NASA/TP-20-

A Treatise on Equivalent-Plate Stiffnesses for 

Stiffened Laminated-Composite Plates and

Plate-Like Lattices 

Michael P. Nemeth

Langley Research Center, Hampton, Virginia

January 2011
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NASA/TP-20-

A Treatise on Equivalent-Plate Stiffnesses for 

Stiffened Laminated-Composite Plates and

Plate-Like Lattices 

Michael P. Nemeth

Langley Research Center, Hampton, Virginia

National Aeronautics and 

Space Administration

Langley research Center

Hampton, Virginia 23681-2199

January 2011
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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

Direct Equilibrium-Compatibility Method for Rectilinear Stiffener Families . . . . . 19

Statical Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Kinematical Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Analysis Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Orthogrid plate with doubly braced bays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Orthogrid plate with singly braced bays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Isosceles-triangle stiffener pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Kagome stiffener pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Basic-Cell Energy-Equivalence Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Analysis Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Orthogrid plate with doubly or singly braced bays. . . . . . . . . . . . . . . . . . . . . 42

Isosceles-triangle and Kagome stiffener patterns . . . . . . . . . . . . . . . . . . . . . . 42

Hexagon-shaped stiffener pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Star-shaped stiffener pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Equivalent-Plate Stiffness for Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Equivalent-Plate Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Appendix A - Equations of First-Order Transverse-Shear Deformation Beam

Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Appendix B - Equivalent-Plate Stiffnesses for a Plate Reinforced with Eccentric 

Stringers, Ribs, and Diagonal Braces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Appendix C - Equivalent-Plate Stiffnesses for a Plate Reinforced with an 

Isosceles-Triangle or Kagome Grid Stiffener Arrangement. . . . . . . . . . . . . . . . . . . 108

Appendix D - Beam-Member Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 112

Appendix E - Mathematica®

 Program for the Basic-Cell Energy-Equivalence

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Appendix F - Equivalent-Plate Stiffnesses for a Plate Reinforced with a 

Hexagon-Shaped Stiffener Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Appendix G - Equivalent-Plate Stiffnesses for a Plate Reinforced with a 

Star-Cell-Shaped Stiffener Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Appendix H - Equivalent-Plate Stiffnesses for a Hexagon-Cell-Core 

Sandwich Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Appendix I - Equivalent-Plate Stiffnesses for a Orthogrid-Core Sandwich Plate . . 139

Appendix J - Equivalent-Plate Stiffnesses for a Star-Cell-Core Sandwich Plate . . . 143
3

List of Tables

1. Stiffener attributes for an eccentric, orthogonal stiffening grid with two diagonal 

 braces per bay (see figure 14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2. Stiffener attributes for an eccentric, orthogonal stiffening grid with one diagonal 

 brace per bay (see figure 16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3. Stiffener attributes for an eccentric, isosceles-triangle stiffener pattern (see figure 17) . 62

4. Basic-cell beam-member attributes for an eccentric, orthogonal stiffener pattern with

 two diagonal braces per bay (see figure 14b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5. Basic-cell beam-member attributes for an eccentric, orthogonal stiffener pattern with

 one diagonal brace per bay (see figure 16b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6. Basic-cell beam-member attributes for a plate stiffened with an eccentric, isosceles-

 triangle stiffener pattern (see figure 17b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7. Basic-cell beam-member attributes for a plate stiffened with an eccentric Kagome 

 stiffener pattern (see figure 18b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8. Basic-cell-member attributes for a plate stiffened with an eccentric, hexagonal 

 stiffener pattern (see figure 21b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9. Basic-cell-member attributes for a plate stiffened with an eccentric, star-shaped 

 stiffener pattern (see figure 23b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4

List of Figures

1. Space Shuttle and external tank structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2. Densely stiffened orthogrid cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3. Isogrid-stiffened cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4. Unidirectionally stiffened plate geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5. Cross-section A-A of a uniformly stiffened panel shown in figure 4b. . . . . . . . . . . . . . . 74

6. Plate and stiffener coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7. Examples of a nonhomogeneous stiffener cross-section . . . . . . . . . . . . . . . . . . . . . . . . . 75

8. Repetitive stiffened-panel element and coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 75

9. Equivalent-plate wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

10. Beam stresses at an arbitrary cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

11. Beam force and moment resultants at an arbitrary cross-section . . . . . . . . . . . . . . . . . . . 77

12. Stiffener beam forces and moments acting on cross-section of repetitive stiffenedpanel element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

13. Plate stress resultants acting on cross-section of equivalent stiffened-panel wall . . . . . . 78

14. Orthogonal stiffener pattern with two diagonal braces per bay and basic cell . . . . . . . . . 79

15. Diamond-shaped stiffener pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

16. Orthogonal stiffener pattern with one diagonal brace per bay and basic cell. . . . . . . . . . 81

17. Isosceles-triangle stiffener pattern and basic cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

18. Kagome stiffener pattern and basic cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

19. Generation of an isosceles-triangle stiffener pattern obtained by translating the 

basic cell defined in figure 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

20. Generation of a Kagome stiffener pattern obtained by translating the basic cell

defined in figure 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

21. Hexagon-shaped stiffener pattern and basic cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5

22. Generation of a hexagonal stiffener pattern obtained by translating the basic

cell defined in figure 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

23. Isosceles-star-shaped stiffener pattern and basic cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

24. Generation of an isosceles-star-shaped stiffener pattern obtained by translating 

the basic cell defined in figure 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

25. Sandwich plate with nonidentical anisotropic face plates and a hexagon-cell core . . . . . 89

26. Sandwich plate with nonidentical anisotropic face plates and an orthogrid core. . . . . . . 90

27. Sandwich plate with nonidentical anisotropic face plates and a star-cell core. . . . . . . . . 91

28. Arbitrary face plate of a sandwich plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6

Summary

 A survey of studies conducted since 1914 on the use of equivalent-plate 

stiffnesses in modeling the overall, stiffness-critical response of stiffened 

plates and shells is presented. Two detailed, comprehensive derivations of 

first-approximation equivalent-plate stiffnesses are also presented that 

are based on the Reissner-Mindlin-type, first-order transverse-shear 

deformation theory for anisotropic plates. First, a derivation based purely 

on static and kinematic equivalence between a stiffened plate and its 

homogenized equivalent is presented, followed by a derivation based on 

equivalence of the strain-energy density. In both derivations, the stiffener 

members are modelled as beams that are shear deformable within and 

transverse to the plane of the plate, consistent with the classical 

continuum mechanics representation of solids. Additionally, each stiffener 

is presumed to be constructed, at most, in a nonhomogeneous manner 

from orthotropic materials with one axis aligned with the stiffener axis 

and the other two axes aligned with the cross-sectional axes. This 

presumption allows the computation of equivalent-plate stiffnesses for 

stiffened panels such as those in which the stiffener caps are reinforced 

with high-strength, pultruded rods. Consistent with a first-approximation 

analysis, inplane bending of the stiffeners and total compatibility between 

the plate skin and the stiffeners are neglected.

 Equivalent-plate stiffness expressions, and a corresponding symbolic 

manipulation computer program, are also presented for several different 

stiffener configurations. These expressions are very general and exhibit 

the full range of anisotropies permitted by the Reissner-Mindlin-type, 

first-order transverse-shear deformation theory for anisotropic plates. 

The expressions presented in the present study were also compared with 

available, previously published results. For the most part, the previously 

published results are for special cases of the general expressions 

presented herein and are almost in complete agreement. Analysis is also 

presented that extends the use of the equivalent-plate stiffness expressions 

to sandwich plates with nonidentical, anisotropic face plates, and 

expressions for equivalent-plate thicknesses are presented.
7

Symbols

 The primary symbols used in the present study are given as follows. Additional symbols are

defined in Tables 1-9.

a dimension of hexagonal beam grid (see figure 21), in.

AS area of prismatic stiffener cross-section (see figure 10), in2

Acell area of basic cell forming beam grid (see equation (39a) and

figures 14, 16-18, 21, and 23), in2

A11, A12, A16, A22, A26, A66 equivalent-plate membrane stiffnesses (see equation (22a)), 

lb/in.

A44, A45, A55 equivalent-plate transverse-shearing stiffnesses (see equation

(22b)), lb/in.

contribution of sandwich-plate core to equivalent-plate 

membrane stiffnesses (see equation (47a)), lb/in.

plate contribution to equivalent-plate membrane stiffnesses

(see equations (23a) and (44a)), lb/in.

contribution of sandwich face plate to equivalent-plate

membrane stiffnesses (see equation (46a)), lb/in.

contribution of sandwich-plate core to equivalent-plate 

transverse-shearing stiffnesses (see equation (47d)), lb/in.

plate contribution to equivalent-plate transverse-shearing 

stiffnesses (see equation (23d)), lb/in.

contribution of sandwich face plate to equivalent-plate 

transverse-shearing stiffnesses (see equation (46d)), lb/in.

 stiffener contribution to equivalent-plate membrane stiffnesses

(see equation (23a)), lb/in.

stiffener contribution to equivalent-plate transverse-shearing 

stiffnesses (see equation (23d)), lb/in.

A11

core, A12core, A16core

,

A22

core, A26core, A66core

A11

plate, A12plate, A16plate

,

A22

plate, A26plate, A66plate

A′11

plate

, A′12

plate

, A′16

plate

,

A′22

plate

, A′26

plate

, A′66

plate

A44

core, A45core, A55core

A44

plate, A45plate, A55plate

A′44

plate

, A′45

plate

, A′55

plate

A11

stiffener, A12stiffener, A16stiffener

,

A22

stiffener, A26stiffener, A66stiffener

A44

stiffener, A45stiffener, A55stiffener
8

b dimension of hexagon-shaped stiffener pattern (see figures 21

and 25), in.

B dimension of star-shaped stiffener pattern (see figures 23

and 27), in.

B11, B12, B16, B22, B26, B66 equivalent-plate coupling stiffnesses (see equation (22a)), lb

plate contribution to equivalent-plate coupling stiffnesses

(see equations (23b) and (44b)), lb

contribution of sandwich face plate to equivalent-plate

 coupling stiffnesses (see equation (46b)), lb

 stiffener contribution to equivalent-plate coupling stiffnesses

(see equation (23b)), lb

c dimension of hexagonal beam grid (see figure 21), in.

[C] beam-stiffener constitutive matrix defined by equations (A25)

and (D1)

[Cp] transformed beam constitutive matrix defined by equation (36)

with elements given in Appendix D

[CP] beam constitutive matrix defined by equation (32)

dS stiffener spacing (see figures 4 and 5), in. 

D11, D12, D16, D22, D26, D66 equivalent-plate bending and twisting stiffnesses (see equation

(22a)), in.-lb

contribution of sandwich-plate core equivalent-plate bending 

stiffnesses (see equation (47c)), in.-lb

plate contribution to equivalent-plate bending stiffnesses

(see equation (23c) and (44c)), in.-lb

contribution of sandwich face plate to equivalent-plate 

bending stiffnesses (see equation (46c)), in.-lb

B11

plate, B12plate, B16plate

,

B22

plate, B26plate, B66plate

B′11

plate

, B′12

plate

, B′16

plate

,

B′22

plate

, B′26

plate

, B′66

plate

B11

stiffener, B12stiffener, B16stiffener

,

B22

stiffener, B26stiffener, B66stiffener

D11

core, D12core, D16core

,

D22

core, D26core, D66core

D11

plate, D12plate, D16plate

,

D22

plate, D26plate, D66plate

D′11

plate

, D′12

plate

, D′16

plate

,

D′22

plate

, D′26

plate

, D′66

plate
9

 stiffener contribution to equivalent-plate bending stiffnesses

(see equation (23c)), in.-lb

e sandwich face plate eccentricity (see figure 28), in.

e1, e2 face plate eccentricities of sandwich plate (see figures 25-27), in.

 beam membrane strain (see equations (11) and (A3))

E, G extensional and shear moduli of isotropic materials, psi

ES, GS effective extensional and shear moduli of nonhomogeneous

stiffeners (see equations (A13) and (A15)), psi

EX, EY, EZ principal extensional moduli for a homogeneous specially

orthotropic material (see equation (A10)), psi

[E] strain-equivalence matrix defined by equations (30)

Eε beam-member strain energy (see equation (31)), lb-in2 

 strain energy density of equivalent plate (see equations (39)), lb

 contribution of plate wall to strain energy density of equivalent

plate (see equations (39)), lb

GXY, GYZ, GXZ principal shear moduli for a homogeneous specially

orthotropic material (see equation (A10)), psi

effective shear moduli for a nonhomogeneous beam

(see equation (A13)), psi

h plate-wall thickness of stiffened plate (see figures 5, 8-9, and

figure 28), in.

hc

 core thicknesses of sandwich plate (see figures 25-27), in.

hS thickness of equivalent-stiffener layer (see figure 9), in.

h1

, h2 face sheet thicknesses of sandwich plate (see figures 25-27), in.

H dimension of star-shaped stiffener pattern (see figures 23

and 27), in.

D11

stiffener, D12stiffener, D16stiffener

,

D22

stiffener, D26stiffener, D66stiffener

eXX

o

Eε

Eε

plate

GXY

S

, GXZ

S
10

IYY, IZZ, IYZ moments and product of inertia of homogeneous beam

(see equations (A8)), in4

effective moments and product of inertia of nonhomogeneous

stiffeners (see equations (A13)), in4

effective moments and product of inertia of nonhomogeneous

stiffeners with respect to the stiffness-weighted centroid (see

 equations (A28)), in4

J torsional constant of homogeneous isotropic beam (see 

equation (A9)), in4

JS effective torsional constant for nonhomogeneous stiffeners 

(see equation (A15)), in4

kY, kZ inplane and transverse shear correction factors for homogeneous 

beams, respectively (see equations (A7))

inplane and transverse effective shear correction factors for 

nonhomogeneous stiffeners (see equation (A24)), respectively

Kplate transverse-shear correction factor for plates (see equation (44d))

L hexagon side length shown in figure 21, in.

LS generic beam-stiffener length (see equation (31)), in.

Lx, Ly beam-grid stiffener spacing (see figures 14-18,21, and 26), in.

MY(X), MZ(X) beam bending moments (see figure 11), in-lb

Mxx, Myy, Mxy equivalent-plate bending stress resultants in (x, y, z) global plate

coordinates (see equations (18) and figure 6), in-lb/in.

MXX, MYY, MXY equivalent-plate bending stress resultants in (X, Y, Z) beam

coordinates (see figures 6 and 13), in-lb/in.

plate-wall bending stress resultants in (x, y, z) global plate

coordinates (see equations (21)), in-lb/in.

plate-wall bending stress resultants (see equations (1)-(9)) in 

beam (X, Y, Z) coordinates, in-lb/in.

IYY

S

, IZZ

S

, IYZ

S

Iηη

S

, I ζζ

S

, Iηζ

S

k Y

S

, k Z

S

Mxx

plate

, Myy

plate

, Mxy

plate

MXX

plate

, MYY

plate

, MXY

plate
11

equivalent-stiffener-layer bending stress resultants 

(see equations (1)-(9)), in-lb/in.

Nxx, Nyy, Nxy equivalent-plate membrane stress resultants in (x, y, z) global 

plate coordinates (see equations (18) and figure 6), lb/in.

NXX, NYY, NXY equivalent-plate membrane stress resultants in (X, Y, Z) beam 

coordinates (see figures 6 and 13),lb/in.

plate-wall membrane stress resultants in (x, y, z) global plate

coordinates (see equations (21)), lb/in.

plate-wall membrane stress resultants (see equations (1)-(9)) in 

(X, Y, Z) beam coordinates, lb/in.

equivalent-stiffener-layer membrane stress resultants 

(see equations (1)-(9)), lb/in.

P(X) beam axial force (see figure 11), lb

Qxz, Qyz equivalent-plate transverse shearing stress resultants in (x, y, z)

global plate coordinates (see equations (18) and figure 6), in-lb

QXZ, QYZ equivalent-plate transverse shearing stress resultants in 

(X, Y, Z) beam coordinates (see figures 6 and 13), in-lb

plate-wall transverse shearing stress resultants in (x, y, z) global

plate coordinates (see equations (21)), lb/in.

plate-wall transverse shearing stress resultants in (X, Y, Z) beam

coordinates (see equations (1)-(9)), lb/in.

equivalent-stiffener-layer transverse shearing stress resultants 

(see equations (1)-(9)), lb/in.

 transformed stiffnesses of laminated-plate theory

(see equations (44)), psi

t, tr, td, ts core wall thicknesses of sandwich plate (see figures 25-27), in.

T(X) beam torque (see figure 11), in-lb

[Tε

], [Tε

]

- 1 strain transformation matrix and its inverse (see equations (17))

MXX

stiffener

, MYY

stiffener

, MXY

stiffener

Nxx

plate

, Nyy

plate

, Nxy

plate

NXX

plate

, NYY

plate

, NXY

plate

NXX

stiffener

, NYY

stiffener

, NXY

stiffener

Qxz

plate, Qyzplate

QXZ

plate, QYZplate

QXZ

stiffener, QYZstiffener

Qij z
12

[Tσ], [Tσ]

- 1 stress transformation matrix and its inverse (see equations (16))

[Tτ

], [Tτ]

- 1 transverse shearing strain and stress transformation matrix and

its inverse (see equations (16) and (17))

u(X), v(X), w(X) beam displacements of points on the reference axis (see

equations (A1), in.

U(X, Y, Z), V(X, Y, Z), displacements of beam material points (see equations

W(X, Y, Z) (A1)), in.

VY(X), VZ(X) beam transverse shearing forces (see figure 11), lb

(x, y, z) global plate coordinates (see figure 4), in.

(x ́, y ́, z ́) local coordinates of sandwich face plate (see figure 28), in.

(X, Y, Z) noncentroidal beam coordinates (see figure 6), in.

, stiffness-weighted eccentricities of nonhomogeneous stiffener

(see figure 6 and equation (A13)), in.

centroidal coordinates of a homogeneous beam with respect to

the (X, Y, Z) beam coordinates (see equations (A8) and 

figure 6), in.

, stiffness-weighted eccentricities of nonhomogeneous stiffener

(see figure 6 and equation (A13)), in.

 plate transverse shearing strains with respect to (x, y, z) 

coordinates (see equations (19) and figure 6)

 plate transverse shearing strains with respect to (X, Y, Z) 

coordinates (see equations (10) and figure 6)

 beam transverse shearing strains (see equations (11) and (A3))

εxx, εyy, γxy, γxz, γyz, εzz plate strains with respect to (x, y, z) coordinates (see equations

(17) and figure 6)

εXX, εYY, γXY, γXZ, γYZ, εZZ plate and beam strains with respect to (X, Y, Z) coordinates

(see equations (10) and figure 6)

 plate membrane strains with respect to (x, y, z) coordinates

(see equations (19) and figure 6)

yS yS

Y, Z

zS zS

γ xz

o

, γ yz

o

γXZ

o

, γYZ

o

ΓXY

o

, ΓXZ

o

εxx

o

, εyy

o

, γ xy

o
13

 plate membrane strains with respect to (X, Y, Z) coordinates

(see equations (10) and figure 6)

{εb} vector of beam strains (see equations (30))

{εp} vector of plate strains referred to (x, y, z) global plate

coordinates (see equations (34))

{εP} vector of plate strains referred to (X, Y, Z) beam coordinates

(see equations (30))

, eccentricities of nonhomogeneous stiffener (see equations

(A27) and (A29)), in.

, eccentricities of nonhomogeneous stiffener (see equations

(A27) and (A29)), in.

plate bending strains with respect to (x, y, z) coordinates 

(see equations (19) and figure 6), in-1 

plate bending strains with respect to (X, Y, Z) coordinates 

(see equations (10)), in-1 

σxx, σxy, σxz, σyy, σyz, σzz beam and plate stresses with respect to (x, y, z) coordinates

(see figure 6), psi

σXX, σXY, σXZ, σYY, σYZ, σZZ beam and plate stresses with respect to (X, Y, Z) coordinates

(see figure 10), psi

τo beam twisting strain (see equations (A2) and (11)), in-1

, inplane and transverse shear parameters (see equations (29))

νXY, νXZ, νYZ major principal Poisson’s ratios for a homogeneous specially

orthotropic material (see equation (A10))

νYX, νZX, νZY minor principal Poisson’s ratios for a homogeneous specially

orthotropic material (see equation (A10))

(ξ, η, ζ) stiffness-weighted centroidal coordinates of nonhomogeneous

beams and centroidal coordinates of homogeneous beams 

(see equations (A27) and figures 6 and 11), in.

φX(X), φY(X), φZ(X) dextral rotations of the beam cross-section about the X-, Y-, and

Z-axes, respectively (see equations (A1))

εXX

o

, εYY

o

, γXY

o

ζS ζS

ηS ηS

κxx

o

, κyy

o

, κxy

o

κ XX

o

, κ YY

o

, κ XY

o

τY

S

τZ

S
14

Φ angle of diagonal stiffeners (see figures 14, 16-18, 21 and 23),

degrees

beam bending strains (see equations (11) and (A2)), in-1

ΨS stiffener angle (see figures 4, 6, and 11), degrees

Introduction

Stiffened plates and shells are common structural forms used in the aerospace industry. For 

example, the Space Shuttle liquid-hydrogen tank utilizes a metallic "orthogrid" construction, as 

shown in figure 1. This type of structural arrangement has two families of uniformly spaced 

stiffeners that intersect at right angles. Stiffened metallic shells with this type of construction are 

often referred to as geometrically orthotropic because their overall equivalent extensional and 

bending stiffnesses are similar to that of a corresponding homogeneous orthotropic material when 

the stiffnesses of the shell skin and stiffeners are homogenized. A similar densely stiffened 

metallic orthogrid cylinder is shown in figure 2. Another common structural arrangement that is 

used for launch vehicles, known as an "isogrid," is shown in figure 3. This type of structural 

arrangement has three identical families of uniformly spaced stiffeners that intersect one another 

at a 60-degree angle to yield a geometrically isotropic shell wall. For this type of metallic 

stiffened shell, the homogenized extensional and bending stiffnesses are like that of an isotropic 

material.

Approximating the overall behavior of stiffened plate and shell structures by using 

homogenized, "equivalent" stiffnesses and "effective" thicknesses has been in use for many years 

(e.g., see references 1-8). In recent times, these plate and shell stiffness representations still see 

utility in early stages of building-block analysis and design approaches that have evolved and are 

used extensively by industry. These simplified approaches are particularly useful for navigating 

the design space rapidly to identify the "first cut" of optimal preliminary designs, which are 

particularly important to the design of lightweight, high-performance launch vehicles such as the 

ARES V, previously under development by NASA. 

The earliest works that utilize equivalent stiffnesses to analyze the behavior of stiffened plates 

appear to be that of Huber.1-3 Originally, this approach was driven by a lack of analysis methods 

that could simulate adequately the discrete nature of stiffened structures and the availability of 

analytical solutions for bending and buckling of homogeneous orthotropic plates. Similarly, the 

earliest corresponding works for stiffened shells appears to be that published by Flugge4

 in 1932 

and 15 years later by van der Neut.9

 As time passed and physical understanding matured, theories 

for analyzing stiffened structures by using equivalent stiffnesses and thickness continued to 

receive attention. For example, Dale and Smith7 used equivalent stiffnesses to study the buckling 

behavior of compression-loaded sandwich plates with an orthogrid core. In 1946, Smith et. al.8

presented an improved formulation that accounts for variations in the neutral-surface position 

associated with local interactions between a plywood plate and a stiffener. Similarly, in 1947, 

χY

o

, χZ

o
15

Pfluger10 presented an improved theory and applied it to the buckling of stiffened plates. Pfluger’s 

work is considered an improvement to Huber’s much earlier work because it includes a more 

accurate treatment of the shearing stresses. One of the earliest works that uses an effective plate 

thickness for designing stiffened plates was given in 1948 by Gomza & Seide.11 In their study, 

the effective plate thickness is obtained by adding the stiffener cross-sectional area, divided by 

the stiffener spacing, to the plate thickness. An early work that developed equivalent-plate 

stiffnesses for an isotropic corrugated sheet sandwiched between two isotropic flat skins was 

presented by Libove and Hubka12 in 1951. Basically, formulas for equivalent-plate elastic 

constants were presented, for use with available symmetric-sandwich plate theories13-15 that 

include inplane dilatation and shearing, pure bending and twisting, and transverse shearing. 

