Buckling, postbuckling and progressive failure analysis of - SOAR

BUCKLING, POSTBUCKLING AND PROGRESSIVE FAILURE ANALYSIS OF HYBRID COMPOSITE SHEAR WEBS USING A CONTINUUM DAMAGE MECHANICS MODEL

A Dissertation by

Javier Herrero

Master of Science, Wichita State University, 2000

Aeronautical Engineer, Polytechnic University of Madrid, 1997

Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of

Wichita State University in partial fulfillment of

the requirements for the degree of Doctor of Philosophy

December 2007

© Copyright 2007 by Javier Herrero

All Rights Reserved

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BUCKLING, POSTBUCKLING AND PROGRESSIVE FAILURE ANALYSIS OF HYBRID COMPOSITE SHEAR WEBS USING A CONTINUUM DAMAGE MECHANICS MODEL

The following faculty have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Aerospace Engineering

__________________________ Charles Yang, Committee Chair ____________________________ James Locke, Committee Member ____________________________ Walter Horn, Committee Member _____________________________ John Tomblin, Committee Member _______________________________ Hamid Lankarani, Committee Member

Accepted for the College of Engineering _____________________________ Zulma Toro-Ramos, Dean Accepted for the Graduate School _______________________________ Susan Kovar, Dean

iv

DEDICATION

To my dear wife, Melissa, and my joyful children, Grant and Isabel, for their silent sacrifices

v

ACKNOWLEDGEMENTS

I would like to thank my advisors, James Locke and Charles Yang, for their many years

of thoughtful, patient guidance and support. I also thank Ismael Heron for his generous advice

and coaching, and to my friend and mentor Everett Cook. I would also like to extend my

gratitude to the members of my committee Walter Horn, John Tomblin, Bert Smith and Hamid

Lankarani. I also want to thank to the Sandia National Laboratories for funding the experimental

portion of this research, to Juan Felipe Acosta, Tom Hermann and Sanjay Sharma of the National

Institute for Aviation Research for their cooperation, and to Tai Vuong of the Boeing Company

for his technical remarks to my work.

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ABSTRACT

This dissertation presents an innovative analysis methodology to enhance the design of

composite structures by extending their work range into the postbuckling regime. This objective

is accomplished by using the numerical simulation capabilities of nonlinear finite element

analysis combined with continuum damage mechanics models to simulate the onset of failure

and the subsequent material properties degradation. A complete analysis methodology is

presented with increasing levels of complexity. The methodology is validated by correlation of

analytical results with experimental data from a set of hybrid carbon/epoxy glass/epoxy

composite panels tested under shear loading using a picture frame fixture.

TABLE OF CONTENTS

Chapter Page

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1 INTRODUCTION ........................................................................................................................1

2 OVERVIEW OF BUCKLING, POSTBUCKLING AND PROGRESSIVE FAILURE..............7

2.1 General requirements and key features of the analysis and simulation ......................7

2.2 Preliminary observations about nonlinearity, buckling and post-buckling ................9

2.3 Linearized buckling solution.....................................................................................11

2.4 Nonlinear solution.....................................................................................................12

2.5 Mode-based failure criteria and progressive failure analysis ...................................14

3 LITERATURE REVIEW ...........................................................................................................16

4 SCOPE AND OBJECTIVES......................................................................................................24

5 EXPERIMENTAL STUDY........................................................................................................27

5.1 Data reduction of strain gage readings......................................................................28

5.2 Data reduction of the ARAMIS optical system measurements ................................35

5.2.1 Laminate G - 20 degree data reduction..........................................................36

5.2.2 Laminate H - 45 degree data reduction..........................................................41

5.2.3 Laminate E - 90 degree data reduction ..........................................................46

5.2.4 Laminate F - 0 degree data reduction ............................................................51

5.3 Comparison plot of ARAMIS data vs. strain gage data............................................54

5.4 Experimental determination of buckling loads. Spencer-Walker method ................58

TABLE OF CONTENTS (continued)

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5.4.1 Southwell plots for struts under compression................................................58

5.4.2 Spencer-Walker plots for plates.....................................................................60

5.4.3 Spencer-Walker plots for the panels tested....................................................62

6 FINITE ELEMENT MODEL.....................................................................................................68

6.1 Modeling the composite panels ................................................................................70

6.2 Modeling the picture frame arms..............................................................................71

6.3 Mechanical properties of the laminates tested ..........................................................72

7 ANALYTICAL CLASSICAL SOLUTIONS.............................................................................75

8 LINEARIZED BUCKLING ANALYSIS ..................................................................................89

8.1 Formulation...............................................................................................................89

8.2 Analysis results .........................................................................................................90

8.2.1 Laminate F - 0 degree with the direction of the load.....................................91

8.2.2 Laminate G – 20 degree with the direction of the load..................................94

8.2.3 Laminate H – 45 degree with the direction of the load..................................97

8.2.4 Laminate E – 90-degree with the direction of the load................................100

8.2.5 Linearized buckling analysis. Summary of results. .....................................103

9 NONLINEAR BUCKLING AND POSTBUCKLING (WITHOUT FAILURE) ....................107

9.1 Formulation.............................................................................................................107


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9.1.1 Historical approaches...................................................................................107

9.1.2 Time approximation and Newton-Raphson method ....................................109

9.2 Effect of initial imperfections in the nonlinear solution .........................................112

9.2.1 Effects of initial imperfections in the ANSYS model .................................112

9.2.2 Effects of initial imperfections in the ABAQUS model ..............................116

9.3 Effect of the sub-stepping in the solution ...............................................................118

9.4 Correlation of FE model strains with strain gage readings.....................................118

9.4.1 Laminate G – 20 degree with the direction of the load................................123

9.4.2 Laminate H – 45 degree with the direction of the load................................127

9.4.3 Laminate E – 90 degree with the direction of the load ................................130

9.4.4 Laminate F – 0 degree with the direction of the load ..................................133

10 PROGRESSIVE FAILURE ANALYSIS...............................................................................137

10.1 Failure prediction in the analysis of deeply postbuckled panels.............................137

10.1.1 Material nonlinearities .................................................................................138

10.1.2 Delamination growth and decohesion elements ..........................................139

10.2 Overview of the progressive failure analysis (PFA)...............................................139

10.3 Failure Criteria. Damage activation functions ........................................................140

10.4 Material degradation model in the FE simulation...................................................144


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10.4.1 Damage initiation.........................................................................................144

10.4.2 Evolution of the damage variables for each mode.......................................145

10.4.3 Maximum degradation and element removal ..............................................151

10.4.4 Viscous regularization algorithm.................................................................152

10.5 Results of the nonlinear model with progressive failure analysis...........................155

10.5.1 Fiber tensile failure evolution ......................................................................156

10.5.2 Fiber compressive failure evolution.............................................................159

10.5.3 Matrix tensile failure evolution....................................................................161

10.5.4 Matrix compressive failure evolution ..........................................................164

10.6 Summary of results. Nonlinear model with progressive failure analysis ...............170

10.7 Comparison of the Hashin’s failure functions with other failure criteria ...............172

11 CONCLUSIONS AND RECOMENDATIONS.....................................................................177

12 REFERENCES .......................................................................................................................185

LIST OF FIGURES

Figure Page

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1. Figure 1.1 Rectangular isotropic plate subjected to pure shear stresses [2] .......................2

2. Figure 1.2 Kuhn’s diagonal-tension beam..........................................................................3

3. Figure 2.1 Response of a thin plate or shell under out-of-plane loading............................9

4. Figure 2.2 Schematic nonlinear versus linearized responses............................................11

5. Figure 2.3 Newton-Raphson limitation (left) and Arc-Length methodology (right)........13

6. Figure 5.1 Schematic diagram of picture frame fixture and panel geometry. ..................27

7. Figure 5.2 Picture frame test setup. ..................................................................................29

8. Figure 5.3 Laminate 20 deg. Strains measured on the tension diagonal...........................30

9. Figure 5.4 Laminate 20 deg. Strains measured on the compression diagonal. .................30





14. Figure 5.9 Laminate 0 deg. Strains measured on the tension diagonal.............................33


16. Figure 5.11 Comparison of all laminates. Strains on tension diagonal, front face ...........34

17. Figure 5.12 Comparison of all laminates. Strains on compression diagonal, front face ..34

18. Figure 5.13 ARAMIS reference axes for strain measurement..........................................35

LIST OF FIGURES (continued)

Figure Page

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19. Figure 5.14 Laminate 20 deg, front face, out of plane displacement ( )lbsP 1.39746= . .36

20. Figure 5.15 Laminate 20 deg, front face, xxε strain, ( )lbsP 1.39746= ...........................36

21. Figure 5.16 Laminate 20 deg, front face, yyε strain, ( )lbsP 1.39746= ...........................37

22. Figure 5.17 Laminate 20 deg, front face, xyε strain, ( )lbsP 1.39746= ...........................37

23. Figure 5.18 xxε strains, compression diag., front face, different load levels. ..................38

24. Figure 5.19 xxε vs load, front face compression diag., around center of panel................38

25. Figure 5.20 yyε strains on the tension diagonal for different load levels. ........................39

26. Figure 5.21 yyε vs load - tension diagonal, locations around center of panel. .................40

27. Figure 5.22 Laminate 45 deg front face, out of plane displacement ( )lbsP 65.33139= .41

28. Figure 5.23 Laminate 45 deg, front face, xxε strain, ( )lbsP 65.33139= . .......................41

29. Figure 5.24 Laminate 45 deg, front face, yyε strain, ( )lbsP 65.33139= . .......................42

30. Figure 5.25 Laminate 45 deg, front face, xyε strain, ( )lbsP 65.33139= . .......................42

31. Figure 5.26 xxε strains on the compression diagonal for different load levels. ...............43

32. Figure 5.27 xxε vs load. Compression diag., locations around center of panel................43




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35. Figure 5.30 Laminate 90 deg front face, out of plane displacement ( )lbsP 9.31044= ...46

36. Figure 5.31 Laminate 90 deg, front face, xxε strain, ( )lbsP 4.23525= . .........................46

37. Figure 5.32 Laminate 90 deg, front face, yyε strains, ( )lbsP 9.31044= .........................47

38. Figure 5.33 Laminate 90 deg, front face, xyε strains, ( )lbsP 9.31044= .........................47

39. Figure 5.34 xxε on the compression diagonal for different load levels. ...........................48

40. Figure 5.35 xxε vs. load - compression diag., locations around center of panel. .............48

41. Figure 5.36 yyε on the tension diagonal for different load levels.....................................49

42. Figure 5.37 yyε on the tension diagonal for different load levels.....................................50

43. Figure 5.38 xxε on the compression diagonal for different load levels. ...........................51

44. Figure 5.39 xxε vs. load - compression diag., locations around center of panel. .............52



47. Figure 5.42 Lam 20 degree - strains measured at different locations of the tension and

the compression diagonal vs. actuator load. ARAMIS vs. strain gage..............................54

48. Figure 5.43 Laminate 45 degree - strains measured at different locations of the tension

and the compression diagonal vs. actuator load. ARAMIS vs. strain gage. ......................55

49. Figure 5.44 Laminate 90 degree - strains measured at different locations of the tension

and the compression diagonal vs. actuator load. ARAMIS vs. strain gage. ......................56


Figure Page

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50. Figure 5.45 Laminate 0 degree - strains measured at different locations of the tension and

the compression diagonal vs. actuator load. ARAMIS vs. strain gage..............................57

51. Figure 5.46 - Laminate F (0 degree): ARAMIS measured out of plane displacement vs.

load (above), and same data in Spencer-Walker format (below).......................................64

52. Figure 5.47 - Laminate G (20 degree): ARAMIS measured out of plane displacement vs.


53. Figure 5.48 - Laminate H (45 degree): ARAMIS measured out of plane displacement vs.


54. Figure 5.49 - Laminate E (90 degree): ARAMIS measured out of plane displacement vs.


55. Figure 6.1 ABAQUS FE model view. Picture frame and panel .......................................69

56. Figure 6.2 ANSYS section plot of the laminate for the case of zero degree with the

direction of the load. ..........................................................................................................73

57. Figure 7.1 Laminate 0 degree. Lay-up and stiffness matrices ..........................................76

58. Figure 7.2 Laminate 20 degrees. Lay-up and stiffness matrices.......................................77

59. Figure 7.3 Laminate 45 degrees. Lay-up and stiffness matrices.......................................78

60. Figure 7.4 Laminate 90 degree. Lay-up and stiffness matrices ........................................79

61. Figure 7.5 Laminate 0 degree. Buckling load calculation, two-term Ritz approx............80

62. Figure 7.6 Laminate 20 degrees. Buckling load calculation, two-term Ritz approx. .......81



Figure Page

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65. Figure 8.1 Laminate 0 degree, section plot.......................................................................92

66. Figure 8.2 First mode, buckling load lbsP 838,21= . Out of plane displacement ..........92

67. Figure 8.3 Second mode, buckling load lbsP 522,22= . Out of plane displacement......93

68. Figure 8.4 Third mode, buckling load lbsP 456,47= . Out of plane displacement ........93

69. Figure 8.6 First mode, buckling load lbsP 681,15= . Out of plane displacement ...........96

70. Figure 8.7 Second mode, buckling load lbsP 335,16= . Out of plane displacement ......96

71. Figure 8.8 Third mode, buckling load lbsP 489,34= . Out of plane displacement.........97

72. Figure 8.9 Laminate section plot. Lam 45 deg. ................................................................98

73. Figure 8.10 First mode, buckling load lbsP 498,9= . Out of plane displacement ..........99

74. Figure 8.11 Second mode, buckling load lbsP 147,10= . Out of plane displacement ....99

75. Figure 8.12 Third mode, buckling load lbsP 083,18= . Out of plane displacement .....100

76. Figure 8.13 Laminate section plot. Lam 90 deg. ............................................................101

77. Figure 8.14 First mode, buckling load lbsP 605,4= . Out of plane displacement ........102

78. Figure 8.15 Second mode, buckling load lbsP 865,5= . Out of plane displacement ....102

79. Figure 8.16 Third mode, buckling load lbsP 788,10= . Out of plane displacement .....103

80. Figure 8.17 Rotation of the buckling nodal line. Second mode......................................105


Figure Page

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81. Figure 8.18 Rotation of the buckling nodal lines. First mode ........................................106

82. Figure 9.1 Replication of the first mode with an out of plane perturbation force in the

middle node. Out of plane displacement plot. Laminate 0 degree...................................113

83. Figure 9.2 Replication of the second mode with a couple of perturbation forces, opposite

in sign on symmetric locations. Out of plane displacement plot. Lam 0 deg ..................114

84. Figure 9.3 Perturbation with a couple of forces to induce the second mode ..................115

85. Figure 9.4 Lam 20 deg ABAQUS strains in the first Gaussian integration point in front

and back layers of the laminate FE at the center of the panel..........................................117

86. Figure 9.5 Elements under the footprint of the strain gage. ABAQUS model mesh

element size 0.25 in, compressed diagonal front face .....................................................120

87. Figure 9.6 Integration points in ABAQUS S4 element, 3 per layer (nonlinear finite strain

laminated shell) ................................................................................................................120

88. Figure 9.7 Lam 20 deg ANSYS results. Normalized strains on compressed diagonal. .124

89. Figure 9.8 Laminate 20 deg ANSYS results. Normalized strains on tension diagonal. .124

90. Figure 9.9 Laminate 20 deg, ABAQUS results. Normalized strains on compressed

diagonal, on front and back faces. ...................................................................................125

91. Figure 9.10 Lam. 20 deg. Out of plane displacements on front face. Top to down:

ARAMIS surface readings, ANSYS results, ABAQUS results (5 mm = 0.1969 in) ......126

92. Figure 9.11 Lam 20 deg ANSYS. Normalized strains on compressed diagonal. ...........127

93. Figure 9.12 Lam 45 deg ANSYS. Normalized strains on tension diagonal. ..................128

94. Figure 9.13 Lam 45 deg, ABAQUS. Normalized strains critxx εε / on compressed


Figure Page

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diagonal on front and back faces. ....................................................................................128

95. Figure 9.14 Lam. 45 deg. Out of plane displ. on front face. Top to bottom: ARAMIS

surface readings, ANSYS and ABAQUS results (7 mm = 0.2756 in) ............................129

96. Figure 9.15 Lam 90 deg ANSYS. Normalized strains on compressed diagonal. ...........130

97. Figure 9.16 Lam 90 deg ANSYS. Normalized strains on tension diagonal. ..................131

98. Figure 9.17 Lam 90 deg, ABAQUS. Normalized strains on compressed diagonal, on

front and back faces. ........................................................................................................131

99. Figure 9.18 Laminate 90 degree. Out of plane displacement. Top to bottom: ARAMIS

surface readings, ANSYS results, ABAQUS results (12 mm = 0.4724 in).....................132

100. Figure 9.19 Lam 0 deg ANSYS. Normalized strains on compressed diagonal. .............133

101. Figure 9.20 Lam 0 deg ANSYS. Normalized strains on tension diagonal. ....................134

102. Figure 9.21 Laminate 0 deg, ABAQUS. Normalized strains on compressed diagonal, on

front and back faces. ........................................................................................................134

103. Figure 9.22 Laminate 0 degree. Out of plane displacement. (a) Nonlinear Analysis

ANSYS last load step. (b) Nonlinear Analysis ABAQUS last load step.........................135

104. Figure 10.2 Bilinear strain softening law in terms of equivalent magnitudes ................148

105. Figure 10.3 Damage variable as a function of equivalent displacement. .......................150

106. Figure 10.4 Effect of the viscous regularization coefficient in the nonlinear solution.

Strains on compressed diagonal at the center, front and back faces (20 degree panel) ...154

107. Figure 10.5 Fiber tensile failure index. First occurrence of damage, load step 9. Actuator

load ( )lbsP 799,13= .......................................................................................................157


Figure Page

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108. Figure 10.6 Fiber tensile failure index, load step 12, ( )lbsP 163,17= ..........................157

109. Figure 10.7 Fiber tensile failure index, load step 14, ( )lbsP 325,20= .........................158



112. Figure 10.10 Fiber compressive failure index, load step 14 ( )lbsP 325,20= ...............160

113. Figure 10.11 Fiber compressive failure index, load step 15, ( )lbsP 465,27= ..............160

114. Figure 10.12 Fiber compressive failure index, load step 16, ( )lbsP 433,30= ..............161

115. Figure 10.13 Matrix tensile failure. First occurrence of damage, load step 7. Actuator

load ( )lbsP 497,11= .......................................................................................................162

116. Figure 10.14 Matrix tensile failure, load step 12. ( )lbsP 163,17= ................................162

117. Figure 10.15 Matrix tensile failure, load step 14. ( )lbsP 325,20= ...............................163

118. Figure 10.16 Matrix tensile failure. End of step 15. ( )lbsP 465,27= ...........................163

119. Figure 10.17 Matrix tensile failure. End of step 16. ( )lbsP 434,30= ..........................164

120. Figure 10.18 Matrix compressive failure, load step 9 ( )lbsP 799,13= ..........................165

121. Figure 10.19 Matrix compressive failure, load step 12 ( )lbsP 163,17= .......................165

122. Figure 10.20 Matrix compressive failure, load step 14 ( )lbsP 325,20= ......................166



Figure Page

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125. Figure 10.23 Matrix compr. failure, load step 14. Failure index at plies 1, 2, 3 & 4 .....168

126. Figure 10.24 Matrix compr failure load step 14. Failure index at plies 5, 6, 7 & 8 .......169

127. Figure 10.25 Strain on compression diagonal vs. actuator load with PFA results. ........171

128. Figure 10.26 Hashin’s failure indices, load step 14, layer 8 (front face) ( )lbsP 325,20= .

Left to right and top to bottom: fiber tension, matrix tension, fiber compression, matrix

compression. ....................................................................................................................174

129. Figure 10.27 All non mode based failure indices, load step 14, layer 8 (front face)

( )lbsP 325,20= . Left to right and top to bottom: Tsai Wu, Tsai Hill, Maximum Stress,

Maximum Strain. .............................................................................................................175

130. Figure 10.28 All plies Tsai Hill failure index, load step 14, layer 8 ( )lbsP 325,20= ..176

LIST OF TABLES

Table Page

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1. Table 1.1 Description of FE models ...................................................................................6

2. Table 3.1 List of abbreviations used in Table 3.2.............................................................21

3. Table 3.2 Summary of publications in buckling, postbuckling and progressive failure

(abbreviations described in Table 3.1)...............................................................................22

4. Table 3.2 (continued) Summary of publications in buckling, postbuckling and progressive

failure (abbreviations in Table 3.1)....................................................................................23

5. Table 5.1 Summary of results from Spencer-Walker data reduction of ARAMIS

measured out of plane displacement. .................................................................................63

6. Table 6.1 FE modeling features. Element types, boundary conditions and loads ............69

7. Table 6.2 Comparison of results. Different picture frame modeling options (ANSYS) ..72

8. Table 6.3 Mechanical properties of individual plies (British Units) ................................74

9. Table 7.1 Summary of results. Buckling load calculation, two-term Ritz approx............84

10. Table 8.1 Eigenvalues and buckling loads vs. level of load applied to the model ...........91

11. Table 8.2 Eigenvalues and buckling loads vs. level of load applied to the model ...........95

12. Table 8.4 Eigenvalues and buckling loads vs. level of load applied to the model .........101

13. Table 8.5 First three buckling loads for each laminate...................................................104

14. Table 9.1 Estimation of the perturbation force. First mode. Laminate 0 degree. ...........113

15. Table 9.2 Perturbation force estimation. Second Mode. Laminate 0 degree. .................114

16. Table 9.3 Effect of initial imperfections on overall results (laminate 0 deg) .................115

17. Table 9.4 Initial imperfections: estimated vs. adopted for FE correlation......................117

LIST OF TABLES (continued)

Table Page

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18. Table 9.5 Loading schedules in the ANSYS model .......................................................121

19. Table 9.6 Buckling load. Comparison of results.............................................................122

20. Table 10.2 Hashin’s failure criteria expressions.............................................................141

21. Table 10.3 – Strength material properties used in Hashin’s functions [4].......................143

22. Table 10.4 Equivalent displacement and equivalent stress for each failure mode .........149

23. Table 10.5 Energy Released constants [31]....................................................................151

24. Table 10.6 Output damage indices [21]..........................................................................155

25. Table 10.7 Loading schedule, nonlinear analysis with PFA...........................................156

26. Table 10.8 Load levels at occurrence of first damage for each failure mode.................170

27. Table 10.9 Laminate 20 deg .Summary of buckling loads .............................................172

28. Table 10.10 Laminate 20 deg . Final design load carrying capability............................172

29. Table 10.11 Failure criteria compared in the NL-FA-Model. ........................................173

1

CHAPTER ONE

1INTRODUCTION

The industrial use of composite materials has continued to increase steadily over the last three

decades with important developments in the aerospace, wind energy, automotive and sports

industries. Outstanding mechanical properties (strength and stiffness) combined with low weight

make them the material of choice for multiple applications. In the context of the aerospace and

wind industries, it has been common since the 1970’s to use composite materials for primary

structural members; e.g., components in which structural failure leads to catastrophic loss. The

shear webs of main structural components, such as spars and ribs in aircraft wings or wind

blades, are made of composite laminates in a large number of cases.

