Flange Termination Angle, Finite-Element andElasticity Results Using15 Eigenvalues and 100 Collocation points. . . . . . . . 52

Figure 12 . Skin-Stiffener Stresses for 90 Flange Termina tion Angle Computed From theElasticity Solution using Three Mesh Refinements,15 Eigenvalues, and 100Collocation points. . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Figure 13. Skin-Stiffener Stresses for45O Flange Termination Angle Computed From theElasticity Solution using Three Mesh Refinements,15 Eigenvalues, and 10 0Collocation points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . 55

Figure 14. Elasticity Solution of Skin-Stiffener Stressesfor 90" Flange Termina tion Angleas Function of the N umber Eigenvalues Using100 Collocation points. . . , . . . 58

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Figure 15. Elasticity Solution of S kin-Stiffener Stresses for4 5 OFlange Termination Angleas Function of the Number E igenvalues Using100 Collocation points. . . . . . . 59

Figure 16. Elasticity Solu tionof Skin-Stiffener Stresses for 90" Flange Te rmination A ngleas Function of the Nu mber Collocation po ints Using 15 Eigenvalues. . . , . , . 60

Figure 17. Elas ticity Solution of Skin-S tiffener Stresses for 45O Flange Ter min atio n Angleas Function of the Num ber Collocation points Using15 Eigenvalues. . . . . . . 61

Figure 18. Skin-Stiffener Stresses for90" Flange Termination Angle Using Boundary DataFrom the Finite-Element Analysis o f a90" and 45" Flange Term ination Angles. 63

Figure 19. Skin-S tiffener Stresses for 45O Flange Te rm ina tion Angle Using Bo unda ry DataFrom the Finite-Element Analysis of a 90" and 45" Flange Term ination Angles. 64

Figure 20. Skin-Stiffener Interfac e Peeling Stress,o,,, Throughout the Flange TerminationRegion as a Function of Adhesive Thickness. . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 21. Skin-Stiffener Interface Shearing Stress,T ~ ~ ,Throughout the Flange TerminationReg ion as a Function of Adhesive T hickness. . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 22. Single Stiffener Plate Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 23. No nlin ea r Skin-Stiffener Analy sis Geome try. . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 24. Structure - Substructure - 'Local Analysis Procedure. . . . . . . . . . . . . . . . . . . . 75Figure 25. EAL Finite-Element D iscretization o f a Stiffened Plate Configu ration.. . . . . . . 78

Figure 26. Out-of-Plane De formation , EAL, STAGS, and Experim ental Res ults (STAGS andExperime ntal Results are From [IO]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Figure 27. 1/4 Sym metry Stiffened Plate Discretization and M esh Refinements.. . . . . . . 82Figure 28. Displacement Convergence Study in a Stiffened Composite Plate Nonline ar

Figure 29. Rotation Convergence Study in a Stiffened Composite P late Nonlinear Analysis

Analysis w ith 10 psi Pressure (Data Plotted at z=b/3). . , , . . . . . . . . . . . . . . 83

with 10 psi Pressu re (Data Plotted at z= b/3). . . . . . . . . . . . . . . , . . . . . . . . . 84

Figure 30. Structuralto Substructural Analysis Procedure and Schematic of SubstructureDiscretization. . . . . . . . . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 31. Transverse Deformation, v, for Structure and Substructure Models, NonlinearAnalysis at 10 psi Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 32. Skin-Stiffener Stresses, ox , or, an d T,,, in the Local Region; EALVerses PE2D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 33. Skin-Stiffener Stresses Along the E ntire Skin-Flange Interface Length in Pres-sure Loaded Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 34. Skin-Stiffener Stresses i n the L ocal Flange Ter mina tion Region in PressureLoaded Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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Figure 35 . Distribution of Skin-Stiffener Interface Stresses Along the Entire Flange Length,Linear and No nlinear Analyses Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 36 . Variation of Skin-Stiffener Interface P eeling and Shearing Stresses in the Lo calFlange Term ination Region. . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . , . . 101

Figure 37. Distribution o fox Through the S kin and Skin-Flange Thicknesses A long theCollocation Boundaries. . . . . . . . . . . , . . . . . . , . . . . . . . . . . . . . . . . . . , . . 102

Figure 38 . Peeling and Sh earing Stress Intensity Factors as a F unction of Plate Pressure. 10 5

Figure 39 . Out-of-Plane Deformation for a Stiffened Composite Plate.. . . . , . . . . . . . . . 10 7

Figure 40 . Influence of S oftening the Flange on the Skin-Stiffener Interface Peeling,oy,and Shearing, T,,,. , Stresses. . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . , . . . . 11 1

Figure 41 . Influence of S hortening the W eb on th e S kin-Stiffener Interface Peeling,oY, an dShearing, T,,,., Stresses. . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . 11 2

Figure 42. Influence of Flange Tapering on the Skin-Stiffener Interface Peeling,oy, andShearing, T~ , Stresses. . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . , , . . . . 114Figure 43 . Influence o f Thickening the Flange on the Skin-Stiffener Interface Peeling,B ~ .

and Shearing, l w , Stresses. . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . , . . 11 5Figure 44. Influence o f Softening and T apering the Flangeon the Skin-Stiffener Interface

Peeling, by, and Shearing, T r y , Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 45 . Influence o f Thickening and Tapering the Flange on the Sk in-Stiffener Interface

Peeling, cy , and Shearing, T ~ ,Stresses. . . . . . . . . . . . . . . . . . . . . . .. . . . . 11 7Figure 46 . Influence o f Thickening and Softening the Flange on the Skin-Stiffener Interface

Peeling, c y , and Shearing, T , ~ ,Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 9

Figure 47. Influence o f Thickening, Softening, an d Tapering the Flange o n the Skin-Stiffener Interface Peeling and Shearing Stresses. . . . . . . . . . . . . . . . . . . . 120

Figure 48 . Skin-Stiffener Interface Peeling and Shearing S tresses for90' and 15O FlangeTermination Angles. . . .*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 49 . Out-of-Plane D eform ation Response of the Various Stiffened-Plate Configura-. . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . 123tions (Plotted at z =O).

Figure A.1. Lamina and Laminate Material Coordinates Nomenclature.. . . . . , . . . , . 14 2Figure 6.1 . General B imaterial Composite Wedge Geometry. . . . . . . . . . . . . . . . . . . , . 14 9Figure C.1. Two-Dimensional lsoparametric Elements Coordinates System... . . . . , 168

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List of Tables

Table 1. Maximum Stresses a t E=0.025 as Computed From the ElasticitySolution U sing Three Finite-Element Me sh Refinements. . . . . . . . . . 56

Tab le 2. Stiffener Configurations. . . . . . , . . . . . . . , . . . . . . . . . . . . . . . . . . , 109

Table 3. Stiffener Parametric Study Results. . . . . , . . . . . . . . . . . . . . . . . . . . 127Table D. l . Laminate Material Properties. . . . . . . . . . . . . . . . . . . . . . . 177

Table 0.2. The First 20 Eigenvalues for Material System Combination no. 1. . 179

Table D.3. The First 20 Eigenvalues for Material System Combination no. 2. . 18 0

X

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I. Introduction

In recent years there has been a dramatic increase in the use of compos ite materials in

aircraft structures. Compared to metallic structures, the cost benefits, performance increase,

and weight reductions which can be realized are substantial.As with metallic structures, one

of the p rim ary components in aircra fl structures is the skin-stiffener combination. In trad itiona l

me tallic structures the stiffener is attached to the skin by rivets. In severe loading conditions,

such as those that o ccu r during postbuckling, the rivets prov ide a site for yielding o f the m etal.

As a result, failure in the form o f local yielding may occur but the failure does not necessarilycripple the structure as a whole. O n the othe r hand, composite m aterials are brittle and do

not yield. Holes and othe r geom etric discontinuities are sites for high stresses that u ltimately

cause the failure o f the structure as a whole. Therefore, the use of rivets as the m ethodof

stiffener attachment is less attractive with composites thanit has been with metals. A more

common method of stiffener-to-skin attachment in composite structures is by an adhesive

secondary bonding or by the cocuring of the stiffener and skin.

