July 2010 

NASA/TP–2010-216726 

Nondimensional Parameters and Equations 

for Nonlinear and Bifurcation Analyses of 

Thin Anisotropic Quasi-Shallow Shells 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
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July 2010 

NASA/TP–2010-216726 

Nondimensional Parameters and Equations 

for Nonlinear and Bifurcation Analyses of 

Thin Anisotropic Quasi-Shallow Shells 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Equations for Nonlinear Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Kinematic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Stress Resultants and Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Nonlinear Equilibrium Equations and Boundary Conditions. . . . . . . . . . . . . . . . 25

Nonlinear Compatibility Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Nondimensional Fields Equations and Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Nondimensional Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Nondimensional Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Symmetrically laminated shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Generally laminated shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Nondimensional Equilibrium Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Nondimensional Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Nondimensional Compatibility Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Nondimensional Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Nondimensional Stress-Function Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Field Equations in terms of W and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Virtual Work in terms of W and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Complementary Virtual Work in terms of W and F . . . . . . . . . . . . . . . . . . . . . 61

Equations for Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2

Nondimensional Bifurcation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Equations for the Primary Equilibrium Path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Equations for Adjacent Equilibrium Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Variational Principle for Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Nondimensional Stress-Function Formulation for Bifurcation . . . . . . . . . . . . . . . . . 84

Virtual Work in terms of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Complementary Virtual Work in terms of and . . . . . . . . . . . . . . . . . . . 90

Values of the Nondimensional Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Results for Angle-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Results for Quasi-Isotropic Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Results for Unbalanced, Unsymmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . 99

Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

W

(1)

F

(1)

W

(1)

F

(1)
3

List of Tables

1. Lamina Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m 

laminates (m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3. Values of and for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m 

laminates (m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4. Values of and for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m,

and (-θ/+θ)m laminates (m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m 

laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m

laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7. Values of for (+θ/-θ)S laminates and -γb for (-θ/+θ)S laminates . . . . . 113

8. Values of for (+θ/-θ)S laminates and -δb for (-θ/+θ)S laminates. . . . . 114

9. Values of for (-θ/+θ) laminates and -e16 for (+θ/-θ) laminates . . . 114

10. Values of for (-θ/+θ) laminates and -e26 for (+θ/-θ) laminates . . . 115

11. Values of for [(+θ/-θ)m]S and -γb for [(-θ/+θ)m]S P-100/3502

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

12. Values of for [(+θ/-θ)m]S and -δb for [(-θ/+θ)m]S P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

β = D12 + 2D66

D11D22

a 22

a 11

1

4 D11

D22

1

4

νm = − a12

a11a22

νb = D12

D11D22

μ = 2a 12 + a 66

2 a 11a 22

h

12 a11a 22D11D22

1/4

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4
4

13. Values of for (-θ/+θ)m and -e16 for (+θ/-θ)m]S P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

14. Values of for (-θ/+θ)m and -e26 for (+θ/-θ)m]S P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

15. Values of for [(±45/0/90)m]S, [(±45/0/90)m]A, [(0/90/±45)m]S, 

[(0/90/±45)m]A, (±45/0/90)m, and (0/90/±45)m laminates (m = 1, 2, ...) . . . . . . . . . . . . 117

16. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates . . . . . . . . . . . . . . 118

17. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates . . . . . . . . . . . . . . 118

18. Values of for (±45/0/90)m unsymmetric laminates. . . . . . . . . . . . . . . . . . . . . . 118

19. Values of for (0/90/±45)m unsymmetric laminates. . . . . . . . . . . . . . . . . . . . . . 119

20. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates. . . . . . . . . 119

21. Values of for [(0/90±45)m]S and [(0/90/±45)m]A laminates . . . . . . . . . 119

22. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

23. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates. . . . . . . . . 120

24. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates. . . . . . . . . 120

25. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

νm = − a12

a11a22

D11

D22

1

4

D11

D22

1

4

D11

D22

1

4

D11

D22

1

4

β = D12 + 2D66

D11D22

β = D12 + 2D66

D11D22

β = D12 + 2D66

D11D22

νb = D12

D11D22

νb = D12

D11D22

νb = D12

D11D22
5

26. Values of for [(±45/0/90)m]S laminates . . . . . . . . . . . . . . . . . . . . . . . . . 121

27. Values of for [(0/90/±45)m]S laminates . . . . . . . . . . . . . . . . . . . . . . . . . 121

28. Values of for (±45/0/90)m unsymmetric laminates . . . . . . . . . . . . . . . . 122

29. Values of for (0/90/±45)m unsymmetric laminates . . . . . . . . . . . . . . . . 122

30. Values of for [(±45/0/90)m]S laminates. . . . . . . . . . . . . . . . . . . . . . . . . 122

31. Values of for [(0/90/±45)m]S laminates. . . . . . . . . . . . . . . . . . . . . . . . . 123

32. Values of for (±45/0/90)m unsymmetric laminates. . . . . . . . . . . . . . . . 123

33. Values of for (0/90/±45)m unsymmetric laminates. . . . . . . . . . . . . . . . 123

34. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates. . . . . . 124

35. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates. . . . . . 124

36. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

37. Values of for (±45/0/90)m unsymmetric laminates . . . . . . . . . . . . . . . 125

38. Values of for (0/90/±45)m unsymmetric laminates . . . . . . . . . . . . . . . 125

γ b = D16

D11

3 D22

1/4

γ b = D16

D11

3 D22

1/4

γ b = D16

D11

3 D22

1/4

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4

δ b = D26

D11D22

3 1/4

δ b = D26

D11D22

3 1/4

δ b = D26

D11D22

3 1/4

h

12 a11a 22D11D22

1/4

h

12 a11a 22D11D22

1/4

h

12 a11a 22D11D22

1/4

e 11 = B11

a11

D11

1/2

e 11 = B11

a11

D11

1/2
6

39. Values of and for (±45/0/90)m unsymmetric

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

40. Values of and for (0/90/±45)m unsymmetric 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

41. Values of and for [(±45/0/90)m]A laminate . . . 126

42. Values of and for [(0/90/±45)m]A laminates. . . 126

43. Values of and for (±45/0/90)m and 

(0/90/±45)m unsymmetric laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

44. Values of for (±45/0/90)m unsymmetric laminates . . . . . . . . . . . . . . . 127

45. Values of for (0/90/±45)m unsymmetric laminates . . . . . . . . . . . . . . . 127

46. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates . . . . . . . . . 128

47. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates. . . . . . . . . . . . . . . 128

48. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...) . . . . . . . . . . 129

49. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...). . . . . 129

50. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates . . . . . . . . . 130

51. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...) . . . . . 130

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

e 22 = B22

a22

D22

1/2

e 22 = B22

a22

D22

1/2

β = D12 + 2D66

D11D22

D11

D22

1

4

a 22

a 11

1

4

νm = − a12

a11a22

νb = D12

D11D22

μ = 2a 12 + a 66

2 a 11a 22
7

52. Values of for (+θ/0/90)m and -γm for (-θ/0/90)m laminates 

(m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

53. Values of for (+θ/0/90)m and -δm for (-θ/0/90)m laminates 

(m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

54. Values of for (+θ/0/90) and -γb for (-θ/0/90) three-ply laminates . . 132

55. Values of for (+θ/0/90) and -δb for (-θ/0/90) three-ply laminates . . 132

56. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates. . . . . . . 133

57. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates . . . . . . . . 133

58. Values of and for (+θ/0/90) and 

(-θ/0/90) three-ply laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

59. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates . . . . . . . . 134

60. Values of for (+θ/0/90) and -e16 for (-θ/0/90) three-ply 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

61. Values of for (+θ/0/90) and -e26 for (-θ/0/90) three-ply

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

62. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates . . . . . . . . . . . . 136

63. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates. . . . . . . 136

64. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates. . . . . . . 137

γm = − a26

a11a22

3 1/4

δm = − a16

a11

3

a22

1/4

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4

h

12 a11a 22D11D22

1/4

e 11 = B11

a11

D11

1/2

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 22 = B22

a22

D22

1/2

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

D11

D22

1

4

β = D12 + 2D66

D11D22

νb = D12

D11D22
8

65. Values of for (+θ/0/90)m and -γb for (-θ/0/90)m P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

66. Values of for (+θ/0/90)m and -δb for (-θ/0/90)m P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

67. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates. . . . 138

68. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates. . . . . . 139

69. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates. . . . . . 139

70. Values of and for (+θ/0/90)m and (-θ/0/90)m

P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

71. Values of for (+θ/0/90)m and -e16 for (-θ/0/90)m P-100/3502

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

72. Values of for (+θ/0/90)m and -e26 for (-θ/0/90)m P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4

h

12 a11a 22D11D22

1/4

e 11 = B11

a11

D11

1/2

e 22 = B22

a22

D22

1/2

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4
9

List of Figures

1. Coordinate system and unit-magnitude base-vector fields for points of 

undeformed shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

2. Sign convention for applied loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3. Fiber orientation of an arbitrary lamina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4. Effects of lamina material properties on nondimensional flexural orthotropy 

parameter β for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply 

laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5. Effects of lamina material properties on parameter coefficients in equations (52a) 

and (55) for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply

laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6. Effects of lamina material properties on Poisson’s ratios defined by equations 

(52e) and (59d) for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m 

angle-ply laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7. Effects of lamina material properties on nondimensional membrane orthotropy parameter 

μ for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates 

(m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8. Effects of lamina material properties on Batdorf-Stein-parameter coefficient in 

equations (45) and (48) for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m 

angle-ply laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9. Effects of lamina material properties on nondimensional flexural anisotropy 

parameters γb for [(+θ/−θ)m]S laminates and -γb for [(-θ/+θ)m]S laminates. . . . . . . . 145

10. Effects of lamina material properties on nondimensional flexural anisotropy 

parameters δb for [(+θ/−θ)m]S laminates and -δb for [(-θ/+θ)m]S laminates . . . . . . . 146

11. Effects of lamina material properties on flexural anisotropy parameters γb

 and 

δb for [(+θ/-θ)m]S and -γb and -δb for [(-θ/+θ)m]S symmetric angle-ply laminates, 

respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

12. Effects of lamina material properties on nondimensional load-path eccentricity parameters

-e16 and +e16 defined by equation (75b) for (+θ /-θ)m and (-θ /+θ)m antisymmetric 

angle-ply laminates, respectively (m = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10

13. Effects of lamina material properties on nondimensional load-path eccentricity parameters

-e26 and +e26 defined by equation (75f) for (+θ /-θ)m and (-θ /+θ)m antisymmetric 

angle-ply laminates, respectively (m = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

14. Effects of lamina material properties on nondimensional load-path eccentricity parameters 

e16 and e26 defined by equations (75) for (-θ /+θ)m and -e16 and -e26 for 

(+θ /-θ)m antisymmetric angle-ply laminates, respectively (m = 1). . . . . . . . . . . . . . . . 148

15. Effects of number of plies on flexural anisotropy parameters γb for [(+θ/−θ)m]S 

and -γb for [(-θ/+θ)m]S P-100/3502 symmetric angle-ply laminates . . . . . . . . . . . . . . 148

16. Effects of number of plies on flexural anisotropy parameters δb for [(+θ/−θ)m]S 

and -δb for [(-θ/+θ)m]S P-100/3502 symmetric angle-ply laminates . . . . . . . . . . . . . . 149

17. Effects of lamina material properties on flexural anisotropy parameters γb and δb 

for [(+θ/-θ)m]S and -γb and -δb for [(-θ/+θ)m]S P-100/3502 symmetric angle-ply 

laminates, respectively (m = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

18. Effects of number of plies on nondimensional load-path eccentricity parameters -e16 and 

+e16 defined by equation (75e) for (+θ /-θ)m and (-θ /+θ)m P-100/3502 antisymmetric 

angle-ply laminates, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

19. Effects of number of plies on nondimensional load-path eccentricity parameters -e26 and 

+e26 defined by equation (75f) for (+θ /-θ)m and (-θ /+θ)m P-100/3502 antisymmetric 

angle-ply laminates, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

20. Effects of number of plies on nondimensional load-path eccentricity parameters e16 and 

e26 defined by equation (75) for (-θ /+θ)m P100/3502 antisymmetric angle-ply 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

21. Effects of number of plies on parameter coefficients in equation (55) for P-100/3502 

quasi-isotropic laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

22. Effects of number of plies on the nondimensional flexural orthotropy parameter β 

for P-100/3502 quasi-isotropic laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

23. Effects of number of plies on the nondimensional Poisson’s ratio νb

 for P-100/3502 

quasi-isotropic laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

24. Effects of number of plies on the nondimensional flexural anisotropy parameter γb

for P-100/3502 quasi-isotropic laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

25. Effects of number of plies on the nondimensional flexural anisotropy parameter δb

for P-100/3502 quasi-isotropic laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11

26. Effects of number of plies on Batdorf-Stein-parameter coefficients in equations (45) 

and (48) for P-100/3502 quasi-isotropic laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

27. Effects of number of plies on nondimensional load-path eccentricity parameter e11 

defined by equations (75b) for P-100/3502 quasi-isotropic laminates . . . . . . . . . . . . . 154

28. Effects of number of plies on nondimensional load-path eccentricity parameters e12 and 

e66 defined by equations (75) for P-100/3502 quasi-isotropic laminates . . . . . . . . . . . . 155

29. Effects of number of plies on nondimensional load-path eccentricity parameters e16 and 

e26 defined by equations (75) for P-100/3502 quasi-isotropic laminates. . . . . . . . . . . . . 155

30. Effects of number of plies on nondimensional load-path eccentricity parameter e22 

defined by equations (75d) for P-100/3502 quasi-isotropic laminates . . . . . . . . . . . . . . 156

31. Effects of lamina material properties on nondimensional flexural orthotropy parameter

β for (+θ /0/90) and (-θ /0/90) unbalanced, unsymmetric three-ply laminates . . . . . . 156

32. Effects of lamina material properties on parameter coefficient in equation (55) for

(+θ /0/90) and (-θ /0/90) unbalanced, unsymmetric three-ply laminates . . . . . . . . . . . 157

33. Effects of lamina material properties on parameter coefficient in equation (52a) for

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...). . . . . . . 157

34. Effects of lamina material properties on Poisson’s ratio defined by equation (52e) for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...). . . . . . . 158

35. Effects of lamina material properties on Poisson’s ratio defined by equation (59d) for 

(+θ /0/90) and (-θ /0/90) unbalanced, unsymmetric three-ply laminates. . . . . . . . . . . 158

36. Effects of lamina material properties on nondimensional membrane orthotropy parameter 

μ for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...). . 159

37. Effects of lamina material properties on nondimensional membrane anisotropy 

parameters γm and -γm for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates, respectively (m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

38. Effects of lamina material properties on nondimensional membrane anisotropy 

parameters δm and -δm for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates, respectively (m = 1, 2, ...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

39. Effects of lamina material properties on nondimensional membrane anisotropy 

parameters γm and δm for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates (m = 1, 2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12

40. Effects of lamina material properties on nondimensional flexural anisotropy parameters

γb and -γb for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates,

respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

41. Effects of lamina material properties on nondimensional flexural anisotropy parameters

δb and -δb for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates,

respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

42. Effects of lamina material properties on nondimensional flexural anisotropy parameters

γb and δb for (+θ /0/90)m unbalanced, unsymmetric laminates (m = 1) . . . . . . . . . . . . 162

43. Effects of lamina material properties on Batdorf-Stein-parameter coefficients in

equations (45) and (48) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

44. Effects of lamina material properties on nondimensional load-path eccentricity parameter 

e11 defined by quation (75b) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates, respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

45. Effects of lamina material properties on nondimensional load-path eccentricity parameters

e12 and e66 defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, 

unsymmetric laminates, respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

46. Effects of lamina material properties on nondimensional load-path eccentricity parameter 

e22 defined by quation (75d) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

laminates, respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

47. Effects of lamina material properties on nondimensional load-path eccentricity parameters

e16 and -e16 defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, 

unsymmetric laminates, respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48. Effects of lamina material properties on nondimensional load-path eccentricity parameters

e26 and -e26 defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, 

unsymmetric laminates, respectively (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

49. Effects of lamina material properties on nondimensional load-path eccentricity parameters

e16 and e26 defined by equations (75) for (+θ /0/90)m unbalanced, unsymmetric 

laminates (m = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

50. Effects of number of plies on parameter coefficients in equation (55) for (+θ /0/90)m 

and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates . . . . . . . . . . . . . . . . 166

51. Effects of number of plies on the orthotropy parameter β for (+θ /0/90)m and

(-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . 166
13

52. Effects of number of plies on Poisson’s ratio defined by equation (59d) for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates . . . . . . 167

53. Effects of number of plies on flexural anisotropy parameters γb and -γb

 for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates, 

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

54. Effects of number of plies on flexural anisotropy parameters δb and -δb for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates,

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

55. Effects of number of plies on flexural anisotropy parameters γb and δb for 

(+θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . 168

56. Effects of number of plies on Batdorf-Stein-parameter coefficients in equations (45) 

and (48) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

57. Effects of number of plies on nondimensional load-path eccentricity parameter e11 

defined by equation (75b) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

58. Effects of number of plies on nondimensional load-path eccentricity parameter e22 

defined by equation (75b) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

59. Effects of number of plies on nondimensional load-path eccentricity parameters e12 and 

e66 defined by equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

P-100/3502 laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

60. Effects of number of plies on nondimensional load-path eccentricity parameters e16 and 

-e16 defined by equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

P-100/3502 laminates, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

61. Effects of number of plies on nondimensional load-path eccentricity parameters e26 and 

-e26 defined by equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric 

P-100/3502 laminates, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

62. Effects of number of plies on nondimensional load-path eccentricity parameters e16 and 

e26 defined by equation (75) for (+θ /0/90)m unbalanced, unsymmetric P-100/3502

laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
14

Summary

A comprehensive development of nondimensional parameters and equations for nonlinear 

and bifurcation analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory 

for thin anisotropic shells, is presented. A complete set of field equations for geometrically 

imperfect shells that includes kinematic equations, isothermal constitutive equations for generally 

laminated shells, equilibrium equations, boundary conditions, and the compatibility equation is 

presented in terms of general lines-of-curvature coordinates. In addition, the corresponding 

virtual work statement is presented. A systematic nondimensionalization of these equations is 

developed, several new nondimensional parameters are defined, and a comprehensive stressfunction formulation is presented that includes variational principles for equilibrium and 

compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and 

a comprehensive set of bifurcation equations are given that include the effects of pre-bifurcation 

rotations, which are commonly neglected. These bifurcation equations also include a stressfunction formulation with variational principles for equilibrium and compatibility of the adjacent 

equilibrium states.

An extensive collection of tables and figures is presented that shows the effects of lamina 

material properties and stacking sequence on the nondimensional parameters. In particular, 

results are presented for nine lamina material systems and several stacking sequences, and are 

independent of the shell geometry. These stacking sequences include balanced symmetric angleply laminates, balanced antisymmetric angle-ply laminates, symmetric quasi-isotropic laminates, 

antisymmetric quasi-isotropic laminates, and unsymmetric quasi-isotropic laminates. Results are 

also given for unbalanced, unsymmetric laminates composed of perpendicular unidirectional 

plies aligned with the shell coordinate curves and angle plies.

Introduction

A common structural element of aerospace vehicles is the thin-walled shell. Often, aerospace 

shell structures are tailored from fiber-reinforced, laminated-composite materials to reduce 

structural weight, increase strength and stiffness, and improve one or more performance 

characteristics of a vehicle. In a tailoring process, it is desirable to know the "landscape" of the 

design space so that a designer can assess the sensitivity of a candidate optimal design to the 

variations in structural characteristics that may occur in a manufacturing process or in a design 

change introduced to accommodate some other vehicle attribute. This point is particularly true for 

laminated-composite structures, which offer a much larger design space than metals because of 

the plethora of material systems available and the laminate constructions that are possible. As a 

result, nondimensional parameters are sometimes used to navigate the design space. In particular, 

many different laminate constuctions may correspond to the same set of nondimensional 

parameters, and the relative magnitudes of the parameters can be used to identify special cases in 

which one or more parameters are negligible. This correspondance effectively reduces the 

dimensionality of the design space to something that can be managed by designers. As a result, 

nondimensional parameters also provide insight into the development of scaling technology used 

to reduce the cost of experimental validation and certification of large-scale structures.
15

Several early works have been published that use nondimensional parameters to characterize 

structural behavior and to faciliate design. For example, Seydel used nondimensional parameters 

to characterize the shear-buckling behavior of orthogonally stiffened long, flat plate strips made 

of metal or plywood in the early 1930s.1, 2 In his study, the parameters were identified by modeling 

the stiffened plate strip as a homogeneous orthotropic plate and by using the corresponding 

differential equation for buckling, derived by Huber around 1923.3 Seydel’s parameters and 

results appeared in a compilation of design technology for airplane design in 1935.4 Similarly, in 

1946, Cozzone and Melcon5

 presented column buckling results in terms of nondimensional 

parameters. Their work was driven by the need to address the numerous new aluminum and steel 

alloys being used in airplane design. Perhaps one of the most well known nondimensional 

parameters to appear in the 1940s is the "Z" parameter introduced by Batdorf, which characterizes 

the effects of length, thickness, and radius of curvature on the linear bifurcation buckling of 

isotropic cylinders.6-10 

Around 1950, Thielemann11 presented an in-depth study of the buckling behavior of generally 

orthotropic plates subjected to compression and shear loads. This class of plates is similar to 

symmetrically laminated plates in that they exhibit anisotropy in the form of coupling between 

pure bending and twisting deformations. More specifically, Thielemann presented numerous 

results for specially orthotropic plates in terms of one of Seydel’s nondimensional parameters, and 

introduces two additional nondimensional parameters to characterize the effects of anisotropy on 

the buckling of generally orthotropic plates. In 1952, Wittrick12 used nondimensional parameters 

to simplify the buckling design of rectangular specially orthotropic plates subjected to 

compression loads. A similar study was presented by Shuleshko in 1957.13 In 1960, Thielemann14

presented another in-depth study that focuses on the buckling and postbuckling behavior of 

specially orthotropic, thin-walled cylinders with initial geometric imperfections and subjected to 

compression, shear, and internal pressure. In this reference, the nonlinear and buckling responses 

are obtained by using a single equilibrium equation and a single strain-compatibility equation. 

Several nondimensional parameters, in addition to those used for specially orthotropic plates, that 

are needed to characterize the response trends are also identified. Similar studies were presented 

by Geier15 in 1965 and Seggelke and Geier16 in 1967 that also addresses the effects of stiffener 

eccentricity on the buckling response of cylinders. 

As the weight saving potential of fiber-reinforced and fabric-reinforced plastics started 

becoming apparent, the use of nondimensional parameters to characterize common response 

trends and behavior also increased. In 1968, Brukva17 presented buckling results, in terms of 

nondimensional parameters, for specially orthotropic plates subjected to axial compression and 

with various combinations of clamped and simply supported edges. The results of his study were 

aimed at understanding the behavior of plates made from a glass-reinforced plastic material. In 

1968-1970, Khot18, 19 and Khot and Venkayya20 used nondimensional parameters to characterize 

the imperfection sensitivity of generally laminated, fiber-reinforced shells subjected to axial 

compression and with small initial geometric imperfections. In these references, the nonlinear and 

buckling responses are obtained by using a single equilibrium equation and a single straincompatibility equation. These two equations were shown to be significantly more complicated 

than the corresponding equations for generally orthotropic plates. Nondimensional parameters 
16

were also identified that represent the anisotropies associated with coupling between membrane 

dilatation and distortion, coupling between pure bending and twisting deformations, and coupling 

between membrane and bending deformations. The relative magnitude of these parameters 

provide measures of the relative importance of each type of coupling, and when some coupling 

terms can be deemed negligible, analytical solutions can sometimes be obtained. Also in 1970, 

Johns21 presented a review of results for the shear buckling of isotropic, specially orthotropic, and 

generally orthotropic rectangular plates. This review gives numerous generic results, useful for 

design, in terms of Seydel’s nondimensional parameters and nondimensional buckling 

coefficients. Nondimensional parameters and results are also given that characterize the effects 

of edge rotational restraint on the buckling resistance. A similar in-depth study with numerous 

general results was given by Housner & Stein in 1975.22 

Later, in 1977, Wiggenraad23 used nondimensional parameters and buckling coefficients to 

study the effects of anistropy associated with coupling between pure bending and twisting 

deformations on the buckling of rectangular symmetrically laminated plates subjected to 

compression and shear loads. The nondimensional parameters used to characterize the anisotropy 

are different than those presented by Thielemann. A large number of results are presented in this 

reference that illustrate the sensitivity of the buckling response to variations in the 

nondimensional orthotropy and anisotropy parameters. An extensive study of the generic 

buckling and vibration behavior of specially orthotropic rectangular and circular plates was 

presented by Oyibo24 in 1981 that uses affine transformations. This approach yields the plate 

response in terms of two independent nondimensional parameters, referred to as the generalized 

Poisson’s ratio and the generalized rigidity ratio by Oyibo. The generalized rigidity ratio for 

rectangular plates is the reciprocal of the parameter defined by Seidel1,2 in the 1930s. Likewise, a 

substantial amount of generic buckling-design data was presented by Fogg25 in 1982 for laminated 

composite plates and curved panels in terms of nondimensional parameters and by Brunelle and 

Oyibo26 in 1983 for specially orthotropic rectangular plates. Oyibo27-29 and Stein30 also presented a 

substantial amount of generic results for flutter and postbuckling, respectively, of specially 

orthotropic rectangular plates in 1983. Also in 1983, Nemeth31 presented nondimensional 

parameters that characterize the effects of anistropy associated with coupling between pure 

bending and twisting deformations on the buckling of rectangular symmetrically laminated plates 

subjected to compression loads. The parameters given in this reference are identical to those 

previously given by Wiggenraad.23 A couple of years later, Oyibo and Berman32 and Nemeth33, 34

published indentical forms of the anisotropy parameters that are different from, but similar to, 

those previously given by Wiggenraad23 and by Nemeth.31 These nondimensional parameters have 

been used extensively by Nemeth35-42 to develop generic design data for buckling of laminated 

composite plates subjected to various loading conditions. Moreover, the nondimensionl 

parameters given in these references are used to quantify just how "close" various families of 

quasi-isotropic laminate are to being isotropic. In 1985, Stein43,44 extended his earlier work to 

include plates subjected to shear and combined loads. Likewise, Brunelle45- 47 extended his earlier 

work in 1985 by examining the values of his nondimensional parameters for several orthotropic 

materials, and in 1986 by using nondimensional parameters to develop similarity rules for 

buckling and vibration and by deriving nondimensional parameters and equations for large 

deflections of specially orthotropic plates. Vibration analysis and generic results for specially 

orthotropic circular plates were also given by Oyibo and Brunelle48 in 1985. Additional generic 
17

results for the bending, buckling, large deflection, postbuckling, and linear and nonlinear 

vibration of symmetrically and unsymmetrically laminated, angle-ply and cross-ply rectangular 

plates were presented by Yang, Shieh, Liu, and Kuo in 1987 through 1989.49-56 Moreover, the 

work presented in references 49-52 and 55 is based on the Reissner-Mindlin first-order 

transverse-shear-deformation plate theory. Brunelle and Shin presented detailed studies of the 

postbuckling behavior of rectangular specially orthotropic plates, also in 1989, by using affine 

transformations and nondimensional parameters to obtain generic response trends.57-59 

In the 1990s, Nemeth60, 61 extended his previous work to develop nondimensional parameters 

and equations for linear bifurcation buckling of symmetrically laminated shallow shells with 

double curvature. The analysis presented in these references uses a single equilibrium equation 

and a single strain-compatibility equation, and the nondimensionalization procedure used was 

heavily influenced by the previous work done by Stein30 on postbuckling of specially orthotropic 

plates. In addition to the nondimensional parameters presented in references 30 and 34, 

parameters that characterize the anisotropy associated coupling between membrane dilatation 

and distortion were given, and analogues of the Batdorf "Z" parameter were derived. In addition, 

generalizations of Donnell’s and Batdorf’s equations for cylinder buckling (see references 6, 7, 

and 10 ) were presented. Also in the 1990s, Radloff et. al.62 developed a nondimensional buckling 

analysis for symmetrically laminated trapezoidal plates subjected to uniaxial compression. A 

study of the generic bending, buckling, and vibration behavior of antisymmetric angle-ply 

laminates was presented by Lee and Yang63 in 1996. In 2000, Nemeth and Smeltzer64 presented 

formulas for the attenuation length of a bending boundary layer in generally laminated shells. In 

this reference, the attenuation length of the bending boundary layer is characterized by two 

nondimensional parameters; that is, one for the shell orthotropy and one for the general 

anisotropy. Values of these two parameters are also presented for nine different lamina material 

systems and several laminate stacking sequences. Later, in 2001, Hilburger et.al.65 used 

nondimensional parameters, based on the Reissner-Mindlin first-order shear-deformation plate 

theory, to obtain scaling laws for a representative portion of a blended-wing-body transport 

aircraft. 

From 2002-2008, Weaver and his colleagues66-75 made extensive use of 

nondimensionalization procedures and parameters to gain insight into the behavior of laminatedcomposite structures. In particular, "knockdown" factors that account for the effects of flexuraltwist and extension-twist anisotropies on the buckling of compression-loaded cylindrical shells 

are given in references 66-68. The results presented in reference 67 indicate that the importance 

of flexural-twist anisotropy depends strongly on the cylinder curvature. A similar finding was 

obtained by Nemeth60, 61 for doubly curved shells subjected to shear loading. Also, design-oriented 

approximate solutions for compression-loaded long plates, in terms of nondimensional 

parameters, are given in references 70 and 74 for buckling and in references 69 and 72 for 

postbuckling. Moreover, a design-oriented approximate solution for compression-loaded, 

generally laminated cylindrical shells is given in reference 71, that uses the reduced bending 

stiffnesses obtained when the partially inverted form of the constitutive equations is used. In 2007, 

Weaver and Nemeth76 presented bounds on the nondimensional parameters that govern 

symmetrically laminated plate buckling behavior, which provide insight into the potential gains 

in buckling resistance that are possible through laminate tailoring and composite-material 
18

development. Similarly, in 2008, Weaver and Nemeth77 presented design-oriented 

nondimensional buckling interaction curves for specially orthotropic plates subjected to 

combined loads. These curves represent a broad range of plate-bending orthotropy and inherently 

indicate the corresponding design sensitivities. 

Recently, Mittelstedt and Beerhorst78 presented nondimensional buckling curves for specially 

orthotropic compression-loaded plates with finite length and elastically supported edges. These 

results are expressed in terms of the reciprocal of Seidel’s orthotropy parameter and a 

nodimensional measure of edge restraint, and are used for the design of stiffened panels. Also 

recently, Nemeth and Mikulas79 presented simple formulas and results for use in determining the 

buckling resistance and stiffness design of laminated-composite cylinders subjected to 

compression loads. Their work is based on the nondimensional parameters and equations given 

in reference 60. One noteworthy aspect of this work is that the validation of the simple formulas 

presented is simplified significantly by establishing a simple parametric relationship between two 

of the nondimensional parameters governing the response. 

The literature discussed previously in the present study indicates clearly the potential for 

simplifying and unifying design criteria for laminated-composite structures by using 

nondimensional parameters and equations. Although a lot has been done in this regard to develop 

generic design technology, the task is monumental and much more remains to be done, 

particularly for shell structures. Thus, one goal of the present study is to extend the 

nondimensionalization procedure given in references 60 and 61 for geometrically perfect, 

symmetrically laminated, quasi-shallow shells to include generally laminated quasi-shallow 

shells, with initial geometric imperfections, undergoing small strains and moderately small 

rotations. Herein, the term, "quasi-shallow shell," is used (e.g., see Brush and Almroth80, p. 143), 

to denote shallow shell panels that are relatively flat and nonshallow shells that exhibit 

deformations that are rapidly varying functions of the reference-surface coordinates. These 

equations, and the corresponding nondimensional parameters, should be useful in the 

development of generic design technology that represents the effects of geometry and laminate 

construction on the imperfection sensitivity of shells subjected to destabilizing loads. Another 

goal is to present a collection of data for the nondimensional parameters presented subsequently 

that shows the effects of lamina material properties and laminate stacking sequence on their 

values and that are independent of the shell geometry. To accomplish these goals, equations of 

quasi-shallow shell theory that govern the nonlinear deformations of geometrically imperfect 

shells are presented first. Then, nondimensionalization of the kinematic equations, constitutive 

equations, equilibrium equations, boundary conditions, compatibility equation, and virtual work 

is presented and several new nondimensional parameters are defined. Next a nondimensional 

stress-function formulation of the nonlinear boundary-value problem is presented that yields 

extensions to the Donnell-Stein equations given previously in references 60 and 61. In addition, 

nondimensional stress-function formulations of the principles of virtual work and complementary 

virtual work are given. Nondimensional bifurcation equations follow that also include kinematic 

equations, constitutive equations, equilibrium equations, boundary conditions, the compatibility 

equation, and the virtual work associated with equilibrium states adjacent to a primary 

equilibrium path. Then, a nondimensional stress-function formulation of the boundary-eigenvalue 

problem is presented that includes variational principles for equilibrium and compatibility. For all 

these analytical developments, an extensive list of symbols is given in the Appendix. Finally an 
19

extensive collection of nondimensional-parameter data is presented for nine lamina material 

systems and several laminate stacking sequences that should be useful to design-technology 

developers.

Equations for Nonlinear Deformations

The basic equations for nonlinear deformations of doubly curved quasi-shallow (e.g., see 

Brush and Almroth80, p. 196) shells with uniform thickness h are presented subsequently in terms 

of the orthogonal, lines-of-curvature, curvilinear coordinates (ξ1,ξ2,ζ) that are depicted in figure 1 

for a generic shell reference surface A. Associated with each point p of the reference surface, 

with coordinates (ξ1,ξ2,0), are three perpendicular, unit-magnitude vector fields The 

vectors and are tangent to the ξ1

 -and ξ2-coordinate curves, respectively, and reside in the 

tangent plane at the point p. The vector is tangent to the ζ-coordinate curve at point p and 

perpendicular to the tangent plane. This class of parametric coordinates permits substantial 

simplification of the shell equations and has many practical applications. 

Lines-of-curvature coordinates form an orthogonal coordinate mesh and are identified by 

examining how the vectors change as the coordinate curves are traversed by an 

infinitesimal amount. In particular, at every point q that is infinitesimally close to point p there is 

another set of vectors with similar attributes; that is, the vectors and are 

orthogonal and tangent to the ξ1 -and ξ2-coordinate curves at q, respectively, and reside in the 

tangent plane at the point q. Likewise, vector is tangent to the ζ-coordinate curve at point q 

and perpendicular to the tangent plane at point q. Next, consider the finite portion of the tangent 

plane at point p shown in figure 1. Because of the identical properties of the vectors 

at every point of the surface, an identical, corresponding planar region exists at point q. Therefore, 

the vectors at point q can be obtained by moving the vectors at point p

to point q. In addition, the plane region at point p moves into coincidence with the corresponding 

plane region at point q as the surface is traversed from point p to point q. During this process, the 

plane region at point p undergoes roll, pitch, and yaw (rotation about the normal line to the 

surface) motions. The roll and pitch motions are caused by surface twist (torsion) and curvature, 

respectively. The yaw motion is associated with the geodesic curvature of the surface curve 

traversed in going from point p to q. When a line-of-curvature coordinate curve is traversed in 

going from point p to q, the planar region at point p undergoes only pitch and yaw motions as it 

moves into coincidence with the corresponding region at point q. Rolling motion associated with 

local surface torsion does not occur. 

These shell equations used subsequently in the present study are relatively well known (e.g., 

see the classic paper by Sanders81) and are referred to commonly as the equations of DonnellMushtari-Vlasov shell theory (see the textbooks by Brush & Almroth80 and Novozhilov82). 

Moreover, these equations are based on the fundamental assumptions of classical Love-Kirchhoff 

e1, e2, and n.

e1 e2

n

e1, e2, and n

e1, e2, and n e1 e2

n

e1, e2, and n

e1, e2, and n e1, e2, and n
20

shell theory, which neglects transverse-shear flexibility. First, the kinematic equations are 

presented, which include the displacement-field distribution, and the strain-displacement 

relations. Then, the stress resultants, constitutive equations, virtual work, and the work-conjugate 

nonlinear equilibrium equations and corresponding boundary conditions are presented, followed 

by the strain compatibility equation. These equations represent a simple, approximate 

representation of nonlinear shell behavior that has seen wide practical application. For each group 

of equations, the generalization to include a known distribution of initial geometric imperfection 

is given. The imperfection is manifested as a "small" displacement normal to the idealized shell 

reference surface.

Kinematic Equations

In the Donnell-Mushtari-Vlasov theory of quasi-shallow shells, the components of the 

displacement vector field of the material points comprising a shell are denoted by U1(ξ1,ξ2,ζ), 

U2(ξ1,ξ2,ζ), and U3(ξ1,ξ2,ζ), where (ξ1, ξ2, ζ) are orthogonal curvilinear coordinates for points of 

three-dimensional Euclidean space. In addition, the coordinates are defined for , 

, and (see figure 2), where h is shell thickness. Similarly, the 

displacement components of points of the two-dimensional shell reference surface, defined by ζ

= 0, are denoted by u1

(ξ1, ξ2), u2(ξ1, ξ2), and w(ξ1, ξ2), where (ξ1, ξ2) are orthogonal curvilinear 

Gaussian coordinates for the reference surface. These surface displacements are usually 

measured with respect to a given geometrically perfect, idealized shell reference surface. 

To analyze to response of a shell with relatively small initial geometric imperfections, 

measured with respect to the idealized shell reference surface, Donnell83 (see p. 349) introduced 

an "imperfection" function wI

(ξ1, ξ2). This imperfection function represents a distribution of small 

deviations in the ζ-coordinate direction, measured perpendicular to the tangent plane at each 

point of the shell reference surface, for an unloaded shell that is stress and strain free. Under the 

application of loads, the shell normal displacement associated with deformation from the 

idealized configuration is given by wI

(ξ1, ξ2) + w(ξ1, ξ2). The corresponding relationships between 

the three-dimensional displacement-field components and the surface-displacement-field 

components of the Donnell-Mushtari-Vlasov quasi-shallow shell theory are given by

 (1)

 (2)

 (3)

where

 (4)

a1 ≤ ξ 1 ≤ b1

a2 ≤ ξ 2 ≤ b2 − h

2 ≤ ζ ≤ h2

U1 ξ 1, ξ 2, ζ = u1 ξ 1, ξ 2 + ζ ψ1 ξ 1, ξ 2 + β1

I

ξ 1, ξ 2

U2 ξ 1, ξ 2, ζ = u2 ξ 1, ξ 2 + ζ ψ2 ξ 1, ξ 2 + β2

I

ξ 1, ξ 2

U3 ξ 1, ξ 2, ζ = w ξ 1, ξ 2 + wI ξ 1, ξ 2

β1

I

= – 1

A1

∂wI

∂ξ 1
21

and

 (5)

are fields that define the initial stress- and strain-free rotation of a material line element that is 

tangent to the shell reference surface, at a given point of the reference surface. Similarly, 

and are fields that define the net rotation of a material line element that is perpendicular 

to the shell reference surface, at a given point of the reference surface, with respect to the 

undeformed idealized configuration. The symbols A1 and A2 are the coefficients of the first 

fundamental form of the shell reference surface that are defined by

 (6)

where ds is the differential arc length between two infinitesimally neighboring points of the 

surface.

The normal-strain fields for a three-dimensional shell body are denoted by ε11(ξ1,ξ2,ζ), 

ε22(ξ1,ξ2,ζ), and ε33(ξ1,ξ2,ζ), and the corresponding shearing strains are denoted by γ12(ξ1,ξ2,ζ), 

γ13(ξ1,ξ2,ζ), and γ23(ξ1,ξ2,ζ). The relationships between the three-dimensional shell strains and the 

reference-surface strains in the Donnell-Mushtari-Vlasov theory are given by

 (7a)

 (7b)

 (7c)

 (7d)

 (7e)

 (7f)

where are the membrane reference-surface strains; are the changes in 

surface curvature; is the change in surface torsion; and are the transverse shearing 

strains. The strain expressions result from substituting equations (1)-(3) into the corresponding 

strain-displacement relations of the theory of elasticity (see Novozhilov84, pp. 56-60) and 

simplifying the results according to the presumptions of the Donnell-Mushtari-Vlasov theory. 

This process yields

β2

I

= – 1

A2

∂wI

∂ξ 2

ψ1 + β1

I

ψ2 + β2

I

ds 2= A1dξ 1

2

+ A2dξ 2

2

ε11 ξ 1, ξ 2, ζ = ε11

o

ξ 1, ξ 2 + ζ κ11

o

ξ 1, ξ 2

ε22 ξ 1, ξ 2, ζ = ε22

o

ξ 1, ξ 2 + ζ κ22

o

ξ 1, ξ 2

ε33 ξ 1, ξ 2, ζ = 0

γ 12 ξ 1, ξ 2, ζ = γ 12

o

ξ 1, ξ 2 + ζ κ12

o

ξ 1, ξ 2

γ 13 ξ 1, ξ 2, ζ = γ 13

o

ξ 1, ξ 2

γ 23 ξ 1, ξ 2, ζ = γ 23

o

ξ 1, ξ 2

ε11

o

, ε11

o

, and γ 12

o

κ11

o

and κ22

o

κ12

o

γ 13

o

and γ 23

o
22

 (8a)

 (8b)

 (8c)

 (9a)

 (9b)

 (9c)

 (10a)

 (10b)

where

 (11a)

 (11b)

are fields that define the rotation of a material line element that is tangent to the imperfect-shell 

reference surface, at a given point of the reference surface. The symbols R1 and R2 represent the 

principal radii of curvature of the shell reference surface along the ξ1 and ξ2 coordinate directions, 

respectively. Expressions for the rotations, ψ1 and ψ2, are obtained in the Donnell-MushtariVlasov thin-shell theory by enforcing the presumption that the transverse shearing strains are 

negligible compared to the other strains. This consideration gives ψ1 = β1 and ψ2 = β2. As pointed 

out by Donnell83 (see p. 349), strain-like terms associated with the "imperfection" function wI are 

subtracted from the corresponding terms associated with w + wI to obtain equations (8)-(10). This 

subtraction process enforces the requirement of a strain-free state in the absence of applied loads.