Traditional metal structures in the aerospace industry have historically been optimized for

minimum weight and validated by well-established analysis methods that are based on a vast

amount of research and testing. In the case of shear resistant webs, conventional aluminum

designs are commonly optimized by the application of the Incomplete Diagonal Tension Theory

(IDTT) first described by Wagner [1] in 1929 and further extended by Kuhn [2] in the 1950’s. As

shown in Figure 1.1, when an isotropic rectangular plate is subjected to pure shear stresses along

the edges, tension and compression stresses exist in the plate. These stresses are equal in

magnitude to the shear stress, and inclined at an angle of 45 degrees. The compressive stresses

on the plate are still resisted immediately after buckling (Figure 1.1 (a)) and well into deep

postbuckling (Figure 1.1 (b)). Wagner’s theory for thin plates under shear proposed an assumed

fully developed postbuckling fold pattern with the thin web modeled as a series of evenly

distributed ribbons or cables carrying tension load only. The method also includes vertical and

2

horizontal stiffeners to which the ribbons are connected. These stiffening members are sized to

remain unbuckled through the entire buckling range of the plate such that the boundary

conditions remain unchanged. This approach is referred to as the theory of “pure diagonal

tension” (PDT).

Figure 1.1 Rectangular isotropic plate subjected to pure shear stresses [2]

Kuhn extended Wagner’s PDT theory to “incomplete diagonal tension” (IDT) [2], which

proposes to limit the buckling of the plate such that the diagonal tension field is not fully

developed as shown in Figure 1.1 (b). Both theories consider the web as inclined tension

members in a frame, but IDT also accounts for the compressive stresses in the plate and its

reinforcing contribution to the stiffeners. Therefore, IDT utilizes the stable postbuckling

behavior of plates in shear to achieve additional load carrying capability and potential weight

3

reductions. Figure 1.2 shows Kuhn’s experimental beam developing a field of diagonal tension

[2]. As mentioned above, the stiffeners dominate the boundary condition of the web so that all of

the buckling is local to the sheet. This progression of the buckling also guarantees that the field

of stresses developed in sheet and stiffener in the post-buckling regime are below the yield stress

of the material and the crippling stress of the stiffeners. And perhaps even more important, this

type of design results in stable postbuckling behavior.

Figure 1.2 Kuhn’s diagonal-tension beam

Composite laminates can be tailored in stiffness or strength, by varying the fiber

directions, which permits a high level of optimization. For a laminated composite design similar

to Kuhn’s diagonal-tension beam design, the function of the vertical stiffeners in the metal

4

design is assumed by the plies with fibers in the vertical direction; whereas, the shear and

diagonal tension field are reacted by plies with fibers at 45-degree angles [3, 4]. Based on this

approach, a commonly adopted conservative design solution is to prescribe a composite lay-up,

which is shear resistant in strength and also exhibits a buckling load well above the expected

range of loading during service. This approach does not take advantage of postbuckling load

carrying capability.

Methods of design and analysis for composite shear webs and hybrid metal-composite

laminates such as Glare [5] have been developed for the past two decades. The basic idea of

design methodologies that utilize the postbuckling strength of composite shear webs is to allow

the panel to carry stresses that are well beyond the buckling load while limiting the allowable

stresses such that no material damage occurs. This is a conservative approach because the final

failure of a composite structure can occur at a load greater than that associated with the first-ply

failure; i.e., first occurrence of damage. In the particular case of the Boeing 787 aircraft, the aft

fuselage skin panels are allowed to buckle under torsion conditions that react the critical loading

on the vertical fin. These panels are designed to prevent (1) Inter-Fiber Failure (IFF) until

loading considerably higher than the Design Limit Load (DLL) and (2) Fiber Failure (FF) until

the final structural collapse (well above 150% of DLL).

The application of this postbuckling design philosophy to laminated composite structures

is the general objective of this research. This dissertation presents an analysis methodology to

validate the design of laminated composite shear webs. It is similar to the IDTT approach

because behavior beyond the initial linear range, including postbuckling, is accounted for in the

analysis. Furthermore, the present work accounts for strength, buckling occurrence and the

progression and onset of damage and its evolution. In order to examine the effects of material

5

damage, four models were developed with increasing levels of complexity ranging from a linear

model with no damage to a fully nonlinear model with progressive failure analysis. The

methodology is validated by correlation with experimental data from a set of four composite

laminated shear panels.

This dissertation presents results from four different finite element (FE) models. The first

model was used to obtain a linearized buckling solution, which provides the buckling critical

loads and corresponding buckling modes as the solution of an eigenvalue problem. A nonlinear

solution was used for the second model, which included finite von Kármán strains and large out-

of-plane displacements. This model was utilized to characterize the response throughout the

complete postbuckling regime. No material nonlinearity or progressive failure analysis was

included in this model. This model (as well as the third and fourth models) used a linear

combination of the buckling modes obtained from the first model to simulate the shape of an

initial imperfection.

The third model differs from the second model because mode-based failure criteria were

included. These criteria were evaluated as a function of load level to determine the critical failure

load and mode for each ply of the laminate. The fourth model used the results from the failure

criteria to model the material damage based on continuum damage mechanics. This resulted in a

progressive failure analysis (PFA) that accounts for material failure and damage progression as a

function of the loading. Table 1.1 describes the four different models used in this dissertation and

their capabilities.

6

Model Abbreviated Name Analysis Capability LE NL NL-FA NL-PFA

Linear eigenvalues X X X X Nonlinear large strains and large displacements

X X X

Failure occurrence X X Failure occurrence and progressive damage

X

Software Used ANSYS ANSYS & ABAQUS ABAQUS ABAQUS

Table 1.1 Description of FE models

7

CHAPTER TWO

2OVERVIEW OF BUCKLING, POSTBUCKLING AND PROGRESSIVE FAILURE

2.1 General requirements and key features of the analysis and simulation

Considering the framework outlined in Chapter 1, the following paragraphs describe the

key features and general requirements for the analytical methods implemented in this

dissertation. The analyses had to predict buckling loads, fully characterize the postbuckling

response (including nonlinear displacements and rotations), predict the onset of material failure,

and describe the postbuckling response conditioned by the post-failure damage propagation.

The commonly used classical laminate theory (CLT) based on the Kirchhoff plate theory

neglects transverse shear stresses. However, composite shear webs can develop significant

transverse shear deformations associated with the out-of-plane deflections in the postbuckling

regime. A method that accounts for first order shear deformation (Reissner-Mindlin plate theory)

was utilized to determine the local ply state of through-the-thickness shear stress.

Appropriate failure criteria were needed to detect the initiation of damage under the

action of postbuckling deflections and stresses. The so-called mode-based failure criteria are

capable of separating different failure modes (fiber tension or compression, matrix tension or

compression, etc.) and were used to detect the onset of failure and as activation functions that

trigger the algorithms that simulate the damage progression [6].

In order to follow the mechanisms that result in the total failure of the structure, material

degradation models were implemented. These models simulated damage propagation and the

updated degradation state of the local material properties. Material degradation models for

composites are, in general, nonlinear material models. Therefore, the post-failure nonlinear

8

material model was a significant feature of the analysis methodology.

Considering the previous point, a procedure to re-establish equilibrium after modifying

the lamina properties was required. This type of operation is sometimes described in the FE

literature as a distinct type of nonlinearity since it involves a change of state in some elements.

Another example of nonlinearity due solely to a change in state is contact elements, which

change to the active state after the prescribed gap is closed. Similarly, layered elements undergo

a discrete change in state when their properties are reduced to reflect the effects of damage.

Although nonlinear finite element analysis is a developed area of simulation, the

progressive failure analysis of nonlinear structures is still an active field of research since failure

theories and damage progression modeling are part of an iterative nonlinear process, with one

being dependent on the other. Comparisons from the World Wide Failure Exercise (WWFE)

[Hinton and Soden, 1998] demonstrated the often poor agreement of existing failure theories

with experimental results. This exercise revealed that even when analyzing simple laminates,

which have been characterized extensively over the past forty years, the predictions using most

theories differ significantly from the experimental observations [7].

The failure theory that more closely approximated the experimental results was Puck’s

Action Plane Strength theory. Matrix failure is analyzed under the hypothesis of brittle fracture,

which is arguably more appropriate for polymeric matrix materials. The method not only

calculates the stresses causing inter fiber failure more realistically than conventional failure

criteria but also predicts the angle in which fracture takes place. Further developments have also

introduced the capability to predict the angle under which fiber band kinking occurs for the fiber

compressive failure. The models developed in this dissertation did utilize a mode based failure

9

criteria that is similar to Puck’s. Cracking and kinking were treated indirectly by degrading the

material properties.

2.2 Preliminary observations about nonlinearity, buckling and post-buckling

The key objectives of the nonlinear analyses of this dissertation were to (1) estimate the

maximum load that the structure could support prior to the onset of material failure and (2)

determine the ultimate load of the structure including the effects of material failure and

degradation.

The first of the above-mentioned objectives essentially consists of determining the pre-

and post-buckling response of the structure. Either with or without material nonlinearities,

structures that undergo buckling will generally follow load-deflection paths similar to the ones

depicted in Figure 2.1.

h

P

Δ

L

P

Δ Δ

P

Δ

Ph /L= 0 h /L= sm a ll h /L= la rge

Postbu ck lin gR eg im e

S nap -th rough

A A '

Figure 2.1 Response of a thin plate or shell under out-of-plane loading

10

The first case, a thin plate with 0=Lh , does not have a collapse point. Because of the

membrane action, the plate increases its stiffness as the displacement grows. For an arch

( largeLh = ) collapse will occur as the load P increases. The most extreme case is commonly

referred to as “snap-through” behavior. This occurs when the structure is evolving from point A

to point A’. The region between A and A’ is dominated by instability since small increments in

displacement beyond point A can generate large fluctuations in the load carrying capability P. If

points A and A’ are in close proximity, the snap-through instability region is small. The buckling

load corresponding to point A may not be as important as what happens between points A and

A’, since the large displacements produced by snap-through can produce structural failure.

For either type of behavior, snap-through or not, the nonlinear response of the structure is

calculated by an incremental analysis, which must include the possibility of decreasing the load

carrying capability as shown in the most right graph of Figure 2.1. The problem statement can be

generically formulated as follows [8]. Let Rt0 be the vector that defines the load distribution

corresponding to the first load step at time 0tt = . At any instant of time τ the load vector is

considered to be proportional to the initial load vector. Therefore,

RR t0βττ = (2.1)

where βτ is the load multiplier for any instant of time τ . For the type of structure and loading

considered in this dissertation, the response as τ increases is of interest. This task requires the

load multiplier βτ to increase or decrease with τ as the structural response is calculated.

The general methodology proposed herein is based on first performing a linearized

buckling analysis to obtain a reasonably good first estimate of the actual buckling load. The

11

linear buckling modes are then used to define the initial imperfection of the structure as a field of

displacements. If imperfections resembling the lowest buckling modes are imposed on the

“perfect” geometry of the model, the load-carrying capacity is much more representative of the

load-carrying capacity of the actual physical structure [8]. The subsequent nonlinear analyses

utilize the initial imperfection as a starting point for the nonlinear postbuckling analysis.

2.3 Linearized buckling solution

The FE commercial packages used in this research utilize a linear buckling solver that

calculates the buckling load factors and the corresponding mode shapes for a structure under the

given load conditions [9]. As shown in Figure 2.2, the solution is based on the assumption that

there exists a buckling point where the primary and the secondary load paths intersect. Before the

point is reached all element stresses change proportionally with the load factor. The lower line in

Figure 2.2 is the nonlinear response. The states labeled as bifurcation points are those in which

the buckling of the structure occurs. As shown, the buckling event for the nonlinear path takes

place at a lower load than that for the linearized path.

Linear Buckling

UnstableRegion

P

Δ

Bifurcation Point

Nonlinear Solution

Secondary Path

BucklingPoint

Figure 2.2 Schematic nonlinear versus linearized responses

12

Analysis for buckling essentially involves determining the distribution of stresses prior to

buckling and the influence of the stresses on the out-of-plane displacement. The geometric

stiffness matrix of the structure reflects the effect of geometric (or displacement) changes on the

element force vector for a known stress state. This matrix and the linear stiffness matrix can be

utilized as a basis for linear buckling analysis, which reduces to an eigenvalue problem. The

eigenvalue closest to zero is the critical load multiplier, and the associated eigenvector gives the

corresponding buckling mode. Negative eigenvalues often are found in the solution, and they

represent potential buckled configurations that the structure could adopt if the applied load is

reversed in direction.

The linear buckling analysis assumes the existence of a bifurcation point where the

primary and the secondary loading paths intersect; see the upper portion of Figure 2.2. At a

bifurcation point, more than one equilibrium position is possible. Since the secondary path is

normally a succession of states of lower elastic energy, the primary path is not usually followed

after the load exceeds this point and the structure is in the postbuckling regime. The slope of the

secondary path at the bifurcation point determines the nature of postbuckling. Positive slope

indicates that the structure will continue to carry load after initial buckling. Negative slope

indicates that the structure will snap through or collapse. Real structures often have geometric

imperfections and loading eccentricities causing the primary path curve and the bifurcation point

to disappear as shown in the nonlinear path of Figure 2.2.

2.4 Nonlinear solution

The curves shown in Figure 2.2 demonstrate linearized buckling, on the linear curve, as

well as nonlinear buckling, which essentially represents a point on the nonlinear load-

13

displacement curve at which the structure becomes unstable. Analysis methods that determine

this type of nonlinear behavior are typically referred to as incremental iterative methods. These

methods start from a known equilibrium point and then determine the increment in

displacements corresponding to an increment in the applied loading.

Some of the methods, such as the popular Newton-Raphson method, can fail to converge

at a zero slope point on the load-displacement curve. This situation is illustrated in Figure 2.3.

To avoid this problem when it occurs, the Arc-Length method, also shown in Figure 2.3, can be

utilized. All results presented herein are based on utilizing the Newton-Raphson method, which

is reviewed in detail in Chapter 9. None of the difficulties illustrated in Figure 2.3 were

encountered for any of the analyses conducted herein.

P

Δ

Newton-Raphson fails here

Δ

PConverged Solution

Spherical arc substep n

1

ii+1

λi

Δλ

Δun

Figure 2.3 Newton-Raphson limitation (left) and Arc-Length methodology (right)

14

2.5 Mode-based failure criteria and progressive failure analysis

The nonlinear postbuckling analysis characterizes the structural load-displacement

response in the absence of failure. That is, the material is assumed to continue carrying load. The

first step in understanding and modeling material failure is to identify failure modes within each

layer of material and their location on the structure.

As described in the literature survey of Chapter 3, the progressive failure analysis of

composite structures usually consists of an approach that as a first step utilizes either the Chang-

Chang or Tsai-Wu failure criteria. Once layer failure is detected, the relevant elastic properties of

the affected element are reduced to zero over a fixed number of steps. This approach is more

sophisticated than the commonly known “ply discount method” for classical laminate strength

analyses and it is a similar method at the element level. Such approaches provide insight on how

the strength degradation evolves, but are unrealistic since post-failure behavior is completely

disregarded [10].

Recent composite damage theories, that model damage evolution more realistically, rely

on continuum constitutive models which feature internal variables that account for the

distribution of microscopic defects that characterizes damage [11]. These are called continuum

damage mechanics (CDM) models, and correspond to the kind of approach used herein. The

CDM model used in this dissertation is supported by the latest version of ABAQUS, as explained

in detail in Chapter 10, and is based on a strain softening material law sized by the volumetric

energy associated with a failure mode. Hashin’s [6] expressions were utilized as failure detection

criteria and activation functions to trigger the constitutive model that simulates damage evolution

at material points.

15

As explained in Chapter 10, Hashin’s criteria were compared with classic single-equation

criteria and were found to be more conservative and more descriptive of the nature of the

damage.

16

CHAPTER THREE

3LITERATURE REVIEW

This chapter presents a review of the most relevant recent works found in a general

literature search involving buckling, postbuckling, and progressive failure analysis. The main

contributions were the formulation of progressive failure models using internal state variables as

proposed by Chang and Chang [12] in 1986, Talreja [13] in 1987, and Chang and Lessard [14] in

1991. Most of the works included in this section use the field variables model for their

progressive damage implementation with different approaches. Table 3.2 shows a summary of

the references found in the field, including type of structure analyzed, loading condition, type of

static solution, whether experimental correlation was presented or not, failure criterion used,

material degradation model used (if any) and the finite element (FE) code used. These fields

reveal trends in the analysis capabilities and the impact that new failure theories have on the

implementation of the damage accumulation algorithms.

The use of commercial FE packages did not become predominant until the turn of the

century, with ABAQUS being the preferred choice for this family of studies. “Ad hoc”

developed codes including specialized NASA applications, such as COMET, have decayed in

usage, since commercial FE software companies have incorporated non-linear analysis

capabilities and assumed the maintenance efforts normally associated with advanced codes. This

change in software usage is coincidental with numerous works that focused on postbuckling

under in-plane shear. It can also be noted that only after 1997 did non-linear analysis become a

common feature, which reflects the maturity in the FE implementation of large displacement

analysis.

17

The types of structure under study are mostly basic assemblies used in aerospace

construction, such as curved and flat panels, with and without holes, and un-stiffened or stiffened

with integral J or blade stiffeners. Papers emphasizing the analysis of more complex buildups

(such as an avionics box or substructures) belong to works focused in practical implementations

of particular projects and were not included in this selection of papers since they do not

constitute a general contribution to the field.

Regarding the failure theory used, the publications can be divided into two general

categories: classic failure theories (maximum stress, maximum strain, Tsai-Hill, Tsai-Wu,

Hoffman, and their modifications), and mode-based failure theories (Hashin, Hashin-Rotem,

Christensen, Chang-Lessard, Puck, and LaRC family). After the results of the World Wide

Exercise in Failure (WWFE) were published in 1998 [7], emphasis was put in developing new

physically based phenomenological failure criteria that incorporated new features such as Puck’s

concept of angle of fracture (applicable to matrix compression), fiber kinking band angle

(applicable to fiber compression) and advances from the field of fracture mechanics (applicable

to matrix tension). The methods of the family labeled as LaRC (Langley Research Center) are

phenomenological failure criteria. The LaRC models were first implemented in progressive

failure analysis by Ambur et al. [11] in 2004. As shown in Table 3.2, mode-based failure criteria

have become the dominant choice.

All of the recent material degradation models, as shown in Table 3.2, rely on a collection

of internal variables to characterize the accumulation of damage after failure has been detected

by the failure criteria. The two general types of models include (1) the Chang & Lessard [14]

model that reduces abruptly to zero (or to a value close to zero to avoid numerical instability) the

different material stiffness properties associated with each failure mode (in Tables 3.1 and 3.2,

18

these models are referred under the abbreviations IRSP and GRSP), and (2) the other group of

models that rely on continuum damage mechanics (CDM) to reduce the stiffness material

properties following constitutive laws that consider the density distribution of cracks over a

volume (these models are referred in Tables 3.1 and 3.2 under the abbreviation CDM with a

suffix that depends on the assumptions made for each implementation). It can be noted that CDM

models became predominant after 2005 when Camanho and Davila very successfully used them

to model the damage progression of delamination damage and the separation of stiffener flanges.

Following Camanho and Davila, the post-failure material degradation model follows a

constitutive law with strain softening that is correlated with the experimental values of the

critical values of energy release rates associated with the three fracture modes IG , IIG , IIIG .

Singh and Kumar [15] presented in 1998 their study of thin laminates for different lay-

ups with a NASTRAN FE model for comparison and failure analysis using the Tsai-Hill

criterion. Huang and Minnetyan [16] published in 1999 a study of J-stiffened panels in the

postbuckling range with an in-house developed FE solver and failure based on maximum stress

and modified Tsai-Hill criterion. In 1999, Sleight performed a general evaluation of the state of

the art progressive failure methodology for several types of structures including the rail-shear

panel, with non-linear analysis, using the maximum strain, Hashin, and Chirstensen criteria. For

compression loading, Davila et al. [17] carried out in 1999 one of the first implementations using

ABAQUS for the non-linear analysis. The study was conducted for notched composite panels

with blade stiffeners and cross stitching with Kevlar fiber, replicating a typical wing box cover

panel. The effect of the Kevlar stitches in the propagation of damage was investigated. The

failure analysis was again mode-based with Hashin’s criterion.

19

The NASA-ICASE group, formed by Jaunky, Ambur, Davila and Hilburger [18],

presented in 2001 their study of panels with and without holes and bead-stiffened panels under

compression and shear well into the postbuckling range. All of the results were obtained using

ABAQUS with failure detected through Hashin’s theory. In 2002 Ambur et al. [19] extended

their work to stiffened panels under shear well into the postbuckling range with ABAQUS and

Hashin’s criterion. They also included the same type of panels but damaged with a notch in an

angle. Zemcik and Las included Puck’s formulation in their analysis in 2005 [20].

The findings of the WWEF had an impact in PFA works as reflected by the NASA-

ICASE group. In 2004 Ambur et al. [18] implemented PFA using ABAQUS and the LaRC02 set

of criteria for panels in compression. Comparisons were made with the same PFA based on

Hashin’s failure criterion and also on Chang-Lesard’s degradation model. In 2005 Zemcik and

Las [20] implemented Puck’s criterion in a PFA, although that work does not fall strictly into the

scope of this dissertation because it used a shock loading.

Since 2005, PFA works predominantly include mode-based phenomenological failure

criteria such as Hashin or the LaRC family. It has also become predominant to use material

degradation models based on continuum damage mechanics. These methods utilize strain

softening laws previously correlated with energy release rates obtained through testing for each

failure mode.

No previous works have covered the study of postbuckled in-plane shear loaded panels

with mode-based failure and material degradation in the progressive failure analysis. Combining

these two advanced features (mode based failure and material degradation) with a nonlinear

postbuckling model is the key contribution of this dissertation. This research also explores the

20

benefits of separating failure for the different modes taking place during the postbuckling of in-

plane shear loaded panels. Such information is expected to prove valuable to the designer in

understanding the damage evolution and its interaction with the instability behavior when the

structure is loaded well into the postbuckling regime.