Because of the differences in material properties and the lack of riveting, the failure char-

acteristics observed with bondedor cocured stiffened compo site skins are qu ite different from

the failures encountered in riveted metallic structures. Failure in stiffened composite skins

occurs in a much more catastrophic manner, being initiated by skin-stiffener separation[I].

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Further, it has been shown [2 ] that stiffened c omposite p anels can fail prem aturely, below the

design load, due to skin-stiffener separation. One such mode of failure has been observedto

initiate at the flange termination region. Figure1 illustrates an examp le of this type of failure.

Because of observations such as shown in Figure 1, it is believed that the stresses in the

flange termination region are high. High skin-stiffener interface stresses can be attributedto

a num ber of factors. F irst, there is the structural inc omp atibility associated with the deforma-

tion o f the skin and deformation o f the stiffener when the skin-stiffener comb ination is sub-

jected to applied loads. This incompatibility is particularly acute in the postbuckling state. In

addition, if the stiffeneris bonded to the skin, the flange termination region leads to a ge-

om etric discontinuity in the structure. This region tends to serve as an area of increased

stress. Finally, in stiffened composite skins the problem is compounded by an additional

complication. In order to gain full advantageof th e tailoribility ofcomposite materials, gener-

ally the stiffener and the skin are constructed of different material layups. Such material dis-

continuity at the interface can lead to significant skin-stiffener interface stresses. Itis th e

primary purpose of the present studyto develop a m ethod by which the stresses in this region

can be accurately determined, and to use the method to investigate the influence of various

stiffener parameters on the stresses in this region. Though the flange termination region is

of primary concern, stresses at all locations along the interface ar e computed.

To predict the skin-stiffener interface stresses, different levels of analytic complexity can

be used. However, it is generally acknowledged that the analytical model should incorporate

a few key features. First, the model should accurately represent the geometric and material

discontinuities associated with the flange termination.As with free edge stresses, these ge-

ometric and material discontinuities can cause the stresses in the flange termination region

to be unbounded. When present, these unbounded stresses can be responsible for the initi-

ation of skin-stiffener separation and should be accounted for. Secondly, since stiffened panels

are most comm only designed to operate at the postbuckling range, the analysis mu st incor-

porate geometrically nonlinear effects. Third, for the analytic tool to be useful in the design

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1. Introduction 3

ORIGINALPAGEBLACKAND WHITE PHOTOGRAPH

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process, it must be sensitive enough to various design parameters, e.g., stiffener geometry

and skin/stiffener material architecture. In addition to these three key factors, there is the is-

su e of computational efficiency or cost. The computational cost should not beso large as to

prohibit its use as a tool for parametric studies or for the design of a stiffened structure. And

finally, the analysis should be such that it can be integrated into any proposed computational

schemes or testbeds [3,4].

Considering the severity of the problem, there have been few studies of skin-stiffener

interface stresses. One of the more notable ones is discussed in [5]. In this investigation the

skin and flange were treated as separate orthotropic plate elements. These elements were

held together via interface forces, the forces being taken as unknowns. The stiffness of the

other stiffener elements, such as the web and the cap, were treated as extensional and rota-

tional springs. The solutionto the problem was formulated using the principle of virtual w ork

and the theorem of minimum potential energy. Although in the postbuckling range the skin

will actually experience moderate to large rotations, the analysis assumed only geometric

linea r deformation theory. Other studies[6-91 have considered similar problems, namely the

adhesively bonded lap joints. In these studies the two adherents were treated as plates under

cylindrical bending and/or inplane loads. In[S-81 the adhesive was modeled as shear spring

only or tension-shear spring combination, and in 191 i t was treated as an elastic layer in which

the stresses did not vary across the adhesive thickness. In all of these studies a solution for

the variation o f shearing and pee ling stresses in the adhesive was obtained. In[9 ] the derived

solution was also a pplied to a simplified stiffened com posite plate geome try. These investi-

gations looked at simplified geo metries an d loading conditions. Hence, although these studies

are important in furthering the understanding o f adhesively bonded components, they cannot

be applied in their present form to the study of stiffened composite aircraft structures. This is

due to the fact that such structures tend to have complex geo metries and loading conditions.

In addition, the above studies did not address the issue of geometric nonlinearities.

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Due to the problems in the previous analytical models discussed above, a more detailed

analysis was undertaken. As stated above, and reiterated here, the following key require-

ments were imposed: a) The model should accurately represent the state of stress near the

point of geometric and material discontinuity; b) The model should be applicable to the ge-

ometric nonlinear range; and c) The model should be sensitive in its stress prediction to var-

ious design parameters, such as stiffener geometry and stiffenerkkin material architecture.

To meet these requirements, an analysis is developed in which the stress predictions are

based on a generalized plane deformation elasticity solution in combination with standard

finite-element calculations. The elasticity solution is val id in the loc alized reg ion ne ar the ter-

mina tion of the stiffener flange. The finite-element calculation s are valid for the stru cture as

a whole, except near the flange termination region. The elasticity solution uses an eigenvalue

expansion of the stress function to predict the stresses. The expansion is applicable in the

flange termination region and satisfies exactly the boundary conditions there. The eigenvalue

expansion is known to within arbitrary, but unknown, coefficients which are associated with

each eigenvalue. The stresses from the finite-element solution and collocation schem e are

used for determining the constants and thus uniquely determining the stresses in the localized

region.

In the next chapters various aspects of the investigation will be discussed. In chapter2 th e

development o f the analytical method and its verification are delineated. The derivations in

this chapter are for geometrically linear analysis only. In chapter3 it is shown how the meth-

odology developed in chapter 2 may be extended to include geometric nonlinearities (Le.,

mod erate rotations). This is followed by a discussion of the application of this extension to the

study of interface stresses in stiffened composite plates. In chapter4, numerical results are

presented. These results highlight the effect of including geom etric non linearities in the

analysis on skin-stiffener interface stresses. In addition, the effect of various stiffener ge-

ometric and material parameters are evaluated. The study ends with some concluding re-

marks and recommendations for future research.

1. Introduction 5

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2. Analytical Method Development

2.1 Analysis Overview and Relevant Literature

To facilitate the following discussion, a skin-stiffener cross-section being studiedis shown

in Figure 2. Note the location of the coordinate system in the figure and the nomenclature

associated with the cross-section. Several coordinate systems will be used hereafter and it

will be importantto differentiate between them. Interest focuseson the com putation of inter-

face stresses at the skin-stiffener interface, along the line y=O. The type of stiffener shown,

a blade stiffener, is only to serve as an example. Other stiffener types such as hat, I, and J

can be studied with the type of analysis being developed. It should be noted that the flange

can terminate at various angles, a, (see Figure 2), relative to the skin. The shaded area,

shown in the e xploded view, is referred to as the local region. This is the region where the

elasticity solution is valid. The region outside of the shaded area, referred to as the global

region, is the region where the finite-element solution is valid. It will be assumed that

Figure 2 represents a cross-section of a stiffened panel that is long enough that the stress

state does not vary in the stiffener direction in the regionof interest. That is, in the nom en-

clature of the figure, the stresses do not vary with thez coordinate.

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Y

t h

+ z axis out of page

Figure 2. Skin-Stiffener Cross-Section Geo metry.