Formulas for equivalent-plate elastic constants associated with coupling between inplane 

dilatation and pure bending and between inplane shearing and twisting were also presented in 

anticipation of the extension of sandwich plate theories to unsymmetric sandwich constructions. 

Experimental results were reported that were characterized as being in "close agreement" with the 

theoretical predictions for the bending and transverse-shear stiffnesses in the direction 

perpendicular to the corrugations and for the twisting stiffness. Later, in 1952, Benscoter and 

MacNeal16 presented an equivalent-plate theory, based on first-order difference equations, for a 

thin multicell low aspect ratio, supersonic wing with straight spars and perpendicular ribs that 

includes transverse-shear deformations. Similarly, Horvay17 presented an equivalent-plate 

formulation for a plate-like grid with a stiffener arrangement forming honeycomb cells in 1952.

What may be the earliest work to provide expressions for equivalent-plate stiffnesses of 

isotropic plates with integral stiffeners was presented by Dow, Libove, and Hubka.18 In this early 

work, 12 independent elastic constants were derived, following a laborious approach, that 

correspond to the A11, A12, A22, A66, B11, B12, B22, B66, D11, D12, D22, and D66 terms appearing in the 

contemporary constitutive equations for laminated-composite plates (see reference 19) that relate 

stress resultants to membrane strains and bending curvatures. The expressions given for the elastic 

constants were obtained by identifying the fundamental repeating element of the stiffened plate 

and then replacing each stiffener in the repeating element with a homogeneous orthotropic plate 

that is perfectly bonded to the skin of the stiffened plate. The strains in the repeating-element 

stiffeners are related to the corresponding plate strains and the strain energy of the repeating 

element is determined in terms of the equivalent-plate strains. Equations relating the equivalentplate strains and the equivalent-plate stress resultants are obtained by differentiating the strain 

energy, which yield the equivalent-plate stiffnesses. Several expressions for an average or 

equivalent-plate thickness were also given that depend on the stiffener arrangement. A somewhat 

related study that focused on the torsional stiffness of orthogonally stiffened plates was presented 

by Crawford and Libove.20 Around the same time period, Hoppmann and his colleagues21-24

conducted experiments to determine the bending and twisting stiffnesses of orthogonally stiffened 

plates, and used these stiffnesses to perform calculations for static bending and vibration, based 

on an orthotropic-plate theory. In 1956, Huffington25 published an analysis for determining the 

equivalent-plate stiffnesses for orthogonally stiffened plates without stiffener eccentricity, with 

respect to the plate skin. Later, in 1957, Bodner26 analyzed buckling of ring-stiffened cylinders 

subjected to hydrostatic pressure by using an equivalent shell-wall thickness and an effective 

moment of inertia for the rings and shell wall combination. In his analysis, the rings were 

presumed to contribute only to the circumferential membrane and bending stiffnesses, and 
16

buckling loads were obtained by treating the stiffened shell as an equivalent orthotropic shell that 

is nonhomogeneous in the radial direction. An effective length between rings was used in the 

calculation of the bending stiffness such that the maximum circumferential stress can be 

computed by using basic ring theory. The direction of ring eccentricity, with respect to the shell 

wall, is not included in the analysis.

During the last fifty years, the equivalent-plate or shell approach was still being used as a firstapproximation method to understand the behavior of eccentrically stiffened and sandwich plates 

and shells, and in simulating the behavior of massive beam-like, plate-like, and shell-like 

reticulated orbiting space structures whose discrete-member analysis was beyond the capability 

of the computing resources available at that time.27-138 For example, Stroud30 presented a derivation 

of five elastic constants associated with pure bending and twisting of corrugation-stiffened panels 

in 1963. These constants were obtained by examining the force-deformation characteristics of a 

repeating corrugation-plate-skin cell. Results of experiments conducted to determine the elastic 

constants showed favorable agreement with the proposed theory. In 1964, Meyer & Bellefante31

presented equivalent-plate elastic constants for inplane dilatation and shearing and pure bending 

of a skin stiffened by an array of stiffeners that enclose equilateral triangles. These constants were 

also obtained by examining the force-deformation characteristics of a repeating cell. Equivalentshell stiffnesses were presented by Sewall, Clary, and Leadbetter32 in 1964 for cylinders with an 

orthogonal arrangement of rings and stringers sandwiched between inner and outer skins. In this 

work, the rings are presumed to make a negligible contribution to the longitudinal inplane and 

bending stiffnesses, the stiffness associated with coupling between longitudinal-circumferential 

dilatation and coupling between longitudinal-circumferential bending, the inplane shearing 

stiffness, and the twisting stiffness. Similarly, the stringers are presumed to make a negligible 

contribution to the circumferential inplane and bending stiffnesses. In contrast, the stringers are 

presumed to contribute to the coupling between longitudinal-circumferential dilatation, the 

coupling between longitudinal-circumferential bending, the inplane shearing stiffness, and the 

twisting stiffness. Block, Card, Mikulas, McElman, and Stein37, 38, 40, 41 presented and applied 

equivalent-shell stiffnesses for orthotropic ring-stiffened corrugated cylinders and ring-andstringer-stiffened cylinders in 1965 and 1966. In this series of papers, a relatively simple strainenergy approach was used to determine stiffnesses that include inplane dilatation and shearing, 

pure bending and twisting, and coupling between inplane dilatation and pure bending and between 

inplane shearing and twisting associated with stiffener eccentricity with respect to shell wall midsurface. Transverse shearing stiffnesses were not considered. Also in 1966, Singer et.al.42

published stiffener expressions similar to those given by Block, Card, Mikulas, McElman, and 

Stein for cylinders with eccentric orthogonal rings and stringers. Their stiffness expressions are 

based on the presumption that the normal strains vary linearly in the skin and stiffeners and are 

equal at the point of contact between the skin and a stiffener. In addition, the stiffeners do not 

contribute to the inplane shearing stiffness, and the torsional stiffness is obtained by adding the 

torsional stiffness of the stiffeners to that of the skin. In 1968, Jones49, 51 presented stiffness 

expressions similar to those given in references 36-39, but for laminated-composite cylinders. 

This work appears to be the first to express the equivalent-shell stiffnesses in the terminology that 

has become standard practice for laminated composites in a large portion of the world. In 1969, 

Soong52-54 presented a derivation of the stiffness expressions similar to those given in references 

37, 38, 40, and 41, based on a strain-energy approach, but for orthotropic cylinders reinforced 

with a balanced pair of spiral stiffeners that make an arbitrary angle with the cylinder generators. 
17

Attempts were made in the 1970s to refine the equivalent-plate stiffnesses for isotropic plates 

and shells with eccentric orthogonal stiffeners. For example, Cusens et.al.58 presented an analysis 

that accounts for the stiffener contributions to the stiffnesses associated with coupling between 

inplane stretching and between anticlastic bending. In their analysis, the coupling effects are 

weighted by the size of the area of contact at the stiffener intersections. Similarly, Nishino et.al.63

modeled the interaction of the shearing stresses between a plate and an eccentrically located 

orthogonal stiffening grid and obtained equivalent-plate twisting stiffnesses that include the 

effects of inplane shearing deformations of the stiffeners. Likewise, Nemeth81 derived equivalentplate stiffnesses for single-layer grids made of beam members with inplane and out-of-plane 

(transverse) shear flexibility in 1979. Stiffness expressions were presented in this study for 

several grid configurations and in-depth comparisons with corresponding results obtained from 

discrete finite element models of the grids are given. In 1979 and 1980, Ko82-84 presented 

equivalent stiffnesses for an isotropic corrugated sheet sandwiched between two isotropic flat 

skins in a symmetric manner, and for a similar symmetric honeycomb-core sandwich plate. These 

stiffnesses are modified forms of the corresponding stiffnesses presented by Libove and Hubka12

in 1951, that account for corrugated walls with nonuniform thickness, and include the effects of 

transverse-shear deformations. This approach was extended by Ko to hat-stiffened panels in 

1991.108 Later, in 1985, Reddy et.al.95 presented equivalent stiffnesses for a circular cylindrical 

shell stiffened by an internal grid. In particular, symmetrically laminated shell walls stiffened 

with rings, stringers, and a pair of identical helical stiffeners making an arbitrary angle with the 

shell generators were considered. Expressions are given that include transverse-shear stiffnesses 

in addition to stiffnesses that account for inplane dilatation and shearing, pure bending and 

twisting, and coupling between inplane dilatation and pure bending and between inplane shearing 

and twisting associated with stiffener eccentricity with respect to shell wall mid-surface. Also in 

1985, Kolpakov97 presented a method for determining the equivalent stiffnesses of elastic 

frameworks that includes an analysis of a planar beam gridwork with an overall negative-valued 

Poisson’s ratio. Deb and Booton102 presented similar equivalent-plate stiffnesses in 1988 for shear 

deformable, orthogonally stiffened isotropic plates with eccentric stiffeners. Also, in 1988, Boot 

& Moore103 presented a detailed list of the factors affecting the validity of equivalent-plate 

stiffnesses, and examined the importance of neglecting the contribution of the stiffeners to the 

coupling between biaxial stretching, for orthotropic plates. In 1989, Bunakov and Protasov105

presented equivalent-continuum stiffnesses for a pair of identical helical stiffeners with a 

rectangular-cross-section attached to a shell. These stiffnesses are based on a micropolar 

continuum model and include transverse-shear and bending stiffnesses of the beam members that 

are associated with deformations within the tangent plane at each point of the shell. In 1990, 

Won106 presented equivalent-plate stiffnesses for isotropic plates reinforced with eccentric, 

regularly spaced pairs of oblique stiffeners. In his analysis, a uniform biaxial stress state is 

presumed to exist at the stiffener joints and is used to obtain a stiffener contribution to equivalentplate stiffness associated with "Poisson coupling" between biaxial deformations. In addition, 

axial stresses in the stiffeners are presumed to develop that resist plate-like inplane shearing and 

twisting deformations, which leads to additional equivalent-plate stiffness contributions. Three 

coupled partial differential equations that govern the bending response are derived, in terms of the 

inplane and out-of-plane displacements, by minimizing the potential energy of the plate-stiffener 

system. An approximate "Huber-type" differential equation (see references 1-3) is also given, in 

terms of the out-of-plane displacement, for an orthotropic plate. This simplified equation, given 
18

in terms of the out-of-plane displacement, represents bending about an "equivalent," inextensible 

neutral surface. The Huber-type equation is obtained by solving the equivalent-plate constitutive 

equations for the inplane stress resultants in terms of the membrane strains, and then substituting 

the resulting expressions into the equivalent-plate constitutive equations for the bending stress 

resultants. This step yields the bending stress resultants in terms of the out-of-plane displacement 

and second-order derivatives of the inplane stress resultants. The approximate "Huber-type" 

differential equation is obtained by substituting the equivalent-plate bending stress resultants into 

the remaining out-of-plane equilibrium equation and then neglecting the second-order derivatives 

of the inplane stress resultants.

 In 1993, Pshenichnov112 published a monograph dealing with reticulated plates and shells, with 

an emphasis on single-layer plate-like and shell-like lattice structures in which the stiffeners are 

not eccentric with respect to the shell middle surface. The equivalent stiffnesses presented are 

based on a classical shell theory (no transverse shear flexibility) and are obtained by using tensor 

transformations to equate beam strains with corresponding shell strains and by equating shell 

stress resultants with transformed beam forces that are uniformly distributed across an equivalent 

shell wall. Although the analysis is based on a classical shell theory, an attempt is made to include 

the effects of stiffener bending in the tangent plane by expressing the beam shearing forces that 

act in the tangent plane in terms of the derivatives of the corresponding beam moments. These 

tangential beam moments are expressed in terms of the beam bending strain, in the usual way, but 

the beam bending strain is obtained in terms of the shell tangential displacements and strains by 

considering deformation associated with rotation about the unit vector normal to the middle 

surface. Although this approach captures tangential stiffener bending effects, the effects cannot 

be represented directly in terms of the shell strains and, as a result, do not enter into the equivalent 

stiffness expressions for plate-like and shell-like lattices.

In 1995, Jaunky et. al.116, 117 presented a refined smeared-stiffener theory for grid-stiffened 

laminated-composite panels, based upon the earlier work presented by Smith et.al.8 for a plywood 

plate with a single central stiffener that has a rectangular cross-section. The refinement presented 

in their work accounts for the variation of the neutral surface caused by interactions between the 

skin and the stiffeners. Results presented by these authors show more accurate equivalent-plate 

stiffness predictions for selected cases, but the analysis used to obtain the equivalent-plate 

stiffnesses is far more involved than the earlier, less refined approach used in earlier works such 

as references 37, 38, 40, and 41. Equivalent-shell stiffnesses for circular cylinders made of 

laminated-composite materials and with rings, stringers, and a pairs of identical geodesic 

stiffeners were presented by Gerhard et.al.118 in 1996 (see pp. 56-75). These equivalent-shell 

stiffnesses were obtained by using strain transformation equations to express the stiffener strains 

in terms of the equivalent-plate strains, and by using force transformation equations to relate the 

force in each stiffener to the equivalent-shell stress resultants. Equivalent stiffnesses for 

laminated-composite flat plates and circular cylindrical shells stiffened by a grid of beams were 

presented by Chen and Tsai119 in 1996. In their study, generally laminated walls stiffened with 

ribs, stringers, and a pair of identical diagonal stiffeners with an arbitrary orientation angle were 

considered. Grid-stiffness expressions are given that include out-of-plane (transverse) and 

inplane shear flexibility of the stiffeners and inplane stiffener bending in addition to the usual 

stiffnesses that account for inplane dilatation and shearing and pure bending and twisting. 

Similarly, Wodesenbet et. al.133 presented an improved smeared-stiffener theory for isogrid-
19

stiffened laminated-composite cylinders in 2003. In this theory, the stiffness contributions of the 

stiffeners are obtained by relating the beam strains to the plate strains with the standard straintransformation equations, and by expressing the shell stress resultants in terms of the beam forces, 

and their moments, that act on each stiffener component within a unit cell (repeating element). 

The equivalent-shell stiffnesses are then obtained by using a superposition of the stress resultant 

for the skin and the stress resultants for the stiffeners within the unit cell that is weighted by the 

corresponding volume fractions for the unit cell.

The literature examined during the present study suggests that treating stiffened plates as an 

equivalent-continuum plate remains a useful design practice, provided the limitations of the 

theory are kept in mind. The surveyed results also indicate that, to a large extent, the derivations 

of equivalent-continuum plate stiffnesses have been ad hoc, the terminology is antiquated, and 

transverse-shear deformations are often neglected. Moreover, the criteria for defining an effective 

thickness of the equivalent-continuum plate appears to be unclear. The major objective of the 

present study is to present two systematic methods for deriving first-approximation, equivalentcontinuum stiffnesses for eccentrically stiffened plates, based on a first-order transverse-shear 

deformation plate theory (e. g., see reference 139). The first method uses equilibrium and 

compatibility in a direct manner for plates reinforced by one or more families of rectilinear 

stiffeners. The second method is closely related and is based on using a basic, repetitive cell of 

the stiffened plate and then defining an equivalence between the strain energy of the basic cell and 

the corresponding equivalent plate. This method is particularly useful for stiffener arrangements 

that are not rectilinear. A second objective is to present a systematic set of equations that can be 

used to determine the equivalent thickness of stiffened plates. Toward these objectives, the 

analysis approach and details for the direct equilibrium-compatibility method are presented first 

for plates stiffened by one or more families of continuous rectilinear stiffeners. This first section 

includes discussions of how statical and kinematical equivalence is achieved between the 

stiffened plate and the equivalent-continuum plate, and equivalent-plate stiffnesses are given for 

several stiffening arrangements. Comparisons of the equivalent-plate stiffnesses with 

corresponding previously published results are also given. Next, the details of the energyequivalence approach are presented, and the method is applied to obtain equivalent-plate 

stiffnesses for several stiffening arrangements. Additional comparisons of the equivalent-plate 

stiffnesses with corresponding previously published results and with the stiffnesses obtained 

herein by using the direct equilibrium-compatibility method are also given. Then, additional 

analysis is presented that shows how to obtain equivalent-plate stiffnesses for sandwich plates 

with two nonidentical, generally laminated face plates and a core made from several arrangements 

of beam stiffeners. Equations are also presented that show several different criteria that can be 

used to select an equivalent thickness of the equivalent-continuum plate, which may be needed to 

perform collateral design calculations.

Direct Equilibrium-Compatibility Method for Rectilinear Stiffener Families

Consider a perfectly flat plate of arbitrary shape, as depicted in figure 4a. Material points of 

the plate are located by the Cartesian coordinates (x, y, z), and z = 0 corresponds to the plate 

midplane. The plate is stiffened by a unidirectional family of identical prismatic beams that 

generally have nonhomogeneous cross-sections and that are equally spaced, as depicted in figure 
20

4b. An example of the plate cross-section A-A indicated in figure 4b is shown in figure 5. In this 

figure, the stiffener spacing is denoted by dS and the effective nonhomogeneous-stiffener 

eccentricity, defined in Appendix A, is given by . The stiffeners are also presumed to be 

perfectly bonded to a laminated-composite skin with thickness h, and make an angle ΨS with the 

x-axis of the plate. Material points of the beam are located by the local, noncentroidal Cartesian 

coordinates (X, Y, Z), and Z = 0 also corresponds to the plate midplane. The geometric 

relationship between the plate and local beam coordinates is shown in figure 6. As indicated in 

Appendix A, the coordinates of the centroidal axis are given by , with respect to the 

(X, Y, Z) coordinate frame. Similarly, the coordinates of the stiffness-weighted centroidal axis of 

nonhomogeneous beams are given by , with respect to the (X, Y, Z) coordinate frame. 

The additional (ξ, η, ζ) coordinate axes shown in figure 6 are used to locate points of the beam 

with respect to the stiffness-weighted centroid.

The stiffeners are modeled in the present study with the Timoshenko-type first-order sheardeformation beam theory presented in Appendix A, based on the local noncentroidal Cartesian 

coordinates (X, Y, Z). In addition, the beam material is presumed to be specially orthotropic with 

respect to the (X, Y, Z) coordinate frame, but may be nonhomogeneous in the cross-sectional 

planes. In particular, the principal axes of orthotropy are aligned with the (X, Y, Z) coordinate 

axes. Moreover, the effects of cross-sectional warping restraint associated with torsion of 

noncircular cross-sections are neglected in the kinematic equations and each cross section is 

presumed to warp in an identical manner. Beam constitutive equations are also presented in 

Appendix A for the nonhomogeneous specially orthotropic material in terms of effective 

engineering constants. Using a nonhomogeneous beam theory permits the modeling of tailored 

beam stiffeners such as those depicted in figure 7. Based on Appendix A, the nonhomogeneous 

beam stiffener is assigned an effective axial modulus ES, an effective shear modulus GS, crosssectional area AS, effective moments of inertia and , effective product of inertia , and 

effective torsional constant JS, in addition to the effective eccentricities , , and . 

Effective shear correction factors for the beam are denoted by and .

For the structural arrangement depicted by figures 4-6, the repetitive element shown in figure 

8 is used herein to represent the essential features of the structure. The stiffener in this repetitive 

element is positioned so that the effective eccentricity . Although a T-shaped stiffener is 

shown in figure 8, the stiffener may have a nonhomogeneous cross-section with an arbitrary 

shape. In the analysis that follows, the repetitive stiffened-plate element is modeled as the 

equivalent-plate element shown in figure 9, which consists of a laminated-composite wall bonded 

perfectly to a single, equivalent-stiffener layer. The reference surface of the equivalent plate is 

selected as the midplane of the plate wall, for convenience. The equivalent-stiffener layer is 

presumed to contribute axial stretching and bending stiffness, inplane shear stiffness, and twisting 

stiffness in the Y-Z plane, consistent with the classical continuum model of deformation. The 

approach that is followed in this section is to first establish an equivalence between the internal 

forces acting on the repetitive stiffened-panel element and the equivalent-plate element. This 

equivalence is referred to herein as statical equivalence and is based on the presumption that the 

zS

X, Y, Z

X, yS, zS

IYY

S

IZZ

S

IYZ

S

yS zS yS zS

k Y

s k Zs

yS = 0
21

variation of stresses across the width of the equivalent-stiffener layer can be neglected for 

relatively small stiffener spacings. The beam stresses and corresponding resultant forces and 

moments, defined relative to the (X, Y, Z) coordinates, that are considered are shown in figures 

10 and 11, respectively. Similarly, the beam forces and moments acting on the repetitive 

stiffened-plate element and the corresponding distributed stress resultants acting on the 

equivalent-plate element are shown in figures 12 and 13, respectively. Then, an equivalence 

between the stiffener strains and the corresponding strains in the equivalent-stiffener layer is 

established, which is referred to herein as kinematical equivalence. The kinematical equivalence 

is based on the presumption that the strain at any point of a stiffener is identical to the 

corresponding strain at the corresponding point in the equivalent-stiffener layer of the equivalentplate element. In addition, it is presumed that the variation of strains across the width of the 

equivalent-stiffener layer can be neglected for relatively small stiffener spacings. With these 

equivalences established, the stiffener-force contributions to the equivalent-plate stress resultants 

are expressed in terms of the equivalent-plate strains, and the corresponding equivalent-plate 

constitutive equations are determined in terms of the plate wall and stiffener properties.

Statical Equivalence

 To establish statical equivalence between the repetitive stiffened-panel and equivalent-plate 

elements, consider the definition of axial stress resultant for the equivalent plate in the X-Y-Z 

coordinate system associated with the stiffener, given by

 (1)

where σXX is the axial stress, h is the plate thickness, and hS is the thickness of the equivalentstiffener layer shown in figure 9. The integration is partitioned to obtain 

 (2)

where

 (3)

Here, refers to the usual definition of the stress resultant for a plate that is given by the first 

integral in equation (2). Using the definition for the axial beam load, P(X), defined by equation 

(A5a) in Appendix A and applying it to the equivalent-stiffener layer of the equivalent-plate 

element, with the presumption that the variation of σXX across the width of the equivalentstiffener layer is negligible, gives

N XX = σ XX dZ − h

2

h

2 + h S

N XX = σ XX dZ − h

2

h

2

+ σ XX dZ h

2

h

2 + h S

≡ N XX

plate

+ N XX

stiffener

N XX

stiffener

= σ XX dZ h

2

h

2 + h S

N XX

plate
22

 (4)

Using equations (3) and (4) gives 

 (5)

and equation (2) gives

 (6)

where dS is the stiffener spacing (see figure 5). Following a similar process of partitioning the 

integrations and relating the two stiffener coordinate systems, the remaining stress resultants 

acting on the edge shown in figure 13 are expressed as

 (7a)

 (7b)

 (7c)

 (7d)

Likewise, applying the definitions of the beam forces VY(X) and VZ(X) and beam moments MY(X) 

and T(X) defined in Appendix A (see figure 11) to the equivalent-stiffener layer of the equivalentplate element and neglecting the variation of the stresses across the width of the equivalentstiffener layer gives

 (8a)

Similarly, 

 (8b)

 (8c)

P = σ XX dYdZ

A S

= dS σ XX dZ h

2

h

2 + h S

N XX

stiffener

= P

dS

N XX = N XX

plate

+ P

dS

N XY = σ XY dZ − h

2

h

2

+ σ XY dZ h

2

h

2 + h S

≡ N XY

plate

+ N XY

stiffener

MXX = σ XXZ dZ − h

2

h

2

+ σ XXZ dZ h

2

h

2 + h S

≡ MXX

plate

+ MXX

stiffener

MXY = σ XYZ dZ − h

2

h

2

+ σ XYZ dZ h

2

h

2 + h S

≡ MXY

plate

+ MXY

stiffener

QXZ = σ XZ dZ − h

2

h

2

+ σ XZ dZ h

2

h

2 + h S

≡ QXZ

plate + QXZstiffener

N XY

stiffener

= VY

dS

MXX

stiffener

= MY

dS

MXY

stiffener

= − T

dS
23

 (8d)

With these results, equations (7a) - (7d) yield

 (9a)

 (9b)

 (9c)

 (9d)

It is worth noting that the beam moment MZ(X), given by equation (A5f), vanishes when the 

variation of σXX across the width of the equivalent-stiffener layer is neglected, consistent with a 

plate theory based on a classical continuum model.