The current literature review found a low number of recent works using the ANSYS

software package versus the ABAQUS package. The practical exploration of the ANSYS

capabilities to support a PFA implementation and the decision to choose ABAQUS to take the

dissertation work to a successful completion can also be considered an added benefit of this

research.

21

Abbreviation Description

EC Experimental correlation

LG Linear geometry (analysis)

NLG Nonlinear geometry (analysis)

NLM Nonlinear material (analysis)

PB Post buckling (analysis)

IRSP Instantaneous reduction of stiffness properties (upon failure onset)

GRSP Gradual reduction of stiffness properties, following a fixed law over a

number of time steps (upon failure onset)

CDM-UCD Continuum damage mechanics model assuming uniform cracks density

CDM-SCD Continuum damage mechanics model assuming statistical cracks density

distribution

CDM- iCG Continuum damage mechanics with constitutive model correlated with the

energy release rates of fracture modes I, II, and III

AVD Artificial viscous damping

DELAM Delamination

ST Shapery theory

Table 3.1 List of abbreviations used in Table 3.2

Table 3.2 Summary of publications in buckling, postbuckling and progressive failure (abbreviations described in Table 3.1)

Year Researchers Analysis Type Structure Type Loading Condition

EC Failure Criterion Material Degradation

FEM Code

1991 Chang & Lesard LG-NLM Plate with a cutout Compression yes Chang (Hashin modified)

IRSP PDHOLE

1992 Minnetyan et al. LG-NLM Plate with a cutout Tension no Not available CODSTRAN Engelstad, Reddy

& Knight NLG-PB Plate with and w/o cutout Compression yes Max stress & Tsai Wu Not available Own code

1995 Shahid & Chang LG Plate Tens, shear yes Hashin modified CDM-UCD PDCOMP

Coats & Harris LG Plate with cutout Tension yes CDM- iCG COMET-FL

1997 Sleight & Knight NLG-PB Stiffened & unstiffened panel with & w/o cutout

Compression yes Max strain, Hashin, Christensen

IRSP & GRSP

COMET

22 Moas & Griffin NLG-PB Curved frame Transverse yes Tsai-Wu, Max strain IRSP Own code

Singh et al NLG-PB Plate Compression no CDM-SCD 1998 Singh et Kumar NLG-PB Plate Shear no Tsai-Hill & Max stress IRSP NASTRAN Gummadi et al NLG-PB Curved panel Transverse no Not available 1999 Huang and

Minnetyan NLG-PB J stiffened panel,stitched Shear yes Modified Tsai-Hill &

Max stress IRSP Own code

Davila Ambur, and McGowan

NLG-PB Stiffened notched panel, stitched

Compression yes Hashin IRSP ABAQUS

Sleight NLG-PB Rail-shear panel, panel with & w/o hole, blade-stiffened panel

Tension, shear compression

yes Max strain, Hashin, Christensen

IRSP COMET

Wang, Lotts, Sleight

NLG-NLM Stiffened panel Tension yes Max Stress, Hashin, Hashin-Rotem

IRSP COMET-AR

Baranski et al NLG-PB Plate Compression no Hashin Not available

Table 3.2 (continued) Summary of publications in buckling, postbuckling and progressive failure (abbreviations in Table 3.1)

Year Researchers Analysis Type

Structure Type Loading Condition

EC Failure Criterion Material Degradation

FEM Code

2000 Knight et al NLG-PB Plate Compression yes Max strain IRSP STAGS 2001 Jaunky, Ambur,

Davila, & Hilburger NLG-PB Panel with&w/o cutouts

Bead-stiffened panel Compression & shear

yes Hashin IRSP ABAQUS

McGowan,Davila Ambur NLG-NLM Notched panel Compression yes Chang (Hashin modif) IRSP ABAQUS Knight, Rankin, Brogan NLG-NLM Plate with a hole Tension no Max strain CDM + AVD STAGS 2002 Ambur, Jaunky Hilburger NLG-PB Stiffened panel Shear yes Hashin IRSP ABAQUS Goyal, Jaunky, Ambur

Johnson NLG-NLM Flat and curved panels Shear &

Compression yes Chang (Hashin

modified) CDM for DELAM, IRSP

ABAQUS

2003 Minnetyan, Zhao, Chamis NLG-NLM Adaptive plate airfoil Tension pressure no Modif Tsai-Hill, Max stress Not available Own code Hyer, Wolford, Knight NLG-NLM Non-circular cylinder Inter pressure no Max stress GRSP STAGS

23 Nagesh NLG-NLM Pressure vessel Inter pressure yes Tsai-Wu GRSP ANSYS

2004 Ambur, Jaunky, Davila NLG-NLM Panel Tens, compr yes LaRC02, Chang-Less. IRSP ABAQUS 2005 Hilburger & Nemeth NLG-NLM Cylind shell reinf hole Compression yes Hashin IRSP STAGS Hermann, Lo

Mamarthupatti NLG-PB Panel Compression no Tsai-Wu, Max Stress Not available ANSYS

Basu, Waas, Ambur NLG-NLM-PB

Panel with hole, double notched panel

Shear, Compression

yes Shapery theory ST ABAQUS

Camanho & Davila NLG-NLM-Delamination

Stiffened rectangular panel

Bending yes Mixed mode quadratic interaction

CDM- iCG ABAQUS

Pinho Iannucci Robinson NLG-NLM Rectangular panel Tens/compress yes LaRC04 CDM- iCG ABAQUS

2006 Miami NLG-NLM- Rectangular panel LaRC02 CDM- iCG ABAQUS

24

CHAPTER FOUR

4SCOPE AND OBJECTIVES

This dissertation presents a progressive failure methodology to support the design of

composite shear webs. Nonlinear finite element (FE) models combined with techniques to

predict the onset of failure and material damage evolution are used to analyze the panels and

expand their working range well into the postbuckling regime. A complete analysis methodology

is laid out with increasing complexity in its approach to the problem. The methodology was

validated by correlation of FE results with experimental data. Regarding other general

considerations that define the scope of this work, this research is concerned with the damage

onset, accumulation, and progression, but it was not concerned with fatigue damage mechanisms.

The type of loading under consideration was static for the analysis and quasi-static for the

testing. The type of structures analyzed herein were flat unstiffened laminates.

The experimental results were recorded using strain gages as well as the latest generation

photo-sensitive digital image system (ARAMIS). The strain and displacement results collected

for the postbuckling response of the panels were compared between ARAMIS and the strain

gages in order to assess consistency and accuracy. Diverse mathematical treatments of the test

data such as the Southwell’s method (in the Spencer-Walker version for buckling of plates) were

applied to extract the experimental values of the critical buckling loads for each specimen.

The analysis methodology is presented with increasing levels of complexity. This

incremental analysis approach was taken (1) in order to independently evaluate each type of

analysis, and (2) to compare results from two commercial FE packages, ANSYS and ABAQUS.

The types of models that were developed are listed as follows: (1) a linear model to estimate the

25

panel buckling load based on eigenvalue analysis, (2) a nonlinear model with von Kármán

nonlinear strains to predict the postbuckled load-displacement and load-strain response, (3) the

nonlinear model from step 2 with mode-based failure indices for each material layer, and (4) the

nonlinear model from step 3 with a continuum damage mechanics materials degradation model

that utilizes the mode-based failure indices as damage activation functions.

The linear models from step 1 and the nonlinear models from step 2 were developed

using ANSYS and ABAQUS. The nonlinear models from steps 3 and 4 were developed using

ABAQUS. The critical loads obtained with the linear models from step 1 were compared with

the experimental buckling loads determined from test results. The nonlinear models from step 2

were used to fully characterize the postbuckling response (without failure) and correlate it with

the response recorded during tests. The high level of correlation obtained validated all aspects of

modeling the structure with finite elements and the results given by the nonlinear solution.

The model utilized for step 3 relied on Hashin’s mode-based failure criteria to provide

failure indices for four different types of failure: fiber tension and compression, and matrix

tension and compression. For one of the laminates (out of the four that were tested) damage

occurrence was assessed for the different failure modes captured in the analysis. In a separate

model, with failure detection only, the Hashin’s mode-based failure criteria were compared to

four classic single equation criteria for composite structures.

For step 4 the progressive failure analysis (PFA) capability was added to the model. The

PFA capability was based on a Continuum Damage Mechanics (CDM) constitutive post-failure

material model. For the same laminate as mentioned above, the results of the complete nonlinear

with PFA analysis were examined to describe the evolution of damage and the damage state

26

corresponding to the final collapse of the structure. The results from the complete analysis were

compared with the nonlinear solution without failure and with the test results.

27

CHAPTER FIVE

5EXPERIMENTAL STUDY

Four different hybrid glass/epoxy-carbon/epoxy panels were tested under in-plane shear

loading as shown in Figure 5.1. Loading, ranging from zero load to the load at panel collapse,

was applied with constant displacement increments of 0.005 mm. Rosette strain gages were

installed at the center point of both sides of the panel following the arrangement shown in Figure

5.1.

Figure 5.1 Schematic diagram of picture frame fixture and panel geometry.

The strain gages collected data through six instrumentation channels. All of the panels

were built using a lay-up sequence consisting of alternating plies of pre impregnated Double

28

Biaxial Glass/Epoxy (denoted as DB) with plies of pre impregnated Unidirectional

Carbon/Epoxy tape (denoted as UC). The basic laminate followed the symmetric sequence:

[ ]DBUCDBUCUCDBUCDB ,,,,,,, or [ ]SUCDBUCDB ,,, . Tests were

performed based on the orientation of the UC carbon layers (0, 20, 45, and 90 degrees) with

respect to the direction of the load; i.e., an angle of 0 degrees corresponds to loading aligned in

the direction of the UC carbon layers.

The apparatus used was a picture frame fixture mounted on an MTS uniaxial load cell, as

shown in Figure 5.2. The frame consists of steel arms of rectangular cross section, with panel

attachments through fasteners, and hinge bolts in all four corners. Based on this setup, the

boundary conditions at the edges of the panel are considered to be fixed-clamped, and the four

arms act as loading edges.

In addition to the strain gage data, a photo-sensitive digital image system (ARAMIS) was

utilized to detect and record the out of plane displacements of the surface of the panels. The

ARAMIS results were compared with the out of plane displacements predicted by FE analyses.

In general terms, the methodology followed for all four specimens was to record the test readings

and attempt to correlate them with the FE results (either ANSYS or ABAQUS).

5.1 Data reduction of strain gage readings

All of the panels exhibited postbuckling strength, and were loaded until final destruction.

The numerical simulations described in subsequent sections will explore different aspects of the

failure analysis to gain further insight. The data recording systems described in this section

collected a vast amount of information during each test run. The following plots present limited

data in an attempt to reveal the onset of buckling for each of the laminates as well as relevant

29

trends and changes in slope. An actual panel in the picture frame fixture is shown in Figure 5.2.

Figure 5.2 Picture frame test setup.

30

0

20

40

60

80

100

120

140

160

180

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Microstrains (mm/mm x 1E-3)

Load Cell Force (kN)

Front FaceBack Face

Figure 5.3 Laminate 20 deg. Strains measured on the tension diagonal.

0

20

40

60

80

100

120

140

160

-9000 -8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000


Load Cell Force (KN)

Back Face Front Face

Figure 5.4 Laminate 20 deg. Strains measured on the compression diagonal.

31

0

20

40

60

80

100

120

140

0 500 1000 1500 2000 2500 3000


Load

Cell

For

ce (k

N) Back Face

Front Face


0

20

40

60

80

100

120

140

-12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000


Load

Cell

For

ce (k

N)

Back FaceFront Face


32

0

20

40

60

80

100

120

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

microstrains (mm/mm x 1E-6)

Loa

d C

ell F

orce

(kN

)

Back FaceFront Face


0

20

40

60

80

100

120

-8000 -6000 -4000 -2000 0 2000 4000 6000


Loa

d C

ell F

orce

(kN

)

Back FaceFront Face


33

0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000


Loa

d C

ell F

orce

(kN

)

Front FaceBack Face


0

20

40

60

80

100

120

140

160

-18000 -15000 -12000 -9000 -6000 -3000 0 3000 6000


Loa

d C

ell F

orce

(kN

)

Front FaceBack Face


34

Strain on tension direction, front face

0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

microstrains (mm/mm 1E-6)

For

ce a

t lo

ad c

ell (

kN)

Laminate 90 degLaminate 0 degLaminate 20 degLaminate 45 deg

Figure 5.11 Comparison of all laminates. Strains on tension diagonal, front face

Strain on compressed diagonal, front face

0

20

40

60

80

100

120

140

160

-3000 -2000 -1000 0 1000 2000 3000 4000 5000

microstrains (mm/mm 1E-6)

For

ce a

t lo

ad c

ell (

kN)

Laminate 90 degLaminate 0 degLaminate 20 degLaminate 45 deg

Figure 5.12 Comparison of all laminates. Strains on compression diagonal, front face

35

5.2 Data reduction of the ARAMIS optical system measurements

The ARAMIS optical system consists of a group of two cameras capturing digital images

of the specimen during the test. It tracks a network of markers attached to the surface of the

specimen, characterizing surface displacements and strains well into the nonlinear range.

The markers are distributed over the specimen and separated from each other by a pitch

several orders of magnitude finer than the characteristic length of the buckling modes exhibited

by the structure. The database recorded by ARAMIS for each specimen was interrogated in

several ways, and the interpretation of the results is presented in this section. Figure 5.13 shows

the axes to which ARAMIS relates strain measurements.

R2.5

12 in

17 in

ARAMIS Reference Axes

45 deg

Loading Direction

Test Panel

Pin JointFrame Member

Rosette Strain Gage

Pin Joint

Pin Joint

Pin Joint

Figure 5.13 ARAMIS reference axes for strain measurement

36

5.2.1 Laminate G - 20 degree data reduction

The ARAMIS database of recordings allows for fringe plotting of displacements and

strains of the surface under observation, as shown in Figures 5.14 through 5.17.

Figure 5.14 Laminate 20 deg, front face, out of plane displacement ( )lbsP 1.39746= .

Figure 5.15 Laminate 20 deg, front face, xxε strain, ( )lbsP 1.39746= .

37

Figure 5.16 Laminate 20 deg, front face, yyε strain, ( )lbsP 1.39746= .

Figure 5.17 Laminate 20 deg, front face, xyε strain, ( )lbsP 1.39746= .

For the correlation of the test results with the FE models, the ARAMIS database was

queried for xxε and yyε strain values on the front face on both diagonals under tension and under

compression. Figures 5.18 and 5.20 show the strains along the whole length of the diagonals for

different load levels. Figures 5.19 and 5.21 show load vs. strain only on points of the diagonals

separated an inch or less from the center of the panel.

38

Laminate 20 deg, Exx strain over compressed diagonal for different load levels

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

0 25 50 75 100 125 150 175 200 225 250 275 300

X position (mm)

Exx

stra

in (%

)

203547617793109124138153167142

Figure 5.18 xxε strains, compression diag., front face, different load levels.

0

20

40

60

80

100

120

140

-1.5 -1 -0.5 0 0.5 1

Exx strain (%)

Act

uato

r loa

d (k

N)

124.65 135.01

141.27 145.45

130.86 149.65

155.97 162.32

166.58 172.97

175.11

Figure 5.19 xxε vs load, front face compression diag., around center of panel.

Curves for locations measured from the corner of the panel (center at 150.00 mm).

39

Figure 5.19 shows strains on points close to the center of the panel and located under the

footprint of the strain gage (125.00 mm to 150.00 mm). Simultaneous observation of Figures

5.18 and 5.19 reveals that strains at some of the locations changed from compression to tension

after the occurrence of buckling. This conclusion is consistent with the mode shape shown in

Figure 5.14.

Observation of Figures 5.20 and 5.21 reveals that all of the points on the tension diagonal

undergo increasing levels of tension with little sensitivity to the occurrence of buckling.

Therefore, the tension diagonal basically behaves like a truss bar under axial loading.

Laminate 20 deg, Eyy strains over tension diagonal for different load levels

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0 50 100 150 200 250 300

Y position (mm)

Eyy

stra

in (%

)

14.8127.0050.1761.4074.0674.0685.5396.76105.36116.83126.14134.98

Figure 5.20 yyε strains on the tension diagonal for different load levels.

40

0

20

40

60

80

100

120

140

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Eyy strain (%)

Act

uato

r loa

d (k

N)

123.563975

139.969694

150.22496

160.476443

174.835777

Figure 5.21 yyε vs load - tension diagonal, locations around center of panel.


41

5.2.2 Laminate H - 45 degree data reduction

The ARAMIS database of recordings allows for fringe plotting of displacements and


Figure 5.22 Laminate 45 deg front face, out of plane displacement ( )lbsP 65.33139=


42

Figure 5.24 Laminate 45 deg, front face, yyε strain, ( )lbsP 65.33139= .

Figure 5.25 Laminate 45 deg, front face, xyε strain, ( )lbsP 65.33139= .

To correlate the test results with the finite element models of the structure, the ARAMIS

database was queried for specific xxε and yyε values on locations on both diagonals under tension

and under compression, as shown in Figures 5.26 and 5.27.

43


-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

0 25 50 75 100 125 150 175 200 225 250 275 300

X position (mm)

Exx

stra

in (%

)

11.2125.2836.7347.9360.1070.5983.2393.01103.98113.99129.97141.66147.41

Figure 5.26 xxε strains on the compression diagonal for different load levels.

0

20

40

60

80

100

120

140

160

-1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80

Exx strain (%)

Act

uato

r loa

d (k

N)

124.27

134.65

140.91

145.09

130.50

149.29

155.59

161.91

166.13

172.47

176.71

Figure 5.27 xxε vs load. Compression diag., locations around center of panel.


44

Figure 5.27 shows strains on points close to the center of the panel, and located under the

footprint of the strain gage (125.00 mm to 150.00 mm). Simultaneous observation of Figures

5.26 and 5.27 reveals that some of the strain gage readings changed from compression to tension

after the occurrence of buckling. This conclusion is consistent with the mode shape shown in

Figure 5.22.




0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 50 100 150 200 250 300

Y position (mm)

Eyy

stra

in (%

)

14.8127.0050.1761.4074.0674.0685.5396.76105.36116.83126.14143.34147.41


45

0

20

40

60

80

100

120

140

160

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Eyy strain (%)

Act

uato

r loa

d (k

N)

150.24

125.63

174.85

137.94

162.54



46

5.2.3 Laminate E - 90 degree data reduction

The ARAMIS database of recordings allows for contour plotting of displacements and


Figure 5.30 Laminate 90 deg front face, out of plane displacement ( )lbsP 9.31044= .


47

Figure 5.32 Laminate 90 deg, front face, yyε strains, ( )lbsP 9.31044= .

Figure 5.33 Laminate 90 deg, front face, xyε strains, ( )lbsP 9.31044= .

To correlate the test results with the finite element model of the structure, the ARAMIS

database was queried for specific xxε and yyε strain values on locations on both diagonals, under

tension and under compression.

48


-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0 25 50 75 100 125 150 175 200 225 250 275 300

X position (mm)

Exx

stra

in (%

)

10263853667992105114124131138

Figure 5.34 xxε on the compression diagonal for different load levels.

0

20

40

60

80

100

120

140

160

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Exx strain (%)

Act

uato

r loa

d (k

N)

125.58 135.83

146.08 156.33

166.58 174.79

Figure 5.35 xxε vs. load - compression diag., locations around center of panel.


49

Figure 5.35 shows strains on points close to the center of the panel and located under the

footprint of the strain gage (125.00 mm to 150.00 mm). Figures 5.34 and 5.35 were extracted

from the same dataset that generated the mode shape shown in Figure 5.30.




-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0 50 100 150 200 250 300

Y position (mm)

Eyy

stra

in (%

)

14.8127.0050.1761.4074.0674.0685.5396.76105.36116.83126.14134.98

Figure 5.36 yyε on the tension diagonal for different load levels.

50

0

20

40

60

80

100

120

140

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Eyy strain (%)

Act

uato

r loa

d (k

N)

123.56

139.97

150.22

160.48

174.84

Figure 5.37 yyε on the tension diagonal for different load levels.


51

5.2.4 Laminate F - 0 degree data reduction

No fringe plots were obtained for this laminate. However, strains xxε and yyε along the

tension and compression diagonals were obtained for the front surface. The data collected was

reduced as shown in Figures 5.38 through 5.41.


-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 25 50 75 100 125 150 175 200 225 250 275 300

X position (mm)

Exx

stra

in (%

)

16

31

46

61

79

95

111

125

140

155

169

167

178

189

199

Figure 5.38 xxε on the compression diagonal for different load levels.

52

0

20

40

60

80

100

120

140

160

180

-2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Exx strain (%)

Act

uato

r loa

d (k

N)

126.92

137.25

147.64

158.10

168.63

Figure 5.39 xxε vs. load - compression diag., locations around center of panel.


Figure 5.39 shows the strains on points close to the center of the panel, and located under

the footprint of the strain gage (125.00 mm to 150.00 mm). Simultaneous observation of Figures

5.38 and 5.39 reveals that none of the strain gage covered points changed from compression to

tension after the occurrence of buckling. Observation of Figures 5.42 and 5.41 reveals that all of

the points on the tension diagonal undergo increasing levels of tension with little sensitivity to

the occurrence of buckling.

53


-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 50 100 150 200 250 300

Y position (mm)

Eyy

stra

in (%

)

15.5330.5861.1678.8494.6194.61110.85124.95140.24155.29169.38166.52


0

20

40

60

80

100

120

140

160

180

200

220

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Eyy strain (%)

Act

uato

r loa

d (k

N)

125.59

135.85

146.10

154.31

164.56



54

5.3 Comparison plot of ARAMIS data vs. strain gage data

The following figures show both the direct readings from the strain gages and the

ARAMIS reduced data. Both sources of measurement are in agreement except for two cases

which are discussed at the end of this section.

0

20

40

60

80

100

120

140

-15000 -10000 -5000 0 5000 10000

Exx strain (microstrains)

Act

uato

r loa

d (k

N)

124.65

135.01

141.27

145.45

130.86

149.65

155.97

162.32

166.58

172.97

175.11

strain gage

0

20

40

60

80

100

120

140

160

0 500 1000 1500 2000 2500 3000 3500 4000

Eyy strain (microstrains)

Act

uato

r loa

d (k

N)

123.56

139.97

150.22

160.48

174.84

strain gage

Figure 5.42 Lam 20 degree - strains measured at different locations of the tension and the

compression diagonal vs. actuator load. ARAMIS vs. strain gage.