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matched along prescribed boundaries with the same boundary quantities w ritten in terms of

the unknown coefficients. This leads toa system of n simultaneous equations from which th e

n unknown coefficients can be determined. The above procedure is sometime referre d to as

the point-matching method. Since there is n o rigorous proof for convergence of the colloca tion

method, it is typically studied by increasing the num ber of terms in the assumed function and

the number of boundary collocation points. To increase accuracy and decrease dependency

in the m anner in which the positions of the collocation points are chosen, it is com mon prac-

tice to use more boundary collocation points than unknown coefficients (28,291. This approach

is known as the overdetermined collocation procedure. Since it leads to an overdetermined

set of equations by which the unknown coefficients are evaluated, the coefficients are deter-

mined in a least-squares sense. Once it is clear that the number of terms in the assumed

function and the nu mbe r and location of the known bounda ry points have little influence on the

numerical results, it is assumed that the procedure has converged. The converged functions

are then assumed to be close to a true representation o f the exact solution within this region.

A number o f investigators have made extensive use of the above described collocation

proced ure. G ross et a1 [12-161 used bo undary collocation in the determ ination o fK, for various

edge crack specimen geometries. In the ir investigation the first coefficient (which is related

to K,) of the Williams stress function,x ,was determined by collocating boundary vales ofxand &.- (n being the no rma l to the boundary) along prescribed boundaries. Carpenter [28]

applied boundary collocation in the de termination of various fracture parameters. Theun-

known coefficients of the truncated function expansion were evaluated from the collocation

of all three stress components ( o x , 0, and cy) obtained from finite-element analysis. In an-

other paper [30] Carpe nter investigated accuracy issues related to the boundary c ollocation

of stresses and/or displacements. Wang and Choi [23] used collocation to study the laminate

free-edge problem.

dn

Other procedures which involve the use of eigenfunction expansion of the stress function

in a localized region include the reciprocal work contour integral (RWCI) method, and the use

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of special singular elements. The RWCl is based on Betti's theorem of elastic bodies. Two

sets of forces (S, and S', ) and displacements (u, and u', ) of the same direction but not the

same magnitude acting at selected points along the body bo undary are in reciprocal equilib-

rium. Use of the theorem provides a scheme by which the boundary value problem discussed

previously may be solved. In addition, the RWCl leads to a path-independent procedure. The

above technique have been used in the computation of K, and/or K,, by Carpenter [17] and

Sinclair et al [31]. In the singular or hybrid element formulation, a special element is devel-

oped which com prises a portion of the localized region where geometric and m aterial dis-

continuities occur. The displacements and stresses within the element boundary are

governed by the exact elasticity solution. The surrounding standard elements of the mesh are

then connected at n nodal points along the boundary of the special element. The coefficients

of the truncated stress function are then determined so as to render continuity (or compat-

ibility) of the n odal displacements at the special element's boundary in an exactor approxi-

mate manner. There are numerous studies which have utilized the above concept. Some of

the m ore relevant ones include work by Wang and Yuan[32] and Jones and C allinan [33].

Clearly, the issue he re is determining the conditions on the boundary of the local region.

In general, it is not important how the boundary conditions are obtained, as long as the in-

formation is accurate. For complex geometries, such as the skin-stiffener cross-section, the

only reasonable m ethod to obtain boundary conditions is with finite-element analysis. Thus the

method here will utilized a finite-element analysis of the cross-section to provide stress in-

formation on the boundary of the localized region. Furthermore, the finite-element analysis o f

the cross-section itself will be coupledto a finite-element analysis of the entire plate. The total

analysis will be of the form of astructure-substructure-local analysis. The remainder of t.his

chapter will be devoted to the local analysis. The governing equations of elasticity, the

eigenvalue expansion solution, and the application of the collocation procedure will be pre-

sented. In addition, the accuracy and convergence of the method are discussed by application

to specific problems.

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2.2 Material model

The analysis here is implemented on the laminate rather than a lamina level. That is, the

skin and stiffener are treated as having homogeneous integrated material properties. The

reason behind this is as follows: It is felt that the interaction between the skin and the stiffener

are controlled more by the o vera ll stiffnesses of the skin and the stiffener than by the stiffness

of the individu al lamina at the skin-stiffener interface. For this reason the integrated ma teria l

properties are used in the present analysis. Furthermore, only symmetric balance laminates

are considered. In obtaining integrated material properties, the lamina principal m aterial co-

ordinates, denoted as the 1-2-3 coordinates, co rrespond t o the transverse, thickness, and fiber

directions, resp ectively. The x-y-z coordinates correspo nd to directions transverse, norma l,

and colinear to the stiffener, respectively (see Figure2). The fiber angle, c p ,measures the

angle between the 3 and z axis, a positive rotation corresponding to rotation of the fiber from

th e z axis toward the x axis.

The we ll known lam ina te constitutive relations i n the 1-2-3 system are written sym bolicallyas ,

-t

E1 = s GI , [I1

d

E, an d G , being the strain a nd stress vectors in the 1-2-3 system and S being the compliance

matrix in the same system. The transformation of stress and strain from the 1-2-3 systemto

the x-y-z system leads to,


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A l l Ai2 A13

A l 2 A22

A13 A23 A33

0 0 O A 4 4 O 0

0 0 0 O A 5 5 O

0 0 0 0 O A s c

whe re the overbar is dropped from the stresses for convenience. The laminate stiffness

components Aij are given by,

n b eing the n umb er of laminae and y, and yk-, being defined as the through-the-thicknesslo-

cations of the laminae interfaces. In inverted form;

c71+tzX = [a ] Gx .

It should be n oted that although each lamina is considered to be anisotropic in the x-y-z sys-

tem, the lam inate constitutive law is that of an orthotropic material, i .e., the sm eared lam inate

properties are orthotropic.

2.3 Elasticity Solution

As mentioned at the outset, the analysis will be developed for the linear case and then

extended to the geom etrically nonlinear analysis case. What follows is the linear analysis

development.


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2.3.1 Development of the Stress Functions

Consider an arbitrary semi-infinite corner composedof two dissimilar orthotropic m aterials

bonded along y=O (or8 = 0), as depicted in Figure 3. If the body in question obeys the fol-

low ing restrictions: a) The dimension in the z direction is much large r than the cross-sectional

dimensions; and b) The external loads on the lateral surface do not vary with z,it is possible

that the stresses, and hence the strains, are independent of the z-coordinate. Such a condi-

tion is referred to as a generalized plane deformation. The endsof the body ma y be subjected

to axial force, P,, twist M,, and m omen ts, both about the x and y axes (i.e.,M, and My). If such

end loads are present, the state of plane deformation will exist at some distance from the ends

in a man ner consistent with St. Vernants principle. Unde r the above conditions the stress

equilibrium equations become,

The st rain-d is placement retat ions a re given by,

C9. e,fl


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where the strains are a function of x and y only. The constitutive relationsof eq. 7 for a

homogenous orthotropic body under consideration can be written in full as,

a? l a12 a13

a12 a22 a23

a13 a2 3 a33

0 0 O a , O 0

0 0 0 O a 5 5 0

0 0 0 0 O a 6 6

c101

The general expressions for the displacement functions are obtained bya series of inte-

grations and differentiations of eqs.8, 9, and 10. The step-by-step de tails are given in [22].

In general, the constitutive relations are writtenin terms of the displacements using eqs. 9.

The integration of threeof these equations, (keeping in m ind that5, is independent of z) and

the satisfaction of the remaining three equations leads to the displacement functions in the

general form,

u = -- B1 a33 z2 - B,yz + U(x,y) + 0 2 z - 0 3 y + u, ,2 C11.al

[ l l . b lv = - - B2a33 z2 + B,xz + V(x,y) + % x - o l z + v, ,2

w = ( B, x + B, y + B3 ) a S 3 z + W(x,y)

+ o q y - 0 2 x + w, 5 C1l . c l

where Bi , ( i = 1,2,3,4), are arbitrary constants of integration,0 , , ( i = 1,2,3) are rigid

body rotations, and u, , v, and w, are rigid body translations. The unknown functions U(x,y),

V(x,y) and W(x,y) must sa tisfy the following conditions:


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- -a - $210, + P 2 2 o y + a23(B,x + B ~ Y+ B3) I [12.b]aY

- + - -au aV - B6f3txy aY ax C12.cl

-aw = P 5 5 % u + B4Y [12.d]ax

where Poare the reduced stiffness coefficients and are given by,

, i,j = 1, 2,4, 5, 6ai3 aj3

a33aij - --Pij -

The compatibility equations are satisfied identically for the above displacement field since it

is derived from the strain-displacement relations. In additionit can be shown [22] that,

The stresses which satisfy a ll the aforem entioned assum ptions can be derived from two stress

functions, F(x,y) and Y(x,y). If the stresses are written in terms of these functions as ,

a2F= - -* 7x y axay *

a2F - a2Fo x = - , o y - -dy 2 ax2

[ lS.a,b,cl

[15.d,el


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then the stress equilibrium eqs. 8.a through 8.c are satisfied identically. The equations gov-

erning F(x,y) and Y(x,y) are obtained by substitution of the above stress relations into eqs.