Kinematical Equivalence

The next step in the analysis is to establish kinematical equivalence so that the stiffener strains 

can be related to the appropriate plate strains in the equivalent-stiffener layer of the equivalentplate element. The general expressions for the strains in a plate, modeled with first-order 

transverse-shear deformation plate theory, are given by (e. g., see reference 139)

 (10a)

 (10b)

 (10c)

 (10d)

 (10e)

 (10f)

QXZ

stiffener

= VZ

dS

N XY = N XY

plate

+

VY

dS

MXX = MXX

plate

+

MY

dS

MXY = MXY

plate

− T

dS

QXZ = QXZ

plate

+

VZ

dS

εXX X, Y, Z = εXX

o

X, Y + Z κ XX

o

X, Y

εYY X, Y, Z = εYY

o

X, Y + Z κ YY

o

X, Y

γXY X, Y, Z = γXY

o

X, Y + Z κ XY

o

X, Y

γXZ X, Y, Z = γXZ

o

X, Y

γYZ X, Y, Z = γYZ

o

X, Y

εZZ X, Y, Z = 0
24

These strains are defined with respect to the reference surface (plate wall midplane) of the 

equivalent-plate element shown in figure 13. Moreover, and are the extensional strains 

of the plate midplane and is the corresponding inplane shearing strain. Likewise, and 

 are bending strains of the plate midplane and is the corresponding twisting strain. The 

symbols and denote the transverse shearing strains. The kinematical equivalence is 

obtained based on the presumption that the strain at any point of a stiffener is identical to the strain 

at the corresponding point in the equivalent-stiffener layer of the equivalent-plate element. In 

addition, it is presumed that bending of the stiffener in the plane parallel to the plate midplane is 

negligible and, as a result, the variation of strains across the width of the equivalent-stiffener layer 

can be neglected. Furthermore, it is presumed that the eccentric stiffener contributes only half of 

the inplane shearing strain and half of the change in surface twist of the equivalent-stiffener layer. 

This presumption is rationalized by noting that the inplane shearing strain and the change in 

surface twist of a plate are deformation measures that have contributions associated with both 

cross-sectional faces of a differential plate element, and that the shearing and twisting 

deformations of the stiffener only act on one face of the equivalent-stiffener layer of the 

equivalent-plate element. Therefore, this last presumption represents an averaging of the stiffener 

contribution to the corresponding overall plate strains. Thus, the kinematical equivalence yields 

the following expressions

 (11a)

 (11b)

 (11c)

 (11d)

 (11e)

 (11f)

In equations (11), is the change in twist of the beam, associated with torsion, and and 

are the beam bending strains associated with the changes in curvature in the X-Z and X-Y planes, 

respectively. The symbols and are the transverse shearing strains in the X-Y and X-Z 

planes, respectively. Substituting these expressions for the strains into the stiffener constitutive 

equations, given by equations (A19)-(A21) and (A25) in Appendix A, gives

 (12a)

εXX

o

εYY

o

γXY

o

κ XX

o

κ YY

o

κ XY

o

γXZ

o

γYZ

o

eXX

o

X = εXX

o

X, Y

χZ

o

X = 0

χY

o

X = κ XX

o

X,Y

τ

o

X = − 1

2κ XY

o

X,Y

ΓXY

o

X = 1

2γXY

o

X, Y

ΓXZ

o

X = γXZ

o

X, Y

τ

o

χY

o

χZ

o

ΓXY

o

ΓXZ

o

P = ESAS εXX

o

+ zS κ XX

o
25

 (12b)

 (12c)

 (12d)

 (12e)

 (12f)

Substituting equations (12) into equations (5) and (8) yields

 (13a)

 (13b)

 (13c)

 (13d)

 (13e)

Equations (13) give the contributions of the stiffeners to the pointwise equivalent-plate stress 

resultants, in the X-Y-Z coordinate system, in terms of the corresponding plate strains. However, 

another issue must be considered. In particular, the shearing stress resultant, Nxy, and twisting 

stress resultant, Mxy, of classical and first-order transverse-shear deformation plate theories are 

based on the symmetries Nxy = Nyx and Mxy = Myx of the corresponding stress resultants acting 

on the cross-sectional faces of a differential plate element. The stiffener is presumed to provide 

no inplane shear stiffness and twisting stiffness to the equivalent-stiffener layer in the inplane 

direction parallel to its axis (e.g., see reference 30, p. 30). Therefore, with regard to the 

constitutive equations, equations (7a) and (7c) must be modified to yield

VY = k Y

S

GSAS

2 γXY

o

+ zSκ XY

o

VZ = k Z

S

GSAS γXZ

o

T = − 1

2 k Y

S

GSASzS γXY

o

+ GSJS κ XY

o

MY = ESASzS εXX

o

+ ESIYY

S

κ XX

o

MZ = ESIYZ

S

κ XX

o

N XX

stiffener

= ESAS

dS

εXX

o

+ zS κ XX

o

N XY

stiffener

= k Y

S

GSAS

2dS

γXY

o

+ zS κ XY

o

MXX

stiffener

= ESASzS

dS

εXX

o

+

ESIYY

S

dS

κ XX

o

MXY

stiffener

= k Y

S

GSASzS

2dS

γXY

o

+

GSJS

2dS

κ XY

o

QXZ

stiffener

= k Z

S

GSAS

dS

γXZ

o
26

 (14a)

 (14b)

These two equations represent an averaging of the stiffener contribution over both cross-sectional 

faces of a differential equivalent-plate element. The constitutive equations of the equivalentstiffener layer become

 (15a)

 (15b)

 (15c)

where it is noted that matrices associated with coupling between inplane and out-of-plane deformations are identical in both equations. It is important to also note that the stiffnesses occupying 

the third row and third column of the matrices in equation (15a) are associated with transverseshear deformations of the beam member in a plane parallel to the plate midplane. As such, these 

stiffnesses are referred to herein as "inplane transverse-shearing stiffnesses." In contrast, the stiffness occupying the second row and second column of the matrix in equation (15c) is associated 

with transverse-shear deformations of the beam member in the direction perpendicular to the plate 

midplane and is referred to herein as an "out-of-plane transverse-shearing stiffness."

The next step in the analysis is to relate the plate stress resultants in the X-Y-Z coordinate 

system to the x-y-z coordinate system shown in figure 6. Using the standard stress and strain 

transformation equations for the rotation of coordinates depicted in figure 6 gives

N XY = N XY

plate

+ 1

2N XY

stiffener

MXY = MXY

plate

+ 1

2MXY

stiffener

NXX

stiffener

NYY

stiffener

1

2NXY

stiffener

=

ESAS

dS

0 0

0 0 0

0 0 k Y

S

GSAS

4dS

εXX

o

εYY

o

γXY

o

+

ESASzS

dS

0 0

0 0 0

0 0 k Y

S

GSASzS

4dS

κ XX

o

κ YY

o

κ XY

o

MXX

stiffener

MYY

stiffener

1

2MXY

stiffener

=

ESASzS

dS

0 0

0 0 0

0 0 k Y

S

GSASzS

4dS

εXX

o

εYY

o

γXY

o

+

ESIYY

S

dS

0 0

0 0 0

0 0 GSJS

4dS

κ XX

o

κ YY

o

κ XY

o

QYZ

stiffener

QXZ

stiffener =

0 0

0 k Z

S

GSAS

dS

γYZ

o

γXZ

o
27

 (16a)

 (16b)

with

 (16c)

and

 (16d)

Additionally,

 (17a)

 (17b)

with

 (17c)

Based on the standard definitions of plate stress results, it follows that

 (18a)

σ XX

σ YY

σ XY

=

cos2

ΨS sin2ΨS 2 sinΨScosΨS

sin2ΨS cos2ΨS – 2 sinΨScosΨS

– sinΨScosΨS sinΨScosΨS cos2ΨS – sin2ΨS

σxx

σyy

σxy

≡ Tσ

σxx

σyy

σxy

σ YZ

σ XZ

= cosΨS − sinΨS

sinΨS cosΨS

σyz

σxz

≡ Tτ

σyz

σxz

Tσ

− 1

=

cos2

ΨS sin2ΨS − 2 sinΨScosΨS

sin2ΨS cos2ΨS 2 sinΨScosΨS

sinΨScosΨS − sinΨScosΨS cos2ΨS – sin2ΨS

Tτ

− 1

= cosΨS sinΨS

− sinΨS cosΨS

εXX

εYY

γXY

=

cos2

ΨS sin2ΨS sinΨScosΨS

sin2ΨS cos2ΨS sinΨScosΨS

– 2 sinΨScosΨS 2 sinΨScosΨS cos2ΨS – sin2ΨS

εxx

εyy

γ xy

≡ Tε

εxx

εyy

γ xy

γYZ

γXZ

= Tτ

γ yz

γ xz

Tε

− 1

=

cos2

ΨS sin2ΨS − sinΨScosΨS

sin2ΨS cos2ΨS sinΨScosΨS

2 sinΨScosΨS − 2 sinΨScosΨS cos2ΨS – sin2ΨS

NXX

NYY

NXY

= Tσ

Nxx

Nyy

Nxy
28

 (18b)

 (18c)

From equations (10) and the corresponding equations in the x-y-z coordinate system, it follows 

from equations (17a) and (17b) that 

 (19a)

 (19b)

 (19c)

Now, using 

 (20a)

 (20b)

 (20c)

with equations (18) it follows that 

 (21a)

MXX

MYY

MXY

= Tσ

Mxx

Myy

Mxy

QYZ

QXZ

= Tτ

Qyz

Qxz

εXX

o

εYY

o

γXY

o

= Tε

εxx

o

εyy

o

γ xy

o

κ XX

o

κ YY

o

κ XY

o

= Tε

κxx

o

κyy

o

κxy

o

γYZ

o

γXZ

o = Tτ

γ yz

o

γ xz

o

NXX

NYY

NXY

=

NXX

plate

NYY

plate

NXY

plate

+

NXX

stiffener

NYY

stiffener

1

2NXY

stiffener

MXX

MYY

MXY

=

MXX

plate

MYY

plate

MXY

plate

+

MXX

stiffener

MYY

stiffener

1

2MXY

stiffener

QYZ

QXZ

= QYZ

plate

QXZ

plate + QYZ

stiffener

QXZ

stiffener

Nxx

Nyy

Nxy

= Tσ

− 1

NXX

plate

NYY

plate

NXY

plate

+ Tσ

− 1

NXX

stiffener

NYY

stiffener

1

2NXY

stiffener

=

Nxx

plate

Nyy

plate

Nxy

plate

+ Tσ

− 1

NXX

stiffener

NYY

stiffener

1

2NXY

stiffener
29

Similarly,

 (21b)

 (21c)

Substituting equations (19) into equations (15) and then substituting the results into equations (21) 

yields 

 (22a)

 (22b)

where

 (23a)

 (23b)

 (23c)

Mxx

Myy

Mxy

=

Mxx

plate

Myy

plate

Mxy

plate

+ Tσ

− 1

MXX

stiffener

MYY

stiffener

1

2MXY

stiffener

Qyz

Qxz

= Qyz

plate

Qxz

plate + Tτ

− 1 QYZ

stiffener

QXZ

stiffener

Nxx

Nyy

Nxy

Mxx

Myy

Mxy

=

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

εxx

o

εyy

o

γ xy

o

κxx

o

κyy

o

κxy

o

Qyz

Qxz

= A44 A45

A45 A55

γ yz

o

γ xz

o

A11 A12 A16

A12 A22 A26

A16 A26 A66

=

A11

plate A12plate A16plate

A12

plate A22plate A26plate

A16

plate A26plate A66plate

+

A11

stiffener A12stiffener A16stiffener

A12

stiffener A22stiffener A26stiffener

A16

stiffener A26stiffener A66stiffener

B11 B12 B16

B12 B22 B26

B16 B26 B66

=

B11

plate B12plate B16plate

B12

plate B22plate B26plate

B16

plate B26plate B66plate

+

B11

stiffener B12stiffener B16stiffener

B12

stiffener B22stiffener B26stiffener

B16

stiffener B26stiffener B66stiffener

D11 D12 D16

D12 D22 D26

D16 D26 D66

=

D11

plate D12plate D16plate

D12

plate D22plate D26plate

D16

plate D26plate D66plate

+

D11

stiffener D12stiffener D16stiffener

D12

stiffener D22stiffener D26stiffener

D16

stiffener D26stiffener D66stiffener
30

 (23d)

with

 (24a)

 (24b)

 (24c)

 (24d)

The constitutive terms with the superscript "plate" are the usual stiffnesses of the ReissnerMindlin-type shear-deformation plate theory that are found in reference 139. The explicit 

expressions for the stiffnesses associated with the stiffeners are given by

 (25a)

 (25b)

A44 A45

A45 A55

= A44

plate A45plate

A45

plate A55plate +

A44

stiffener A45stiffener

A45

stiffener A55stiffener

A11

stiffener A12stiffener A16stiffener

A12

stiffener A22stiffener A26stiffener

A16

stiffener A26stiffener A66stiffener

= Tσ

− 1

ESAS

dS

0 0

0 0 0

0 0 k Y

S

GSAS

4dS

Tε

B11

stiffener B12stiffener B16stiffener

B12

stiffener B22stiffener B26stiffener

B16

stiffener B26stiffener B66stiffener

= Tσ

− 1

ESASzS

dS

0 0

0 0 0

0 0 k Y

S

GSASzS

4dS

Tε

D11

stiffener D12stiffener D16stiffener

D12

stiffener D22stiffener D26stiffener

D16

stiffener D26stiffener D66stiffener

= Tσ

− 1

ESIYY

S

dS

0 0

0 0 0

0 0 1

4

GSJS

dS

Tε

A44

stiffener A45stiffener

A45

stiffener A55stiffener = Tτ

− 1 0 0

0 k Z

S

GSAS

dS

Tτ

A11

stiffener

= ESAS

dS

cos2ΨS cos2ΨS + τY

S

sin2ΨS

A12

stiffener

= ESAS

dS

sin2ΨScos2ΨS 1 − τY

S
31

 (25c)

 (25d)

 (25e)

 (25f)

 (26a)

 (26b)

 (26c)

 (26d)

 (26e)

 (26f)

 (27a)

 (27b)

A16

stiffener

= ESAS

dS

sinΨScosΨS cos2ΨS − τY

S

2

cos2ΨS

A22

stiffener

= ESAS

dS

sin2ΨS sin2ΨS + τY

S

cos2ΨS

A26

stiffener

= ESAS

dS

sinΨScosΨS sin2ΨS +

τY

S

2

cos2ΨS

A66

stiffener

= ESAS

dS

sin2ΨScos2ΨS +

τY

S

4 cos2

2ΨS

B11

stiffener

= ESAS

dS

cos2ΨS zS cos2ΨS + τY

S

zS sin2

ΨS

B12

stiffener

= ESAS

dS

sin2ΨScos2ΨS zS − τY

S

zS

B16

stiffener

= ESAS

dS

sinΨScosΨS zS cos2ΨS − τY

S

zS

2

cos2ΨS

B22

stiffener

= ESAS

dS

sin2ΨS zS sin2ΨS + τY

S

zS cos2

ΨS

B26

stiffener

= ESAS

dS

sinΨScosΨS zSsin2ΨS +

τY

S

zS

2

cos2ΨS

B66

stiffener

= ESAS

dS

zS sin2

ΨScos2ΨS +

τY

S

zS

4 cos2

2ΨS

D11

stiffener

= cos2ΨS

ESIYY

S

dS

cos2ΨS +

GSJS

dS

sin2ΨS

D12

stiffener = sin2

ΨScos2ΨS

ESIYY

S

dS

− GSJS

dS
32

 (27c)

 (27d)

 (27e)

 (27f)

 (28a)

 (28b)

 (28c)

where

 (29a)

is an inplane-shear-deformation parameter associated with the shear flexibility of the stiffeners in 

a direction parallel to the plate midplane, based on a first-order transverse-shear deformation 

beam theory, and

 (29b)

is a corresponding transverse-shear-deformation parameter associated with the shear flexibility of 

the stiffeners in a direction perpendicular to the plate midplane. The effects of inplane-sheardeformation and transverse-shear-deformation of the stiffeners can be neglected in equations 

(25)-(28) by setting and , respectively.

 Expressions like those given by equations (25) and (27) have been given by Heki and Saka50, 56 

for homogeneous isotropic stiffeners with negligible inplane-shear stiffnesses and appear to be in 

agreement for the most part. Some discrepancies exist for the term associated with the twisting 

stiffness. However, the expressions for the stiffnesses associated with membrane anisotropy, 

equations (25c) and (25e), are in complete agreement with those of Heki and Saka. Expressions 

D16

stiffener = sinΨScosΨS

ESIYY

S

dS

cos2ΨS − 1

2

GSJS

dS

cos2ΨS

D22

stiffener = sin2

ΨS

ESIYY

S

dS

sin2ΨS +

GSJS

dS

cos2ΨS

D26

stiffener = sinΨScosΨS

ESIYY

S

dS

sin2ΨS + 1

2

GSJS

dS

cos2ΨS

D66

stiffener

= ESIYY

S

dS

sin2ΨScos2ΨS + 1

4

GSJS

dS

cos22ΨS

A44

stiffener

= ESASτZ

S

dS

sin2ΨS

A45

stiffener

= ESASτZ

S

dS

sinΨScosΨS

A55

stiffener

= ESASτZ

S

dS

cos2ΨS

τY

S

≡ k Y

S

GS

ES

τZ

S

≡ k Z

S

GS

ES

τY

S

= 0 τZ

S

= 0
33

like those given by equations (25) and (27) have also been given by Slinchenko and Verijenko129

for homogeneous isotropic stiffeners with negligible inplane-shear and torsional stiffnesses and 

are in complete agreement. Furthermore, expressions like those given by equations (25)-(27) have 

been given by Won106 for homogeneous isotropic beam members with rectangular cross-sections 

of equal depth and negligible transverse-shear stiffnesses. The expressions given by Won are for 

a pair of oblique stiffener families and include higher-order effects associated with the interaction 

of the plate wall with the stiffeners at the stiffener intersections, which are neglected in a firstapproximation analysis. When these higher-order effects are neglected, and only a single stiffener 

family considered, the expressions given by Won are in complete agreement with the 

corresponding expressions given herein. Pshenichnov112 and Sumec107 have also presented 

expressions like those given by equations (25) and (27) for homogeneous isotropic members with 

negligible transverse-shear stiffnesses and are in complete agreement. 

Analysis Applications

 The equivalent-plate stiffnesses given by equations (25)-(28) represent the homogenization of 

a single family of unidirectional, equally spaced, identical stiffeners that are oriented at an angle 

ΨS with respect to the plate x-axis, as shown in figure 6. Inspection of the analysis reveals that 

the analysis is easily extend to plates stiffened with multiple families of stiffeners by simply 

adding the stiffness contributions of each family. Several examples of applying the analysis are 

presented subsequently for plates with the stiffener arrangements depicted in figures 14-18. In 

these figures, stiffeners aligned with the global plate x- and y-axes are referred to as stringers and 

ribs, respectively. Other stiffeners are referred to as diagonal braces or, simply, as diagonals. All 

stiffeners are eccentric with respect to the plate midplane, unless stated otherwise. The notation 

used for the material and section properties and orientation angle of each stiffener family in these 

examples is given in Tables 1-3. In these tables, the stiffener extensional modulus, shear modulus, 

eccentricities, moment of inertia, torsion constant, and transverse-shear correction factors refer to 

the corresponding effective quantities defined in Appendix A for a nonhomogeneous, specially 

orthotropic beam. Furthermore, quantities associated with the stringers, ribs, and diagonals are 

identified by the lower case subscripts or superscripts "s," "r," and "d," respectively. Wherever 

two families of diagonals are involved, the quantities associated with the two families are 

identified by the subscripts or superscripts "d1," and "d2."

 Orthogrid plate with doubly braced bays. Consider a laminated-composite plate that is 

eccentrically stiffened with the arrangement of stringers, ribs, and braces shown in figure 14a. The 

pockets, or bays, formed by the skin, stringers, and ribs are diagonally braced with two 

nonidentical families of stiffeners. The spacings of the ribs and stringers are denoted by Lx and 

Ly, respectively. The stiffness expressions are obtained from equations (23) and (25)-(28) by 

applying equations (25)-(28) to each family of stiffeners with the attributes given in Table 1. The 

resulting equivalent-plate stiffnesses are given in Appendix B. For this general stiffener 

arrangement, the stiffnesses in Appendix B include a broad range of anisotropies characterized by 

the A16, A26, A45, D16, D26, B11, B12, B16, B22, B26, and B66 equivalent-plate constitutive terms appearing 

in equations (22). It is worth noting that by neglecting the contributions of the ribs in the stiffness 

expressions given in Appendix B, the resulting expressions will then correspond to the diamondshaped stiffener pattern shown in figure 15.
34

 Several corresponding results have been published for stiffened plates and plate-like lattices 

with homogeneous, isotropic stiffeners. In particular, the stiffnesses given in Appendix B reduce 

to those given by Karmakar88 and by Gerhard et. al.118 (see pp. 57-75) for the special case of 

identical braces and members with negligible transverse-shear and torsional stiffnesses. 

Similarly, for the special case of no braces (orthogrids) and members with negligible transverse 

shear stiffnesses, the stiffnesses given in Appendix B reduce to those given by Block, Card, and 

Mikulas.38 Likewise, the membrane stiffnesses given in Appendix B can be obtained from the 

Heki-Saka50, 56 equations that are similar to those given by equations (25), for the special case in 

which the stiffeners have negligible transverse-shear stiffnesses in the plane parallel to the plate. 

Expressions have also been given by Chen and Tsai119 that include orthogrids and grids with only 

diagonal braces as special cases. The equivalent membrane and out-of-plane bending and 

transverse-shearing stiffnesses given by Chen and Tsai are in complete agreement with the 

corresponding results given in Appendix B. However, inplane transverse-shearing stiffnesses are 

presented by Chen and Tsai that disagree with the corresponding stiffnesses given in Appendix 

B. This disagreement appears to be associated with the fact that the stiffnesses given by Chen and 

Tsai are derived by using a method that accounts for combined bending and shearing of the 

stiffeners in the plane parallel to the plate. This combined stiffener bending-shearing action is not 

included in the first-approximation analysis presented herein. Similarly, equivalent-plate 

stiffnesses are given in reference 81 for rectangular orthogrids with two identical diagonal braces 

per bay and homogeneous, isotropic stiffeners. The corresponding stiffnesses presented herein 

for this special case are only in agreement if the terms associated with shear deformation and 

torsion of the stiffeners are neglected. At least part of the discrepancies, particularly that 

associated with inplane shear deformation of the stiffeners, appears to be associated with the 1/2 

appearing in equations (11e) and (14a) being neglected in reference 81. It is important to point out 

that neglecting the 1/2 appearing in equations (11e) and (14a) results in a lack of the symmetry in 

equations (15a) and (15b). 