55

-10

10

30

50

70

90

110

130

150

-15000 -10000 -5000 0 5000 10000

Exx strain (microstains) on compression diagonal

Act

uato

r loa

d (k

N)

124.27

134.65

140.91

145.09

130.50

149.29

155.59

161.91

166.13

172.47

176.71

strain gage

-20

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000 12000

Eyy strain (microstrains) on tension diagonal

Act

uato

r loa

d (k

N)

150.24

125.63

174.85

137.94

162.54

strain gage

Figure 5.43 Laminate 45 degree - strains measured at different locations of the tension and the


56

0

20

40

60

80

100

120

-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0

Exx strain (microstrains) on the compressed diagonal

Act

uato

r loa

d (k

N) 125.58

135.83

146.08

156.33

166.58

174.79

strain gage

0

20

40

60

80

100

120

140

160

0 500 1000 1500 2000 2500 3000 3500 4000

Eyy strain (microstrains) on tension diagonal

Act

uato

r loa

d (k

N) 123.56

139.97

150.22

160.48

174.84

strain gage



57

0

20

40

60

80

100

120

140

160

180

-18000 -16000 -14000 -12000 -10000 -8000 -6000 -4000 -2000 0

Exx strain (microstrains) on the compressed diagonal

Act

uato

r loa

d (k

N) 126.92

137.25

147.64

158.10

168.63

strain gage

0

20

40

60

80

100

120

140

160

180

200

220

0 500 1000 1500 2000 2500 3000

Eyy strain (microstrains) on the tension diagonal (lam 0deg)

Act

uato

r loa

d (k

N)

125.59

135.85

146.10

154.31

164.56

strain gage



The strain gage and ARAMIS data were found to be in reasonably good agreement and

provided an additional check and validation of the strain gage data. The plots corresponding to

58

tension for both the 0 and the 90 degree laminates did not show a clear correlation; however, the

rest of the plots provide the consistency to validate the test data from both the strain gages and

ARAMIS system.

5.4 Experimental determination of buckling loads. Spencer-Walker method

Southwell proposed in 1934 a method to determine the buckling load of struts under

compression from experimental readings of the out of plane displacements measured at the

middle point of the strut [24]. The Southwell method was extended to the buckling of plates;

however, it exhibits important limitations in situations of high nonlinearity [25] such as the

plates considered in this dissertation. Spencer and Walker published in 1975 a method that

constitutes an extension of the Southwell method for rectangular plates with a variety of

boundary conditions and loading.

5.4.1 Southwell plots for struts under compression

Southwell considered a simply supported strut of length L and constant flexural

rigidity EI , with an existing distributed imperfection ( )xw~ along its length and under the action

of a quasi-static compressive load P . The additional transverse deflection is denoted here as

( )xw and it is the magnitude usually measured in practical tests, where x is measured from an

origin at one end of the strut. The values of these functions at 2Lx = are denoted by the

corresponding uppercase sW ' , and the Fourier coefficients by sWi ' . The equilibrium is given by

equation (5.1):

''2''2 ~wwwIV αα −=+ , where EIP=2α (5.1)

with the boundary conditions:

59

( ) ( ) ( ) ( ) 000 '''' ==== LwLwww (5.2)

Assuming that the imperfection shape can be expressed as a Fourier series,

( ) ∑∞

=⎟⎠⎞

⎜⎝⎛=

1sin~~

nn L

xnWxw π (5.3)

which leads to

( ) ∑⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⎟

⎠

⎞⎜⎝

⎛−=

−

Lxn

PP

Wxw nn

πsin1~1

(5.4)

where nP is the buckling load corresponding to the thn buckling mode:

222 LEInPn π= (5.5)

1

1~ −

⎟⎠⎞

⎜⎝⎛ −=

PPWW n

nn (5.6)

Southwell noted that “if P is a considerable fraction of 1P ” then

11

11 1~ −

⎟⎠⎞

⎜⎝⎛ −==

PPWWW (5.7)

so that as 1PP → the fundamental mode predominates. Hence, if 1~~ WW = , then

WPWPW ~

1 −⎟⎠⎞

⎜⎝⎛= (5.8)

Southwell’s brilliant suggestion was to plot W vs ( )PW and obtain CPP =1 as the slope

of the predicted straight-line graph. However, equation (5.8) was not derived for any arbitrary

deflection parameter W of an elastic structure under the action of an arbitrary load parameter P

60

[25]. At most, the analysis shows that for boundary value problems represented by equation

(5.1), the first eigenvalue of the associated homogeneous problem may be obtained from the

slope of the graph. In particular, the behavior of plates in the postbuckling regime is entirely

different from the hyperbolic relationship predicted by equation (5.8).

5.4.2 Spencer-Walker plots for plates

Walker showed that for rectangular plates over a wide range of aspect ratios, boundary

conditions and loading, a good approximation that is valid well into the postbuckling range is:

322 ~

ψψ BAh

Wh

W T

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ (5.9)

where,

TC W

WPP ~

1+−=ψ (5.10)

h is the plate thickness, and A and B are constants [25].

For the particular experiments completed in the present work as CPP →

1~ <<TWW

ψψ <<3

Considering the previous results, Donnell [25] proposed the equation:

( )( )WW

AhWWWP

PC

~21~

112 ++

+=⎟

⎠⎞

⎜⎝⎛ (5.11)

which is a special case of equation (5.9) that is based on using a single term of a double-odd

61

Fourier series for the deflection with 0=B .

The following paragraphs describe the application of the previous concept to plates with

small imperfections and data extending into the postbuckling regime. The “pivot point concept”

will be introduced in equation (5.11) in order to parallel Southwell’s method and obtain a linear

equation whose slope coefficient provides the buckling load.

Equation (5.11) contains three unknown parameters: CPW ,~ , and A . The plot of equation

(5.11) can provide two pieces of information, slope and intercept, so one parameter remains

undetermined in this case. In order to overcome this limitation, the “pivot point concept” is

introduced, which is a generalization of common rectification techniques of fitting empirical

data. The name “pivot point” is given here to any piece of actual or faired-in data which, in the

judgment of the experimenter, is sufficiently accurate to deserve its introduction into an analysis

as if it were an extra equation. For example, the assumption of most experimenters that two-

dimensional phenomena start precisely at ( )0,0 actually constitutes a pivot point. The approach

is to substitute the values at the pivot point into the known equation and solve for one parameter.

This method provides an engineering technique for finding empirical constants in cases where

there are fewer equations than unknown constants. Its name was chosen to emphasize that the

accuracy of the consequent results “pivots” about the accuracy of the chosen pivot points [25].

Introduction of a pivot point ( )** ,WP into the Donnell equation (5.11) results in

( )( )WW

AhWWWP

PC

~21~

11 *2**

*

+++

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (5.12)

A can be eliminated between equations (5.11) and (5.12) to give the plot equation:

62

( ) WHFPH C~

112 −= (5.13)

where,

( ) φ2*2*2 WPPWH −≡

( ) φ22*1 WWH −≡

2**

** 33 F

WWWP

WWPW ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−⎟⎟

⎠

⎞⎜⎜⎝

⎛+≡φ

( )WWWF ++≡ *1

~31

( ) ( )( ) ( )*2*2**

*2**2**

233

33

WWWPWWWP

GWWWPGWWWPF+−+

++−++≡

( )[ ]WWWWG ~232~ ** ++≡

( )[ ]WWWWG ~232~ * ++≡

The proposed method is to assume at first that 121 == FF and to plot 2H versus 1H for

the data points. If a straight line results, its slope and intercept W~ can be measured, the

correction factor 1F estimated, and CP calculated.

5.4.3 Spencer-Walker plots for the panels tested

Equation (5.13) was plotted for the four panels tested (E, F, G, and H). Significant pivot

points were chosen for each panel, and the resulting points were found to very closely fit a

straight line. Resulting plots in Spencer-Walker format are shown in Figures 5.46 through 5.49.

The slopes of the linear regression lines provided the values of buckling loads listed in Table 5.1.

63

Laminate Buckling Load

(lbs) Buckling Load

(kN) Initial imperfection in the middle point (mm)

F (0) 15,598 69.38 0.0019

G (20) 16,366 72.80 -0.0147

H (45) 9,871 43.91 0.7202

E (90) 4,394 19.54 0.0004

Table 5.1 Summary of results from Spencer-Walker data reduction of ARAMIS measured out of

plane displacement.

64

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00

Measured out of plane displacement (mm)

Actu

ator

Loa

d P

(lbs)

y = 15598x - 0.0147

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

-0.00012 -0.0001 -0.00008 -0.00006 -0.00004 -0.00002 0 0.00002

H1 parameter

H2 p

aram

eter

Pivot PointP* = 19551W* = -2.570

Figure 5.46 - Laminate F (0 degree): ARAMIS measured out of plane displacement vs. load

(above), and same data in Spencer-Walker format (below)

65

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

-0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00

Measured out of plane disp W (mm)

Actu

ator

Loa

d (lb

s)

y = 16366x + 5E-05

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

-0.0000012 -0.000001 -0.0000008 -0.0000006 -0.0000004 -0.0000002 0

H1 parameter

H2 p

aram

eter

Pivot PointP* = 16328W* = -0.017

Figure 5.47 - Laminate G (20 degree): ARAMIS measured out of plane displacement vs. load


66

0

5000

10000

15000

20000

25000

0.00 1.00 2.00 3.00 4.00 5.00

Measured out of plane displacement W (mm)

Actu

ator

Loa

d (lb

s)

y = 9871.3x + 0.7202

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006H1 parameter

H2 pa

ram

eter

Pivot PointP* = 27393W* = 2.93

Figure 5.48 - Laminate H (45 degree): ARAMIS measured out of plane displacement vs. load


67

0

5000

10000

15000

20000

25000

30000

35000

-14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00

Measured out of plane displacement W (mm)

Actu

ator

Loa

d P

(lbs)

y = 4393.7x + 0.0004

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

-3.5E-06 -3E-06 -2.5E-06 -2E-06 -1.5E-06 -1E-06 -5E-07 0 5E-07

H1 parameter

H2 p

aram

eter

Pivot PointP* = 4243W* = -0.0123

Figure 5.49 - Laminate E (90 degree): ARAMIS measured out of plane displacement vs. load


68

CHAPTER SIX

6FINITE ELEMENT MODEL

Previous chapters introduced the general requirements for a finite element (FE) analysis

to accurately simulate nonlinear buckling and progressive failure. This chapter describes the

specific modeling features that were introduced in the process of building the models used in this

dissertation. The finite element models used were created and analyzed in ANSYS v.10 and in

ABAQUS v.6.6, which incorporate their own pre/postprocessors. The models included the

features listed in Table 6.1.

As a brief reminder, the general methodology of the analysis portion of this dissertation

can be summarized as follows. First, run the linearized FE analysis to obtain the lowest buckling

loads and modes, where the lowest buckling load should be a reasonably good estimate of the

critical buckling load. As indicated previously, this estimate will only be reliable if the pre-

buckling displacements are small and the stresses vary linearly with the load. Second, using the

lowest mode shapes or the perturbation force method, an initial geometric imperfection can be

defined and introduced in the nonlinear analysis. If initial imperfections are imposed on the

“perfect” geometry of the structure, the load versus deflection results from the model should be a

closer match with the actual physical structure. Third, obtain the nonlinear solution with

imperfection to characterize the structural response in full, with stresses, strains and

displacements. When the solution includes progressive failure analysis (PFA) a cut-off level of

load associated with the first occurring failure of several potential modes of damage will indicate

the maximum design loading.

69

Component ANSYS element

type

ABAQUS

element type

Section

Panel (carbon/Epoxy and

glass/Epoxy)

SHELL 181 S4 (large strain

shell)

Laminated shell, different lay-ups

Picture frame arms (Steel) BEAM 44 B31 (3D beam) Rectangular, 1.1 x 3.0

Table 6.1 FE modeling features. Element types, boundary conditions and loads

Figure 6.1 ABAQUS FE model view. Picture frame and panel

Feature Modeled as

Actuator pull load Forces Fx, Fy on top right corner, equal in magnitude

Picture frame hinges Built by duplicating coincident nodes. Constrain Equation to have same

Ux, Uy, Uz, Rx, and Ry on coincident nodes

Nodal constraints Ux, Uy, Uz set to zero on lower left hinge

70

6.1 Modeling the composite panels

Based on the findings of Hermann et al. [4], the ANSYS SHELL181 is the element of

choice for postbuckling simulation. It is formulated for large rotations and/or large strain

nonlinear applications such as postbuckling.

It is a four-node element with six degrees of freedom per node: Cartesian x, y, z

translations and rotations about the x, y, z axes. Its formulation is based on the first order shear

deformation theory. This element supports both full and reduced integration schemes. The

assumptions and restrictions for the element are as follows [9]. This element works best with a

full Newton-Raphson solution scheme. Shear deflections are included in the element

formulation; however, sections normal to the center plane before deformation are assumed to

remain straight after deformation. For a composite laminate, this translates into the assumption

of no slippage between the element layers. This is standard first order shear deformation plate

theory. The usage of reduced integration (KEYOPTION 3 = 0 in ANSYS SHELL181) is

generally recommended to avoid the occurrence of shear locking, which is common in first order

shear deformation elements. However, SHELL181 includes incompatible modes capability,

which avoids shear locking. For elements that have large strain capability, the output stresses are

true (Cauchy) stresses in the rotated element coordinate system (the element coordinate system

follows the material as it rotates). Strain outputs are the logarithmic or Hencky strains, also in the

rotated element coordinate system.

In ABAQUS 6.6 the S4 element was used to model the laminates. Element type S4 is a

fully integrated, general-purpose, finite-membrane-strain shell element available in

ABAQUS/Standard. The element's membrane response is treated with an assumed strain

71

formulation that gives accurate solutions to in-plane bending problems, is not sensitive to

element distortion, and avoids parasitic locking. Element type S4 does not have hourglass modes

in either the membrane or bending response of the element; hence, the element does not require

hourglass control. The element has four integration locations per element, which makes the

element computationally expensive. S4 can be used for problems prone to membrane- or

bending-mode hourglassing, in areas where greater solution accuracy is required, or for problems

where in-plane bending is expected.

6.2 Modeling the picture frame arms

The element chosen to model the picture frame arms in ANSYS is BEAM44 3-D Elastic

Beam, which allows for large rotations. Shear effects are not included by default (but can be

included as an option). BEAM44 was chosen after running a quick study to compare solutions

using BEAM44 against solutions using BEAM4 Elastic Beam and BEAM188 Finite Strain Beam

as shown in Table 6.2.The BEAM4 elastic beam element is formulated for Classical Beam

Theory and does not allow for finite strains or large displacements. Elements belonging to the

180 series (BEAM188 and BEAM189) allow for large rotations and strains and they include

shear effects by default.

The interpretation of these results is that although the frame arms do not undergo finite

strains since they are very stiff compared to the plate, they still experience large rotations around

the hinges of the fixture for some of the panels. For large rotation cases, the BEAM4 element

presented convergence difficulties. The BEAM188 includes shear effects by default, which

makes the execution cumbersome and results in a very stiff element. The BEAM44 element

proved to have the most robust convergence properties while still not being too stiff. For the

72

stated reasons, the element chosen to model the Picture Frame fixture was the BEAM44.

Buckling Load, Non Linear Analysis (lbs)

Panel BEAM4 BEAM44 BEAM188

Laminate 20 deg 13,545 14,059 27,256

Laminate 45 deg 8,060 8,840 21,731

Laminate 90 deg Not converged 4,526 15,788

Laminate 0 deg Not converged 20,489 47,354

Table 6.2 Comparison of results. Different picture frame modeling options (ANSYS)

In ABAQUS 6.6 the beams were modeled using the B31 two-node linear 3D beam

element. This choice of element did not pose any difficulties.

6.3 Mechanical properties of the laminates tested

As stated previously, the laminates tested consisted of alternating plies of pre impregnated

Double Biax Glass/Epoxy (denoted as DB) with plies of pre impregnated Unidirectional

Carbon/Epoxy tape (denoted as UC). The basic laminate followed the symmetric sequence

[ ]SUCDBUCDB ,,, . For all of the laminates, fibers of the DB layers are oriented at 45 degrees

with respect to the fibers of the UC layers. The material properties of the DB layer were tested

and reported in the ply axes. Therefore, every DB layer was modeled as a single orthotropic layer

in the SHELL 181 (and ABAQUS S4) layered element.

The ANSYS section plot, shown in Figure 6.2, has a fictitious fiber direction for the DB

73

layers parallel to the direction of the fiber in the UC layers. The material properties of the plies

used to build the laminates exhibit the mechanical properties listed in Table 6.3.

Figure 6.2 ANSYS section plot of the laminate for the case of zero degree with the direction of

the load.

Note that in Figure 6.2 the colored lines represent the axes of the ply. The fibers in the

DB layers are oriented at 45 degrees with respect to the ply axes.

74

Double Biax Glass Fabric (12 oz/yd2)

Unidirectional Carbon Fabric (15 oz/yd2)

Material MGDB12 MCUD15

Laminate Thickness (in) 0.152 0.110

E1 (lbf/in2) 1.610E+06 1.846E+07

E2 (lbf/in2) 1.610E+06 1.273E+06

E3 (lbf/in2) 1.610E+05 1.273E+05

V12 0.604 0.328

V23 0.360 0.360

V13 0.360 0.360

V21 0.604 0.023

V32 0.036 0.036

V31 0.036 0.002

G23 (lbf/in2) 1.639E+05 1.639E+05

G31 (lbf/in2) 1.639E+05 1.639E+05

G12 (lbf/in2) 1.461E+06 7.359E+05

Fiber Density (lbm/in3) 0.0939 0.0650

Fiber Density (slug/in3) 2.919E-03 2.021E-03

Nominal Density (lbm/in3) 0.0664 0.0562

Nominal Density (slug/in3) 2.064E-03 1.747E-03

Dry Material Weight (oz/yd2) 12 15

Dry Material Weight (oz/in2) 9.259E-03 1.157E-02

Table 6.3 Mechanical properties of individual plies (British Units)

75

CHAPTER SEVEN

7ANALYTICAL CLASSICAL SOLUTIONS

This chapter presents approximate analytical results for the linear buckling loads for each

of the laminates tested. These solutions were developed as a baseline estimate for the detailed

finite element solutions. It should be noted that until recently, analytical solutions were

considered to be the standard method to calculate the buckling load of composite panels with

different boundary conditions and loading; e.g., four clamped edges under shear load. In fact,

most in-house structural design manuals rely on analytical solutions that have been validated by

test data. For the panels considered herein, the combination of laminate stiffness properties

combined with the boundary conditions and loading does not permit an exact analytical solution;

therefore, an approximate solution has been obtained using the Raleigh-Ritz technique [22]. The

validity of the approximate solution is limited to specially orthotropic laminates, such that all

elements of the B sub-matrix and the D16 and D26 elements of the D sub-matrix are zero, where

B and D are sub-blocks of the laminate stiffness matrix derived from the Classical Laminated

Plate Theory (CLPT).

The CLPT stiffness matrix was calculated for each laminate using the software package

ESAComp, which shows that only one of the panels (laminate 45 degree) included in this

research complies with the specially orthotropic condition. Even so, a two-term Raleigh-Ritz

solution was obtained to gain insight regarding the order of magnitude to expect for the buckling

load. The results are given in the following figures. Figures 7.1 through 7.4 show the stiffness

matrices of each laminate as directly output from ESAComp script. Figures 7.7 through 7.10

show the calculation of the buckling load based on the two-term Raleigh-Ritz solution provided

76

by Reddy [22]. Table 7.1 shows a summary of results obtained from the analytical solution.

Laminate : Laminate_0deg_08-2005

Modified : Mon May 08 20:37:11 2006

Lay-up

(+45a/+45b/+45a/+45b/+45b/+45a/+45b/+45a)

Ply t (mm)

a Double Bias Glass +-45 (12 oz/yd2) 0.329b Unidirectional Carbon (15 oz/yd2) 0.487

n = 8 m_A = 6928 g/m²h = 3.264 mm rho = 2123 kg/m³

Classification

Ply types : Solid;Reinf.Lay-up : Symm.