12.a through 12.e and the elimination ofU, V, an d W by differentiation. For an orthotropic

material these equations become,

a2Y + 8 5 5 7a2y -- - 26, .844 -ax2

[16.b]

The decoupling o f the two equations is a distinct ch aracteristic of orthotropic m aterials (i.e.,

the equations are not uncoupled for anisotropic material). The equation governingF is ho-

mogeneous while the equation forY involves a particular solution.The solution for F an d the

homogeneous equation for Y have the form [22],

Y = Y ( x + u y ) , [17.b]

where p an d u are parameters to be determined. For the loc al region near the vertex of the

bim ateria l cor ner the solutions for F(x,y) and Y(x,y) a re approxima ted in [21] as,

zk+ 2

( h + 1) ( h + 2)F(2) = C '

z("+1)( 6 + 1) 'Y(Z) = D

[18.a]

[18.b]

where,

z = x + J l y ,

z = x + u y ,

[19.a)

[ l9 .b]

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and C and D are arbitrary constants. The substitution of F(x,y) andY (x,y) into eqs. 16.a and

16.b lea ds to,

The first equation is satisfied under the following conditions:

a) p has one of four unique values given by the characteristic equation

[20.b]

Such roots do exist an d they are alway s complex o r imaginary (for detailed discussion see ref.

22). Considering these fou r values of p , F(x,y) is given by,

4 Z i + 22 1 ck ( h + 2)(h + 1) 'F(e)(x,y) =

b) p is arbitrary and,

h = 0 , l

c221

This leads to

F(a)(x,y) = bl x3 + b 2 x 2 y + b,xy 2 + b 4 y3

+ b5x2 + b,xy + b,y2 . C231

The superscripts (e) and (a) designates the eigenvalue expansion and auxiliary solutions,

respectively.

The secon d of eq. 20 is satisfied u nder the conditions:

a) u has one of two unique values given by the characteristic equation,


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This leads to,

[24.b]

from which Y(x,y) becomes,

b) u has an arbitrary value and,

6 = 0 ,

leading to,

Y'(a)(x,y) = b, x + b 9 y .

Finally the par ticula r solution of eq 16.b is taken as,

Y(p)(x,y) = bl,x2 + bll y2 .

In the above the Ck and D, ar e arbitrary comp lex constan ts and b,,i = I, ..., 9 are arbitrary rea l

constan ts. The constan ts b,, and b,, are not comp lete ly arb itrar y, namely

In addition, ?. an d 6 are unknown parame ters at this point. The total solution for the two stress

functions is then written as the sum of the component solutions, Le.,

F(x,y) = F(e)(x,y) + F(a)(x,y) , C28.al

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2.3.2 Expressions for the Stresses and Displacements

The Cartesian components of stress are de rived fromeqs. 15.a thro ug h 15.e as,

where the auxiliary stresses are given by:

o(a)X = 2 b 3 x + 6 b 4 y + 2b7 ,

,(a)Y = 6 b q x + 2 b 2 y + 2b5 ,

,(a) = - 2 b p ~ - 2 b 3 y - b, ,XY

T ( ~ ) = - b8 - 2 bq,-,x ,YZ

bg + 2 b l l Y .p) =xz

[28.b]


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It turns out that it is m ore convenientto impose conditions on the boundary of the localized

region using the polar cylindr ical stresses. Using the cylindrica l coordinatesr - 8 - z (Fig-ure 3), the stress components become,

The general displacement functions u, v and w are given byeqs. 1l. a through 1l.c. The

unknown functions U(x.y), V(x,y) and W(x,y) can now be determined by substitution of the


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stresses eqs. 29.a th rou gh 29.e into eqs. 12.a throu gh 12.e. Integr ation ofof eqs. 12.a, 12.b and

12.e lead s to,

+ d a ) ( x , y ) + u, ,= CkPk ( h + 1)4 Z k + lUO(,y) = k = l[33.b]

*"I

+ W(")(x,y) + w, , C33.clk2

k = lw(x*Y) = Dk rk (6 +

C34.clP44r k = - - uk '

and,

U(') Ow) = Bll (b 3x + 6 b, Y + 2 b, ) + PI, ( 3 bl x + 2 b, y + 2 b,)

+ % ( B l x + 2 B 2 y + 2 B 3 ) x + g(y) ,2

C35.al

V") (x,Y) = P j z(2 b, x + 3 b, y + 2 bT ) + 8 2 2 ( 6 b, X + 2 b, y + 2 b, )

+ & ( 2 B 1 x + B 2 y+ 2 B t ) y + f(x) , [35.b]2

W(a)(x,y) = - [ Pa (b, + 2 b l o x ) + B ~ x ]Y + h(x) . [35.c]

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In addition, U andV must satisfy eqs. 12.c and W eq. 12.d. This leads to

h(X) = p55 b , ~ + Wo . C35.fl

The cons tants U, , V, and W, may be dropped since they represents rigid body motion, terms

which w ere a lready included in the gene ral formulation of the displacement functions (see eq.

11).

2.3.3 Application of the Boundary and Interface Conditions

2.3.3.a The Eigenvalue Solution

An examination of Figures 2 and3 illustrate the conditions that must be appliedto th e sol-

utions to have the solutions satisfy the conditions of the skin-stiffener interface problems.

Specifically referring to Figure3, the surface represented by 8 = a, and 8 = - n are gener-

ally free o f any tractions. If pressure-loaded pa nels are being considered, these surfaces could

be exposed to the norm al pressure traction. However, the magnitudeof this traction relative

to the m agnitude of the stresses generated within the ma teria l is negligible and can be con-

sidered zero. Hence, one condition on the analysis is that the surfaces at 8 = a, an d

8 = - n are traction free. In addition along the lin e8 = 0, the stiffener and skin are joined.It

is the intent of the joining to provide a condition ofno slippage along this line, i.e., the dis-

placements are continuous across the interface. Finally, from stress equilibrium arguments,

the stresses ow ,T, , an d T ~ ,are continuous across the interface. These conditions provide the

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necessary equations for determ ining som e of the constants, and hence the characteristic of,

F(Z) and 'u(Z).

The conditions o f traction-free bo undaries are rep resented by (see Figure 3),

o r ) ( r , a,) = @ ( r , q ) = TZe(1) (r,a,) = o , [36.a]

oe2)(r, - R) = Tre(2) (r , - R) = Tze(2) (r, - T t) = o , [36.b]

where the su perscripts 1 and 2 designate material 1 (flange) and m aterial2 (skin) respectively.

For a perfect skin-to-flange bon d the interface cond itions along8 = 0 require,

C37.d

[37.d]

C37.el

C37.fl

Finally, at the ends of the cross-section (see Figure 3) the following inte gra l conditions ar e

requ ired to be satisfied,

J JA r,dxdy = 0 , J JA . r yZdxdy = 0 , [38.a,b]


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It should be noted that as far as the above end conditions are concerned, it is possibleto

impose kinematic boundary conditions rather than force conditions,or it is possible to impo se

a m ixture of the two. For example, rather than impose the a rea integral of6, over the bodys

ends to equal P,, it can be required that w = e,z, e, be ing an ap pli ed ax ial stra in.