ε11

o

= 1

A1

∂u1

∂ξ 1

+

u2

A1A2

∂A1

∂ξ 2

+ w

R1

+ 1

2 β1

2

+ β1

I

β1

ε22

o

= 1

A2

∂u2

∂ξ 2

+

u1

A1A2

∂A2

∂ξ 1

+ w

R2

+ 1

2 β2

2

+ β2

I

β2

γ 12

o

= 1

A2

∂u1

∂ξ 2

– u2

A1A2

∂A2

∂ξ 1

+ 1

A1

∂u2

∂ξ 1

– u1

A1A2

∂A1

∂ξ 2

+ β1β2 + β1

I

β2 + β2

I

β1

κ11

o

= 1

A1

∂ψ1

∂ξ 1

+ ψ2

A1A2

∂A1

∂ξ 2

κ22

o

= 1

A2

∂ψ2

∂ξ 2

+ ψ1

A1A2

∂A2

∂ξ 1

κ12

o

= 1

A2

∂ψ1

∂ξ 2

– ψ2

A1A2

∂A2

∂ξ 1

+ 1

A1

∂ψ2

∂ξ 1

– ψ1

A1A2

∂A1

∂ξ 2

γ 13

o

= ψ1 − β1

γ 23

o

= ψ2 − β2

β1 = − 1

A1

∂w

∂ξ 1

β2 = − 1

A2

∂w

∂ξ 2
23

Stress Resultants and Constitutive Equations

The linear elastic constitutive equations for a laminated composite shell depend on the 

specific definition for the two-dimensional stress resultants that are used to represent the internal 

stresses, and on the presumed strain distribution. In the Donnell-Mushtari-Vlasov theory, the 

stress resultants are defined by

 (12a)

 (12b)

and

 (13)

where σ11, σ22, σ12, σ13, and σ23 are stresses. Equations (7) define the strain distribution, which is 

expressed in matrix form as

 (14)

The state of stress in a shell is presumed to be a state of plane stress, and is represented by

 (15)

where the subscripted symbols denote the transformed, reduced stiffness coefficients (reduced 

for a state of plane stress) and are found in the well-known book by Jones.85 Note that all 

quantities that appear in equation (15) are functions of the ζ coordinate. The two-dimensional 

constitutive equations for a shell are obtained by substituting equation (14) into equation (15) first 

and then by substituting the resulting expression into equations (12a) and (12b). This procedure 

N11

N22

N12

=

σ11

σ22

σ12

dζ

– h

2

+ h

2

M11

M22

M12

=

σ11

σ22

σ12

ζ dζ

– h

2

+ h

2

Q1

Q2

= σ13

σ23

dζ

– h

2

+ h

2

ε11

ε22

γ 12

=

ε11

o

ε22

o

γ 12

o

+ ζ

κ11

o

κ22

o

κ12

o

σ11

σ22

σ12

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

ε11

ε22

γ 12

Q
24

yields

 (16)

where

 (17)

 (18)

 (19)

In equations (17) through (21), the symbol h denotes the shell thickness and the reference surface 

is the middle surface of the shell. For a laminated-composite shell with thin layers, the layer 

properties are presumed constant and the integrands that are indicated in equations (17) through 

(19) are piecewise constant. Thus, the integrations are replaced by summations of the appropriate 

layer attributes over the number of layers that comprise a specific shell. 

To gain insight into the nondimensionalization process presented herein, and for comparisons 

with other works, it is useful to express the constitutive equations in the partially inverted form

 (20a)

N11

N22

N12

M11

M22

M12

=

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66

ε11

o

ε22

o

γ 12

o

κ11

o

κ22

o

κ12

o

A11 A12 A16

A12 A22 A26

A16 A26 A66

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

dζ

– h

2

+ h

2

B11 B12 B16

B12 B22 B26

B16 B26 B66

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

ζ dζ

– h

2

+ h

2

D11 D12 D16

D12 D22 D26

D16 D26 D66

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

ζ

2

dζ

– h

2

+ h

2

ε11

o

ε22

o

γ 12

o

=

a11 a12 a16

a12 a22 a26

a16 a26 a66

N11

N22

N12

+

b11 b12 b16

b12 b22 b26

b16 b26 b66

κ11

o

κ22

o

κ12

o
25

 (20b)

where

 (21a)

 (21b)

 (21c)

Nonlinear Equilibrium Equations and Boundary Conditions

Equilibrium equations and boundary conditions that are work conjugate to the strains given 

by equations (7)-(10) are obtained by applying the principle of virtual work. The statement of this 

principle for the Donnell-Mushtari-Vlasov theory of quasi-shallow shells is obtained by using 

equations (7), (12), and (13) with the general virtual work statement for a three-dimensional solid 

undergoing "small" strains and "moderately small" rotations (e.g., see Washizu86, pp. 325-327). 

The resulting expression is given by

 (22)

where δWint is the virtual work of the internal stresses and is the virtual work of the external 

surface tractions acting at each point of the shell reference surface A depicted in figure 1. The 

symbol represents the virtual work of the external tractions acting on the curve ∂A that 

encloses the region A, as shown in figure 1. The internal virtual work is given by

 (23a)

In this expression, the virtual strains , etc. are obtained by taking the first variation of the 

strains given by equations (8)-(10). The pointwise external virtual work of the tangential surface 

M11

M22

M12

= −

b11 b12 b16

b12 b22 b26

b16 b26 b66

T

N11

N22

N12

+

d11 d12 d16

d12 d22 d26

d16 d26 d66

κ11

o

κ22

o

κ12

o

a11 a12 a16

a12 a22 a26

a16 a26 a66

=

A11 A12 A16

A12 A22 A26

A16 A26 A66

− 1

b11 b12 b16

b12 b22 b26

b16 b26 b66

= −

A11 A12 A16

A12 A22 A26

A16 A26 A66

− 1

B11 B12 B16

B12 B22 B26

B16 B26 B66

d11 d12 d16

d12 d22 d26

d16 d26 d66

=

D11 D12 D16

D12 D22 D26

D16 D26 D66

−

B11 B12 B16

B12 B22 B26

B16 B26 B66

A11 A12 A16

A12 A22 A26

A16 A26 A66

− 1

B11 B12 B16

B12 B22 B26

B16 B26 B66

δWint

A

A1A2dξ 1dξ 2 = δWext

A

A1A2dξ 1dξ 2 + δWext

B

ds

∂A

δWext

δWext

B

δWint = N11 δε11

o

+ N12 δγ 12

o

+ N22 δε22

o

+ M11 δκ11

o

+ M12 δκ12

o

+ M22 δκ22

o

+ Q1δγ 13

o

+ Q2δγ 23

o

δε11

o
26

tractions q1 and q2 and the normal surface traction q3 is

 (23b)

The surface tractions q1, q2, and q3 are presumed positive-valued in the positive ξ1, ξ2, and ζ

coordinate directions, respectively, as shown in figure 2. The boundary integral in equation (22) 

represents the virtual work of forces per unit length that are applied to the boundary ∂A of the 

region A, and it is implied that the integrand is evaluated on the boundary. The symbol ds 

denotes the boundary differential arc-length coordinate, which is traversed in accordance with the 

surface divergence theorem of Calculus. In the present study, the domain of the surface A is 

given by and , and the boundary curve ∂A consists of four smooth arcs 

given by the constant values of the coordinates ξ1 and ξ2, as depicted in figure 2. For this case, 

the boundary integral is expressed more precisely as

 (23c)

In equation (23c); N, S, and V represent external forces per unit length that are applied normal, 

tangential, and transverse to the given edge, respectively, as shown in figure 2. Likewise, M is a 

moment per unit length with an axis of rotation that is parallel to the given edge, at the given point 

of the boundary. In addition, V contains a contribution due to an applied twisting moment per unit 

length, consistent with the definition of the Kirchhoff shear stress resultant used in classical plate 

and shell theory. 

The equilibrium equations and boundary conditions are obtained by applying "integrationby-parts" formulas, obtained by specialization of the surface divergence theorem, to the first 

integral in equation (22). For two arbitrary differentiable functions and , the 

"integration-by-parts" formulas are given in general form by

 (24a)

 (24b)

δWext = q1δu1 + q2δu2 + q3δw

a1 ≤ ξ 1 ≤ b1 a2 ≤ ξ 2 ≤ b2

δWext

B

ds

∂A

= N ξ 2 δu1 + S ξ 2 δu2 + V ξ 2 δw − M ξ 2

1

A1

∂δw

∂ξ 1 a1

b 1

A2dξ 2

a 2

b 2

+

S ξ 1 δu1 + N ξ 1 δu2 + V ξ 1 δw − M ξ 1

1

A2

∂δw

∂ξ 2 a2

b 2

A1dξ 1

a 1

b 1

f ξ 1,ξ 2 g ξ 1,ξ 2

∂f

∂ξ 1

g dξ 1dξ 2

A

= – f

∂g

∂ξ 1

dξ 1dξ 2

A

+ fg

A2

N • e1 ds

∂A

∂f

∂ξ 2

g dξ 1dξ 2

A

= – f

∂g

∂ξ 2

dξ 1dξ 2

A

+ fg

A1

N • e2 ds

∂A
27

where is the outward unit-magnitude vector field perpendicular to A and ∂A, and and 

are unit-magnitude vector fields that are tangent to the ξ1 and ξ2 coordinate curves, respectively, 

at every point of A and ∂A, as shown in figure 1. Thus, at a given point of ∂A, lies in the surface 

tangent plane at that point. For the specific surface domain given by and 

, and enclosed by the four smooth arcs given by the constant values of the coordinates 

ξ1 and ξ2, equations (24) are expressed as

 (25a)

 (25b)

where

 (25c)

 (25d)

The notation defined by equations (25) is used throughout the present study.

The virtual strains appearing in equation (23a) are given by

 (26a)

 (26b)

 (26c)

 (27a)

N e1 e2

N

a1 ≤ ξ 1 ≤ b1

a2 ≤ ξ 2 ≤ b2

∂f

∂ξ 1

g dξ 1dξ 2

A

= – f

∂g

∂ξ 1

dξ 1dξ 2

A

+ fg a1

b1

dξ 2

a 2

b2

∂f

∂ξ 2

g dξ 1dξ 2

A

= – f

∂g

∂ξ 2

dξ 1dξ 2

A

+ fg a2

b2

dξ 1

a 1

b1

fg a1

b1

≡ f b 1,ξ 2 g b 1,ξ 2 − f a1,ξ 2 g a1,ξ 2

fg a2

b2

≡ f ξ 1,b 2 g ξ 1,b 2 − f ξ 1,a2 g ξ 1,a2

δε11

o

= 1

A1

∂δu1

∂ξ 1

+

δu2

A1A2

∂A1

∂ξ 2

+ δw

R1

− β1 + β1

I 1

A1

∂δw

∂ξ 1

δε22

o

= 1

A2

∂δu2

∂ξ 2

+

δu1

A1A2

∂A2

∂ξ 1

+ δw

R2

− β2 + β2

I 1

A2

∂δw

∂ξ 2

δγ 12

o

= 1

A2

∂δu1

∂ξ 2

– δu2

A1A2

∂A2

∂ξ 1

+ 1

A1

∂δu2

∂ξ 1

– δu1

A1A2

∂A1

∂ξ 2

− β1 + β1

I 1

A2

∂δw

∂ξ 2

− β2 + β2

I 1

A1

∂δw

∂ξ 1

δκ11

o

= 1

A1

∂δψ1

∂ξ 1

+ δψ2

A1A2

∂A1

∂ξ 2
28

 (27b)

 (27c)

 (28a)

 (28b)

By using these virtual strain expressions, applying equations (25) to the first integral in equation 

(22), and enforcing the Fundamental Lemma of the Calculus of Variations (see Reddy87, pp. 107-

108), the equilibrium equations are found to be

 (29a)

 (29b)

 (29c)

 (29d)

 (29e)

where the nonlinear terms are contained in

 (30)

Expanding the derivative terms gives

δκ22

o

= 1

A2

∂δψ2

∂ξ 2

+ δψ1

A1A2

∂A2

∂ξ 1

δκ12

o

= 1

A2

∂δψ1

∂ξ 2

– δψ2

A1A2

∂A2

∂ξ 1

+ 1

A1

∂δψ2

∂ξ 1

– δψ1

A1A2

∂A1

∂ξ 2

δγ 13

o

= δψ1 + 1

A1

∂δw

∂ξ 1

δγ 23

o

= δψ2 + 1

A2

∂δw

∂ξ 2

∂

∂ξ 1

N11A2 + ∂

∂ξ 2

N12A1 − N22

∂A2

∂ξ 1

+ N12

∂A1

∂ξ 2

+ A1A2q1 = 0

∂

∂ξ 1

N12A2 + ∂

∂ξ 2

N22A1 − N11

∂A1

∂ξ 2

+ N12

∂A2

∂ξ 1

+ A1A2q2 = 0

∂

∂ξ 1

Q1A2 + ∂

∂ξ 2

Q2A1 + A1A2 q3 − N11

R1

− N22

R2

+ Pm = 0

∂

∂ξ 1

M11A2 + ∂

∂ξ 2

M12A1 − M22

∂A2

∂ξ 1

+ M12

∂A1

∂ξ 2

− A1A2Q1 = 0

∂

∂ξ 1

M12A2 + ∂

∂ξ 2

M22A1 − M11

∂A1

∂ξ 2

+ M12

∂A2

∂ξ 1

− A1A2Q2 = 0

A1A2Pm = – ∂

∂ξ 1

A2 β1 + β1

I

N11 + β2 + β2

I

N12

– ∂

∂ξ 2

A1 β1 + β1

I

N12 + β2 + β2

I

N22
29

 (31)

and using equations (29a) and (29b) gives the alternate form

 (32)

The boundary conditions that result from enforcing the Fundamental Lemma of the Calculus of 

Variations (see Reddy87, pp. 107-108) consist of two groups. On the edges given by ξ1 = a1 and 

ξ1 = b1, the boundary conditions are

 or (33a)

 or (33b)

 or (33c)

 or (33d)

where

Pm = − 1

A1

∂N11

∂ξ 1

+ 1

A2

∂N12

∂ξ 2

+

N11

A2

1

A1

∂A2

∂ξ 1

+

N12

A1

1

A2

∂A1

∂ξ 2

β1 + β1

I

− 1

A1

∂N12

∂ξ 1

+ 1

A2

∂N22

∂ξ 2

+

N12

A2

1

A1

∂A2

∂ξ 1

+

N22

A1

1

A2

∂A1

∂ξ 2

β2 + β2

I

− N11

1

A1

∂

∂ξ 1

β1 + β1

I

− N22

1

A2

∂

∂ξ 2

β2 + β2

I

− N12

1

A1

∂

∂ξ 1

β2 + β2

I

+ 1

A2

∂

∂ξ 2

β1 + β1

I

Pm = q1 β1 + β1

I

+ q2 β2 + β2

I

– N11

β2 + β2

I

A1

1

A2

∂A1

∂ξ 2

– 1

A1

∂

∂ξ 1

β1 + β1

I

– N22

β1 + β1

I

A2

1

A1

∂A2

∂ξ 1

– 1

A2

∂

∂ξ 2

β2 + β2

I

– N12

1

A2

∂

∂ξ 2

β1 + β1

I

– β1 + β1

I

A1

1

A2

∂A1

∂ξ 2

– N12

1

A1

∂

∂ξ 1

β2 + β2

I

– β2 + β2

I

A2

1

A1

∂A2

∂ξ 1

N11 = N ξ 2 u1 = Δ1 ξ 2

N12 = S ξ 2 u2 = Δ2 ξ 2

Q1 + 1

A2

∂M12

∂ξ 2

– β1 + β1

I

N11 + β2 + β2

I

N12 = V ξ 2 w = Δn ξ 2

M11 = M ξ 2 β1 = Φ ξ2
30

 (33e)

where are applied displacements and is an applied rotation. On the 

edges given by ξ2 = a2 and ξ2 = b2, the boundary conditions are

 or (34a)

 or (34b)

 or (34c)

 or (34d)

where

 (34e)

where are applied displacements and is an applied rotation. In 

addition, "corner conditions" arise that must be satisifed; that is, either M12 or w must be 

specified at the points (a1, a2), (a1, b2), (b1, a2), and (b1, b2).

Nonlinear Compatibility Equation

The compatibility equation of the Donnell-Mushtari-Vlasov theory for a geometrically 

perfect shell is presented in the book by Wempner88 (see p. 616), in terms of tensor analysis. The 

corresponding equation for an imperfect shell is obtained from this equation by replacing the shell 

normal displacement with wI

(ξ1, ξ2) + w(ξ1, ξ2), and then by noting that when w(ξ1, ξ2) = 0 the 

strains vanish and compatibility must be satisfied. The resulting equation for lines-of-curvature 

coordinates, using the notation herein, is given by

 (35)

where

Q1 = 1

A1

∂M11

∂ξ 1

+ 1

A2

∂M12

∂ξ 2

+

M11 − M22

A2

1

A1

∂A2

∂ξ 1

+

2M12

A1

1

A2

∂A1

∂ξ 2

Δ1 ξ 2 , Δ2 ξ 2 , and Δn ξ 2 Φ ξ 2

N22 = N ξ 1 u2 = Δ2 ξ 1

N12 = S ξ 1 u1 = Δ1 ξ 1

Q2 + 1

A1

∂M12

∂ξ 1

– β1 + β1

I

N12 + β2 + β2

I

N22 = V ξ 1 w = Δn ξ 1

M22 = M ξ 1 β2 = Φ ξ1

Q2 = 1

A1

∂M12

∂ξ 1

+ 1

A2

∂M22

∂ξ 2

+

M22 − M11

A1

1

A2

∂A1

∂ξ 2

+

2M12

A2

1

A1

∂A2

∂ξ 1

Δ1 ξ 1 , Δ2 ξ 1 , and Δn ξ 1 Φ ξ 1

1

A1A2

C11 ε11

o

+ C22 ε22

o

+ C12 γ 12

o

+

κ22

o

R1

+

κ11

o

R2

− κ11

o

κ22

o

+ 1

4 κ12

o 2

− κ11

o

κ22

I

− κ11

I

κ22

o

+ 1

2κ12

o

κ12

I

= 0
31

 (36a)

 (36b)

 (36c)

 (37a)

 (37b)

 (37c)

Equation (35) gives the necessary and sufficient conditions for compatible displacements in a 

simply connected domain. For a multiply connected domain, single-valuedness of the 

displacements around each curve enclosing a cutout must be enforced, in addition to equation (35) 

to have compatible displacement fields.

Nondimensional Fields Equations and Parameters

The nondimensionalization procedure used herein follows that given in references 60 and 61 

for symmetrically laminated shells, modified to accomodate generally laminated shells. In 

particular, the only differences in the equations for symmetrically laminated and generally 

laminated shells appear in the constitutive equations. The underyling premise of this procedure is 

to make the field variables and their derivatives quantities with magnitudes on the order of unity, 

to minimize the number of parameters required to characterize the behavior, and to avoid 

introducing a preferential direction, or bias, into the nondimensional equations. This approach is 

intended to enable one to assess the relative importance of terms in the nondimensional field 

equations, and to provide a means for rationalizing similar response characteristics of shells with 

different material composition and geometry. Based on this approach, it follows that it is 

convenient to define the nondimensional normal displacement W for a generally laminated shell 

also by In addition, to facilitate nondimensionalization of the DonnellMushtari-Vlasov equations, it is convenient to introduce the nondimensional arc-length Gaussian 

coordinates of references 60 and 61 given by and , where L1

 and 

C11 ε11

o

= − ∂

∂ξ 1

1

A1

∂A2

∂ξ 1

ε11

o

+

∂

∂ξ 2

A1

A2

∂ε11

o

∂ξ 2

+

1

A2

∂A1

∂ξ 2

ε11

o

C22 ε22

o

= − ∂

∂ξ 2

1

A2

∂A1

∂ξ 2

ε22

o

+

∂

∂ξ 1

A2

A1

∂ε22

o

∂ξ 1

+

1

A1

∂A2

∂ξ 1

ε22

o

C12 γ 12

o

= − ∂

∂ξ 1

1

2

∂γ 12

o

∂ξ 2

+

1

A1

∂A1

∂ξ 2

γ 12

o

− ∂

∂ξ 2

1

2

∂γ 12

o

∂ξ 1

+

1

A2

∂A2

∂ξ 1

γ 12

o

κ11

I

= − 1

A1

∂

∂ξ 1

1

A1

∂wI

∂ξ 1

− 1

A1 A2

2

∂A1

∂ξ 2

∂wI

∂ξ 2

κ22

I

= − 1

A2

∂

∂ξ 2

1

A2

∂wI

∂ξ 2

− 1

A2 A1

2

∂A2

∂ξ 1

∂wI

∂ξ 1

κ12

I

= − A2

A1

∂

∂ξ 1

1

A2

2

∂wI

∂ξ 2

− A1

A2

∂

∂ξ 2

1

A1

2

∂wI

∂ξ 1

W ≡ a11a22D11D22

− 1

4 w.

z1, z2 ξ 1 = L1z1 ξ 2 = L2z2
32

L2 are characteristic dimensions of the reference surface that can be picked to faciliate solution of 

a specific problem. For these coordinates, the surface metric coefficients A1 and A2 are equal to 

unity, which greatly simplifies the shell equations. In the analysis that follows, it is presumed that 

the shell stiffnesses defined in equation (16), and the corresponding compliances, are independent 

of the (ξ1, ξ2) surface coordinates. Unlike previously published studies on nondimensional 

parameters, a complete set of nondimensional field equations are presented subsequently.

Nondimensional Kinematic Equations

First, consider the rotation of the reference surface given by equation (11a). Introducing the 

nondimensional normal displacement and the coordinates into 

equation (11a) gives

 (38)

Defining the nondimensional rotation as 

 (39)

yields

 (40)

Similarly,

 (41)

and

 (42)

Likewise, introducing and the coordinates into equation (8a) 

gives 

 (43)

where is a nondimensional initial geometric imperfection function. By 

defining a nondimensional strain E11, a nondimensional displacement U1 given by

W ≡ a11a22D11D22

− 1

4 w z1, z2

β1 = – 1

L1

a11a22D11D22

1

4 ∂W

∂z1

Ω1 = β1L1 a11a22D11D22

− 1

4

Ω1 = – ∂W

∂z1

Ω2 = β2L2 a11a22D11D22

− 1

4

Ω2 = – ∂W

∂z2

W ≡ a11a22D11D22

− 1

4 w z1, z2

ε11

o

L1

2

a11a22D11D22

1

2

= L1

a11a22D11D22

1

2

∂u1

∂z1

+

L1

2

R1 a11a22D11D22

1

4

W + 1

2

∂W

∂z1

2

+ ∂W

∂z1

∂WI

∂z1

WI ≡ a11a22D11D22

− 1

4 wI
33

 (44)

and introducing the Batdorf-Stein parameter, previously defined in references 60 and 61, as

 (45)

equation (43) is expressed as

 (46a)

Similarly,

 (46b)

 (46c)

where

 (47)

and

 (48)

Furthermore, similar nondimensionalization of the bending strains given by equations (9) yields

 (49a)

 (49b)

U1 ≡ u1L1

a11a22D11D22

1

2

Z1 ≡ L1

2

12R1 a11a22D11D22

1

4

= L1

R1

2

R1

h

h

12 a11a22D11D22

1

4

E11 ≡ ε11

o

L1

2

a11a22D11D22

1

2

= ∂U1

∂z1

+ 12Z1W + 1

2

∂W

∂z1

2

+ ∂W

∂z1

∂WI

∂z1

E22 ≡ ε22

o

L2

2

a11a22D11D22

1

2

= ∂U2

∂z2

+ 12Z2W + 1

2

∂W

∂z2

2

+ ∂W

∂z2

∂WI

∂z2

G12 ≡ γ 12

o

L1L2

a11a22D11D22

1

2

= ∂U1

∂z2

+

∂U2

∂z1

+ ∂W

∂z1

∂W

∂z2

+

∂WI

∂z1

∂W

∂z2

+

∂WI

∂z2

∂W

∂z1

U2 ≡ u2L2

a11a22D11D22

1

2

Z2 ≡ L2

2

12R2 a11a22D11D22

1

4

= L2

R2

2

R2

h

h

12 a11a22D11D22

1

4

K11 ≡ κ11

o

L1

2

a11a22D11D22

1

4

= – ∂

2

W

∂z1

2

K22 ≡ κ22

o

L2

2

a11a22D11D22

1

4

= – ∂

2

W

∂z2

2
34

 (49c)

Nondimensional Constitutive Equations

In deriving a set of nondimensional constitutive equations, it is desirable to keep the number 

of parameters that characterize the material behavior to a minimum. To achieve this goal, and to 

bring clarity to the nondimensionalization procedure, nondimensional constitutive equations 

symmetrically laminated shells are derived first. With these baseline constitutive equations 

established, the approach is extended to obtain nondimensional constitutive equations generally 

laminated shells. The guiding principles for this second case is to develop nondimensional 

constitutive equations for generally laminated shells with as few new nondimensional parameters 

as possible and that retain the nondimensional parameters for symmetrically laminated shells as 

an explicit subset. For example, Khot and Venkayya20 define several nondimensional parameters 

for generally laminated shells in terms of the "reduced" bending stiffnesses defined by equation 

(21c). With their approach, the nondimensional parameters for symmetrically laminated shells do 

not appear explicitly. The signficance of this difference will be indicated subsequently.

Symmetrically laminated shells. For this class of laminates, the first partially inverted 

constitutive equation appearing in equation (20a) reduces to 

 (50)

Multiplying this equation by and using equation (46a) gives

 (51)

This equation, and the others that follow, are simplified further by using the parameters 

 (52a)

 (52b)

 (52c)

K12 ≡ κ12

o

L1L2

a11a22D11D22

1

4

= – 2 ∂

2

W

∂z1∂z2

ε11

o

= a11N11 + a12N22 + a16N12

L1

2

a11a22D11D22

− 1

2

E11 = L1

2 a11N11 + a12N22 + a16N12

a11a22D11D22

1

2

αm ≡ L2

L1

a22

a11

1

4

μ ≡ 2a12 + a66

2 a11a22

γm ≡ − a26

a11a22

3 1

4
35

 (52d)

previously defined in references 60 and 61 and by introducing the generalized Poisson’s ratio 

associated with membrane action defined as

 (52e)

That is, equation (51) becomes

 (53)

Equation (53), and others that follow, are simplified further by introducing the nondimensional 

stress resultants

 (54a)

 (54b)

 (54c)

and using the parameter αb defined in references 60 and 61 as

 (55)

Specifically, equation (53) becomes

 (56a)

Similarly,

δm ≡ − a16

a11

3

a22

1

4

νm ≡ − a12

a11a22

E11 = π2

αm

2

N11L2

2

π2D11D22

− π2νm

N22L1

2

π2D11D22

− π2 δm

αm

L1

L2

D22

D11

1

4 N12L2

2

π2D11D22

3 1 4

N11 ≡ N11L2

2

π2D11D22

N22 ≡ N22L1

2

π2D11D22

N12 ≡ N12L2

2

π2D11D22

3 1 4

αb ≡ L2

L1

D11

D22

1

4

E11 = π2 1

αm

2 N11 − νm N22 − δm

αm

N12

αb
36

 (56b)

 (56c)

Next, consider the constitiutive equation (see equation(16))

 (57)

Using equations (49) and multiplying by gives

 (58)

Equation (58), and others that follow, are simplified further by using equation (55), and by using 

the parameters β, γb

, and δb defined in references 60 and 61 as

 (59a)

 (59b)

 (59c)

In addition, a generalized Poisson’s ratio associated with anticlastic bending action is defined by

 (59d)

Specifically, equation (58) becomes

 (60a)

E22 = π2− νm N11 + αm

2

N22 − αmγm

N12

αb

G12 = π2− δm

αm

N11 − αmγm N22 + 2 μ + νm

N12

αb

M11 = D11κ11

o

+ D12κ22

o

+ D16κ12

o

L2

2

a11a22D11

3

D22

3 − 1

4

M11L2

2

a11a22D11

3

D22

3 1

4

= − L2

2

L1

2

D11

D22

1

2

∂

2

W

∂z1

2 +

D12

D11D22

∂

2

W

∂z2

2 + 2 D16

D11D22

L2

L1

∂

2

W

∂z1∂z2

β ≡ D12 + 2D66

D11D22

γ b ≡ D16

D11

3

D22

1

4

δ b ≡ D26

D11D22

3 1

4

νb ≡ D12

D11D22

M11 ≡ M11L2

2

a11a22D11

3

D22

3 1

4

= − αb

2 ∂

2

W

∂z1

2 + νb

∂

2

W

∂z2

2 + 2αbγ b

∂

2

W

∂z1∂z2
37

Similarly,

 (60b)

 (60c)

Now, equations (56) and (60) have the matrix representations

 (61a)

 (61b)

Generally laminated shells. For this general class of shells, consider the constitutive 

equations given by equations (20a) written in the form

 (62)

by using equations (21). To obtain the desired additional nondimensional parameters, equations 

(46), (49), and (54) are expressed as

M22 ≡ M22L1

2

a11a22D11

3

D22

3 1

4

= − νb

∂

2

W

∂z1

2 + 1

αb

2

∂

2

W

∂z2

2 + 2 δ b

αb

∂

2

W

∂z1∂z2

M12 ≡ M12L1L2

a11a22D11

3

D22

3 1

4

= − αbγ b

∂

2

W

∂z1

2 +

δ b

αb

∂

2

W

∂z2

2 + β − νb

∂

2

W

∂z1∂z2

E11

E22

G12

= π2

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

N11

N22

N12

αb

M11

M22

M12

= −

αb

2

νb αbγ b

νb

1

αb

2

δ b

αb

αbγ b

δ b

αb

β − νb

2

∂

2

W

∂z1

2

∂

2

W

∂z2

2

2 ∂

2

W

∂z1∂z2

ε11

o

ε22

o

γ 12

o

=

a11 a12 a16

a12 a22 a26

a16 a26 a66

N11

N22

N12

−

a11 a12 a16

a12 a22 a26

a16 a26 a66

B11 B12 B16

B12 B22 B26

B16 B26 B66

κ11

o

κ22

o

κ12

o
38

 (63)

 (64)

 (65)

With these expressions, matrix equation (62) becomes

 (66)

Inspection of equations (61) and (66) indicates

ε11

o

ε22

o

γ 12

o

= a11a22D11D22

1

2

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

E11

E22

G12

κ11

o

κ22

o

κ12

o

= a11a22D11D22

1

4

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

K11

K22

K12

N11

N22

N12

= π2D11D22

1

L2

2 0 0

0 1

L1

2 0

0 0 1

L1L2

N11

N22

N12

αb

E11

E22

G12

= π2

a11a22

L1

2

0 0

0 L2

2

0

0 0 L1L2

a11 a12 a16

a12 a22 a26

a16 a26 a66

1

L2

2 0 0

0 1

L1

2 0

0 0 1

L1L2

N11

N22

N12

αb

− 1

a11a22D11D22

1

4

L1

2

0 0

0 L2

2

0

0 0 L1L2

a11 a12 a16

a12 a22 a26

a16 a26 a66

B11 B12 B16

B12 B22 B26

B16 B26 B66

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

K11

K22

K12
39

 (67)

Using equation (67) and the identity 

 (68)

with equation (66) yields

 (69a)

where

 (69b)

or

 (69c)

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

= 1

a11a22

L1

2

0 0

0 L2

2

0

0 0 L1L2

a11 a12 a16

a12 a22 a26

a16 a26 a66

1

L2

2 0 0

0 1

L1

2 0

0 0 1

L1L2

a11 a12 a16

a12 a22 a26

a16 a26 a66

=

a11 a12 a16

a12 a22 a26

a16 a26 a66

1

L2

2 0 0

0 1

L1

2 0

0 0 1

L1L2

L2

2

0 0

0 L1

2

0

0 0 L1L2

E11

E22

G12

=

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

π2

N11

N22

N12

αb

−

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

K11

K22

K12

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

= a11a22

D11D22

1

4

L2

2

0 0

0 L1

2

0

0 0 L1L2

B11 B12 B16

B12 B22 B26

B16 B26 B66

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

= a11a22

D11D22

1

4

B11

L2

2

L1

2 B12 B16

L2

L1

B12 B22

L1

2

L2

2 B26

L1

L2

B16

L2

L1

B26

L1

L2

B66
40

Now, consider the constitutive equation 

 (70)

To obtain the desired additional nondimensional parameters, equations (60) are used to get 

 (71)

Using equations (63), (64), and (71), matrix equation (70) is expressed as

 (72)

Inspection of equations (49), (61), and (72) indicates

 (73)

Thus, using equations (69b) and (73) with equation (72) yields

M11

M22

M12

=

B11 B12 B16

B12 B22 B26

B16 B26 B66

ε11

o

ε22

o

γ 12

o

+

D11 D12 D16

D12 D22 D26

D16 D26 D66

κ11

o

κ22

o

κ12

o

M11

M22

M12

= a11a22D11

3

D22

3 1

4

1

L2

2 0 0

0 1

L1

2 0

0 0 1

L1L2

M11

M22

M12

M11

M22

M12

= a11a22

D11D22

1

4

L2

2

0 0

0 L1

2

0

0 0 L1L2

B11 B12 B16

B12 B22 B26

B16 B26 B66

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

E11

E22

G12

+

1

D11D22

L2

2

0 0

0 L1

2

0

0 0 L1L2

D11 D12 D16

D12 D22 D26

D16 D26 D66

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2

K11

K22

K12

αb

2

νb αbγb

νb

1

αb

2

δ b

αb

αbγb

δ b

αb

β − νb

2

= 1

D11D22

L2

2

0 0

0 L1

2

0

0 0 L1L2

D11 D12 D16

D12 D22 D26

D16 D26 D66

1

L1

2 0 0

0 1

L2

2 0

0 0 1

L1L2
41

 (74)

At this point in the analysis, it is convenient to express equation (69c) as

 (75a)

where

 (75b)

 (75c)

 (75d)

 (75e)

 (75f)

 (75g)

are defined as load-path eccentricity parameters. The partially inverted form of the 

nondimensional constitutive equations derived herein is obtained by expressing equation (69a) as

M11

M22

M12

=

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

E11

E22

G12

+

αb

2

νb αbγb

νb

1

αb

2

δ b

αb

αbγb

δ b

αb

β − νb

2

K11

K22

K12

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

=

αmαbe 11 e 12 αme 16

e 12

e 22

αmαb

e 26

αm

αme 16

e 26

αm

e 66

e 11 ≡ B11

a11

D11

1

2

e 12 ≡ B12

a11a22

D11D22

1

4

e 22 ≡ B22

a22

D22

1

2

e 16 ≡ B16

a11

2

D11D22

1

4

e 26 ≡ B26

a22

2

D11D22

1

4

e 66 ≡ B66

a11a22

D11D22

1

4
42

 (76a)

where

 (76b)

Using equation (75a) gives

 (77a)

 (77b)

 (77c)

 (77d)

 (77e)

 (77f)

 (77g)

 (77h)

 (77i)

E11

E22

G12

= π2

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

N11

N22

N12

αb

+

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

∂

2

W

∂z1

2

∂

2

W

∂z2

2

2 ∂

2

W

∂z1∂z2

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

=

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

b 11 b 12 b 16

b 12 b 22 b 26

b 16 b 26 b 66

B 11 = αb

αm

e 11 − νm e 12 − δm e 16

B 12 = 1

αm

2 e 12 − νmαm

αb

e 22 − δm e 26

B 16 = 1

αm

e 16 − νm e 26 − δm e 66

B 21 = αm

2

e 12 − νmαb

αm

e 11 − γme 16

B 22 = αm

αb

e 22 − νme 12 − γme 26

B 26 = αm e 26 − νm e 16 − γme 66

B 61 = αm 2 μ + νm e 16 − δmαb

αm

e 11 − γm e 12

B 62 = 1

αm

2 μ + νm e 26 − δm e 12 − αmγm

αb

e 22

B 66 = 2 μ + νm e 66 − δm e 16 − γm e 26
43

where it is noted that all of equations (77) vanish for symmetrically laminated shells. Now, 

substituting equation (76a) into equation (74), and using equation (75a) yields

 (78a)

where

 (78b)

In expanded form,

 (79a)

 (79b)

 (79c)

 (79d)

 (79e)

M11

M22

M12

= π2

B 11 B 21 B 61

B 12 B 22 B 62

B 16 B 26 B 66

N11

N22

N12

αb

−

d 11 d 12 d 16

d 12 d 22 d 26

d 16 d 26 d 66

∂

2

W

∂z1

2

∂

2

W

∂z2

2

2 ∂

2

W

∂z1∂z2

d 11 d 12 d 16

d 21 d 22 d 26

d 61 d 62 d 66

=

αb

2

νb αbγb

νb

1

αb

2

δ b

αb

αbγb

δ b

αb

β − νb

2

−

αmαbe 11 e 12 αme 16

e 12

e 22

αmαb

e 26

αm

αme 16

e 26

αm

e 66

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

d 11 = αb

2

1 − e 11

2

+ αm

2

2 αbνm

αm

e 11e 12 − e 12

2

+ 2 αbδm

αm

e 11 + γm e 12 e 16 − 2 μ + νm e 16

2

d 12 = νb + νm e 11e 22 + e 12

2

− αb

αm

e 11 +

αm

αb

e 22 e 12 + δm

αb

αm

e 11e 26 + e 12e 16 +

γm

αm

αb

e 16e 22 + e 12e 26 − 2 μ + νm e 16e 26

d 16 = αb γb + δm e 66 − e 16 + νm e 26 e 11 + αm γm e 66 − e 26 + νm e 16 e 12

+ αm δm e 16 + γm e 26 − 2 μ + νm e 66 e 16

d 22 = 1

αb

2 1 − e 22

2 + 1

αm

2 2 αmνm

αb

e 12e 22 − e 12

2

+ 2 αmγm

αb

e 22 + δm e 12 e 26 − 2 μ + νm e 26

2

d 26 = 1

αb

δ b + γme 66 + νm e 16 − e 26 e 22 + 1

αm

δm e 66 − e 16 + νm e 26 e 12

+ 1

αm

δm e 16 + γm e 26 − 2 μ + νm e 66 e 26
44

 (79f)

Equations (76a) and (78a) define a set of partially inverted constitutive equations that are 

expressed in terms of the nondimensional parameters defined in references 60 and 61 for 

symmetrically laminated shells and six new nondimensional parameters defined by equations (75) 

that characterize the anisotropies associated with coupling between membrane and bending 

deformations. This formulation is different from those used in references 18-20 and 71 in that all 

of the nondimensional parameters are defined in terms of the stiffnesses appearing in equation 

(16) and not the reduced stiffnesses appearing in equations (20b), defined by equations (21c). The 

utility of the formulation presented herein is revealed by inspection of equations (77) and (79); 

that is, the equations contain the fewest number of parameters needed to characterize fully the 

load path eccentricity associated with subscripted b-terms appearing in the constitutive equations 

given by equations (20). Moreover, equations (77) and (79) show explicitly the coupling between 

all forms of orthotropy and anisotropy that can occur in a generally laminated shell.