Constit. beh. : AeBoDe

1 a 45°

2 b 45°

3 a 45°

4 b 45°

5 b 45°

6 a 45°

7 b 45°

8 a 45°

Laminate stiffness and compliance matrices


Stiffness matrix

[A] [B] (N/m) (N) [B] [D] (N) (Nm)

1.11183e+008 6.48842e+007 5.81521e+007 0 0 06.48842e+007 1.11183e+008 5.81521e+007 0 0 05.81521e+007 5.81521e+007 6.84933e+007 0 0 0

0 0 0 89.6925 43.183 35.51230 0 0 43.183 89.6925 35.51230 0 0 35.5123 35.5123 44.9573

Compliance matrix

[a] [b] (m/N) (1/N) [b]^T [d] (1/N) (1/(Nm))

1.72658e-008 -4.33301e-009 -1.09802e-008 0 0 0-4.33301e-009 1.72658e-008 -1.09802e-008 0 0 0-1.09802e-008 -1.09802e-008 3.32447e-008 0 0 0

0 0 0 0.0172633 -0.00423774 -0.0102890 0 0 -0.00423774 0.0172633 -0.0102890 0 0 -0.010289 -0.010289 0.0384982

Figure 7.1 Laminate 0 degree. Lay-up and stiffness matrices

77



Lay-up

(+65a/+65b/+65a/+65b/+65b/+65a/+65b/+65a)

Ply t (mm)


n = 8 m_A = 6928 g/m²h = 3.264mm rho = 2123 kg/m³

Classification


Constit. beh. : AfBoDf

1 a 65°

2 b 65°

3 a 65°

4 b 65°

5 b 65°

6 a 65°

7 b 65°

8 a 65°



Stiffness matrix

[A] [B] (N/m) (N)

[B] [D] (N) (Nm)

5.51592e+007 4.61492e+007 2.22196e+007 0 0 04.61492e+007 2.04677e+008 6.68745e+007 0 0 02.22196e+007 6.68745e+007 4.97583e+007 0 0 0

0 0 0 53.0057 34.216 16.51760 0 0 34.216 144.313 37.89050 0 0 16.5176 37.8905 35.9903

Compliance matrix

[a] [b] (m/N) (1/N)

[b]^T [d] (1/N) (1/(Nm))

2.32956e-008 -3.30494e-009 -5.96087e-009 0 0 0-3.30494e-009 9.1798e-009 -1.08617e-008 0 0 0-5.96087e-009 -1.08617e-008 3.7357e-008 0 0 0

0 0 0 0.0234117 -0.00377252 -0.006772990 0 0 -0.00377252 0.0101844 -0.008990710 0 0 -0.00677299 -0.00899071 0.0403591

Figure 7.2 Laminate 20 degrees. Lay-up and stiffness matrices

78



Lay-up

(90a/90b/90a/90b/90b/90a/90b/90a)

Ply t (mm)


n = 8 m_A = 6928 g/m²h = 3.264mm rho = 2123 kg/m³

Classification

Ply types : Solid;Reinf.Lay-up : Symm.;Cross-ply

Constit. beh. : AsBoDs

1 a 90°

2 b 90°

3 a 90°

4 b 90°

5 b 90°

6 a 90°

7 b 90°

8 a 90°



Stiffness matrix

[A] [B] (N/m) (N) [B] [D] (N) (Nm)

4.02228e+007 1.95404e+007 0 0 0 01.95404e+007 2.72831e+008 0 0 0 0

0 0 2.31494e+007 0 0 00 0 0 40.3704 21.4804 00 0 0 21.4804 182.42 00 0 0 0 0 23.2547

Compliance matrix

[a] [b] (m/N) (1/N)

[b]^T [d] (1/N) (1/(Nm))

2.57577e-008 -1.84479e-009 0 0 0 0-1.84479e-009 3.7974e-009 0 0 0 0

0 0 4.31976e-008 0 0 00 0 0 0.0264263 -0.00311178 00 0 0 -0.00311178 0.00584829 00 0 0 0 0 0.043002

Figure 7.3 Laminate 45 degrees. Lay-up and stiffness matrices

79



Lay-up

(-45a/-45b/-45a/-45b/-45b/-45a/-45b/-45a)

Ply t (mm)


n = 8 m_A = 6928 g/m²h = 3.264mm rho = 2123 kg/m³

Classification


Constit. beh. : AfBoDe

1 a -45°

2 b -45°

3 a -45°

4 b -45°

5 b -45°

6 a -45°

7 b -45°



Stiffness matrix

[A] [B] (N/m) (N) [B] [D] (N) (Nm)

1.11183e+008 6.48842e+007 -5.81521e+007 0 0 06.48842e+007 1.11183e+008 -5.81521e+007 0 0 0

-5.81521e+007 -5.81521e+007 6.84933e+007 0 0 00 0 0 89.6925 43.183 -35.51230 0 0 43.183 89.6925 -35.51230 0 0 -35.5123 -35.5123 44.9573

Compliance matrix

[a] [b] (m/N) (1/N) [b]^T [d] (1/N) (1/(Nm))

1.72658e-008 -4.33301e-009 1.09802e-008 0 0 0-4.33301e-009 1.72658e-008 1.09802e-008 0 0 01.09802e-008 1.09802e-008 3.32447e-008 0 0 0

0 0 0 0.0172633 -0.00423774 0.0102890 0 0 -0.00423774 0.0172633 0.010289

Figure 7.4 Laminate 90 degree. Lay-up and stiffness matrices

80

P 1.866 104×=

P Nxy 12⋅ 2⋅:=a22 526.588=

Nxy 1.1 103×=a12 0.16=

a11 59.149=Nxya11 a22⋅

a12:=

a223791.532 D11⋅

a4

4227.255 D12 2 D66⋅+( )⋅

a2 b2⋅+

3791.532 D22⋅

b4+:=

a1223.107

a b⋅:=

a11537.181 D11⋅

a4

324.829 D12 2 D66⋅+( )⋅

a2 b2⋅+

537.181 D22⋅

b4+:=

Ritz two terms approximation (Reddy, [8])

D66 395.004=

D12 379.414=

D22 788.056=D66 Di66 0.223⋅ 39.4⋅:=

D11 788.056=D22 Di22 0.223⋅ 39.4⋅:=

D12 Di12 0.223⋅ 39.4⋅:=

D11 Di11 0.223⋅ 39.4⋅:=

Transform Matrix D to English Units

Di66 44.9573:=

Di22 89.6925:=

Di12 43.1830:=b 12:=

a 12:=Di11 89.6925:=

Matrix D of the laminate in SI units:

Laminate 0 degrees

Figure 7.5 Laminate 0 degree. Buckling load calculation, two-term Ritz approx.

81

P 1.838 104×=

P Nxy 12⋅ 2⋅:=a22 507.214=

Nxy 1.083 103×=a12 0.16=

a11 59.529=Nxya11 a22⋅

a12:=

a223791.532 D11⋅

a4

4227.255 D12 2 D66⋅+( )⋅

a2 b2⋅+

3791.532 D22⋅

b4+:=

a1223.107

a b⋅:=

a11537.181 D11⋅

a4

324.829 D12 2 D66⋅+( )⋅

a2 b2⋅+

537.181 D22⋅

b4+:=


D66 316.215=

D12 300.629=

D22 1.268 103×=D66 Di66 0.223⋅ 39.4⋅:=

D11 465.721=D22 Di22 0.223⋅ 39.4⋅:=

D12 Di12 0.223⋅ 39.4⋅:=

D11 Di11 0.223⋅ 39.4⋅:=


Di66 35.990:=

Di22 144.313:=

Di12 34.216:=b 12:=

a 12:=Di11 53.006:=


Laminate 20 degrees

Figure 7.6 Laminate 20 degrees. Buckling load calculation, two-term Ritz approx.

82

P 1.828 104×=

P Nxy 12⋅ 2⋅:=a22 490.449=

Nxy 1.077 103×=a12 0.16=

a11 60.894=Nxya11 a22⋅

a12:=

a223791.532 D11⋅

a4

4227.255 D12 2 D66⋅+( )⋅

a2 b2⋅+

3791.532 D22⋅

b4+:=

a1223.107

a b⋅:=

a11537.181 D11⋅

a4

324.829 D12 2 D66⋅+( )⋅

a2 b2⋅+

537.181 D22⋅

b4+:=


D66 230.679=

D12 188.731=

D22 1.603 103×=D66 Di66 0.223⋅ 39.4⋅:=

D11 354.702=D22 Di22 0.223⋅ 39.4⋅:=

D12 Di12 0.223⋅ 39.4⋅:=

D11 Di11 0.223⋅ 39.4⋅:=


Di66 26.2547:=

Di22 182.420:=

Di12 21.4804:=b 12:=

a 12:=Di11 40.3704:=


Laminate 45 degrees


83

P 1.866 104×=

P Nxy 12⋅ 2⋅:=a22 526.588=

Nxy 1.1 103×=a12 0.16=

a11 59.149=Nxya11 a22⋅

a12:=

a223791.532 D11⋅

a4

4227.255 D12 2 D66⋅+( )⋅

a2 b2⋅+

3791.532 D22⋅

b4+:=

a1223.107

a b⋅:=

a11537.181 D11⋅

a4

324.829 D12 2 D66⋅+( )⋅

a2 b2⋅+

537.181 D22⋅

b4+:=


D66 395.004=

D12 379.414=

D22 788.056=D66 Di66 0.223⋅ 39.4⋅:=

D11 788.056=D22 Di22 0.223⋅ 39.4⋅:=

D12 Di12 0.223⋅ 39.4⋅:=

D11 Di11 0.223⋅ 39.4⋅:=


Di66 44.9573:=

Di22 89.6925:=

Di12 43.183:=b 12:=

a 12:=Di11 89.6925:=


Laminate 90 degrees


84

Laminate Buckling Load (lbs)

0 degree 18,660

20 degree 18,380

45 degree 18,280

90 degree 18,660

Table 7.1 Summary of results. Buckling load calculation, two-term Ritz approx.

Inspection of Table 7.1 reveals the limited validity of this kind of solution. It is useful to

provide a first cut of the value of the buckling load. The details of this approximate analysis are

listed in the following paragraphs.

The governing equation for a specially orthotropic laminate under shear load, 0xyN , is

given as

( )yx

wN

yw

Dyx

wDD

xw

D xy ∂∂∂

=∂∂

+∂∂

∂++

∂∂ 0

20

40

4

22220

4

661240

4

11 222 (7.1)

This problem does not permit the Navier solution; therefore, an approximate solution has been

obtained using the Rayleigh Ritz method. For the case of the clamped plate (like the panels

considered herein) the total potential energy expression for the case of in-plane shear loading is:

85

( )

dxdyy

wx

wN

yw

D

yxw

Dyw

xw

Dxw

Dw

xy

b a

⎥⎥⎦

⎤⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=Π ∫ ∫

0002

20

2

22

2

02

6620

2

20

2

12

2

20

2

110 00

2

4221

(7.2)

The minimum total potential energy principle requires that 0=Πδ , which results in

( ) ( ) ( )( )

dxdyyw

xw

yw

xw

Nyw

yw

D

yxw

yxw

Dyw

xw

xw

yw

Dxw

xw

D

dxdywqMMM

xy

b a

xyxyyyyyxxxx

b a

⎥⎥⎦

⎤⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂∂∂

+∂∂

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=

=−++=

∫ ∫

∫ ∫

000002

02

20

2

22

02

02

6620

2

20

2

20

2

20

2

1220

2

20

2

110 0

0111

0 0

4

0

δδδ

δδδδ

δδγδεδε

(7.3)

A Ritz approximation of the following form is assumed

( ) ( ) ( )yxcyxWyxw ij

m

i

n

jijmn ,,,

1 10 ϕ∑∑

= =

=≈ (7.4)

where

( ) ( ) ( )yYxXyx iiij =,ϕ (7.5)

with

( ) ( )2121

1,1 ⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛=

++

by

byyY

ax

axxX

j

j

i

i (7.6)

or

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )yyyyxY

xxxxxX

jjjjjj

iiiiii

λλαλλλλαλλ

coscoshsinhsincoscoshsinhsin−+−=−+−=

(7.7)

86

for mi ,...,2,1= ; nj ,...,2,1= . The parameters iλ and iα of the previous equations are obtained

from solving the characteristic equation, and come defined as:

( )2

12,853.7,730.4 21πλλλ +≈== iaaa iL

21,99922.0,0178.1 21 >=== iforiααα L

Substituting the Ritz approximation into the total potential energy equation results in:

( )

pqqpj

iq

pji

xyq

pj

i

qpjiq

pj

ib a

cdxdyYdx

dYdydY

Xdy

dYXY

dxdX

Ndy

YdX

dyYd

XD

dydY

dxdX

dydY

dxdX

DDYdx

XdY

dxXd

D

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

⎪⎩

⎪⎨⎧

⎢⎢⎣

⎡+++⎟

⎟⎠

⎞⎜⎜⎝

⎛= ∫ ∫

02

2

2

2

22

66122

2

2

2

110 0220

(7.8)

When the functions ( )xX i and ( )yYj are used, at least two terms are required because

the coefficient of 0xyN is zero for 1== nm ; other coefficients are zero for 2,1 == nm and

1,2 == nm . Using the approximation

( ) ( ) ( ) ( ) ( )yYxXcyYxXcyxw 222211110 , +≈ (7.9)

with the expressions

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

ax

ax

ax

axxX 73.4cos73.4cosh0178.173.4sinh73.4sin1 (7.10)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

ax

ax

ax

axxX 853.7cos853.7cosh9992.0853.7sinh853.7sin2

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

by

by

by

byyY 73.4cos73.4cosh0178.173.4sinh73.4sin1

87

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

by

by

by

byyY 853.7cos853.7cosh9992.0853.7sinh853.7sin2

results in

( ) 0107.23181.5372829.324181.53722

0112236612113 =−⎥⎦

⎤⎢⎣⎡ +++ cNcD

baDD

abD

ab

xy

( ) 0107.23181.5372255.4227532.379111

0222236612113 =−⎥⎦

⎤⎢⎣⎡ +++ cNcD

baDD

abD

ab

xy

or in matrix form,

⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

00

22

11

22120

120

11

cc

aaNaNa

xy

xy (7.11)

where

( ) 0107.23181.5372829.324181.53722

01122366122211411 =−⎥⎦

⎤⎢⎣⎡ +++= cNcD

bDD

baD

aa xy ab

a 107.2312 =

For a nontrivial solution, the determinant of the coefficient matrix should be zero:

( ) 0012122211 =− xyNaaaa

Solving for the buckling load ( )0xyN ,

221112

0 1 aaa

N xy ±= (7.12)

The ± sign indicates that the shear buckling load can be either positive or negative. For

an isotropic square plate, we would have:

ba =

88

( ) DDDDD =+== 66122211 2

and the shear buckling loads are:

20 176

aDN xy ±=

89

CHAPTER EIGHT

8LINEARIZED BUCKLING ANALYSIS

8.1 Formulation

The algorithm to obtain the linearized buckling load is described in this section.

Considering the matrix form of the system of equations assembled by the finite element (FE)

solver algorithm, the tangent stiffness matrices at times t and ( )tt Δ+ are denoted as tK and

ttK Δ+ , and the corresponding vectors of externally applied loads are tR and ttR Δ+ . In linearized

buckling analysis, it is assumed that at any time τ

( )tttt KKKK −+= Δ+λτ

and the applied load vector is:

( )tttt RRRR −+= Δ+λτ

where λ is a scaling factor and those values of λ which are greater than one are of interest. At

collapse or buckling the tangent stiffness matrix is singular and hence the condition for

calculating λ is

( ) 0det =τK

or, equivalently

0=φτK

where φ is a nonzero vector. Combining the previous equations yields the eigenproblem

( )φλφ tttt KKK Δ+−=

90

The eigenvalues iλ , for ni ,....1= give the buckling loads and the eigenvectors iφ

represent the corresponding buckling modes. It is assumed that the matrices tK and ttK Δ+ are

both positive definite but in general the difference matrix ( )ttt KK Δ+− is indefinite; hence, the

eigenproblem will have both positive and negative solutions and some eigenvalues will be

negative. Negative eigenvalues can be ignored, and the smallest positive eigenvalues are the only

ones desired. The last equation can be rewritten as

( )φγφ ttt KK =Δ+

λλγ 1−

=

The eigenvalues iγ are all positive and usually only the smallest values 1γ , 2γ … are of

interest. Namely, 1γ corresponds to the smallest value of λ in the original problem. Once 1γ is

known, 1λ is also known and the buckling load is given by:

( )ttttbuckling RRRR −+= Δ+1λ

Similarly, the buckling loads associated with the remaining iγ for 1>i can be evaluated.

In practice, most frequently 0=t (initial configuration with 00 =R ) and tΔ is the first load step

with tRΔ ; however, these equations are applicable to any load step prior to buckling.

8.2 Analysis results

The linearized ANSYS buckling analysis executed the eigenbuckling solution using

block Lanzcos, and the first three eigenvalues and buckling modes were requested. Once the

lowest eigenvalues are known, the buckling load was calculated as

91

appiicr FP λ=,

where iλ is the eigenvalue i ,

icrP , is the critical or buckling load associated with the eigenvalue i

appF is the load applied to the model in order to run the linearized analysis.

It should be noted that ANSYS does not automatically perform the initial linear static

analysis necessary to determine the geometric stiffness matrix. Therefore, the analyst should first

run the linear static solution for the model with the option “Calculate pre-stress effects” turned

on. Once the linear solution is known, the eigenbuckling analysis should be ran from the same

folder using the Block Lanzcos factorization scheme.

8.2.1 Laminate F - 0 degree with the direction of the load

The lay-up sequence of this laminate is shown in Figure 8.1. All of the plies of the

laminate form the same angle with the direction of the load. Table 8.1 lists the results of the

linearized buckling analysis for this panel.

Load Applied for Analysis (lbs)

First Eigenvalue/ Buckling Load (lbs)

Second Eigenvalue/ Buckling Load (lbs)

Third Eigenvalue/ Buckling Load (lbs)

2.1838 2.2522 31.214 1,000

21,838 22,522 31,214

21.838 22.522 4.7456 10,000

21,838 22,522 47,456

1.0919 1.1261 2.3728 20,000

21,838 22,522 47,456

Table 8.1 Eigenvalues and buckling loads vs. level of load applied to the model

92

Figure 8.1 Laminate 0 degree, section plot.

Note that in Figure 8.1 the colored lines represent the axes of the ply. The fibers in the

DB layers are oriented at 45 degrees with respect to the ply axes..

From the previous table it can be concluded that the level of the load applied to the model

for the linearized analysis has little or no influence on the calculated critical buckling loads.

Figure 8.2 First mode, buckling load lbsP 838,21= . Out of plane displacement

93

Figure 8.3 Second mode, buckling load lbsP 522,22= . Out of plane displacement

Figure 8.4 Third mode, buckling load lbsP 456,47= . Out of plane displacement

94

From the observation of the previous three figures, as well as the behavior of the panels

during the test, it can be concluded that the third and subsequent modes are unrealistic and do not

add any value to the analysis. Therefore, only the first three buckling modes are requested for the

linearized analyses.

The first buckling mode exhibits a central out of plane narrow bubble together with two

very shallow bubbles on both sides that are opposite in direction. This shape denotes how the

presence of the unidirectional carbon plies in the direction of the load reduces the relative

stiffness of the laminate in the compression direction at -45 degrees with respect to the horizontal

axis, allowing the formation of the “negative” bubble on both sides (blue) and allowing the

laminate “to wrinkle” in that direction. The second mode is antisymmetric with two identical

bubbles in opposite directions.

The buckling load associated with the second mode is very close to the one of the first

mode. Therefore, the possibility of either the first or second mode occurring first and dominating

the actual test displacements should be considered.

8.2.2 Laminate G – 20 degree with the direction of the load

The lay-up sequence of this laminate is shown in Figure 8.5. All of the plies of the

laminate form the same angle with the direction of the load. Table 8.2 lists the results of the

linearized buckling analysis for this panel. Note that in Figure 8.5 the colored lines represent the

axes of the ply. Fibers in the DB layers are oriented at 45 degrees with respect to the ply axes.

95

Figure 8.5 Laminate 20 degree section plot.

Load Applied for Analysis (lbs)

First Eigenvalue/ Buckling Load

Second Eigenvalue/ Buckling Load

Third Eigenvalue/ Buckling Load

0.0002 157.07 163.42 100

0.02 15,707 16,342

0.0406 15.681 16.335 1,000

40,63 15,681 16,335

1.5681 1.6335 3.4489 10,000

15,681 16,335 34,489

1.5681 1.6335 3.4489 15,000

15,681 16,335 34,489


Table 8.2 shows that the level of the load applied to the model for the linearized analysis

has little or no influence on the calculated critical buckling loads. However, it can still be

observed that, as the applied load gets closer to the buckling load, some spurious solutions

96

present at lower levels tend to disappear, which is always desirable.



97


As mentioned for the previous panel, only the first three buckling modes were requested

in the linearized analyses. All of the previous general comments for describing the mode shapes

of the 0 degree panel are also applicable for the 20 degree panel.

It can also be noted that the 20 degree lay-up produces a rotation in the orientation of the

buckling “bubbles” of the mode shapes. The nodal line that divides the semi sine waves in the

second mode appears rotated about 20 degrees in the positive direction. The same observation is

valid for the first and third modes.

Compared to the 0 degree laminate the 20 degree laminate has a decrease in the buckling

load of 40%.

8.2.3 Laminate H – 45 degree with the direction of the load

The lay-up sequence of this laminate is shown in Figure 8.9. Table 8.3 lists the results of

98

the linearized buckling analysis for this panel. Note that in Figure 8.9 the colored lines represent

the axes of the ply. The fibers in the DB layers are oriented at 45 degrees with respect to the ply

axes.

Figure 8.9 Laminate section plot. Lam 45 deg.

Load Applied for

Analysis (lbs)

First Eigenvalue/

Buckling Load

Second Eigenvalue/

Buckling Load

Third Eigenvalue/

Buckling Load

0.0041 16.012 20.508 100

0.41 1,601 2,051

9.4980 10.174 18.083 1,000

9,498 10,174 18,083

0.94980 1.0174 1.8083 10,000

9,498 10,174 18,083


Table 8.3 shows that the level of the load applied to the model for the linearized analysis

99

has little or no influence on the calculated critical buckling loads. It can still be observed that, as

the applied load gets closer to the buckling load, some spurious solutions present at lower levels

disappear.



100


Only the first three buckling modes were requested in the linearized analyses. All of the

general comments for describing the mode shapes of the 0 and 20 degree panels are also

applicable for the 45 degree panel. The 45 degree lay-up causes a rotation in the orientation of

the buckling “bubbles” of the mode shapes. The nodal line that divides the semi sine waves in

the second mode appears rotated about 45 degrees in the positive direction. This same

observation is valid for the first and third modes. It also can be concluded that the 45 degree

laminate has a decrease in the buckling load of 130% with respect to the 0 degree laminate.

8.2.4 Laminate E – 90-degree with the direction of the load

The lay-up sequence of this laminate is shown in Figure 8.13. Table 8.4 lists the results of

the linearized buckling analysis for this panel. Note that in Figure 8.13 the colored lines

101

represent the axes of the ply. Fibers in the DB layers are oriented at 45 degrees with respect to

the ply axes.

Figure 8.13 Laminate section plot. Lam 90 deg.

Load Applied for

Analysis (lbs)

First Eigenvalue/

Buckling Load

Second Eigenvalue/

Buckling Load

Third Eigenvalue/

Buckling Load

0.0002 46.049 58.651 100

0.02 4,605 5,865

4.6049 5.8652 10.788 1,000

4,605 5,865 10,788

0.4605 0.5865 1.0788 10,000

4,605 5,865 10,788


As mentioned for the previous laminates, the level of the load applied to the model for

the linearized analysis has no influence in the calculated critical buckling loads. As the applied

102

load gets closer to the buckling load, some spurious solutions present at lower levels disappear.



103


Only the first three buckling modes were requested in the linearized analyses. All of the

general comments for describing the mode shapes of the previous panels are also applicable for

the 90 degree panel. The 90 degree lay-up produces a rotation in the orientation of the buckling

“bubbles” of the mode shapes. In this panel all of the relative stiffness is now oriented

perpendicular to the direction of the load. It also can be concluded that the 90 degree laminate

divided the buckling load by four with respect to the 0 degree laminate.

8.2.5 Linearized buckling analysis. Summary of results.

Table 8.5 summarizes the results of the linear buckling analysis, high lighting the first

buckling load, which is the one that is critical for design.

104

First Mode Second Mode Third Mode Laminate (degrees with the direction of the load)

lbs kN lbs kN lbs kN

0 degree 21,838 97.135 22,525 100.191 47,456 211.084

20-degree 15,681 69.749 16,335 72.658 34,489 153.407

45-degree 9,498 42.247 10,147 45.134 18,083 80.433

90-degree 4,605 20.483 5,865 26.087 10,788 47.985

Table 8.5 First three buckling loads for each laminate

The trend in the first column indicates that the variation of the angle of the laminate,

particularly of the carbon fibers, with the direction of the load dramatically decreases the critical

buckling load.