The su bstitution of stresses, eqs. 31.a through 32.e, and d isplacements, eqs. 33.a throu gh

35.f, in to the traction-free boun dary conditions, eqs. 36.a and 36.b, and stress an d displace -

me nt continuity cond itions, eqs. 37.a throug h 37.f, p laces certain cond itions on the stress

functions, F(x,y) and Y(x,y). Ho wever, following the su bstitution of stresses and displacem ents

into eqs. 36.a through 37.f and the application of variable separationto the resulting ex-

pressions, it is evident that the conditions on the eigenvalue expansion part of the solution

separate from the cond itions on the auxiliary and p articular parts of the solution. Therefore,

the c onditions o n the eigen value expansion part of the solution (Le., F(.)(x,y) and @(x,y)) are

treated sepa rately from the conditions relatedto the p articular and auxiliary part of the tota l

so lution (Le., F()(x,y), y)(x,y) an d @)(x,y)). The impositionof traction-free and traction and

displacement continuity conditionson F(*)(x,y) an d yb0)(x,y) lea dsto the eigenvalue problem

associated with 6 and h. Hence these p arameters are determined uniquely for each problem.

The co nditio ns o n F((x,y), w()(x,y), and $p)(x,y) for the s pec ific skin-stiffener con figura tion a re

discussed later, while the conditions on the eigenvalue expansion are discussed next.

The traction-free boun dary conditions associated with W(x,y) a re given by eqs.36 as,

and the traction and displacement continuity conditions by eqs. 37 as,

W()(X,O) = W(*)(X,O) .

Equations 39.a through 39.c leadto the following relations associated with yb(x,y):


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C40.al

[40.d]

where rk is defined by eq. 34.c.eigenvalue problem for6. This set of equations can be written symbolically as,

The above 4 simultaneous set of equations forms the

For a nontrivial solution the values of6 are given by,

I Q ( a l , 6 ) I = 0 . [41 .b]

+ 4

The eigenvector, 6 , consists of the tw o eigenvectors,

2, respectively, i.e.,

and D(*), relatedto materials 1 and

where both and consist of two constants each.

If 6 is complex, solutions occur in complex conjugate pairs of the form,

6 = y k i $ .

However, in ord er that w(x,y) be finite at the origin,

C41 . c l

[4 l .d]

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- l < y .

The eigenvalue problem for6 is described in greater detail in Appendix6.

141. e l

The boundary conditions associated withF (x,y) a re giv en by eqs. 36 as

$) ( r, a,) = T$)(r, all = o , C42.al

ob2) (r , - n) = 7::) (r, - n) = o . [ 42. b]

The traction and disp lacem ent continuity conditions associated with F(x,y) at the interface

(6 = 0) are given by eqs. 37,

oil) (r,O) = oh2)(r,o) , C43.al

~ # ( r , o ) = #(r,o) , [43.b]

U()(X,O) = u(2)(x,o) , C43.cl

(I) (x,O) = J2)(x,O) . [43.d]

Equations 42.a through 43.d lead to the following8 simultaneous equations associated with

F(*)(x,y):

4

k = lI; ~ r ) r h ( c o sal + p k(1 ) s in ul)(h+2) = 0 t C44.al

4

k = lI; C f ) r h ( & I cos u, - sin ul)( co s ul + pi) sin u1)(+) = 0 , C44.bl


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A detailed description of the eigenva lue prob lem forh is given in AppendixB.

The eigenvalue problems associated with)i and 6 do not occur in a standard form and hence

requ ire special procedures in order to determine the eigenvalues. Two methods were used in

the present investigation, both methods involve the computationof the characteristic ex-

pressions (in closed-form or numerically) which is equal to the d etem inant of the part icular

ma trix of interest, i.e.,Q of eq. 41.b or A of eq. 45.b. The first method, M ull er [34,35], ope rate s

on a complex characteristic equa tion to find the roots. Once a root is foundit i s eliminated (or

deflated) from the characteristic expression. This method is particularly well suited for com-

plex root computation (see ref. 35). The second method is based on the secant technique for

simultaneous nonlinear equations [36]. Here the real and imaginary partsof the determinant

are treated as a set of two simultaneous equations, with the two unknowns being9 an d E, or

y and p .

Using the relations between the stresses and the stress functions, eqs. 15.a through 15.e,

the eigenfunction expansion of the stresses can be written [23]. For the nth rea l eigenv alue6,

an d An, these stresses take the form,

[46.b]

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Whereas, for the nth com plex eigenvalues 6, an d h, the stresses take the following form:

[47.d]

where, i = 1,2 corresponding to material 1 (flange) and material 2 (skin). In addition,

c, , c', , d, and d', ar e unknown rea l coefficients, whereas, c# and d # ,i = 1 , 2 , are known

quantities of the no rmalized nth eigenvector asso ciated with m aterials1 an d 2 respectively.

The reader is referr ed to AppendixB for a detailed discussion o f the ap propriate eigenfunction

represe ntation for a real and complex eigenvalue. Next the applicationof the boundary and

interface conditions on the auxiliary and particular portions of the solution is discussed.

2.3.3.b The Remaining Part of the Solution

In previous sections we saw that the imposition of traction-free boundary and interface

conditions on the eigenva lue expansion part of the so lution for F(x,y) and Y(x,y) le d to the

eigenvalue problems for h and 6, respectively. In this section these con ditions are enforced

on the other parts of these solutions (i.e. F(d(x,y) , and Y(P) (x,y)). The imposition of the

boundary and interface conditions are implem ented in the context of the skin-stiffener geom-


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etry as discussed in section 2.3.2. The traction and displacement continuity conditions are

given by eqs. 37 . These conditions requ ire that,

u()(x,O) = l i2)(X,O) , [48.a]

V(l)(X,O) = J2)(X,O) , [48.b]

r$)(x,O) = Tg)(X,O) , [48.c]

rt:)(x,O) = @X.O) , [48.d]

o?) (x ,O) = oy(x,o) . [48.e]

Here we chose to use the Cartesian components of stress. Considering eqs. 1l. a through 1 l.c

and 29.a through 30.e the above relations become:

[49.b]

2. Analytical Me thod Development 32

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6 b(,')x + 2 b p ) = 6 b i2 )x + 2 bi2) . C49.fl

Matching coefficients o f the same power of x,y, andz we arrive at the following relations:

i = 1,2,3 , C50.al(1 ) (1) = (2) (2)a33 Bi a33 I

BY) = Bi2) , C50.bl

oy) 2 ai (2) 9 i = 12 I C5o.cl

where the superscripts 1 and 2 correspond to material 1 (flange) and material 2 (skin). Next

the traction-free conditions are appliedto the skin an d flange free surfaces. These conditions

for the skin, eq. 36.b, require that,

Consideration of eqs. 29 through 32 in conjunction with the above conditions leads t o the fol-

lowing relations:


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Equating coefficients o f the same power o fr yields

b[2) = 0 , i = 1,2,5,6,8,10

Finally, considering eq. 50.d togethe r with eq. 53 leads to

bll) = b[2) = 0 , i = 1,2,5,6,8,10 .

[52.a]

[52.b]

C52.Cl

C531

C541

Based on the abo ve results, the a uxiliary stress components, eqs. 30, reduce to

OF)= 2 b 3 x + 6 b 4 y + 2 b 7 , C55.al

.(a)Y = 0 , [55.b]

K55.cl

[55.d]

Further information can be obtained by considering the traction-free conditions for the

flange (material 1). These conditions are

(1 106 (r,al) = 0 , [56.a]

L56.b)

2. Analytical Method Oevelopment 34

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The use of these con ditions, tog ether with eqs. 32 and55, leads to the following relations:

6 r n 2 [ rn bL1) + n b i ) ] + 2 b y ) n 2 = 0 , C57.al

- 2 r n [ b L 1 ) ( 2 r n 2 - n 2 ) + 3 b k ) r n n I - 2 r n n b i ) = 0 , [57.b]

- n b r ) - 2b!,\)rn2 = 0 , C57.cl

where, rn = cos(a,) , n = sin(a,) . Equating coefficientsof the same power of r leads to

bl) = 0 , i = 3,4,7,9,11 . C581

Finally, considering eqs. 50.e through 50.9 in conjunction with eqs. 27.b, 35, 54 and58 leads

to

bL2) = 0 , [59.d]


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where,

Finally, considering all the relations derived in this section, the auxiliary stresses for the skin

and stiffener are as follows:

[61.b]

[61 .c,d]

[6l.e,f]

The stresses in material 1 (flange) and ma terial2 (skin) are given by eqs. 46.a through 47.e.