Nondimensional Equilibrium Equations

In this section of the present study, nondimensional equilibrium equations are obtained by 

direct nondimensionalization of the equilibrium equations given by equations (29). Specifically, 

introducing the coordinates into equation (29a) gives

 (80)

Multiplying by and using equations (54) gives

 (81)

where

 (82)

Similarly, equation (29b) becomes

 (83)

where

d 66 = 1

2 β − νb − 2 μ + νm e 66

2

+ 2νm e 16e 26 − e 16

2

− e 26

2

+ 2 δm e 16 + γm e 26 e 66

z1, z2

1

L1

∂N11

∂z1

+ 1

L2

∂N12

∂z2

+ q1 = 0

L1L2

2

π2D11D22

∂N11

∂z1

+ 1

αb

∂N12

∂z2

+ q 1 = 0

q 1 ≡ q1L1L2

2

π2D11D22

1

αb

∂N12

∂z1

+

∂N22

∂z2

+ q 2 = 0
45

 (84)

Next, introducing the coordinates into equation (29d) gives

 (85)

Multiplying equation (85) by and then using equations (60) gives

 (86)

where

 (87)

Similarly, equation (29e) becomes

 (88)

where

 (89)

Likewise, introducing the coordinates into equation (29c) gives

 (90)

Multiplying equation (90) by and using equations (87) and (89) gives

 (91)

where

q 2 ≡ q2L1

2

L2

π2D11D22

z1, z2

1

L1

∂M11

∂z1

+ 1

L2

∂M12

∂z2

– Q1 = 0

L1L2

2

a11a22D11

3

D22

3 − 1

4

∂M11

∂z1

+

∂M12

∂z2

– Q1 = 0

Q1 ≡ Q1L1L2

2

a11a22D11

3

D22

3 1

4

∂M12

∂z1

+

∂M22

∂z2

– Q2 = 0

Q2 ≡ Q2L1

2

L2

a11a22D11

3

D22

3 1

4

z1, z2

1

L1

∂Q1

∂z1

+ 1

L2

∂Q2

∂z2

+ q3 – N11

R1

– N22

R2

+ Pm = 0

L1

2

L2

2

a11a22D11

3

D22

3 − 1

4

∂Q1

∂z1

+

∂Q2

∂z2

+ q 3 + L1

2

L2

2

a11a22D11

3

D22

3 − 1

4

Pm – N11

R1

– N22

R2

= 0
46

 (92)

Next, using equations (45), (48), and (54) with equation (91) gives

 (93)

where

 (94)

The specific form of Pm is obtained by first introducing the coordinates into equation 

(30), which gives

 (95)

By using equations (39)-(42), (54), and (94); and noting that 

 (96a)

 (96b)

equation (95) yields

 (97a)

Expanding the derivatives of the bracketed terms and using equations (81) and (83) give the 

alternate form

q 3 ≡ q3L1

2

L2

2

a11a22D11

3

D22

3 1

4

∂Q1

∂z1

+

∂Q2

∂z2

+ q 3 – π212 N11Z1 + N22Z2 + Pm = 0

Pm ≡ PmL1

2

L2

2

a11a22D11

3

D22

3 1

4

z1, z2

Pm = – 1

L1

∂

∂z1

β1 + β1

I

N11 + β2 + β2

I

N12

– 1

L2

∂

∂z2

β1 + β1

I

N12 + β2 + β2

I

N22

β1

I

= – 1

L1

a11a22D11D22

1

4 ∂WI

∂z1

β2

I

= – 1

L2

a11a22D11D22

1

4 ∂WI

∂z2

Pm = π2 ∂

∂z1

N11

∂

∂z1

W + WI +

N12

αb

∂

∂z2

W + WI

+ π2 ∂

∂z2

N12

αb

∂

∂z1

W + WI + N22

∂

∂z2

W + WI
47

 (97b)

Nondimensional Boundary Conditions

In terms of the nondimensional coordinates, the boundary conditions are defined at the edges 

given by constant values of z1 and z2. On the edges given by z1 = a1 / L1 and z1 = b1 / L1, the 

boundary conditions given by equations (33) become

 or (98a)

 or (98b)

 

or (98c)

 or (98d)

where Q1 is given by equation (86). On the edges given by z2 = a2 / L2 and z2 = b2 / L2, the 

boundary conditions specified by equations (34) become

 or (99a)

 or (99b)

Pm = − π2q 1

∂

∂z1

W + WI + q 2

∂

∂z2

W + WI

+ π2N11

∂

2

∂z1

2 W + WI + N22

∂

2

∂z2

2 W + WI + 2 N12

αb

∂

2

∂z1∂z2

W + WI

N11 = N z2 L2

2

π2D11D22

≡ N z2

U1 = Δ1 z2 L1

a11a22D11D22

1

2

≡ Δ1 z2

N12 = S z2 L2

2

π2D11D22

3 1 4 ≡ S z2 U2 = Δ2 z2 L2

a11a22D11D22

1

2

≡ Δ2 z2

Q1 +

∂M12

∂z2

+ π2N11

∂

∂z1

W + WI + π2

αb

N12

∂

∂z2

W + WI = V z2 L1L2

2

a11a22D11

3

D22

3

1

4

≡ V z2

W = Δn z2

a11a22D11D22

1

4

≡ Δn z2

M11 = M z2 L2

2

a11a22D11

3

D22

3

1

4

≡ M z2 – ∂W

∂z1

= Φ z2 L1

a11a22D11D22

1

4

≡ Φ z2

N22 = N z1 L1

2

π2D11D22

≡ N z1

U2 = Δ2 z1 L2

a11a22D11D22

1

2

≡ Δ2 z1

N12 = S z1 L2

2

π2D11D22

3 1 4 ≡ S z1 U1 = Δ1 z1 L1

a11a22D11D22

1

2

≡ Δ1 z1
48

 

or (99c)

 or (99d)

where Q2 is given by equation (88). 

Nondimensional Compatibility Equation

Nondimensional compatibility equations for geometrically imperfect shells are obtained by 

introducing the nondimensional arc-length Gaussian coordinates into equations (36) and 

(37). This step gives

 (100a)

 (100b)

 (100c)

 (101a)

 (101b)

 (101c)

Using equations (100), it follows that the first three terms of equation (35) becomes

Q2 +

∂M12

∂z1

+ π2

αb

N12

∂

∂z1

W + WI + π2N22

∂

∂z2

W + WI = V z1 L1

2

L2

a11a22D11

3

D22

3

1

4

≡ V z1

W = Δn z1

a11a22D11D22

1

4

≡ Δn z1

M22 = M z1 L1

2

a11a22D11

3

D22

3

1

4

≡ M z1 – ∂W

∂z2

= Φ z1 L2

a11a22D11D22

1

4

≡ Φ z1

z1, z2

C11 ε11

o

= 1

L2

2

∂

2

ε11

o

∂z2

2

C22 ε22

o

= 1

L1

2

∂

2

ε22

o

∂z1

2

C12 γ 12

o

= − 1

L1L2

∂

2

γ 12

o

∂z1∂z2

κ11

I

= − 1

L1

2

∂

2

wI

∂z1

2

κ22

I

= − 1

L2

2

∂

2

wI

∂z2

2

κ12

I

= − 2

L1L2

∂

2

wI

∂z1∂z2
49

 (102)

and using the definitions in equations (46) with the previous expression gives 

 (103)

Similarly, substituting into equations (101) and then using the results 

with equations (45), (48), and (49) gives the second part of equation (35) as

 (104)

Substituting equations (103) and (104) into equation (35) gives the nondimensional compatibility 

equation

 (105)

Nondimensional Virtual Work

A nondimensional form of the principle of virtual work given by equation (22) is obtained by 

by first taking the variation of the nondimensional displacements given by equations (44), (47), 

and the definition ; and strains given by equations (46) and (49). This step 

produces

 (106a)

1

A1A2

C11 ε11

o

+ C22 ε22

o

+ C12 γ 12

o

= 1

L2

2

∂

2

ε11

o

∂z2

2 + 1

L1

2

∂

2

ε22

o

∂z1

2 − 1

L1L2

∂

2

γ 12

o

∂z1∂z2

1

A1A2

C11 ε11

o

+ C22 ε22

o

+ C12 γ 12

o

= a11a22D11D22

1

2

L1

2

L2

2

∂

2

E11

∂z2

2 +

∂

2

E22

∂z1

2 − ∂

2

G12

∂z1∂z2

wI = a11a22D11D22

1

4 WI

κ22

o

R1

+

κ11

o

R2

− κ11

o

κ22

o

+ 1

4 κ12

o 2

− κ11

o

κ22

I

− κ11

I

κ22

o

+ 1

2κ12

o

κ12

I

=

− a11a22D11D22

1

2

L1

2

L2

2 12 Z1

∂

2

W

∂z2

2 + Z2

∂

2

W

∂z1

2

+

a11a22D11D22

1

2

L1

2

L2

2

∂

2

W

∂z1∂z2

2

− ∂

2

W

∂z1

2

∂

2

W

∂z2

2 − ∂

2

W

∂z1

2

∂

2

WI

∂z2

2 − ∂

2

W

∂z2

2

∂

2

WI

∂z1

2 + 2

∂

2

W

∂z1∂z2

∂

2

WI

∂z1∂z2

∂

2

E11

∂z2

2 +

∂

2

E22

∂z1

2 − ∂

2

G12

∂z1∂z2

= 12 Z1

∂

2

W

∂z2

2 + Z2

∂

2

W

∂z1

2 +

∂

2

W

∂z1

2

∂

2

W

∂z2

2 − ∂

2

W

∂z1∂z2

2

+

∂

2

W

∂z1

2

∂

2

WI

∂z2

2 +

∂

2

W

∂z2

2

∂

2

WI

∂z1

2 − 2

∂

2

W

∂z1∂z2

∂

2

WI

∂z1∂z2

w ≡ a11a22D11D22

1

4 W

δU1 ≡ δu1L1

a11a22D11D22

1

2
50

 (106b)

 (106c)

 (107a)

 (107b)

 (107c)

 (108a)

 (108b)

 (108c)

Using these definitions and the definitions in equations (54) and (60) with the internal virtual work 

per unit area given by equation (23a) and requiring the transverse shearing strains to vanish gives

 (109a)

Similarly, the external work per unit area given by equation (23b) becomes

 (109b)

δU2 ≡ δu2L2

a11a22D11D22

1

2

δW ≡ δw

a11a22D11D22

1

4

δE11 ≡ δε 11

o

L1

2

a11a22D11D22

1

2

= ∂δU1

∂z1

+ 12Z1 δW + ∂W

∂z1

+

∂WI

∂z1

∂δW

∂z1

δE22 ≡ δε22

o

L2

2

a11a22D11D22

1

2

= ∂δU2

∂z2

+ 12Z2 δW + ∂W

∂z2

+

∂WI

∂z2

∂δW

∂z2

δG12 ≡ δγ 12

o

L1L2

a11a22D11D22

1

2

= ∂δU1

∂z2

+

∂δU2

∂z1

+ ∂W

∂z2

+

∂WI

∂z2

∂δW

∂z1

+ ∂W

∂z1

+

∂WI

∂z1

∂δW

∂z2

δK11 ≡ δκ11

o

L1

2

a11a22D11D22

1

4

= – ∂

2

δW

∂z1

2

δK22 ≡ δκ22

o

L2

2

a11a22D11D22

1

4

= – ∂

2

δW

∂z2

2

δK12 ≡ δκ12

o

L1L2

a11a22D11D22

1

4

= – 2

∂

2

δW

∂z1∂z2

δWint ≡ δWint L1

2

L2

2

a11a22

1

2

D11D22

=

π2

N11 δE11 +

π2

αb

N12 δG12 + π2N22 δE22 + M11 δK11 + M12 δK12 + M22 δK22

δWext ≡ δWext L1

2

L2

2

a11a22

1

2

D11D22

= π2q 1 δU1 + π2q 2 δU2 + q 3δW
51

Multiplying equation (22) by gives the principle of virtual work in 

nondimensional form as

 (110)

where A is the nondimensional domain given by and , ∂A denotes 

the boundary of A, ds denotes the nondimensional arc-length coordinate for ∂A, and where

 (111)

The explicit expression for virtual work of the applied loads acting on ∂A is obtained from 

equation (23c) as follows. From equations (38)-(42) it follows that

 (112a)

 (112b)

Introducing the nondimensional coordinates (z1, z2) and substituting equations (106) and (112) 

into equation (111), and making use of the notation for the nondimensional boundary loads 

defined in equations (98) and (99) yields

 (113)

L1L2

a11a22

1

2

D11D22

δWint

A

dz1dz2 = δWext

A

dz1dz2 + δWext

B

ds

∂A

a 1

L1

≤ z1 ≤ b 1

L1

a 2

L2

≤ z2 ≤ b 2

L2

δWext

B

ds

∂A

≡ L1L2

a11a22

1

2

D11D22

δWext

B

ds

∂A

δβ1 = – 1

L1

a11a22D11D22

1

4 ∂δW

∂z1

δβ2 = – 1

L2

a11a22D11D22

1

4 ∂δW

∂z2

δWext

B

ds

∂A

= π2N z2 δU1 +

π2

αb

S z2 δU2 + V z2 δW − M z2

∂δW

∂z1 a1

L1

b1

L1

dz2

a2

L2

b2

L2

+

π2

αb

S z1 δU1 + π2N z1 δU2 + V z1 δW − M z1

∂δW

∂z2 a2

L2

b2

L2

dz1

a1

L1

b1

L1
52

Nondimensional Stress-Function Formulation

The stress-function formulation of the Donnell-Mushtari-Vlasov equations is often used to 

facilitiate solution of practical problems by reducing the number of unknown functions to two. 

These two unknowns are the normal displacement w(ξ1, ξ2) and a stress function F(ξ1, ξ2). In 

particular, the stress resultants N11, N22, and N12 are defined in terms of derivatives of a stress 

function such that the equilibrium equations given by equations (29a) and (29b) are satisfied 

identically. As a result, the compatibility equation given by equation (35) must be satisfied. 

Similarly, moment equilibrium equations (21d) and (21e) are used to eliminate the transverse 

shear stress resultants in the force equilibrium equation given by equation (29c). In addition, N11

and N22 in the force equilibrium equation are expressed in terms of the stress function. Next, the 

constitutive equations given by equations (20a), the bending strains given by equations (8), and 

the rotations defined by equations (10), neglecting transverse shearing deformations, is used to 

express the membrane strains in terms of w(ξ1, ξ2) and F(ξ1, ξ2). Likewise, equations (20b) give the 

bending stress resultants in terms of w(ξ1, ξ2) and F(ξ1, ξ2). With these modified constitutive 

equations, the compatibility equation and the transverse force equilibrium equation are expressed 

as two coupled nonlinear partial differential equations in terms of the normal displacement 

w(ξ1, ξ2) and the stress function F(ξ1, ξ2). A similar process that uses the nondimensional field 

equations derived herein directly is presented in this section first. Then, the corresponding 

expressions for the virtual work and complementary virtual work, are presented that are useful for 

for solving boundary-value problems by direct variational methods. A similar similar approach 

has been given by Zhang and Matthews89 that uses the basic integral forms that are used to derive 

the principles of virtual work and complementary virtual work (see Washizu86, pp. 18-32 for a 

general treatment). In the last part of this section, simplifications to the stress-function 

formulation are presented for cases of practical importance. 

Field Equations in terms of W and F

The nondimensional stress-function formulation is obtained by first using equations (86) and 

(88) to reduce the number of independent equilibrium equations to three force balance equations 

by eliminating Q1

 and Q2. This step yields

 (114)

for the force balance, in the direction normal to the tangent plane at a given point of the shell 

reference surface, given by equation (93). Next, a stress function is defined that satisfies the 

tangential-equilibrium equations, equations (81) and (83), indentically. Let denote 

the stress function defined by

 (115a)

∂

2

M11

∂z1

2 + 2 ∂

2

M12

∂z1∂z2

+

∂

2

M22

∂z2

2 + q 3 – π2

12 N11Z1 + N22Z2 + Pm = 0

F = F z1, z2

π2N11 = ∂

2

F

∂z2

2 − π2

q 1 dz1
53

 (115b)

 (115c)

such that equations (81) and (83) are satisfied identically. Equation (114) becomes

 (116a)

where PT is a known function given by

 (116b)

In addition, equation (97a) becomes

 (117a)

where D

q

 is a linear differential operator defined as

 (117b)

and L is a bilinear differential operator defined as

 (117c)

The next step in the stress-function formulation is to express the partially inverted 

constitutive equations, equations (76a) and (78a), in terms of the stress function. This action 

yields

π2N22 = ∂

2

F

∂z1

2 − π2

q 2 dz2

π2

αb

N12 = − ∂

2

F

∂z1∂z2

∂

2

M11

∂z1

2 + 2 ∂

2

M12

∂z1∂z2

+

∂

2

M22

∂z2

2 + q 3 – 12 Z1

∂

2

F

∂z2

2 + Z2

∂

2

F

∂z1

2 + Pm = PT

PT = − π212 Z1 q 1 dz1 + Z2 q 2 dz2 = PT z1, z2

Pm = − DqW + WI + L F, W + WI

Dq

W + WI ≡ π2 ∂

∂z1

q 1 dz1

∂

∂z1

W + WI + π2 ∂

∂z2

q 2 dz2

∂

∂z2

W + WI

L F, W + WI ≡ ∂

2

F

∂z2

2

∂

2

∂z1

2 W + WI +

∂

2

F

∂z1

2

∂

2

∂z2

2 W + WI − 2 ∂

2

F

∂z1∂z2

∂

2

∂z1∂z2

W + WI
54

 (118)

where

 (119)

and

 (120)

where

 (121)

Substituting equations (120) into equation (116a), and using equations (116b) and (117), give the 

stress-function form of the transverse equilibrium equation as

 (122a)

where Db, Dc

, and DB are linear differential operators defined as

E11

E22

G12

=

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

∂

2

F

∂z2

2

∂

2

F

∂z1

2

− ∂

2

F

∂z1∂z2

+

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

∂

2

W

∂z1

2

∂

2

W

∂z2

2

2 ∂

2

W

∂z1∂z2

−

p1

p2

p3

p1

p2

p3

=

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

π2q 1 dz1

π2q 2 dz2

0

M11

M22

M12

=

B 11 B 21 B 61

B 12 B 22 B 62

B 16 B 26 B 66

∂

2

F

∂z2

2

∂

2

F

∂z1

2

− ∂

2

F

∂z1∂z2

−

d 11 d 12 d 16

d 12 d 22 d 26

d 16 d 26 d 66

∂

2

W

∂z1

2

∂

2

W

∂z2

2

2 ∂

2

W

∂z1∂z2

−

m1

m2

m3

m1

m2

m3

=

B 11 B 21 B 61

B 12 B 22 B 62

B 16 B 26 B 66

π2q 1 dz1

π2q 2 dz2

0

Db W + 12 Dc F − DB F =

L F, W + WI − Dq

W + WI + q 3 − PT − ∂

2

m1

∂z1

2 + 2 ∂

2

m3

∂z1∂z2

+

∂

2

m2

∂z2

2
55

 (122b)

 (122c)

and

 (122d)

Since the tangential equilibrium equations are satisfied identically, the tangential strains must 

satisfy the compatibility equation given by equation (105). By using the operators defined by 

equations (117c) and (122c), the compatibility equation given by equation (105) is expressed as 

 (123a)

where, in particular, 

 (123b)

Substituting equations (118) into equation (123a) gives the nondimensional compatibility 

equation

 (124a)

where

 (124b)

Expressions for the tangential displacements U1 and U2 are obtained from the nondimensional 

strain-displacement relations, equations (46); that is

 (125)

Db W ≡ d 11

∂

4

W

∂z1

4 + 4d 16

∂

4

W

∂z1

3

∂z2

+ 2 d 12 + 2d 66

∂

4

W

∂z1

2

∂z2

2 + 4d 26

∂

4

W

∂z1∂z2

3 + d 22

∂

4

W

∂z2

4

Dc F ≡ Z1

∂

2

F

∂z2

2 + Z2

∂

2

F

∂z1

2

DB F ≡ B 21

∂

4

F

∂z1

4 + 2B 26 − B 61

∂

4

F

∂z1

3

∂z2

+ B 11 + B 22 − 2B 66

∂

4

F

∂z1

2

∂z2

2

+ 2B 16 − B 62

∂

4

F

∂z1∂z2

3 + B 12

∂

4

F

∂z2

4

∂

2

E11

∂z2

2 +

∂

2

E22

∂z1

2 − ∂

2

G12

∂z1∂z2

= 12 Dc W − 1

2L W, W + 2WI

1

2L W, W + 2WI = ∂

2

W

∂z1

2

∂

2

W

∂z2

2 − ∂

2

W

∂z1∂z2

2

+

∂

2

W

∂z1

2

∂

2

WI

∂z2

2 +

∂

2

W

∂z2

2

∂

2

WI

∂z1

2 − 2

∂

2

W

∂z1∂z2

∂

2

WI

∂z1∂z2

Dm F + DB W − 12 Dc W

+ 1

2L W, W + 2WI = ∂

2

p1

∂z2

2 +

∂

2

p2

∂z1

2 − ∂

2

p3

∂z1∂z2

Dm F ≡ αm

2 ∂

4

F

∂z1

4 + 2αmγm

∂

4

F

∂z1

3

∂z2

+ 2μ

∂

4

F

∂z1

2

∂z2

2 + 2 δm

αm

∂

4

F

∂z1∂z2

3 +

1

αm

2

∂

4

F

∂z2

4

∂U1

∂z1

= E11 − 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1
56

 (126)

 (127)

Substituting equations (118) into these three expressions gives

 (128)

 (129)

 (130)

The nondimensional tangential displacements are represented in terms of W and F, to within a 

rigid-body motion, by the integrals of these three equations.

Next, the desired form of the boundary conditions are obtained by using equations (86) and 

(88) to express the nondimensional transverse shear stress resultants in terms of the 

nondimensional bending stress resultants, by using equations (115) to express the nondimensional 

membrane stress resultants in terms of F, and by using equations (120) to express the 

nondimensional bending stress resultants in term of W and F. On the edges given by z1 =

a1 / L1 and z1 = b1 / L1, the boundary conditions given by equations (98) become

 or (131a)

 or (131b)

∂U2

∂z2

= E22 − 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

∂U1

∂z2

+

∂U2

∂z1

= G12 − ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1

∂U1

∂z1

= 1

αm

2

∂

2

F

∂z2

2 − νm

∂

2

F

∂z1

2 +

δm

αm

∂

2

F

∂z1∂z2

+ B 11

∂

2

W

∂z1

2 + B 12

∂

2

W

∂z2

2 + 2B 16

∂

2

W

∂z1∂z2

− p1 − 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1

∂U2

∂z2

= − νm

∂

2

F

∂z2

2 + αm

2 ∂

2

F

∂z1

2 + αmγm

∂

2

F

∂z1∂z2

+ B 21

∂

2

W

∂z1

2 + B 22

∂

2

W

∂z2

2 + 2B 26

∂

2

W

∂z1∂z2

− p2 − 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

∂U1

∂z2

+

∂U2

∂z1

= − δm

αm

∂

2

F

∂z2

2 − αmγm

∂

2

F

∂z1

2 − 2 μ + νm

∂

2

F

∂z1∂z2

+ B 61

∂

2

W

∂z1

2 + B 62

∂

2

W

∂z2

2

+ 2B 66

∂

2

W

∂z1∂z2

− p3 − ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1

∂

2

F

∂z2

2 = π2

N z2 + π2q 1 dz1

z 1 = constant

U1 = Δ1 z2

− ∂

2

F

∂z1∂z2

= π2

αb

S z2 U2 = Δ2 z2
57

or (131c)

 

or (131d)

On the edges given by z2 = a2 / L2 and z2 = b2 / L2, the boundary conditions given by equations 

(99) become

 

or (132c)

 

or (132d)

Virtual Work in terms of W and F

The virtual work statement given by equations (107) - (110) is expressed in terms of the stress 

function F and the displacement W by using equations (115) and the integration-by-parts 

B 21

∂

3

F

∂z1

3 + 2B 26 − B 61

∂

3

F

∂z1

2

∂z2

+ B 11 − 2B 66

∂

3

F

∂z1∂z2

2 + 2B 16

∂

3

F

∂z2

3 − d 11

∂

3

W

∂z1

3 − 4d 16

∂

3

W

∂z1

2

∂z2

− d 12 + 4d 66

∂

3

W

∂z1∂z2

2 − 2d 26

∂

3

W

∂z2

3 +

∂

2

F

∂z2

2 − π2

q 1 dz1

∂

∂z1

W + WI

− ∂

2

F

∂z1∂z2

∂

∂z2

W + WI = V z2 +

∂m1

∂z1

+ 2

∂m3

∂z2

W = Δn z2

B 11

∂

2

F

∂z2

2 + B 21

∂

2

F

∂z1

2 − B 61

∂

2

F

∂z1∂z2

− d 11

∂

2

W

∂z1

2 − d 12

∂

2

W

∂z2

2 − 2d 16

∂

2

W

∂z1∂z2

= M z2 + m1

– ∂W

∂z1

= Φ z2

2B 26

∂

3

F

∂z1

3 + B 22 − 2B 66

∂

3

F

∂z1

2

∂z2

+ 2B 16 − B 62

∂

3

F

∂z1∂z2

2 + B 12

∂

3

F

∂z2

3 − 2d 16

∂

3

W

∂z1

3

− d 12 + 4d 66

∂

3

W

∂z1

2

∂z2

− 4d 26

∂

3

W

∂z1∂z2

2 − d 22

∂

3

W

∂z2

3 +

∂

2

F

∂z1

2 − π2

q 2 dz2

∂

∂z2

W + WI

− ∂

2

F

∂z1∂z2

∂

∂z1

W + WI = V z1 +

∂m2

∂z2

+ 2

∂m3

∂z1

W = Δn z1

B 12

∂

2

F

∂z2

2 + B 22

∂

2

F

∂z1

2 − B 62

∂

2

F

∂z1∂z2

− d 12

∂

2

W

∂z1

2 − d 22

∂

2

W

∂z2

2 − 2d 26

∂

2

W

∂z1∂z2

= M z1 + m2

– ∂W

∂z2

= Φ z1
58

formulas, equations (25), specialized for the nondimensional coordinates (z1, z2). In particular, 

applying equations (25) to the terms , , and 

appearing in , , and , respectively, and using equations (81), 

(83), and (108) yield the variational statement

 (133)

where 

 (134a)

 (134b)

where the subscripts and superscripts on the braces indicate the integrand evaluated at the upper 

limit minus the integrand evaluated at the lower limit, and where

 (135a)

 (135b)

π2N11

∂δU1

∂z1

π2N22

∂δU2

∂z2

π2

αb

N12

∂δU1

∂z2

+

∂δU2

∂z1

π2N11 δE11 π2N22 δE22

π2

αb

N12 δG12

q 3 − π212 Z1 Z2 0 N δW − M TδK

A

− π2Ω + ΩI

T

N δΩ dz1dz2 + I 1

B

+ I 2

B

= 0

I1

B

= π2N z2 − N11 δU1 +

π2

αb

S z2 − N12 δU2 + V z2 δW − M z2

∂δW

∂z1 a1

L 1

b 1

L 1

dz2

a 2

L 2

b 2

L 2

I2

B

= π2

αb

S z1 − N12 δU1 + π2N z1 − N22 δU2 + V z1 δW − M z1

∂δW

∂z2 a2

L 2

b 2

L 2

dz1

a 1

L 1

b 1

L 1

N T= N11 N22

N12

αb

M T= M11 M22 M12
59

 (135c)

 (136a)

 (136b)

 (136c)

 (136d)

In particular, the last term of the integrand in equation (133) becomes

 (136e)

Typically, the stress function is selected to satisfy the tangential boundary conditions and, as a 

result, the boundary integrals reduce to

 (137a)

N = N11

N12

αb

N12

αb

N22

Ω T= − ∂W

∂z1

− ∂W

∂z2

ΩI

T

= − ∂WI

∂z1

− ∂WI

∂z2

δΩ T= − ∂δW

∂z1

− ∂δW

∂z2

δK T= − ∂

2

δW

∂z1

2 − ∂

2

δW

∂z2

2 − 2

∂

2

δW

∂z1∂z2

π2Ω + ΩI

T

N δΩ = π2N11

∂

∂z1

W + WI +

π2

αb

N12

∂

∂z2

W + WI

∂δW

∂z1

+

π2

αb

N12

∂

∂z1

W + WI + π2N22

∂

∂z2

W + WI

∂δW

∂z2

I1

B

= V z2 δW − M z2

∂δW

∂z1 a1

L 1

b 1

L 1

dz2

a 2

L 2

b 2

L 2
60

 (137b)

for that special case. Next, it is convenient to express equation (120) as

 (138)

where

 (139a)

 (139b)

 (139c)

 (139d)

 (139e)

By using these equations, and noting that equations (115) give

 (140a)

with

 (140b)

equation (133) is expressed as

I2

B

= V z1 δW − M z1

∂δW

∂z2 a2

L 2

b 2

L 2

dz1

a 1

L 1

b 1

L 1

M = B T∂F + d K − m

K T= − ∂

2

W

∂z1

2 − ∂

2

W

∂z2

2 − 2

∂

2

W

∂z1∂z2

B =

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

d =

d 11 d 12 d 16

d 12 d 22 d 26

d 16 d 26 d 66

m T= m1 m2 m3

∂F T= ∂

2

F

∂z2

2

∂

2

F

∂z1

2 − ∂

2

F

∂z1∂z2

π2N = ∂F − π2q

q T= q 1 dz1 q 2 dz2 0
61

(141)

where

 (142a)

 (142b)

Equations (141) and (137) constitute the virtual work in terms of the two unknowns, the stress 

function F and the displacement W, and can be used as an alternative to equilibrium equation 

(122a). 

Complementary Virtual Work in terms of W and F

The complementary virtual work is obtained by considering the work of incompatible 

tangential strains and violated kinematic boundary conditions generated when the shell is 

subjected to statically admissible internal stresses. The statically admissible stress resultants are 

denoted by , , and ; and as a result satisfy the equilibrium equations

 (143a)

 (143b)

the boundary conditions

 (144a)

q 3 − 12 Z1 Z2 0 ∂F − π2q δW − ∂F TB + K Td − m TδK

A

− Ω + ΩI

T

∂F − π2q δΩ dz1dz2 + I1

B

+ I2

B

= 0

∂F ≡

∂

2

F

∂z2

2 − ∂

2

F

∂z1∂z2

− ∂

2

F

∂z1∂z2

∂

2

F

∂z1

2

q ≡

q 1 dz1 0

0 q 2 dz2

π2δN*11 π2δN*22

π2

αb

δN*12

∂δN*11

∂z1

+ 1

αb

∂δN*12

∂z2

= 0

1

αb

∂δN*12

∂z1

+

∂δN*22

∂z2

= 0

δN*11 = 0
62

 (144b)

on the edges given by z1 = a1 / L1 and z1 = b1 / L1, and the boundary conditions

 (145a)

 (145b)

the edges given by z2 = a2 / L2 and z2 = b2 / L2. For this case, the nondimensional form of the 

principle of complementary virtual work is stated as

 (146)

where

 (147a)

and

 (147b)

Using the integration-by-parts formulas, specialized for the nondimensional coordinates (z1

, z2), 

to eliminate the derivatives of the tangential displacements and enforcing equations (143), 

equation (146) is transformed into

δN*12 = 0

δN*12 = 0

δN*22 = 0

δW*int

A

dz1dz2 + δW*

B

= 0

δW*int = E11 − ∂U1

∂z1

− 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1

π2δN*11

+ E22 − ∂U2

∂z2

− 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

π2δN*22

+ G12 − ∂U1

∂z2

− ∂U2

∂z1

− ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1

π2

αb

δN*12

δW*

B

= U1 − Δ1 z1

π2

αb

δN*12 + U2 − Δ2 z1 π2δN*22

a2

L2

b 2

L2

dz1

a1

L1

b1

L1

+ U1 − Δ1 z2 π2δN*11 + U2 − Δ2 z2

π2

αb

δN*12

a1

L1

b 1

L1

dz2

a2

L2

b2

L2
63

 (148)

where

 (149a)

and

 (149b)

It is important to note that when only traction boundary conditions are specified, vanishes. 

Next, a virtual stress function is defined such that equations (143) are satisfied identically; that is,

 (150a)

 (150b)

 (150c)

Substituting equations (150) into equation (149a) gives

δW*int

A

dz1dz2 − δW*

B

= 0

δW*int = E11 − 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1

π2δN*11

+ E22 − 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

π2δN*22

+ G12 − ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1

π2

αb

δN*12

δW*

B

= Δ1 z1

π2

αb

δN*12 + Δ2 z1 π2δN*22

a2

L2

b 2

L2

dz1

a1

L1

b1

L1

+ Δ1 z2 π2δN*11 + Δ2 z2

π2

αb

δN*12

a1

L1

b 1

L1

dz2

a2

L2

b2

L2

δW*

B

π2δN*11 = ∂

2

δF

∂z2

2

π2δN*22 = ∂

2

δF

∂z1

2

π2

αb

δN*12 = − ∂

2

δF

∂z1∂z2
64

 (151a)

where 

 (151b)

 (151c)

 (151d)

 (151e)

and where is defined by equation (136a). Likewise, equation (149b) becomes

 (152)

Next, equation (118) is expressed as

δW*int = E T− 12W Z1 Z2 0 − 1

2 Ω T

Ω − 2 ΩI ∂δF

E T= E11 E22 G12

Ω ≡

− ∂W

∂z1

0 − ∂W

∂z2

0 − ∂W

∂z2

− ∂W

∂z1

ΩI ≡

− ∂WI

∂z1

0 − ∂WI

∂z2

0 − ∂WI

∂z2

− ∂WI

∂z1

∂δF T

= ∂

2

δF

∂z2

2

∂

2

δF

∂z1

2 − ∂

2

δF

∂z1∂z2

Ω

δW*

B

= Δ2 z1

∂

2

δF

∂z1

2 − Δ1 z1

∂

2

δF

∂z1∂z2 a2

L2

b 2

L2

dz1

a1

L1

b1

L1

+ Δ1 z2

∂

2

δF

∂z2

2 − Δ2 z2

∂

2

δF

∂z1∂z2 a1

L1

b 1

L1

dz2

a2

L2

b2

L2
65

 (153)

where

 (154a)

 (154b)

and where , , and are defined by equations (139). Substituting equation (153) into 

equation (151a) gives the desired variational principle

 (155)

Equations (148), (152), and (155) constitute the principle of complementary virtual work for 

the stress-function formulation. Further integration by parts of equations (148) and (149), with the 

virtual stress resultants given by equations (150), yields

 (156)

E = a ∂F − B K − p

a ≡

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

p T

= p1 p2 p3

B ∂F K

δW*int = ∂F Ta − K TB T− p T− 12W Z1 Z2 0

− 1

2 Ω T

Ω − 2 ΩI ∂δF

∂

2

E22

∂z1

2 +

∂

2

E11

∂z2

2 − ∂

2

G12

∂z1∂z2

− 12Dc W + 1

2L W, W + 2WI δF

A

dz1dz2 +

+ − Δ1 z 2

∂

2

δF

∂z2

2 + Δ2 z 2

∂

2

δF

∂z1∂z2

+ S22

∂δF

∂z1

− 1

2S12

∂δF

∂z2

+ 1

2

∂S12

∂z2

− ∂S22

∂z1

δF

a1

L1

b 1

L1

dz2

a2

L2

b 2

L2

+ − Δ2 z 1

∂

2

δF

∂z1

2 + Δ1 z 1

∂

2

δF

∂z1∂z2

+ S11

∂δF

∂z2

− 1

2S12

∂δF

∂z1

+ 1

2

∂S12

∂z1

− ∂S11

∂z2

δF

a2

L2

b 2

L2

dz1

a1

L1

b1

L1

= 0
66

where

 (157a)

 (157b)

 (157c)

and the integrand of the double integral is the nondimensional compatibility equation given by 

equations (123) and (122c). When only traction boundary conditions are applied to the shell, 

equations (144) and (145) must be satisfied by δF. This task is done by noting that only the second 

derivatives of δF are required to be unique in order to obtain unique stress resultants. Thus, it is 

convenient to enforce δF = 0 on the shell boundary, to require on the edges z1 =

a1 / L1 and z1 = b1 / L1, and to require on the edges z2 = a2 / L2 and z2 = b2 / L2. For 

these choices, equation (156) reduces to

 (158)

For the case of displacement boundary conditions, the boundary integrals in equation (156) can 

be converted, by further integration by parts, into expressions that require satisfaction of the 

tangential-strain-displacement relations and continuity of the displacements and their derivatives 

on the boundary when the Fundamental Lemma of the Calculus of Variations is enforced.

Equations for Special Cases 

Significant simplification of the equations for stress-function formulation can be obtained for 

several cases of practical importance. For example, when the tangential surface tractions q1 and 

q2 are negligible, the transverse equilibrium equation given by equation (122a) reduces to

 (159)

and the compatibility equation given by equation (124a) reduces to

 (160)

S11 = E11 − 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1

S22 = E22 − 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

S12 = G12 − ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1

∂δF

∂z1

= 0

∂δF

∂z2

= 0

∂

2

E22

∂z1

2 +

∂

2

E11

∂z2

2 − ∂

2

G12

∂z1∂z2

− 12Dc W + 1

2L W, W + 2WI δF

A

dz1dz2 = 0

Db W + 12 Dc F − DB F = L F, W + WI + q 3

Dm F + DB W − 12 Dc W = 1

2L W, W + 2WI
67

Likewise, the boundary conditions on the edges z1 = a1 / L1 and z1 = b1 / L1, given by equations 

(131), reduce to 

 or (161a)

 or (161b)

 

 

or (161c)

 

or (161d)

On the edges z2 = a2 / L2 and z2 = b2 / L2, given by equations (132), reduce to

 or (162a)

 or (162b)

 

or (162c)

∂

2

F

∂z2

2 = π2

N z2 U1 = Δ1 z2

− ∂

2

F

∂z1∂z2

= π2

αb

S z2 U2 = Δ2 z2

B 21

∂

3

F

∂z1

3 + 2B 26 − B 61

∂

3

F

∂z1

2

∂z2

+ B 11 − 2B 66

∂

3

F

∂z1∂z2

2 + 2B 16

∂

3

F

∂z2

3 − d 11

∂

3

W

∂z1

3 − 4d 16

∂

3

W

∂z1

2

∂z2

− d 12 + 4d 66

∂

3

W

∂z1∂z2

2 − 2d 26

∂

3

W

∂z2

3 +

∂

2

F

∂z2

2

∂

∂z1

W + WI

− ∂

2

F

∂z1∂z2

∂

∂z2

W + WI = V z2

W = Δn z2

B 11

∂

2

F

∂z2

2 + B 21

∂

2

F

∂z1

2 − B 61

∂

2

F

∂z1∂z2

− d 11

∂

2

W

∂z1

2 − d 12

∂

2

W

∂z2

2 − 2d 16

∂

2

W

∂z1∂z2

= M z2

– ∂W

∂z1

= Φ z2

∂

2

F

∂z1

2 = π2

N z1 U2 = Δ2 z1

− ∂

2

F

∂z1∂z2

= π2

αb

S z1 U1 = Δ1 z1

2B 26

∂

3

F

∂z1

3 + B 22 − 2B 66

∂

3

F

∂z1

2

∂z2

+ 2B 16 − B 62

∂

3

F

∂z1∂z2

2 + B 12

∂

3

F

∂z2

3 − 2d 16

∂

3

W

∂z1

3

− d 12 + 4d 66

∂

3

W

∂z1

2

∂z2

− 4d 26

∂

3

W

∂z1∂z2

2 − d 22

∂

3

W

∂z2

3 +

∂

2

F

∂z1

2

∂

∂z2

W + WI

− ∂

2

F

∂z1∂z2

∂

∂z1

W + WI = V z1

W = Δn z1
68

 

 or (162d)

where the tangential displacements are obtained from equations (128)-(130) with p1 = p2 = p3 = 0. 

For this special case, with q1 = q2 = 0, the virtual work given by equation (141) reduces to

 (163)

and the complementary virtual work term given by equation (155) becomes

 (164)

When the tangential surface tractions q1 and q2 are negligible and the shell is symmetrically 

laminated, the transverse equilibrium equation given by equation (159) reduces to

 (165)

and the compatibility equation given by equation (160) reduces to

 (166)

with the additional simplification

 (167)

Likewise, the boundary conditions on the edges z1 = a1 / L1 and z1 = b1 / L1

, given by equations 

(161), reduce to 

 or (168a)

B 12

∂

2

F

∂z2

2 + B 22

∂

2

F

∂z1

2 − B 62

∂

2

F

∂z1∂z2

− d 12

∂

2

W

∂z1

2 − d 22

∂

2

W

∂z2

2 − 2d 26

∂

2

W

∂z1∂z2

= M z1

– ∂W

∂z2

= Φ z1

q 3 − 12 Z1 Z2 0 ∂F δW − ∂F TB + K Td δK

A

− Ω + ΩI

T

∂F δΩ dz1dz2 + I1

B

+ I2

B

= 0

δW*int = ∂F Ta − K TB T− 12W Z1 Z2 0 − 1

2 Ω T

Ω − 2 ΩI ∂δF

Db W + 12 Dc F = L F, W + WI + q 3

Dm F − 12 Dc W + 1

2L W, W + 2WI = 0

Db W = αb

2 ∂

4

W

∂z1

4 + 4αbγ b

∂

4

W

∂z1

3

∂z2

+ 2β ∂

4

W

∂z1

2

∂z2

2 + 4 δ b

αb

∂

4

W

∂z1∂z2

3 + 1

αb

2

∂

4

W

∂z2

4

∂

2

F

∂z2

2 = π2

N z2 U1 = Δ1 z2
69

 or (168b)

 or (168c)

 or (168d)

On the edges z2 = a2 / L2 and z2 = b2 / L2, the boundary conditions given by equations (162) reduce 

to

 or (169a)

 or (169b)

 or (169c)

 or (169d)

where the tangential displacements are obtained from

 (170)

 (171)

 (172)

The virtual work given by equation (163) reduces to

− ∂

2

F

∂z1∂z2

= π2

αb

S z2 U2 = Δ2 z2

− αb

2 ∂

3

W

∂z1

3 − 4αbγb

∂

3

W

∂z1

2

∂z2

− 2β − νb

∂

3

W

∂z1∂z2

2 − 2

δ b

αb

∂

3

W

∂z2

3

+

∂

2

F

∂z2

2

∂

∂z1

W + WI − ∂

2

F

∂z1∂z2

∂

∂z2

W + WI = V z2

W = Δn z2

− αb

2 ∂

2

W

∂z1

2 − νb

∂

2

W

∂z2

2 − 2αbγb

∂

2

W

∂z1∂z2

= M z2 – ∂W

∂z1

= Φ z2

∂

2

F

∂z1

2 = π2

N z1 U2 = Δ2 z1

− ∂

2

F

∂z1∂z2

= π2

αb

S z1 U1 = Δ1 z1

− 2αbγb

∂

3

W

∂z1

3 − 2β − νb

∂

3

W

∂z1

2

∂z2

− 4

δ b

αb

∂

3

W

∂z1∂z2

2 − 1

αb

2

∂

3

W

∂z2

3

+

∂

2

F

∂z1

2

∂

∂z2

W + WI − ∂

2

F

∂z1∂z2

∂

∂z1

W + WI = V z1

W = Δn z1

− νb

∂

2

W

∂z1

2 − 1

αb

2

∂

2

W

∂z2

2 − 2

δ b

αb

∂

2

W

∂z1∂z2

= M z1 – ∂W

∂z2

= Φ z1

∂U1

∂z1

= 1

αm

2

∂

2

F

∂z2

2 − νm

∂

2

F

∂z1

2 +

δm

αm

∂

2

F

∂z1∂z2

− 12Z1W − 1

2

∂W

∂z1

2

− ∂W

∂z1

∂WI

∂z1

∂U2

∂z2

= − νm

∂

2

F

∂z2

2 + αm

2 ∂

2

F

∂z1

2 + αmγm

∂

2

F

∂z1∂z2

− 12Z2W − 1

2

∂W

∂z2

2

− ∂W

∂z2

∂WI

∂z2

∂U1

∂z2

+

∂U2

∂z1

= − δm

αm

∂

2

F

∂z2

2 − αmγm

∂

2

F

∂z1

2 − 2 μ + νm

∂

2

F

∂z1∂z2

− ∂W

∂z1

∂W

∂z2

− ∂WI

∂z1

∂W

∂z2

− ∂WI

∂z2

∂W

∂z1
70

 (173)

and the complementary virtual work term given by equation (164) becomes

 (174)

Nondimensional Bifurcation Equations

Bifurcation analysis presumes the existence of a known continuous set of primary 

equilibrium states, called the primary or fundamental equilibrium path, for a geometrically perfect 

shell whose continuity is manifested by a continuously varying loading parameter . In addition, 

each primary equilibrium state is presumed to be governed by a linear boundary-value problem. 