For this structure, it is recommended that the linearized analysis be performed with an

applied load close in value to the smallest buckling load. This will prevent the output of spurious

solutions in the analysis. Since the value of the buckling load is unknown in advance, a first

exploratory analysis should be ran to search for meaningful values of the lowest buckling load.

Table 8.5 reveals that the first and second buckling loads are very close in value. Thus, it

cannot be assumed in advance that the configuration adopted immediately after buckling by the

actual tested structure will resemble the first linearized buckling mode. Small variations in the

initial geometric imperfection can trigger the onset of either the first or the second mode.

105

Laminate 0 deg Laminate 20 deg


Figure 8.17 Rotation of the buckling nodal line. Second mode

106



Figure 8.18 Rotation of the buckling nodal lines. First mode

107

CHAPTER NINE

9NONLINEAR BUCKLING AND POSTBUCKLING (WITHOUT FAILURE)

This section describes the development, solution, and results of the nonlinear finite

element modeling for the panels and experiments described in Chapter 5. The correlation of the

model results with the test results is also presented. No failure detection or damage evolution was

included in these models. The focus was solely to evaluate the capabilities of the nonlinear

model to predict the response of the structure in the postbuckling regime. A different model and

analysis that includes failure prediction and damage evolution are presented in Chapter 10. Two

nonlinear models were constructed: one using ANSYS (version 10, release 2006) and a second

one using ABAQUS (version 6.6 release 2007), and results from both models are presented and

compared herein.

9.1 Formulation

9.1.1 Historical approaches

The nonlinear partial differential equations governing composite laminates of arbitrary

geometries and boundary conditions (BC) cannot be solved exactly. Approximate analytical

solutions to the large deflection theory in von Kármán’s sense have been developed in the past

for laminated composite plates. These solutions are limited to rectangular or cylindrical

geometries and in most cases ignore the effects of shear deformation [22]. The Rayleigh-Ritz

method, Galerkin method, perturbation method, and double series method are limited to simple

geometries and BC. The use of approximate numerical methods facilitates the solution of these

equations for general geometries and BC. Among the numerical methods available for the

solution of nonlinear differential equations defined over arbitrary domains, the finite element

108

method is the most practical and robust computational technique [22].

Historically, two distinct approaches have been followed in developing nonlinear finite

element models of laminated structures. The first approach is based on the laminate theory, in

which the 3-D elasticity equations are reduced to 2-D equations through certain kinematic

assumptions and homogenization through the thickness. In the nonlinear formulation based on

small strains and moderate rotations, the geometry of the structure is supposed to remain

unchanged during the loading, and the geometric nonlinearity in the form of von Kármán’s

strains is included. The elements based on such assumptions are commonly referred to as

“laminated elements”.

The second approach is based on the 3-D continuum formulation, where any kinematic

assumptions are directly introduced through the spatial finite element approximation. Nonlinear

strains are included and the equations are derived in an incremental form. This formulation

accounts for geometric changes that occurred during the previous increment of loading and the

geometry is updated between load increments. Finite elements based on this formulation are

called “continuum elements”.

There are two incremental continuum formulations that are used to determine the

deformation and stress states in continuum problems [22]: first, the total Lagrangian formulation,

and second, the updated Lagrangian formulation. In these formulations, the geometry of the

structure for the current load increment is determined from a previously known configuration. In

the total Lagrangian formulation, all of the quantities are referred to a fixed (often the

undeformed) configuration and changes in the displacement and stresses fields are determined

with respect to the reference configuration. The strain and stress measures used in this approach

109

are the Green-Lagrange strain tensor and the 2nd Piola-Kirchhoff stress tensor.

In the updated Lagrangian formulation, the geometry of the structure from the previous

increment is updated using the deformation computed in the current increment, and the updated

configuration is used as the reference configuration for the next increment. The stresses and

strain measures used in this approach are the Cauchy stresses and Almansi infinitesimal strains.

9.1.2 Time approximation and Newton-Raphson method

The principal of virtual work is applied by means of the weak form of the equations of

motion for a composite plate with the first order shear deformation theory. These expressions are

not shown herein and are available in the literature [22]. They contain the first derivatives of the

dependent variables (displacements of the middle plane of the laminate and rotations about x and

y). Therefore, the dependent variables can be approximated by Lagrange interpolation functions.

Substituting those functions into the virtual work statement, the finite element equations of

motion can be obtained, which in compact form are given as:

[ ]{ } { }eee FK =Δ (9.1)

The discretized equations, for an instant of time 1+= Stt are

{ }( )[ ]{ } { } 1,11 +++ =ΔΔ SSSS FK (9.2)

Equation (9.2) represents a system of nonlinear algebraic equations that must be solved

by an iterative method. The method that has proven best for this class of problems is Newton-

Raphson, which is based on Taylor’s series [8]. The Newton-Raphson method considers the

Taylor’s series expansion of equation (9.2) about the known solution, { }SΔ . Suppose that

equation (9.2) is to be solved for the generalized displacement vector { } 1+Δ S at time 1+St . Since

110

the coefficient matrix { }( )[ ]1+Δ SK depends on the unknown solution, the equations are solved

iteratively. To formulate the equations to be solved at the 1+r st iteration by the Newton-

Raphson method, it is assumed that the solution at the r th iteration { }rS 1+Δ is known. Then

define

{ } { }( ) { }( )[ ]{ } { } 01,111 =−ΔΔ=Δ ++++ SSSSS FKR (9.3)

where { }R is called the residual, which is a nonlinear function of the unknown solution { } 1+Δ S .

Expanding { }R in a Taylor’s series about { }rS 1+Δ results in

{ } { }( ) { }( )[ ]{ } { }{ } { } { }( )+Δ−Δ⎥

⎦

⎤⎢⎣

⎡Δ∂

∂+Δ=Δ= +

++

++++

rS

rS

r

S

rSSS

RRKR 111

11110

{ }{ } { } { } { }( ) K+Δ−Δ⎥

⎦

⎤⎢⎣

⎡Δ∂Δ∂

∂+ +

++

+

2

111

1

2

!21 r

SrS

r

S

R

{ } { }( )[ ]{ } { }( ) 00 211 =Δ+ΔΔ+= ++ δδ OKR r

STr

S (9.4)

where ( ).O denotes higher order terms in { }Δδ and [ ]TK is known as the tangent stiffness

matrix (or geometric stiffness matrix):

{ }( )[ ] { }{ }

r

S

rS

T RK1

1+

+ ⎥⎦

⎤⎢⎣

⎡Δ∂

∂≡Δ

{ } { }( )[ ]{ } { } 1,111 ++++ −ΔΔ= SSrS

rS

rS FKR (9.5)

Equation (9.5) can be applied to a typical element. In other words, the coefficient matrix

{ }( )[ ]rSK 1+Δ can be assembled after the element tangent stiffness matrices and force residual

111

vectors are computed. The assembled equations are then solved for the incremental displacement

vector after imposing the boundary and initial conditions of the problem,

{ } { }( )[ ] { }rS

rS

T RK 1

1

1 +

−

+ −Δ−=Δδ (9.6)

The total displacement vector is obtained from

{ } { } { }Δ+Δ=Δ +++ δr

SrS 1

11 (9.7)

The element tangent stiffness matrix is evaluated using the latest known solution, while

the residual vector contains contributions from the latest known solution in computing element

matrix { }( )[ ]{ }rS

rSK 11 ++ ΔΔ and from the previous time step solution in computing element vector

{ } 1, +SSF . After assembly and imposition of the boundary conditions, the linearized system of

equations is solved for { }Δδ .

At the beginning of the iteration (i.e. 0=r ) it is assumed that { } 00 =Δ so that the

solution at the first iteration is the linear solution, because the nonlinear stiffness matrix reduces

to the linear one. The iteration process is continued until the difference between { }rS 1+Δ and

{ } 11

++Δ r

S reduces to a pre-selected error tolerance. The error criterion is of the form (the subscript

1+S in the quantities has been omitted):

ε<ΔΔ−Δ ∑∑=

+

=

+N

I

rI

N

I

rI

rI

1

21

1

21 (9.8)

In equation (9.8) N is the total number of nodal generalized displacements in the finite

element mesh, and ε is the error tolerance. The Newton-Raphson method fails to predict the

nonlinear equilibrium path through the limit points where the tangent matrix [ ]TK becomes

112

singular and the iteration process diverges. Riks suggested a procedure to predict the nonlinear

equilibrium path through limit points. Such a procedure provides the Newton-Raphson method

with a technique to control progress along the equilibrium path.

9.2 Effect of initial imperfections in the nonlinear solution

9.2.1 Effects of initial imperfections in the ANSYS model

As previously stated, the presence of initial geometric imperfections or load eccentricities can

greatly affect the response of the actual structure. The linear mode shapes represent alternative

possible configurations of equilibrium beyond the bifurcation point. Thus, it is realistic to

assume that the state of displacements that the structure will tend to adopt after buckling will

either resemble one of the linear mode shapes or a combination of the linear mode shapes.

Therefore, the mode shapes associated with the lowest buckling loads were used to define an

initial geometric imperfection. Since the linear mode shapes only describe aspect ratios but not

absolute displacements, the mode shapes have to be scaled to represent true imperfections. A

scaling factor was used to provide a maximum displacement of 1% to 5% of the laminate

thickness for any point of the structure.

ANSYS provides two ways of introducing initial imperfections: (1) to read-in a file with

prescribed initial displacements on the nodes of the model based on the mode shapes obtained

from the linear analysis and (2) to initially apply an out of plane perturbation force (or system of

forces) that slightly deforms the structure into a shape that resembles the desired mode shape.

The latter is the procedure followed herein for the ANSYS solution. To find the appropriate

magnitude of the perturbation force, a set of preliminary linear static analyses was completed

using the perturbation force as the only applied force, in order to tune its value to produce a

113

maximum displacement of 5% of the thickness of the laminate. The results are shown in Tables

9.1 and 9.2. Figures 9.1 and 9.2 show the modal shapes obtained using this method. Due to the

proximity of the first two buckling modes, there is a possibility of the structure transitioning

quickly into the second mode. Both initial imperfections (first and second mode) were introduced

separately into the nonlinear analysis for all of the laminates to explore possible differences; it

made no difference with regards to the occurrence of buckling and the developed postbuckling

configuration. Figure 9.3 shows the perturbation forces to induce the second mode.

Thickness: ( )int 124.0= , 5% of thickness: ( )inu 0062.0max =

Perturbation Force (lbs) Maximum displacement (in)

2.0 0.002587

4.0 0.005175

5.0 0.006469

Table 9.1 Estimation of the perturbation force. First mode. Laminate 0 degree.

Figure 9.1 Replication of the first mode with an out of plane perturbation force in the middle

node. Out of plane displacement plot. Laminate 0 degree.

114

Thickness of the laminate: ( )int 124.0= 5% of thickness ( )inu 0062.0max =

Perturbation Force (lbs) Maximum displacement (in)

5.0 0.002939

10.0 0.005877

11.0 0.006465

Table 9.2 Perturbation force estimation. Second Mode. Laminate 0 degree.

Figure 9.2 Replication of the second mode with a couple of perturbation forces, opposite in sign

on symmetric locations. Out of plane displacement plot. Lam 0 deg

The introduction of initial imperfections proved to be a main consideration in order to

achieve convergence for the nonlinear analysis. However, as shown in the results of Table 9.3,

once convergence is achieved the value of the perturbation force was found to have no influence

on the overall results of the analysis.

115

Figure 9.3 Perturbation with a couple of forces to induce the second mode

The findings of this research indicated that if difficulties are found with the convergence

of the non-linear solution then a preliminary exploration should be performed to identify values

of the perturbation force that produce a converged solution..

Perturbation Force (lbs)

Convergence Maximum displacement (lbs)

0.0 NO -----

1.0 NO -----

3.0 NO -----

5.0 YES 0.42366

10.0 YES 0.42505

15.0 YES 0.43360

Table 9.3 Effect of initial imperfections on overall results (laminate 0 deg)

116

It is also important to mention that the ANSYS nonlinear solution can be carried out with

a single load step with automatic sub-stepping, or with multiple load step files, where each file is

executed with automatic sub-stepping. The multiple file option was the one preferred in this

dissertation since it provided a better definition of the response. The introduction of the initial

perturbation force in the multiple file method should be implemented by applying the force in the

first step file and withdrawing it from the subsequent files.

9.2.2 Effects of initial imperfections in the ABAQUS model

For the ABAQUS model, the initial imperfections were introduced by importing a field

of nodal displacements corresponding to the first two mode shapes obtained from the linear

buckling solution. None of the ABAQUS nonlinear models showed convergence difficulties with

any of the imperfection levels introduced, a robust behavior that differed greatly form that

exhibited by the ANSYS solution. Figure 9.4 shows the effect of the magnitude of the initial

imperfection on the response of the ABAQUS nonlinear model.

The results shown in Figure 9.4 reveal that smaller imperfections were associated with

higher buckling loads with no substantial differences in the overall shape of the curve. The

curves plotted in Figure 9.4 are the strains in the compressed diagonal, measured on the outer

surface plies of the same finite element, located close to the center of the laminate. The

magnitude of the initial imperfection also affected the initial separation of the two curves in the

linear regime.

The magnitude of the imperfection was tuned by trial and error to achieve the best

correlation with the test results. As an initial value for this process, the initial imperfection value

117

provided by the intercept of the Spencer Walker plot was utilized. Table 9.4 shows the final

value of imperfection adopted to generate the results presented in the following paragraphs.

0

20

40

60

80

100

120

140

-12500 -10500 -8500 -6500 -4500 -2500 -500 1500εxx strain (microstrains)

Act

uato

r loa

d (k

N)

p1_2_microstrain front facep2_2_microstrain back faceABAQUS imperfection = 0.0025 (in) front faceABAQUS imperfection = 0.0025 (in) back faceABAQUS imperfection = 0.0020 (in) front faceABAQUS imperfection = 0.0020 (in) back faceABAQUS imperfection = 0.0030 (in) front faceABAQUS imperfection = 0.0030 (in) back face

Figure 9.4 Lam 20 deg ABAQUS strains in the first Gaussian integration point in front and back

layers of the laminate FE at the center of the panel

Laminate Spencer-Walker estimated (in)

Best correlation ABAQUS-Test (in)

20 degree -0.0147 -0.0030

45 degree 0.0001 0.0350

90 degree 0.7280 0.0065

0 degree 0.0019 0.0065

Table 9.4 Initial imperfections: estimated vs. adopted for FE correlation

118

9.3 Effect of the sub-stepping in the solution

In the case of the ANSYS nonlinear model, the multiple load step file solution provided

more output points and therefore, better definition of the response in the areas of the nonlinear

curve where the variables change more rapidly. The linear eigenvalue analysis revealed that the

first two modes are very close for the panels under study. It was observed that in the ANSYS

model, limitations in the maximum number of sub-steps, which is an adjustable parameter in the

inputs, could arbitrarily trigger a dominance of either the first or second mode. This type of

variation was never found in the ABAQUS models. Therefore, a previous exploration of ANSYS

solution for different values of this parameter was performed to assure the independence of the

results from the sub-stepping. In general, not limiting the number of sub-steps is recommended.

In the case of the ABAQUS nonlinear model, a robust algorithm automatically chooses the

number of sub-steps and no sensitivity to sub-stepping was noted in the solution.

9.4 Correlation of FE model strains with strain gage readings

The plots presented in this section compare the principal strains provided by the FE

models with the strain gage data of the four test panels. The correlation was investigated at the

central region of the panel where the strain gage rosettes were located back to back on the outer

surfaces of each panel. The location of the rosettes is described in Chapter 5. The normal arms of

the rosettes were oriented in the direction of the load and normal to it. The test principal strains

were calculated from the readings of the three arms of each rosette. The lay-up sequences of the

four test panels are described in Chapter 8.

The footprint of the strain gage arms stretched over an inch from the center of the panel.

Thus, the readings represent the integration or average of the strains at the underlying surface,

119

which were provided by the finely discretized ARAMIS readings. The strains in the direction of

the tension and compression diagonals were extracted from the FE model at the central region of

the panel. In the ANSYS model, the principal strains at the center node of the panel were

requested, and were calculated by the solver as an average of the strains in the four elements

sharing the central node. In the ABAQUS model however, the strains were obtained at the first

Gaussian integration point of the top and bottom plies, of the most representative element located

under the footprint of the strain gage. Figure 9.5 shows in blue the elements located under the

strain gage.

As an operational note, Table 9.5 shows the loading schedule for the solution in ANSYS

and the settings used (multiple load step files which have to be produced first using the command

“Write Load Step File”). The Load Step File contains the perturbation force and applied loads

with constraint and application points and that information does not need to be included in the

general model file. The ABAQUS analysis automates all of the load stepping and none of the file

write operations described for ANSYS are needed.

Table 9.6 shows the values of the buckling load for the different panels obtained from the

different methods used in this dissertation. Every value originated from a totally different source

and it can be observed that the results are in very good agreement.

120

Figure 9.5 Elements under the footprint of the strain gage. ABAQUS model mesh element size

0.25 in, compressed diagonal front face

Figure 9.6 Integration points in ABAQUS S4 element, 3 per layer (nonlinear finite strain

laminated shell)

121

Laminate Load Step No.

Perturbation Force (lbs)

Load at End of Load Step (lbs)

No. of Sub Steps

Max No. of Sub Steps

1 5.00 6,000 10 20

G 2 0.00 12,000 10 20

(20 deg) 3 0.00 14,000 10 20

4 0.00 15,000 10 20

5 0.00 16,000 10 20

6 0.00 18,000 10 20

7 0.00 20,000 10 20

8 0.00 25,000 10 20

1 2.00 3,000 50 100

H 2 0.00 4,000 50 100

(45 deg) 3 0.00 6,000 50 100

4 0.00 10,000 50 100

5 0.00 16,000 50 100

6 0.00 18,000 50 100

7 0.00 25,000 50 100

1 2.00 3,000 50 100

E 2 0.00 6,000 50 100

(90 deg) 3 0.00 10,000 50 100

4 0.00 14,000 50 100

5 0.00 17,000 50 100

6 0.00 20,000 50 100

1 10.00 5,000 50 100

F 2 0.00 8,000 50 100

(0 deg) 3 0.00 12,000 50 100

4 0.00 16,000 50 100

5 0.00 18,000 50 100

6 0.00 20,000 50 100

7 0.00 25,000 50 100

Table 9.5 Loading schedules in the ANSYS model

122

Laminate Analysis Buckling Load (kN)

Buckling Load (lbs)

Runtime (min)

G Linear Eigenvalue Analysis 72.66 16,336 0.01

(20 deg) Nonlinear Analysis ANSYS 63.10 14,186 25.55

Nonlinear Analysis ABAQUS 68.40 15,338 10.55

Experimental (Spencer-Walker plot) 72.80 16,367 Not applicable

H Linear Eigenvalue Analysis 45.13 10,146 0.01




E Linear Eigenvalue Analysis 20.48 4,604 0.01




F Linear Eigenvalue Analysis 97.14 21,839 0.01




Note: running times could be at times an arbitrary measure, since FE software licensed to a

network computer is affected not only by machine capacity but also network traffic.

Table 9.6 Buckling load. Comparison of results

123

9.4.1 Laminate G – 20 degree with the direction of the load

Figures 9.7 and 9.8 show the correlation of the ANSYS non-dimensional principal strains

on the front face with the test data (both tension and compression diagonals) as a function of the

non-dimensional load. Both plots show good correlation with the test results.

Figure 9.7 shows good agreement over the entire linear segment, which represents the in-

plane behavior, and also throughout the change in slope, which indicates the occurrence of

buckling at a load level of 14,186 lbs. The model displayed a higher stiffness than the test article

in the postbuckling regime. Figure 9.8 corresponds to the strain measured in the direction of the

load with good correlation over the whole range of data, but this figure does not clearly reveal

the occurrence of buckling.

Figure 9.9 corresponds to the ABAQUS normalized principal strains versus non-

dimensional load on the compressed diagonal, measured on both sides of the center of the panel,

back to back. The strains on the back face show good correlation even for the postbuckling

regime. This plot represents the transition from compression to tension, and the strain reaches the

positive domain with some minor disparity between analysis and strain gage results after the

buckling event. In general, the ABAQUS model more accurately simulated the response,

including the postbuckling range.

124

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-4.00 -3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00

Epsilon/Epsilon_cr

P/Pc

rit

Laminate 20 deg - TestLaminate 20 deg - ANSYS

Figure 9.7 Lam 20 deg ANSYS results. Normalized strains on compressed diagonal.

0

0.5

1

1.5

2

2.5

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Epsilon/Epsilon_crit

P/Pc

rit

Laminate 20 deg - TestLaminate 20 deg - ANSYS

Figure 9.8 Laminate 20 deg ANSYS results. Normalized strains on tension diagonal.

125

0.00

0.50

1.00

1.50

2.00

2.50

-5.50 -4.50 -3.50 -2.50 -1.50 -0.50 0.50

Non dimensional strain εxx/εcrit

Non

dim

ensio

nal a

ctua

tor l

oad

P/Pc

rit

Principal Compression Strain, Front Face

Principal Compression Strain, Back Face

ABAQUS (Imp = 0.003) Back Face Compressed Diagonal

ABAQUS (Imp = 0.003) Front Face Compressed Diagonal

Figure 9.9 Laminate 20 deg, ABAQUS results. Normalized strains on compressed diagonal, on

front and back faces.

Figure 9.10 shows the out of plane displacements measured on the front face of the

laminate at the end of the final load step. The correlation is good although both the ANSYS and

ABAQUS models show a mild out of plane displacement towards the corners of the laminate

that is not present in the ARAMIS plot. Both models correlated well in displacements, and for

the 20 degree laminate the rotation of the buckling nodal line is noticeable.

126

Figure 9.10 Lam. 20 deg. Out of plane displacements on front face. Top to down: ARAMIS

surface readings, ANSYS results, ABAQUS results (5 mm = 0.1969 in)

127

9.4.2 Laminate H – 45 degree with the direction of the load

In this case, the ANSYS model provided worse correlation with the test results than the

previous laminate, except over the linear regime as shown in Figures 9.11 and 9.12. The ANSYS

model displayed a higher stiffness in the postbuckling regime than the test article. Figure 9.13

shows the ABAQUS strain correlation, which in general is closer to the test than the ANSYS

solution. The ABAQUS model also showed a stiffer response that the actual panel. This model

exhibited higher sensitivity to the magnitude of the initial imperfection than the 20 degree

laminate.