These stresses are given in term s of the u nknown coefficients cn,cfn,dn,df,, and

B,, i = 1,2,3,4. For the specific skin-stiffener problem these coefficients are determined by a

collocation procedure. Thisis discussed next.


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2.4 Collocation Procedure

The eigenfunction expansion elasticity solution is validfor a semi-infinite domain. The

particular boundary value problem associated with the skin-stiffener geometry is solved by

assum ing that the solution is valid i n the finite dom ain repre sented by the loc alized flange

termina tion region. Furthermore, the solution is assumedto be represented by a finite num ber

o f ter m s in the series. The unkn own coefficients c, and c', in eqs. 46.a throu gh 46.c and 47.a

through 47.c are determined using the boundary collocation technique. Referring to

Figure 2, the boundary of the local reg ion is bounded by contour ABCDEFA. In the actualstructure analysed, bo undaries AF, AB, BC, and ED are traction free, whereas boundaries CD

and EF are subjected to both normal, on, and tangential, z, tractions. O bviously in the nom en-

clature of the problem, 0 , = 0 , an d t, = tTy on CD and EF, and on = cry an d t, = ty on BC and

DE. By the developmentof the elasticity solution, the traction-free conditions on FA and AB

are already satisfied. The collocation procedure is usedto satisfy the traction-free conditions

on boundaries BC and DE, and to match the normal and tangential tractions o n boundaries

CD and EF as determ ined by the global finite-element analysis. It should be noted that i n theboundary collocation procedure, the normal and shear stresses, on an d t, , along the contour

BCDEFare written in term sof the unk nown coefficients, c, a nd c', ina truncated e igenfunction

expansion. These stresses are matched with the same stress components calculated by the

finite-element analysis on the contour. Althoughit is possible to collocate other responses,

such as three componen ts of stress, the strains, the displacements, etc., it is felt that matching

the normal and shear stresses on the boundary is the best choice. The main reason for this

is the fact that enforcement of the force equilibrium conditions on the local region as a finite

body involves on ly the no rma l and shear traction on the boundaryof the body.

For the specific problem here, the collocation at m points around boundary BCDEF leads

to 2m simultaneous equations from which the 2n unknown coefficients are determined. The


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use of more collocations points (i.e., m > n) leads to an overdetermined set of equations.

Solution o f these eq uations p roduce s the 2n unkn own coefficients c, and c', wh ich satisfy all

boundary conditions ina least-squares sense. If the original set of equations is represented

by 9

-+ -+

S = A C . [62.a]

then the least squares solution for is [37],

+c = ( A ~ A ) -'AT: . [62.b]

In the above, A is a 2m x 2n known matrix (for which m> n ), s is a vector of length 2mconsisting ofa 2m known boundary stress quantities, and isa vector of length 2n consisting

o f 2n unkn ow n coeffi cien ts, c, and c', , of the truncated eigenfunction. The elements of matrix

A involve m ate rial properties and the coordinates of the collocation points. Once these coef-

ficients are determined, the stresses in the localized region can be written.For a converged

eigenfunction, these stresses a re assumed t o represent the true stress field in this region. In

a later section convergence is studied by varying the number of terms (o r eigenvalues) in the

truncated function expansion and by varying the number of collocation points.

2.5 Global Finite-Element Analysis

To facilitate the ap plication of the loca l elasticity solution to the skin-stiffener problem , a

finite-element program was implemented which incorporated the generalized plane deforma-

tion assumption. This finite-element formulation is consistent with the elasticity solution de-

veloped ea rlier. The finite-element program was developed for two reasons. The primary

reason was to provide boundary conditions for the eigenvalue expansion local elasticitysol-


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ution. The second rea son was t o check the results of the eigenvalue expans ion. As indicated

in the introduction, the methodology was developed for geome trically lin ear problems, and

then extended to geometrically nonlinear problems. Thus the finite-element program imple-

mentation was very important in the first step of the solution strategy developm ent.

A body which conforms t o the genera lized plane deformation assum ption may be analysed

using a two-dimensiona l finite-element model. In this section a short des cription of the finite-

element program (PE2D) is given. The PE2D program developed is based on th e FEM2D

finite-element program,[38], with the appropriate modificationsto meet the need of the pres-

ent investigation. For a more detailed discussion the re ader is referred to Appendix C.

The generalized plane deformation elasticity finite-element model is based on the dis-

placem ent field given by eqs. 1l.a through 1 l.c for homogeneous anisotropic bodies for which

stresses do not vary along the generator (Le., the z axis). This displacement field can be

written in vectorial form as,

where;

and,

+- + +u = u , + u ,

V,(X,Y,Z) = - -B2a33z2+ B,xz ,2


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Here the terms associated with the rigid body rotation and translation are omitted. The un-

known functions U(x,y), V(x,y) and W(x,y), are approximated according to the finite-element

method. Further, it may be shown that the unknown constantsB i , ( i = 1,2,3,4) are related

to the body's kinematic end conditions, that is:

In the stiffene d skin structu ral context, e, andIC , are the axial extension and curvature in the

z direction, and K, is the twist curvature about thez axis. The coefficientB, is related to th e

inplane twist about the y-axis and is of no consequence in the structure considered h ere and

is therefore set to zero. The strain vector,E, is given by,

-+++& , = E + c0 ,

where;

ET -- { - au . a ; 0 ; -m . a . -a u + - } ,av+ax ay dY I ax I ay ax

(eo + K, Y) ; x ;K x z

2- -

L

Since only orthotropic media are being considered, the constitutive relation isas given before

(see eq, 6.a) as

-+ox =


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Considering eqs. 6.a a nd 65, it may be concluded that the problem of determining V(x,y) and

U(x,y) decouples from the problem of determining W(x,y). In addition, since U,V and W are

functions which depend on x and y alone, only a two-dimensional finite-element model is re-

quired. The finite-element mod el is derived via the variation formulation in a standard m anner.

The conditions for this mod el consist of specified overall kinema tic conditions (Le., e,, K,, and

K, specified ) and force and/or kinema tic conditions on the boundary in the x-y plane.It

should be pointed out that because of the generalized plane deformation assumption, the

specified o ve ra ll kinem atic cond itions e,, K, and K, do notivary along the the z-axis of the

body.

2.6 Verification of the Analytical Model

In this section attention is givento a results relevant to the verification of the line ar analysis

model. In addition, the fidelity of the local-globalelasticity-finite-element analysis is demon-

strated. By fidelity is meant the accurate representation of stresses by the elasticity solutionwithin the localized region and the smooth transitionto the global region. Whereas the con-

vergence of the finite-element method has been discussed in [39], there is no rigorous proof

for the convergence of the collocation method and its applicationto the present problem. For

the problem here the accuracy o f such a procedure w ill depend on a numbe r of factors such

as: a) the accu racy of the bound ary conditions; b) the num ber of eigenvalues used in the

eigenfunction expansion: and c) the num ber of collocation data points. In the following section

these issues will be addressed.