In the present study, each primary equilibrium state, determined by the specific value of , is 

represented by the displacement fields , , and . Here, the 

superscript (0) denotes quantites associated with the primary equilibrium states prior to 

bifurcation. Bifurcation analysis also presumes the existence of a critical value of , denoted by 

, for which one or more solutions to the corresponding nonlinear boundary-value problem 

intersect the primary equilibrium path. Therefore, in the "small" neighborhood of a bifurcation, 

the shell response is represented by the displacement fields

 (175a)

 (175b)

 (175c)

where and the superscript (1) denotes quantites associated with equilibrium states that 

are adjacent to the unique primary equilibrium state at the bifurcation point, given by . It 

is important to note that although defines a unique point of the primary equilibrium path, 

q 3 − 12 Z1 Z2 0 ∂F δW − K Td δK

A

− Ω + ΩI

T

∂F δΩ dz1dz2 + I1

B

+ I2

B

= 0

δW*int = ∂F Ta − 12W Z1 Z2 0 − 1

2 Ω T

Ω − 2 ΩI ∂δF

p

p

U1

(0)

z1, z2, p U2

(0)

z1, z2, p W

(0)

z1, z2, p

p

pcr

U1 = U1

(0)

+ εU1

(1)

U2 = U2

(0)

+ εU2

(1)

W = W

(0)

+ εW

(1)

ε << 1

p = pcr

p = pcr
71

the displacement fields , , and are not unique; that is, more than one adjacent 

equilibrium state may correspond to the same value of . Bifurcation points of this type are 

typically referred to as points of compound bifurcation. 

Equations for the Primary Equilibrium Path

The rotation and strain fields associated with each primary equilibrium state are obtained 

directly from linearization of equations (40), (42), (46), and (49); which gives

 (176a)

 (176b)

 (177a)

 (177b)

 (177c)

 (178a)

 (178b)

 (178c)

The corresponding constitutive equations are obtained from equations (76) and (78) and are given 

by

U1

(1)

U2

(1)

W

(1)

pcr

Ω1

(0)

= – ∂W

(0)

∂z1

Ω2

(0)

= – ∂W

(0)

∂z2

E11

(0)

= ∂U1

(0)

∂z1

+ 12Z1W

(0)

E22

(0)

= ∂U2

(0)

∂z2

+ 12Z2W

(0)

G12

(0)

= ∂U1

(0)

∂z2

+

∂U2

(0)

∂z1

K11

(0)

= – ∂

2

W

(0)

∂z1

2

K22

(0)

= – ∂

2

W

(0)

∂z2

2

K12

(0)

= – 2 ∂

2

W

(0)

∂z1∂z2
72

 (179)

and

 (180)

where are membrane stress resultants and are bending stress 

resultants associated with each primary equilibrium state. The equilibrium equations governing 

each primary equilibrium state are obtained from linearization of equations (81), (83), (86), (88), 

and (93) and are given by

 (181a)

 (181b)

 (181c)

 (181d)

 (181e)

where and are transverse shear stress resultants associated with each primary equilibrium 

state and the surface tractions q1, q2, and q3 have been scaled by the loading parameter so that a 

E11

(0)

E22

(0)

G12

(0)

= π2

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

N11

(0)

N22

(0)

N12

(0)

αb

+

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

∂

2

W

(0)

∂z1

2

∂

2

W

(0)

∂z2

2

2 ∂

2

W

(0)

∂z1∂z2

M11

(0)

M22

(0)

M12

(0)

= π2

B 11 B 21 B 61

B 12 B 22 B 62

B 16 B 26 B 66

N11

(0)

N22

(0)

N12

(0)

αb

−

d 11 d 12 d 16

d 12 d 22 d 26

d 16 d 26 d 66

∂

2

W

(0)

∂z1

2

∂

2

W

(0)

∂z2

2

2 ∂

2

W

(0)

∂z1∂z2

N11

(0)

, N22

(0)

, and N12

(0)

M11

(0)

, M22

(0)

, and M12

(0)

∂N11

(0)

∂z1

+ 1

αb

∂N12

(0)

∂z2

+ pq 1 = 0

1

αb

∂N12

(0)

∂z1

+

∂N22

(0)

∂z2

+ pq 2 = 0

∂M11

(0)

∂z1

+

∂M12

(0)

∂z2

– Q1

(0)

= 0

∂M12

(0)

∂z1

+

∂M22

(0)

∂z2

– Q2

(0)

= 0

∂Q1

(0)

∂z1

+

∂Q2

(0)

∂z2

+ pq 3 – π212 N11

(0)

Z1 + N22

(0)

Z2 = 0

Q1

(0)

Q2

(0)
73

unique solution is associated with each value of .

The boundary conditions associated with each primary equilibrium state are also obtained 

from linearization of equations (98) and (99), with the applied edge tractions and displacements 

also scaled by the loading parameter . Thus, the boundary conditions on the edges z1 = a1 / L1

and z1 = b1 / L1, given by equations (98), become

 or (182a)

 or (182b)

 or (182c)

 or (182d)

where is given by equation (181c). On the edges z2 = a2 / L2 and z2 = b2 / L2, the boundary 

conditions given by equation (99) become

 or (183a)

 or (183b)

 or (183c)

 or (183d)

where is given by equation (181d). 

Equations (176) - (183) constitute a family of linear boundary-value problems whose 

solutions depend on the loading parameter and the relative magnitudes of the loads. The 

relative magnitudes of the loads are given by the specific values selected for the surface tractions 

q1

, q2, and q3 and the edge tractions or displacements specified in equations (182) and (183). The 

family of solutions is represented by 

p

p

N11

(0)

= pN z2 U1

(0)

= pΔ1 z2

N12

(0)

= pS z2 U2

(0)

= pΔ2 z2

Q1

(0)

+

∂M12

(0)

∂z2

= pV z2 W

(0)

= pΔn z2

M11

(0)

= pM z2 – ∂W

(0)

∂z1

= pΦ z2

Q1

(0)

N22

(0)

= pN z1 U2

(0)

= pΔ2 z1

N12

(0)

= pS z1 U1

(0)

= pΔ1 z1

Q2

(0)

+

∂M12

(0)

∂z1

= pV z1 W

(0)

= pΔn z1

M22

(0)

= pM z1 – ∂W

(0)

∂z2

= pΦ z1

Q2

(0)

p
74

 (184a)

 (184b)

 (184c)

 (185a)

 (185b)

 (185c)

 (186a)

 (186b)

 (186c)

 (187a)

 (187b)

 (187c)

 (188a)

 (188b)

which are generally transcendental functions of the loading parameter .

Equations for Adjacent Equilibrium Paths

The equations governing adjacent equilibrium paths at a bifurcation point of the primary 

equilibrium path are obtained by substituting equations (175) into the equations for the nonlinear 

boundary-value problem of the idealized, geometrically perfect shell, and then noting that all 

U1

(0)

= U1

(0)

z1, z2, p

U2

(0)

= U2

(0)

z1, z2, p

W

(0)

= W

(0)

z1, z2, p

E11

(0)

= E11

(0)

z1, z2, p

E22

(0)

= E22

(0)

z1, z2, p

G12

(0)

= G12

(0)

z1, z2, p

N11

(0)

= N11

(0)

z1, z2, p

N22

(0)

= N22

(0)

z1, z2, p

N12

(0)

= N12

(0)

z1, z2, p

M11

(0)

= M11

(0)

z1, z2, p

M22

(0)

= M22

(0)

z1, z2, p

M12

(0)

= M12

(0)

z1, z2, p

Q1

(0)

= Q1

(0)

z1, z2, p

Q2

(0)

= Q2

(0)

z1, z2, p

p
75

resulting equations for the primary equilibrium path are satisfied identically. In particular, 

substituting equations (175) into equations (40) and (42) gives

 (189a)

 (189b)

and substituting equations (175) into equations (46) and (49) gives

 (190a)

 (190b)

 (190c)

 (191a)

 (191b)

 (191c)

where the symbol is used to denote terms with magnitudes that are at most second order 

in the small parameter ε. In these equations,

 (192a)

 (192b)

 (193a)

 (193b)

Ω1 = Ω1

(0)

+ εΩ1

(1)

Ω2 = Ω2

(0)

+ εΩ2

(1)

E11 = E11

(0)

+ εE11

(1)

+ O ε

2

E22 = E22

(0)

+ εE22

(1)

+ O ε

2

G12 = G12

(0)

+ εG12

(1)

+ O ε

2

K11 = K11

(0)

+ εK11

(1)

K22 = K22

(0)

+ εK22

(1)

K12 = K12

(0)

+ εK12

(1)

O ε

2

Ω1

(1)

≡ – ∂W

(1)

∂z1

Ω2

(1)

≡ – ∂W

(1)

∂z2

E11

(1)

≡ ∂U1

(1)

∂z1

+ 12Z1W

(1)

+ ∂W

(0)

∂z1

∂W

(1)

∂z1

E22

(1)

≡ ∂U2

(1)

∂z2

+ 12Z2W

(1)

+ ∂W

(0)

∂z2

∂W

(1)

∂z2
76

 (193c)

 (194a)

 (194b)

 (194c)

Next, substituting equations (175c) and (193) into equations (76) and (78) implies the expansions

 (195a)

 (195b)

 (195c)

 (195d)

 (195e)

 (195f)

and the constitutive equations

 (196)

and

G12

(1)

≡ ∂U1

(1)

∂z2

+

∂U2

(1)

∂z1

+ ∂W

(0)

∂z1

∂W

(1)

∂z2

+ ∂W

(0)

∂z2

∂W

(1)

∂z1

K11

(1)

≡ – ∂

2

W

(1)

∂z1

2

K22

(1)

≡ – ∂

2

W

(1)

∂z2

2

K12

(1)

≡ – 2 ∂

2

W

(1)

∂z1∂z2

N11 = N11

(0)

+ εN11

(1)

+ O ε

2

N22 = N22

(0)

+ εN22

(1)

+ O ε

2

N12 = N12

(0)

+ εN12

(1)

+ O ε

2

M11 = M11

(0)

+ εM11

(1)

+ O ε

2

M22 = M22

(0)

+ εM22

(1)

+ O ε

2

M12 = M12

(0)

+ εM12

(1)

+ O ε

2

E11

(1)

E22

(1)

G12

(1)

= π2

1

αm

2 − νm − δm

αm

− νm αm

2

− αmγm

− δm

αm

− αmγm 2 μ + νm

N11

(1)

N22

(1)

N12

(1)

αb

+

B 11 B 12 B 16

B 21 B 22 B 26

B 61 B 62 B 66

∂

2

W

(1)

∂z1

2

∂

2

W

(1)

∂z2

2

2 ∂

2

W

(1)

∂z1∂z2
77

 (197)

where are membrane stress resultants and are bending stress 

resultants associated with the adjacent equilibrium states. The equilibrium equations governing 

the adjacent equilibrium states are obtained by substituting equations (175c) and (195) into 

equations (81), (83), (86), (88), and (94) and then enforcing equations (181) and neglecting terms 

of second order and higher and nonlinear terms associated with the primary equilibrium states. 

The resulting equations are given by

 (198a)

 (198b)

 (198c)

 (198d)

 (198e)

where it is noted that

 (199a)

M11

(1)

M22

(1)

M12

(1)

= π2

B 11 B 21 B 61

B 12 B 22 B 62

B 16 B 26 B 66

N11

(1)

N22

(1)

N12

(1)

αb

−

d 11 d 12 d 16

d 12 d 22 d 26

d 16 d 26 d 66

∂

2

W

(1)

∂z1

2

∂

2

W

(1)

∂z2

2

2 ∂

2

W

(1)

∂z1∂z2

N11

(1)

, N22

(1)

, and N12

(1)

M11

(1)

, M22

(1)

, and M12

(1)

∂N11

(1)

∂z1

+ 1

αb

∂N12

(1)

∂z2

= 0

1

αb

∂N12

(1)

∂z1

+

∂N22

(1)

∂z2

= 0

∂M11

(1)

∂z1

+

∂M12

(1)

∂z2

– Q1

(1)

= 0

∂M12

(1)

∂z1

+

∂M22

(1)

∂z2

– Q2

(1)

= 0

∂Q1

(1)

∂z1

+

∂Q2

(1)

∂z2

– π212 N11

(1)

Z1 + N22

(1)

Z2

+ π2 ∂

∂z1

N11

(0) ∂W

(1)

∂z1

+

N12

(0)

αb

∂W

(1)

∂z2

+ N11

(1) ∂W

(0)

∂z1

+

N12

(1)

αb

∂W

(0)

∂z2

+ π2 ∂

∂z2

N12

(0)

αb

∂W

(1)

∂z1

+ N22

(0) ∂W

(1)

∂z2

+

N12

(1)

αb

∂W

(0)

∂z1

+ N22

(1) ∂W

(0)

∂z2

= 0

Q1 = Q1

(0)

+ εQ1

(1)

+ O ε

2
78

 (199b)

such that and are transverse shear stress resultants associated with the adjacent equilibrium 

states. The loading parameter enters the equations for the adjacent equilibrium states through the 

displacement and the membrane stress resultants To further simplify 

matters, sometimes it is presumed that the pre-bifurcation displacement is mildy varying such 

that the corresponding pre-bifurcation rotations, given by the derivatives of in equation 

(198e), are negligible. For this case, equation (198e) reduces to

 (200a)

Expanding the derivatives of the bracketed terms in this equation and using equations (198a) and 

(198b) give the alternate form

 (200b)

Setting Z1

 = Z2 = 0 in this equation yields the bifurcation equation that is comonly cited for 

buckling of flat plates.

The boundary conditions associated with the adjacent equilibrium states are also obtained 

from equations (98) and (99). In particular, substituting equations (175), (195), and (199) into 

equations (98) and using equations (182), the boundary conditions on the edges z1 = a1 / L1 and 

z1 = b1 / L1, given by equations (98), become

 or (201a)

 or (201b)

Q2 = Q2

(0)

+ εQ2

(1)

+ O ε

2

Q1

(1)

Q2

(1)

W

(0)

N11

(0)

, N22

(0)

, and N12

(0)

.

W

(0)

W

(0)

∂Q1

(1)

∂z1

+

∂Q2

(1)

∂z2

– π212 N11

(1)

Z1 + N22

(1)

Z2

+ π2 ∂

∂z1

N11

(0) ∂W

(1)

∂z1

+

N12

(0)

αb

∂W

(1)

∂z2

+ π2 ∂

∂z2

N12

(0)

αb

∂W

(1)

∂z1

+ N22

(0) ∂W

(1)

∂z2

= 0

∂Q1

(1)

∂z1

+

∂Q2

(1)

∂z2

– π212 N11

(1)

Z1 + N22

(1)

Z2 − pπ2q 1

∂W

(1)

∂z1

+ q 2

∂W

(1)

∂z2

+ π2N11

(0) ∂

2

W

(1)

∂z1

2 + N22

(0) ∂

2

W

(1)

∂z2

2 + 2 N12

(0)

αb

∂

2

W

(1)

∂z1∂z2

= 0

N11

(1)

= 0 U1

(1)

= 0

N12

(1)

= 0 U2

(1)

= 0
79

 or (201c)

 or (201d)

where is given by equation (198c). If nonlinear pre-bifurcation rotations and are 

neglected, equation (201c) reduces to

 or (201e)

Similarly, on the edges z2

 = a2 / L2 and z2 = b2 / L2, the boundary conditions given by equation 

(99) become

 or (202a)

 or (202b)

 or (202c)

 or (202d)

where is given by equation (198d). If nonlinear pre-bifurcation rotations are neglected, then 

equation (202c) reduces to

 or (202e)

Inspection of equations (189)-(202) indicates a system of homogeneous differential 

equations and homogeneous boundary conditions that depend on the loading parameter through 

the displacement and the membrane stress resultants , generally in a 

transcendental manner. Thus, the equations for the adjacent equilibrium states constitute a 

nonlinear boundary-eigenvalue problem. 

Q1

(1)

+

∂M12

(1)

∂z2

+ π2N11

(0) ∂W

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z2

+ π2N11

(1) ∂W

(0)

∂z1

+ π2

αb

N12

(1) ∂W

(0)

∂z2

= 0 W

(1)

= 0

M11

(1)

= 0 ∂W

(1)

∂z1

= 0

Q1

(1)

Ω1

(0)

Ω2

(0)

Q1

(1)

+

∂M12

(1)

∂z2

+ π2N11

(0) ∂W

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z2

= 0 W

(1)

= 0

N22

(1)

= 0 U2

(1)

= 0

N12

(1)

= 0 U1

(1)

= 0

Q2

(1)

+

∂M12

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z1

+ π2N22

(0) ∂W

(1)

∂z2

+ π2

αb

N12

(1) ∂W

(0)

∂z1

+ π2N22

(1) ∂W

(0)

∂z2

= 0 W

(1)

= 0

M22

(1)

= 0 ∂W

(1)

∂z2

= 0

Q2

(1)

Q2

(1)

+

∂M12

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z1

+ π2N22

(0) ∂W

(1)

∂z2

= 0 W

(1)

= 0

p

W

(0)

N11

(0)

, N22

(0)

, and N12

(0)
80

Variational Principle for Bifurcation

A variational principle for bifurcation is obtained by noting that equations (198a), (198b), and 

(198e) represent pointwise summations of internal forces in the three coordinate directions, for 

the adjacent equilibrium states. Likewise, the traction boundary conditions in equations (201) and 

(202) represent pointwise summations of internal forces acting at the edges given by constant values of z1

 and z2, respectively. Thus, a statement of the corresponding virtual work is given by

 (203)

where A is the nondimensional domain given by and , and 

 

 (204a)

 (204b)

 (204c)

δW

(1)

= − δW

(1)

A

dz1dz2 + δW1

(1) B

a2

L2

b2

L2

dz1

a1

L1

b 1

L1

+ δW2

(1) B

a1

L1

b1

L1

dz2

a2

L2

b 2

L2

a 1

L1

≤ z1 ≤ b 1

L1

a 2

L2

≤ z2 ≤ b 2

L2

δW

(1)

= ∂N11

(1)

∂z1

+ 1

αb

∂N12

(1)

∂z2

δU1 + 1

αb

∂N12

(1)

∂z1

+

∂N22

(1)

∂z2

δU2 +

∂Q1

(1)

∂z1

+

∂Q2

(1)

∂z2

– π212 N11

(1)

Z1 + N22

(1)

Z2 + π2 ∂

∂z1

N11

(0) ∂W

(1)

∂z1

+

N12

(0)

αb

∂W

(1)

∂z2

+ N11

(1) ∂W

(0)

∂z1

+

N12

(1)

αb

∂W

(0)

∂z2

+ π2 ∂

∂z2

N12

(0)

αb

∂W

(1)

∂z1

+ N22

(0) ∂W

(1)

∂z2

+

N12

(1)

αb

∂W

(0)

∂z1

+ N22

(1) ∂W

(0)

∂z2

δW

δW1

(1) B

= N12

(1)

δU2 + N22

(1)

δU2 − M22

(1) ∂δW

∂z2

+ Q2

(1)

+

∂M12

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z1

+ π2N22

(0) ∂W

(1)

∂z2

+ π2

αb

N12

(1) ∂W

(0)

∂z1

+ π2N22

(1) ∂W

(0)

∂z2

δW

δW2

(1) B

= N11

(1)

δU2 + N12

(1)

δU2 − M11

(1) ∂δW

∂z1

+ Q1

(1)

+

∂M12

(1)

∂z2

+ π2N11

(0) ∂W

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z2

+ π2N11

(1) ∂W

(0)

∂z1

+ π2

αb

N12

(1) ∂W

(0)

∂z2

δW
81

In equations (204) δU1, δU2, and δW are arbitrary nondimensional virtual displacement fields 

along the z1, z2, and ζ directions, respectively. Integrating by parts, by using equations (25) specialized for the nondimensional coordinates (z1

, z2), gives

 (205a)

where: 

 (205b)

 (205c)

 (205d)

δW

(1)

A

dz1dz2 = − δWint

(1)

+ δW′int

(1)

A

dz1dz2 + I 1

(1)B

+ I 2

(1)B

− M12

(1)

δW a2

L 2

b 2

L 2

a1

L1

b 1

L1

δWint

(1)

=

π2

N11

(1) ∂δU1

∂z1

+ 12Z1δW + ∂W

(0)

∂z1

∂δW

∂z1

+ π2N22

(1) ∂δU2

∂z2

+ 12Z2δW + ∂W

(0)

∂z2

∂δW

∂z2

+

π2 N12

(1)

αb

∂δU1

∂z2

+

∂δU2

∂z1

+ ∂W

(0)

∂z1

∂δW

∂z2

+ ∂W

(0)

∂z2

∂δW

∂z1

− M11

(1) ∂

2

δW

∂z1

2 − 2M12

(1) ∂

2

δW

∂z1∂z2

− M22

(1) ∂

2

δW

∂z2

2

δW′int

(1)

= π2N11

(0) ∂W

(1)

∂z1

∂δW

∂z1

+

N12

(0)

αb

∂W

(1)

∂z2

∂δW

∂z1

+ ∂W

(1)

∂z1

∂δW

∂z2

+ N22

(0) ∂W

(1)

∂z2

∂δW

∂z2

I 1

B (1)

= π2N11

(1)

δU1 + π2 N12

(1)

αb

δU2 − M11

(1) ∂δW

∂z1

a2

L 2

b 2

L 2

+ Q1

(1)

+

∂M12

(1)

∂z2

+ π2N11

(0) ∂W

(1)

∂z1

+

N12

(0)

αb

∂W

(1)

∂z2

+ N11

(1) ∂W

(0)

∂z1

+

N12

(1)

αb

∂W

(0)

∂z2

δW

a1

L1

b 1

L1

dz2
82

 (205e)

The symbol denotes the evalution of at discontinuities of the 

boundary curve ∂A and are commonly called corner conditions. Next, noting that the boundary 

conditions given by equations (201) and (202) are homogeneous and enforcing the conditions that 

the virtual displacements satisfy the kinematic boundary conditions and the kinematic relations 

given by equations (192)-(194) results in the following form for equation (205a):

 (206a)

with

 (206b)

 (206c)

where

 (207a)

 (207b)

I 2

B (1)

= π2 N12

(1)

αb

δU1 + π2N22

(1)

δU2 − M22

(1) ∂δW

∂z2

a1

L 1

b 1

L 1

+ Q2

(1)

+

∂M12

(1)

∂z1

+ π2 N12

(0)

αb

∂W

(1)

∂z1

+ N22

(0) ∂W

(1)

∂z2

+

N12

(1)

αb

∂W

(0)

∂z1

+ N22

(1) ∂W

(0)

∂z2

δW

a2

L2

b 2

L2

dz1

M12

(1)

δW a2

L 2

b 2

L 2

a1

L1

b 1

L1

M12

(1)

δW

δWint

(1)

+ δW′int

(1)

A

dz1dz2 = 0

δWint

(1)

= π2N11

(1)

δE11

(1)

+

N12

(1)

αb

δG12

(1)

+ N22

(1)

δE22

(1)

+ M11

(1)

δK11

(1)

+ M12

(1)

δK12

(1)

+ M22

(1)

δK22

(1)

δW′int

(1)

= π2N11

(0)

Ω1

(1)

δΩ1

(1)

+

N12

(0)

αb

Ω2

(1)

δΩ1

(1)

+ Ω1

(1)

δΩ2

(1)

+ N22

(0)

Ω2

(1)

δΩ2

(1)

δΩ1

(1)

≡ – ∂δW

(1)

∂z1

= δ Ω1

(1)

δΩ2

(1)

≡ – ∂δW

(1)

∂z2

= δ Ω2

(1)
83

 (208a)

 (208b)

 (208c)

 (209a)

 (209b)

 (209c)

where δ denotes the variational operator of the Calculus of Variations. A convenient matrix form 

of this variational statement is given by

 (210)

where

 (211a)

 (211b)

δE11

(1)

≡ ∂δU1

(1)

∂z1

+ 12Z1δW

(1)

+ ∂W

(0)

∂z1

∂δW

(1)

∂z1

= δ E11

(1)

δE22

(1)

≡ ∂δU2

(1)

∂z2

+ 12Z2δW

(1)

+ ∂W

(0)

∂z2

∂δW

(1)

∂z2

= δ E22

(1)

δG12

(1)

≡ ∂δU1

(1)

∂z2

+

∂δU2

(1)

∂z1

+ ∂W

(0)

∂z1

∂δW

(1)

∂z2

+ ∂W

(0)

∂z2

∂δW

(1)

∂z1

= δ G12

(1)

δK11

(1)

≡ – ∂

2

δW

(1)

∂z1

2 = δ K11

(1)

δK22

(1)

≡ – ∂

2

δW

(1)

∂z2

2 = δ K22

(1)

δK12

(1)

≡ – 2 ∂

2

δW

(1)

∂z1∂z2

= δ K12

(1)

π2N

(1)

T

δE

(1)

+ M

(1)

T

δK

(1)

+ π2Ω

(1) T

N

(0)

δΩ

(1)

A

dz1dz2 = 0

N

(1) T

= N11

(1)

N22

(1) N12

(1)

αb

M

(1) T

= M11

(1)

M22

(1)

M12

(1)
84

 (211c)

 (212a)

 (212b)

 (212c)

 (213a)

 (213b)

 (213c)

 (213d)

Nondimensional Stress-Function Formulation for Bifurcation

Like for the nonlinear boundary-value problem described previously herein, the stressfunction formulation of the Donnell-Mushtari-Vlasov bifurcation equations is also used to 

facilitiate solution of practical problems by reducing the number of unknown functions to two. 

These two unknowns for this case are the normal displacement and a corresponding 

stress function and the procedure for obtaining the corresponding equations is the same 

as that previously described herein for the nonlinear boundary-value problem. Subsequently, the 

reduction of the boundary-eigenvalue problem to two coupled partial differential equations is 

presented, along with the corresponding boundary conditions. Then, the corresponding 

expressions for the virtual work and complementary virtual work, are presented that are useful for 

N

(0)

= N11

(0) N12

(0)

αb

N12

(0)

αb

N22

(0)

= N

(0)

p

Ω

(0) T

= Ω1

(0)

Ω2

(0)

= Ω

(0)

p

T

Ω

(1) T

= Ω1

(1)

Ω2

(1)

δΩ

(1) T

= δΩ1

(1)

δΩ2

(1)

E

(1) T

= E11

(1)

E22

(1)

G12

(1)

δE

(1) T

= δE11

(1)

δE22

(1)

δG12

(1)

K

(1) T

= K11

(1)

K22

(1)

K12

(1)

δK

(1) T

= δK11

(1)

δK22

(1)

δK12

(1)

W

(1)

z1, z2

F

(1)

z1, z2 ,
85

for solving boundary-value problems by direct variational methods. 

Following the definitions given by equations (115), let denote the stress 

function defined by

 (214a)

 (214b)

 (214c)

such that equations (198a) and (198b) are satisfied identically and 

 (214d)

Using equations (181), (198a) - (198d), and (214), the transverse equilibrium equation given by 

equation (198e) becomes

 (215)

where

 (216)

Next, substituting equations (214) into equations (197), and then substituting the result into 

equation (215) yields the nondimensional stress-function form of transverse equilibrium equation 

given by 

 (217)

F

(1)

= F

(1)

z1, z2

π2N11

(1)

= ∂

2

F

(1)

∂z2

2

π2N22

(1)

= ∂

2

F

(1)

∂z1

2

π2

αb

N12

(1)

= − ∂

2

F

(1)

∂z1∂z2

F = F

(0)

+ εF

(1)

+ O ε

2

∂

2

M11

(1)

∂z1

2 + 2 ∂

2

M12

(1)

∂z1∂z2

+

∂

2

M22

(1)

∂z2

2 – 12 ∂

2

F

(1)

∂z2

2 Z1 +

∂

2

F

(1)

∂z1

2 Z2 − pπ2

q 1

∂W

(1)

∂z1

+ q 2

∂W

(1)

∂z2

+ L F

(1)

, W

(0)

+ π2N11

(0)

p ∂

2

W

(1)

∂z1

2 +

2π2

αb

N12

(0)

p ∂

2

W

(1)

∂z1∂z2

+ π2N22

(0)

p ∂

2

W

(1)

∂z2

2 = 0

L F

(1)

, W

(0)

= ∂

2

F

(1)

∂z2

2

∂

2

W

(0)

∂z1

2 +

∂

2

F

(1)

∂z1

2

∂

2

W

(0)

∂z2

2 − 2 ∂

2

F

(1)

∂z1∂z2

∂

2

W

(0)

∂z1∂z2

Db W

(1)

+ 12Dc F

(1)

− DB F

(1)

− L F

(1)

, W

(0)

p + pπ2q 1

∂W

(1)

∂z1

+ q 2

∂W

(1)

∂z2

− π2N11

(0)

p ∂

2

W

(1)

∂z1

2 +

2

αb

N12

(0)

p ∂

2

W

(1)

∂z1∂z2

+ N22

(0)

p ∂

2

W

(1)

∂z2

2 = 0
86

where

 (218a)

 (218b)

 (218c)

The nondimensional compatibility equation needed is obtained by substituting equations 

(175c) and (190) into equation (105), and then retaining only terms that are first order in the small 

parameter ε. The resulting equation is given by

 (219)

Next, substituting equations (214) into equations (196), and then substituting the result into 

equation (219) yields the nondimensional stress-function form of the compatibility equation as

 (220)

where

 (221a)

 (221b)

The boundary conditions associated with the adjacent equilibrium states are obtained by 

using equations (197), (198c), (198d), and (214) with equations (201) and (202). The boundary 

conditions on the edges z1

 = a1 / L1 and z1 = b1 / L1, given by equations (201), become

 or (222a)

Db W

(1)

= d 11

∂

4

W

(1)

∂z1

4 + 4d 16

∂

4

W

(1)

∂z1

3

∂z2

+ 2 d 12 + 2d 66

∂

4

W

(1)

∂z1

2

∂z2

2 + 4d 26

∂

4

W

(1)

∂z1∂z2

3 + d 22

∂

4

W

(1)

∂z2

4

Dc F

(1)

= Z1

∂

2

F

(1)

∂z2

2 + Z2

∂

2

F

(1)

∂z1

2

DB F

(1)

= B 21

∂

4

F

(1)

∂z1

4 + 2B 26 − B 61

∂

4

F

(1)

∂z1

3

∂z2

+ B 11 + B 22 − 2B 66

∂

4

F

(1)

∂z1

2

∂z2

2

+ 2B 16 − B 62

∂

4

F

(1)

∂z1∂z2

3 + B 12

∂

4

F

(1)

∂z2

4

∂

2

E11

(1)

∂z2

2 +

∂

2

E22

(1)

∂z1

2 − ∂

2

G12

(1)

∂z1∂z2

= 12Dc W

(1)

+ L W

(0)

p , W

(1)

Dm F

(1)

+ DB W

(1)

− 12 Dc W

(1)

= L W

(0)

p , W

(1)

Dm F

(1)

≡ αm

2 ∂

4

F

(1)

∂z1

4 + 2αmγm

∂

4

F

(1)

∂z1

3

∂z2

+ 2μ

∂

4

F

(1)

∂z1

2

∂z2

2 + 2 δm

αm

∂

4

F

(1)

∂z1∂z2

3 +

1

αm

2

∂

4

F

(1)

∂z2

4

L W

(0)

p , W

(1)

= ∂

2

W

(0)

∂z2

2

∂

2

W

(1)

∂z1

2 + ∂

2

W

(0)

∂z1

2

∂

2

W

(1)

∂z2

2 − 2 ∂

2

W

(0)

∂z1∂z2

∂

2

W

(1)

∂z1∂z2

∂

2

F

(1)

∂z2

2 = 0 U1

(1)

= 0
87

 or (222b)

 

 or (222c)

 

 or (222d)

Similarly, on the edges z2 = a2 / L2 and z2 = b2 / L2, the boundary conditions given by equation 

(202) become

 or (223a)

 or (223b)

 

 or (223c)

 or (223d)

In these equations, the tangential displacements must be expressed in terms of the normal 

displacement and the stress function. Expressions for the tangential displacements and 

are obtained from the nondimensional strain-displacement relation, equations (193); that is

∂

2

F

(1)

∂z1∂z2

= 0 U2

(1)

= 0

B 21

∂

3

F

(1)

∂z1

3 + 2B 26 − B 61

∂

3

F

(1)

∂z1

2

∂z2

+ B 11 − 2B 66

∂

3

F

(1)

∂z1∂z2

2 + 2B 16

∂

3

F

(1)

∂z2

3 − d 11

∂

3

W

(1)

∂z1

3 − 4d 16

∂

3

W

(1)

∂z1

2

∂z2

− d 12 + 4d 66

∂

3

W

(1)

∂z1∂z2

2 − 2d 26

∂

3

W

(1)

∂z2

3 + π2

N11

(0) ∂W

(1)

∂z1

+ π2

αb

N12

(0) ∂W

(1)

∂z2

+

∂

2

F

(1)

∂z2

2

∂W

(0)

∂z1

− ∂

2

F

(1)

∂z1∂z2

∂W

(0)

∂z2

= 0

W

(1)

= 0

B 11

∂

2

F

(1)

∂z2

2 + B 21

∂

2

F

(1)

∂z1

2 − B 61

∂

2

F

(1)

∂z1∂z2

− d 11

∂

2

W

(1)

∂z1

2 − d 12

∂

2

W

(1)

∂z2

2 − 2d 16

∂

2

W

(1)

∂z1∂z2

= 0 ∂W

(1)

∂z1

= 0

∂

2

F

(1)

∂z1

2 = 0 U2

(1)

= 0

∂

2

F

(1)

∂z1∂z2

= 0 U1

(1)

= 0

2B 26

∂

3

F

(1)

∂z1

3 + B 22 − 2B 66

∂

3

F

(1)

∂z1

2

∂z2

+ 2B 16 − B 62

∂

3

F

(1)

∂z1∂z2

2 + B 12

∂

3

F

(1)

∂z2

3 − 2d 16

∂

3

W

(1)

∂z1

3 − d 22

∂

3

W

(1)

∂z2

3

− d 12 + 4d 66

∂

3

W

(1)

∂z1

2

∂z2

− 4d 26

∂

3

W

(1)

∂z1∂z2

2 + π2

αb

N12

(0) ∂W

(1)

∂z1

+ π2N22

(0) ∂W

(1)

∂z2

+

∂

2

F

(1)

∂z1

2

∂W

(0)

∂z2

− ∂

2

F

(1)

∂z1∂z2

∂W

(0)

∂z1

= 0

W

(1)

= 0

B 12

∂

2

F

(1)

∂z2

2 + B 22

∂

2

F

(1)

∂z1

2 − B 62

∂

2

F

(1)

∂z1∂z2

− d 12

∂

2

W

(1)

∂z1

2 − d 22

∂

2

W

(1)

∂z2

2 − 2d 26

∂

2

W

(1)

∂z1∂z2

= 0 ∂W

(1)

∂z2

= 0

U1

(1)

U2

(1)
88

 (224a)

 (224b)

 (224c)

Substituting equations (214) into equation (196) and the result into these three expressions gives

 (225a)

 (225b)

 (225c)

The nondimensional displacements are represented in terms of and , to within a rigid-body 

motion, by the integrals of these three equations.

Virtual Work in terms of and 

Equations (217) and (220) are the nondimensional forms of the equations governing 

pointwise equilibrium normal to the tangent plane and compatibility, respectively. For some 

problems, it is more useful to use a stress-function formulation of the variational principle given 

by equation (210) instead of equation (217). First, by using equations (214), equation (211a) is 

written as

∂U1

(1)

∂z1

= E11

(1)

− 12Z1W

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z1

∂U2

(1)

∂z2

= E22

(1)

− 12Z2W

(1)

− ∂W

(0)

∂z2

∂W

(1)

∂z2

∂U1

(1)

∂z2

+

∂U2

(1)

∂z1

= G12

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z2

− ∂W

(0)

∂z2

∂W

(1)

∂z1

∂U1

(1)

∂z1

= 1

αm

2

∂

2

F

(1)

∂z2

2 − νm

∂

2

F

(1)

∂z1

2 +

δm

αm

∂

2

F

(1)

∂z1∂z2

+ B 11

∂

2

W

(1)

∂z1

2 + B 12

∂

2

W

(1)

∂z2

2 + 2B 16

∂

2

W

(1)

∂z1∂z2

− 12Z1W

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z1

∂U2

(1)

∂z2

= − νm

∂

2

F

(1)

∂z2

2 + αm

2 ∂

2

F

(1)

∂z1

2 + αmγm

∂

2

F

(1)

∂z1∂z2

+ B 21

∂

2

W

(1)

∂z1

2 + B 22

∂

2

W

(1)

∂z2

2 + 2B 26

∂

2

W

(1)

∂z1∂z2

− 12Z2W

(1)

− ∂W

(0)

∂z2

∂W

(1)

∂z2

∂U1

(1)

∂z2

+

∂U2

(1)

∂z1

= − δm

αm

∂

2

F

(1)

∂z2

2 − αmγm

∂

2

F

(1)

∂z1

2 − 2 μ + νm

∂

2

F

(1)

∂z1∂z2

+

B 61

∂

2

W

(1)

∂z1

2 + B 62

∂

2

W

(1)

∂z2

2 + 2B 66

∂

2

W

(1)

∂z1∂z2

− ∂W

(0)

∂z1

∂W

(1)

∂z2

− ∂W

(0)

∂z2

∂W

(1)

∂z1

F

(1)

W

(1)

W

(1)

F

(1)
89

 (226a)

such that

 (226b)

Similarly, by using equations (194), (207)-(209), and (214), equations (196) and (197) yield

 (227)

and

 (228)

where , , and are defined by equations (154a), (139b), and (139c), respectively. 

Noting that equation (206b) can be expressed as

 (229)

it follows that 

 (230)

The variational principle specified by equation (206a) is given in terms of the stress function 

and normal displacement by using equation (230) and

 (231)

The boundary-eigenvalue problem is then posed with the resulting variational principle, the 

compatibility equation given by equation (220), and the appropriate boundary conditions. The 

variational principle enforces equilibrium in the direction normal to the shell reference surface.

π2N

(1)

= ∂F

(1)

≡ ∂

2

F

(1)

∂z2

2

∂

2

F

(1)

∂z1

2 − ∂

2

F

(1)

∂z1∂z2

T

δ ∂F

(1)

= ∂δF

(1)

≡ ∂

2

δF

(1)

∂z2

2

∂

2

δF

(1)

∂z1

2 − ∂

2

δF

(1)

∂z1∂z2

T

δE

(1)

= a ∂δF

(1)

− B δK

(1)

M

(1)

= B

T

∂F

(1)

+ d K

(1)

a B d

δWint

(1)

= ∂F

(1) T

δE

(1)

+ M

(1) T

δK

(1)

δWint

(1)

= ∂F

(1) T

a ∂δF

(1)

− ∂F

(1) T

B δK

(1)

F

(1)

W

(1)

δW′int

(1)

= π2Ω

(1) T

N

(0)

δΩ

(1)
90

Complementary Virtual Work in terms of and 

In lieu of using compatibilty equation (220), a variational principle is obtained from the 

complementary virtual work given by

 (232a)

with

 (232b)

 (232c)

and where are statically admissible virtual stress resultants 

associated with the corresponding incompatible tangential strains defined by equations (193). 