 Bunakov and Protasov105 presented equivalent-continuum stiffnesses for the stiffener 

arrangement shown in figure 14a, but without stringers and ribs and with identical diagonals with 

a rectangular cross-section. These stiffnesses are based on a micropolar continuum model of 

deformation and include transverse-shear and bending stiffnesses of the beam members 

associated with deformations in a plane parallel to the plate midplane. When the inplane bending 

stiffnesses of the stiffeners and the micro-rotation normal to the plate midplane are neglected, the 

stiffnesses given in reference 105, which include inplane and out-of-plane transverse-shear 

stiffnesses of the beam members, are in complete agreement with the stiffnesses given in 

Appendix B. It is important to point out that the expressions for the stiffnesses A66 and D66 were 

obtained from the micropolar theory by noting that both of the asymmetric inplane shearing strain 

measures and the asymmetric twisting strain measures used in reference 105 reduce to one half 

the inplane shearing strain and one half the twisting strain, respectively, when the micro-rotations 

normal to the plate midplane are neglected. The stiffnesses A66 and D66 were then obtained by 

relating the average of the asymmetric stress resultants to the corresponding strain.

 Orthogrid plate with singly braced bays. Equations (23) and (25)-(28) were also applied to 

a laminated-composite plate that is stiffened with eccentric stringers, ribs, and nonidentical 

diagonal braces, as shown in figure 16a. For this case, alternating bays, enclosed by stringers and 

ribs, are braced with a single diagonal stiffener, and the spacing of the ribs and stringers is denoted 
35

by Lx and Ly, respectively. The stiffness expressions are obtained from equations (23) and (25)-

(28) by applying equations (25)-(28) to each family of stiffeners with the attributes given in Table 

2. The resulting equivalent-plate stiffnesses are also given in Appendix B. For the special case of 

homogeneous, isotropic stiffeners, identical braces, and members with negligible inplane 

transverse-shear stiffnesses, the stiffnesses given in Appendix B reduce to those given by Reddy 

et. al.95 and by Chen and Tsai.119 However, the out-of-plane transverse-shear stiffnesses of Reddy 

et. al. do not agree with those in Appendix B or with those given by Chen and Tsai, and appear to 

be in error. Likewise, for the special case of square bays with identical braces and members with 

negligible inplane and out-of-plane transverse-shear and inplane bending stiffnesses, the 

stiffnesses given in Appendix B reduce to those given by Hefzy & Nayfeh.90, 99

 Isosceles-triangle stiffener pattern. The analysis of this section was also applied to a plate 

stiffened with an eccentric grid that has pockets that are isosceles triangles, as shown in figure 

17a. The notation used for the material and section properties and orientation angle of each 

stiffener family is given in Table 3, and the horizontal spacing of the triangles and the spacing of 

the stringers are denoted by Lx

 and Ly, respectively. For this stiffening grid, the two families of 

diagonal beams are nonidentical and the stiffeners intersect at the vertices of the triangular 

pockets. The resulting equivalent-plate stiffnesses obtained by applying equations (23) and (25)-

(28) are given in Appendix C. Expressions are also given in Appendix C for the special case of 

pockets that are equilateral triangles and members with identical properties. This particular 

stiffener configuration is often referred to as an isogrid because of the isotropic nature of its 

equivalent-plate inplane and bending stiffnesses. Similarly, equivalent-plate stiffnesses are given 

in reference 81 for this special case with homogeneous, isotropic stiffeners that are in complete 

agreement with those presented herein, except for the stiffness contributions associated with 

inplane shear deformation and the twisting stiffness D66. The discrepancy also appears to be 

associated with the 1/2 appearing in equations (11e) and (14a) being neglected in reference 81. 

The stiffnesses for this special case with homogeneous, isotropic stiffeners that have negligible 

transverse-shear stiffnesses are in complete agreement with the corresponding stiffnesses given 

by Hefzy & Nayfeh.99 

 Kagome stiffener pattern. Equations (23) and (25)-(28) were also applied to a plate stiffened 

with a similar eccentric stiffening grid presented by Wodesenbet et.al.133 and shown in figure 18a. 

This type of stiffening configuration is sometimes referred to as a Kagome pattern.140, 141 The 

notation used for the material and section properties and orientation angle of each stiffener family 

is the same as that given in Table 3. For this stiffener arrangement, the two families of diagonal 

beams are nonidentical and the horizontal stiffeners do not intersect the diagonal stiffeners at the 

triangle vertices. The resulting equivalent-plate stiffnesses are identical to those given in 

Appendix C for the isosceles-triangle stiffener pattern. For the special case in which the beam 

members are made of a homogeneous, isotropic material, have rectangular cross-sections, and 

negligible inplane transverse-shear and torsional stiffnesses, the stiffnesses given in Appendix C 

reduce to those given by Vasiliev et. al.131 In addition, for the case in which all stiffeners have the 

same properties and all stiffeners have negligible transverse-shear, bending, and torsional 

stiffnesses, the stiffnesses given in Appendix C reduce to those given by Heki50 and by 

Wodesenbet et.al.133 Moreover, the stiffnesses given in Appendix C reduce to those given by 

Wang et. al.138 for the special case of negligible transverse-shear and torsional stiffnesses.
36

Basic-Cell Energy-Equivalence Method

In this method, a basic repeating cell of the stiffened plate, such as the cell shown in figure 

14b, is identified. The criterion for selecting the basic cell used herein is that it be a simple unit 

that will generate a stiffened structure of a given species by translation over the plate midplane, 

without any overlapping. Each basic cell contains a finite number of stiffener members. Consider 

a single, arbitrary beam stiffener. Applying the kinematical equivalence described by equations 

(11), the beam strains are expressed in terms of the equivalent-plate strains, in the (X, Y, Z) 

coordinates shown in figure 6, by

 (30a)

where

 (30b)

 (30c)

and

 (30d)

In these equations, the uppercase subscript, P, denotes reference to the (X, Y, Z) coordinates. The 

matrix defined by equation (30d) is referred to herein as the strain-equivalence matrix. By using 

these equations, the strain energy of the beam, given by (A25) and (A26), is expressed as

 (31)

with

 (32)

and where LS is the beam-member length. Performing the matrix multiplication gives

ε b = E ε P

ε b

T

= eXX

o

ΓXY

o

ΓXZ

o

χZ

o

χY

o

τ

o

ε P

T

= εXX

o

εYY

o

γXY

o

κ XX

o

κ YY

o

κ XY

o

γYZ

o

γXZ

o

E =

1 0 0 0 0 0 0 0

0 0 1

2 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 − 1

2 0 0

Eε = 1

2 ε P

T

C P ε P dX

LS

C P = E

T

C E
37

 (33)

The elements of this matrix are based on the notation used in Appendix A for the effective 

engineering constants of a nonhomogeneous, specially orthotropic beam. In particular, the 

nonhomogeneous beam stiffener is assigned an effective axial modulus ES, an effective shear 

modulus GS, cross-sectional area AS, effective moments of inertia and , effective product 

of inertia , and effective torsional constant JS, in addition to the effective eccentricities , , 

 and . Effective shear correction factors for the beam are denoted by and .

 Next, the plate strains in the (X, Y, Z) beam coordinates are defined in terms of the global plate 

(x, y, z) coordinates shown in figure 6 as follows

 (34a)

with

 (34b)

and where the lowercase subscript, p, denotes reference to the (x, y, z) coordinates of the plate. 

The transformation sub-matrices appearing in equation (34a) are defined by equations (17). In 

addition,

 (34c)

C P =

ESA S 0 0 ESA SzS 0 0 0 0

0 0 0 0 0 0 0 0

0 0 1

4k Y

S

GSA S 0 0 1

4k Y

S

GSA SzS 0 0

ESA SzS 0 0 ESIYY

S 0 0 0 0

0 0 0 0 0 0 0 0

0 0 1

4k Y

S

GSA SzS 0 0 1

4GSJS 0 − 12k Z

S

GSA SyS

0 0 0 0 0 0 0 0

0 0 0 0 0 − 1

2k Z

S

GSA SyS 0 k Z

S

GSA S

IYY

S

IZZ

S

IYZ

S

yS zS

yS zS k Y

s k Zs

ε P =

Tε

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0 Tε

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0 Tτ

ε p

ε p

T

= εxx

o

εyy

o

γ xy

o

κxx

o

κyy

o

κxy

o

γ yz

o

γ xz

o

ε P

T

= ε p

T

Tε

T 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0 Tε

T 0 0

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0 Tτ

T
38

The beam strain energy defined by equation (31) is expressed in terms of the global plate 

coordinate system by

 (35)

where dX = dX(x, y) and

 (36)

The explicit expressions for the elements of this matrix are given in Appendix D.

 The basic presumption of this energy-based homogenization method is that the basic repetitive 

cell is small enough, compared to the overall structural dimensions, so that the strain-energy 

density of every beam member within it can be represented adequately by a constant value. Based 

upon this discretization concept, the integrand of equation (35) is a constant and, as a result, the 

beam-member strain energy is given by

 (37)

The beam-member strain energy is now completely characterized by the plate strains and beam 

properties. The dependence on the strains is expressed functionally by

 (38)

 The "average" two-dimensional strain-energy density for the basic equivalent-plate cell, 

with area Acell, is given by

 (39a)

where

 (39b)

Eε = 1

2 ε p

T

Cp ε p dX

LS

Cp =

Tε

T 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0 Tε

T 0 0

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0 Tτ

T

C P

Tε

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0 Tε

0 0

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0 Tτ

Eε = LS

2 ε p

T

Cp ε p

Eε = Eε ε xx

o

, ε yy

o

, γxy

o

, κxx

o

, κyy

o

, κxy

o

, γxz

o

, γyz

o

Eε

Eε = Eε

plate

+

1

AcellΣ Eε

members

= Eε ε xx

o

, ε yy

o

, γxy

o

, κxx

o

, κyy

o

, κxy

o

, γxz

o

, γyz

o

2Eε

plate

= Nxx

plate

ε xx

o

+ Nyy

plate

ε yy

o

+ Nxy

plate

γxy

o

+ Mxx

plate

κxx

o

+ Myy

plate

κyy

o

+ Mxy

plate

κxy

o

+ Qxz

plate

γxz

o

+ Qyz

plate

γyz

o
39

This expression represents an equivalent pointwise function for the contribution of all beam 

members within the basic, repetitive cell to the equivalent-plate strain energy density, plus the 

contribution of the plate wall. The strain energy density in this expression is given in terms of all 

stiffener properties, stiffener orientations, and wall properties, in addition to the global plate 

strains. The next step in this method is to relate equation (39a) to the general form for an 

anisotropic plate, based on the first-order transverse-shear deformation theory presented herein. 

In particular, the strain-energy density of an anisotropic plate is given by

 (40)

By using the anisotropic-plate constitutive equations

 (41a)

and

 (41b)

the strain energy density given by equation (40) becomes a quadratic function of the plate strains. 

Therefore, it follows that the symmetric constitutive matrices are given by

 (42a)

2Eε = Nxxε xx

o

+ Nyyε yy

o

+ Nxyγxy

o

+ Mxxκxx

o

+ Myyκyy

o

+ Mxyκxy

o

+ Qxzγxz

o

+ Qyzγyz

o

Nxx

Nyy

Nxy

Mxx

Myy

Mxy

=

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

εxx

o

εyy

o

γ xy

o

κxx

o

κyy

o

κxy

o

Qyz

Qxz

= A44 A45

A45 A55

γ yz

o

γ xz

o

A11 A12 A16

• A22 A26

symmetric • A66

=

∂

2

Eε

∂εxx

o

∂εxx

o

∂

2

Eε

∂εxx

o

∂εyy

o

∂

2

Eε

∂εxx

o

∂γ xy

o

• ∂

2

Eε

∂εyy

o

∂εyy

o

∂

2

Eε

∂εyy

o

∂γ xy

o

symmetric • ∂

2

Eε

∂γ xy

o

∂γ xy

o
40

 (42b)

 (42c)

 (42d)

Inspection of equation (39) indicates that the contributions of the stiffeners in the basic cell to the 

constitutive equations are given by

 (43a)

B11 B12 B16

• B22 B26

symmetric • B66

=

∂

2

Eε

∂εxx

o

∂κxx

o

∂

2

Eε

∂εxx

o

∂κyy

o

∂

2

Eε

∂εxx

o

∂κxy

o

• ∂

2

Eε

∂εyy

o

∂κyy

o

∂

2

Eε

∂εyy

o

∂κxy

o

symmetric • ∂

2

Eε

∂γ xy

o

∂κxy

o

D11 D12 D16

• D22 D26

symmetric • D66

=

∂

2

Eε

∂κxx

o

∂κxx

o

∂

2

Eε

∂κxx

o

∂κyy

o

∂

2

Eε

∂κxx

o

∂κxy

o

• ∂

2

Eε

∂κyy

o

∂κyy

o

∂

2

Eε

∂κyy

o

∂κxy

o

symmetric • ∂

2

Eε

∂κxy

o

∂κxy

o

A44 A45

symmetric A55

=

∂

2

Eε

∂γ yz

o

∂γ yz

o

∂

2

Eε

∂γ yz

o

∂γ xz

o

symmetric

∂

2

Eε

∂γ xz

o

∂γ xz

o

A11

stiffener

A12

stiffener

A16

stiffener

• A22

stiffener

A26

stiffener

symmetric • A66

stiffener

= 1

AcellΣ

members

∂

2

Eε

∂εxx

o

∂εxx

o

∂

2

Eε

∂εxx

o

∂εyy

o

∂

2

Eε

∂εxx

o

∂γ xy

o

• ∂

2

Eε

∂εyy

o

∂εyy

o

∂

2

Eε

∂εyy

o

∂γ xy

o

symmetric • ∂

2

Eε

∂γ xy

o

∂γ xy

o

= 1

AcellΣ LS

members C11

p

C12

p

C13

p

• C22

p

C23

p

symmetric • C33

p
41

 (43b)

 (43c)

 (43d)

Using equations (43), the constitutive equations for the equivalent plate are given by equations 

(22) and (23).

Analysis Application

 Several examples of applying the basic-cell energy-equivalence method are presented 

subsequently for plates with the stiffener arrangements depicted in figures 14-24. In these figures, 

stiffeners aligned with the global plate x- and y-axes are referred to as stringers and ribs, 

respectively. Other stiffeners are referred to as diagonal braces or, simply, as diagonals. All 

stiffeners are eccentric with respect to the plate midplane, and each family of stiffeners has 

different properties unless stated otherwise. The notation used for the material and section 

B11

stiffener

B12

stiffener

B16

stiffener

• B22

stiffener

B26

stiffener

symmetric • B66

stiffener

= 1

AcellΣ

members

∂

2

Eε

∂εxx

o

∂κxx

o

∂

2

Eε

∂εxx

o

∂κyy

o

∂

2

Eε

∂εxx

o

∂κxy

o

• ∂

2

Eε

∂εyy

o

∂κyy

o

∂

2

Eε

∂εyy

o

∂κxy

o

symmetric • ∂

2

Eε

∂γ xy

o

∂κxy

o

= 1

AcellΣ LS

members C14

p

C15

p

C16

p

• C25

p

C26

p

symmetric • C36

p

D11

stiffener

D12

stiffener

D16

stiffener

• D22

stiffener

D26

stiffener

symmetric • D66

stiffener

= 1

AcellΣ

members

∂

2

Eε

∂κxx

o

∂κxx

o

∂

2

Eε

∂κxx

o

∂κyy

o

∂

2

Eε

∂κxx

o

∂κxy

o

• ∂

2

Eε

∂κyy

o

∂κyy

o

∂

2

Eε

∂κyy

o

∂κxy

o

symmetric • ∂

2

Eε

∂κxy

o

∂κxy

o

= 1

AcellΣ LS

members C44

p

C45

p

C46

p

• C55

p

C56

p

symmetric • C66

p

A44

stiffener A45stiffener

symmetric A55

stiffener = 1

AcellΣ

members

∂

2

Eε

∂γ yz

o

∂γ yz

o

∂

2

Eε

∂γ yz

o

∂γ xz

o

symmetric

∂

2

Eε

∂γ xz

o

∂γ xz

o

= 1

AcellΣ LS

members

C77

p

C78

p

symmetric C88

p
42

properties and orientation angle of each stiffener member of the basic cell used in these examples 

is given in Tables 4-9. In these tables, the member designation denotes numbers associated with 

the beginning and end of the members comprising a basic cell, as shown in figures 14b, 16b, 17b, 

18b, 21b, and 23b. Also, the stiffener extensional modulus, shear modulus, eccentricity, moment 

of inertia, torsion constant, and transverse-shear correction factors, refer to the corresponding 

effective quantities defined in Appendix A for a nonhomogeneous, specially orthotropic beam. 

Quantities associated with the stringers, ribs, and diagonals are identified by the subscripts or 

superscripts "s," "r," and "d," respectively. Wherever two families of diagonals are involved, the 

quantities associated with the two families are identified by the subscripts or superscripts "d1," 

and "d2." For the examples presented subsequently, the Mathematica® computer program142

presented in Appendix E was used to obtain the equivalent-plate stiffnesses.

 Orthogrid plate with doubly or singly braced bays. The basic-cell energy-equivalence 

method was applied to the orthogonally stiffened plates with diagonally braced bays shown in 

figures 14a and 16a. The basic cell for the stiffener grid in figure 14a is shown in figure 14b and 

consists of four beam members with the basic-cell beam-member attributes given in Table 4. The 

basic cell for the stiffener grid in figure 16a is shown in figure 16b and consists of six beam 

members with the basic-cell beam-member attributes given in Table 5. The ends of each beam 

member are indicated in figures 14b and 16b by the filled red circular symbols. For both stiffener 

grids, the rib and stringer spacings are denoted by Lx and Ly, respectively, and the length of the 

diagonals is given by . The dashed rectangle shown in figures 14a and 14b 

indicates the perimeter of the basic repetitive cell and has the area Acell = LxLy. Similarly, the 

shaded rectangle shown in figures 16a and 16b indicates the basic repetitive cell, with the area 

Acell = 4LxLy. Applying the analysis described in this section to these two basic cells yields the 

same equivalent plate stiffnesses given in Appendix B that were obtained by using the Direct 

Equilibrium-Compatibility Method presented herein. 

 Isosceles-triangle and Kagome stiffener patterns. The basic-cell energy-equivalence method 

was also applied to a plate stiffened with the stiffener arrangements shown in figures 17 and 18. 

The basic cell for the grid in figure 17a is shown in figure 17b and consists of three stiffeners with 

the properties given in Table 6. Likewise, the basic cell for the Kagome grid in figure 18a is 

shown in figure 18b and consists of four beam members with the properties given in Table 7. The 

ends of each beam member are also indicated in figures 17b and 18b by the filled red circular 

symbols. For both stiffener grids, the horizontal spacing of the diagonals and the stringer spacing 

are denoted by Lx and Ly

, respectively. The dashed hexagon shown in figures 17a and 17b and 

the dashed rectangle shown in figures 18a and 18b indicates the perimeter of each basic repetitive 

cell and has the area Acell = Lx

Ly and 2LxLy, respectively, and the length of the diagonals is given 

by . The shaded region surrounding each cell perimeter indicates how the cell 

relates to the surrounding structure, for clarity. In addition, figures 19 and 20 indicate how the 

basic cells in figures 17b and 18b are translated to generate the stiffened-plate structure, 

respectively. Applying the analysis described in this section to the basic cells with the attributes 

given in Tables 6 and 7 also yields the same equivalent plate stiffnesses given in Appendix C that 

were obtained by using the Direct Equilibrium-Compatibility Method presented herein. 

Ld = Lx

2

+ Ly

2

L = Lx

2

+ Ly

2
43

 Hexagon-shaped stiffener pattern. Another stiffened plate considered in the present study is 

shown in figure 21, in which the plate wall is reinforced with a hexagonal grid of stiffeners from 

three families. The basic cell for this structure is shown in figure 21b and consists of five 

stiffeners with the basic-cell beam-member attributes given in Table 8. The ends of each member 

are indicated in the figure by the filled red circular symbols, and the length of the ribs is different 

from the length of the diagonals. The dashed hexagon shown in figures 21a and 21b indicates the 

perimeter of the basic repetitive cell and has the area Acell = 2a(b + c). The shaded region 

surrounding the cell perimeter indicates how the cell relates to the surrounding structure. Figure 

22 shows how the basic cell in figure 21b is translated to generate the stiffened-plate structure. 

Applying the analysis described in this section to the basic cell yields the equivalent plate 

stiffnesses given in Appendix F. Similar membrane stiffnesses have been given by Heki50 for a 

hexagonal stiffener arrangement in which all beam members forming a hexagon have the same 

length and are made of a homogeneous, isotropic material. The stiffnesses presented in Appendix 

F do not agree with those in reference 50. For the special case of homogeneous, isotropic 

members with the same length and identical properties and negligible inplane and out-of-plane 

transverse-shear stiffnesses, the stiffnesses given in Appendix F are in complete agreement with 

those given by Hefzy and Nayfeh.99 Similarly, equivalent-plate stiffnesses are given in reference 

81 for the special case of a homogeneous, isotropic single-layer grid comprised of members with 

the same length and identical properties. The stiffnesses presented herein are in agreement with 

those in reference 81 for this special case, except for the stiffness contributions associated with 

inplane shear deformation. This discrepancy also appears to be associated with the 1/2 appearing 

in equations (11e) and (14a) being neglected in reference 81. 

 Star-shaped stiffener pattern. The analysis was also applied to a plate stiffened with the starshaped stiffener arrangement shown in figures 23. For this case, the star shapes are formed from 

two oppositely directed isosceles triangles with coincident centroids. The basic cell for this 

structure is shown in figure 23b and consists of twelve stiffeners from three families with the 

basic-cell beam-member attributes given in Table 9. The ends of each member are indicated in the 

figure by the filled red circular symbols. The dashed hexagon shown in figures 23a and 23b 

indicates the perimeter of the basic repetitive cell and has the area Acell = 4BH/3, where B and H 

are the base and height of either isosceles triangle forming the star shape. The shaded region 

surrounding the cell perimeter indicates how the cell relates to the surrounding structure. Figure 

24 shows how the basic cell in figure 23b is translated to generate the stiffened-plate structure. 

Applying the analysis described in this section to the basic cell yields the equivalent plate 

stiffnesses given in Appendix G.