In order to study convergence and accuracy of the current model, the elasticity results are

compared with results from a finite-element analysis in which the mesh was refined twice in

the localized region. The results are compared for a particular problem. The use of a finite-


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element analysis for comparison is due to the lack of another analysis of this particular

problem . Two flange geome tries are considered in this phaseof the study, a 90" and a 45"

flange termination angle, i.e.,a, 90" and a, = 45" in Figures2 an d 3. Two angles were con-

sidered at this stage so as to make conclusions regarding m odel verification more general.

The flange and the skin are constructed of an 8-ply quasi-isotropic( f 45/0/90), lam ina te, and

an 8-ply orthotropic( f 45/90,), lamina te, respe ctively. Ma te ria l prop erties ar e give n in Ap-

pendix D. The finite-element discretization for 90" and 45" flange angle skin-stiffener geom-

etries are show n in Figures 4a and 5a, respectively, for what is referred to as the coarse m esh.

Subsequent me sh refinements of the localized region a re shown in Figures 4b,c and 5b,c. Due

to the geo metric and material symmetry, only one-half of the structure cross-sectionis mod-

eled. The p articular problem conside red for the verification study is shown Figures 4.a and 5.a,

namely the stiffened plate subjected to a pure bending momentM. The particular loading was

chosen to illustrate the computational method because this loading produce peeling and

shearing stresses at the skin-stiffener interface that are approximately the same orderof

magnitude. It should be mentioned that for this problemT~~ an d T~ are identically zero.

Therefore, attention is focused onox , oy, and T ~ .The exact nature of the skin-stiffener inter-

face stresses near the flange terminus depends on the the value of q,, the real part of the firsteigenv alue (see eq. 45.d). These stresses are unbound ed if -1 c q,.

In the collocation procedure, the normal,on, and tangential, T,, stresses are collocated

along the closed co ntour ABCDEFA shown in Figures 4 and5 by the heavy line. The stresses

as computed by the finite-element analysis of the entire cross section are used to provide

collocation data on the contour. In reality, collocation takes place only along boundaries BC,

CD, DE, and EF, since the conditions o f stress-free bo undaries along A 6 and FA are satisfied

exactly by the e lasticity solution. B ound aries BC and DE were taken as stress-free faces of the

skin and flange, respectively, whereas the stresses along the internal boundaries CD and EF

are those determined by the finite-element analysis. These stresses on the boundary are

calculated exactly within the finite-element context by postprocessing the finite-element dis-


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( a ) Coarse Mesh

I

( b ) Refined Mesh

* E D

( c )Fine

Mesh

Figure 4. Loading and Finite-Element Discretizations of Skin-Stiffener SO' Flange TerminationAngle.


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(a) Coarse Mesh

E 0

( b ) Re f in ed Mesh

e D

( c ) Fine Mesh

Figure 5. Loading and Finite-Element Discretizations of Skin-Stiffener 4 5 O Flange TerminationAngle.


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denoted as is, EY, an d TV Throughout the study, the stresses, and infact, distances, have

been normalized in a m anner relevant to the particular problem.

Figures 6 through 11 show the variation of the skin-stiffener interlaminar stresses,a,, ay,

and T, as a function of distance along the interface for the90" and 45" flange termination an-

gles. Both stresses determined by the finite-element analysis and those calculated using the

truncated eigenfunctions, eqs. 46 an d 47, are shown. The elasticity resu lts shown in the figures

were generated using 15 eigenvalues and 100 collocation points. The eigenvalues for both

geometries are given in Appendix D. Each figure illustrates calculated values of a particular

stress component for each of the three different meshes (Le., coarse, refined, and fine). The

data point closest to the vertex for which finite-element stress data is plotted is atZ = 0.025.

It should be noted that X goes from 0 to 3 because the skin and flange are of equal thickness,

t, , and the length of the local region,L, is 1.5 times the combined thickness of the flange and

skin, 2t,. The scale o f the vertical axes was taken t o account for the value of the no rmalized

stress at Z = 0.025.

A number of interesting observations can be made from these figures. In general, for both

flange termination angles, at sufficient distances from the flange termination vertex, the

finite-element and the elasticity solutions show excellent agreemen t for all three com ponents

of stress. The point at which this agreement can be categorized as being excellent mo ves

closer and closer to the vertex with increases in the mesh refinement. As will be discussed

shortly, there is very little difference between the elasticity solution resulting from collocating

data from a coarse mesh and the elasticity solution resulting from collocating data from the

fine mesh. In essence, for either flange termination angle, the elasticity solution does not

change from one mesh to the next and hence the use of the co arse m esh is sufficient. What

the figures are showing is that the elasticity solution coincides with the solution the finite-

elemen t analysis is appearing to converge to. It should also be noted thatay and fy computed

by the finite-elements adjacent to the interface ar e discontinuous across the interface in some

region near the vertex. The situation is worst forT~ than for ay. This discontinuity relatesto


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Ii

iI

P Q

rv)W

zW

L

o m 0 Lo 0 L o oL o 0 1 0 I\ Lo

tbx

Iv)

nW

LWK

,-o m 0 ' Lo 0 L o om 0 1 0 I\ Lo 017 7 7

tbx

Iv)

:Wv)KU00

0 m 0 0 m 0m c u 0 P m cu

r)

01

7

0

cn-w

\X

cu

c

0

cn

-w

\X

I bx

2. Analytic81 Method Development 47

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I b>'

I

I b>'

v)+

\X

03

Iv)WIWv)[rU

0

cu

0 c

003 cu c 0 -

v)-+\

X

fmii

I


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m

0M 0 -

I

I b2WCmm

01 e"

\7 x

I ;b- I

0

Iv)

Wv)aa00

E!

ii3

m0


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,-

0 Lo 0 L o oL o N 0 b Lo 01

I b"

I(I)w5

w

LLw

n

a

0 Lo 0 L o oL o N 0 b Lo 01

I b"

I(I)w5w(I)

o m 0 m 0 m oLo (u 0 P I n cu7 - -

r)

N

0

r)

cn-+

\X

UcQ

r"

\, x

0

0

I b"


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Iv)

W5WzLL

r) 01 - 0 -I

I bh

Iv)W5DwzI&Wa

r)

01 +J"

\, x

0

r)

cn0 1 - e

\7 - x

0-I

I bzI

Iv)W5Wv)aa00

0

cr)

VIN-C,

\

. - x

d- N 7 0 c

I

US

4!?a

iim

2. Analytical Method Doveloprnent 51

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W02

J YL L m

a z

c- iz zw w1 1w w >J J C

w w ow w i =C k - "z z 5L L L L W

P 9

Lul

Iv)wIWz

L1,-

d

Iv)wI

d- 0 -

I

01

0

ul-+

\

X

m

0m cu c 0 -

I

I t-"


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the inherent inaccuracy of the finite-element data near points of stress singularity. Although

the region of inaccuracy shrinks with increases in mesh density near this point, the discrep-

ancy between the two values of stress near the vertex is quite significant. This characteristic

oc cur s for both the 90 and the 45" flange termin ation angles.

To illustrate that the stresses as computed by the coarse mesh are sufficiently accurate to

serve as boundary conditions for the elasticity solution, Figures 12 and 13 illustrate the vari-

ation withi7 of the three stresses Cx, Cy, and yv, as generated by the elasticity solution for the

tw o flange geom etries. Each figure illustrates the three componentsof stress and each stress

component is represented by three relations, each relation corresponding to the collocation

of stress data from the different finite-element meshes.It is immediately evident from the fig-

ures that for each stress component for both flange angles, the results of collocating stress

data from the three different me sh densities coincide. This indicates that in order to obtain a

good a pproximation o f the state of stress in the localized region using the elasticity solution,

only the coarse mesh need to be used. This is somewhat expected since at some distance

away from the pointof singularity the co arse mesh wil l yield a converged setof stresses. That

is, the values of the stresses which were used for collocation on the boundariesCD and EF

were approximately the same for all m eshes. This supports the computation effectivenessof

the current procedure, since only a coarse finite-element mesh is requ ired to produce highly

accura te stresses in the localized region. From the figures it is hard to distinguish between

the magnitudes of the stresses at i7 = 0.025. Hence, the values of E x , E y, and TY at these

points, for the three meshes, are given in Table 1. It is observed that the percent difference

in these values is negligible, again emp hasizing the power of this methodology.