Equation (232c) represents the virtual work that occurs when the tangential displacements fail to 

satisfy the geometric boundary conditions specified by equations (201a,b) and (202a,b). If the 

geometric constraints are forced to be satisfied and the strains are forced to be compatible, 

pointwise, then the resulting work must be zero, as stated by equation (232a). Integrating 

equation (232b) by parts, by using equations (25) specialized for the nondimensional coordinates 

(z1, z2), gives

W

(1)

F

(1)

δW*int

(1)

A

dz1dz2 + δW*

(1) B

= 0

δW*int

(1)

= E11

(1)

− ∂U1

(1)

∂z1

− 12Z1W

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z1

π2δN*11

+ E22

(1)

− ∂U2

(1)

∂z2

− 12Z2W

(1)

− ∂W

(0)

∂z2

∂W

(1)

∂z2

π2δN*22

+ G12

(1)

− ∂U1

(1)

∂z2

− ∂U2

(1)

∂z1

− ∂W

(0)

∂z1

∂W

(1)

∂z2

− ∂W

(0)

∂z2

∂W

(1)

∂z1

π2

αb

δN*12

δW*

B (1)

= π2U1

(1) δN*12

αb

+ U2

(1)

δN*22 a2

L 2

b 2

L 2

dz1

a1

L 1

b 1

L 1

+ π2U1

(1)

δN*11 + U2

(1) δN*12

αb a1

L 1

b 1

L 1

dz2

a2

L 2

b 2

L 2

π2δN*11, π2δN*22, and π2

αb

δN*12
91

 (233)

The virtual stress resultants are required to be statically admissible; that is they satisfy equilibrium 

equations (198a,b) and the force boundary conditions in equations (201a,b) and (202a,b). This 

requirement yields , , and . In addition, the boundary 

integrals in equations (232c) and (233) cancel. Thus, equation (232a) reduces to

 (234)

where the eqauations

 (235a)

 (235b)

 (235c)

have been used. Equation (234) is an integral statement of compatibility of the tangential strains 

δW*int

(1)

A

dz1dz2 = E11

(1)

− 12Z1W

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z1

π2δN*11

A

+ E22

(1)

− 12Z2W

(1)

− ∂W

(0)

∂z2

∂W

(1)

∂z2

π2δN*22 + G12

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z2

− ∂W

(0)

∂z2

∂W

(1)

∂z1

π2

αb

δN*12 dz1dz2

+ π2 ∂δN*11

∂z1

+ π2

αb

∂δN*12

∂z2

U1

(1)

+ π2

αb

∂δN*12

∂z1

+ π2 ∂δN*22

∂z2

U2

(1)

A

dz1dz2

− π2U1

(1)

δN*11 + U2

(1) δN*12

αb a1

L 1

b 1

L 1

dz2

a2

L 2

b 2

L 2

− π2U1

(1) δN*12

αb

+ U2

(1)

δN*22 a2

L 2

b 2

L 2

dz1

a1

L 1

b 1

L 1

δN*11 → δN11

(1)

δN*22 → δN22

(1)

δN*11 → δN11

(1)

δW*int

(1)

A

dz1dz2 = E11

(1)

− 12Z1W

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z1

∂

2

δF

(1)

∂z2

2

A

+ E22

(1)

− 12Z2W

(1)

− ∂W

(0)

∂z2

∂W

(1)

∂z2

∂

2

δF

(1)

∂z1

2 − G12

(1)

− ∂W

(0)

∂z1

∂W

(1)

∂z2

− ∂W

(0)

∂z2

∂W

(1)

∂z1

∂

2

δF

(1)

∂z1∂z2

dz1dz2 = 0

π2δN11

(1)

= ∂

2

δF

(1)

∂z2

2

π2δN22

(1)

= ∂

2

δF

(1)

∂z1

2

π2

αb

δN12

(1)

= − ∂

2

δF

(1)

∂z1∂z2
92

associated with adjacent equilibrium states. Further integration by parts of equation (234) yields 

 (236)

where the integrand is the compatibility equation given by equation (219).

A convenient matrix representation of equation (234) is given by

 (237)

where is defined by equation (212b) and (192), and

 (238)

Using constitutive equation (196), the integrand of equation (237) is expressed as

 (239)

δW*int

(1)

A

dz1dz2 =

∂

2

E11

(1)

∂z2

2 +

∂

2

E22

(1)

∂z1

2 − ∂

2

G12

(1)

∂z1∂z2

− 12Dc W

(1)

− L W

(0)

p , W

(1)

δF

(1)

dz1dz2

A

+ boundary terms

E

(1) T

− 12W

(1)

Z1 Z2 0 − Ω

(1) T

Ω

(0)

∂δF

(1)

A

dz1dz2 = 0

Ω

(1)

Ω

(0)

≡

− ∂W

(0)

∂z1

0 − ∂W

(0)

∂z2

0 − ∂W

(0)

∂z2

− ∂W

(0)

∂z1

E

(1) T

− 12W

(1)

Z1 Z2 0 − Ω

(1) T

Ω

(0)

∂δF

(1)

= ∂F

(1) T

a − K

(1) T

B T

− 12W

(1)

Z1 Z2 0 − Ω

(1) T

Ω

(0)

∂δF

(1)
93

Values of the Nondimensional Parameters

An extensive collection of tables and figures is presented in this section that shows the effects 

of lamina material properties and stacking sequence on the fundamental nondimensional 

parameters, and in some cases, their associated coefficients. In particular, results are presented for 

the nine lamina material systems given in Table 1 and for several stacking sequences. The fiber 

orientation angle θ of an arbitrary lamina is depicted in figure 3. This angle is defined as the angle 

between the line tangent to the ξ1-coordinate curve and the line tangent to the fiber at a given point 

(ξ1, ξ2, ζ) of a shell. The stacking sequences considered include balanced symmetric angle-ply 

laminates, balanced antisymmetric angle-ply laminates, symmetric quasi-isotropic laminates, 

antisymmetric quasi-isotropic laminates, and unsymmetric quasi-isotropic laminates. Results are 

also given for unbalanced, unsymmetric laminates composed of perpendicular plies aligned with 

the two surface coordinates and a single family of angle plies. For every laminate considered, the 

total thickness h is calculated based on the number of plies and a ply thickness of 0.005 in.

Results for Angle-Ply Laminates

Results showing the effects of the fiber orientation angle θ on the values of the 

nondimensional parameters, or their associated coefficients, for [(+θ/-θ)m]S and [(-θ/+θ)m]S

symmetric laminates and for (+θ/-θ)m and (-θ/+θ)m antisymmetric laminates are presented in 

Tables 2-14 and figures 4-20. The results shown in the figures were computed for one-degree 

increments of θ. In each of figures 4-14, nine curves are shown that correspond to the nine 

material system given in Table 1, and the results indicate that the P-100/3502 laminates generally 

possess the most extreme values of the nondimensional parameters, or their associated 

coefficients. In contrast, the effect of fiber angle is generally the most benign for the boronaluminum laminates. As a consequence of the extreme values exhibited, results for laminates 

made of the P-100/3502 material are presented in Tables 11-14, and curves are shown in figures 

15-20 that correspond to different values of the stacking sequence number m.

Results showing the effect of the fiber orientation angle θ on the values of the flexural 

orthotropy parameter β are shown in figure 4 and Table 2 for the symmetric and antisymmetric 

angle-ply laminates. Each of these laminates has the same value of β for a given material system, 

regardless of the number of plies (m = 1, 2, ...). For all laminates, the larger values of β occur in 

the range 30 degrees ≤ θ ≤ 60 degrees, with the maximum at θ = 45 degrees. For all the results, 

0.250 ≤ β ≤ 2.76.

Values of the orthotropy coefficients (a22/a11)

1/4 and (D11/D22)1/4 that appear in the 

nondimensional parameters αm and αb, respectively, are given in Table 3 and shown in figure 5 

as a function of the fiber angle θ. For these particular laminates, both coefficients have the same 

value for a given material system and a given value of θ. In addition, these values are independent 

of the number of laminate plies. The results presented in figure 5 show monotonic reductions in 

the values of the coefficients with increasing values of θ. Moreover, the larger values of the 

coefficientsfor each laminate occurs for θ < 45 degrees and the smaller values for θ > 45 degrees. 

All values of these coefficients are between 0.342 and 2.93.
94

The effects of material system and fiber orientation on the values for the generalized 

Poisson’s ratios νm and νb are given in Table 4 and shown in figure 6. For these symmetric and 

antisymmetric angle-ply laminates, the Poisson’s ratios possess identical values for a given 

material system and fiber angle, and these values are independent of the number of plies. Figure 

6 indicates that all Poisson’s ratios are positive valued and the largest value is less than 0.9. For 

all laminates, the larger values occur in the range 30 degrees ≤ θ ≤ 60 degrees, with the maximum 

at θ = 45 degrees. 

Results showing the effect of the fiber orientation angle θ on the values of the membrane 

orthotropy parameter μ are shown in figure 7 and Table 5 for the symmetric and antisymmetric 

angle-ply laminates. Each of these laminates also has the same value of μ for a given material 

system, regardless of the number of plies. For all laminates, the larger values of μ occur in the 

ranges θ < 10 degrees and θ > 80 degrees, and the smallest occur in the range 30 degrees ≤ θ ≤

60 degrees, with the minimum at θ = 45 degrees. Altogether, -1 < μ < 5.5.

Values of the coefficient that appears in the nondimensional Batdorf-Stein 

parameters Z1 and Z2 are given in Table 6 and shown in figure 8 as a function of the fiber angle 

θ. Each of these laminates also has the same value of this coefficient for a given material system 

and fiber angle, regardless of the number of plies. All values of the coefficient are between 0.4 

and 1.0, with the smaller values in the range 30 degrees ≤ θ ≤ 60 degrees.

The effects of material system and fiber orientation on the values of the flexural anisotropy 

parameters γb and δb for (+θ/-θ)S and (-θ/+θ)S four-ply symmetric laminates are given in Tables 

7 and 8, respectively, and shown in figure 9-10, respectively. The values of these two parameters 

are between -0.015 and 0.7 for the (+θ/-θ)S laminates, and between -0.7 and 0.015 for the (-θ/+θ)S

laminates. Although it is not shown herein for all material systems considered, the magnitudes of 

these parameters diminish in the [(+θ/-θ)m]S and [(-θ/+θ)m]S symmetric laminates as the number 

of plies increases. For the (+θ/-θ)m and (-θ/+θ)m antisymmetric laminates, γb and δb are 

identically equal to zero for all values of θ and m. A parametric plot of γb and δb is presented 

in figure 11 for the (+θ/-θ)S and (-θ/+θ)S symmetric laminates, where θ is the parameter. Each 

curve in this figure is traversed counterclockwise as θ increases from 0 to 90 degrees, and each 

curve is symmetric about a line passing through the points of the curves that correspond to θ = 

45 degrees. Moreover, each curve begins and ends at the origin where θ = 0 and 90 degrees, 

consistent with a lack of flexural anisotropy. The unfilled circular symbols correspond to 

sequential values of θ in 15-degree increments. The difference in shape of the parametric curves 

indicates that the lamina material properties have a moderate effect on the relative proportions of 

the two anisotropy parameters, with respect to the fiber angle.

Values of the only nonzero load-path eccentricity parameters, e16 and e26, are given for the 

(+θ/-θ) and (-θ/+θ) two-ply antisymmetric laminates in Tables 9 and 10, respectively, and are 

shown in figures 12 and 13, respectively, as a function of the fiber angle θ. The corresponding 

parametric plot, with θ as the parameter, is shown in figure 14. Each parametric curve in this 

figure is also traversed counterclockwise as θ increase from 0 to 90 degrees, and each curve is 

h

12 a 11a 22D11D22

1

4
95

also symmetric about a line passing through the points of the curves that correspond to θ = 45 

degrees. The unfilled circular symbols in this figure also correspond to sequential values of θ in 

15-degree increments. The values of these two parameters are between -0.015 and 1.78 for the (-

θ/+θ) laminates, and between -1.78 and 0.015 for the [+θ/-θ] laminates. The difference in shape 

of the parametric curves shown in figure 14 indicates that the lamina material properties has a 

moderate effect on the relative proportions of the two load-path eccentricity parameters, with 

respect to the fiber angle. For the symmetric angle-ply laminates, all load-path eccentricity 

parameters are identically equal to zero.

The combined effects of fiber orientation and number of plies on the values of the flexural 

anisotropy parameters γb and δb are given in Tables 11 and 12, respectively, and shown in figures 

15-16, respectively, for [(+θ/-θ)m]S and [(-θ/+θ)m]S symmetric laminates made of the P-100/3502 

material. The corresponding parametric plot, with θ as the parameter, is shown in figure 17. Each 

parametric curve in this figure is also traversed counterclockwise as θ increases from 0 to 90 

degrees, and each curve is also symmetric about a line passing through the points of the curves 

that correspond to θ = 45 degrees. In going from four to twenty-four plies, the maximum 

magnitude of these anisotropy parameters is reduced by 83%. Likewise, in going from four to 

forty-eight plies, the maximum magnitude of these anisotropy parameters is reduced by 92%. The 

similar shape of the parametric curves indicates that the number of plies has a relatively small 

effect on the relative proportions of the two anisotropy parameters, with respect to the fiber angle.

The combined effects of fiber orientation and number of plies on the values of the load-path 

eccentricity parameters e16 and e26 are given in Tables 13 and 14, respectively, and shown in 

figure 18-19, respectively, for the (+θ/-θ)m and (-θ/+θ)m antisymmetric laminates made of the P100/3502 material. The corresponding parametric plot, with θ as the parameter, is shown in figure 

20. Each parametric curve in this figure is also traversed counterclockwise as θ increase from 0 

to 90 degrees, and each curve is also symmetric about a line passing through the points of the 

curves that correspond to θ = 45 degrees. In going from two to twenty-four plies, the maximum 

magnitude of these anisotropy parameters is reduced by 92%. Likewise, in going from two to 

forty-eight plies, the maximum magnitude of these anisotropy parameters is reduced by 96%. The 

similar shape of the parametric curves for these parameters also indicates that the number of plies 

has no significant effect on their relative proportions, with respect to the fiber angle.

Results for Quasi-Isotropic Laminates

Results showing the effects of the number of plies, and their order, on the values of the 

nondimensional parameters, and associated coefficients, for quasi-isotropic laminates made of the 

nine lamina material systems are presented in Tables 15-45. Additionally, the results for the P100/3502 laminates are shown figures 21-30. In particular, results are presented in these Tables 

and figures for [(±45/0/90)m]S and [(0/90/±45)m]S symmetric laminates, [(±45/0/90)m]A and 

[(0/90/±45)m]A antisymmetric laminates (antisymmetry is indicated by the subscript A in the 

stacking sequence notation herein), and (±45/0/90)m and (0/90/±45)m unsymmetric laminates, for 

sequential integer values of m from 1 to 8. Likewise, three or four curves that connect symbols 

are shown in figures 21-30. In most of the figures, the dashed and solid blue curves correspond to 
96

results for the [(±45/0/90)m]S and [(±45/0/90)m]A laminates and the [(0/90/±45)m]S and [(0/90/

±45)m]A laminates, respectively. The dashed and solid gray curves correspond to results for the 

(±45/0/90)m laminates and the (0/90/±45)m laminates, respectively. For each of these laminates, 

(a22/a11)

1/4 = 1 and μ = 1, regardless of the lamina material system and number of plies. Similarly, 

the values of the generalized membrane Poisson’s ratio νm are independent of the number of plies. 

The variation of νm with material system is presented in Table 15 and ranges from 0.272 to 0.325.

The effects of lamina material properties and number of plies on the coefficient (D11/D22)

1/4 

that appears in the nondimensional parameter αb are presented in Tables 16-19. Specifically, the 

values given in Table 16 are for the [(±45/0/90)m]S and [(±45/0/90)m]A laminates, the values in 

Table 17 are for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates, the values in Table 18 are for the 

(±45/0/90)m laminates, and the values in Table 19 are for the (0/90/±45)m laminates. The results 

for each of these laminates made of the P-100/3502 material are shown in figure 21. The results 

in Tables 16 and 17 for the symmetric and antisymmetric laminates, and in Table 19 for the (0/

90/±45)m unsymmetric laminates, indicate that the values for (D11/D22)

1/4 approach unity, the value 

for a homogeneous isotropic material, from above as the number of plies increases, regardless of 

the lamina material system, as shown in figure 21 for the corresponding P-100/3502 laminates. 

However, the results in Table 18 and figure 21 for the (±45/0/90)m unsymmetric laminates show 

convergence from below to unity as the number of plies increases. Altogether, these results are 

bounded by the values (D11/D22)

1/4 = 0.772 and 1.30.

Results that show the effects of lamina material properties and number of plies on the 

nondimensional parameter β are presented in Table 20 for the [(±45/0/90)m]S and [(±45/0/90)m]A

laminates, in Table 21 for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates, and in Table 22 for the 

(±45/0/90)m and (0/90/±45)m laminates. The results for each of these laminates made of the P-100/

3502 material are shown in figure 22. The results in Table 20 for the [(±45/0/90)m]S and [(±45/0/

90)m]A laminates, and in Table 22 for both families of unsymmetric laminates, indicate that the 

values for β approach unity, the value for a homogeneous isotropic material, from above as the 

number of plies increases, regardless of the lamina material system. This trend is shown in figure 

22 for the corresponding P-100/3502 laminates. However, the results in Table 21 and figure 22 

for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates show convergence from below to unity as the 

number of plies increases. Altogether, 0.266 ≤ β ≤ 2.22, and the unsymmetric laminates exhibit 

significantly faster convergence to unity than the other laminates with increasing number of plies. 

This convergence characteristic is explained by noting the basic unit forming each stacking 

sequence is repeated more for the unsymmetric laminates, for a given number of plies, thus 

approaching the homogeneity of an isotropic material faster.

Results that show the effects of lamina material properties and number of plies on the 

generalized Poisson’s ratio νb are presented in Table 23 for the [(±45/0/90)m]S and [(±45/0/90)m]A

laminates, in Table 24 for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates, and in Table 25 for the 

(±45/0/90)m and (0/90/±45)m laminates. The results for each of these laminates made of the P-100/

3502 material are shown in figure 23. The results in Table 23 for the [(±45/0/90)m]S and [(±45/0/

90)m]A laminates, and in Table 25 for both families of unsymmetric laminates, indicate that the 

values for νb approach the corresponding value of νm given in Table 15, from above as the 
97

number of plies increases, regardless of the lamina material system. This trend is shown in figure 

23 for the corresponding P-100/3502 laminates. In contrast, the results in Table 24 and figure 23 

for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates show convergence from below to the 

corresponding value of νm as the number of plies increases. Altogether, 0.091 ≤ νb ≤ 0.716, and 

the unsymmetric laminates exhibit significantly faster convergence than the other laminates with 

increasing number of plies. 

The effects of lamina material properties and number of plies on the flexural anisotropy 

parameters γb and δb

 are presented in Tables 26 and 30 for the [(±45/0/90)m]S laminates, 

respectively; in Tables 27 and 31 for the [(0/90/±45)m]S laminates, respectively; in Tables 28 and 

32 for the (±45/0/90)m laminates, respectively; and in Tables 29 and 33 for the (0/90/±45)m 

laminates, respectively. Values of γb and δb for each of these laminates made of the P-100/3502 

material are shown in figures 24 and 25, respectively. In contrast to the previous figures, results 

for the [(±45/0/90)m]S and [(0/90/±45)m]S symmetric laminates are indicated in these two figures 

by dashed and solid black lines, respectively. The values of γb and δb for the [(±45/0/90)m]A and 

[(0/90/±45)m]A antisymmetric laminates are equal to zero and are indicated in the two figures by 

the solid blue line. Unlike the previous corresponding results, the results in Tables 26-33, and 

figures 24 and 25, indicate convergence to a value of zero, the value for a homogeneous isotropic 

material, from above for the [(±45/0/90)m]S, [(0/90/±45)m]S, and (±45/0/90)m laminates and from 

below for the (0/90/±45)m laminates. Altogether, -0.209 ≤ γb ≤ 0.351, -0.351 ≤ δb ≤ 0.253, and the 

unsymmetric laminates exhibit significantly faster convergence than the other laminates with 

increasing number of plies. 

Values of the coefficient that appears in the nondimensional Batdorf-Stein 

parameters Z1 and Z2 are given in Table 34 for the [(±45/0/90)m]S and [(±45/0/90)m]A laminates, 

in Table 35 for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates, and in Table 36 for the (±45/0/

90)m and (0/90/±45)m laminates. The results for each of these laminates made of the P-100/3502 

material are shown in figure 26. The results in Tables 34 and 36 indicate that the values for the 

coefficient for each lamina material system approach the corresponding value of , where 

νm is given in Table 15, from above as the number of plies increases. This trend is shown in figure 

26 for the corresponding P-100/3502 laminates. In contrast, the results in Table 35 and figure 26 

for the [(0/90/±45)m]S and [(0/90/±45)m]A laminates show convergence from below to the 

corresponding value of as the number of plies increases. For all cases considered, the 

values for are between 0.875 and 1.09, and the unsymmetric laminates exhibit 

significantly faster convergence than the other laminates with increasing number of plies.

Values of the load-path eccentricity parameter e11 are given for the (±45/0/90)m and (0/90/

±45)m laminates in Tables 37 and 38, respectively, and are shown in figure 27 as a function of the 

number of plies for the laminates made of the P-100/3502 material. The values of this parameter 

for the symmetric and antisymmetric quasi-isotropic laminates are equal to zero, and are indicated 

in figure 27 by the solid blue line. These results indicate that the values of e11 are negative and 

approach zero from below with increasing number of plies for all the laminates except the (±45/

h

12 a 11a 22D11D22

1

4

1− νm

2

1− νm

2

h

12 a 11a 22D11D22

1

4
98

0/90)m laminates made from the Boron-Aluminum and the Boron-epoxy materials. These two 

exceptions approach zero from above with increasing number of plies. Moreover, the values of 

e11 for the (0/90/±45)m laminates are significantly larger than the values for the corresponding 

(±45/0/90)m laminates, and exhibit a slower convergence rate. This difference in convergence 

rates is illustrated in figure 27 for the laminates made of the P-100/3502 material. 

Values of the load-path eccentricity parameters e12 and e66 are given for the (±45/0/90)m and 

(0/90/±45)m laminates in Tables 39 and 40, respectively, and are shown in figure 28 as a function 

of the number of plies for the laminates made of the P-100/3502 material. The values of these two 

parameters for the symmetric and antisymmetric quasi-isotropic laminates are equal to zero, and 

are indicated in figure 28 by the solid blue line. These results indicate that the values of e12 and 

e66 are negative and approach zero from below with increasing number of plies for all the 

(±45/0/90)m laminates. However, the values of e12 and e66 for the (0/90/±45)m laminates are 

positive and approach zero from above with increasing number of plies. Moreover, the values of 

the parameters for a given (±45/0/90)m laminate are the negative of the corresponding (0/90/±45)m

laminate. Thus, both laminate constructions exhibit the same convergence rate as illustrated in 

figure 28 for the laminates made of the P-100/3502 material. 

The effects of lamina material properties and number of plies on the load-path eccentricity 

parameters e16 and e26 are given for the [(±45/0/90)m]A and [(0/90/±45)m]A antisymmetric 

laminates in Tables 41 and 42, respectively, and for the (±45/0/90)m and (0/90/±45)m 

unsymmetric laminates in Table 43. The corresponding results are shown in figure 29 as a 

function of the number of plies for the laminates made of the P-100/3502 material. These results 

indicate that the values of e16 and e26 are equal and negative and approach zero from below with 

increasing number of plies for all the laminates. In addition, all the laminate families exhibit 

nearly the same convergence rate, as depicted in figure 29. The values of the parameters for the 

corresponding symmetric laminates are identically equal to zero and are depicted in figure 29 by 

the black solid line.

Values of the load-path eccentricity parameter e22 are given for the (±45/0/90)m and 

(0/90/±45)m laminates in Tables 44 and 45, respectively, and are shown in figure 30 as a function 

of the number of plies for the laminates made of the P-100/3502 material. The values of this 

parameter for the symmetric and antisymmetric quasi-isotropic laminates are equal to zero, and 

are indicated in figure 30 by the solid blue line. These results indicate that the values of e22 are 

positive and approach zero from above with increasing number of plies for all the laminates 

except the (±45/0/90)m laminates made from the Boron-Aluminum and the Boron-epoxy 

materials. These two exceptions approach zero from above with increasing number of plies. 

Moreover, the values of e22 for the (0/90/±45)m laminates are significantly smaller than the values 

for the corresponding (±45/0/90)m laminates, and exhibit a faster convergence rate, as is illustrated 

in figure 30 for the laminates made of the P-100/3502 material. 
99

Results for Unbalanced, Unsymmetric Laminates

Results showing the effects of the fiber orientation angle θ on the values of the 

nondimensional parameters, and associated coefficients, for (+θ/0/90)m and (-θ/0/90)m laminates 

are presented in Tables 46-72 and figures 31-62. In each of figures 31-49, nine curves are shown 

for one-degree increments of θ that correspond to the nine material systems given in Table 1, and 

the results also indicate that the P-100/3502 laminates generally possess the most extreme values 

of the nondimensional parameters, or their associated coefficients. As seen for the angle-ply and 

quasi-isotropic laminates examined herein, the effect of fiber angle is generally the most benign 

for the boron-aluminum laminates. As a consequence, results for laminates made of the P-100/

3502 material are presented in Tables 62-72, and curves are shown in figures 50-62 that 

correspond to different values of the stacking sequence number m.

Values of the nondimensional flexural orthotropy parameter β for the (+θ/0/90)m and 

(-θ/0/90)m laminates vary with the number of plies forming a given laminate. The effects of the 

fiber orientation angle θ on the values of β are shown in figure 31 and given in Table 46 for the 

(+θ/0/90) and (-θ/0/90) three-ply laminates. For all three-ply laminates, the larger values of β

generally occur in the range 40 degrees ≤ θ ≤ 65 degrees, with the maximums between θ = 50 

and 55 degrees. Overall, 0.064 ≤ β ≤ 1.23 for the three-ply laminates.

Values of the orthotropy coefficients (D11/D22)

1/4 and (a22/a11)1/4 that appear in the 

nondimensional parameters αb and αm, respectively, are given in Tables 47 and 48, respectively, 

and shown in figures 32 and 33, respectively, as a function of the fiber angle θ. For these families 

of unbalanced and unsymmetric laminates, the two coefficients generally have different values for 

a given material system and a given value of θ, unlike the symmetric angle-ply laminates 

examined herein. In addition, the values of (a22/a11)

1/4 are independent of the number of laminate 

plies, whereas the values of (D11/D22)

1/4 depend on the number of plies. Thus, the results presented 

in figure 32 are for (+θ/0/90) and (-θ/0/90) three-ply laminates. The general trend shown in 

figures 32 and 33 is a monotonic reduction in the values of the coefficients with increasing values 

of θ. Moreover, the coefficient (D11/D22)

1/4 exhibits the the most pronounced reductions with 

increasing values of θ. Altogether, 0.478 ≤ (D11/D22)

1/4 ≤ 1.02 for the three-ply laminates and 

0.845 ≤ (a22/a11)

1/4 ≤ 1.18 for these laminates with any number of plies.

The effects of material system and fiber orientation on the values of the generalized Poisson’s 

ratios νm and νb

 are given in Tables 49 and 50, respectively, and shown in figures 34 and 35, 

respectively. For these families of unbalanced and unsymmetric laminates, the two coefficients 

also generally have different values for a given material system and a given value of θ. In addition, 

the values for νm are independent of the number of laminate plies, whereas the values of νb

depend on the number of plies. Thus, the results presented in figure 35 are for (+θ/0/90) and (-θ/

0/90) three-ply laminates. The results given for these parameters indicate that all Poisson’s ratios 

are positive valued and the largest value is less than 0.4. For all laminates, the larger values of 

both Poisson’s ratios occur in the range 40 ≤ θ ≤ 65 degrees. Moreover, the results indicate 0.008 

≤ νb

 ≤ 0.385 for the three-ply laminates and 0.009 ≤ νm ≤ 0.242 for these laminates with any 

number of plies.
100

Results showing the effect of the fiber orientation angle θ on the values of the membrane 

orthotropy parameter μ are shown in figure 36 and given in Table 51 for the two families of 

unbalanced, unsymmetric laminates. Each of these laminates also has the same value of μ for a 

given material system, regardless of the number of plies. For all laminates, the larger values of μ

occur in the ranges θ < 10 degrees and θ > 80 degrees, and the smallest occur in the range 30 

degrees ≤ θ ≤ 60 degrees, with the minimum at θ = 45 degrees. Altogether, 1.23 < μ < 16.9.

The effects of material system and fiber orientation on the values for the membrane 

anisotropy parameters γm and δm for the (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...) are 

given in Tables 52 and 53, respectively, and shown in figure 37 and 38, respectively. The values 

of these two parameters are between -0.048 and 2.06 for the (+θ/0/90)m laminates, and between 

-2.06 and 0.048 for the (-θ/0/90)m laminates. The largest magnitudes of γm and δm occur for θ > 

65 and θ < 25 degrees, respectively. A parametric plot of γm and δm is presented in figure 39 for 

the (+θ/0/90)m laminates, where θ is the parameter. Each curve in this figure is traversed 

clockwise as θ increases from 0 to 90 degrees, and each curve is symmetric about a line passing 

through the points of the curves that correspond to θ = 45 degrees. The unfilled circular symbols 

correspond to sequential values of θ in 15-degree increments. The difference in shape of the 

parametric curves indicates that the lamina material properties have a moderate effect on the 

relative proportions of the two anisotropy parameters, with respect to the fiber angle.

The values of the flexural anisotropy parameters γb

 and δb for (+θ/0/90)m and (-θ/0/90)m

laminates generally vary with the number of plies, for a given material system and fiber angle θ. 

To gain insight into the nature of these parameters, the effects of material system and fiber 

orientation on the values of the flexural anisotropy parameters γb and δb for the (+θ/0/90) and 

(-θ/0/90) three-ply laminates are given in Tables 54 and 55, respectively, and shown in figures 40 

and 41, respectively. The values of γb are between -0.008 and 0.499 for the (+θ/0/90) laminates, 

and between -0.499 and 0.008 for the (-θ/0/90) laminates. The corresponding magnitudes of δb

are generally smaller than the corresponding magnitudes of γb. In particular, the values of δb are

between -0.007 and 0.349 for the (+θ/0/90) laminates, and between -0.349 and 0.007 for the 

(-θ/0/90) laminates. In addition, the larger magnitudes of γb and δb occur in the ranges 40 < θ < 

50 degrees and 55 < θ < 65 degrees, respectively. A parametric plot of γb and δb is presented in 

figure 42 for the (+θ/0/90) three-ply laminates, where θ is the parameter. Each curve in this figure 

is traversed counterclockwise as θ increases from 0 to 90 degrees, and each curve is not 

symmetric about a line passing through the points of the curves that correspond to θ = 45 degrees. 

The unfilled circular symbols correspond to sequential values of θ in 15-degree increments. The 

difference in shape of the parametric curves indicates that the lamina material properties have 

only a moderate effect on the relative proportions of the two anisotropy parameters, with respect 

to the fiber angle.

Values of the coefficient that appears in the nondimensional Batdorf-Stein 

parameters Z1 and Z2 are given in Table 56 and shown in figure 43 as a function of the fiber angle 

θ for the (+θ/0/90) and (-θ/0/90) three-ply laminates. Unlike the symmetric and antisymmetric 

angle-ply laminates examined herein, the value of this parameter for the (+θ/0/90)m and (-θ/0/90)m

laminates depends on the number of plies, for a given material system and fiber angle. All values 

of this coefficient for the three-ply laminates are between 0.911 and 1.48, with the smaller values 

h

12 a 11a 22D11D22

1

4
101

in the range 10 < θ < 25 degrees.

The (+θ/0/90)m and (-θ/0/90)m laminates exhibit all six load-path eccentricity parameters, 

which vary with the number of plies for a given material system and fiber angle θ. Values of e11

are presented in Table 57 and shown in figure 44 for the corresponding three-ply laminates. All 

values of this parameter are between -0.715 and 0, and the magnitudes diminish as θ increases. 

Corresponding values of e12 and e66 are presented in Table 58 and shown in figure 45 for the threeply laminates. For this case, the values of both parameters are identical and are between -0.272 

and 0. The largest magnitudes occur for values of 30 < θ < 60 degrees. The effects of material 

system and fiber angle on e22 are presented in Table 59 and shown in figure 46 for the three-ply 

laminates. These results indicate a monotonic reduction in e22 as θ increases. In addition, 0 ≤ e22

≤ 0.928. 

Values of the load-path eccentricity parameters, e16 and e26, are given for the three-ply 

laminates in Tables 60 and 61, respectively, and shown in figures 47 and 48, respectively, as a 

function of the fiber angle θ. These results indicate that the values of these two parameters are 

nonpositive and, for the most part, negative for the (+θ/0/90) laminates. In contrast, the values of 

these two parameters are nonnegative and, for the most part, positive for the (-θ/0/90) laminates. 

In addition, the magnitude of e26 is generally larger than the corresponding magnitude of e16 for 

values of θ > 45 degrees, and vice versa. In particular, 0.316 ≤ | e16| ≤ 0 and 0.395 ≤ | e26 | ≤ 0. The 

corresponding parametric plot, with θ as the parameter, is shown in figure 49 for the (+θ/0/90) 

laminates. Each parametric curve in this figure is also traversed counterclockwise as θ increases 

from 0 to 90 degrees, and the unfilled circular symbols correspond to sequential values of θ in 

15-degree increments. Examination of this figure indicates that the curves are generally not 

symmetric about a line passing through the points of the curves that correspond to θ = 45 degrees. 

The amount of asymmetry is also influenced slightly by the lamina material properties.

The combined effects of fiber orientation and number of plies on the values of the parameter 

coefficient (D11/D22)

1/4 for (+θ/0/90)m and (-θ/0/90)m laminates made of the P-100/3502 material 

are indicated in Table 62 and shown in figure 50. These results indicate a monotonic reduction in 

the value of (D11/D22)

1/4 as θ increases and a coalescence of the curves as the number of plies 

increases. Associated with this coalescence of the curves is an increase in the value of the 

coefficient and a reduction in the variation of the values as the fiber angle varies. Similar results 

are presented in Table 63 and figure 51 for the flexural orthotropy parameter β, and in Table 64 

and figure 52 for the generalized Poisson’s ratio νb

. The results for these two parameters also 

show a coalescence of the curves as the number of plies increases, but the magnitudes of each 

parameter generally decreases as the number of plies increases. Moreover, the curves coalesce to 

a curve that is symmetric about the line given by θ = 45 degrees. 

The combined effects of fiber orientation and number of plies on the values of the flexural 

anisotropy parameters γb and δb are given in Tables 65 and 66, respectively, and shown in figures 

53 and 54 respectively, for (+θ/0/90)m and (-θ/0/90)m laminates made of the P-100/3502 material. 

The corresponding parametric plot, with θ as the parameter, is shown in figure 55. Each 

parametric curve in this figure is also traversed counterclockwise as θ increases from 0 to 90 
102

degrees, and the curves for laminates with m ≤ 3 are generally not symmetric about a line passing 

through the points of the curves that correspond to θ = 45 degrees. However, the parametric 

curves coalesce to a curve that exhibits symmetry about the line corresponding to θ = 45 degrees 

as the number of plies increases. In going from three to twenty-four plies, the curves shown in 

these figures coalesce and the maximum magnitude of γb and δb are reduced by 55% and 36%, 

respectively. 

Values of the coefficient are given in Table 67 and shown in figure 56 as a 

function of the fiber angle θ and number of plies for the (+θ/0/90)m and (-θ/0/90)m laminates 

made of the P-100/3502 material. These results indicate substantial reductions in the values of this 

coefficient as the number of plies increases for θ > 35 degrees. In going from three to twenty-four 

plies, the maximum magnitude is reduced by approximately 32%.

The effects of fiber orientation and number of plies on the values of the load-path eccentricity 

parameters e11 and e22 are given in Tables 68 and 69, respectively, and shown in figures 57 and 

58, respectively, for the (+θ/0/90)m and (-θ/0/90)m laminates made of the P-100/3502 material. 

The values of e11 are nonpositive and negative for the most part, the values of e22 are nonnegative 

and positive for the most part. The curves shown in each figure coalesce to a single curve with 

substantially smaller magnitudes as the number of plies increases. In going from three to twentyfour plies, the maximum magnitudes of e11 and e22 are reduced by approximately 89% and 85%, 

respectively. Similar results for the load-path eccentricity parameters e12 and e66 are given in 

Table 70 and shown in figure 59. For this case the values of these two parameters are identical 

and are nonpositive and negative for the most part. The curves shown in figure 59 also coalesce 

as the number of plies increases, with a reduction in the maximum magnitude of approximately 

90%.

 Values of the load-path eccentricity parameters e16 and e26 are given in Tables 71 and 72, 

respectively, and shown in figure 60 and 61, respectively, for the (+θ/0/90)m and (-θ/0/90)m

laminates made of the P-100/3502 material as a function of fiber orientation and number of plies. 

The corresponding parametric plot for the (+θ/0/90)m laminates, with θ as the parameter, is shown 

in figure 62. Each parametric curve in this figure is also traversed counterclockwise as θ 

increases from 0 to 90 degrees. In going from three to twenty-four plies, the maximum magnitude 

of e16 and e26 are reduced by approximately 90% and the parametric curves coalesce to a curve 

that is symmetric about a line corresponding to θ = 45 degrees. 

Concluding Remarks

A comprehensive development of nondimensional parameters and equations for nonlinear 

and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory 

for thin anisotropic shells, has been presented. A complete set of field equations for geometrically 

imperfect shells that includes kinematic equations, isothermal constitutive equations for generally 

laminated shells, equilibrium equations, boundary conditions, the compatibility equation, and the 

h

12 a 11a 22D11D22

1

4
103

virtual work has been presented in terms of lines-of-curvature coordinates. A systematic 

nondimensionalization of these equations has been developed, several new nondimensional 

parameters have been defined, and a comprehensive stress-function formulation has been 

presented that includes variational principles for equilibrium and compatibility. Bifurcation 

analysis was also applied to the nondimensional nonlinear field equations and a comprehensive 

set of bifurcation equations have been given that include the effects of pre-bifurcation rotations, 

which are commonly neglected. These bifurcation equations also include a stress-function 

formulation with variational principles for equilibrium and compatibility of the adjacent 

equilibrium states.

An extensive collection of tables and figures has been presented that show the effects of 

lamina material properties and stacking sequence on the nondimensional parameters. In 

particular, results are presented for nine lamina material systems and several stacking sequences. 

These stacking sequences include balanced symmetric angle-ply laminates, balanced 

antisymmetric angle-ply laminates, symmetric quasi-isotropic laminates, antisymmetric quasiisotropic laminates, and unsymmetric quasi-isotropic laminates. Results are also given for 

unbalanced, unsymmetric laminates composed of perpendicular unidirectional plies aligned with 

the shell surface coordinate curves and angle plies. For each laminate configuration, the numerical 

range of each nondimensional parameter, or the associated coefficient, has been given. These 

numerical values provide reasonable estimates to boundaries of the nondimensional design space 

for a wide range of practical laminates and highly tailored laminates. Overall, the analysis and 

results should be of great interest to researchers developing structural design technology.

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vol. 43, 2006, pp. 6978-6997.
109

73. Weaver, P. M.: Physical Insight into the Buckling Phenomena of Composite Structures. 

Proceedings of the AIAA/ASME/ASCE/AHS/ASC 47th Structures, Structural Dynamics and 

Materials Conference, 2006, AIAA Paper No. 2006-2031.

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Rectangular Plates with Flexural/Twist Anisotropy. Proceedings of the Royal Society A, vol. 

462, 2006, pp. 59-73.

75. Weaver, P. M.: Anisotropic Elastic Tailoring in Laminated Composite Plates and Shells. 

Buckling and Postbuckling of Structures-Experimental, Analytical and Numerical Studies, B. 

G. Falzon and M. H. Aliabadi, eds., Imperial College Press, 2008, pp. 177-224.

76. Weaver, P. M.; and Nemeth, M. P.: Bounds on Flexural Properties and Buckling Response for 

Symmetrically Laminated Composite Plates. Journal of Engineering Mechanics, vol. 133, no. 

11, 2007, pp. 1178-1191.

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Plates Under Combined Loading. AIAA Journal, vol. 46, no. 9, 2008, pp. 2391-2396.

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Composite Plates Braced by Omega-Stringers. Composite Structures, vol. 88, 2009, pp. 424-

435.

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and Stiffness Design of Compression-Loaded Laminated-Composite Cylinders. NASA/TP2009-215778, 2009.

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Mathematics, vol. 21, no. 1, 1963, pp. 21-36.

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Inc., 1999.

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1982.

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John Wiley & Sons, Inc., 2002.
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558.

Table 1. Lamina Properties

* The symbols L and T denote the longitudinal fiber and transverse matrix directions of a specially orthotropic lamina, respectively.

Lamina 

property*

Material Systems

Boron-Al S-glassepoxy

Kevlar 

49-epoxy

IM7/

5260

AS4/

3502

AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

EL, Msi 33 7.5 11.02 22.1 18.5 20.01 29.58 20.35 53.5

ET

, Msi 21 1.7 0.8 1.457 1.64 1.30 2.68 1.16 0.73

νLT 0.23 0.25 0.34 0.258 0.30 0.30 0.23 0.29 0.31

GLT

, Msi 7.0 0.80 0.33 0.860 0.87 1.03 0.81 0.61 0.76
111

Table 2. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m laminates 

(m = 1, 2, ...) 

Table 3. Values of and for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m 

laminates (m = 1, 2, ...)