Equivalent-Plate Stiffnesses for Sandwich Plates

 Inspection of the analysis presented herein reveals that the stiffnesses given in Appendices B, 

C, F, and G can also be used to obtain the equivalent-plate stiffnesses for sandwich plates with 

two nonidentical anisotropic face plates, such as those shown in figures 25-27. This task is 

accomplished by using equations (23) based on two plate members, neither of which are located 

at z = 0. The stiffeners forming the core are presumed to be made of a homogeneous orthotropic 

or isotropic material. For convenience, the equivalent-plate reference plane is taken as the 
44

midplane of the core layer formed by the stiffeners such that all core anisotropies associated with 

coupling between membrane and bending actions vanish. The contribution of the face plates to 

the equivalent-plate stiffnesses are obtained by applying the integral definitions of the plate 

stiffnesses, in accordance with figure 28. In particular, the stiffnesses of a face plate are given, 

with respect to the global plate (x, y, z) coordinates shown in figure 28, by

 (44a)

 (44b)

 (44c)

 (44d)

where h is the face-plate thickness, e is the face-plate eccentricity, are the transformed 

lamina stiffnesses of first-order shear-deformation laminated-plate theory (see reference 139), 

and Kplate is a transverse-shear correction factor. For the core, transverse-shear correction factors 

are used for the beam members. Next, the local (x ́, y ́, z ́) coordinates shown in figure 28 are 

used to obtain

 (45a)

 (45b)

A11

plate A12plate A16plate

A12

plate A22plate A26plate

A16

plate A26plate A66plate

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

dz

e − h

2

e + h

2

B11

plate B12plate B16plate

B12

plate B22plate B26plate

B16

plate B26plate B66plate

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

z dz

e − h

2

e + h

2

D11

plate D12plate D16plate

D12

plate D22plate D26plate

D16

plate D26plate D66plate

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

z

2

dz

e − h

2

e + h

2

A44

plate A45plate

A45

plate A55plate = Kplate Q44 Q45

Q45 Q55

dz

e − h

2

e + h

2

Qij z

A11

plate A12plate A16plate

A12

plate A22plate A26plate

A16

plate A26plate A66plate

=

A′11

plate

A′12

plate

A′16

plate

A′12

plate

A′22

plate

A′26

plate

A′16

plate

A′26

plate

A′66

plate

B11

plate B12plate B16plate

B12

plate B22plate B26plate

B16

plate B26plate B66plate

= e

A′11

plate

A′12

plate

A′16

plate

A′12

plate

A′22

plate

A′26

plate

A′16

plate

A′26

plate

A′66

plate

+

B′11

plate

B′12

plate

B′16

plate

B′12

plate

B′22

plate

B′26

plate

B′16

plate

B′26

plate

B′66

plate
45

 (45c)

 (45d)

where and 

 (46a)

 (46b)

 (46c)

 (46d)

 

Equations (46) represent the plate stiffnesses of a given face plate that are calculated in the usual 

way (see references 19 and 139) in which z ́ = 0 represents the plate midplane. With these definitions, the equivalent-sandwich-plate stiffnesses are given by

D11

plate D12plate D16plate

D12

plate D22plate D26plate

D16

plate D26plate D66plate

= e2

A′11

plate

A′12

plate

A′16

plate

A′12

plate

A′22

plate

A′26

plate

A′16

plate

A′26

plate

A′66

plate

+ 2e

B′11

plate

B′12

plate

B′16

plate

B′12

plate

B′22

plate

B′26

plate

B′16

plate

B′26

plate

B′66

plate

+

D′11

plate

D′12

plate

D′16

plate

D′12

plate

D′22

plate

D′26

plate

D′16

plate

D′26

plate

D′66

plate

A44

plate A45plate

A45

plate A55plate = A′44

plate

A′45

plate

A′45

plate

A′55

plate

z = z′ + e

A′11

plate

A′12

plate

A′16

plate

A′12

plate

A′22

plate

A′26

plate

A′16

plate

A′26

plate

A′66

plate

=

Q11 z′ Q12 z′ Q16 z′

Q12 z′ Q22 z′ Q26 z′

Q16 z′ Q26 z′ Q66 z′

dz′

− h

2

+ h

2

B′11

plate

B′12

plate

B′16

plate

B′12

plate

B′22

plate

B′26

plate

B′16

plate

B′26

plate

B′66

plate

=

Q11 z′ Q12 z′ Q16 z′

Q12 z′ Q22 z′ Q26 z′

Q16 z′ Q26 z′ Q66 z′

z′ dz′

− h

2

+ h

2

D′11

plate

D′12

plate

D′16

plate

D′12

plate

D′22

plate

D′26

plate

D′16

plate

D′26

plate

D′66

plate

= =

Q11 z′ Q12 z′ Q16 z′

Q12 z′ Q22 z′ Q26 z′

Q16 z′ Q26 z′ Q66 z′

z′

2

dz′

− h

2

+ h

2

A′44

plate

A′45

plate

A′45

plate

A′55

plate = Kplate Q44 z′ Q45 z′

Q45 z′ Q55 z′ dz′

− h

2

+ h

2
46

 (47a)

 (47b)

 (47c)

 (47d)

 

 The analysis of the present study was applied to the sandwich plates with two nonidentical 

anisotropic face plates shown in figures 25-27. For these examples, the eccentricities of the face 

plates, plates no. 1 and 2, are given by and , with respect to the 

A11 A12 A16

A12 A22 A26

A16 A26 A66

=

A′11

plate no. 1

A′12

plate no. 1

A′16

plate no. 1

A′12

plate no. 1

A′22

plate no. 1

A′26

plate no. 1

A′16

plate no. 1

A′26

plate no. 1

A′66

plate no. 1

+

A′11

plate no. 2

A′12

plate no. 2

A′16

plate no. 2

A′12

plate no. 2

A′22

plate no. 2

A′26

plate no. 2

A′16

plate no. 2

A′26

plate no. 2

A′66

plate no. 2

+

A11

core A12core A16core

A12

core A22core A26core

A16

core A26core A66core

B11 B12 B16

B12 B22 B26

B16 B26 B66

=

B′11

plate no. 1

B′12

plate no. 1

B′16

plate no. 1

B′12

plate no. 1

B′22

plate no. 1

B′26

plate no. 1

B′16

plate no. 1

B′26

plate no. 1

B′66

plate no. 1

+

B′11

plate no. 2

B′12

plate no. 2

B′16

plate no. 2

B′12

plate no. 2

B′22

plate no. 2

B′26

plate no. 2

B′16

plate no. 2

B′26

plate no. 2

B′66

plate no. 2

+ eplate no. 1

A′11

plate no. 1

A′12

plate no. 1

A′16

plate no. 1

A′12

plate no. 1

A′22

plate no. 1

A′26

plate no. 1

A′16

plate no. 1

A′26

plate no. 1

A′66

plate no. 1

+ eplate no. 2

A′11

plate no. 2

A′12

plate no. 2

A′16

plate no. 2

A′12

plate no. 2

A′22

plate no. 2

A′26

plate no. 2

A′16

plate no. 2

A′26

plate no. 2

A′66

plate no. 2

D11 D12 D16

D12 D22 D26

D16 D26 D66

=

D′11

plate no. 1

D′12

plate no. 1

D′16

plate no. 1

D′12

plate no. 1

D′22

plate no. 1

D′26

plate no. 1

D′16

plate no. 1

D′26

plate no. 1

D′66

plate no. 1

+

D′11

plate no. 2

D′12

plate no. 2

D′16

plate no. 2

D′12

plate no. 2

D′22

plate no. 2

D′26

plate no. 2

D′16

plate no. 2

D′26

plate no. 2

D′66

plate no. 2

+ eplate no. 1 2

A′11

plate no. 1

A′12

plate no. 1

A′16

plate no. 1

A′12

plate no. 1

A′22

plate no. 1

A′26

plate no. 1

A′16

plate no. 1

A′26

plate no. 1

A′66

plate no. 1

+ eplate no. 2 2

A′11

plate no. 2

A′12

plate no. 2

A′16

plate no. 2

A′12

plate no. 2

A′22

plate no. 2

A′26

plate no. 2

A′16

plate no. 2

A′26

plate no. 2

A′66

plate no. 2

+ 2eplate no. 1

B′11

plate no. 1

B′12

plate no. 1

B′16

plate no. 1

B′12

plate no. 1

B′22

plate no. 1

B′26

plate no. 1

B′16

plate no. 1

B′26

plate no. 1

B′66

plate no. 1

+ 2eplate no. 2

B′11

plate no. 2

B′12

plate no. 2

B′16

plate no. 2

B′12

plate no. 2

B′22

plate no. 2

B′26

plate no. 2

B′16

plate no. 2

B′26

plate no. 2

B′66

plate no. 2

+

D11

core D12core D16core

D12

core D22core D26core

D16

core D26core D66core

A44 A45

A45 A55

= A′44

plate no. 1

A′45

plate no. 1

A′45

plate no. 1

A′55

plate no. 1 +

A′44

plate no. 2

A′45

plate no. 2

A′45

plate no. 2

A′55

plate no. 2 +

A44

core A45core

A45

core A55core

e1 = − 1

2 h1 + hc e2 = 12 h2 + hc
47

midplane of the core, respectively, where hc is the depth of the core and h1 and h2 are the 

thicknesses of plates no. 1 and 2, respectively. The sandwich plate shown in figure 25 has a 

regular hexagon-cell core composed of two member types, each with a rectangular cross-section. 

One member type is aligned with the y-axis of the plate, as shown in figure 25, and is referred to 

herein as a rib. The other member type makes an angle with the x-axis and is referred to herein 

as a diagonal. The core has the same layout as the hexagonal stiffener grid shown in figure 21a 

and with the dimensions , , and . The geometrical properties of the face 

plates and the core are also shown in figure 25, and the section properties of the core are given in 

Appendix H. The core contributions to the equivalent-plate stiffnesses in equations (47) are given 

in Appendix H and are obtained by specializing the general stiffnesses given in Appendix F for a 

hexagonal stiffener arrangement. For the special case in which the face plates are made of the 

same isotropic material, the core is made of a second isotropic material, the thicknesses of the core 

members are identical (ts

 = td = t), the face plates have identical thicknesses (h1 = h2 = h), and shear 

deformation is neglected, the stiffnesses in Appendix H reduce to those given by Kalamkarov110

(see p. 205), with the exception of the contribution of the torsional stiffnesses of the core 

members.

 The sandwich plate shown in figure 26 has a regular "orthogrid" core with the same layout as 

the stiffening arrangement shown in figure 14a, but without the diagonal braces. The geometrical 

properties of the face plates and the core are shown figure 26, and the relationship of these 

properties to the orthogonal stiffeners is given in Table 4 and Appendix I. The core contributions 

to equations (47) are given in Appendix I and are obtained by specializing the general stiffnesses 

given in Appendix B. Expressions are also given in Appendix I for the special case in which the 

two face plates and the core are made of three different homogeneous orthotropic materials. 

When these expressions are further reduced for face plates with identical thicknesses (h1 = h2) and 

shear deformation is neglected, the stiffnesses reduce to those given by Kalamkarov110 (see pp. 

200-201), with the exception of the torsional stiffnesses of the core members.

 A sandwich plate with a star-shaped, or star-cell, core is shown in figure 26 which has the same 

layout as the stiffening arrangement shown in figure 23a. The relationship of the geometrical 

properties to the core stiffeners is given in Table 9 and Appendix J. Specializing the general 

stiffnesses in Appendix G yields the core contribution to equations (47) given in Appendix J. 

Expressions are also given in Appendix J for the special case in which the star shapes forming the 

core are composed of equilateral triangles with coincident centroids. 

Equivalent-Plate Thickness

The equivalent-plate thickness, h + hS, shown in figure 9, is often needed to perform auxiliary 

design computations. However, for the analysis presented herein, the results reveal that this 

thickness is not defined uniquely. For example, enforcing equivalence between the area of the 

stiffener and the area of the equivalent-stiffener layer gives the equivalent-stiffener layer 

thickness as 

a

L = 3

2

b

L = 12

c

L = 1
48

 (48)

In contrast, the stiffness-weighted first moment of stiffener area about the y-axis, defined by 

equation (A13e), is given by 

 (49)

Applying this equation to the equivalent-stiffener layer is generally complicated for an arbitrary 

nonhomogeneous stiffener. For the simpler case of a homogeneous stiffener, equation (49) gives 

the equivalence statement

 (50)

which is solved to obtain hS. Likewise, the stiffness-weighted second moment of stiffener area 

about the y-axis, defined by equation (A13f), is given by 

 (51)

Applying this equation to the equivalent-stiffener layer, for the simpler case of a homogeneous 

stiffener, gives 

 (52)

which is solved to obtain hS. Similarly, applying the formula from elasticity theory for torsion of 

a homogeneous, isotropic beam with a rectangular cross-section given in reference 144 (see p. 89) 

to the equivalent-stiffener layer gives the transcendental equation 

 (53)

Thus, equations (48), (50), (52), and (53) give four different values for the thickness of the 

equivalent-stiffener layer, but other criteria based on equations (A13) exist. In particular, equation 

(48) results from equivalence based on membrane action, and equation (50) results from 

equivalence based on coupling between membrane and bending action. Equation (52) results 

from equivalence based on bending action, and equation (53) results from equivalence based on 

twisting action. The appropriate formula to use for hS should be based on the dominant physical 

characteristic response involved in a particular auxiliary computation.

h S = AS

dS

1

ES

EX Z dYdZ

A S

= ASzS

h S h S + h = 2ASzS

dS

1

ES

EX Z2dYdZ

A S

= IYY

S

h S

3

+ 3h

2 h S

2

+ 3h2

4 h S = 3

dS

IYY

h S

3

dS

3 1 − 64

3π5

h S

dS

tanh πdS

2h S

= JS
49

Concluding Remarks

 A survey of studies conducted since 1914 on the use of equivalent-plate stiffnesses in modeling 

the overall, stiffness-critical response of stiffened plates and shells, and two detailed, 

comprehensive derivations of first-approximation equivalent-plate stiffnesses have been 

presented. The derivations are based on the Reissner-Mindlin-type, first-order transverse-shear 

deformation theory for anisotropic plates. The first derivation is based purely on static and 

kinematic equivalence between a stiffened plate and its homogenized equivalent. The second 

derivation is based on equivalence of the strain-energy density and is more amenable to 

complicated stiffener arrangements such as the hexagonal and star-shaped stiffening 

arrangements presented herein. In both derivations, the stiffener members are modelled as beams 

that are shear deformable within and transverse to the midplane of the plate, consistent with the 

classical continuum model of solid mechanics. In general, each stiffener is prismatic and may 

have a nonhomogeneous cross-section made from specially orthotropic materials. This general 

formulation allows the computation of equivalent-plate stiffnesses for stiffened panels such as 

those in which the stiffener caps are reinforced with high-strength, pultruded rods. Consistent 

with a first-approximation theory, inplane bending of the stiffeners and total compatibility 

between the plate skin and the stiffeners are neglected. Both methods presented herein have been 

shown to yield identical results.

 Equivalent-plate stiffness expressions, and a corresponding symbolic manipulation computer 

program, have also presented for several different stiffener configurations. These expressions are 

very general and exhibit the full range of anisotropies permitted by the Reissner-Mindlin-type, 

first-order transverse-shear deformation theory for anisotropic plates. The expressions presented 

in the present study were also compared with available, previously published results. For the most 

part, the previously published results are for special cases of the general expressions presented 

herein. These previously published results and are almost in complete agreement with the 

corresponding results contained herein and plausible reasons for the discrepancies have been 

given. Analysis has also been presented that extends the use of the equivalent-plate stiffness 

expressions to sandwich plates with nonidentical, anisotropic face plates. In addition, several 

criteria for obtaining an equivalent-plate thickness for stiffened plates have been presented.

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Table 1. Stiffener attributes for an eccentric, orthogonal stiffening grid with two diagonal braces 

 per bay (see figure 14)

Table 1. Concluded

 

Stiffener 

family

Spacing, 

dS

Angle, ΨS,

degrees cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

Stringers Ly 0 1 0 As

Ribs Lx 90 0 1 Ar

Diagonals #1 Φ Ad1

Diagonals #2 180 - Φ Ad2

Stiffener 

family

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective shear 

modulus,

GS

Stringers Is Js Es Gs

Ribs Ir Jr Er Gr

Diagonals #1 Id1 Jd1 Ed1 Gd1

Diagonals #2 Id2 Jd2 Ed2 Gd2

zS zS

zs zs

zr zr

LxLy

Ld

Lx

Ld

Ly

Ld

zd1 zd1

LxLy

Ld

− Lx

Ld

Ly

Ld

zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

r k Zr

k Y

d1 k Zd1

k Y

d2 k Zd2

L d = L x

2 + L y2
61

Table 2. Stiffener attributes for an eccentric, orthogonal stiffening grid with one diagonal brace 

 per bay (see figure 16)

Table 2. Concluded

 

Stiffener 

family

Spacing, 

dS

Angle, ΨS,

degrees cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

Stringers Ly 0 1 0 As

Ribs Lx 90 0 1 Ar

Diagonals #1 Φ Ad1

Diagonals #2 180 - Φ Ad2

Stiffener 

family

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective shear 

modulus,

GS

Stringers Is Js Es Gs

Ribs Ir Jr Er Gr

Diagonals #1 Id1 Jd1 Ed1 Gd1

Diagonals #2 Id2 Jd2 Ed2 Gd2

zS zS

zs zs

zr zr

2LxLy

Ld

Lx

Ld

Ly

Ld

zd1 zd1

2LxLy

Ld

− Lx

Ld

Ly

Ld

zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

r k Zr

k Y

d1 k Zd1

k Y

d2 k Zd2

L d = L x

2 + L y2
62

Table 3. Stiffener attributes for an eccentric, isosceles-triangle stiffener pattern (see figure 17)

Table 3. Concluded

Stiffener 

family

Spacing, 

dS

Angle, ΨS,

degrees cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

Stringers Ly 0 1 0 As

Diagonals #1 Φ Ad1

Diagonals #2 180 - Φ Ad2

Stiffener 

family

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, ES

Effective shear 

modulus,

GS

Stringers Is Js Es Gs

Diagonals #1 Id1 Jd1 Ed1 Gd1

Diagonals #2 Id2 Jd2 Ed2 Gd2

zS zS

zs zs

LxLy

L

Lx

2L

Ly

L zd1 zd1

LxLy

L − Lx

2L

Ly

L zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

d1 k Zd1

k Y

d2 k Zd2
63

Table 4. Basic-cell beam-member attributes for an eccentric, orthogonal stiffener pattern with

 two diagonal braces per bay (see figure 14b)

Table 4. Concluded

 

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle, ΨS,

degrees cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

1-2 , 0 , 0 Lx 0 1 0 As

3-4 0, 0, Ly 90 0 1 Ar

5-6 , , Ld Φ Ad1

7-8 , , Ld 180 - Φ Ad2

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus,

 ES

Effective shear 

modulus,

GS

1-2 Is Js Es Gs

3-4 Ir Jr Er Gr

5-6 Id1 Jd1 Ed1 Gd1

7-8 Id2 Jd2 Ed2 Gd2

zS zS

− Lx

2

Lx

2 zs zs

− Ly

2

Ly

2 zr zr

− Lx

2 − Ly

2

Lx

2

Ly

2

Lx

Ld

Ly

Ld

zd1 zd1

Lx

2 − Ly

2 − Lx

2

Ly

2 − Lx

Ld

Ly

Ld

zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

r k Zr

k Y

d1 k Zd1

k Y

d2 k Zd2

L d = L x

2 + L y2
64

Table 5. Basic-cell beam-member attributes for an eccentric, orthogonal stiffener pattern with

 one diagonal brace per bay (see figure 16b)

Table 5. Concluded

 and Acell = 4LxLy 

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle,

ΨS,

degrees 

cosΨS sinΨS

Area, 

As

Effective 

eccentricity, 

Effective 

eccentricity, 

1-2 -Lx

, 0 Lx, 0 2Lx 0 1 0 As

3-4 0, -Ly 0, Ly 2Ly 90 0 1 Ar

5-6 -Lx, -Ly Lx, Ly 2Ld Φ Ad1

5-7 -Lx, -Ly Lx, -Ly 2Lx 0 1 0 As

5-8 -Lx, -Ly -Lx, Ly 2Ly 90 0 1 Ar

7-8 Lx, -Ly -Lx, Ly 2Ld 180 - Φ Ad2

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus,

 ES

Effective shear 

modulus,

GS

1-2 Is Js Es Gs

3-4 Ir Jr Er Gr

5-6 Id1 Jd1 Ed1 Gd1

5-7 Is Js Es Gs

5-8 Ir Jr Er Gr

7-8 Id2 Jd2 Ed2 Gd2

zS zS

zs zs

zr zr

Lx

Ld

Ly

Ld

zd1 zd1

zs zs

zr zr

− Lx

Ld

Ly

Ld

zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

r k Zr

k Y

d1 k Zd1

k Y

s k Zs

k Y

r k Zr

k Y

d2 k Zd2

L d = L x

2 + L y2
65

Table 6. Basic-cell beam-member attributes for a plate stiffened with an eccentric, isoscelestriangle stiffener pattern (see figure 17b)

Table 6. Continued

 

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle,

ΨS,

degrees 

cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

1-2 , 0 , 0 Lx 0 1 0 As

3-4 , , L Φ Ad1

5-6 , , L 180 - Φ Ad2

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective shear 

modulus,

GS

1-2 Is Js Es Gs

3-4 Id1 Jd1 Ed1 Gd1

5-6 Id2 Jd2 Ed2 Gd2

zS zS

− Lx

2

Lx

2 zs zs

− Lx

4 − Ly

2

Lx

4

Ly

2

Lx

2L

Ly

L zd1 zd1

Lx

4 − Ly

2 − Lx

4

Ly

2 − Lx

2L

Ly

L zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

d1 k Zd1

k Y

d2 k Zd2

L = L x

2 + L y2
66

Table 7. Basic-cell beam-member attributes for a plate stiffened with an eccentric Kagome 

stiffener pattern (see figure 18b)

Table 7. Continued

 

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle,

ΨS,

degrees 

cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

1-2 , , Lx 0 1 0 As

3-4 , , Lx 0 1 0 As

5-6 , -Ly , Ly L Φ Ad1

7-8 , -Ly , Ly L 180 - Φ Ad2

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective shear 

modulus,

GS

1-2 Is Js Es Gs

3-4 Is Js Es Gs

5-6 Id1 Jd1 Ed1 Gd1

7-8 Id2 Jd2 Ed2 Gd2

zS zS

− Lx

2 − Ly

2

Lx

2 − Ly

2 zs zs

− Lx

2

Ly

2

Lx

2

Ly

2 zs zs

− Lx

2

Lx

2

Lx

2L

Ly

L zd1 zd1

Lx

2 − Lx

2 − Lx

2L

Ly

L zd2 zd2

IYY

S

k Y

S k ZS

k Y

s k Zs

k Y

s k Zs

k Y

d1 k Zd1

k Y

d2 k Zd2

L = L x

2 + L y2
67

Table 8. Basic-cell-member attributes for a plate stiffened with an eccentric, hexagonal stiffener 

pattern (see figure 21b)

Table 8. Continued

 

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle,

ΨS,

degrees 

cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

2-1 -Φ Ad2

2-3 -180 + Φ Ad1

2-5 c 90

0 1

Ar

5-4 Φ Ad1

5-6 180 - Φ Ad2

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, 

JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective shear 

modulus,

GS

2-1 Id2 Jd2 Ed2 Gd2

2-3 Id1 Jd1 Ed1 Gd1

2-5 Ir Jr Er Gr

5-4 Id1 Jd1 Ed1 Gd1

5-6 Id2 Jd2 Ed2 Gd2

zS zS

0, − c

2 a

2, − b + c

2

L

2

a

L − b

L zd2 zd2

0, − c

2 − a

2, − b + c

2

L

2 − a

L − b

L zd1 zd1

0, − c

2 0, c2 zr zr

0, c

2 a

2, b + c

2

L

2

a

L

b

L zd1 zd1

0, c

2 − a

2, b + c

2

L

2 − a

L

b

L zd2 zd2

IYY

S

k Y

S k ZS

k Y

d2 k Zd2

k Y

d1 k Zd1

k Y

r k Zr

k Y

d1 k Zd1

k Y

d2 k Zd2

L = a2 + b2
68

Table 9. Basic-cell-member attributes for a plate stiffened with an eccentric, star-shaped stiffener 

pattern (see figure 23b)

Member 

designation

Starting 

coordinates, 

(x, y)

Ending 

coordinates, 

(x, y)

Length, 

LS

Angle,

ΨS,

degrees 

cosΨS sinΨS

Area, 

AS

Effective 

eccentricity, 

Effective 

eccentricity, 

1-12 L/3 - Φ Ad2

1-2 L/3 Φ Ad1

3-2 B/3 0 1 0 As

3-4 L/3 180 − Φ Ad2

5-4 L/3 Φ Ad1

5-6 B/3 180 -1 0 As

7-6 L/3 180 − Φ Ad2

7-8 L/3 180 + Φ Ad1

9-8 B/3 180 0 1 As

9-10 L/3 − Φ Ad2

11-10 L/3 180 + Φ Ad1

11-12 B/3 0 1 0 As

zS zS

B

3 , 0 B2 , − H

3

B

2L − H

L zd2 zd2

B

3 , 0 B2 , H

3

B

2L

H

L zd1 zd1

B

6 , H

3

B

2 , H

3 zs zs

B

6 , H

3 0, 2H3 − B2L

H

L zd2 zd2

− B

6 , H3 0, 2H3

B

2L

H

L zd1 zd1

− B

6 , H3 − B2 , H3 zs zs

− B

3 , 0 − B2 , H3 − B2L

H

L zd2 zd2

− B

3 , 0 − B2 , − H3 − B2L − HL zd1 zd1

− B

6 , − H3 − B2 , − H3 zs zs

− B

6 , − H3 0, − 2H3

B

2L − H

L zd2 zd2

B

6 , − H

3 0, − 2H3 − B2L − HL zd1 zd1

B

6 , − H

3

B

2 , − H

3 zs zs
69

Table 9. Concluded

Member 

designation

Effective 

moment of 

inertia, 

Torsion 

constant, JS

Shear 

correction 

factor, 

Shear 

correction 

factor, 

Effective 

modulus, 

ES

Effective 

shear 

modulus,

GS

1-12 Id2 Jd2 Ed2 Gd2

1-2 Id1 Jd1 Ed1 Gd1

3-2 Is Js Es Gs

3-4 Id2 Jd2 Ed2 Gd2

5-4 Id1 Jd1 Ed1 Gd1

5-6 Is Js Es Gs

7-6 Id2 Jd2 Ed2 Gd2

7-8 Id1 Jd1 Ed1 Gd1

9-8 Is Js Es Gs

9-10 Id2 Jd2 Ed2 Gd2

11-10 Id1 Jd1 Ed1 Gd1

11-12 Is Js Es Gs

IYY

S

k Y

S k ZS

k Y

d2 k Zd2

k Y

d1 k Zd1

k Y

s k Zs

k Y

d2 k Zd2

k Y

d1 k Zd1

k Y

s k Zs

k Y

d2 k Zd2

k Y

d1 k Zd1

k Y

s k Zs

k Y

d2 k Zd2

k Y

d1 k Zd1

k Y

s k Zs
70

(a) Space Shuttle 

(b) Internal structure of external tank 

Figure 1. Space Shuttle and external tank structure. 