Figure 9.14 shows the out of plane displacements for the front face of the laminate for all:

test panels, ANSYS and ABAQUS. The correlation is very good and the nodal line rotation is

easily observable.

0.00

1.00

2.00

3.00

-6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00

ε/εcrit

P/Pc

rit

Laminate 45 deg ExperimentalLaminate 45 deg ANSYS

Figure 9.11 Lam 20 deg ANSYS. Normalized strains on compressed diagonal.

128

0.00

1.00

2.00

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

ε/εcrit

P/Pc

rit

Laminate 45 deg ExperimentalLaminate 45 deg ANSYS

Figure 9.12 Lam 45 deg ANSYS. Normalized strains on tension diagonal.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0

Non-dimensional Exx strain (microstrains)

Non

-dim

ensio

nal a

ctua

tor l

oad

(kN

)

Principal compression strain, front facePrincipal compression strain, back faceABAQUS (Imp=0.035) front face compressed digonal ABAQUS (Imp=0.035) back face compressed digonal

Figure 9.13 Lam 45 deg, ABAQUS. Normalized strains critxx εε / on compressed diagonal on

front and back faces.

129

Figure 9.14 Lam. 45 deg. Out of plane displ. on front face. Top to bottom: ARAMIS surface

readings, ANSYS and ABAQUS results (7 mm = 0.2756 in)

130

9.4.3 Laminate E – 90 degree with the direction of the load

The ANSYS model in this case provided good correlation with the test results over the

postbuckling regime as shown in Figures 9.15 and 9.16, but large discrepancies occur around the

buckling event and the linear region.

Figure 9.17 shows the ABAQUS strain correlation, which in general is closer to the test

in the linear range. The ABAQUS model also shows a stiffer response than the actual panel.

Figure 9.18 shows a comparison of the out of plane displacements for the front face, where a

large semi-wave dominates the postbuckling configuration of the laminate.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-8.00 -7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00ε/εcrit

P/Pc

rit

Laminate 90 deg - Experimental

Laminate 90 deg - ANSYS


131

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 1.00 2.00 3.00 4.00 5.00 6.00

P/P

crit

Laminate 90 deg - ExperimentalLaminate 90 deg - ANSYS

ε/εcrit


0.00

1.00

2.00

3.00

4.00

-8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00

Normalized strain εxx/εcrit

Nor

mali

zed

actu

ator

load

P/P

crit

p2_2/epsilon_crit (Back Face)ABAQUS EEP1 Elem 1628 (imp = 0.0065) Front FaceABAQUS EEP1 Elem 1628 (imp = 0.0065) Back Facep1_2/epsilon_crit (front Face0

Figure 9.17 Lam 90 deg, ABAQUS. Normalized strains on compressed diagonal, on front and

back faces.

132

Figure 9.18 Laminate 90 degree. Out of plane displacement. Top to bottom: ARAMIS surface

readings, ANSYS results, ABAQUS results (12 mm = 0.4724 in)

133

9.4.4 Laminate F – 0 degree with the direction of the load

The ANSYS model provided good correlation with the test results over the tension

diagonal and worse correlation for the compression diagonal except for the linear regime as

shown in Figures 9.19 and 9.20. Figure 9.20 shows overall higher stiffness in the FEM model

with respect to the test panel in the post buckling regime. The reason for the disparity of the

ANSYS model is explained in the paragraph below Figure 9.22.

Figure 9.21 shows the ABAQUS strain correlation, which in general is closer to the test

than ANSYS. The ABAQUS model also exhibited a stiffer response than the actual test panel.

Figure 9.22 shows the out of plane displacement comparison for the front face of the laminate.

No ARAMIS snapshot was recorded during the test, and therefore was not available.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00Normalized strains (ε/εcr)

Non

dim

ensio

nal f

orce

at l

oad

cell

(P/P

cr)

Laminate 0 deg - Experiment

Laminte 0 deg - ANSYS


134

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

-8.00 -7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00Normalized strain (ε/ε crit)

Non

dim

ensio

nal f

orce

at l

oad

cell

(P/P

cr)

Laminate 0 deg - ExperimentLaminte 0 deg - ANSYS


-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00

Normalized strain (ε/εcr)

Non

dim

ensio

nal a

ctua

tor l

oad

(P/P

crit)

p1_2/epsilon_crit (front Face0p2_2/epsilon_crit (Back Face)ABAQUS EEP1 Elem 1628 (imp = 0.0065) Front FaceABAQUS EEP1 Elem 1628 (imp = 0.0065) Back Face

Figure 9.21 Laminate 0 deg, ABAQUS. Normalized strains on compressed diagonal, on front

and back faces.

135

Figure 9.22 Laminate 0 degree. Out of plane displacement. (a) Nonlinear Analysis ANSYS last

load step. (b) Nonlinear Analysis ABAQUS last load step

136

Figure 9.22 shows a discrepancy between the mode shape developed in the ANSYS

model and the ABAQUS model at the end of the postbuckling regime. The shape exhibited by

ANSYS corresponds to the first buckling mode, whereas the ABAQUS model developed the

second mode. The most likely cause for this disparity is the proximity between the first and

second linear buckling modes (see Chapter 8 linearized analysis results:

( )lbsModeFirst 838,21= , ( )lbsModeSecond 522,22= ). In the case of a structure with

proximity between the first two critical load levels, it can buckle with deflections arbitrarily

following either mode depending on small eccentricities or load fixture bending effects.

An ARAMIS picture of the deformed surface was not recorded during the test for this

panel. However, the better correlation displayed in Figure 9.21 by the ABAQUS model and the

rest of the ARAMIS data analyzed in Section 5.2.4 suggest that the ABAQUS model described

the buckling more realistically, and therefore the actual test panel most likely developed the

second mode shape.

137

CHAPTER TEN

10PROGRESSIVE FAILURE ANALYSIS

10.1 Failure prediction in the analysis of deeply postbuckled panels

With the incorporation of composite materials into commercial transport airplanes during

the last decade, civil certification requirements have significantly increased. Regarding the static

strength, certification requires showing that the structure does not present any mode of material

damage before reaching design limit load (either matrix or fiber damage, fiber-matrix de-

bonding, or delamination). Beyond limit load, no fiber damage should occur before the ultimate

load (1.5 times design limit load) since the structure has to be able to carry ultimate load for at

least three seconds and the load will be resisted mostly by the fibers. For damage tolerance the

certification requirement is satisfied by loading a deliberately previously damaged structure up to

limit load levels. The components should have undergone barely visible impact damage (BVID)

as well as embedded defects (ED) before static strength testing. The ED are intended to resemble

potential manufacturing defects.

To validate a design for those requirements, an analytical model and/or a test program are

typically carried out to establish a database of allowables. For components that are subjected to

buckling and postbuckling, the simulation requires the incorporation of progressive failure

analysis (PFA). In the same way that a classic thin web aluminum design relies on the elastic

limit of the aluminum to set the maximum level of buckling and diagonal tension allowed in the

structure, PFA is required to find the design point in the postbuckling regime beyond which the

composite shear web should not be loaded. Furthermore, the extension of PFA beyond failure

and its continuation until final collapse reveals areas where damage propagates and shows the

138

important regions of the structure which actually resist the load as the damage progresses. There

are three regions in which stress gradients will potentially initiate damage: on the nodal lines of

the buckling shape, on the crest of the buckling semi-waves, and close to the boundary of the

fixture. Damage over the nodal lines and over the crests of the semi-waves could be sustained by

the laminate for some time, with the continued ability to carry load effectively as long as the

undamaged areas would continue to resist the load working as a tension truss, and the section in

the direction perpendicular to the load can still carry some compression. However, failure over

the boundaries would lead directly to structural collapse, since that damage would detach the

panel from the picture frame and the boundary condition could no longer be imposed, causing

the panel to deform into a collapsed configuration.

As previously mentioned, the buckling and postbuckling FE models with PFA have been

approached incrementally (with increasing levels of complexity) in this dissertation. This

approach is not only advised in the literature [8], it is also required since having a validated

nonlinear model is considered to be a prerequisite to performing PFA. As shown in Table 1.1,

the models in this chapter differ from those in Chapter 9 due to the addition of the PFA.

10.1.1 Material nonlinearities

The response of the undamaged material is assumed to be linear elastic, and the model is

valid with composites for which damage initiates without a large amount of plastic deformation.

Nonlinearities in the material before failure are therefore not considered, but the level of

correlation between the FE model and test data shown in Chapter 9 proves that pre-failure

material nonlinearities can be ignored. The Hashin's initiation criteria are used to predict the

onset of damage, and the damage evolution law used herein is based on the energy dissipated

139

during the damage process and linear material softening. The post- failure effects are included in

an ABAQUS Continuum Damage Mechanics Model (CDM) as explained in the following

section. Nonlinearities that occur just before failure are not characterized in this model.

10.1.2 Delamination growth and decohesion elements

Delamination is a common cause of failure due to an impact, a manufacturing defect, or

stress gradients near geometric discontinuities such as ply drop-offs, stiffener terminations and

flanges, bonded and bolted joints, and access holes. The panels tested and modeled in this

research did not exhibit any of these geometric features that would require modeling for

decohesive interface failure. Also, no internal damage or manufacturing defect was assumed. To

model the growth of such damage, the use of Decohesion Elements [10] is the best technique. It

was never the intent of this research to model delamination; therefore, is not analyzed herein as a

separate class of failure and will not be distinguished from a generic matrix failure.

10.2 Overview of the progressive failure analysis (PFA)

As described in the literature survey of Chapter 3, during the last two decades the

progressive failure analysis of composites has been approached utilizing either the Chang-Chang

or Tsai-Wu failure criteria. Once failure is detected in a ply of an element, the relevant elastic

properties are reduced to zero instantaneously, or over a fixed number of steps. This approach

can be described as the equivalent, at the element level, of the commonly known “ply discount

method” for the classical strength analysis of an entire laminate. These types of models were

referred to in Chapter 3 as IRSP (instantaneous reduction of stiffness properties) and GRSP

(gradual reduction of stiffness properties). Such approaches provide some insight on how the

strength degradation evolves, but are unrealistic since post-failure behavior is disregarded.

140

In order to model damage evolution more realistically, an innovative composite damage

approach [27, 29] was utilized; this approach relies on using a continuum constitutive model

which can feature internal variables that characterize the damage. These types of models are

called Continuum Damage Mechanics (CDM) models [27].

The CDM model used in this dissertation corresponds to the one proposed by

Matzenmiller et al. [29] and it is supported by ABAQUS. The FE implementation of this model

uses a strain softening material law sized by the volumetric energy associated with a failure

mode (area under the stress-stain curve). Since the FE model evaluates the volumetric energy

over an element or group of elements, the results are mesh-sensitive. To alleviate this mesh

dependency, the formulation used herein introduces a characteristic element length into the

material softening law.

In this dissertation, the NL-PFA-Model incorporated Hashin’s failure criteria with a

CDM post-failure constitutive model to simulate the evolution of damage. The parameters

required by the model for this purpose have a clear physical meaning, and can be determined by

standard tests. For this dissertation work, the values used were found in the literature. All PFA

analyses are performed using ABAQUS version 6.6 with the functions *DAMAGE

EVOLUTION, *DAMAGE INITATION, and *DAMAGE STABILIZATION.

10.3 Failure Criteria. Damage activation functions

Damage initiation refers to the onset of degradation at a material point. The damage

initiation criteria used in this dissertation was based on Hashin's theory [6]. These criteria

consider four failure modes for the fiber and matrix constituents. Namely, the different damage

initiation mechanisms considered are: fiber tension, fiber compression, matrix tension, and

141

matrix compression. Denoting by ijσ the components of the local stress tensor in the ply material

axes, the initiation criteria have the general forms shown in Table 10.2. Failure occurs if any of

the indices ije exceed unity.

Failure Mode Range of

Application

Failure Index

Fiber tension ( )011 ≥σ 212

211 ⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= LT

tf SX

eσ

ασ

Fiber compression ( )011 ≤σ 211 ⎟⎠⎞

⎜⎝⎛= C

cf X

e σ

Matrix tension ( )022 ≥σ 212

222 ⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= LT

tm SY

e σσ

Matrix compression ( )022 ≤σ 21222

2222 1

22⎟⎠⎞

⎜⎝⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛= LCT

C

Tcm SYS

YS

eσσσ

Symbol Description

TX Tensile strength in the fiber direction

CX Compressive strength in the fiber direction

TY Tensile strength in the direction perpendicular to the fibers

CY Compressive strength in the direction perpendicular to the fibers

LS Longitudinal shear strength

TS Transverse shear strength

α Coefficient determining the contribution of the shear stress to the fiber tensile initiation criterion

Table 10.2 Hashin’s failure criteria expressions

142

Various failure criteria developed subsequent to the World Wide Failure Exercise

(WWFE) of 1998 [10], incorporate the capability to detect the angle of fracture for the matrix

compression failure. Criteria such as Puck and Schurman [7] and the LaRC series (LaRC04

being the latest [10]) provide such an angle. However, in this dissertation it was considered that

such information did not provide any additional insight or value for the analysis of composite

panels in post buckling. Hashin’s criteria do not provide any information regarding the angle of

fracture. The adequacy of the PFA approach used herein, CDM with Hashin’s initiation

functions, was assessed in light of the comparison of results with test data and with the nonlinear

model without PFA. Results obtained using the current methodology are presented and discussed

later in this chapter.

As a general observation, all of the failure criteria and degradation models result in an

approximation of the damage progression. The WWFE results do indicate that Puck’s method

produces better results than most, particularly for cases involving combined compression

(perpendicular to the fibers) and shear. However, most plate elements provide a poor estimate of

the interlaminar and intralaminar stresses, which makes the 3D Puck criteria impossible to apply.

The initiation criteria presented above can be specialized to obtain the model proposed in

Hashin and Rotem (1973) by setting 0.0=α and 2CT YS = , or the model proposed in Hashin

(1980) by setting 0.1=α . For the models considered for this dissertation, the Hashin option was

used. Table 10.3 shows the different strength values (obtained from the literature [4]) that were

used in the criteria functions.

143

Symbol Description Unidirectional carbon

tape 15 oz/yrd 2

Biaxial glass fabric 12

oz/yrd 2

TX Tensile strength in the fiber

direction

229,924 16,297

CX Compressive strength in the

fiber direction

136,682 16,297

TY Tensile strength in the

direction perpendicular to the

fibers

4,960 16,297

CY Compressive strength in the

direction perpendicular to the

fibers

14,881 16,297

LS Longitudinal shear strength 10,678 17,780

TS Transverse shear strength 1,000,000 10,000

Table 10.3 – Strength material properties used in Hashin’s functions [4]

144

10.4 Material degradation model in the FE simulation

10.4.1 Damage initiation

Damage initiation refers to the onset of degradation at a material point. In ABAQUS the

damage initiation criteria for composites are based on Hashin’s criteria which were introduced in

the previous section. For every step and sub-step (time increment) of the nonlinear FE solution,

the components 122211 ˆ,ˆ,ˆ σσσ of the effective stress tensorσ̂ at every material point are calculated

and used to re-evaluate the initiation criteria. The effective stress tensor is assumed to be the

stress acting over the area of a section that still remains undamaged, and it is computed from the

relationship:

σσ M=ˆ (10.1)

where σ is the nominal or applied stress over the entire section area, including the damaged and

undamaged portions. The matrix M is the damage operator [21,29]

( )

( )

( )⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

=

s

m

f

d

d

d

M

1100

01

10

001

1

(10.2)

and mf dd , and sd are internal (damage) variables that characterize fiber, matrix, and shear

damage. These damage variables are calculated at each integration point for all plies of each

laminated element and are derived from the damage variables tm

cf

tf ddd ,, and c

md , corresponding

to the four modes discussed in the previous section, according to expressions (10.3).

145

( ) ( ) ( ) ( )cm

tm

cf

tfs

cm

tm

m

cf

tf

f

ddddd

ifd

ifdd

ifd

ifdd

−−−−−=

⎪⎩

⎪⎨⎧

<

≥=

⎪⎩

⎪⎨⎧

<

≥=

11111

0ˆ0ˆ

0ˆ

0ˆ

22

22

11

11

σ

σ

σ

σ

(10.3)

Prior to any damage initiation and evolution, the damage operator M , is equal to the

identity matrix, so σσ =ˆ . Once damage initiation and evolution have occurred for at least one

mode, the damage operator becomes significant in the criteria for damage initiation of other

modes. The effective stressσ̂ , is intended to represent the stress acting over the damaged area

that effectively resists the internal forces.

An output variable is associated with each initiation criterion index (fiber tension, fiber

compression, matrix tension and matrix compression) to indicate whether the criterion has been

met. Each of the failure indices are calculated at the integration point of each ply of each

composite plate element. A value of the index of 1.0 or higher indicates that the initiation

criterion has been met. If a damage initiation model is defined without associating an evolution

law, as was done herein for the third model built, the initiation criteria will affect only the

outputs. Therefore, these criteria can be used to evaluate the propensity of the panels to undergo

damage without modeling the progressive damage process. The NL-FA-Model, discussed in

Chapter 9, was developed in this fashion. The sole purpose of the NL-FA-Model was to compare

Hashin’s criteria with classic one-equation criteria such as Tsai-Wu and Tsai-Hill. The current

model (referred to as NL-PFA-Model) includes both the initiation criteria and the material

degradation algorithm to obtain the full PFA.

10.4.2 Evolution of the damage variables for each mode

146

This section presents details regarding the post-damage constitutive model (see Eqs. 10.2

and 10.3). The current PFA algorithm considers the post-damage behavior as a material

constitutive model with strain softening. Prior to damage initiation the material is linearly elastic

with the stiffness matrix of a plane stress orthotropic material. Thereafter, the response of the

material is computed from the constitutive equations of Matzenmiller et al. [29]:

εσ dC= (10.4)

where ε is the strain and dC is the damaged elasticity matrix, which has the form [21,29]

( ) ( ) ( )( ) ( ) ( )

( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−−

−−−

=

GDd

EdEdd

EddEd

DC

s

mmf

mff

d

100

0111

01111

2212

1211

υ

υ

(10.5)

where ( ) ( ) 2112111 υυmf ddD −−−=

The parameter fd reflects the current state of fiber damage, md reflects the current state

of matrix damage, and sd reflects the current state of shear damage. The stiffness 1E is the

Young's modulus in the fiber direction, 2E is the Young's modulus in the matrix direction, G is

the shear modulus, and 12υ and 21υ are Poisson's ratios. The damage variables fd , md , and sd

are derived from damage variables cm

tm

cf

tf dandddd ,, corresponding to the four damage modes

using the expressions (10.3).

In order to simulate the post failure evolution in the most realistic fashion, a linear strain

softening model was used. Some strain softening models that have been proposed in the

literature are compared in Figure 10.1. This family includes linear elastic-perfectly plastic; linear

147

elastic-linear softening; linear elastic-progressive softening; linear elastic-regressive softening,

and Needleman [30]. The main characteristic of all of the softening models is that the material

continues to transfer load after the onset of damage ( 0ε in Figure 10.1). For loading

corresponding to each failure mode (fiber tension and compression, matrix tension and

compression) after the constituents have reached their respective strengths dictated by the

Hashin’s failure functions, the stiffnesses are gradually reduced to zero following the softening

law. The area under the stress-strain curve is the respective fracture energy associated with that

failure mode. The values of CG for the different failure modes can be obtained from simple

experiments or from existing data in the literature.

Figure 10.1 Different strain softening models. The suffix of each intersection with the

horizontal axis labels each curve

To alleviate mesh dependency during the material softening, a characteristic element

length is introduced into the formulation; the resulting strain softening constitutive law is

expressed as a stress-displacement relation. The damage variable will evolve such that the stress-

148

displacement behaves linearly, as shown in Figure 10.2, in each of the four failure modes. The

positive slope of the stress-displacement curve prior to damage initiation corresponds to linear

elastic material; the negative slope after damage initiation is achieved by evolution of the

respective damage variables following the equations of Table 10.4.

Figure 10.2 Bilinear strain softening law in terms of equivalent magnitudes

Equivalent displacement and stress for each of the four damage modes are defined as

149

follows:

Failure Mode Range of

Application

Equivalent displacement and equivalent

stress

Fiber tension ( )0ˆ11 ≥σ ( )212

211 εαεδ += Cft

eq L

Cfteq

fteq Lδ

εταεσσ 12121111 +

=

Fiber compression ( )0ˆ11 <σ 11εδ −= Cfceq L

Cfceq

fceq Lδ

σεσσ 111111 −−−−

=

Matrix tension ( )0ˆ 22 ≥σ ( )212

222 εεδ += Cmt

eq L

Cmteq

mteq Lδ

ετεσσ 12122222 +

=

Matrix compression ( )0ˆ 22 ≤σ ( )212

222 εεδ +−= Cmc

eq L

Cmceq

mceq Lδ

ετεσσ 12122222 +−

=

Table 10.4 Equivalent displacement and equivalent stress for each failure mode

For elements with a plane stress formulation the characteristic length CL is computed as

the square root of the integration point area. The symbol in the equations above represents

the Mc-Cauley bracket operator, which is defined for every ℜ∈x as

( )2

xxx

+= or { }xx ,0max=

After damage initiation (i.e., 0eqeq δδ ≥ ) the damage variable for a particular mode is given

150

by

( )( )0

0

eqf

eqeq

eqeqf

eqdδδδδδδ

−

−= (10.6)

where 0eqδ is the initial equivalent displacement at which the initiation criterion for that mode

was met and feqδ is the displacement at which the material is completely damaged in this failure

mode. The above relation is presented graphically in Figure 10.3.

Figure 10.3 Damage variable as a function of equivalent displacement.

The values of 0eqδ for the various modes depend on the elastic stiffness and the strength

parameters specified as part of the damage initiation definition. Unloading from a partially

damaged state, such as point B in Figure 10.2, occurs along a linear path toward the origin in the

plot of equivalent stress vs. equivalent displacement; this same path is followed back to point B

upon reloading.

151

The energy dissipated due to failure, CG , which corresponds to the area of the triangle

OAC in Figure 10.2, must be specified for each failure mode. The values of feqδ for the various

modes depend on the respective CG values. Since the composite laminates under study are

hybrids of Glass/Epoxy with Carbon/Epoxy, the NL-PFA-Model needs different values of CG

for each material as well as for each failure mode. Table 10.3 shows the different values used,

which were found in the literature [31]. Some of the values for Glass/Epoxy were not found in

the literature, and were estimated from the values known for the carbon/epoxy material system,

in proportion to the relation between the stiffness moduli of the carbon/epoxy system to the

glass/epoxy system.