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .697 .561 .312 .368 .403 .478 .250 .319 .279

5 .715 .596 .390 .447 .469 .551 .324 .406 .460

10 .768 .699 .617 .674 .661 .757 .539 .659 .954

15 .851 .860 .966 1.02 .954 1.06 .880 1.04 1.57

20 .957 1.06 1.38 1.41 1.30 1.40 1.30 1.47 2.09

25 1.08 1.28 1.77 1.77 1.64 1.72 1.72 1.86 2.42

30 1.19 1.48 2.09 2.06 1.92 1.96 2.08 2.16 2.60

35 1.29 1.65 2.30 2.26 2.12 2.13 2.33 2.36 2.70

40 1.36 1.75 2.42 2.36 2.24 2.23 2.47 2.46 2.74

45 1.38 1.79 2.46 2.40 2.28 2.26 2.51 2.50 2.76

50 1.36 1.75 2.42 2.36 2.24 2.23 2.47 2.46 2.74

55 1.29 1.65 2.30 2.26 2.12 2.13 2.33 2.36 2.70

60 1.19 1.48 2.09 2.06 1.92 1.96 2.78 2.16 2.60

65 1.08 1.28 1.77 1.77 1.64 1.72 1.72 1.86 2.42

70 .957 1.06 1.38 1.41 1.30 1.40 1.30 1.47 2.09

75 .851 .860 .966 1.02 .954 1.06 .880 1.04 1.57

80 .768 .699 .617 .674 .661 .757 .539 .659 .954

85 .715 .596 .390 .447 .469 .551 .324 .406 .460

90 .697 .561 .312 .368 .403 .478 .250 .319 .279

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 1.12 1.45 1.93 1.97 1.83 1.98 1.82 2.05 2.93

5 1.12 1.44 1.92 1.96 1.82 1.97 1.82 2.04 2.90

10 1.11 1.43 1.89 1.93 1.80 1.92 1.80 2.00 2.79

15 1.11 1.40 1.83 1.86 1.74 1.85 1.76 1.93 2.56

20 1.10 1.36 1.74 1.75 1.65 1.73 1.68 1.81 2.24

25 1.08 1.30 1.61 1.61 1.54 1.59 1.57 1.66 1.92

30 1.07 1.24 1.45 1.45 1.41 1.43 1.43 1.48 1.63

35 1.05 1.16 1.29 1.29 1.26 1.28 1.28 1.31 1.38

40 1.02 1.08 1.14 1.14 1.13 1.13 1.14 1.15 1.17

45 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

50 .977 .927 .878 .879 .888 .883 .881 .873 .852

55 .957 .863 .774 .775 .791 .782 .780 .765 .725

60 .939 .809 .689 .688 .712 .697 .698 .675 .615

65 .924 .767 .623 .620 .650 .629 .637 .604 .522

70 .912 .736 .576 .570 .604 .577 .595 .552 .446

75 .903 .714 .546 .538 .575 .542 .570 .519 .391

80 .898 .700 .529 .519 .557 .520 .556 .500 .359

85 .894 .692 .521 .509 .548 .508 .550 .491 .345

90 .893 .690 .519 .507 .546 .505 .549 .489 .342

β = D12 + 2D66

D11D22

a 22

a 11

1

4 D11

D22

1

4
112

Table 4. Values of and for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and 

(-θ/+θ)m laminates (m = 1, 2, ...)

Table 5. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m laminates 

(m = 1, 2, ...)

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .183 .119 .092 .066 .089 .076 .069 .069 .036

5 .189 .131 .118 .092 .111 .101 .094 .098 .097

10 .206 .164 .193 .168 .175 .170 .165 .182 .263

15 .233 .217 .309 .282 .272 .272 .278 .309 .475

20 .267 .283 .447 .414 .388 .388 .418 .453 .654

25 .305 .354 .578 .538 .503 .497 .559 .586 .770

30 .342 .421 .684 .636 .598 .583 .678 .688 .836

35 .374 .475 .756 .703 .666 .642 .761 .755 .872

40 .396 .510 .796 .741 .706 .676 .809 .791 .889

45 .403 .522 .809 .753 .719 .687 .824 .803 .895

50 .396 .510 .796 .741 .706 .676 .809 .791 .889

55 .374 .475 .756 .703 .666 .642 .761 .755 .872

60 .342 .421 .684 .636 .598 .583 .678 .688 .836

65 .305 .354 .578 .538 .503 .497 .559 .586 .770

70 .267 .283 .447 .414 .388 .388 .418 .453 .654

75 .233 .217 .309 .282 .272 .272 .278 .309 .475

80 .206 .164 .193 .168 .175 .170 .165 .182 .263

85 .189 .131 .118 .092 .111 .101 .094 .098 .097

90 .183 .119 .092 .066 .089 .076 .069 .069 .036

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 1.70 2.11 4.41 3.23 3.08 2.40 5.43 3.91 4.08

5 1.64 1.98 3.50 2.71 2.65 2.10 4.22 3.12 2.63

10 1.50 1.65 2.08 1.75 1.82 1.48 2.43 1.85 1.09

15 1.30 1.26 1.07 .973 1.09 .902 1.25 .931 .230

20 1.08 .898 .413 .420 .543 .450 .518 .328 -.255

25 .873 .592 -.021 .037 .154 .122 .031 -.072 -.522

30 .697 .355 -.304 -.219 -.114 -.104 -.292 -.331 -.665

35 .564 .186 -.478 -.378 -.285 -.247 -.493 -.486 -.740

40 .481 .086 -.570 -.463 -.379 -.326 -.601 -.567 -.777

45 .453 .053 -.599 -.490 -.409 -.351 -.635 -.593 -.788

50 .481 .086 -.570 -.463 -.379 -.326 -.601 -.567 -.777

55 .564 .186 -.478 -.378 -.285 -.247 -.493 -.486 -.740

60 .697 .355 -.304 -.219 -.114 -.104 -.292 -.331 -.665

65 .873 .592 -.021 .037 .154 .122 .031 -.072 -.522

70 1.08 .898 .413 .420 .543 .450 .518 .328 -.255

75 1.30 1.26 1.07 .973 1.09 .902 1.25 .931 .230

80 1.50 1.65 2.08 1.75 1.82 1.48 2.43 1.85 1.09

85 1.64 1.98 3.50 2.71 2.65 2.10 4.22 3.12 2.63

90 1.70 2.11 4.41 3.23 3.08 2.40 5.43 3.91 4.08

νm = − a12

a11a22

νb = D12

D11D22

μ = 2a 12 + a 66

2 a 11a 22
113

Table 6. Values of for [(+θ/-θ)m]S, [(-θ/+θ)m]S, (+θ/-θ)m, and (-θ/+θ)m 

laminates (m = 1, 2, ...)

Table 7. Values of for (+θ/-θ)S laminates and -γb

 for (-θ/+θ)S laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .983 .993 .996 .998 .996 .997 .998 .998 .999

5 .982 .991 .993 .996 .994 .995 .996 .995 .995

10 .979 .986 .981 .986 .985 .985 .986 .983 .965

15 .973 .976 .951 .960 .962 .962 .960 .951 .880

20 .964 .959 .895 .911 .922 .921 .908 .891 .757

25 .952 .935 .816 .843 .865 .868 .829 .810 .638

30 .940 .907 .730 .771 .801 .813 .735 .726 .549

35 .927 .880 .655 .711 .746 .767 .648 .656 .490

40 .918 .860 .606 .672 .708 .737 .588 .612 .457

45 .915 .853 .588 .658 .695 .726 .566 .596 .446

50 .918 .860 .606 .672 .708 .737 .588 .612 .457

55 .927 .880 .655 .711 .746 .767 .648 .656 .490

60 .940 .907 .730 .771 .801 .813 .735 .726 .549

65 .952 .935 .816 .843 .865 .868 .829 .810 .638

70 .964 .959 .895 .911 .922 .921 .908 .891 .757

75 .973 .976 .951 .960 .962 .962 .960 .951 .880

80 .979 .986 .981 .986 .985 .985 .986 .983 .965

85 .982 .991 .993 .996 .994 .995 .996 .995 .995

90 .983 .993 .996 .998 .996 .997 .998 .998 .999

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 .032 .069 .115 .116 .105 .113 .110 .123 .184

10 .063 .136 .228 .230 .208 .222 .219 .243 .356

15 .090 .199 .333 .335 .303 .322 .322 .355 .496

20 .112 .255 .425 .424 .386 .405 .414 .449 .589

25 .127 .300 .496 .493 .452 .469 .488 .519 .642

30 .133 .332 .543 .538 .497 .512 .538 .565 .670

35 .131 .348 .568 .562 .522 .536 .564 .589 .683

40 .119 .347 .574 .569 .527 .544 .567 .595 .688

45 .099 .329 .561 .558 .515 .536 .549 .585 .686

50 .075 .295 .528 .530 .483 .512 .509 .557 .676

55 .049 .248 .474 .482 .431 .471 .443 .510 .657

60 .025 .193 .394 .410 .358 .410 .352 .437 .621

65 .006 .137 .293 .317 .269 .330 .243 .338 .557

70 -.007 .088 .186 .213 .177 .238 .136 .225 .449

75 -.012 .049 .094 .119 .098 .149 .055 .122 .296

80 -.012 .024 .035 .053 .044 .079 .010 .051 .140

85 -.007 .009 .009 .018 .015 .032 -.003 .015 .042

90 0 0 0 0 0 0 0 0 0

h

12 a11a22D11D22

1/4

γ b = D16

D11

3 D22

1/4
114

Table 8. Values of for (+θ/-θ)S laminates and -δb for (-θ/+θ)S laminates 

Table 9. Values of for (-θ/+θ) laminates and -e16 for (+θ/-θ) laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.007 .009 .009 .018 .015 .032 -.003 .015 .042

10 -.012 .024 .035 .053 .044 .079 .010 .051 .140

15 -.012 .049 .094 .119 .098 .149 .055 .122 .296

20 -.007 .088 .186 .213 .177 .238 .136 .225 .449

25 .006 .137 .293 .317 .269 .330 .243 .338 .557

30 .025 .193 .394 .410 .358 .410 .352 .437 .621

35 .049 .248 .474 .482 .431 .471 .443 .510 .657

40 .075 .295 .528 .530 .483 .512 .509 .557 .676

45 .099 .329 .561 .558 .515 .536 .549 .585 .686

50 .119 .347 .574 .569 .527 .544 .567 .595 .688

55 .131 .348 .568 .562 .522 .536 .564 .589 .683

60 .133 .332 .543 .538 .497 .512 .538 .565 .670

65 .127 .300 .496 .493 .452 .469 .488 .519 .642

70 .112 .255 .425 .424 .386 .405 .414 .449 .589

75 .090 .199 .333 .335 .303 .322 .322 .355 .496

80 .063 .136 .228 .230 .208 .222 .219 .243 .356

85 .032 .069 .115 .116 .105 .113 .110 .123 .184

90 0 0 0 0 0 0 0 0 0

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 .038 .081 .134 .135 .122 .131 .128 .143 .213

10 .074 .160 .268 .269 .243 .260 .256 .286 .426

15 .107 .236 .405 .403 .364 .386 .387 .430 .650

20 .134 .307 .549 .538 .484 .508 .527 .581 .898

25 .154 .371 .702 .675 .604 .624 .680 .740 1.16

30 .164 .423 .860 .805 .716 .727 .845 .899 1.41

35 .163 .457 1.00 .914 .808 .807 1.00 1.04 1.61

40 .149 .466 1.09 .978 .860 .852 1.12 1.12 1.74

45 .125 .446 1.10 .980 .855 .852 1.12 1.13 1.77

50 .094 .397 1.01 .912 .787 .803 1.00 1.05 1.71

55 .060 .326 .835 .783 .667 .710 .789 .897 1.55

60 .030 .246 .624 .614 .515 .583 .552 .694 1.31

65 .007 .170 .415 .434 .359 .439 .338 .482 1.01

70 -.008 .105 .240 .270 .222 .298 .173 .292 .686

75 -.014 .058 .115 .144 .118 .179 .066 .148 .388

80 -.014 .028 .042 .063 .052 .092 .012 .059 .167

85 -.008 .011 .010 .020 .018 .037 -.004 .017 .049

90 0 0 0 0 0 0 0 0 0

δ b = D26

D11D22

3 1/4

e 16 = B16

a 11

2

D11D22

1/4
115

Table 10. Values of for (-θ/+θ) laminates and -e26 for (+θ/-θ) laminates 

Table 11. Values of for [(+θ/-θ)m]S and -γb for [(-θ/+θ)m]S P-100/3502 laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.008 .011 .010 .020 .018 .037 -.004 .017 .049

10 -.014 .028 .042 .063 .052 .092 .012 .059 .167

15 -.014 .058 .115 .144 .118 .179 .066 .148 .388

20 -.008 .105 .240 .270 .222 .298 .173 .292 .686

25 .007 .170 .415 .434 .359 .439 .338 .482 1.01

30 .030 .246 .624 .614 .515 .583 .552 .694 1.31

35 .060 .326 .835 .783 .667 .710 .789 .897 1.55

40 .094 .397 1.01 .912 .787 .803 1.00 1.05 1.71

45 .125 .446 1.10 .980 .855 .852 1.12 1.13 1.77

50 .149 .466 1.09 .978 .860 .852 1.12 1.12 1.74

55 .163 .457 1.00 .914 .808 .807 1.00 1.04 1.61

60 .164 .423 .860 .805 .716 .727 .845 .899 1.41

65 .154 .371 .702 .675 .604 .624 .680 .740 1.16

70 .134 .307 .549 .538 .484 .508 .527 .581 .898

75 .107 .236 .405 .403 .364 .386 .387 .430 .650

80 .074 .160 .268 .269 .243 .260 .256 .286 .426

85 .038 .081 .134 .135 .122 .131 .128 .143 .213

90 0 0 0 0 0 0 0 0 0

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 12

0 0 0 0 0 0 0 0

5 .184 .092 .061 .046 .037 .031 .015

10 .356 .178 .119 .089 .071 .059 .030

15 .496 .248 .165 .124 .099 .083 .041

20 .589 .294 .196 .147 .118 .098 .049

25 .642 .321 .214 .160 .128 .107 .053

30 .670 .335 .223 .167 .134 .112 .056

35 .683 .342 .228 .171 .137 .114 .057

40 .688 .344 .229 .172 .138 .115 .057

45 .686 .343 .229 .171 .137 .114 .057

50 .676 .338 .225 .169 .135 .113 .056

55 .657 .328 .219 .164 .131 .109 .055

60 .621 .310 .207 .155 .124 .103 .052

65 .557 .278 .186 .139 .111 .093 .046

70 .449 .225 .150 .112 .090 .075 .037

75 .296 .148 .099 .074 .059 .049 .025

80 .140 .070 .047 .035 .028 .023 .012

85 .042 .021 .014 .011 .008 .007 .004

90 0 0 0 0 0 0 0

e 26 = B26

a 22

2

D11D22

1/4

γ b = D16

D11

3 D22

1/4
116

Table 12. Values of for [(+θ/-θ)m]S and -δb for [(-θ/+θ)m]S P-100/3502 laminates 

Table 13. Values of for (-θ/+θ)m and -e16 for (+θ/-θ)m]S P-100/3502 

laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 12

0 0 0 0 0 0 0 0

5 .042 .021 .014 .011 .008 .007 .004

10 .140 .070 .047 .035 .028 .023 .012

15 .296 .148 .099 .074 .059 .049 .025

20 .449 .225 .150 .112 .090 .075 .037

25 .557 .278 .186 .139 .111 .093 .046

30 .621 .310 .207 .155 .124 .103 .052

35 .657 .328 .219 .164 .131 .109 .055

40 .676 .338 .225 .169 .135 .113 .056

45 .686 .343 .229 .171 .137 .114 .057

50 .688 .344 .229 .172 .138 .115 .057

55 .683 .342 .228 .171 .137 .114 .057

60 .670 .335 .223 .167 .134 .112 .056

65 .642 .321 .214 .160 .128 .107 .053

70 .589 .294 .196 .147 .118 .098 .049

75 .496 .248 .165 .124 .099 .083 .041

80 .356 .178 .119 .089 .071 .059 .030

85 .184 .092 .061 .046 .037 .031 .015

90 0 0 0 0 0 0 0

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 12 m = 24

0 0 0 0 0 0 0 0 0

5 .213 .107 .071 .053 .043 .036 .018 .009

10 .426 .213 .142 .106 .085 .071 .035 .018

15 .650 .325 .217 .163 .130 .108 .054 .027

20 .898 .449 .299 .225 .180 .150 .075 .037

25 1.16 .581 .387 .290 .232 .194 .097 .048

30 1.41 .705 .470 .352 .282 .235 .117 .059

35 1.61 .806 .537 .403 .322 .269 .134 .067

40 1.74 .869 .580 .435 .348 .290 .145 .072

45 1.77 .887 .591 .443 .355 .296 .148 .074

50 1.71 .854 .570 .427 .342 .285 .142 .071

55 1.55 .774 .516 .387 .310 .258 .129 .065

60 1.31 .653 .435 .327 .261 .218 .109 .054

65 1.01 .504 .336 .252 .201 .168 .084 .042

70 .686 .343 .229 .171 .137 .114 .057 .029

75 .388 .194 .129 .097 .078 .065 .032 .016

80 .167 .084 .056 .042 .033 .028 .014 .007

85 .049 .024 .016 .012 .010 .008 .004 .002

90 0 0 0 0 0 0 0 0

δ b = D26

D11D22

3 1/4

e 16 = B16

a 11

2

D11D22

1/4
117

Table 14. Values of for (-θ/+θ)m and -e26 for (+θ/-θ)m]S P-100/3502 

laminates 

 

 Table 15. Values of for [(±45/0/90)m]S, [(±45/0/90)m]A, [(0/90/±45)m]S, 

 [(0/90/±45)m]A, (±45/0/90)m, and (0/90/±45)m laminates (m = 1, 2, ...) 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 12 m = 24

0 0 0 0 0 0 0 0 0

5 .049 .024 .016 .012 .010 .008 .004 .002

10 .167 .084 .056 .042 .033 .028 .014 .007

15 .388 .194 .129 .097 .078 .065 .032 .016

20 .686 .343 .229 .171 .137 .114 .057 .029

25 1.01 .504 .336 .252 .201 .168 .084 .042

30 1.31 .653 .435 .327 .261 .218 .109 .054

35 1.55 .774 .516 .387 .310 .258 .129 .065

40 1.71 .854 .570 .427 .342 .285 .142 .071

45 1.77 .887 .591 .443 .355 .296 .148 .074

50 1.74 .869 .580 .435 .348 .290 .145 .072

55 1.61 .806 .537 .403 .322 .269 .134 .067

60 1.41 .705 .470 .352 .282 .235 .117 .059

65 1.16 .581 .387 .290 .232 .194 .097 .048

70 .898 .449 .299 .225 .180 .150 .075 .037

75 .650 .325 .217 .163 .130 .108 .054 .027

80 .426 .213 .142 .106 .085 .071 .035 .018

85 .213 .107 .071 .053 .043 .036 .018 .009

90 0 0 0 0 0 0 0 0

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

.281 .272 .325 .299 .303 .284 .323 .312 .316

e 26 = B26

a 22

2

D11D22

1/4

νm = − a12

a11a22
118

Table 16. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates 

Table 17. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates 

Table 18. Values of for (±45/0/90)m unsymmetric laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 1.01 1.04 1.07 1.07 1.06 1.06 1.07 1.07 1.08

2 1.01 1.03 1.04 1.04 1.04 1.04 1.04 1.05 1.05

3 1.01 1.02 1.03 1.03 1.03 1.03 1.03 1.03 1.04

4 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.03 1.03

5 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02

6 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02

7 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.02 1.02

8 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 1.03 1.10 1.14 1.14 1.14 1.14 1.14 1.15 1.16

2 1.01 1.04 1.06 1.06 1.06 1.06 1.06 1.06 1.07

3 1.01 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.04

4 1.01 1.02 1.03 1.03 1.03 1.03 1.03 1.03 1.03

5 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.03

6 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02

7 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02

8 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .956 .868 .804 .802 .814 .805 .810 .797 .772

2 .989 .966 .950 .949 .952 .950 .951 .948 .942

3 .995 .985 .977 .977 .979 .978 .978 .977 .974

4 .997 .991 .987 .987 .988 .987 .988 .987 .985

5 .998 .995 .992 .992 .992 .992 .992 .992 .991

6 .999 .996 .994 .994 .995 .994 .994 .994 .993

7 .999 .997 .996 .996 .996 .996 .996 .996 .995

8 .999 .998 .997 .997 .997 .997 .997 .997 .996

D11

D22

1

4

D11

D22

1

4

D11

D22

1

4
119

Table 19. Values of for (0/90/±45)m unsymmetric laminates 

Table 20. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates 

Table 21. Values of for [(0/90±45)m]S and [(0/90/±45)m]A laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 1.05 1.15 1.24 1.25 1.23 1.24 1.24 1.26 1.30

2 1.01 1.04 1.05 1.05 1.05 1.05 1.05 1.06 1.06

3 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.03

4 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02

5 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

6 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01

7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01

8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 1.28 1.57 2.02 1.98 1.90 1.89 2.05 2.04 2.22

2 1.14 1.27 1.45 1.44 1.40 1.40 1.46 1.46 1.53

3 1.09 1.17 1.29 1.28 1.26 1.26 1.30 1.30 1.33

4 1.07 1.13 1.21 1.21 1.19 1.19 1.22 1.22 1.25

5 1.05 1.10 1.17 1.16 1.15 1.15 1.17 1.17 1.19

6 1.04 1.08 1.14 1.13 1.13 1.12 1.14 1.14 1.16

7 1.04 1.07 1.12 1.12 1.11 1.11 1.12 1.12 1.14

8 1.03 1.06 1.10 1.10 1.09 1.10 1.11 1.11 1.12

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .757 .571 .344 .362 .398 .406 .327 .334 .266

2 .874 .770 .640 .651 .671 .676 .631 .634 .594

3 .915 .843 .752 .759 .774 .777 .745 .747 .719

4 .936 .881 .810 .816 .827 .830 .805 .807 .785

5 .949 .904 .847 .851 .861 .862 .842 .844 .826

6 .957 .920 .871 .875 .883 .885 .868 .869 .854

7 .963 .931 .889 .892 .899 .901 .886 .887 .874

8 .968 .939 .903 .906 .912 .913 .900 .901 .889

D11

D22

1

4

β = D12 + 2D66

D11D22

β = D12 + 2D66

D11D22
120

Table 22. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric laminates 

Table 23. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates 

Table 24. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 1.00 1.04 1.10 1.10 1.09 1.10 1.09 1.11 1.14

2 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01

3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .371 .452 .663 .617 .596 .569 .672 .654 .716

2 .325 .356 .475 .440 .434 .412 .477 .463 .489

3 .310 .327 .421 .390 .387 .366 .422 .409 .426

4 .303 .313 .396 .366 .365 .345 .396 .383 .397

5 .298 .304 .381 .352 .352 .332 .380 .368 .380

6 .296 .299 .371 .343 .344 .324 .371 .358 .369

7 .293 .295 .365 .336 .338 .318 ,364 .352 .361

8 .292 .292 .360 .332 .333 .314 .358 .346 .355

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .203 .135 .108 .091 .106 .092 .100 .093 .074

2 .241 .199 .206 .185 .196 .180 .201 .192 .183

3 .254 .222 .243 .221 .229 .212 .239 .229 .224

4 .261 .234 .263 .239 .247 .229 .259 .249 .245

5 .265 .241 .275 .251 .257 .240 .271 .261 .259

6 .268 .246 .283 .259 .265 .247 .280 .269 .268

7 .270 .250 .289 .264 .270 .252 .286 .275 .275

8 .271 .253 .293 .268 .274 .256 .290 .279 .280

β = D12 + 2D66

D11D22

νb = D12

D11D22

νb = D12

D11D22
121

Table 25. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric laminates 

Table 26. Values of for [(±45/0/90)m]S laminates 

Table 27. Values of for [(0/90/±45)m]S laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .283 .283 .357 .329 .329 .311 .353 .345 .360

2 .281 .273 .327 .301 .304 .286 .325 .314 .319

3 .281 .272 .326 .299 .303 .284 .324 .312 .317

4 .281 .272 .326 .299 .303 .284 .323 .312 .316

5 .281 .272 .325 .299 .303 .284 .323 .312 .316

6 .281 .272 .325 .299 .303 .284 .323 .312 .316

7 .281 .272 .325 .299 .303 .284 .323 .312 .316

8 .281 .272 .325 .299 .303 .284 .323 .312 .316

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .036 .114 .182 .182 .170 .176 .178 .189 .217

2 .015 .045 .069 .069 .065 .067 .067 .071 .080

3 .009 .027 .042 .042 .039 .041 .040 .043 .048

4 .006 .020 .030 .030 .028 .029 .029 .031 .035

5 .005 .015 .023 .023 .022 .023 .022 .024 .027

6 .004 .013 .019 .019 .018 .019 .018 .020 .022

7 .003 .011 .016 .016 .015 .016 .016 .017 .019

8 .003 .009 .014 .014 .013 .014 .013 .014 .016

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .010 .028 .039 .039 .038 .039 .038 .040 .044

2 .008 .023 .033 .034 .032 .033 .032 .034 .038

3 .006 .018 .026 .026 .025 .026 .025 .027 .030

4 .005 .014 .021 .021 .020 .021 .020 .022 .024

5 .004 .012 .017 .018 .017 .017 .017 .018 .020

6 .003 .010 .015 .015 .014 .015 .014 .015 .017

7 .003 .009 .013 .013 .012 .013 .013 .014 .015

8 .003 .008 .012 .012 .011 .012 .011 .012 .013

νb = D12

D11D22

γ b = D16

D11

3 D22

1/4

γ b = D16

D11

3 D22

1/4
122

Table 28. Values of for (±45/0/90)m unsymmetric laminates 

Table 29. Values of for (0/90/±45)m unsymmetric laminates 

Table 30. Values of for [(±45/0/90)m]S laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .048 .165 .280 .283 .260 .277 .268 .295 .351

2 .011 .036 .054 .055 .051 .054 .053 .056 .064

3 .005 .016 .023 .024 .022 .023 .023 .024 .027

4 .003 .009 .013 .013 .012 .013 .013 .013 .015

5 .002 .006 .008 .008 .008 .008 .008 .009 .010

6 .001 .004 .006 .006 .005 .006 .006 .006 .007

7 .001 .003 .004 .004 .004 .004 .004 .004 .005

8 .001 .002 .003 .003 .003 .003 .003 .003 .004

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.043 -.124 -.181 -.182 -.172 -.180 -.176 -.187 -.209

2 -.011 -.033 -.049 -.049 -.047 -.049 -.048 -.051 -.056

3 -.005 -.015 -.022 -.022 -.021 -.022 -.022 -.023 -.026

4 -.003 -.009 -.013 -.013 -.012 -.013 -.012 -.013 -.015

5 -.002 -.005 -.008 -.008 -.008 -.008 -.008 -.008 -.009

6 -.001 -.004 -.006 -.006 -.005 -.006 -.006 -.006 -.007

7 -.001 -.003 -.004 -.004 -.004 -.004 -.004 -.004 -.005

8 -.001 -.002 -.003 -.003 -.003 -.003 -.003 -.003 -.004

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .037 .123 .207 .207 .191 .200 .202 .216 .253

2 .015 .047 .075 .075 .070 .073 .073 .078 .089

3 .009 .028 .044 .044 .042 .044 .043 .046 .052

4 .007 .020 .031 .031 .029 .031 .030 .032 .037

5 .005 .016 .024 .024 .023 .024 .023 .025 .028

6 .004 .013 .020 .020 .018 .019 .019 .020 .023

7 .004 .011 .016 .017 .016 .016 .016 .017 .019

8 .003 .009 .014 .014 .013 .014 .014 .015 .017

γ b = D16

D11

3 D22

1/4

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4
123

Table 31. Values of for [(0/90/±45)m]S laminates 

Table 32. Values of for (±45/0/90)m unsymmetric laminates 

Table 33. Values of for (0/90/±45)m unsymmetric laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .011 .034 .050 .051 .048 .051 .048 .052 .059

2 .008 .025 .037 .038 .036 .037 .036 .039 .043

3 .006 .019 .028 .028 .027 .028 .027 .029 .032

4 .005 .015 .022 .022 .021 .022 .021 .023 .025

5 .004 .012 .018 .018 .017 .018 .018 .019 .021

6 .003 .010 .015 .016 .015 .015 .015 .016 .018

7 .003 .009 .013 .014 .013 .013 .013 .014 .016

8 .003 .008 .012 .012 .011 .012 .012 .012 .014

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .043 .124 .181 .182 .172 .180 .176 .187 .209

2 .011 .033 .049 .049 .047 .049 .048 .051 .056

3 .005 .015 .022 .022 .021 .022 .022 .023 .026

4 .003 .009 .013 .013 .012 .013 .012 .013 .015

5 .002 .005 .008 .008 .008 .008 .008 .008 .009

6 .001 .004 .006 .006 .005 .006 .006 .006 .007

7 .001 .003 .004 .004 .004 .004 .004 .004 .005

8 .001 .002 .003 .003 .003 .003 .003 .003 .004

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.048 -.165 -.280 -.283 -.260 -.277 -.268 -.295 -.351

2 -.011 -.036 -.054 -.055 -.051 -.054 -.053 -.056 -.064

3 -.005 -.016 -.023 -.024 -.022 -.023 -.023 -.024 -.027

4 -.003 -.009 -.013 -.013 -.012 -.013 -.013 -.013 -.015

5 -.002 -.006 -.008 -.008 -.008 -.008 -.008 -.009 -.010

6 -.001 -.004 -.006 -.006 -.005 -.006 -.006 -.006 -.007

7 -.001 -.003 -.004 -.004 -.004 -.004 -.004 -.004 -.005

8 -.001 -.002 -.003 -.003 -.003 -.003 -.003 -.003 -.004

δ b = D26

D11D22

3 1/4

δ b = D26

D11D22

3 1/4

δ b = D26

D11D22

3 1/4
124

Table 34. Values of for [(±45/0/90)m]S and [(±45/0/90)m]A laminates 

Table 35. Values of for [(0/90/±45)m]S and [(0/90/±45)m]A laminates 

Table 36. Values of for (±45/0/90)m and (0/90/±45)m unsymmetric laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .993 1.03 1.06 1.07 1.06 1.06 1.07 1.07 1.09

2 .976 .994 .999 1.01 1.00 1.01 1.00 1.01 1.01

3 .970 .983 .980 .988 .984 .990 .981 .985 .988

4 .968 .978 .971 .979 .976 .982 .972 .976 .978

5 .966 .975 .965 .974 .971 .977 .967 .970 .972

6 .965 .972 .962 .970 .968 .974 .963 .967 .968

7 .964 .971 .960 .968 .966 .971 .961 .964 .965

8 .964 .970 .958 .966 .964 .970 .959 .963 .963

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .931 .916 .878 .889 .891 .898 .875 .882 .875

2 .944 .935 .905 .914 .915 .922 .904 .908 .903

3 .949 .944 .917 .926 .927 .933 .917 .921 .916

4 .952 .948 .923 .933 .933 .939 .923 .927 .924

5 .953 .951 .928 .937 .937 .942 .928 .932 .928

6 .954 .953 .930 .939 .939 .945 .931 .935 .931

7 .955 .954 .933 .941 .941 .947 .933 .937 .934

8 .956 .955 .934 .943 .943 .948 .934 .938 .936

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .962 .981 .990 1.00 .993 1.00 .988 .999 1.01

2 .960 .963 .948 .957 .955 .961 .949 .953 .952

3 .960 .963 .946 .955 .953 .959 .947 .951 .949

4 .960 .962 .946 .954 .953 .959 .946 .950 .949

5 .960 .962 .946 .954 .953 .959 .946 .950 .949

6 .960 .962 .946 .954 .953 .959 .946 .950 .949

7 .960 .962 .946 .954 .953 .959 .946 .950 .949

8 .960 .962 .946 .954 .953 .959 .946 .950 .949

h

12 a11a22D11D22

1/4

h

12 a11a22D11D22

1/4

h

12 a11a 22D11D22

1/4
125

Table 37. Values of for (±45/0/90)m unsymmetric laminates 

Table 38. Values of for (0/90/±45)m unsymmetric laminates 

Table 39. Values of and for (±45/0/90)m unsymmetric 

laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .025 -.020 -.008 -.020 -.021 -.039 .011 -.014 -.015

2 .012 -.009 -.003 -.008 -.009 -.016 .004 -.006 -.006

3 .008 -.006 -.002 -.005 -.006 -.010 .003 -.004 -.004

4 .006 -.004 -.002 -.004 -.004 -.008 .002 -.003 -.003

5 .005 -.003 -.001 -.003 -.003 -.006 .002 -.002 -.002

6 .004 -.003 -.001 -.003 -.003 -.005 .001 -.002 -.002

7 .003 -.002 -.001 -.002 -.002 -.004 .001 -.002 -.002

8 .003 -.002 -.001 -.002 -.002 -.004 .001 -.001 -.001

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.127 -.277 -.417 -.408 -.387 -.389 -.418 -.424 -.468

2 -.066 -.151 -.236 -.231 -.218 -.220 -.236 -.241 -.269

3 -.044 -.103 -.162 -.158 -.149 -.150 -.161 -.165 -.185

4 -.033 -.078 -.122 -.120 -.113 -.114 -.122 -.125 -.140

5 -.027 -.062 -.098 -.096 -.091 -.092 -.098 -.100 -.113

6 -.022 -.052 -.082 -.080 -.076 -.076 -.082 -.084 -.094

7 -.019 -.045 -.070 -.069 -.065 -.066 -.070 -.072 -.081

8 -.017 -.039 -.062 -.060 -.057 -.057 -.062 -.063 -.071

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.079 -.151 -.256 -.246 -.229 -.226 -.262 -.261 -.297

2 -.039 -.074 -.123 -.118 -.110 -.108 -.126 -.124 -.140

3 -.026 -.049 -.082 -.078 -.073 -.072 -.084 -.083 -.093

4 -.020 -.037 -.061 -.059 -.055 -.054 -.063 -.062 -.070

5 -.016 -.030 -.049 -.047 -.044 -.043 -.050 -.050 -.056

6 -.013 -.025 -.041 -.039 -.037 -.036 -.042 -.041 -.046

7 -.011 -.021 -.035 -.034 -.031 -.031 -.036 -.035 -.040

8 -.010 -.018 -.031 -.029 -.028 -.027 -.031 -.031 -.035

e 11 = B11

a11

D11

1/2

e 11 = B11

a11

D11

1/2

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4
126

Table 40. Values of and for (0/90/±45)m unsymmetric 

laminates 

Table 41. Values of and for [(±45/0/90)m]A laminates 

Table 42. Values of and for [(0/90/±45)m]A laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .079 .151 .256 .246 .229 .226 .262 .261 .297

2 .039 .074 .123 .118 .110 .108 .126 .124 .140

3 .026 .049 .082 .078 .073 .072 .084 .083 .093

4 .020 .037 .061 .059 .055 .054 .063 .062 .070

5 .016 .030 .049 .047 .044 .043 .050 .050 .056

6 .013 .025 .041 .039 .037 .036 .042 .041 .046

7 .011 .021 .035 .034 .031 .031 .036 .035 .040

8 .010 .018 .031 .029 .028 .027 .031 .031 .035

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.014 -.044 -.070 -.070 -.066 -.068 -.068 -.073 -.083

2 -.007 -.021 -.033 -.033 -.031 -.032 -.032 -.034 -.039

3 -.005 -.014 -.022 -.022 -.020 -.021 -.021 -.022 -.025

4 -.003 -.010 -.016 -.016 -.015 -.016 -.016 -.017 -.019

5 -.003 -.008 -.013 -.013 -.012 -.013 -.012 -.013 -.015

6 -.002 -.007 -.011 -.011 -.010 -.010 -.010 -.011 -.012

7 -.002 -.006 -.009 -.009 -.009 -.009 -.009 -.009 -.011

8 -.002 -.005 -.008 -.008 -.007 -.008 -.008 -.008 -.009

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.013 -.039 -.058 -.058 -.055 -.058 -.056 -.060 -.067

2 -.007 -.020 -.030 -.030 -.028 -.030 -.029 -.031 -.034

3 -.004 -.013 -.020 -.020 -.019 -.020 -.020 -.021 -.023

4 -.003 -.010 -.015 -.015 -.014 -.015 -.015 -.016 -.018

5 -.003 -.008 -.012 -.012 -.012 -.012 -.012 -.013 -.014

6 -.002 -.007 -.010 -.010 -.010 -.010 -.010 -.011 -.012

7 -.002 -.006 -.009 -.009 -.008 -.009 -.009 -.009 -.010

8 -.002 -.005 -.008 -.008 -.007 -.008 -.008 -.008 -.009

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4
127

Table 43. Values of and for (±45/0/90)m and (0/90/±45)m

unsymmetric laminates 

Table 44. Values of for (±45/0/90)m unsymmetric laminates 

Table 45. Values of for (0/90/±45)m unsymmetric laminates 

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.027 -.084 -.131 -.131 -.123 -.129 -.127 -.136 -.155

2 -.014 -.041 -.063 -.063 -.059 -.062 -.061 -.065 -.073

3 -.009 -.028 -.042 -.042 -.039 -.041 -.041 -.043 -.048

4 -.007 -.021 -.031 -.031 -.030 -.031 -.030 -.032 -.036

5 -.005 -.017 -.025 -.025 -.024 -.025 -.024 -.026 -.029

6 -.005 -.014 -.021 -.021 -.020 -.020 -.020 -.022 -.024

7 -.004 -.012 -.018 -.018 -.017 -.018 -.017 -.018 -.021

8 -.003 -.010 -.016 -.016 -.015 -.015 -.015 -.016 -.018

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 .127 .277 .417 .408 .387 .389 .418 .424 .468

2 .066 .151 .236 .231 .218 .220 .236 .241 .269

3 .044 .103 .162 .158 .149 .150 .161 .165 .185

4 .033 .078 .122 .120 .113 .114 .122 .125 .140

5 .027 .062 .098 .096 .091 .092 .098 .100 .113

6 .022 .052 .082 .080 .076 .076 .082 .084 .094

7 .019 .045 .070 .069 .065 .066 .070 .072 .081

8 .017 .039 .062 .060 .057 .057 .062 .063 .071

m

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

1 -.025 .020 .008 .020 .021 .039 -.011 .014 .015

2 -.012 .009 .003 .008 .009 .016 -.004 .006 .006

3 -.008 .006 .002 .005 .006 .010 -.003 .004 .004

4 -.006 .004 .002 .004 .004 .008 -.002 .003 .003

5 -.005 .003 .001 .003 .003 .006 -.002 .002 .002

6 -.004 .003 .001 .003 .003 .005 -.001 .002 .002

7 -.003 .002 .001 .002 .002 .004 -.001 .002 .002

8 -.003 .002 .001 .002 .002 .004 -.001 .001 .001

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4

e 22 = B22

a22

D22

1/2

e 22 = B22

a22

D22

1/2
128

Table 46. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates 

Table 47. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .680 .435 .157 .177 .220 .229 .138 .144 .064

5 .689 .450 .176 .196 .239 .247 .158 .164 .085

10 .713 .491 .234 .253 .293 .300 .216 .222 .147

15 .752 .556 .326 .343 .379 .386 .310 .315 .247

20 .801 .641 .447 .462 .492 .498 .433 .438 .380

25 .855 .736 .589 .601 .624 .629 .578 .582 .537

30 .907 .834 .740 .748 .763 .768 .731 .736 .708

35 .951 .923 .886 .891 .896 .902 .879 .886 .877

40 .981 .992 1.01 1.01 1.09 1.02 1.00 1.02 1.03

45 .993 1.03 1.10 1.10 1.09 1.10 1.09 1.11 1.15

50 .985 1.04 1.13 1.14 1.11 1.14 1.11 1.15 1.22

55 .959 1.01 1.10 1.12 1.09 1.12 1.08 1.13 1.23

60 .918 .941 1.01 1.03 1.01 1.05 .973 1.04 1.15

65 .868 .856 .868 .896 .878 .932 .821 .896 .994

70 .815 .762 .695 .730 .728 .784 .645 .717 .776

75 .767 .672 .525 .564 .579 .634 .475 .537 .538

80 .728 .600 .384 .425 .457 .506 .337 .387 .332

85 .703 .552 .293 .335 .376 .422 .248 .290 .196

90 .695 .536 .262 .303 .349 .393 .217 .256 .148

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02

5 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02

10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

15 .997 .991 .988 .987 .988 .987 .988 .987 .986

20 .991 .975 .965 .965 .966 .965 .966 .964 .960

25 .984 .955 .936 .935 .938 .935 .938 .933 .927

30 .976 .931 .900 .899 .904 .900 .903 .897 .886

35 .967 .903 .859 .858 .866 .860 .863 .854 .838

40 .957 .874 .815 .813 .824 .816 .820 .808 .785

45 .948 .845 .769 .767 .781 .770 .776 .760 .730

50 .938 .817 .724 .722 .739 .726 .733 .713 .674

55 .929 .791 .684 .680 .701 .685 .695 .670 .622

60 .922 .769 .649 .645 .668 .649 .663 .633 .576

65 .915 .751 .622 .616 .642 .620 .638 .604 .538

70 .910 .738 .602 .596 .623 .598 .621 .583 .511

75 .906 .728 .589 .582 .610 .583 .609 .569 .493

80 .903 .721 .582 .574 .602 .573 .603 .561 .484

85 .901 .717 .578 .569 .598 .568 .600 .557 .479

90 .901 .716 .577 .568 .597 .567 .599 .556 .478

β = D12 + 2D66

D11D22

D11

D22

1

4
129

Table 48. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...)