External tank

Solid-rocket 

booster

Orbiter

Liquid-oxygen tank

Liquid-hydrogen tank
71

Figure 2. Densely stiffened orthogrid cylinder.
72

Figure 3. Isogrid-stiffened cylinders.
73

(a) Unidirectionally Stiffened Plate

(b) Family of beams within a "small" rectangular plate region

dS

Figure 4. Unidirectionally stiffened plate geometry.

x-axis

y-axis

z-axis

y-axis

x-axis

 

ΨS

A

A
74

dS

Stiffener spacing, dS

Plate wall

Stiffener

Thickness, h Plate 

midplane

Figure 5. Cross-section A-A of a uniformly stiffened panel shown in figure 4b.

zS

Z-axis

Y-axis

X-axis

x-axis

z-axis

y-axis

Plate midplane

Stiffness-weighted

Generic stiffener

 

ΨS

Figure 6. Plate and stiffener coordinate systems.

 

η-axis

 

ζ-axis

 

ξ-axis

zS

yS

centroidal axis

Stiffener reference axis
75

Figure 7. Examples of a nonhomogeneous stiffener cross-section.

Plate

High-modulus

pultuded rod

Orthotropic

materials

dS

h Z

z, Z, and ζ axes

η-axis Plate

midplane

Figure 8. Repetitive stiffened-panel element and coordinate systems.

Y-axis

zS

dS

h Plate

midplane

hS

Equivalent-stiffener layer

Plate

Figure 9. Equivalent-plate wall.
76

Figure 10. Beam stresses at an arbitrary cross-section.

η

ζ dAS

Generic point, P

Cross-section area, AS

η-axis

ζ-axis

ξ-axis

Y-axis

Z-axis

X-axis

Y

σXZ

σXY

σXX

Stiffness-weighted

centroidal axis Z

zS

yS
77

Z-axis

Y-axis

X-axis

x-axis

z-axis

y-axis

Plate midplane

Stiffness-weighted

Generic stiffener

ΨS

η-axis

ζ-axis

ξ-axis

centroidal axis

Figure 11. Beam force and moment resultants at an arbitrary cross-section.

P(X)

VY(X)

VZ(X)

T(X)

MY(X)

MZ(X)

AS

yS

zS
78

P VY

T MY

MZ

Figure 12. Stiffener beam forces and moments acting on cross-section of repetitive 

dS

stiffened-panel element.

Plate midplane

X-axis

Y-axis

Z-axis

VZ

QXZ

Figure 13. Plate stress resultants acting on cross-section of equivalent 

stiffened-panel wall.

Plate midplane and

equivalent plate wall

reference surface

Equivalent-stiffener layer

N XX 

NXY

 

MXY

dS

X-axis Y-axis

Z-axis

 

MXX
79

8 4 6

1 2

5 3 7

Figure 14. Orthogonal stiffener pattern with two diagonal braces per bay and basic cell. 

(a) Stiffener arrangement

x-axis

y-axis

Ly

Lx

(b) Basic Cell

Rib

Stringer

Diagonal

x-axis

Ly

1

2

Ly

1

2

y-axis

Lx

1

2 Lx

1

2

 

Φ Φ

Basic cell

perimeter

braces
80

x-axis

y-axis

Ly

Lx Stringer

Braces

Figure 15. Diamond-shaped stiffener pattern. 
81

8 6

1 2

Figure 16. Orthogonal stiffener pattern with one diagonal brace per bay and basic cell. 

(a) Stiffener arrangement

x-axis

y-axis

Ly

Lx

(b) Basic Cell

Rib

Stringer

x-axis

y-axis

 

Φ Φ

Basic cell

perimeter

5 3 7

Ly

Ly

Lx

4

Lx

Diagonal

braces
82

x-axis

y-axis

Ly

Lx Stringer

Braces

Figure 17. Isosceles-triangle stiffener pattern and basic cell. 

(a) Stiffener arrangement

(b) Basic cell and adjacent structure

Basic cell

Triangle centroid

Ly

Ly

1

3

Ly

Ly

1

3

Lx

1

2 Lx

1

2

Φ Φ

L

ds

4

2

6

1

3 5

perimeter

ds = LxLy

L

Acell = LxLy

y-axis

x-axis
83

x-axis

y-axis

Ly

Lx

Stringer

Braces

Figure 18. Kagome stiffener pattern and basic cell. 

(a) Stiffener arrangement

(b) Basic cell and adjacent structure

Basic cell

Ly

1

2

Lx

1

2 Lx

1

2

Φ Φ L

dS

perimeter

4

6

Ly

1

2

8

5 7

1 2

3

y-axis

x-axis

Ly

1

2

Ly

1

2 Acell = 2LxLy

Lx

1 L2

1 L2

dS = LxLy

L
84

Figure 19. Generation of an isosceles-triangle stiffener pattern obtained by 

translating the basic cell defined in figure 17. 

Figure 20. Generation of a Kagome stiffener pattern obtained by translating the basic

cell defined in figure 18. 
85

x-axis

c Ly

b

y-axis

y-axis

x-axis

Figure 21. Hexagon-shaped stiffener pattern and basic cell. 

(a) Stiffener arrangement

(b) Basic cell and adjacent structure

3 1

2

4

5

6

Basic cell

perimeter

Φ

c

a

b

L

Acell = 2a b + c

1 c2

Rib

Diagonals

Lx = a

1 a2

(b + c) 1

2
86

Figure 22. Generation of a hexagonal stiffener pattern obtained by translating the basic

cell defined in figure 21. 
87

y-axis

x-axis

Figure 23. Isosceles-star-shaped stiffener pattern and basic cell. 

(a) Stiffener arrangement

(b) Basic cell and adjacent structure

11

2

1

6

Basic cell

perimeter

Φ

1 H3

x-axis

y-axis

12

5

4

3

7

8

9

10

1 H3

1 H3

1 H3

2 H3

1 B21 B2

Centroid of

isosceles

triangles

Centroid of

hexagon

1 B6

1 B3

L

Acell = 4

3BH

2 H3

B

H

Stringer

Diagonals
88

Figure 24. Generation of an isosceles-star-shaped stiffener pattern obtained by

translating the basic cell defined in figure 23. 
89

x-axis

z-axis y-axis

h1

h2

hc

hc

1

2

e1

e2 x-axis

Plate no. 1

Plate no. 2

z-axis

Hexagon-cell core

hc

1

2

hc

1

2

h2

1

2

h1

1

2

(a) Cut-away view of sandwich plate construction

(b) Plate cross-section

tr

td

c b

L

Figure 25. Sandwich plate with nonidentical anisotropic face plates and a hexagon-cell core.

Plate no. 2

Diagonal

Rib
90

(a) Cut-away view of sandwich plate construction

Figure 26. Sandwich plate with nonidentical anisotropic face plates and an orthogrid core.

e1

e2

x-axis

Plate no. 1

Plate no. 2

z-axis

Orthogrid core

hc

1

2

hc

1

2

h2

1

2

h1

1

2

tr

(b) Plate cross-section

h1

h2

hc

x-axis

y-axis z-axis

hc

1

2

tr

ts

Plate no. 2

Plate no. 1

Ly

Lx

Lx
91

(a) Cut-away view of sandwich plate construction

Figure 27. Sandwich plate with nonidentical anisotropic face plates and a star-cell core.

e1

e2

x-axis

Plate no. 1

Plate no. 2

z-axis

Star-cell core

hc

1

2

hc

1

2

h2

1

2

h1

1

2

(b) Plate cross-section

h1

h2

hc

x-axis

y-axis

z-axis

hc

1

2

t

Plate no. 2

Plate no. 1

2 H3

B
92

x-axis

x ́-axis

z ́-axis

z-axis

z ́

z

h

e

h/2

Figure 28. Arbitrary face plate of a sandwich plate.

Face plate
93

Appendix A

Equations of First-Order Transverse-Shear Deformation Beam Theory

Let (X, Y, Z) denote the coordinates of the material particles forming a beam stiffener, with 

respect to the noncentroidal coordinate system shown in figure 6. The displacement of any 

material particle in the beam is given by the X-, Y-, and Z-components U(X, Y, Z), V(X, Y, Z), and 

W(X, Y, Z), respectively. Consistent with Euler-Bernoulli and Timoshenko’s first-order 

transverse-shear-deformation beam theories, these displacement components are expressed as

 (A1a)

 (A1b)

 (A1c)

where u(X), v(X), and w(X) are the displacements of the corresponding point on the beam stiffener 

reference axis, and φX(X), φY(X), and φZ(X) are the dextral rotations of the beam cross-section 

about the X-, Y-, and Z-axes, respectively. For these kinematics, the cross-sectional planes are 

presumed to be planes of elastic symmetry; that is, the beam material is, at most, monoclinic with 

respect to the longitudinal X axis (see Lekhnitskii,143 p. 271 and p. 293). The effects of crosssectional warping restraint associated with torsion of noncircular cross-sections are neglected in 

these kinematic equations and each cross section is presumed to warp in an identical manner. 

Substituting these displacement expressions into the linear strain-displacement relations for a 

three-dimensional elastic solid gives

 (A2a)

 (A2b)

 (A2c)

 (A2d)

where

 (A3a)

 (A3b)

 (A3c)

U X, Y, Z = u X – YφZ X + ZφY X

V X, Y, Z = v X – ZφX X

W X, Y, Z = w X + YφX X

εXX X, Y, Z = eXX

o

X + Y χZ

o

X + Z χY

o

X

εYY X, Y, Z = εZZ X, Y, Z = γYZ X, Y, Z = 0

γXY X, Y, Z = ΓXY

o

X − Z τ

o

X

γXZ X, Y, Z = ΓXZ

o

X + Y τ

o

X

eXX

o

X = ∂u

∂X

ΓXY

o

X = ∂v

∂X – φZ

ΓXZ

o

X = ∂w

∂X + φY
94

 (A3d)

 (A3e)

 (A3f)

The symbol represents the change in twist of the beam, associated with torsion. The 

symbols and represent the changes in curvature in the X-Z and X-Y planes, 

respectively. The symbols and represent the transverse shear deformations in the 

X-Y and X-Z planes, respectively. In first-order shear-deformation beam theory, the set u(X), 

v(X), w(X), φX(X), φY(X), and φZ(X) are the primary unknowns. In classical Euler-Bernoulli beam 

theory, the transverse shear deformations are presumed negligible, which yields

 (A4a)

 (A4b)

and eliminates two of the unknowns. In addition, equations (A3e) and (A3f) yield

 (A4c)

 (A4d)

In the Euler-Bernoulli and first-order shear-deformation beam theories, all stresses are 

presumed to be negligible except σXX, σXY, and σXZ. These stresses are shown in figure 10. The 

corresponding resultant internal forces are defined, with respect to the (X, Y, Z) coordinates, as 

 (A5a)

 (A5b)

 (A5c)

τ

o

X = ∂φX

∂X

χY

o

X = ∂φY

∂X

χZ

o

X = – ∂φZ

∂X

τ

o

X

χY

o

X χZ

o

X

ΓXY

o

X ΓXZ

o

X

φY = − ∂w

∂X

φZ = ∂v

∂X

χY

o

X = − ∂

2

w

∂X2

χZ

o

X = − ∂

2

v

∂X2

P X = σ XX dYdZ

A S

VY X = σ XY dYdZ

A S

VZ X = σ XZ dYdZ

A S
95

 (A5d)

 (A5e)

 (A5f)

where AS denotes the cross-sectional area of the beam stiffener. For these definitions, the 

corresponding positive valued moments are shown in figure 11. 

The constitutive equations for a homogeneous, isotropic beam are obtained by substituting the 

beams strains and stresses into the constitutive equations for a general three-dimensional body. 

The result is

 (A6a)

 (A6b)

 (A6c)

where E and G denote Young’s extensional modulus and the shear modulus, respectively. 

Substituting equations (A2) into (A6), and then substituting the results into equations (A5) gives

 (A7a)

 (A7b)

 (A7c)

 (A7d)

 (A7e)

 (A7f)

T X = Yσ XZ – Zσ XY dYdZ

A S

MY X = Z σ XX dYdZ

A S

MZ X = Y σ XX dYdZ

A S

σ XX = EεXX

σ XY = GγXY

σ XZ = GγXZ

P = EAS eXX

o

+ YχZ

o

+ ZχY

o

VY = k YGAS ΓXY

o

− Z τ

o

VZ = k ZGAS ΓXZ

o

+ Y τ

o

T = GAS k ZY ΓXZ

o

− k YZ ΓXY

o

+ GI pτ

o

MY = EASZ eXX

o

+ EIYZ χZ

o

+ EIYY χY

o

MZ = EASY eXX

o

+ EIZZ χZ

o

+ EIYZχY

o
96

where are the coordinates of the cross-section centroid, kY and kZ are transverse-shear 

correction factors, and it is noted that 

 (A8a)

 (A8b)

 (A8c)

 (A8d)

 (A8e)

 (A8f)

 

The symbols IYY and IZZ represent the area moments of inertia with respect to the noncentroidal 

(X, Y, Z) coordinate system, and IYZ denotes the corresponding product of inertia. The symbol IP 

denotes the polar moment of inertia, with respect to the origin of the coordinate system. To 

account for torsion of noncircular cross-sections, based on the St. Venant Theory of Torsion, 

equation (A7d) is expressed as

 (A9)

where J is the torsion constant of elasticity theory.144 

 A set of constitutive equations for nonhomogeneous beams, similar to equations (A7) and to 

those given by Allen and Haisler145 (see pp. 171-187), are obtained as follows. First, the beam 

materials are presumed to be, at most, specially orthotropic with respect to the beam coordinate 

system. The material constitutive equations are given by

Y, Z

Y dYdZ

A S

= ASY

Z dYdZ

A S

= ASZ

Z2dYdZ

A S

≡ IYY

Y2dYdZ

A S

≡ IZZ

YZ dYdZ

A S

≡ IYZ

Y2+ Z2dYdZ

A S

= IP

T = GAS k ZY ΓXZ

o

− k YZ ΓXY

o

+ GJ τ

o
97

 (A10)

where EX, EY, and EZ are the principal Young’s moduli; νXY, νXZ, and νYZ are the major Poisson’s 

ratios; νYX, νZX, and νZY are the minor Poisson’s ratios; and GXY, GYZ, and GXZ are the principal shear 

moduli. For a nonhomogeneous cross-section, these material parameters are functions of the Y 

and Z cross-sectional coordinates. Based on equations (A2) and the presumption that σYY, σZZ, 

and σYZ are negligible, equation (A10) reduces to

 (A11a)

or

 (A11b)

Substituting equations (A2) into (A11), and then substituting the results into equations (A5) gives

 (A12a)

εXX

εYY

εZZ

γYZ

γXZ

γXY

=

1

EX

− νYX

EY

− νZX

EZ

0 0 0

− νXY

EX

1

EY

− νZY

EZ

0 0 0

− νXZ

EX

− νYZ

EY

1

EZ

0 0 0

0 0 0 1

GYZ

0 0

0 0 0 0 1

GXZ

0

0 0 0 0 0 1

GXY

σ XX

σ YY

σ ZZ

σ YZ

σ XZ

σ XY

εXX

0

0

0

γXZ

γXY

=

1

EX

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1

GXZ

0

0 0 0 0 0 1

GXY

σ XX

0

0

0

σ XZ

σ XY

σ XX

0

0

0

σ XZ

σ XY

=

EX 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 GXZ 0

0 0 0 0 0 GXY

εXX

0

0

0

γXZ

γXY

P X = EX dYdZ

A S

eXX

o

+ EX Y dYdZ

A S

χZ

o

+ EX Z dYdZ

A S

χY

o
98

 (A12b)

 (A12c)

 (A12d)

 (A12e)

 (A12f)

By defining the effective quantities

 (A13a)

 (A13b)

 (A13c)

 (A13d)

 (A13e)

 (A13f)

VY X = GXY dYdZ

A S

ΓXY

o

− GXYZ dYdZ

A S

τ

o

VZ X = GXZ dYdZ

A S

ΓXZ

o

+ GXZY dYdZ

A S

τ

o

T X = GXZY dYdZ

A S

ΓXZ

o

− GXYZ dYdZ ΓXY

o

A S

+

GXZY2

+ GXYZ2dYdZ τ

o

A S

MY X = EXZ dYdZ

A S

eXX

o

+ EX YZ dYdZ

A S

χZ

o

+ EX Z2dYdZ

A S

χY

o

MZ X = EXY dYdZ

A S

eXX

o

+ EX Y2dYdZ

A S

χZ

o

+ EX YZ dYdZ

A S

χY

o

ES = 1

AS

EX dYdZ

A S

GXZ

S

= 1

AS

GXZ dYdZ

A S

GXY

S

= 1

AS

GXY dYdZ

A S

yS =

EX Y dYdZ

A S

ASES

zS =

EX Z dYdZ

A S

ASES

IYY

S

=

EX Z2

dYdZ

A S

ES
99

 (A13g)

 (A13h)

 (A13i)

 (A13j)

 

 (A13k)

equations (A12) are expressed as

 (A14a)

 (A14b)

 (A14c)

 (A14d)

 (A14e)

 (A14f)

where ES is the effective extensional modulus of the nonhomogeneous beam stiffener; and 

IZZ

S

=

EX Y2

dYdZ

A S

ES

IYZ

S

=

EXYZ dYdZ

A S

ES

IP

S

=

Y2

GXZ + Z2GXY dYdZ

A

GXZ

S

GXY

S

ys =

GXZY dYdZ AS

ASGXZ

S

zS =

GXYZ dYdZ

A S

ASGXY

S

P = ESAS eXX

o

+ ySχZ

o

+ zSχY

o

VY = k YGXY

S

AS ΓXY

o

− zS τ

o

VZ = k ZGXZ

S

AS ΓXZ

o

+ yS τ

o

T = k ZGXZ

S

ASySΓXZ

o

− k YGXY

S

ASzS ΓXY

o

+ GXZ

S

GXY

S

IP

S

τ

o

MY = ESASzS eXX

o

+ ESIYZ

S

χZ

o

+ ESIYY

S

χY

o

MZ = ESASySeXX

o

+ ESIZZ

S

χZ

o

+ ESIYZ

S

χY

o

GXZ

S
100

 are effective shear moduli; are stiffness-weighted first moments of area; 

 and are effective, stiffness-weighted moments of inertia; is the corresponding 

stiffness-weighted product of inertia; and is a stiffness-weighted polar moment of inertia. It is 

noteworthy to indicate that for homogeneous orthotropic materials with the principal axes aligned 

with the beam axes , , , , , , , 

, and . To account for torsion of noncircular cross-sections based on 

the St. Venant Theory of Torsion, equation (A14d) is expressed as

 (A15a)

where JS is an effective torsion constant that based on elasticity theory, and generally depends on 

the material composition in addition to the cross-section geometry, and

 (A15b)

For a stiffener with a rectangular cross-section of height h and thickness t, made of a 

homogeneous orthotropic material, Leknitskii143 gives an expression for the torsional stiffness that 

is equivalent to

 (A16)

The strain energy of a beam is obtained from the general definition given by the theory of 

elasticity; that is,

 (A17)

where VS denotes the volume of the material particles forming the beam stiffener. For the 

prismatic beams considered herein, the volume is the product of the cross-sectional area AS and 

the length LS. Using equations (A2) and (A5) with (A17) gives the beam strain energy as

 (A18)

For the analysis presented herein, it is convenient to define a vector of beam forces and moments 

by

GXY

S

yS, zS, yS, and zS

IYY

S

IZZ

S

IYZ

S

IP

S

ES = EX GXZ

S

= GXZ GXY

S

= GXY yS = yS = Y zS = zS = Z IYY

S

= IYY IZZ

S

= IZZ

IYZ

S

= IYZ IP

S

= GXZIZZ + GXYIYY

T = k ZGXZ

S

ASySΓXZ

o

− k YGXY

S

ASzS ΓXY

o

+ GSJSτ

o

GS ≡ GXZ

S

GXY

S

JS = GXZ

GXY

1

2

t

3

h

3 1 − 96

π5

t

h

GXZ

GXY

1

2 1 − −1 p

p

5 tanh pπh

2t

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞

Eε = 1

2 σ XXεXX + σ YYεYY + σ ZZεZZ + σ XYγXY+ σ XZγXZ+ σ YZ γYZ dXdYdZ

VS

Eε = 1

2 PeXX

o

+ MZχZ

o

+ MYχY

o

+ VYΓXY

o

+ VZΓXZ

o

+ Tτ

o

dX

LS
101

 (A19)

and strains by 

 (A20)

where the superscript "T" denotes matrix transposition. Then, equations (A7) and (A9) yield

 (A21)

where 

 (A22)

is the constitutive matrix for a homogeneous beam made of isotropic material. For a 

heterogeneous beam made of, at most, specially orthotropic materials, equations (A14) and (A15) 

yield

 (A23)

To obtain a matrix equation similar in form to equation (A22), it is convenient to introduce the 

stiffness-weight shear correction factors

 (A24a)

 (A24b)

F

T

= P VY VZ MZ MY T

ε b

T

= eXX

o

ΓXY

o

ΓXZ

o

χZ

o

χY

o

τ

o

F = C ε b

C =

EAS 0 0 EASY EASZ 0

• k YGAS 0 0 0 − k YGASZ

• • k ZGAS 0 0 k ZGASY

• • • EIZZ EIYZ 0

• • • • EIYY 0

symmetric • • • • GJ

C =

ESAS 0 0 ESASyS ESASzS 0

• k YGXY

S

AS 0 0 0 − k YGXY

S

ASzS

• • k ZGXZ

S

AS 0 0 k ZGXZ

S

ASyS

• • • ESIZZ

S

ESIYZ

S

0

• • • • ESIYY

S

0

symmetric • • • • GSJS

k Y

S

= k Y

GXY

S

GXZ

S

k Z

S

= k Z

GXZ

S

GXY

S
102

such that equation (A23) becomes

 (A25)

Note that for a homogeneous beam, equation (A25) has the same structure as equation (A22). By 

using equations (A11) - (A15), the strain energy is expressed as 

 (A26)

In equation (A25), define the coordinates of a stiffness-weighted centroidal axis, 

shown in figures 6 and 11, which is coincident with the centroidal axis for prismatic beams with 

homogeneous cross sections. By defining the stiffness-weighted-centroidal coordinates (ξ, η, ζ) 

shown in figures 10 and 11, the effective quantities defined by equations (A13) yield

 (A27a)

 (A27b)

 (A27c)

 (A27d)

 (A27e)

C =

ESAS 0 0 ESASyS ESASzS 0

• k Y

S

GSAS 0 0 0 − k Y

S

GSASzS

• • k Z

S

GSAS 0 0 k Z

S

GSASyS

• • • ESIZZ

S

ESIYZ

S

0

• • • • ESIYY

S

0

symmetric • • • • GSJS

Eε = 1

2 F T

ε b dX

LS

= 1

2 ε b

T

C ε b dX

LS

X, yS, zS

ES = 1

AS

EX dηdζ

A S

GXZ

S

= 1

AS

GXZ dηdζ

A S

GXY

S

= 1

AS

GXY dηdζ

A S

ηS =

EX η dηdζ

A S

ASES

= 0

ζS =

EX ζ dηdζ

A S

ASES

= 0
103

 (A27f)

 (A27g)

 (A27h)

 (A27j)

 

 (A27k)

with the stiffness-weighted-centroidal moments and product of inertia 

 (A28a)

 (A28b)

 (A28c)

and the shear-stiffness-weighted first moments of area

 (A29a)

 (A29b)

For beams with homogeneous cross sections, the shear-stiffness-weighted first moments of area 

vanish and the stiffness-weighted-centroidal moments and product of inertia become the 

corresponding centroidal moments and product of inertia. 