Energy Released (lbs/in)

G1+

G1-

G2+

G2-

G6

Failure Mode Fiber Fracture Tension

Fiber Fracture Compression

Matrix Fracture Tension

Matrix Fracture

Compression

Shear Fracture

T300/934 512.929 446.922 1.313 4.339 2.626

Glass/Epoxy 743.747 648.0365 1.904 6.292 3.809

Table 10.5 Energy Released constants [31]

10.4.3 Maximum degradation and element removal

ABAQUS provides control on how elements with severe damage are treated. By default,

the upper bound to all damage variables at a material point is 0.1max =d . This upper bound can

be reduced; nevertheless in the NL-PFA-Model the default option was used. By default, an

152

element is removed (deleted) once damage variables for all failure modes at all material points

reach maxd . For a removed element the output variable STATUS is set to zero for the element,

and no resistance occurs due to subsequent deformation. However, the element still remains in

the ABAQUS model and is visible in the post-processing visualization module.

10.4.4 Viscous regularization algorithm

The PFA process of the NL-PFA-Model evaluated the stresses at the end of each load step of the

nonlinear static solution, over each integration point within each ply of each element, and

modified the material properties in the damaged plies. When that loop was completed at the end

of the load step, the stresses were necessarily different due to changes in material properties and

element status, and therefore needed to be redistributed. A new equilibrium nonlinear solution

was obtained right after the PFA algorithm before moving on to the next load step. Material

models exhibiting softening behavior and stiffness degradation often lead to severe convergence

difficulties in implicit FE solutions, such as ABAQUS/Standard used herein. To overcome some

of these convergence difficulties, the viscous regularization scheme, available in

ABAQUS/Standard, was activated. This technique causes the tangent stiffness matrix of the

softening material to be positive for sufficiently small time increments. In this regularization

scheme a viscous damage variable is defined by the evolution equation (10.7):

( )vv ddd −=η1& (10.7)

with ( )vv ddtdd =&

where t represents time, η is the viscosity coefficient representing the relaxation time of the

153

viscous system, and d is the damage variable evaluated in the inviscid model before

regularization. The damaged response of the viscous material is given as:

εσ dC= (10.8)

In expression (10.8) the damaged elasticity matrix, dC , is computed using viscous values

of damage variables for each failure mode. Using viscous regularization with a small value of the

viscosity parameter, small compared to the characteristic time increment, usually helps improve

the rate of convergence of the model in the softening regime, without compromising results. The

basic idea is that the solution of the viscous system relaxes to that of the inviscid case as

∞→ηt , where t represents time.

Different values of viscous coefficients can be specified for the different failure modes

commonly referred to with the notation mcmtfcft ηηηη ,,, for viscosity coefficients for fiber

tension, fiber compression, matrix tension, and matrix compression failure modes respectively.

Alternatively, the algorithm allows for defining just one viscous coefficient to be used for all

failure modes. In the NL-PFA-Model built herein, just one viscous coefficient was defined.

The NL-PFA-Model was solved several times with different values of the viscous

coefficient to explore the effect of the viscous regularization on the overall solution. Figure 10.4

shows the strains calculated at the center of the 20 degree panel, on the front and back faces, for

an initial out of plane displacement imperfection of 0.003 in. at the middle point for different

viscous coefficients. The results of this experiment show that for higher viscosities the solution

resembles that of the nonlinear analysis without progressive failure.

Comparing the PFA solution with the experimental results of the test it can be noted that

154

the PFA solution does a better job at predicting the panel behavior in the linear regime and even

more accurately predicts the occurrence of buckling. However, it exhibits lower stiffness than the

tested panels after the buckling event, thus providing a conservative solution. For values of

viscosity higher than 100, the response of the model remained unchanged and became insensitive

to further increases. For that reason, a value of 100 for the viscous coefficient was finally

adopted.

0

20

40

60

80

100

120

140

-15000 -13000 -11000 -9000 -7000 -5000 -3000 -1000 1000 3000

εxx strain (microstrains)

Act

uat

or lo

ad (

kN)

Strain Gage Front FaceStrain Gage Back FaceNonlinear no PFA (Imp = 0.003) front faceNonlinear no PFA (Imp = 0.003) back faceeta = 0.5 front faceeta = 0.5 back faceeta = 1.5 front faceeta = 1.5 back faceeta = 10.0 front faceeta = 10.0 back faceeta = 100.0 front faceeta = 100.0 back face

Figure 10.4 Effect of the viscous regularization coefficient in the nonlinear solution. Strains

on compressed diagonal at the center, front and back faces (20 degree panel)

155

10.5 Results of the nonlinear model with progressive failure analysis

The PFA scheme was introduced in the nonlinear model through the functions

*DAMAGE EVOLUTION, *DAMAGE INITATION, and *DAMAGE STABILIZATION. At

every integration point (one per ply, see Figure 9.2) the four damage indices defined in Hashin’s

functions are calculated and tracked as a history output. The model provides contour plots for the

four damage indices as shown in Figures 10.5 through 10.24. The output variables represented

are listed in Table 10.6. As mentioned above, only the 20 degree laminate model was expanded

with the PFA capability.

Variable name Definition

HSNFTCRT Hashin’s tensile fiber initiation criteria at integration points

HSNFCCRT Hashin’s compressive fiber initiation criteria at integration points

HSNMTCRT Hashin’s tensile matrix initiation criteria at integration points

HSNMCCRT Hashin’s compressive matrix initiation criteria at integration points

Table 10.6 Output damage indices [21]

The nonlinear analysis with PFA simulated a ramp loading divided into 16 steps to help

convergence and make sure that enough intermediate points were recorded. The loading schedule

is indicated in Table 10.7.

156

Step Incremental Load (lbs) Step Incremental

Load (lbs) Step Incremental Load (lbs) Step

Incremental Load (lbs)

1 1000 5 1000 9 1000 13 1000 2 1000 6 1000 10 1000 14 2000 3 4000 7 1000 11 1000 15 7000 4 1000 8 1000 12 1000 16 3000

Table 10.7 Loading schedule, nonlinear analysis with PFA

10.5.1 Fiber tensile failure evolution

Figure 10.5 shows the first fiber tensile (FT) damage, which took place during step 9,

with the FT index reaching value of 1. An inspection of the failure index over all plies of the

laminate revealed that plies 1 and 8 are the most severely affected by this failure mode.

However, the elements affected were located under the picture frame fixture and do not represent

a relevant failure occurrence conducive to final collapse. The first failure worth consideration is

shown in Figure 10.8, during step 15, with ( )lbsP 04.465,27= . Failure at that load level was

worth considering due to the area of damage located on the right hand side of the lower corner. It

can be observed that the panel’s inability to carry any load in the damaged area generates a shift

in the load path to the right side, which results in further severe damage developing on the right

crest of the mode shape.

157

Figure 10.5 Fiber tensile failure index. First occurrence of damage, load step 9. Actuator

load ( )lbsP 799,13=

Figure 10.6 Fiber tensile failure index, load step 12, ( )lbsP 163,17=

158



159


10.5.2 Fiber compressive failure evolution

Figure 10.10 shows the fiber compressive (FC) damage at the end of load step 14. An

inspection of the index failure over all plies of the laminate revealed that plies 1 and 8 were the

most severely affected by this failure mode. The elements affected were located under the picture

frame fixture, and the failure index was still below one and remained so until load step 16.

Failure was not declared until reaching a load level of ( )lbsP 58.433,30= . It can be inferred

from the plots that FC was not a dominant type of failure and it did not drive the final structural

collapse.

160

Figure 10.10 Fiber compressive failure index, load step 14 ( )lbsP 325,20=

Figure 10.11 Fiber compressive failure index, load step 15, ( )lbsP 465,27=

161

Figure 10.12 Fiber compressive failure index, load step 16, ( )lbsP 433,30=

10.5.3 Matrix tensile failure evolution

Figure 10.13 shows the first matrix tensile (MT) damage, which occurred during step 7.

An inspection of the failure index for all plies of the laminate revealed that plies 1 and 8 were the

most severely affected by this failure mode. However, the elements affected were located under

the picture frame fixture, and did not represent a relevant onset of damage. The first failure worth

consideration is shown in Figure 10.17 during step 15, with a load level of ( )lbsP 04.465,27= .

162

Figure 10.13 Matrix tensile failure. First occurrence of damage, load step 7. Actuator load

( )lbsP 497,11=

Figure 10.14 Matrix tensile failure, load step 12. ( )lbsP 163,17=

163

Figure 10.15 Matrix tensile failure, load step 14. ( )lbsP 325,20=

Figure 10.16 Matrix tensile failure. End of step 15. ( )lbsP 465,27=

164

Figure 10.17 Matrix tensile failure. End of step 16. ( )lbsP 434,30=

10.5.4 Matrix compressive failure evolution

Figure 10.18 shows the first matrix compressive (MC) damage, which occurred during

load step 9. The elements affected were located under the picture frame fixture, and did not

represent relevant damage. The first failure worth consideration is shown in Figure 10.20 at the

end of step 14 with ( )lbsP 60.324,20= . This was the most massive type of failure and the

analysis showed two bands of severe damage, formed in areas of the buckled shape under high

stress gradients. This mode of failure most likely drove the final collapse of the structure. Since

MC failure was driving the collapse of the panel, failure continued to develop ply-by-ply as

shown in Figures 10.23 and 10.24. These reflect the MC failure index at load step 14 at the

middle integration point of each layer. Damage is present to a considerable extent in layer 4,

165

which is near the middle surface of the 8 layer laminate with layers1 and 8 on the front and back

faces respectively.

Figure 10.18 Matrix compressive failure, load step 9 ( )lbsP 799,13=


166



167


Figure 10.23 Matrix compr. failure, load step 14. Failure index at plies 1, 2, 3 & 4

168

Figure 10.24 Matrix compr failure load step 14. Failure index at plies 5, 6, 7 & 8

169

170

10.6 Summary of results. Nonlinear model with progressive failure analysis

As mentioned above, only the model for the 20 degree laminate was solved with the

extended PFA capability. For each failure mode, a threshold load level was found in which

severe onset of damage was detected as shown in Table 10.8. A structural design seeking to

preserve the integrity of the panels can only develop postbuckling stresses at load levels

lower than the lowest of the damage levels found for each failure mode.

Load level at first damage Output Variable Failure mode

lbs kN

HSNFTCRT Fiber tension 465,27 17.122

HSNFCCRT Fiber compression 434,30 37.135

HSNMTCRT Matrix tension 465,27 17.122

HSNMCCRT Matrix compression 325,20 41.90

Table 10.8 Load levels at occurrence of first damage for each failure mode

Figure 10.25 shows the comparison of strains at the middle point of the laminate, on

the compressed diagonal, for readings on the front and back faces. The load levels for the

different types of damage are represented as horizontal lines. It can be concluded that the

damage level that limits the postbuckling load carrying capability is the level corresponding

to matrix compressive failure with the values shown in Table 10.8. Table 10.9 shows the

final summary of buckling loads, and Table 10.10 shows the final design load carrying

capability.

05

101520253035404550556065707580859095

100105110115120125130135140

-13000 -11000 -9000 -7000 -5000 -3000 -1000 1000 3000

εxx strain (microstrains)

Act

uat

or lo

ad (

kN)

Strain Gage Front FaceStrain Gage Back FaceNonlinear no PFA (Imp = 0.003) front faceNonlinear no PFA (Imp = 0.003) back faceNonlinear with PFA (eta = 100.0, imp=0.003) front faceNonlinear with PFA (eta = 100.0, imp=0.003) back faceFiber tension failure levelFiber compression faliure levelMatrix tension failure levelMatrix compression failure level

Figure 10.25 Strain on compression diagonal vs. actuator load with PFA results.

171

172

Analysis Buckling Load (kN) Runtime (min)

Linear Eigenvalue Analysis 72.66 0.01

Nonlinear Analysis ANSYS 63.10 25.55

Nonlinear Analysis ABAQUS 68.40 10.55

Experimental 72.80 Not applicable

Nonlinear Analysis ABAQUS with Progressive Failure

67.50 105.50

Table 10.9 Laminate 20 deg .Summary of buckling loads

Analysis Limiting Load to avoid laminate damage(kN)

Nonlinear Analysis ABAQUS with Progressive Failure

90.41

Table 10.10 Laminate 20 deg . Final design load carrying capability

10.7 Comparison of the Hashin’s failure functions with other failure criteria

In order to compare the Hashin’s functions with the most commonly used one-

equation failure criteria, a separate ABAQUS model was built, the NL-FA-Model, in which

damage evolution is ignored and only the failure criteria are considered. In this model,

several failure criteria were evaluated, at each load level, element, and ply. Within each ply,

the S4 element under a Simpson’s rule formulation uses three integration points (top of layer,

middle plane, and bottom of layer). The failure criteria evaluated are listed in Table 10.11.

173

Failure criterion Mode Based

Hashin Fiber Tension Fiber Compression Matrix Tension Matrix Compression

YES

Tsai Wu NO

Tsai Hill NO

Maximum Stress NO

Maximum Strain NO

Table 10.11 Failure criteria compared in the NL-FA-Model.

The NL-FA-Model was solved only for the 20 degree laminate. First failure was

consistently detected at load step 14 ( ( )lbsP 60.324,20= ) with the NL-PFA-Model. For that

load step ply number 8 located on the front face was the most severely damaged as shown in

Figures 10.26 through 10.28. Any failure evaluation subsequent to load step 14 is not

realistic since this model does not consider damage evolution, but simply the state of stresses

at the load step and the resulting failure index calculated from the stress values. Therefore,

the failure indices can take values larger than one. In all of the contour plots, the value of

unity is displayed in red color to denote failure. Values higher than one are plotted in grey.

Hashin’s criteria proved to be the most conservative, flagging failure before all of the non

mode based criteria, with the matrix compression failure as the most critical.

Figure 10.26 Hashin’s failure indices, load step 14, layer 8 (front face) ( )lbsP 325,20= . Left to right and top to bottom: fiber tension, matrix tension, fiber compression, matrix compression.

174

Figure 10.27 All non mode based failure indices, load step 14, layer 8 (front face) ( )lbsP 325,20= . Left to right and top to bottom: Tsai Wu, Tsai Hill, Maximum Stress, Maximum Strain.

175

Figure 10.28 All plies Tsai Hill failure index, load step 14, layer 8 ( )lbsP 325,20=

176

177

CHAPTER ELEVEN

11CONCLUSIONS AND RECOMENDATIONS

This dissertation presented a new methodology to support the design of composite shear

webs by expanding the working range of the structure well into the postbuckling regime. This

objective was accomplished by using the numerical simulation capabilities of nonlinear finite

element analysis combined with material failure criteria and a damage evolution model. The

method was validated by correlation of strain and displacements results with data from

laboratory experiments carried out on four composite panels built with hybrid carbon-

epoxy/glass-epoxy lay-ups and different orientations.

A literature review revealed the absence of updated progressive failure analyses (PFA)

for the type of structure and loading under consideration (composite hybrid flat panels under in-

plane shear) and also showed how composite failure detection and the simulation of damage

evolution are an active field of research. This review guided the study towards choosing a mode

based failure theory together with a material degradation model formulated using continuum

damage mechanics (CDM).

A review of the solutions provided by classic analytical methods to predict the onset of

instability and the value of the buckling load was also carried out. Important limitations were

found with regards to the practical applicability of classic analytical solutions. For the kind of

structure, loading and boundary conditions of this study, the analytical solutions found were only

applicable for the 45 degree laminate. Furthermore, the case under study herein does not accept

an exact analytical solution but rather an approximation using Raleigh-Ritz technique [22]. There

is a method for obtaining an exact solution but published results are rare. The Rayleigh-Ritz

178

analytical buckling load was somewhat higher than the lowest buckling load found through the

linearized FE analysis and close to the buckling load of the second FE mode. The specially

orthotropic analytical solution should be viewed as a very approximate solution for the lowest

buckling load of the structure.

The aforementioned set of composite laminated panels were tested in a picture frame

shear loading fixture. The test panels differed in the orientation of the laminate with respect to

the direction of the load. Four orientations were considered: 0, 20, 45, and 90 degrees. The test

results were recorded using different means, which include strain gages as well as a photo-

sensitive digital image system (ARAMIS). The strain gage data (due to the size of the gages)

represented more of an average; whereas, the ARAMIS data represents detailed localized strains

(and displacements). The data collected describing the postbuckling response of the test panels

was compared between the different data sources in order to access the consistency of the results.

In general both set of data were found to be in agreement.

Mathematical treatments of the test data, such as the Southwell’s method and its Spencer-

Walker extension to nonlinear plates [25,32], were applied to extract the experimental values of

the critical buckling loads for each specimen. This method proved to be a robust way to extract

the critical buckling loads from the experimental data. Good consistency was found between the

values obtained using Spencer-Walker and other graphical methods to determine the

experimental buckling loads. The recommendation from this research is to rely on the Spencer-

Walker method for the experimental determination of the buckling load.

Regarding the numerical analysis methodology, it was approached with increasing levels

of complexity. First, a linearized eigenvalue solution was obtained (using the ANSYS v.10

179

commercial package) to provide a first approximation of the values of critical buckling loads.

The critical loads obtained with the linearized analysis were compared with the experimental

buckling loads from the Spencer-Walker method, and good correlation was obtained.

Next, nonlinear FE models (without failure) were built for the four different panels, to

fully simulate the postbuckling response and correlate it with the experimental results. Two sets

of models were developed: one set with ABAQUS v.6.6 and one set with ANSYS v.10.

As reported in the literature, the presence of initial imperfections was found to have a

significant impact. Two different methods of including the initial imperfection in the FE model

were explored. Both methods produced the same results. In ANSYS the imperfection was

introduced by means of an out of plane perturbation force. The appropriate magnitude of such

force was calculated from a series of FE static solutions in which the perturbation force was

varied in order to find the value that produced an out of plane displacement at the center of the

specimen ranging between 0% and 5% of the total thickness of the laminate. In ABAQUS, the

imperfection was introduced as an initial displacement field obtained from a linear combination

of the lowest buckling modes from the linearized analysis. Linear combinations of the first two

modes were used, and the effect of introducing additional higher modes in the linear combination

was found to be negligible.

Correlation of experimental and finite element displacements and strains at the center of

the panel were used to judge the accuracy of the models. The FE results exhibited good

correlation with the test results, with the 20 degree laminate model showing the closest

agreement between experimental and FE data. In an attempt to produce better agreement

between the data, a range of initial imperfections were introduced into the FE model. Future

180

research to obtain more refined correlations is recommended, including a physical exploration of

the picture frame fixture to detect particular nuances such as bedding effects or eccentricities that

might provide additional insight. Furthermore, initial imperfections should be measured and

characterized for each test panel.

Both software packages, ANSYS and ABAQUS, provided good results. However, the

robustness in the convergence of the solutions together with superior pre and post processing

capabilities and somewhat better correlations with test made ABAQUS v6.6 the package of

choice for the implementation of the progressive failure analysis (PFA) in the nonlinear model.

This implementation was only carried out for the 20 degree laminate model. To implement the

nonlinear model with PFA capabilities, two different models were created: (1) one with failure

detection capability only, and (2) one with failure detection and a damage evolution algorithm.

The first model (with failure detection capability only) included Hashin’s theory of

failure for reinforced composites. The results obtained based on Hashin’s failure criteria were

compared to classic single-equation criteria for composite structures; namely, comparisons were

made with the maximum stress, maximum strain, Tsai-Hill, and Tsai-Wu criteria. Besides

providing information on failure for different modes, Hashin’s proved to be the most

conservative criteria.

In the second model the damage evolution algorithm was added to the nonlinear model to

complete the progressive failure analysis capability. Different aspects of the model were

discussed such as the constitutive laws used to model material degradation once failure had been

detected. Since such a constitutive model is not activated until failure has been detected at the

element level for at least one mode, the Hashin’s criteria also act as functions for the activation

181

of the material degradation model.

The material degradation model is introduced in the FE solution as a constitutive model,

and it relies on the use of artificial viscous damping in the element’s stiffness properties to

mitigate the convergence difficulties in the nonlinear solution after the introduction of the PFA.

The effect of different levels of damping on the solution was evaluated. A single value of

damping was used for both materials (unidirectional carbon/epoxy and biaxial glass/epoxy) and

also for both fiber and matrix constituents. A recommendation derived from this study is that the

damping level should be chosen such that variations in the damping constant do not greatly

affect the solution. A potential continuation of this research should include the effect of using

different damping constants for each material and each failure mode.

The results of the complete nonlinear analysis with PFA were compared with the

nonlinear solution without failure and with the test results. The model with PFA produced an

improved FE-experimental correlation in the linear range and around the buckling region;

whereas, no improvements were found in the high load level postbuckling region. In that region

the stiffness of the specimen was underestimated by the nonlinear model with PFA, which

resulted in a more conservative approach than the nonlinear only model without PFA.

As a potential continuation of this research, the differences between the nonlinear-PFA

model and the test results can be further reduced by using test values of the critical energy

release constants, which can be obtained by standard fracture toughness tests of the specific

composite material systems used to build the specimens. The values used in this dissertation

were found in the open literature.

The evolution of the four different failure modes was examined in order to visualize the

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load paths leading to the final collapse of the structure. The most massive mode of failure was

found to be matrix compression, which was fully developed over the structure for a load level in

which the fiber tensile failure had barely started. To further validate the predictions of damage

evolution presented in this dissertation, an ultrasound scanning of the test panels to map the

existence and nature of damage would have been very valuable. This line of work is

recommended as a continuation of this research.

From the perspective of establishing a design methodology, the current nonlinear-PFA

analysis provides enough information to determine postbuckling load levels with no damage

versus those with severe damage. The different levels of load corresponding to the occurrence of

damage, for each of the failure modes tracked, were plotted together with the load-strain

response curves of the 20 degree panel to establish the maximum allowable load at the first

occurrence of failure.

In order to further improve the analysis models and correlation the following

recommendations are made. First, the entire test fixture should be modeled using 3D shell

elements, with the properties of the exact aluminum alloy used to build it, and the *FASTENER

feature available in ABAQUS should be utilized to model the stiffness of the picture frame

fasteners.

Next, it is recommended to carry out test of additional panels. Following the

recommendation of the Federal Aviation Administration (FAA), at least 3 identical specimens of

each panel should be tested This approach provides more consistent test results. It is also

recommended that a large variety of potential initial imperfections be investigated. Different

linear combinations of buckling modes (up to the third buckling mode) should be introduced in

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the nonlinear model to potentially improve the correlation with test results. Finally, panels with

initial damage should be tested and analyzed to evaluate the effect of various types of damage on

postbuckling strength and stability.

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Buckling, postbuckling and progressive failure analysis of - SOAR