Table 49. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...) 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 1.04 1.11 1.16 1.16 1.15 1.16 1.15 1.17 1.18

5 1.04 1.11 1.15 1.15 1.15 1.16 1.14 1.15 1.16

10 1.04 1.10 1.12 1.13 1.13 1.14 1.11 1.13 1.12

15 1.03 1.09 1.09 1.10 1.10 1.11 1.09 1.10 1.08

20 1.03 1.07 1.07 1.08 1.08 1.09 1.06 1.07 1.05

25 1.02 1.06 1.05 1.06 1.06 1.07 1.04 1.05 1.03

30 1.02 1.04 1.03 1.04 1.04 1.05 1.03 1.03 1.02

35 1.01 1.03 1.02 1.03 1.03 1.03 1.02 1.02 1.01

40 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

45 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

50 .994 .986 .990 .988 .988 .986 .992 .990 .994

55 .989 .973 .980 .976 .975 .971 .983 .979 .987

60 .983 .959 .968 .962 .960 .955 .972 .967 .978

65 .978 .945 .954 .946 .944 .938 .959 .952 .967

70 .974 .931 .936 .927 .926 .919 .942 .934 .951

75 .969 .919 .915 .906 .906 .899 .921 .912 .928

80 .966 .908 .891 .884 .887 .880 .897 .887 .897

85 .964 .901 .871 .867 .872 .865 .876 .866 .862

90 .964 .899 .862 .860 .866 .860 .867 .858 .845

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .179 .095 .048 .033 .051 .038 .040 .033 .009

5 .182 .097 .053 .037 .055 .041 .046 .037 .012

10 .188 .106 .064 .046 .064 .048 .059 .046 .017

15 .197 .117 .076 .057 .076 .058 .074 .057 .022

20 .207 .130 .086 .067 .087 .067 .086 .066 .025

25 .218 .143 .095 .075 .097 .076 .095 .073 .028

30 .228 .153 .100 .081 .105 .082 .101 .078 .029

35 .235 .161 .104 .085 .110 .086 .106 .082 .030

40 .240 .166 .106 .088 .113 .089 .108 .084 .030

45 .242 .168 .107 .088 .114 .090 .109 .084 .030

50 .240 .166 .106 .088 .113 .089 .108 .084 .030

55 .235 .161 .104 .085 .110 .086 .106 .082 .030

60 .228 .153 .100 .081 .105 .082 .101 .078 .029

65 .218 .143 .095 .075 .097 .076 .095 .073 .028

70 .207 .130 .086 .067 .087 .067 .086 .066 .025

75 .197 .117 .076 .057 .076 .058 .074 .057 .022

80 .188 .106 .064 .046 .064 .048 .059 .046 .017

85 .182 .097 .053 .037 .055 .041 .046 .037 .012

90 .179 .095 .048 .033 .051 .038 .040 .033 .009

a 22

a 11

1

4

νm = − a12

a11a22
130

Table 50. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates 

Table 51. Values of for (+θ/0/90)m and (-θ/0/90)m laminates (m = 1, 2, ...) 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .179 .092 .046 .032 .049 .037 .038 .031 .008

5 .182 .097 .052 .038 .055 .042 .045 .038 .015

10 .190 .110 .071 .056 .072 .060 .064 .057 .036

15 .202 .131 .102 .086 .101 .087 .095 .087 .069

20 .218 .158 .142 .124 .137 .123 .136 .128 .112

25 .235 .188 .189 .169 .180 .165 .184 .175 .164

30 .251 .219 .239 .217 .225 .209 .234 .225 .220

35 .266 .246 .287 .262 .268 .251 .283 .274 .275

40 .275 .268 .328 .301 .304 .286 .324 .315 .325

45 .279 .279 .357 .328 .328 .311 .351 .345 .363

50 .276 .280 .368 .338 .336 .320 .359 .357 .385

55 .268 .268 .358 .329 .325 .312 .346 .349 .385

60 .255 .247 .327 .299 .296 .286 .312 .318 .357

65 .238 .218 .279 .252 .253 .244 .261 .269 .303

70 .222 .187 .221 .196 .203 .194 .202 .209 .229

75 .206 .158 .164 .141 .154 .143 .146 .149 .149

80 .194 .134 .118 .095 .113 .100 .100 .099 .080

85 .186 .119 .087 .065 .086 .073 .070 .067 .035

90 .183 .114 .077 .055 .077 .063 .060 .055 .019

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 1.75 2.73 8.58 6.54 5.54 4.93 9.56 8.42 16.9

5 1.73 2.68 8.06 6.24 5.32 4.76 8.91 7.92 14.9

10 1.68 2.55 6.84 5.49 4.78 4.33 7.42 6.74 11.2

15 1.61 2.36 5.51 4.61 4.10 3.78 5.85 5.44 7.98

20 1.52 2.15 4.38 3.81 3.46 3.24 4.57 4.34 5.80

25 1.43 1.95 3.53 3.17 2.93 2.79 3.63 3.52 4.41

30 1.36 1.78 2.94 2.70 2.53 2.44 2.99 2.94 3.53

35 1.28 1.65 2.56 2.39 2.25 2.20 2.58 2.57 3.00

40 1.24 1.57 2.35 2.21 2.09 2.06 2.35 2.36 2.72

45 1.23 1.54 2.28 2.15 2.04 2.01 2.28 2.29 2.63

50 1.24 1.57 2.35 2.21 2.09 2.06 2.35 2.36 2.72

55 1.28 1.65 2.56 2.39 2.25 2.20 2.58 2.57 3.00

60 1.35 1.78 2.94 2.70 2.53 2.44 2.99 2.94 3.53

65 1.43 1.95 3.53 3.17 2.93 2.79 3.63 3.52 4.41

70 1.52 2.15 4.38 3.81 3.46 3.24 4.57 4.34 5.80

75 1.60 2.36 5.51 4.61 4.10 3.78 5.85 5.44 7.98

80 1.68 2.55 6.84 5.49 4.78 4.33 7.42 6.74 11.2

85 1.73 2.68 8.06 6.24 5.32 4.76 8.91 7.92 14.9

90 1.75 2.73 8.58 6.54 5.54 4.93 9.56 8.42 16.9

νb = D12

D11D22

μ = 2a 12 + a 66

2 a 11a 22
131

Table 52. Values of for (+θ/0/90)m and -γm for (-θ/0/90)m laminates (m = 1, 2, ...) 

Table 53. Values of for (+θ/0/90)m and -δm for (-θ/0/90)m laminates (m = 1, 2, ...) 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.022 -.004 -.019 .008 .001 .023 -.045 .001 .036

10 -.039 .000 -.010 .032 .016 .054 -.052 .023 .094

15 -.048 .014 .033 .077 .053 .100 -.012 .073 .173

20 -.046 .043 .105 .142 .110 .159 .062 .146 .265

25 -.034 .084 .196 .223 .184 .231 .157 .235 .366

30 -.011 .137 .298 .315 .270 .312 .265 .336 .474

35 .019 .197 .410 .415 .365 .401 .382 .444 .590

40 .055 .263 .530 .522 .467 .495 .507 .560 .716

45 .092 .329 .657 .635 .573 .593 .640 .684 .855

50 .128 .393 .792 .752 .682 .692 .781 .815 1.01

55 .158 .448 .933 .871 .790 .788 .931 .953 1.19

60 .180 .489 1.08 .986 .890 .874 1.09 1.09 1.39

65 .188 .506 1.21 1.09 .969 .936 1.24 1.23 1.62

70 .181 .490 1.32 1.14 1.00 .954 1.36 1.33 1.86

75 .156 .432 1.33 1.11 .957 .894 1.41 1.34 2.06

80 .116 .326 1.17 .934 .782 .719 1.26 1.17 2.05

85 .061 .176 .716 .550 .451 .409 .786 .715 1.45

90 0 0 0 0 0 0 0 0 0

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 .061 .176 .716 .550 .451 .409 .786 .715 1.45

10 .116 .326 1.17 .934 .782 .719 1.26 1.17 2.05

15 .156 .432 1.33 1.11 .957 .894 1.41 1.34 2.06

20 .181 .490 1.32 1.14 1.00 .954 1.36 1.33 1.86

25 .188 .506 1.21 1.09 .969 .936 1.24 1.23 1.62

30 .180 .489 1.08 .986 .890 .874 1.09 1.09 1.39

35 .158 .448 .933 .871 .790 .788 .931 .953 1.19

40 .128 .393 .792 .752 .682 .692 .781 .815 1.011

45 .092 .329 .657 .635 .573 .593 .640 .684 .855

50 .055 .263 .530 .522 .467 .495 .507 .560 .716

55 .019 .197 .410 .415 .365 .401 .382 .444 .590

60 -.011 .137 .298 .315 .270 .312 .265 .336 .474

65 -.034 .084 .196 .223 .184 .231 .157 .235 .366

70 -.046 .043 .105 .142 .110 .159 .062 .146 .265

75 -.048 .014 .033 .077 .053 .100 -.012 .073 .173

80 -.039 .000 -.010 .032 .016 .054 -.052 .023 .094

85 -.022 -.004 -.019 .008 .001 .023 -.045 .001 .036

90 0 0 0 0 0 0 0 0 0

γm = − a26

a11a22

3 1/4

δm = − a16

a11

3

a22

1/4
132

Table 54. Values of for (+θ/0/90) and -γb for (-θ/0/90) three-ply laminates 

Table 55. Values of for (+θ/0/90) and -δb for (-θ/0/90) three-ply laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 .022 .049 .070 .070 .067 .068 .070 .072 .079

10 .043 .097 .139 .139 .132 .134 .138 .143 .156

15 .061 .141 .205 .204 .193 .198 .203 .210 .231

20 .075 .179 .265 .264 .250 .256 .262 .272 .301

25 .084 .209 .318 .316 .298 .307 .313 .327 .365

30 .087 .230 .359 .359 .336 .348 .353 .372 .420

35 .083 .239 .387 .387 .360 .376 .378 .403 .463

40 .075 .236 .398 .400 .368 .389 .384 .417 .491

45 .062 .220 .388 .392 .357 .384 .369 .411 .499

50 .046 .193 .356 .364 .327 .359 .332 .382 .484

55 .030 .158 .303 .314 .279 .316 .274 .331 .441

60 .015 .120 .235 .249 .218 .257 .204 .262 .369

65 .004 .083 .162 .177 .154 .191 .132 .185 .275

70 -.004 .052 .096 .110 .096 .127 .071 .113 .175

75 -.008 .029 .047 .059 .051 .075 .028 .057 .092

80 -.007 .014 .017 .026 .023 .038 .005 .023 .037

85 -.004 .005 .004 .008 .008 .015 -.002 .007 .010

90 0 0 0 0 0 0 0 0 0

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.004 .003 .001 .003 .003 .005 -.001 .002 .002

10 -.007 .008 .006 .009 .009 .013 .002 .007 .008

15 -.007 .017 .017 .020 .020 .026 .011 .019 .020

20 -.004 .032 .036 .040 .039 .047 .028 .039 .042

25 .003 .051 .064 .068 .066 .075 .056 .068 .074

30 .013 .076 .100 .105 .100 .110 .092 .105 .116

35 .027 .104 .143 .147 .140 .149 .135 .149 .164

40 .041 .132 .187 .190 .180 .190 .180 .194 .216

45 .055 .157 .229 .231 .218 .228 .222 .237 .266

50 .067 .176 .263 .263 .248 .257 .257 .272 .308

55 .076 .186 .283 .283 .265 .275 .277 .294 .338

60 .079 .186 .287 .287 .267 .278 .281 .298 .349

65 .077 .175 .272 .272 .253 .263 .266 .284 .337

70 .069 .153 .240 .240 .222 .232 .234 .250 .301

75 .057 .123 .192 .193 .178 .187 .187 .201 .244

80 .040 .086 .134 .134 .124 .130 .130 .140 .171

85 .021 .044 .069 .069 .064 .067 .067 .072 .088

90 0 0 0 0 0 0 0 0 0

γ b = D16

D11

3 D22

1/4

δ b = D26

D11D22

3 1/4
133

Table 56. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates 

Table 57. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .982 .984 .978 .978 .979 .977 .980 .977 .973

5 .982 .984 .971 .973 .975 .974 .972 .970 .957

10 .982 .982 .957 .962 .967 .967 .957 .956 .929

15 .981 .980 .945 .953 .960 .960 .945 .944 .913

20 .980 .979 .942 .950 .957 .958 .942 .941 .911

25 .980 .981 .947 .955 .961 .962 .949 .947 .921

30 .980 .986 .961 .967 .971 .972 .963 .961 .941

35 .981 .994 .980 .986 .987 .989 .983 .981 .968

40 .982 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00

45 .983 1.01 1.03 1.04 1.03 1.04 1.03 1.04 1.04

50 .984 1.03 1.06 1.07 1.06 1.07 1.06 1.07 1.09

55 .985 1.04 1.09 1.10 1.09 1.10 1.09 1.11 1.15

60 .986 1.05 1.12 1.14 1.11 1.14 1.11 1.14 1.20

65 .988 1.06 1.15 1.17 1.14 1.17 1.13 1.17 1.25

70 .989 1.07 1.18 1.20 1.16 1.20 1.15 1.20 1.30

75 .990 1.08 1.20 1.22 1.19 1.23 1.17 1.23 1.35

80 .992 1.09 1.23 1.25 1.21 1.26 1.19 1.26 1.40

85 .993 1.09 1.25 1.27 1.23 1.27 1.22 1.29 1.45

90 .993 1.09 1.26 1.28 1.23 1.28 1.23 1.30 1.48

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 -.167 -.438 -.578 -.584 -.562 -.585 -.560 -.594 -.640

5 -.165 -.436 -.584 -.588 -.564 -.588 -.566 -.600 -.660

10 -.158 -.429 -.596 -.596 -.568 -.591 -.578 -.613 -.697

15 -.146 -.417 -.599 -.598 -.565 -.590 -.580 -.618 -.715

20 -.130 -.397 -.587 -.586 -.552 -.578 -.565 -.608 -.706

25 -.111 -.369 -.557 -.558 -.524 -.553 -.532 -.578 -.673

30 -.090 -.332 -.510 -.515 -.481 -.513 -.483 -.533 -.621

35 -.068 -.289 -.450 -.458 -.427 -.461 -.421 -.473 -.555

40 -.047 -.241 -.381 -.392 -.364 -.400 -.350 -.404 -.479

45 -.028 -.191 -.308 -.321 -.296 -.332 -.276 -.329 -.397

50 -.013 -.144 -.233 -.248 -.227 -.262 -.203 -.253 -.313

55 -.002 -.101 -.164 -.178 -.162 -.194 -.136 -.181 -.231

60 .004 -.065 -.104 -.117 -.106 -.133 -.081 -.118 -.156

65 .007 -.039 -.059 -.069 -.062 -.083 -.041 -.068 -.093

70 .007 -.020 -.028 -.035 -.032 -.046 -.016 -.034 -.047

75 .005 -.009 -.011 -.015 -.014 -.022 -.004 -.014 -.019

80 .002 -.003 -.003 -.005 -.005 -.008 .000 -.004 -.006

85 .001 -.001 .000 -.001 -.001 -.002 .000 -.001 -.001

90 0 0 0 0 0 0 0 0 0

h

12 a11a 22D11D22

1/4

e 11 = B11

a11

D11

1/2
134

Table 58. Values of and for (+θ/0/90) and (-θ/0/90) threeply laminates 

Table 59. Values of for (+θ/0/90) and (-θ/0/90) three-ply laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.002 -.003 -.005 -.005 -.005 -.005 -.005 -.005 -.006

10 -.007 -.013 -.020 -.020 -.019 -.018 -.021 -.021 -.023

15 -.016 -.029 -.045 -.044 -.041 -.041 -.046 -.046 -.052

20 -.027 -.049 -.078 -.075 -.070 -.070 -.080 -.079 -.090

25 -.039 -.071 -.115 -.111 -.104 -.103 -.118 -.117 -.134

30 -.050 -.093 -.153 -.148 -.138 -.136 -.156 -.156 -.178

35 -.060 -.112 -.186 -.180 -.168 -.166 -.190 -.190 -.218

40 -.066 -.125 -.212 -.205 -.190 -.188 -.216 -.216 -.249

45 -.068 -.131 -.225 -.218 -.201 -.200 -.229 -.231 -.268

50 -.066 -.128 -.224 -.217 -.200 -.199 -.228 -.230 -.272

55 -.060 -.117 -.208 -.201 -.185 -.185 -.211 -.215 -.258

60 -.050 -.099 -.179 -.173 -.158 -.159 -.180 -.185 -.227

65 -.039 -.076 -.140 -.136 -.123 -.125 -.140 -.145 -.182

70 -.027 -.053 -.097 -.095 -.086 -.087 -.097 -.101 -.129

75 -.016 -.032 -.057 -.056 -.051 -.052 -.057 -.060 -.077

80 -.007 -.015 -.026 -.026 -.023 -.024 -.026 -.027 -.035

85 -.002 -.004 -.007 -.006 -.006 -.006 -.006 -.007 -.009

90 0 0 0 0 0 0 0 0 0

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 .181 .555 .803 .816 .772 .818 .768 .834 .928

5 .182 .554 .803 .815 .771 .816 .768 .833 .928

10 .184 .552 .801 .812 .769 .812 .769 .831 .925

15 .187 .547 .796 .805 .762 .802 .766 .825 .918

20 .189 .537 .784 .790 .748 .783 .758 .811 .902

25 .190 .519 .761 .764 .724 .754 .739 .786 .874

30 .188 .493 .724 .725 .686 .712 .707 .746 .831

35 .181 .457 .673 .671 .635 .655 .659 .692 .770

40 .170 .412 .606 .602 .570 .585 .597 .622 .693

45 .155 .358 .528 .522 .495 .505 .522 .541 .603

50 .135 .300 .442 .435 .412 .419 .438 .452 .505

55 .112 .240 .352 .346 .328 .332 .351 .360 .403

60 .088 .181 .266 .260 .246 .248 .266 .271 .305

65 .064 .128 .186 .182 .172 .173 .187 .190 .214

70 .043 .083 .118 .115 .109 .110 .119 .120 .137

75 .025 .046 .065 .063 .060 .060 .066 .066 .075

80 .011 .020 .028 .027 .026 .026 .028 .028 .031

85 .003 .005 .007 .007 .006 .006 .007 .007 .007

90 0 0 0 0 0 0 0 0 0

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 22 = B22

a22

D22

1/2
135

Table 60. Values of for (+θ/0/90) and -e16 for (-θ/0/90) three-ply laminates 

Table 61. Values of for (+θ/0/90) and -e26 for (-θ/0/90) three-ply laminates 

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 -.018 -.036 -.051 -.051 -.048 -.049 -.051 -.052 -.057

10 -.034 -.072 -.104 -.102 -.097 -.098 -.104 -.106 -.121

15 -.048 -.105 -.156 -.153 -.144 -.146 -.156 -.160 -.185

20 -.059 -.133 -.203 -.199 -.187 -.189 -.203 -.208 -.241

25 -.066 -.154 -.239 -.234 -.220 -.224 -.237 -.246 -.284

30 -.068 -.166 -.261 -.257 -.240 -.246 -.257 -.268 -.309

35 -.065 -.169 -.266 -.263 -.246 -.254 -.261 -.275 -.316

40 -.058 -.162 -.255 -.254 -.238 -.247 -.248 -.265 -.306

45 -.047 -.146 -.231 -.232 -.216 -.228 -.222 -.240 -.279

50 -.035 -.124 -.196 -.199 -.185 -.198 -.185 -.205 -.240

55 -.023 -.099 -.154 -.159 -.147 -.161 -.143 -.163 -.194

60 -.011 -.073 -.112 -.117 -.109 -.123 -.100 -.120 -.145

65 -.003 -.050 -.073 -.079 -.073 -.086 -.062 -.080 -.098

70 .003 -.031 -.042 -.047 -.044 -.055 -.032 -.047 -.058

75 .006 -.017 -.020 -.025 -.023 -.031 -.013 -.023 -.029

80 .006 -.008 -.007 -.011 -.010 -.016 -.002 -.009 -.011

85 .003 -.003 -.002 -.003 -.003 -.006 .001 -.003 -.003

90 0 0 0 0 0 0 0 0 0

θ,

deg

Material Systems

Boron/Al S-glass/

Epoxy

Kevlar49/

Epoxy IM7/5260 AS4/3502 AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

0 0 0 0 0 0 0 0 0 0

5 .003 -.003 -.001 -.003 -.003 -.005 .001 -.002 -.002

10 .005 -.007 -.006 -.008 -.008 -.012 -.002 -.007 -.008

15 .006 -.016 -.016 -.019 -.019 -.025 -.010 -.018 -.020

20 .003 -.028 -.034 -.038 -.036 -.044 -.027 -.036 -.040

25 -.003 -.046 -.060 -.065 -.062 -.071 -.052 -.064 -.072

30 -.011 -.069 -.096 -.100 -.095 -.105 -.087 -.101 -.113

35 -.023 -.095 -.138 -.142 -.134 -.145 -.129 -.145 -.164

40 -.035 -.122 -.185 -.187 -.176 -.187 -.176 -.193 -.221

45 -.047 -.146 -.231 -.232 -.216 -.228 -.222 -.240 -.279

50 -.058 -.166 -.270 -.269 -.250 -.262 -.262 -.282 -.333

55 -.065 -.176 -.297 -.294 -.271 -.283 -.288 -.310 -.374

60 -.068 -.177 -.305 -.301 -.275 -.288 -.297 -.319 -.395

65 -.066 -.166 -.290 -.286 -.261 -.272 -.283 -.305 -.386

70 -.060 -.145 -.254 -.250 -.227 -.237 -.247 -.266 -.345

75 -.049 -.115 -.199 -.196 -.178 -.187 -.194 -.209 -.273

80 -.035 -.079 -.133 -.133 -.121 -.127 -.130 -.140 -.182

85 -.018 -.040 -.066 -.066 -.061 -.064 -.064 -.069 -.087

90 0 0 0 0 0 0 0 0 0

e 16 = B16

a 11

2

D11D22

1/4

e 26 = B26

a 22

2

D11D22

1/4
136

Table 62. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

Table 63. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 1.02 1.14 1.16 1.17 1.18 1.18 1.18 1.18

5 1.02 1.14 1.16 1.17 1.17 1.18 1.18 1.18

10 1.00 1.13 1.15 1.16 1.17 1.17 1.17 1.17

15 .986 1.11 1.14 1.15 1.16 1.16 1.16 1.16

20 .960 1.10 1.12 1.13 1.14 1.14 1.14 1.14

25 .927 1.07 1.10 1.11 1.12 1.12 1.12 1.12

30 .886 1.04 1.07 1.08 1.09 1.09 1.09 1.09

35 .838 1.01 1.04 1.05 1.06 1.06 1.06 1.06

40 .785 .973 1.007 1.02 1.02 1.03 1.03 1.03

45 .730 .936 .972 .984 .990 .993 .995 .996

50 .674 .901 .938 .951 .957 .960 .962 .963

55 .622 .869 .907 .920 .927 .930 .932 .933

60 .576 .842 .881 .894 .900 .904 .906 .907

65 .538 .820 .859 .872 .879 .882 .884 .885

70 .511 .802 .842 .855 .862 .865 .867 .868

75 .493 .790 .829 .843 .849 .853 .855 .856

80 .484 .782 .821 .835 .841 .844 .846 .847

85 .479 .777 .816 .830 .836 .839 .841 .842

90 .478 .776 .815 .828 .834 .838 .840 .841

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 .064 .066 .067 .068 .068 .068 .068 .068

5 .085 .083 .083 .083 .083 .083 .083 .083

10 .147 .131 .129 .128 .128 .128 .128 .128

15 .247 .208 .202 .200 .199 .198 .198 .198

20 .380 .306 .294 .290 .289 .288 .287 .287

25 .537 .415 .397 .391 .388 .387 .386 .385

30 .708 .524 .498 .489 .485 .483 .482 .481

35 .877 .618 .584 .572 .567 .565 .563 .562

40 1.03 .684 .642 .629 .623 .619 .617 .616

45 1.15 .713 .665 .650 .643 .639 .637 .635

50 1.22 .698 .648 .632 .624 .620 .618 .617

55 1.23 .642 .593 .577 .570 .567 .564 .563

60 1.15 .553 .509 .495 .489 .486 .484 .482

65 .994 .444 .408 .397 .392 .389 .388 .387

70 .776 .330 .304 .295 .292 .290 .289 .288

75 .538 .226 .209 .204 .201 .200 .199 .199

80 .332 .144 .134 .131 .130 .129 .129 .128

85 .196 .091 .086 .085 .084 .084 .084 .084

90 .148 .073 .070 .069 .069 .068 .068 .068

D11

D22

1

4

β = D12 + 2D66

D11D22
137

Table 64. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

Table 65. Values of for (+θ/0/90)m and -γb

 for (-θ/0/90)m P-100/3502 laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 .008 .009 .009 .009 .009 .009 .009 .009

5 .015 .014 .014 .014 .014 .014 .014 .014

10 .036 .030 .029 .029 .029 .029 .029 .029

15 .069 .055 .053 .052 .052 .052 .052 .052

20 .112 .088 .084 .082 .082 .081 .081 .081

25 .164 .124 .118 .116 .115 .114 .114 .114

30 .220 .160 .151 .148 .147 .146 .146 .145

35 .275 .191 .179 .175 .174 .173 .172 .172

40 .325 .212 .199 .194 .192 .191 .190 .190

45 .363 .222 .206 .201 .199 .197 .197 .196

50 .385 .217 .200 .195 .193 .191 .191 .190

55 .385 .198 .182 .177 .175 .174 .173 .172

60 .357 .168 .154 .150 .148 .147 .146 .146

65 .303 .132 .121 .117 .116 .115 .114 .114

70 .229 .095 .086 .084 .083 .082 .082 .081

75 .149 .060 .055 .053 .053 .052 .052 .052

80 .080 .033 .030 .029 .029 .029 .029 .029

85 .035 .015 .014 .014 .014 .014 .014 .014

90 .019 .009 .009 .009 .009 .009 .009 .009

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 0 0 0 0 0 0 0 0

5 .079 .056 .052 .051 .050 .050 .050 .050

10 .156 .109 .102 .099 .098 .098 .097 .097

15 .231 .159 .148 .144 .142 .141 .141 .140

20 .301 .201 .186 .181 .179 .178 .177 .177

25 .365 .234 .216 .210 .207 .206 .205 .204

30 .420 .256 .234 .227 .224 .222 .221 .221

35 .463 .263 .239 .232 .228 .226 .225 .224

40 .491 .256 .231 .223 .219 .217 .216 .215

45 .499 .235 .210 .202 .198 .196 .195 .195

50 .484 .201 .178 .171 .168 .166 .165 .165

55 .441 .160 .141 .135 .132 .131 .130 .129

60 .369 .117 .102 .098 .096 .095 .094 .094

65 .275 .077 .067 .064 .063 .062 .061 .061

70 .175 .045 .039 .037 .036 .036 .036 .035

75 .092 .022 .019 .018 .018 .018 .018 .017

80 .037 .009 .008 .007 .007 .007 .007 .007

85 .010 .002 .002 .002 .002 .002 .002 .002

90 0 0 0 0 0 0 0 0

νb = D12

D11D22

γ b = D16

D11

3 D22

1/4
138

Table 66. Values of for (+θ/0/90)m and -δb for (-θ/0/90)m P-100/3502 laminates 

Table 67. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 0 0 0 0 0 0 0 0

5 .002 .002 .002 .002 .002 .002 .002 .002

10 .008 .007 .007 .007 .007 .007 .007 .007

15 .020 .018 .018 .017 .017 .017 .017 .017

20 .042 .037 .036 .035 .035 .035 .035 .035

25 .074 .063 .062 .061 .061 .061 .061 .061

30 .116 .097 .094 .093 .093 .093 .093 .093

35 .164 .135 .131 .130 .129 .128 .128 .128

40 .216 .173 .167 .165 .164 .164 .163 .163

45 .266 .206 .198 .195 .194 .194 .193 .193

50 .308 .229 .220 .217 .215 .215 .214 .214

55 .338 .240 .230 .226 .225 .224 .223 .223

60 .349 .237 .226 .223 .221 .220 .220 .219

65 .337 .220 .210 .206 .205 .204 .204 .203

70 .301 .191 .182 .179 .178 .177 .176 .176

75 .244 .152 .144 .142 .141 .140 .140 .140

80 .171 .105 .100 .098 .098 .097 .097 .097

85 .088 .054 .051 .050 .050 .050 .050 .050

90 0 0 0 0 0 0 0 0

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 .973 .989 .995 .997 .998 .999 .999 .999

5 .957 .971 .977 .979 .980 .981 .981 .981

10 .929 .940 .944 .946 .947 .948 .948 .948

15 .913 .917 .921 .922 .923 .924 .924 .924

20 .911 .906 .909 .910 .911 .911 .911 .911

25 .921 .904 .905 .906 .907 .907 .907 .907

30 .941 .908 .907 .907 .907 .907 .907 .907

35 .968 .913 .910 .909 .909 .909 .909 .909

40 1.00 .919 .913 .912 .911 .911 .911 .911

45 1.04 .924 .916 .914 .913 .912 .912 .912

50 1.09 .928 .917 .914 .913 .912 .911 .911

55 1.15 .931 .917 .913 .911 .911 .910 .910

60 1.20 .933 .917 .912 .910 .909 .909 .908

65 1.25 .936 .918 .913 .911 .910 .909 .909

70 1.30 .942 .923 .918 .916 .914 .914 .913

75 1.35 .956 .937 .931 .929 .927 .927 .926

80 1.40 .982 .962 .956 .953 .952 .951 .951

85 1.45 1.02 .995 .989 .986 .985 .984 .984

90 1.48 1.04 1.01 1.01 1.00 1.00 1.00 1.00

δ b = D26

D11D22

3 1/4

h

12 a11a22D11D22

1/4
139

Table 68. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

Table 69. Values of for (+θ/0/90)m and (-θ/0/90)m P-100/3502 laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 -.640 -.291 -.191 -.142 -.114 -.095 -.081 -.071

5 -.660 -.299 -.196 -.147 -.117 -.097 -.083 -.073

10 -.697 -.314 -.206 -.153 -.122 -.102 -.087 -.076

15 -.715 -.318 -.208 -.155 -.124 -.103 -.088 -.077

20 -.706 -.308 -.201 -.150 -.119 -.099 -.085 -.074

25 -.673 -.286 -.186 -.138 -.110 -.092 -.078 -.069

30 -.621 -.254 -.165 -.122 -.097 -.081 -.069 -.061

35 -.555 -.217 -.140 -.104 -.083 -.069 -.059 -.051

40 -.479 -.177 -.113 -.084 -.067 -.055 -.047 -.041

45 -.397 -.137 -.087 -.064 -.051 -.043 -.036 -.032

50 -.313 -.100 -.063 -.046 -.037 -.031 -.026 -.023

55 -.231 -.067 -.042 -.031 -.025 -.020 -.018 -.015

60 -.156 -.041 -.026 -.019 -.015 -.013 -.011 -.009

65 -.093 -.023 -.014 -.010 -.008 -.007 -.006 -.005

70 -.047 -.011 -.007 -.005 -.004 -.003 -.003 -.002

75 -.019 -.004 -.003 -.002 -.002 -.001 -.001 -.001

80 -.006 -.001 -.001 -.001 .000 .000 .000 .000

85 -.001 .000 .000 .000 .000 .000 .000 .000

90 0 0 0 0 0 0 0 0

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 .928 .527 .361 .274 .220 .184 .158 .138

5 .928 .527 .361 .273 .220 .184 .158 .138

10 .925 .525 .360 .273 .219 .183 .157 .138

15 .918 .521 .357 .270 .217 .182 .156 .137

20 .902 .512 .351 .266 .214 .179 .153 .134

25 .874 .496 .340 .258 .207 .173 .149 .130

30 .831 .472 .323 .245 .197 .164 .141 .124

35 .770 .437 .300 .227 .183 .153 .131 .115

40 .693 .394 .270 .204 .164 .137 .118 .103

45 .603 .343 .235 .178 .143 .120 .103 .090

50 .505 .287 .197 .149 .120 .100 .086 .075

55 .403 .229 .157 .119 .096 .080 .069 .060

60 .305 .173 .119 .090 .072 .060 .052 .045

65 .214 .122 .084 .063 .051 .043 .037 .032

70 .137 .078 .053 .040 .032 .027 .023 .020

75 .075 .042 .029 .022 .018 .015 .013 .011

80 .031 .018 .012 .009 .007 .006 .005 .005

85 .007 .004 .003 .002 .002 .001 .001 .001

90 0 0 0 0 0 0 0 0

e 11 = B11

a11

D11

1/2

e 22 = B22

a22

D22

1/2
140

Table 70. Values of and for (+θ/0/90)m and (-θ/0/90)m

P-100/3502 laminates 

Table 71. Values of for (+θ/0/90)m and -e16 for (-θ/0/90)m P-100/3502

laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 0 0 0 0 0 0 0 0

5 -.006 -.003 -.002 -.001 -.001 -.001 -.001 -.001

10 -.023 -.012 -.008 -.006 -.005 -.004 -.003 -.003

15 -.052 -.026 -.018 -.013 -.011 -.009 -.008 -.007

20 -.090 -.045 -.030 -.023 -.018 -.015 -.013 -.011

25 -.134 -.066 -.044 -.033 -.026 -.022 -.019 -.016

30 -.178 -.086 -.057 -.043 -.034 -.029 -.024 -.021

35 -.218 -.103 -.068 -.051 -.041 -.034 -.029 -.026

40 -.249 -.114 -.076 -.057 -.045 -.038 -.032 -.028

45 -.268 -.119 -.078 -.059 -.047 -.039 -.033 -.029

50 -.272 -.116 -.076 -.057 -.045 -.038 -.032 -.028

55 -.258 -.105 -.069 -.051 -.041 -.034 -.029 -.026

60 -.227 -.088 -.058 -.043 -.034 -.029 -.025 -.021

65 -.182 -.068 -.044 -.033 -.026 -.022 -.019 -.016

70 -.129 -.047 -.031 -.023 -.018 -.015 -.013 -.011

75 -.077 -.027 -.018 -.013 -.011 -.009 -.008 -.007

80 -.035 -.012 -.008 -.006 -.005 -.004 -.003 -.003

85 -.009 -.003 -.002 -.001 -.001 -.001 -.001 -.001

90 0 0 0 0 0 0 0 0

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 0 0 0 0 0 0 0 0

5 -.057 -.029 -.020 -.015 -.012 -.010 -.008 -.007

10 -.121 -.061 -.041 -.031 -.025 -.021 -.018 -.015

15 -.185 -.093 -.062 -.047 -.037 -.031 -.027 -.023

20 -.241 -.120 -.080 -.060 -.048 -.040 -.035 -.030

25 -.284 -.139 -.093 -.070 -.056 -.047 -.040 -.035

30 -.309 -.149 -.099 -.075 -.060 -.050 -.043 -.037

35 -.316 -.149 -.099 -.074 -.059 -.050 -.042 -.037

40 -.306 -.140 -.093 -.069 -.056 -.046 -.040 -.035

45 -.279 -.124 -.082 -.061 -.049 -.041 -.035 -.030

50 -.240 -.102 -.067 -.050 -.040 -.033 -.029 -.025

55 -.194 -.079 -.052 -.039 -.031 -.026 -.022 -.019

60 -.145 -.056 -.037 -.027 -.022 -.018 -.016 -.014

65 -.098 -.036 -.024 -.018 -.014 -.012 -.010 -.009

70 -.058 -.021 -.014 -.010 -.008 -.007 -.006 -.005

75 -.029 -.010 -.007 -.005 -.004 -.003 -.003 -.002

80 -.011 -.004 -.003 -.002 -.002 -.001 -.001 -.001

85 -.003 -.001 -.001 -.001 .000 .000 .000 .000

90 0 0 0 0 0 0 0 0

e 12 = B12

a11a22

D11D22

1/4

e 66 = B66

a11a22

D11D22

1/4

e 16 = B16

a 11

2

D11D22

1/4
141

Table 72. Values of for (+θ/0/90)m and -e26 for (-θ/0/90)m P-100/3502 

laminates 

θ,

deg

Stacking sequence number, m

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

0 0 0 0 0 0 0 0 0

5 -.002 -.001 -.001 -.001 .000 .000 .000 .000

10 -.008 -.004 -.003 -.002 -.002 -.001 -.001 -.001

15 -.020 -.010 -.007 -.005 -.004 -.003 -.003 -.002

20 -.040 -.020 -.013 -.010 -.008 -.007 -.006 -.005

25 -.072 -.035 -.023 -.018 -.014 -.012 -.010 -.009

30 -.113 -.055 -.036 -.027 -.022 -.018 -.016 -.014

35 -.164 -.077 -.051 -.039 -.031 -.026 -.022 -.019

40 -.221 -.101 -.067 -.050 -.040 -.033 -.029 -.025

45 -.279 -.124 -.082 -.061 -.049 -.041 -.035 -.030

50 -.333 -.141 -.093 -.070 -.056 -.046 -.040 -.035

55 -.374 -.152 -.100 -.075 -.060 -.050 -.042 -.037

60 -.395 -.153 -.101 -.075 -.060 -.050 -.043 -.037

65 -.386 -.144 -.094 -.070 -.056 -.047 -.040 -.035

70 -.345 -.125 -.082 -.061 -.049 -.040 -.035 -.030

75 -.273 -.097 -.063 -.047 -.038 -.031 -.027 -.023

80 -.182 -.064 -.042 -.031 -.025 -.021 -.018 -.015

85 -.087 -.031 -.020 -.015 -.012 -.010 -.008 -.007

90 0 0 0 0 0 0 0 0

e 26 = B26

a 22

2

D11D22

1/4

Shell reference surface, A 

+ ξ1 direction 

+ ξ2 direction 

Figure 1. Coordinate system and unit-magnitude base-vector fields for points 

+ ζ direction 

e1

n

Boundary curve, ∂A 

e1 Tangent plane at

 of undeformed shell. 

arbitrary point of ∂A

p

Portion of the tangent plane at point p

e2

N

e2
142

+ ξ1 direction 

+ ξ2 direction 

Figure 2. Sign convention for applied loads. 

+ ζ direction 

N(ξ2) 

S(ξ2) 

V(ξ2) 

N(ξ1) 

S(ξ1) 

V(ξ1) 

q1(ξ1, ξ2)

q2(ξ1, ξ2)

q3(ξ1, ξ2)

M(ξ1) 

M(ξ2) 

Edge ξ1 = b1

Edge ξ2 = b2

Edge ξ1 = a1

Edge ξ2 = a2

+ ξ1 direction

+ ξ2 direction 

Fiber angle, θ

(ξ1, ξ2, 0)

(ξ1, ξ2, ζ)

Shell reference surface

Arbitrary lamina

Fiber

Figure 3. Fiber orientation of an arbitrary lamina. 

+ ζ direction

Line tangent to ξ1

+ ξ2 direction 

+ ξ1 direction

coordinate curve

Line tangent to 

fiber at (ξ1, ξ2, ζ)

at (ξ1, ξ2, ζ)
143

Fiber angle, θ, deg

0 15 30 45 60 75 90

0

1

2

3

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Flexural

orthotropy

parameter,

β = D12 + 2D66

D11D22

Figure 4. Effects of lamina material properties on nondimensional flexural orthotropy parameter β for [(+θ /-θ)m]S, 

[(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates (m = 1, 2, ...).

Fiber angle, θ, deg

0 15 30 45 60 75 90

1.2

2.4

3.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

0

1.6

2.8

Parameter

coefficients,

 and

Figure 5. Effects of lamina material properties on parameter coefficients in equations (52a) and (55) for [(+θ /-θ)m]S, 

[(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates (m = 1, 2, ...).

2.0

0.8

0.4

a22

a11

1

4

D11

D22

1

4
144

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.2

0.6

1

 0

0.4

0.8

Poisson’s

ratios,

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 6. Effects of lamina material properties on Poisson’s ratios defined by equations (52e) and (59d) for [(+θ /-θ)m]S, 

[(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates (m = 1, 2, ...).

and

νm = − a12

a11a22

νb = D12

D11D22

Fiber angle, θ, deg

0 15 30 45 60 75 90

1

3

6

 -1

2

5

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 7. Effects of lamina material properties on nondimensional membrane orthotropy parameter μ for 

[(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates (m = 1, 2, ...).

 0

4

μ = 2a12 + a66

2 a11a22

Membrane

orthotropy

parameter,
145

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.2

0.6

1.0

 0

0.4

0.8

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 8. Effects of lamina material properties on Batdorf-Stein-parameter coefficient in equations (45) and (48) 

for [(+θ /-θ)m]S, [(-θ /+θ)m]S, (+θ /-θ)m, and (-θ /+θ)m angle-ply laminates (m = 1, 2, ...).

1.2

Parameter

 coefficient,

h

12 a11a22D11D22

1

4

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0.2

0.6

1.0

0

0.4

0.8

Flexural

anisotropy

parameter,

S-glass-epoxy Boron-Aluminum

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

γb = D16

D11

3

D22

1/4

Figure 9. Effects of lamina material properties on nondimensional flexural anisotropy parameters γb for [(+θ/−θ)m]S

laminates and -γb for [(-θ/+θ)m]S laminates.
146

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0.2

0.6

1.0

0

0.4

0.8

Flexural

anisotropy

parameter,

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5 P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

δb = IM7/5260

D26

D11D22

3 1/4

Figure 10. Effects of lamina material properties on nondimensional flexural anisotropy parameters δb for [(+θ/-θ)m]S

laminates and -δb for [(-θ/+θ)m]S laminates. 

0.2

0.6

 0

0.4

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 11. Effects of lamina material properties on flexural anisotropy parameters γb and δb for [(+θ/-θ)m]S 

and -γb and -δb for [(-θ/+θ)m]S symmetric angle-ply laminates, respectively (m = 1).

0.8

- 0.2

- 0.2 0 0.2 0.4 0.6 0.8

+θ

θ = 45 degrees

γb = D16

D11

3

D22

1/4

δb = D26

D11D22

3 1/4

Flexural

anisotropy

parameter,

Flexural anisotropy parameter,
147

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.4

1.2

 0

0.8

1.6

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 12. Effects of lamina material properties on nondimensional load-path eccentricity parameters -e 16 and +e 16 

defined by equation (75b) for (+θ /-θ)m and (-θ /+θ)m antisymmetric angle-ply laminates, respectively (m = 1).

2.0

- 0.4

Load-path

eccentricity

e 16 = B16

a11

2

D11D22

1

4

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.4

1.2

 0

0.8

1.6

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 13. Effects of lamina material properties on nondimensional load-path eccentricity parameters -e 26 and +e 26 

defined by equation (75f) for (+θ /-θ)m and (-θ /+θ)m antisymmetric angle-ply laminates, respectively (m = 1). 