IYY

S

= Iηη

S

+ ASzS

2

IZZ

S

= I ζζ

S

+ ASyS

2

IYZ

S

= Iηζ

S

+ ASySzS

yS = ηS + yS

zS = ζS + zS

Iηη

S

=

EX ζ

2

dηdζ

A S

ES

I ζζ

S

=

EX η2

dηdζ

A S

ES

Iηζ

S

=

EX ηζ dηdζ

A S

ES

ζS =

GXYζ dηdζ

A S

ASGXY

S

ηS =

GXZη dηdζ

A S

ASGXZ

S
104

Appendix B

Equivalent-Plate Stiffnesses for a Plate Reinforced with Eccentric Stringers, 

Ribs, and Diagonal Braces

The expressions presented in this appendix are for a general laminated-composite plate that is 

stiffened with stringers, ribs, and diagonal braces, as shown in figure 14 or figure 16. The 

stringers and ribs are eccentric with respect to the plate midplane and the pockets formed by the 

skin, stringers, and ribs are diagonally braced with two nonidentical families of eccentric 

stiffeners. The notation used for the material and section properties and orientation angle of each 

stiffener family is given in Tables 1 and 2 for the stiffener arrangements shown in figures 14 and 

16, respectively. In particular, the subscripts and superscripts "s" and "r" refer to quantities 

associated with the stringers and ribs, respectively. The subscripts and superscripts "d1" and "d2" 

refer to quantities associated with the two nonidentical families of diagonals. The stiffness 

expressions are obtained from equations (23) and (25)-(28) by applying equations (25)-(28) to 

each family of stiffeners with the attributes given in Tables 1 and 2. The resulting equivalent-plate 

stiffnesses are given as follows where n = 1 and 1/2 for the stiffener arrangements shown in 

figures 14 and 16, respectively. In the expressions that follow, the stiffener extensional modulus, 

shear modulus, eccentricity, moment of inertia, torsion constant, and transverse-shear correction 

factors refer to the corresponding effective quantities defined in Appendix A for a 

nonhomogeneous, specially orthotropic beam. In addition, the length of the diagonal stiffeners is 

given by . 

 (B1)

 (B2)

 (B3)

 (B4)

 (B5)

Ld = Lx

2

+ Ly

2

A11 = A11

plate

+

EsAs

Ly

+

nLx

3

Ld

3

Ly

Ed1Ad1 1 + τY

d1 Ly

2

Lx

2 + Ed2Ad2 1 + τY

d2 Ly

2

Lx

2

A12 = A12

plate

+

nLxLy

Ld

3 Ed1Ad1 1 − τY

d1

+ Ed2Ad2 1 − τY

d2

A16 = A16

plate

+

nLx

2

Ld

3 Ed1Ad1 1 − τY

d1

2

1 − Ly

2

Lx

2 − Ed2Ad2 1 − τY

d2

2

1 − Ly

2

Lx

2

A22 = A22

plate

+

ErAr

Lx

+

nLy

3

Ld

3

Lx

Ed1Ad1 1 + τY

d1 Lx

2

Ly

2 + Ed2Ad2 1 + τY

d2 Lx

2

Ly

2

A26 = A26

plate

+

nLy

2

Ld

3 Ed1Ad1 1 − τY

d1

2

1 − Lx

2

Ly

2 − Ed2Ad2 1 − τY

d2

2

1 − Lx

2

Ly

2
105

 (B6)

 (B7)

 (B8)

 (B9)

 (B10)

 (B11)

 (B12)

 (B13)

 (B14)

A66 = A66

plate

+

EsAsτY

s

4Ly

+

ErArτY

r

4Lx

+

nLxLy

Ld

3 Ed1Ad1 1 + τY

d1

4

Lx

Ly

− Ly

Lx

2

+ Ed2Ad2 1 + τY

d2

4

Lx

Ly

− Ly

Lx

2

B11 = B11

plate

+

EsAszs

Ly

+

nLx

3

Ld

3

Ly

Ed1Ad1 zd1 + τY

d1

zd1

Ly

2

Lx

2 + Ed2Ad2 zd2 + τY

d2

zd2

Ly

2

Lx

2

B12 = B12

plate

+

nLxLy

Ld

3 Ed1Ad1 zd1 − τY

d1

zd1 + Ed2Ad2 zd2 − τY

d2

zd2

B16 = B16

plate

+

nLx

2

Ld

3 Ed1Ad1 zd1 − τY

d1

zd1

2

1 − Ly

2

Lx

2 − Ed2Ad2 zd2 − τY

d2

zd2

2

1 − Ly

2

Lx

2

B22 = B22

plate

+

ErArzr

Lx

+

nLy

3

Ld

3

Lx

Ed1Ad1 zd1 + τY

d1

zd1

Lx

2

Ly

2 + Ed2Ad2 zd2 + τY

d2

zd2

Lx

2

Ly

2

B26 = B26

plate

+

nLy

2

Ld

3 Ed1Ad1 zd1 − τY

d1

zd1

2

1 − Lx

2

Ly

2 − Ed2Ad2 zd2 − τY

d2

zd2

2

1 − Lx

2

Ly

2

B66 = B66

plate

+

EsAsτY

s

zs

4Ly

+

ErArτY

r

zr

4Lx

+

nLxLy

Ld

3 Ed1Ad1 zd1 +

τY

d1

zd1

4

Lx

Ly

− Ly

Lx

2

+ Ed2Ad2 zd2 +

τY

d2

zd2

4

Lx

Ly

− Ly

Lx

2

D11 = D11

plate

+

EsIs

Ly

+

nLx

3

Ld

3

Ly

Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

Ly

2

Lx

2

D12 = D12

plate

+

nLxLy

Ld

3 Ed1Id1 + Ed2Id2 − Gd1Jd1 − Gd2Jd2
106

 (B15)

 (B16)

 (B17)

 (B18)

 (B19)

 (B20)

 (B21)

where the inplane-shear-deformation parameters are defined as

 (B22)

 (B23)

 (B24)

 (B25)

D16 = D16

plate

+ n

Lx

2

Ld

3 Ed1Id1 − Ed2Id2 +

Gd2Jd2 − Gd1Jd1

2

1 − Ly

2

Lx

2

D22 = D22

plate

+

ErIr

Lx

+

nLy

3

LxLd

3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

Lx

2

Ly

2

D26 = D26

plate

+ n

Ly

2

Ld

3 Ed1Id1 − Ed2Id2 +

Gd1Jd1 − Gd2Jd2

2

Lx

2

Ly

2 − 1

D66 = D66

plate

+ 1

4

GsJs

Ly

+

GrJr

Lx

+

nLxLy

Ld

3 Ed1Id1 + Ed2Id2 +

nLxLy

4Ld

3 Gd1Jd1 + Gd2Jd2

Lx

Ly

− Ly

Lx

2

A44 = A44

plate

+

ErArτZ

r

Lx

+

nLy

LxLd

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

A45 = A45

plate

+ n

Ed1Ad1τZ

d1

− Ed2Ad2τZ

d2

Ld

A55 = A55

plate

+

EsAsτZ

s

Ly

+

nLx

LyLd

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

τY

s

≡ k Y

s

Gs

Es

τY

r

≡ k Y

r

Gr

Er

τY

d1

≡ k Y

d1

Gd1

Ed1

τY

d2

≡ k Y

d2

Gd2

Ed2
107

and the transverse-shear-deformation parameters are defined as

 (B26)

 (B27)

 (B28)

 (B29)

Setting these shear-deformation parameters equal to zero eliminates the effects of stiffener shear 

deformation. For the special case of no diagonal braces, the stiffnesses reduce to

 (B30)

 (B31)

 (B32)

 (B33)

τZ

s

≡ k Z

s

Gs

Es

τZ

r

≡ k Z

r

Gr

Er

τZ

d1

≡ k Z

d1

Gd1

Ed1

τZ

d2

≡ k Z

d2

Gd2

Ed2

A11 A12 A16

A12 A22 A26

A16 A26 A66

=

A11

plate A12plate A16plate

A12

plate A22plate A26plate

A16

plate A26plate A66plate

+

EsAs

Ly

0 0

0 Er

Ar

Lx

0

0 0 EsAsτY

s

4Ly

+

ErArτY

r

4Lx

B11 B12 B16

B12 B22 B26

B16 B26 B66

=

B11

plate B12plate B16plate

B12

plate B22plate B26plate

B16

plate B26plate B66plate

+

EsAs

Ly

zs 0 0

0 Er

Ar

Lx

zr 0

0 0 EsAsτY

s

4Ly

zs +

ErArτY

r

4Lx

zr

D11 D12 D16

D12 D22 D26

D16 D26 D66

=

D11

plate D12plate D16plate

D12

plate D22plate D26plate

D16

plate D26plate D66plate

+

EsIs

Ly

0 0

0 Er

Ir

Lx

0

0 0 1

4

GsJs

Ly

+

GrJr

Lx

A44 A45

A45 A55

= A44

plate A45plate

A45

plate A55plate +

ErArτZ

r

Lx

0

0 EsAsτZ

s

Ly
108

Appendix C

Equivalent-Plate Stiffnesses for a Plate Reinforced with an Isosceles-Triangle 

or Kagome Grid Stiffener Arrangement

The expressions presented in this appendix are for a general laminated-composite plate that is 

stiffened with stringers and two nonidentical families of diagonal braces, as shown in figure 17 or 

figure 18. The stringers and diagonal braces are eccentric with respect to the plate midplane and 

the pockets formed by the skin and stiffeners shown in figure 17 are isosceles triangles. The 

pockets shown in figure 18 for the Kagome grid are isosceles triangles and hexagons. The 

notation used for the material and section properties and orientation angle of each stiffener family 

is given in Table 3 for both stiffener arrangements. In particular, the subscripts and superscripts 

"s" refer to quantities associated with the stringers, and the subscripts and superscripts "d1" and 

"d2" refer to quantities associated with the two nonidentical families of diagonals. The stiffness 

expressions presented subsequently are obtained from equations (23) and (25)-(28) by applying 

equations (25)-(28) to each family of stiffeners with the attributes given in Table 3. In the 

expressions that follow, the stiffener extensional modulus, shear modulus, eccentricity, moment 

of inertia, torsion constant, and transverse-shear correction factors refer to the corresponding 

effective quantities defined in Appendix A for a nonhomogeneous, specially orthotropic beam. In 

addition, the length of the diagonal stiffeners is given by .

 (C1)

 (C2)

 (C3)

 (C4)

 (C5)

 (C6)

L = Lx

2

+ Ly

2

A11 = A11

plate

+

EsAs

Ly

+

Lx

3

16L3Ly

Ed1Ad1 1 + 4τY

d1 Ly

2

Lx

2 + Ed2Ad2 1 + 4τY

d2 Ly

2

Lx

2

A12 = A12

plate

+

LxLy

4L3 Ed1Ad1 1 − τY

d1

+ Ed2Ad2 1 − τY

d2

A16 = A16

plate

+

Lx

2

8L3 Ed1Ad1 1 − τY

d1

2

1 − 4Ly

2

Lx

2 − Ed2Ad2 1 − τY

d2

2

1 − 4Ly

2

Lx

2

A22 = A22

plate

+

Ly

3

L

3

Lx

Ed1Ad1 1 +

τY

d1

4

Lx

2

Ly

2 + Ed2Ad2 1 +

τY

d2

4

Lx

2

Ly

2

A26 = A26

plate

+

Ly

2

2L3 Ed1Ad1 1 − τY

d1

2

1 − Lx

2

4Ly

2 − Ed2Ad2 1 − τY

d2

2

1 − Lx

2

4Ly

2

A66

stiffener

= EsAsτY

s

4Ly

+

LxLy

4L3 Ed1Ad1 1 + τY

d1 Lx

2

16Ly

2 1 − 4Ly

2

Lx

2

2

+ Ed2Ad2 1 + τY

d2 Lx

2

16Ly

2 1 − 4Ly

2

Lx

2

2
109

 (C7)

 (C8)

 (C9)

 (C10)

 (C11)

 ( C12)

 (C13)

 (C14)

 (C15)

 (C16)

B11 = B11

plate

+

EsAszs

Ly

+

Lx

3

16L3Ly

Ed1Ad1 zd1 + 4τY

d1

zd1

Ly

2

Lx

2 + Ed2Ad2 zd2 + 4τY

d2

zd2

Ly

2

Lx

2

B12 = B12

plate

+

LxLy

4L3 Ed1Ad1 zd1 − τY

d1

zd1 + Ed2Ad2 zd2 − τY

d2

zd2

B16 = B16

plate

+

Lx

2

8L3 Ed1Ad1 zd1 − τY

d1

zd1

2

1 − 4Ly

2

Lx

2 − Ed2Ad2 zd2 − τY

d2

zd2

2

1 − 4Ly

2

Lx

2

B22 = B22

plate

+

Ly

3

L

3

Lx

Ed1Ad1 zd1 +

τY

d1

zd1

4

Lx

2

Ly

2 + Ed2Ad2 zd2 +

τY

d2

zd2

4

Lx

2

Ly

2

B26 = B26

plate

+

Ly

2

2L3 Ed1Ad1 zd1 − τY

d1

zd1

2

1 − Lx

2

4Ly

2 − Ed2Ad2 zd2 − τY

d2

zd2

2

1 − Lx

2

4Ly

2

B66 = B66

plate

+

EsAsτY

s

4Ly

zs +

LxLy

4L3 Ed1Ad1 zd1 + τY

d1

zd1

Lx

2

16Ly

2 1 − 4Ly

2

Lx

2

2

+ Ed2Ad2 zd2 + τY

d2

zd2

Lx

2

16Ly

2 1 − 4Ly

2

Lx

2

2

D11 = D11

plate

+

EsIs

Ly

+

Lx

3

16LyL

3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

4Ly

2

Lx

2

D12 = D12

plate

+

LxLy

4L3 Ed1Id1 + Ed2Id2 − Gd1Jd1 − Gd2Jd2

D16 = D16

plate

+

Lx

2

8L3 Ed1Id1 − Ed2Id2 + 1

2 Gd2Jd2 − Gd1Jd1 1 − 4Ly

2

Lx

2

D22 = D22

plate

+

Ly

3

L3Lx

Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

Lx

2

4Ly

2
110

 (C17)

 (C18)

 (C19)

 (C20)

 (C21)

where the inplane-shear-deformation parameters are defined as

 (C22)

 (C23)

 (C24)

and the transverse-shear-deformation parameters are defined as

 (C25)

 (C26)

 (C27)

Setting these shear-deformation parameters equal to zero eliminates the effects of stiffener shear 

deformation.

D26 = D26

plate

+

Ly

2

2L3 Ed1Id1 − Ed2Id2 + 1

2 Gd2Jd2 − Gd1Jd1 1 − Lx

2

4Ly

2

D66 = D66

plate

+

GsJs

4Ly

+

LxLy

4L3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

Lx

2

16Ly

2 1 − 4Ly

2

Lx

2

2

A44 = A44

plate

+

Ly

LLx

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

A45 = A45

plate

+

1

2L

Ed1Ad1τZ

d1

− Ed2Ad2τZ

d2

A55 = A55

plate

+

EsAsτZ

s

Ly

+

Lx

4LLy

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

τY

s

≡ k Y

s

Gs

Es

τY

d1

≡ k Y

d1

Gd1

Ed1

τY

d2

≡ k Y

d2

Gd2

Ed2

τZ

s

≡ k Z

s

Gs

Es

τZ

d1

≡ k Z

d1

Gd1

Ed1

τZ

d2

≡ k Z

d2

Gd2

Ed2
111

 For the special case in which the triangle-shaped pockets enclosed by the stiffeners form 

equilateral triangles and the properties of all stiffeners are identical, Lx = L and and 

the stiffnesses reduce to 

 (C28)

 (C29)

 (C30)

 (C31)

Ly = 3L

2

A11 A12 A16

A12 A22 A26

A16 A26 A66

=

A11

plate A12plate A16plate

A12

plate A22plate A26plate

A16

plate A26plate A66plate

+

3EsAs

4L

3 + τY

s

1 − τY

s

0

1 − τY

s

3 + τY

s

0

0 0 1 + τY

s

B11 B12 B16

B12 B22 B26

B16 B26 B66

=

B11

plate B12plate B16plate

B12

plate B22plate B26plate

B16

plate B26plate B66plate

+

3EsAs

4L

3zs + τY

s

zs zs − τY

s

zs 0

zs − τY

s

zs 3zs + τY

s

zs 0

0 0 zs + τY

s

zs

D11 D12 D16

D12 D22 D26

D16 D26 D66

=

D11

plate D12plate D16plate

D12

plate D22plate D26plate

D16

plate D26plate D66plate

+

3 EsIs

4L

3 + GsJs

EsIs

1 − Gs

Js

EsIs

0

1 − Gs

Js

EsIs

3 + Gs

Js

EsIs

0

0 0 1 + GsJs

EsIs

A44 A45

A45 A55

= A44

plate A45plate

A45

plate A55plate +

3EsAsτZ

s

L

1 0

0 1
112

Appendix D

Beam-Member Stiffness Coefficients

 In the present study, it was found to be convenient to express equation (A25) as

 (D1)

such that equation (36) yields the following nonzero elements of the symmetric matrix :

 (D2)

 (D3)

 (D4)

 (D5)

 (D6)

 (D7)

 (D8)

 (D9)

 (D10)

C =

ESAS 0 0 ESASyS ESASzS 0

• τY

S

ESAS 0 0 0 − τY

S

ESASzS

• • τZ

S

ESAS 0 0 τZ

S

ESASyS

• • • ESIZZ

S

ESIYZ

S

0

• • • • ESIYY

S

0

symmetric • • • • GSJS

Cp

C11

p

= ESAS cos2ΨS cos2ΨS + τY

S

sin2ΨS

C12

p

= ESAS sin2ΨS cos2ΨS 1 − τY

S

C13

p

= ESAS sinΨS cosΨS cos2ΨS − 1

2τY

S

cos2ΨS

C14

p

= ESAS cos2ΨS zS cos2ΨS + τY

S

zS sin2

ΨS

C15

p

= ESAS sin2ΨScos2ΨS zS − τY

S

zS

C16

p

= ESAS sinΨS cosΨS z cos2ΨS − 1

2τY

S

zS cos2ΨS

C22

p

= ESAS sin2ΨS sin2ΨS + τY

S

cos2ΨS

C23

p

= ESAS sinΨS cosΨS sin2ΨS + 1

2τY

S

cos2ΨS

C24

p

= ESAS sin2ΨS cos2ΨS zS − τY

S

zS
113

 (D11)

 (D12)

 (D13)

 (D14)

 (D15)

 (D16)

 (D17)

 (D18)

 (D19)

 (D20)

 (D21)

 (D22)

 (D23)

 (D24)

 (D25)

 (D26)

C25

p

= ESAS sin2ΨS zS sin2ΨS + τY

S

zS cos2

ΨS

C26

p

= ESAS sinΨS cosΨS zS sin2ΨS + 1

2τY

S

zS cos2ΨS

C33

p

= ESAS sin2ΨS cos2ΨS + 1

4τY

S

cos22ΨS

C34

p

= ESAS sinΨScosΨS zS cos2ΨS − 1

2τY

S

zS cos2ΨS

C35

p

= ESAS sinΨScosΨS zS sin2ΨS + 1

2τY

S

zS cos2ΨS

C36

p

= ESAS zS sin2ΨScos2ΨS + 1

4τY

S

zS cos2

2ΨS

C44

p

= cos2ΨS ESIYY

S

cos2ΨS + GSJS sin2ΨS

C45

p

= sin2ΨS cos2ΨS ESIYY

S

− GSJS

C46

p

= sinΨScosΨS ESIYY

S

cos2ΨS − 1

2GSJS cos2ΨS

C47

p

= sin2ΨScosΨS ESASτZ

S

yS

C48

p

= cos2ΨSsinΨS ESASτZ

S

yS

C55

p

= sin2ΨS cos2ΨS GSJS + sin2ΨS ESIYY

S

C56

p

= sinΨS cosΨS ESIYY

S

sin2ΨS + 1

2 GSJS cos2ΨS

C57

p

= − sin2ΨS cosΨS ESASτZ

S

yS

C58

p

= − cos2ΨS sinΨS ESASτZ

S

yS

C66

p

= ESIYY

S

sin2ΨS cos2ΨS + 1

4GSJS cos2

2ΨS
114

 (D27)

 (D28)

 (D29)

 (D30)

 (D31)

where

 (D32)

is an inplane-shear-deformation parameter and

 (D33)

is a transverse-shear-deformation parameter. Setting these shear-deformation parameters equal 

to zero eliminates the effects of stiffener shear deformation.