At this pointit is appropriate to depart from the discussion of the accuracy of the method

and illus trate some of the physical results that can be addressed from the analysis. Table1

indicates that in terms of the peel stress, the 45" flange termination angle is better than the

90" flange termination angle. Ati7 = 0.025, the norm alized peeling stress i s 0.290, while for the

45" angle, the peeling stress is 0.192, 33% less. On the other hand, the shear stress is about


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I

xI t-"

I b"

2. A ~ l y t i u lMothod Dovobprmnt

cn-w

\

X

54

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M

M DJ 7 0 -I

r"

\, x

0

M

0U) d- M DJ 7 0 -

I

I bh

0 Ln 0 Ln 0 m 0LD cu 0 P Ln

I b"

"-+

\X

"-+

\X


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PIC .

lbh

C3v)

lb"

( D a mm m m0 0 0. . .r r r

PIC .

0)mCaii0v)d

Ibh

lb"


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the same for the two different flange geometries. On balance, the 45" angle would be better

of the two. This coincides with intuition but the methodology developed here allows for the

behavior of the two different angles to be quantified.

Next, the issue of the convergence of the truncated eigenfunction expansion, as related to

the number of eigenvalues and collocation data points, is addressed. Figures 14 and 15 illus-

trate the variation withE o f Cx,Cy,and TXvas computed by the truncated stress functions (eqs.

46 and 47) for 90 and 45" flange termination angles. The multiple data on each plot corre-

spond to stress calculations produced by 5, 10 and 15 eigenvalues in the truncated

eigenfunctions. In these co mputations 100 collocation points from the coa rse mesh were used.

As can b e seen in the figures, convergen ce of the truncated stress functions forZy and TY

occurs between 10 and 15 eigenvalues. That is, the stress com putation for 15 eigenvalues is

bracketed between the computation for5 eigenvalues and the computation for 10 eigenvalues.

On the oth er hand, this convergence is not completely evident for0.. There seems to be a

slight increase in this stress value with an increasing num berof eigenvalues. There does not

seem t o be a bracketing effect with increases in the num ber of eigenvalues, as was observed

for the other stress components. However, on a percentage basis, the increase of stress is

minimal. Hence it is felt that the use of between 10 to 15 eigenvalues leads to convergence

of the eigenfunctions in eqs. 46 and 47. Figures 16 an d 17 illustrate data for the same stress

components in which the number o f collocation points were varied. In particular, the number

of collocation points was doubled.For these computations 15 eigenvalues and a coarse mesh

were used in the stress computations. F rom Figures 16 and 17 it is apparent that the nu mber

of collocation points has a minimal (or no) effect. This is true provided that the number of

collocation points are approximately3 to 4 times larger than the number of coefficients in the

truncated eigenfunctions. It should be noted that the number of collocation points for the90"

flange termination angle, Figure 16, is different than the number of collocation points for the

45" flange geome try, Figure 17, because of slight differences in geom etry.


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rl 7 0 7I

u)

-+

\

X

I

P

I b

2. Andytic8l Method Developmont

0In 0 In 0 in 0cu 0 P In cu

58

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M

01

0

u)+

\X

aQ

0 m 0 m 0 m om cu 0 P m cv

I bx


u)+J

\

X

59

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a d - M 01 7 0 7

I

M

a 4"

\7 x

0

>rI t-"

-

v)m l -I- z

z gO aa

22 0O FI - '

cu

0

0 In 0 In 0 In 0In 0 P In cu

"4

\

X

2. Analytkd Method Development

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I

III

m e M 01 r- 0 r-I

v)4

\X

M

0 4 4"

\X

I bzI

0 In 0 In 0 In 0m cu 0 I? Lo cv

I b"

2. Analytical Method Devolopment

m

0

61

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r) 01 7 0

- I d

w w0 00 01 1

+ I -z zw w1 1

w w- I Jw w

I 1w wk Ez zL L L C

o ma *

z z0 0

0 0w wlnln- 2 4

0 0

m m

III1IIIIII

I

I bhm

r"

\A X

0

o m 0 In 0 m 0m 0

I b"


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2 - l

w w0 0

0 0S I

I - +z zw w1 1

w wI 1

w w

Y Y

t kz zL L L L

I n 0

+ a ,

z z0 0

0 0w wW W4 4

0 0

m m

IIIIIIIIII

r) 01 7- 0

xI bX

01 7- 0 -I

I bx

r)

01 r"

\, x

0

rl

01 -r"

\7 - x

0

-0

Lo 0 In 0 L o o0

Lo cu 0 P Lo cu

u)

-e

\X


I bx

64

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of the adhesive layer on the peeling and shearing stresses. In what follows, the interface

stresses are computed using finite-element analyses for two adhesive layer thicknesses.

These results are compared to results for the case for which a zero bond line thickness is

assumed. The data presented provide important information as to whether the elasticitysol-

ution is conservative or nonconservative in the calculation of skin-stiffener interface stresses

when an adhesive laye r is present. One may postulate that an elasticity solution in which the

two adherents are assumed to have a perfect bond of zero thickness will leadto noncon-

servative stress calculations. This is a con sequence of the fact that the assu mption o f perfect

bond reduces the flexibility o f the interface, as compared to if the adhesive la yer was present.

Hence it is important to know how large an erro r is introduced as a consequence of the as-

sumption made. To determine this effect, the same skin-stiffener geometry and loading con-

ditions which ar e shown in Figure 4 are analysed. Two adhesive layer thicknesses are

evaluated, h- = 0.05 an d = 0.10. The thicker adhesive layer represents the upper limit of

the bond line thickness. T he thinner laye r thickness is the nom inal bond line thickness. The

results from these analyses are compared to stress data for - = 0 layer thickness as com-

puted by the finite-element program and as computed by the local elasticity analysis. The

t, t,

ht,

adhesive ma terial properties are given in AppendixD. Mesh refinement in the localized regio n

corresponds to the fine mesh of Figure 4.c. A coarser mesh is used outside the local region.

However, element size in the coarse region is such that the adhesive elements aspect ratio

does not exceed 1:4. Figure 20 shows the character of the interface peeling stress cy in the

localized region. Figure21 displays similar information for the interface shearing stressty

Each figure consists o f tw o plots, one illustrating the stresses at the interface as computed

by elements in the flange, and the other illustrating the same stresses but evaluated by ele-

ments in the skin. The finite-element stress data is denoted by symbols, whereas, the local

elasticity results are indicated as solid curves. As in previous figures the stresses and dis-

tance along the interface have been normalized.


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0 0 0

E

n.-nY

Lo707

C

Q

.-h

Y

In0

I b"

0Lo0

C0mal

oc

-

I

Ln 0 Ln 0 Ln0 04 +

I

Ca0

cmaecba

0N

t

iiam

I bA


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c-Yv)

cn

.-

.-nY

0U) 0 Lo 0 Ln

%-- 0 0.-I

cn0

.-Y

>rI kX

ia

0

mCaE05Ua0c03zif


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In general, the results of Figure 20 suggest that there is a slight increase in the m agnitude

of the peeling stress, cry , between j?. = 0.025 to Z = 1 as the bond line thickness increases.

However, the increase relative to the zero bond line results is quite small, something o n the

order of less than 5% . In general, the results of Figure20 indicate that even in the present

of adhesive layer, the peel stresses tend to become unbounded as the flange terminus is ap-

proached. The unbounded nature of the peel stress with an adhesive is in qualitative agree-

ment with the finding of [9]. The conclusions regarding the behavior of the shear stress are

no t so clear. As has been shown in Figures 6 and 11, the elasticity solution and the finite-

element results for zero thickness bond line are in disagreement near the flange termin ation

point. Further, the

19890008528

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