2.0

- 0.4

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,
148

0.4

1.2

 0

0.8

1.6

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 14. Effects of lamina material properties on nondimensional load-path eccentricity parameters e 16 and e 26 

defined by equation (75) for (-θ /+θ)m and -e 16 and -e 26 for (+θ /-θ)m antisymmetric angle-ply laminates, 

2.0

- 0.4

- 0.4 0 0.4 0.8 1.2 1.6 2.0

+θ

θ = 45 degrees

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,

e 16 = B16

a11

2

D11D22

1

4

Load-path eccentricity parameter,

respectively (m = 1).

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0.2

0.6

0

0.4

0.8

Flexural

anisotropy

parameter,

Figure 15. Effects of number of plies on flexural anisotropy parameters γb for [(+θ/−θ)m]S and -γb for [(-θ/+θ)m]S 

P-100/3502 symmetric angle-ply laminates.

m = 1

m = 12

2

3

4

5

γ 6 b = D16

D11

3

D22

1/4
149

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0.2

0.6

0

0.4

0.8

Flexural

anisotropy

parameter,

Figure 16. Effects of number of plies on flexural anisotropy parameters δb for [(+θ/−θ)m]S and -δb for [(-θ/+θ)m]S 

P-100/3502 symmetric angle-ply laminates.

m = 1

m = 12

2

3

4

5

δb = 6 D26

D11D22

3 1/4

0.2

0.6

 0

0.4

Figure 17. Effects of lamina material properties on flexural anisotropy parameters γb and δb for [(+θ/-θ)m]S 

and -γb and -δb for [(-θ/+θ)m]S P-100/3502 symmetric angle-ply laminates, respectively (m = 1).

0.8

- 0.2

- 0.2 0 0.2 0.4 0.6 0.8

m = 1

2

3

4

m = 12

5

6

δb = D26

D11D22

3 1/4

γb = D16

D11

3

D22

Flexural anisotropy parameter, 1/4

Flexural

anisotropy

parameter,
150

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.4

1.2

 0

0.8

1.6

Figure 18. Effects of number of plies on nondimensional load-path eccentricity parameters -e 16 and +e 16 

defined by equation (75e) for (+θ /-θ)m and (-θ /+θ)m P-100/3502 antisymmetric angle-ply laminates, respectively.

2.0

- 0.4

m = 1

m = 24

12

2

3 4

5 7

e 16 = B16

a11

2

D11D22

1

4

8

6

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.4

1.2

2.0

 0

0.8

1.6

Figure 19. Effects of number of plies on nondimensional load-path eccentricity parameters -e 26 and +e 26 

defined by equation (75f) for (+θ /-θ)m and (-θ /+θ)m P-100/3502 antisymmetric angle-ply laminates, respectively.

- 0.4

m = 1

m = 24

12

2

3 4

5

6

e 26 = B26

a22

2

D11D22

1

4

8 7

Load-path

eccentricity

parameter,
151

0.4

1.2

2.0

 0

0.8

1.6

Figure 20. Effects of number of plies on nondimensional load-path eccentricity parameters e 16 and e 26 

defined by equation (75) for (-θ /+θ)m P100/3502 antisymmetric angle-ply laminates.

- 0.4

- 0.4 0 0.4 0.8 1.2 1.6 2.0

m = 1

2

3

m = 24

6

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,

e 16 = B16

a11

2

D11D22

1

4

Load-path eccentricity parameter,

Number of plies

0 8 16 24 32 40 48

0.5

0

1.0

1.5

Figure 21. Effects of number of plies on parameter coefficients in equation (55) for P-100/3502 quasi-isotropic

laminates.

Parameter

 coefficient, 

D11

D22

1

4

[(0/90/+45/-45)m]S and [(0/90/+45/-45)m]A 

(0/90/+45/-45)m

[(+45/-45/0/90)m]S and [(+45/-45/0/90)m]A 

(+45/-45/0/90)m
152

Number of plies

0 8 16 24 32 40 48

0.8

0

1.6

2.4

Figure 22. Effects of number of plies on the nondimensional flexural orthotropy parameter β for P-100/3502 

[(0/90/+45/-45)m]S and [(0/90/+45/-45)m]A 

(0/90/+45/-45)m and (+45/-45/0/90)m

[(+45/-45/0/90)m]S and [(+45/-45/0/90)m]A 

0.4

2.0

1.2

Flexural

orthotropy

parameter,

β = D12 + 2D66

D11D22

quasi-isotropic laminates.

Number of plies

0 8 16 24 32 40 48

0.4

0

0.6

1.0

Figure 23. Effects of number of plies on the nondimensional Poisson’s ratio νb for P-100/3502 

[(0/90/+45/-45)m]S and [(0/90/+45/-45)m]A 

(0/90/+45/-45)m and (+45/-45/0/90)m

[(+45/-45/0/90)m]S and [(+45/-45/0/90)m]A 

0.2

0.8

quasi-isotropic laminates.

Poisson’s

ratio,

νb = D12

D11D22
153

Number of plies

0 8 16 24 32 40 48

0

-0.4

0.4

Figure 24. Effects of number of plies on the nondimensional flexural anisotropy parameter γb for P-100/3502 

[(0/90/+45/-45)m]S 

[(+45/-45/0/90)m]S 

-0.2

0.2

quasi-isotropic laminates.

(0/90/+45/-45)m

(+45/-45/0/90)m

[(0/90/+45/-45)m]A and [(+45/-45/0/90)m]A

Flexural

anisotropy

parameter,

γb = D16

D11

3

D22

1/4

Number of plies

0 8 16 24 32 40 48

0

-0.4

0.4

Figure 25. Effects of number of plies on the nondimensional flexural anisotropy parameter δb for P-100/3502 

[(0/90/+45/-45)m]S 

[(+45/-45/0/90)m]S 

-0.2

0.2

quasi-isotropic laminates.

(0/90/+45/-45)m

(+45/-45/0/90)m

[(0/90/+45/-45)m]A and [(+45/-45/0/90)m]A

Flexural

anisotropy

parameter,

δb = D26

D11D22

3 1/4
154

Number of plies

0 8 16 24 32 40 48

0

1.0

1.5

Figure 26. Effects of number of plies on Batdorf-Stein-parameter coefficients in equations (45) and (48) for P-100/3502 

[(0/90/+45/-45)m]S and [(0/90/+45/-45)m]A 

(0/90/+45/-45)m and (+45/-45/0/90)m

[(+45/-45/0/90)m]S and [(+45/-45/0/90)m]A 

0.5

quasi-isotropic laminates.

Parameter

 coefficient,

h

12 a11a22D11D22

1

4

Number of plies

0 8 16 24 32 40 48

-0.8

 0

0.2

Figure 27. Effects of number of plies on nondimensional load-path eccentricity parameter e 11 defined 

[(0/90/+45/-45)m]S, [(0/90/+45/-45)m]A,

(+45/-45/0/90)m

[(+45/-45/0/90)m]S, and [(+45/-45/0/90)m]A 

-0.4

by equations (75b) for P-100/3502 quasi-isotropic laminates.

(0/90/+45/-45)m

-0.2

-0.6

e 11 = B11

a11

D11

1

2

Load-path

eccentricity

parameter,
155

Number of plies

0 8 16 24 32 40 48

-0.4

 0.2

0.4

Figure 28. Effects of number of plies on nondimensional load-path eccentricity parameters e 12 and e 66 defined by 

[(0/90/+45/-45)m]S, [(0/90/+45/-45)m]A,

(+45/-45/0/90)m

[(+45/-45/0/90)m]S, and [(+45/-45/0/90)m]A 

equations (75) for P-100/3502 quasi-isotropic laminates.

(0/90/+45/-45)m

0

-0.2

and

e 12 = B12

a11a22

D11D22

1

4

e 66 = B66

a11a22

D11D22

1

4

Load-path

eccentricity

parameters,

Number of plies

0 8 16 24 32 40 48

-0.2

 0

0.1

Figure 29. Effects of number of plies on nondimensional load-path eccentricity parameters e 16 and e 26 defined 

[(0/90/+45/-45)m]A

[(+45/-45/0/90)m]A 

-0.1

by equations (75) for P-100/3502 quasi-isotropic laminates.

(0/90/+45/-45)m and (+45/-45/0/90)m

and

[(0/90/+45/-45)m]S and [(+45/-45/0/90)m]S

e 16 = B16

a11

2

D11D22

1

4

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameters,
156

Number of plies

0 8 16 24 32 40 48

-0.2

0.6

Figure 30. Effects of number of plies on nondimensional load-path eccentricity parameter e 22 defined by 

[(0/90/+45/-45)m]S, [(0/90/+45/-45)m]A,

(+45/-45/0/90)m

[(+45/-45/0/90)m]S, and [(+45/-45/0/90)m]A 

0.2

equations (75d) for P-100/3502 quasi-isotropic laminates.

(0/90/+45/-45)m

0.4

0

e 22 = B22

a22

D22

1

2

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

0

0.2

1.4

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5 P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Flexural

orthotropy

parameter,

β = D12 + 2D66

D11D22

Figure 31. Effects of lamina material properties on nondimensional flexural orthotropy parameter β for (+θ /0/90) 

and (-θ /0/90) unbalanced, unsymmetric three-ply laminates.

1.2

1.0

0.4

0.6

0.8
157

Fiber angle, θ, deg

0 15 30 45 60 75 90

1.2

Boron-Aluminum S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

0

0.6

1.0

Parameter

 coefficient,

Figure 32. Effects of lamina material properties on parameter coefficient in equation (55) for (+θ /0/90) 

and (-θ /0/90) unbalanced, unsymmetric three-ply laminates.

0.8

0.4

0.2

D11

D22

1

4

Fiber angle, θ, deg

0 15 30 45 60 75 90

1.1

1.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

0.8

1.0

Parameter

 coefficient,

Figure 33. Effects of lamina material properties on parameter coefficient in equation (52a) (+θ /0/90)m 

and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...).

0.9

a22

a11

1

4
158

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.1

0.2

0.3

 0

Poisson’s

ratio,

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5 P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 34. Effects of lamina material properties on Poisson’s ratio defined by equation (52e) for (+θ /0/90)m 

and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...).

νm = − a12

a11a22

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.2

0.5

 0

0.3

0.4

Poisson’s

ratio,

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 35. Effects of lamina material properties on Poisson’s ratio defined by equation (59d) for (+θ /0/90) 

and (-θ /0/90) unbalanced, unsymmetric three-ply laminates.

νb = D12

D11D22

0.1
159

Fiber angle, θ, deg

0 15 30 45 60 75 90

8

20

0

12

16

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 36. Effects of lamina material properties on nondimensional membrane orthotropy parameter μ for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1, 2, ...).

 4

μ = 2a12 + a66

2 a11a22

Membrane

orthotropy

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.5

2.5

 -0.5

1.0

2.0

Boron-Aluminum S-glass-epoxy

IM7/PETI-5 P-100/3502

Kevlar 49-epoxy AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 37. Effects of lamina material properties on nondimensional membrane anisotropy parameters γm and -γm for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1, 2, ...).

 0

γ 1.5 m = − a26

a11a22

3 1/4

Membrane

anisotropy

parameter,
160

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.5

2.5

 -0.5

1.0

2.0

Boron-Aluminum S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 38. Effects of lamina material properties on nondimensional membrane anisotropy parameters δm and -δm for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1, 2, ...).

 0

1.5

δm = − a16

a11

3

a22

1/4

Membrane

anisotropy

parameter,

0.5

2.0

 0

1.5

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 39. Effects of lamina material properties on membrane anisotropy parameters γm and δm for (+θ /0/90)m 

unbalanced, unsymmetric laminates (m = 1, 2, ...).

2.5

- 0.5

- 0.5 0 0.5 1.5 2.0 2.5

+θ

θ = 45 degrees

δm = − a16

a11

3

a22

1/4

γm = − a26

a11a22

3 1/4

1.0

1.0

Membrane

anisotropy

parameter,

Membrane anisotropy parameter,
161

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0.1

0.4

0.6

0

0.3

0.5

Flexural

anisotropy

parameter,

Boron-Aluminum S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260 0.2

γb = D16

D11

3

D22

1/4

Figure 40. Effects of lamina material properties on nondimensional flexural anisotropy parameters γb and -γb for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0.1

0.4

0.6

0

0.3

0.5

Flexural

anisotropy

parameter,

Boron-Aluminum S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

0.2

δb = D26

D11D22

3 1/4

Figur 41. Effects of lamina material properties on nondimensional flexural anisotropy parameters δb and -δb for 

(+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).
162

0.2

0.4

 0

0.3

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 42. Effects of lamina material properties on flexural anisotropy parameters γb and δb for (+θ /0/90)m 

unbalanced, unsymmetric laminates (m = 1).

0.5

- 0.1

- 0.1 0 0.2 0.3 0.4 0.5

+θ

δb = D26

D11D22

3 1/4

0.1

0.1

γb = D16

D11

3

D22

Flexural anisotropy parameter, 1/4

Flexural

anisotropy

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.8

1.4

 0.6

1.0

1.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 43. Effects of lamina material properties on Batdorf-Stein-parameter coefficients in equations (45) and (48) 

for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates (m = 1).

1.6

Parameter

 coefficient,

h

12 a11a22D11D22

1

4
163

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.6

-0.4

0

 -0.8

-0.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

Figure 44. Effects of lamina material properties on nondimensional load-path eccentricity parameter e 11 

defined by equation (75b) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

0.2

- 1.0

IM7/5260

e 11 = B11

a11

D11

1

2

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0

 -0.3

-0.1

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy

AS4/3502

AS4/3501-6

Boron-epoxy

Figure 45. Effects of lamina material properties on nondimensional load-path eccentricity parameters e 12 and e 66 

defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

0.1

- 0.4

IM7/5260

and

e 12 = B12

a11a22

D11D22

1

4

e 66 = B66

a11a22

D11D22

1

4

Load-path

eccentricity

parameters,
164

Fiber angle, θ, deg

0 15 30 45 60 75 90

0.4

0.6

1.0

0

0.8

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

AS4/3502

Kevlar 49-epoxy and IM7/5260

Boron-epoxy

Figure 46. Effects of lamina material properties on nondimensional load-path eccentricity parameter e 22 

defined by equation (75d) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

1.2

- 0.2

0.2

AS4/3501-6

e 22 = B22

a22

D22

1

2

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0

 -0.4

-0.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

Kevlar 49-epoxy and IM7/5260

AS4/3502

AS4/3501-6

Boron-epoxy

Figure 47. Effects of lamina material properties on nondimensional load-path eccentricity parameters e 16 and -e 16 

defined by equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

0.1

- 0.5

 -0.3

e 16 = B16

a11

2

D11D22

1

4

Load-path

eccentricity

parameter,
165

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.3

-0.1

0

 -0.4

-0.2

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

AS4/3502

AS4/3501-6

Boron-epoxy

Figure 48. Effects of lamina material properties on nondimensional load-path eccentricity parameters e 26 and -e 26 

defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric laminates, respectively (m = 1).

0.1

- 0.5

Kevlar 49-epoxy and IM7/5260

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,

0

-0.1

Boron-Aluminum

S-glass-epoxy

IM7/PETI-5

P-100/3502

AS4/3502 Kevlar 49-epoxy

AS4/3501-6

Boron-epoxy

IM7/5260

Figure 49. Effects of lamina material properties on nondimensional load-path eccentricity parameters e 16 and e 26 

defined by equations (75) for (+θ /0/90)m unbalanced, unsymmetric laminates (m = 1).

0.1

- 0.5

- 0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

+θ

θ = 45 degrees

- 0.4

- 0.3

- 0.2

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,

e 16 = B16

a11

2

D11D22

1

4

Load-path eccentricity parameter,
166

Fiber angle, θ, deg

0 15 30 45 60 75 90

0

0.6

1.2

0.8

1.4

Figure 50. Effects of number of plies on parameter coefficients in equation (55) for (+θ /0/90)m and (-θ /0/90)m

unbalanced, unsymmetric P-100/3502 laminates.

m = 8

m = 1

2

3

Parameter

coefficient,

D11

D22

1

4

1.0

0.4

0.2

Fiber angle, θ, deg

0 15 30 45 60 75 90

0

0.6

1.2

0.8

1.4

Figure 51. Effects of number of plies on the orthotropy parameter β for (+θ /0/90)m and (-θ /0/90)m 

unbalanced, unsymmetric P-100/3502 laminates.

m = 8

m = 1

2

3

1.0

0.4

0.2

Flexural

orthotropy

parameter,

β = D12 + 2D66

D11D22
167

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0.2

0.4

0.5

Figure 52. Effects of number of plies on Poisson’s ratio defined by equation (59d) for (+θ /0/90)m and (-θ /0/90)m 

unbalanced, unsymmetric P-100/3502 laminates.

m = 1

m = 8

2

3

0.3

0.1

0

Poisson’s

ratio,

νb = D12

D11D22

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0.2

0.5

0

0.3

0.6

Flexural

anisotropy

parameter,

Figure 53. Effects of number of plies on flexural anisotropy parameters γb and -γb for (+θ /0/90)m and (-θ /0/90)m 

unbalanced, unsymmetric P-100/3502 laminates, respectively.

m = 1

m = 8

2

3

0.1

γb = D16

D11

3

D22

1/4

0.4
168

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0.2

0.5

0

0.3

0.6

Flexural

anisotropy

parameter,

Figure 54. Effects of number of plies on flexural anisotropy parameters δb and -δb for (+θ /0/90)m and (-θ /0/90)m 

unbalanced, unsymmetric P-100/3502 laminates, respectively.

m = 1

m = 8

2

3

0.1

δb = D26

D11D22

3 1/4

0.4

0.2

0.4

 0

0.3

Figure 55. Effects of the number of plies on flexural anisotropy parameters γb and δb for (+θ /0/90)m 

unbalanced, unsymmetric P-100/3502 laminates.

0.5

- 0.1

- 0.1 0 0.2 0.3 0.4 0.5

δb = D26

D11D22

3 1/4

0.1

0.1

m = 1

2

3

m = 8 

γb = D16

D11

3

D22

Flexural anisotropy parameter, 1/4

Flexural

anisotropy

parameter,
169

Fiber angle, θ, deg

0 15 30 45 60 75 90

0

0.6

1.2

0.8

1.6

Figure 56. Effects of number of plies on Batdorf-Stein-parameter coefficients in equations (45) and (48) 

for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates.

m = 8

m = 1

2

3

Parameter

coefficient,

1.0

0.4

0.2

1.4

h

12 a11a22D11D22

1

4

Fiber angle, θ, deg

0 15 30 45 60 75 90

-1

-0.6

0

-0.8

-0.4

0.2

Figure 57. Effects of number of plies on nondimensional load-path eccentricity parameter e 11 defined by 

equation (75b) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates.

m = 1

m = 8

2 3

-0.2

4 5

6

7

e 11 = B11

a11

D11

1

2

Load-path

eccentricity

parameter,
170

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.2

0.2

1.0

0

0.4

1.2

Figure 58. Effects of number of plies on nondimensional load-path eccentricity parameter e 22 defined by 

equation (75d) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates.

m = 8

m = 1

2

3 0.6

4

5

6

e 22 = B22

a22

D22

1

2

0.8

7

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

-1

-0.6

0

-0.8

-0.4

0.2

Figure 59. Effects of number of plies on nondimensional load-path eccentricity parameters e 12 and e 66 

defined by equations (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates.

m = 1

m = 8

2

3

-0.2

6 5 4

and

e 12 = B12

a11a22

D11D22

1

4

e 66 = B66

a11a22

D11D22

1

4

Load-path

eccentricity

parameters,
171

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0

 -0.4

-0.2

0.1

- 0.5

 -0.3

m = 1

m = 8

2 3

4 6 5

Figure 60. Effects of number of plies on nondimensional load-path eccentricity parameters e 16 and -e 16 defined by 

equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates, respectively.

e 16 = B16

a11

2

D11D22

1

4

Load-path

eccentricity

parameter,

Fiber angle, θ, deg

0 15 30 45 60 75 90

-0.1

0

 -0.4

-0.2

0.1

- 0.5

 -0.3

m = 1

m = 8

2

3

5 4

6

Figure 61. Effects of number of plies on nondimensional load-path eccentricity parameters e 26 and -e 26 defined 

by equation (75) for (+θ /0/90)m and (-θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates, respectively.

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,
172

0

-0.1

Figure 62. Effects of number of plies on nondimensional load-path eccentricity parameters e 16 and e 26 defined 

by equations (75) for (+θ /0/90)m unbalanced, unsymmetric P-100/3502 laminates.

0.1

- 0.5

- 0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

+θ

- 0.4

- 0.3

- 0.2

m = 1

2

3

4

5

6

m = 8

e 16 = B16

a11

2

D11D22

1

4

e 26 = B26

a22

2

D11D22

1

4

Load-path

eccentricity

parameter,

Load-path eccentricity parameter,
173

Appendix

Symbols

A domain of the orthogonal Gaussian coordinates (ξ1, ξ2) for

the shell reference surface, 

a1, a2, b1, b2 constants that define the domain of the shell reference surface

Gaussian coordinates; i. e., and 

a11, a12, a16, a22, a26, a66 membrane compliances defined by equation (21a), in./lb

A domain of the nondimensional orthogonal Gaussian coordinates

(z1, z2) for the shell reference surface, 

A1(ξ1, ξ2), A2(ξ1, ξ2) metric coefficients of shell reference surface defined by

equation (11)

A11, A12, A16, A22, A26, A66 membrane stiffnesses defined by equation (17), lb/in.

[a ] matrix of constitutive constants defined by equation (154a)

b11, b12, b16, b22, b26, b66 constitutive constants defined by equation (21b), in.

B11, B12, B16, B22, B26, B66 coupling stiffnesses defined by equation (18), lb

b11, b12, b16, b22, b26, b66 constitutive constants defined by equation (69c)

B11, B12, B16, B21, B22, constitutive constants defined by equation (77)

B26, B61, B62, B66

[B ] matix of constitutive constants defined by equation (139b)

ds differential arc length defined in equation (11), in.

d11, d12, d16, d22, d26, d66 reduced bending stiffnesses defined by equation (21c), in-lb

d11, d12, d16, d22, d26, d66 constitutive constants defined by equations (79)

[d ] matix of constitutive constants defined by equation (139c)

D11, D12, D16, D22, D26, D66 bending stiffnesses defined by equation (19), in-lb

a1, b1 × a2, b2

a1 ≤ ξ 1 ≤ b1 a2 ≤ ξ 2 ≤ b2

a1

L1

, b 1

L1

× a2

L2

, b 2

L2
174

Db( ), Dc( ), DB( ), Dm( ), Dq( ) nondimensional linear differential operators defined by 

equations (122b), (122c), (122d), (124b), and (117b),

respectively

EL, ET, GLT lamina moduli, psi

E11(z1, z2), E22(z1, z2), G12(z1, z2) nondimensional membrane strain fields defined by 

equations (46)

{E} vector of nondimensional membrane strains defined by

equation (151b)

vector of nondimensional membrane strains associated with 

adjacent equilibrium states and defined by equation (213a)

e11, e12, e16, e22, e26, e66 nondimensional load-path eccentricity parameters defined by

equations (75)

nondimensional membrane-strain fields associated with the 

primary equilibrium path and defined by equations (177)

nondimensional membrane-strain fields associated with 

adjacent equilibrium states and defined by equations (193)

F(z1

, z2) nondimensional stress function defined by equations (115)

nondimensional stress function associated with adjacent 

equilibrium states and defined by equations (214)

h shell thickness, in.

boundary integrals defined by equations (134)

boundary integrals defined by equations (205)

K11(z1

, z2), K22(z1, z2), K12(z1, z2) nondimensional bending strain fields defined by 

equations (49)

nondimensional bending-strain fields associated with the 

primary equilibrium path and defined by equations (178)

E

(1)

E11

(0)

z 1, z 2, p , E22

(0)

z 1, z 2, p ,

G12

(0)

z 1, z 2, p

E11

(1)

z 1, z 2 , E22

(1)

z 1, z 2 , G12

(1)

z 1, z 2

F

(1)

z1, z2

I1

B

, I2

B

I1

(1)B

, I2

(1)B

K11

(0)

z 1, z 2, p , K22

(0)

z 1, z 2, p

K12

(0)

z 1, z 2, p
175

nondimensional bending-strain fields associated with 

adjacent equilibrium states and defined by equations (194)

{K} vector of nondimensional bending strains defined by

equation (139a)

vector of nondimensional bending strains associated with 

adjacent equilibrium states and defined by equation (213c)

L1

, L2 characteristic dimensions used for scaling the (ξ1, ξ2) Gaussian

coordinates

L( , ) nondimensional bilinear differential operator defined by 

equation (117c)

M(ξ1

) moment per unit length applied to edges ξ2 = a2 and ξ2 = b2, 

as shown in figure 2, lb

M(ξ2

) moment per unit length applied to edges ξ1 = a1 and ξ1 = b1, 

as shown in figure 2, lb

M11(ξ1, ξ2), M22(ξ1

, ξ2), M12(ξ1, ξ2) bending stress resultants defined by equation (12b), lb

nondimensional loading applied to edges ξ2 = a2 and ξ2 = b2

and defined by equations (99)

nondimensional loading applied to edges ξ1

 = a1 and ξ1 = b1

and defined by equations (98)

m1, m2, m3 nondimensional functions defined by equation (121)

M11(z1

, z2), M22(z1, z2), M12(z1, z2) nondimensional bending stress resultants defined by 

equations (60)

nondimensional bending stress resultants associated with the 

primary equilibrium path (see equations (181) and (195))

nondimensional bending stress resultants associated with 

adjacent equilibrium states (see equations (195))

{m} vector of nondimensional functions defined by equation (139d)

{M} vector of nondimensional stress resultants defined by

equation (135b)

K11

(1)

z1, z2 , K22

(1)

z1, z2 , K12

(1)

z1, z2

K

(1)

M z1

M z2

M11

(0)

z 1, z 2, p , M22

(0)

z 1, z 2, p ,

M12

(0)

z 1, z 2, p

M11

(1)

z 1, z 2 , M22

(1)

z 1, z 2 , M12

(1)

z 1, z 2
176

vector of nondimensional stress resultants associated with 

adjacent equilibrium states (see equations (211))

N(ξ1) loads applied to edges ξ2 = a2 and ξ2 = b2, as shown in

figure 2, lb/in.

N(ξ2) loads applied to edges ξ1 = a1 and ξ1 = b1, as shown in

figure 2, lb/in.

N11(ξ1, ξ2), N22(ξ1, ξ2), N12(ξ1, ξ2) membrane stress resultants defined by equation (12a), lb/in.

N11(z1, z2), N22(z1, z2), N12(z1, z2) nondimensional membrane stress resultants defined by 

equations (54)

nondimensional membrane stress resultants associated with the 

primary equilibrium path (see equations (181) and (195))

nondimensional membrane stress resultants associated with 

adjacent equilibrium states (see equations (195))

nondimensional loading applied to edges ξ2 = a2 and ξ2 = b2

and defined by equations (99)

nondimensional loading applied to edges ξ1 = a1 and ξ1 = b1

and defined by equations (98)

{N} vector of nondimensional stress resultants defined by

equation (135a)

[N] matrix of nondimensional stress resultants defined by

equation (135c)

matrix of nondimensional stress resultants associated with the 

primary equilibrium path (see equations (211))

vector of nondimensional stress resultants associated with 

adjacent equilibrium states (see equations (211))

Pm force per unit area defined by equation (30), psi 

Pm nondimensional value of Pm defined by equation (94)

PT nondimensional function defined by equation (116b)

M

(1)

N11

(0)

z 1, z 2, p , N22

(0)

z 1, z 2, p ,

N12

(0)

z 1, z 2, p

N11

(1)

z 1, z 2 , N22

(1)

z 1, z 2 , N12

(1)

z 1, z 2

N z1

N z2

N

(0)

p

N

(1)
177

, nondimensional loading parameter and corresponding value at

bifurcation, respectively

p1, p2, p3 nondimensional functions defined by equation (119)

{p} vector of nondimensional functions defined by equation (154b)

q1

(ξ1, ξ2), q2(ξ1, ξ2), q3(ξ1, ξ2) applied tractions acting on shell reference surface, psi

Q1(ξ1, ξ2), Q2(ξ1, ξ2) transverse-shearing stress resultants defined by equation (13),

lb/in.

transformed reduced stiffnesses for laminae in a state of plane

stress, psi

q1(z1, z2), q2(z1, z2), q3(z1, z2) nondimensional applied tractions acting on shell reference 

surface and defined by equations (82), (84), and (92), 

respectively

Q1(z1, z2), Q2(z1, z2) nondimensional transverse-shear stress resultants defined by 

equations (86)-(89)

nondimensional shear stress resultants associated with the 

primary equilibrium path (see equations (181) and (195))

nondimensional shear stress resultants associated with 

adjacent equilibrium states (see equations (195))

vector of nondimensional functions defined by equation (140b)

matrix of nondimensional functions defined by equation (142b)

R1, R2 principal radii of curvature of the shell reference surface, in.

S(ξ1) loads applied to edges ξ2 = a2 and ξ2 = b2, as shown in

figure 2, lb/in.

S(ξ2

) loads applied to edges ξ1 = a1 and ξ1 = b1, as shown in

figure 2, lb/in.

nondimensional loading applied to edges ξ2 = a2 and ξ2 = b2

and defined by equations (99)

p pcr

Q11, Q12, Q16, Q22, Q26, Q66

Q1

(0)

z 1, z 2, p , Q2

(0)

z 1, z 2, p

Q1

(1)

z1, z2 , Q2

(1)

z1, z2

q

q

S z1
178

nondimensional loading applied to edges ξ1 = a1 and ξ1 = b1

and defined by equations (98)

U1(ξ1,ξ2

,ζ), U2(ξ1,ξ2,ζ), U3(ξ1,ξ2,ζ) components of the displacement vector field of the material

points comprising a shell, in.

u1

(ξ1, ξ2), u2(ξ1, ξ2), w(ξ1, ξ2) displacement components of points of the two-dimensional shell

reference surface defined by ζ = 0 

U1(z1, z2), U2(z1, z2

) nondimensional displacement fields defined by equations (44)

and (47), respectively 

nondimensional displacement fields associated with the primary

equilibrium path and defined by equations (175)

nondimensional displacement fields associated with adjacent

equilibrium states and defined by equations (175)

V(ξ1) loads applied to edges ξ2 = a2 and ξ2 = b2, as shown in

figure 2, lb/in.

V(ξ2) loads applied to edges ξ1

 = a1 and ξ1 = b1, as shown in

figure 2, lb/in.

nondimensional loading applied to edges ξ2 = a2 and ξ2 = b2

and defined by equations (98)

nondimensional loading applied to edges ξ1 = a1 and ξ1 = b1

and defined by equations (99)

W(z1

, z2) nondimensional displacement defined by 

wI

(ξ1, ξ2) distribution of small geometric deviations in the ζ-coordinate 

direction, measured perpendicular to the tangent plane at each

point of the shell reference surface

WI

(z1, z2) nondimensional geometric imperfection function defined by

(z1

, z2) Nondimensional orthogonal Gaussian coordinates for shell 

reference surface given by and 

Z1, Z2 Batdorf-Stein parameters defined by equations (45) and (48)

S z2

U1

(0)

z 1, z 2, p , U2

(0)

z 1, z 2, p ,

W

(0)

z1, z 2, p

U1

(1)

z1, z2 , U2

(1)

z1, z2 , W

(1)

z1, z2

V z1

V z2

w ≡ a11a22D11D22

1

4 W

wI ≡ a11a22D11D22

1

4 WI

ξ 1 = L1z1 ξ 2 = L2z2
179

αb, αm nondimensional stiffness-weighted aspect ratios defined by 

equations (55) and (52a), respectively

β nondimensional flexural orthotropy parameter defined by 

equation (59a)

β1(ξ1, ξ2), β2(ξ1, ξ2) fields defining rotation of material line elements tangent to the

shell reference, defined by equations (10)

fields defining rotation of material line elements tangent to the

shell reference surface associated with "small" initial geometric

imperfections

γb nondimensional flexural-twist anisotropy parameter defined by

equation (59b)

γm nondimensional membrane anisotropy parameter defined by

equation (52c)

γ12(ξ1,ξ2,ζ), γ13(ξ1,ξ2,ζ), γ23(ξ1,ξ2,ζ) shearing-strain fields for a three-dimensional shell body

tangential, membrane shearing-strain fields of shell reference

surface

transverse-shearing-strain fields of shell reference

surface

δ variational operator of the Calculus of Variations

δb nondimensional flexural-twist anisotropy parameter defined by

equation (59c)

δm nondimensional membrane anisotropy parameter defined by

equation (52d)

δE11(z1, z2), δE22(z1, z2), δG12(z1, z2) nondimensional virtual membrane strain fields defined by 

equations (107)

virtual membrane strains associated with adjacent equilibrium

states and defined by equations (208)

vector of nondimensional virtual membrane strains associated

with adjacent equilibrium states and defined by equation (213b)

δF nondimensional virtual stress function (see equations (150))

β1

I

ξ 1, ξ 2 , β2

I

ξ 1, ξ 2

γ 12

o

ξ 1, ξ 2

γ 13

o

ξ 1, ξ 2 , γ 23

o

ξ 1, ξ 2

δE11

(1)

z 1, z 2 , δE22

(1)

z 1, z 2 , δG12

(1)

z 1, z 2

δE

(1)
180

δK11(z1, z2), δK22(z1, z2), δK12(z1, z2) nondimensional virtual bending strain fields defined by 

equations (108)

virtual bending strains associated with adjacent equilibrium

states and defined by equations (209)

{δK} vector of nondimensional virtual bending strains defined by 

equation (136d)

vector of nondimensional virtual bending strains associated

with adjacent equilibrium states and defined by equation (213d)

 nondimensional virtual stress resultants used in equations (143)

δu1(ξ1, ξ2), δu2(ξ1, ξ2), δw(ξ1, ξ2) virtual-displacement fields of the two-dimensional shell

reference surface defined by ζ = 0 

δU1(z1, z2), δU2(z1, z2), δW(z1, z2) nondimensional virtual-displacement fields defined by 

equations (106)

δW nondimensional radial-displacement field at buckling (see

equations (1) and (2))

δWext external virtual work per unit area defined by equation (23b), 

lb/in.

external virtual work per unit length defined by equations (23c),

lb

δWint internal virtual work per unit area defined by equation (23a), 

lb/in.

 nondimensional virtual work associated with adjacent 

equilibrium states (see equations (203))

δWiext nondimensional external virtual work per unit area defined by

equation (109b)

nondimensional virtual work per unit area associated

with adjacent equilibrium states and defined by equation (204a)

external virtual work per unit length defined by equations (113)

nondimensional virtual work per unit length associated

with adjacent equilibrium states and defined by equations (204)

δK11

(1)

z 1, z 2 , δK22

(1)

z 1, z 2 , δK12

(1)

z 1, z 2

δK

(1)

δN*11, δN*22, δN*12

δWext

B

δW

(1)

δW

(1)

δWext

B

δW1

(1) B

, δW2

(1) B
181

complementary virtual work defined by equationa (147b) and

(149b), respectively

δWint nondimensional internal virtual work per unit area defined by

equation (109a)

nondimensional internal virtual work per unit area associated

with adjacent equilibrium states and defined by equation (206b)

 nondimensional internal virtual work per unit area associated

with adjacent equilibrium states and defined by equation (206c)

nondimensional complementary internal virtual work per unit

area defined by equations (147a) and (149a), respectively

 nondimensional complementary internal virtual work per unit

area associated with adjacent equilibrium states and defined by

equation (234)

 nondimensional complementary virtual work associated with

adjacent equilibrium states and defined by equation (232c)

virtual membrane-strain fields of shell reference surface

virtual transverse-shearing-strain fields of shell reference

surface

fields defining virtual changes in shell reference-surface 

curvature and torsion 

δψ1(ξ1

, ξ2), δψ2(ξ1, ξ2) virtual rotation fields of the shell reference surface 

(see equations (27) and (28))

virtual rotations associated with adjacent equilibrium

states and defined by equations (207)

{δΩ} vector of nondimensional virtual rotations defined by 

equation (136c)

δW*

B

, δW*

B

δWint

(1)

δW′int

(1)

δW*int, δW*int

δW*int

(1)

δW*

(1) B

δε11

o

ξ 1, ξ 2 , δε22

o

ξ 1, ξ 2 ,

δγ 12

o

ξ 1, ξ 2

δγ 13

o

ξ 1, ξ 2 , δγ 23

o

ξ 1, ξ 2

δκ11

o

ξ 1, ξ 2 , δκ22

o

ξ 1, ξ 2 ,

δκ12

o

ξ 1, ξ 2

δΩ1

(1)

z 1, z 2 , δΩ2

(1)

z 1, z 2
182

vector of virtual rotations associated with adjacent equilibrium

states and defined by equations (212c)

Δ1(ξ1), Δ2(ξ1), Δn(ξ1) displacements applied to edges ξ2 = a2 and ξ2 = b2; positive in

the positive ξ1, ξ2, and ζ directions, respectively, in.

Δ1(ξ2), Δ2(ξ2), Δn(ξ2) displacements applied to edges ξ1 = a1 and ξ1 = b1; positive in

the positive ξ1, ξ2, and ζ directions, respectively, in.

nondimensional displacements applied to edges ξ2 = a2 and

ξ2

 = b2 and defined by equations (99)

nondimensional displacements applied to edges ξ1 = a1 and 

ξ1 = b1 and defined by equations (98) 

ε "small" parameter used in bifurcation analysis, see 

equations (175)

ε11(ξ1,ξ2,ζ), ε22(ξ1,ξ2,ζ), ε33(ξ1,ξ2,ζ) normal-strain fields for a three-dimensional shell body

tangential, membrane normal-strain fields of shell reference 

surface

θ lamina fiber angle (see figure 3), degrees 

fields defining changes in shell reference-surface curvature and

torsion 

μ nondimensional orthotropy parameter defined by equation (52b)

νLT lamina major Poisson's ratio

νb, νm generalized laminate Poisson's ratios associated with membrane

and bending action, respectively (see equations (52e) and (59d))

(ξ1, ξ2) orthogonal Gaussian coordinates for shell reference surface

(ξ1, ξ2, ζ) orthogonal curvilinear coordinates for points of threedimensional Euclidean space

ρ stiffness-weighted radius-to-thickness ratio defined by 

equations (10) and (20)

shell stresses, psi

δΩ

(1)

Δ1 z1 , Δ2 z1 , Δn z1

Δ1 z2 , Δ2 z2 , Δn z2

ε11

o

ξ 1, ξ 2 , ε22

o

ξ 1, ξ 2

κ11

o

ξ 1, ξ 2 , κ22

o

ξ 1, ξ 2 , κ12

o

ξ 1, ξ 2

σ11, σ12, σ22, σ13, σ22
183

Φ(ξ1) rotation applied to edges ξ2 = a2 and ξ2 = b2; positive about

the positive ξ1 direction

Φ(ξ2) rotation applied to edges ξ1 = a1 and ξ1 = b1; positive about

the negative ξ2 direction

 nondimensional rotation applied to edges ξ2 = a2 and ξ2 = b2

and defined by equation (99d)

nondimensional rotation applied to edges ξ1

 = a1 and ξ1 = b1

and defined by equation (98d)

ψ1(ξ1, ξ2), ψ2(ξ1, ξ2) fields defining rotations of material line elements perpendicular 

to the shell reference surface

Ω1

(z1, z2), Ω2(z1, z2) nondimensional rotation fields defined by equations (38)-(42)

{Ω} vector of nondimensional rotations defined by equation (136a)

[Ω] matrix of nondimensional rotations defined by equation (151c)

{ΩI

} vector of nondimensional rotations associated with an initial 

geometric imperfection and defined by equation (136b)

[ΩI

] matrix of nondimensional rotations associated with an initial 

geometric imperfection and defined by equation (151d)

nondimensional rotation fields associated with the primary

equilibrium path and defined by equations (176)

nondimensional rotation fields associated with adjacent

equilibrium states and defined by equations (192)

 vector of nondimensional rotation fields associated with the 

primary equilibrium path and defined by equations (212a)

matrix of nondimensional rotation fields associated with the 

primary equilibrium path and defined by equations (237)

vector of nondimensional rotation fields associated with

adjacent equilibrium states and defined by equations (212b)

∂A general boundary curve enclosing the reference surface domain

Φ z1

Φ z2

Ω1

(0)

z 1, z 2, p , Ω2

(0)

z 1, z 2, p

Ω1

(1)

z 1, z 2 , Ω2

(1)

z 1, z 2

Ω

(0)

p

Ω

(0)

p

Ω

(1)
184

{∂F} vector of nondimensional stress-function derivatives defined by

equation (139e)

[∂F] matrix of nondimensional stress-function derivatives defined by

equation (142a)

vector of nondimensional stress-function derivatives associated

with adjacent equilibrium states and defined by equation (211a)

{∂δF} vector of nondimensional virtual-stress-function derivatives 

defined by equation (151e)

vector of nondimensional virtual-stress-function derivatives 

associated with adjacent equilibrium states and defined by 

equation (226b)

∂F

(1)

∂δF

(1)
REPORT DOCUMENTATION PAGE Form Approved

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 4. TITLE AND SUBTITLE

Nondimensional Parameters and Equations for Nonlinear and Bifurcation 

Analyses of Thin Anisotropic Quasi-Shallow Shells 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Nemeth, Michael P.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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14. ABSTRACT

A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of 

quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of 

field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic 

nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a 

comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. 

Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation 

equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material 

properties and stacking sequence on the nondimensional parameters.

15. SUBJECT TERMS

Buckling; Shell mechanics; Structural design; Stability analysis; Nondimensional parameters

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01- 07 - 2010