C67

p

= − 1

2sinΨScos2ΨS ESASτZ

S

yS

C68

p

= − 1

2cosΨScos2ΨS ESASτZ

S

yS

C77

p

= sin2ΨS ESASτZ

S

C78

p

= sinΨS cosΨS ESASτZ

S

C88

p

= cos2ΨS ESASτZ

S

τY

S ≡ k Y

S

GS

ES

τZ

S ≡ k Z

S

GS

ES
115

Appendix E

Mathematica®

 Program for the Basic-Cell Energy-Equivalence Method

 The computer program142 used in the present study to determine the contribution of the stiffeners 

to the equivalent-continuum plate stiffnesses, based on the basic-cell energy-equivalence method 

described herein, is presented subsequently. The input used in this program for the orthogonal 

stiffener arrangement with diagonal braces shown in figure 14 is given in Table 4 and the output 

is presented in Appendix B, where n = 1. IGRID = 1 corresponds to the special case in which the 

stiffening grid has no diagonal braces and the stringers are generally different from the ribs (see 

figure 14a). In contrast, IGRID = 2 corresponds to the general case in which the stiffening grid 

has two nonidentical families of diagonal braces in addition to the stringers and ribs. For IGRID 

= 3, all diagonal braces are identical. Similarly, the input used in the program for the orthogonal 

stiffener arrangement with diagonal braces shown in figure 16 is given in Table 5 and the output 

is presented in Appendix B, where n = 1/2. IGRID = 4 corresponds to the general case in which 

the stiffening grid has two nonidentical families of diagonal braces in addition to the stringers and 

ribs. For IGRID = 5, all diagonal braces are identical.

 The input used in this program for the isosceles-triangle stiffener arrangement shown in figure 

17 is given in Table 6 and the output is presented in Appendix C. IGRID = 6 corresponds to the 

general case in which the stiffening grid has two nonidentical families of diagonal braces that are 

different from the stringers (see figure 17a). IGRID = 7 corresponds to the special case of an 

equilateral-triangle grid with identical braces and stringers. Similarly, the input used in this 

program for the Kagome stiffener arrangement, with two nonidentical families of diagonal braces 

that are different from the stringers, shown in figure 18 (IGRID = 8) is given in Table 7 and the 

output was found to be identical to that presented in Appendix C for the isosceles-triangle 

stiffener arrangement. 

 The input used in the program for the hexagon-shaped stiffener arrangement shown in figure 

21 is given in Table 8 and the output is presented in Appendix F. IGRID = 9 corresponds to the 

general case in which one pair of opposite sides have a different length and different properties 

than the remaining sides, and the properties of each family of diagonal members are different. 

IGRID = 10 corresponds to the special case of an equilateral hexagon-shaped stiffener 

arrangement with identical side lengths and identical diagonal members. 

 The input used in the program for the star-shaped stiffener arrangement shown in figure 23 is 

given in Table 9 and the output is presented in Appendix G. IGRID = 11 corresponds to the 

general case in which the star shape is made from identical isosceles triangles with coincident 

centroids, and the two families of diagonal members have different properties. IGRID = 12 

corresponds to the special case in which the star shape is made from identical equilateral triangles 

with coincident centroids, and with identical diagonal members. 

 IGRID = 13 corresponds to the hexagon-shaped core of the stiffened plate shown in figure 25. 

For this case, the core is made of isosceles hexagons formed by beams with rectangular cross 

sections, and is centered on plane z = 0, as shown in figure 25. The basic cell for this case is 

identical to that shown in figure 21b. Likewise, IGRID = 14 is for the special case in which the 
116

core is made of isotropic beams with the same length. The output for these cases is presented in 

Appendix H. 

 IGRID = 15 corresponds to the star-shaped core of the stiffened plate shown in figure 27. For 

this case, the core is made of isosceles-star shapes formed by beams with rectangular cross 

sections, and is centered on plane z = 0, as shown in figure 27. The basic cell for this case is 

identical to that shown in figure 23b. Likewise, IGRID = 16 is for the special case in which the 

core is made of isotropic beams with the same length. The output for these cases is presented in 

Appendix J.

 In the computer program listing that follows, the user-defined input is given by the single red 

line that indicates a selection for the IGRID variable. The program is then executed by using the 

"evaluate notebook" Mathematica®

 command. For each of the stiffener arrangements defined by 

IGRID, the (x, y) coordinates of the starting point and ending point of members forming a given 

basic cell are arranged in row vectors, in ascending order. For example, the starting and ending 

points, or nodes, of the members in the basic cell shown in figure 14b correspond to the filled red 

circles and are numbered from 1 to 8. The corresponding row vectors for the coordinates of the 

nodes, specified in ascending order from 1 to 8, are

 x = {-Lx/2, Lx/2, 0, 0, -Lx/2, Lx/2, Lx/2, -Lx/2}

 y = { 0, 0, -Ly/2, Ly/2, -Ly/2, Ly/2, -Ly/2, Ly/2}

Next, the starting and ending node number and the geometric and material properties of each 

beam member forming a given basic cell are specified in a table. For example, the table for the 

basic cell shown in figure 14b is given by

PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,Ly},

 {5,6,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld},

 {7,8,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld}}

Each row in this table corresponds to one beam member of the basic cell shown in figure 14b. The 

nomenclature follows that given in Table 4 and Appendix D. 

 Upon forming the geometry and material properties of the specified basic cell, the strain energy 

of each member in the basic cell is computed by using equation (37) and then summed and divided 

by the area of the basic cell, as indicated by equation (39a). To obtain this equivalent-continuum 

strain-energy density, cosΨS and sinΨS for each beam member in the basic cell are computed by 

using the nodal coordinates the length of each member. This calculation enables formation of the 

transformation matrix given by equation (34a). Then the beam constitutive matrix given by 

equation (D1) is formed, the strain-equivalence matrix given by equation (30d) is formed, and the 

calculations indicated by equations (32) and (36) are performed. It is important to note that the 

matrix elements of equation (D1) are the effective, noncentroidal values. All the calculations are 

completed within a "DO LOOP" that corresponds to the summation symbol in equation (39a). 

Upon forming the equivalent-continuum strain-energy density, the contributions of the stiffeners 

to the equivalent-plate stiffnesses are calculated following equations (43). The code that performs 
117

these calculations is given as follows.

(************************************************************************)

(* LIST OF STIFFENER ARRANGEMENTS *)

(* *)

(* IGRID=1 Orthogonal Grid of ribs and stringers *)

(* 2 Orthogonal Grid of ribs and stringers with two *)

(* nonidentical braces per bay *)

(* 3 Orthogonal Grid of ribs and stringers with two *)

(* identical braces per bay *)

(* 4 Orthogonal Grid of ribs and stringers with nonidentical *)

(* braces and one brace per bay *)

(* 5 Orthogonal Grid of ribs and stringers with identical *)

(* braces and one brace per bay *)

(* 6 Isosceles-Triangle Grid with nonidentical diagonals *)

(* 7 Equilateral-Triangle Grid with identical members *)

(* 8 KAGOME Grid depicted in figure 18 with *)

(* nonidentical diagonals *)

(* 9 Isosceles-Hexagon Grid with nonidentical diagonals *)

(* 10 Equilateral-Hexagon Grid with identical diagonals *)

(* 11 Isosceles-star-cell Grid with nonidentical diagonals *)

(* 12 Equilateral-star-cell Grid with identical diagonals *)

(* 13 Isosceles-Hexagon Core made of rectangular-cross-section *)

(* beams and with identical diagonals *)

(* 14 Equilateral-Hexagon Core made of rectangular-cross-section *)

(* isotropic beams *)

(* 15 Isosceles-star-cell Core with identical diagonals *)

(* 16 Equilateral-star-cell Core with identical isotropic members*)

(************************************************************************)

IGRID = 3; (* SELECT GRID-STIFFENER ARRANGEMENT *)

If[IGRID == 1, { 

 

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers = 2, CellArea = Lx Ly,

 x = {-Lx/2, Lx/2, 0, 0},

 y = { 0, 0, -Ly/2, Ly/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,Ly} } }];

If[IGRID == 2, { 

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers = 4, CellArea = Lx Ly,

 x = {-Lx/2, Lx/2, 0, 0, -Lx/2, Lx/2, Lx/2, -Lx/2},

 y = { 0, 0, -Ly/2, Ly/2, -Ly/2, Ly/2, -Ly/2, Ly/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,Ly},

 {5,6,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld},

 {7,8,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld} } }];

If[IGRID==3,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=4,CellArea=Lx Ly,

 x={-Lx/2, Lx/2, 0, 0, -Lx/2, Lx/2, Lx/2, -Lx/2},

 y={ 0, 0, -Ly/2, Ly/2, -Ly/2, Ly/2, -Ly/2, Ly/2},
118

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,Ly},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld},

 {7,8,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld} } }];

If[IGRID==4,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=6,CellArea=4Lx Ly,

 x={-Lx, Lx, 0, 0, -Lx, Lx, Lx, -Lx},

 y={ 0, 0, -Ly, Ly, -Ly, Ly, -Ly, Ly},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,2Lx},

 {5,7,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,2Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,2Ly},

 {5,8,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,2Ly},

 {5,6,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,2Ld},

 {7,8,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,2Ld} } }];

If[IGRID==5,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=6,CellArea=4Lx Ly,

 x={-Lx, Lx, 0, 0, -Lx, Lx, Lx, -Lx},

 y={ 0, 0, -Ly, Ly, -Ly, Ly, -Ly, Ly},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,2Lx},

 {5,7,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,2Lx},

 {3,4,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,2Ly},

 {5,8,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,2Ly},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,2Ld},

 {7,8,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,2Ld} } }];

If[IGRID==6,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=3,CellArea=Lx Ly,

 x={-Lx/2, Lx/2, -Lx/4, Lx/4, Lx/4, -Lx/4},

 y={ 0, 0, -Ly/2, Ly/2, -Ly/2, Ly/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld},

 {5,6,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld} } }];

If[IGRID==7,{

 (* SET PROPERTY EQUALITIES FOR IDENTICAL MEMBERS AND SPECIAL CASE CRITERIA *)

 Lx=L,Ly=Sqrt[3]/2 L,Ld=L,Ad=As,Ed=Es,Gd=Gs,Jd=Js,

 YBd=YBs,Y2Bd=Y2Bs,ZBd=ZBs,Z2Bd=Z2Bs,IYd=IYs,IZd=IZs,IYZd=IYZs,

 TauYd=TauYs,TauZd=TauZs,

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=3,CellArea=Lx Ly,

 x={-Lx/2, Lx/2, -Lx/4, Lx/4, Lx/4, -Lx/4},

 y={ 0, 0, -Ly/2, Ly/2, -Ly/2, Ly/2},
119

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld} } }];

If[IGRID==8,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=4,CellArea=2 Lx Ly,

 x={-Lx/2, Lx/2, -Lx/2, Lx/2, -Lx/2, Lx/2, Lx/2, -Lx/2},

 y={-Ly/2, -Ly/2, Ly/2, Ly/2, -Ly, Ly, -Ly, Ly},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {3,4,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,Lx},

 {5,6,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,2L},

 {7,8,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,2L} } }];

If[IGRID==9,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=5,CellArea=2*a*(b+c),

 x={ a/2, 0, -a/2, a/2, 0, -a/2},

 y={ -(b+c)/2, -c/2, -(b+c)/2, (b+c)/2, c/2, (b+c)/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{2,1,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,L/2},

 {2,3,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,L/2},

 {2,5,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,c},

 {5,4,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,L/2},

 {5,6,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,L/2} } }];

If[IGRID==10,{

 (* SET PROPERTY EQUALITIES FOR IDENTICAL MEMBERS AND SPECIAL CASE CRITERIA *)

 a=Sqrt[3]/2 L,b=L/2,c=L,

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=5,CellArea=2*a*(b+c),

 x={ a/2, 0, -a/2, a/2, 0, -a/2},

 y={ -(b+c)/2, -c/2, -(b+c)/2, (b+c)/2, c/2, (b+c)/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{2,1,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,3,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,5,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,c},

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2} } }];

If[IGRID==11,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=12,CellArea=4 B H/3,

 x={B/3, B/2, B/6, 0, -B/6, -B/2, -B/3, -B/2, -B/6, 0, B/6, B/2},

 y={ 0, H/3, H/3, 2 H/3, H/3, H/3, 0, -H/3, -H/3, -2 H/3, -H/3, -H/3},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,12,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld/3},
120

 {1,2,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld/3},

 {3,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {3,4,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld/3},

 {5,4,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld/3},

 {5,6,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {7,6,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2,Ld/3},

 {7,8,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld/3},

 {9,8,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {9,10,Ed2,Ad2,YBd2,Y2Bd2,ZBd2,Z2Bd2,IYd2,IZd2,IYZd2,Gd2,TauYd2,TauZd2,Jd2, Ld/3},

 {11,10,Ed1,Ad1,YBd1,Y2Bd1,ZBd1,Z2Bd1,IYd1,IZd1,IYZd1,Gd1,TauYd1,TauZd1,Jd1,Ld/3},

 {11,12,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3} } }];

If[IGRID==12,{

 H=Sqrt[3] L/2,B= L,Ld=L,

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=12,CellArea=4 B H/3,

 

 x={B/3, B/2, B/6, 0, -B/6, -B/2, -B/3, -B/2, -B/6, 0, B/6, B/2},

 y={ 0, H/3, H/3, 2 H/3, H/3, H/3, 0, -H/3, -H/3, -2 H/3, -H/3, -H/3},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 PROPS={{1,12,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {1,2,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {3,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {3,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {5,6,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {7,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {7,8,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {9,8,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {9,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd, Ld/3},

 {11,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {11,12,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3} } }];

If[IGRID==13,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=5,CellArea=2*a*(b+c),

 x={ a/2, 0, -a/2, a/2, 0, -a/2},

 y={ -(b+c)/2, -c/2, -(b+c)/2, (b+c)/2, c/2, (b+c)/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 Ad=hc td,

 Ar=hc tr,

 ZBd=0,

 Z2Bd=0,

 ZBr=0,

 Z2Br=0,

 IYd=hc^3 td/12,

 IZd=td^3 hc/12,

 IYZd=0,

 IYr=hc^3 tr/12,

 IZr=tr^3 hc/12,

 IYZr=0,

 PROPS={{2,1,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,3,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,5,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,c},

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2} } }];

If[IGRID==14,{

 a=Sqrt[3]/2 L,b=L/2,c=L,

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=5,CellArea=2*a*(b+c),
121

 x={ a/2, 0, -a/2, a/2, 0, -a/2},

 y={ -(b+c)/2, -c/2, -(b+c)/2, (b+c)/2, c/2, (b+c)/2},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 Ad=hc td,

 Ar=hc tr,

 ZBd=0,

 Z2Bd=0,

 ZBr=0,

 Z2Br=0,

 IYd=hc^3 td/12,

 IZd=td^3 hc/12,

 IYZd=0,

 IYr=hc^3 tr/12,

 IZr=tr^3 hc/12,

 IYZr=0,

 Ed=EE,Er=EE,Gd=G,Gr=G,TauYd=k G/EE, TauYr=k G/EE,TauZd=k G/EE,TauZr=k G/EE,

 Jd=td^3 hc/3 (1-Kd),Jr=tr^3 hc/3 (1-Kr),

 PROPS={{2,1,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,3,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {2,5,Er,Ar,YBr,Y2Br,ZBr,Z2Br,IYr,IZr,IYZr,Gr,TauYr,TauZr,Jr,c},

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2},

 {5,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,L/2} } }];

If[IGRID==15,{

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=12,CellArea=4 B H/3,

 x={B/3, B/2, B/6, 0, -B/6, -B/2, -B/3, -B/2, -B/6, 0, B/6, B/2},

 y={ 0, H/3, H/3, 2 H/3, H/3, H/3, 0, -H/3, -H/3, -2 H/3, -H/3, -H/3},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 As = ts hc,Ad = td hc,IYs = ts hc^3/12,IYd = td hc^3/12,

 ZBd=0,ZBs=0,Z2Bd=0,Z2Bs=0,Js = ts^3 hc/3 (1-Ks),Jd = td^3 hc/3 (1-Kd),

 PROPS={{1,12,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {1,2,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {3,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {3,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {5,6,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {7,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {7,8,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {9,8,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {9,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd, Ld/3},

 {11,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {11,12,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3} } }];

If[IGRID==16,{

 B=L,H=Sqrt[3] L/2,

 (* DEFINE THE NUMBER OF MEMBERS, THE AREA OF THE BASIC CELL, AND *)

 (* THE SEQUENTIAL GLOBAL (x, y) NODAL COORDINATES AS ROW VECTORS *)

 NMembers=12,CellArea=4 B H/3,

 x={B/3, B/2, B/6, 0, -B/6, -B/2, -B/3, -B/2, -B/6, 0, B/6, B/2},

 y={ 0, H/3, H/3, 2 H/3, H/3, H/3, 0, -H/3, -H/3, -2 H/3, -H/3 ,-H/3},

 (* DEFINE A TABLE OF BEAM-MEMBER BEGINNING AND ENDING NODES AND *)

 (* PROPERTIES FOR THE BASIC CELL *)

 Ld=L,

 As = t hc,Ad = t hc,IYs = t hc^3/12,IYd = t hc^3/12,

 ZBd=0,ZBs=0,Z2Bd=0,Z2Bs=0,Js = t^3 hc/3 (1-KK),Jd = t^3 hc/3 (1-KK),

 Es=EE,Ed=EE,Gs=G,Gd=G,TauYd=k G/EE, TauYs=k G/EE,TauZd=k G/EE,TauZs=k G/EE,

 PROPS={{1,12,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {1,2,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {3,2,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {3,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},
122

 {5,4,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {5,6,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {7,6,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {7,8,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {9,8,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3},

 {9,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd, Ld/3},

 {11,10,Ed,Ad,YBd,Y2Bd,ZBd,Z2Bd,IYd,IZd,IYZd,Gd,TauYd,TauZd,Jd,Ld/3},

 {11,12,Es,As,YBs,Y2Bs,ZBs,Z2Bs,IYs,IZs,IYZs,Gs,TauYs,TauZs,Js,B/3} } }];

(************************************************************************)

(* BEGIN A DO-LOOP THAT COMPUTES THE STRAIN ENERGY CONTRIBUTION OF EACH *)

(* BEAM MEMBER TO THE STRAIN ENERGY OF THE EQUIVALENT CONTINUUM *)

(* *)

(* THE CURRENT BEAM MEMBER IN THE LOOP IS IDENTIED BY THE INDEX K *)

(************************************************************************)

(* INITIALIZE THE STRAIN-ENERGY OF THE EQUIVALENT CONTINUUM *)

StrainEnergy = 0;

Do[

(* COMPUTE GLOBAL COORDINATES OF THE CURRENT BEAM MEMBER IJ *)

 xI = x[[ PROPS[[K, 1]] ]];

 xJ = x[[ PROPS[[K, 2]] ]];

 yI = y[[ PROPS[[K, 1]] ]];

 yJ = y[[ PROPS[[K, 2]] ]];

(* COMPUTE THE LENGTH OF THE CURRENT BEAM MEMBER *)

 LengthIJ = PROPS[[K, 16]];

(* COMPUTE m = COS(psi) and n = SIN(psi) FOR THE CURRENT BEAM MEMBER *)

 m = (xJ - xI)/LengthIJ;

 n = (yJ - yI)/LengthIJ;

(* COMPUTE THE PROPERTIES OF THE CURRENT BEAM MEMBER *)

EX=PROPS[[K,3]];

A=PROPS[[K,4]];

Ybar=PROPS[[K,5]];

Y2bar=PROPS[[K,6]];

Zbar=PROPS[[K,7]];

Z2bar=PROPS[[K,8]];

IYY=PROPS[[K,9]];

IZZ=PROPS[[K,10]];

IYZ=PROPS[[K,11]];

G=PROPS[[K,12]];

TauY=PROPS[[K,13]];

TauZ=PROPS[[K,14]];

J=PROPS[[K,15]];

(* COMPUTE THE STIFFNESS MATRIX FOR THE CURRENT BEAM MEMBER; see equation (D1) *)

C11 = EX A;

C14 = EX A Ybar;

C15 = EX A Zbar;

C22 = EX A TauY;

C26 = -EX A TauY Z2bar;

C33 = EX A TauZ;

C36 = EX A TauZ Y2bar;

C44 = EX IZZ;

C45 = EX IYZ;

C55 = EX IYY;

C66 = G J;

 Cmatrix = { { C11, 0, 0, C14, C15, 0},

 { 0, C22, 0, 0, 0, C26},

 { 0, 0, C33, 0, 0, C36},

 { C14, 0, 0, C44, C45, 0},

 { C15, 0, 0, C45, C55, 0},

 { 0, C26, C36, 0, 0, C66} };
123

(* COMPUTE THE TRANSFORMED STIFFNESS MATRIX FOR THE CURRENT BEAM MEMBER; see equations (30)-(36) *)

 Ematrix = { { 1, 0, 0, 0, 0, 0, 0, 0},

 { 0, 0, 1/2, 0, 0, 0, 0, 0},

 { 0, 0, 0, 0, 0, 0, 0, 1},

 { 0, 0, 0, 0, 0, 0, 0, 0},

 { 0, 0, 0, 1, 0, 0, 0, 0},

 { 0, 0, 0, 0, 0, -1/2, 0, 0} };

 CPmatrix = Transpose[Ematrix].Cmatrix.Ematrix;

 Transformationmatrix = { 

 { m^2, n^2, m*n, 0, 0, 0, 0, 0},

 { n^2, m^2, m*n, 0, 0, 0, 0, 0},

 { -2*m*n, 2*m*n, m^2 - n^2, 0, 0, 0, 0, 0}, 

 { 0, 0, 0, m^2, n^2, m*n, 0, 0},

 { 0, 0, 0 n^2, m^2, m*n, 0, 0},

 { 0, 0, 0, -2*m*n, 2*m*n, m^2 - n^2, 0, 0}, 

 { 0, 0, 0, 0, 0, 0, m, -n},

 { 0, 0, 0, 0, 0, 0, n, m} }; 

 Cpmatrix = Transpose[Transformationmatrix].CPmatrix.Transformationmatrix;

(* COMPUTE STRAIN ENERGY CONTRIBUTION OF THE CURRENT BEAM MEMBER; see equation (37) *)

 StrainList = {epxx, epyy, gamxy, kapxx, kapyy, kapxy, gamyz, gamxz};

 MemberStrainEnergy = LengthIJ/2 StrainList.Cpmatrix.StrainList;

(* ADD THE STRAIN-ENERGY CONTRIBUTION OF THE CURRENT BEAM MEMBER TO THE *)

(* CONTINUUM STRAIN ENERGY; see equation (39a) *) 

 StrainEnergy = StrainEnergy + MemberStrainEnergy,

 (* CLOSE DO-LOOP *) {K, 1, NMembers}]

(* COMPUTE THE TOTAL STRAIN ENERGY DENSITY OF THE EQUIVALENT CONTINNUM; see equation (39a) *)

 StrainEnergyDensity = StrainEnergy/CellArea;

(* COMPUTE THE A-MATRIX OF FIRST-ORDER TRANSVERSE-SHEAR DEFORMATION PLATE THEORY; see equation (43a) *)

Print["Astiffener11 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, epxx] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener12 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, epyy] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener16 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, gamxy] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener22 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epyy, epyy] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener26 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epyy, gamxy] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener66 = ", FullSimplify[Factor[ D[StrainEnergyDensity, gamxy, gamxy] ] ] ];

Print[" "]; Print[" "]; Print[" "];

(* COMPUTE THE B-MATRIX OF FIRST-ORDER TRANSVERSE-SHEAR DEFORMATION PLATE THEORY; see equation (43b) *)

Print["Bstiffener11 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, kapxx] ] ] ];

Print[" "]; Print[" "];

Print["Bstiffener12 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, kapyy] ] ] ];

Print[" "]; Print[" "];

Print["Bstiffener16 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epxx, kapxy] ] ] ];

Print[" "]; Print[" "];
124

Print["Bstiffener22 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epyy, kapyy] ] ] ];

Print[" "]; Print[" "];

Print["Bstiffener26 = ", FullSimplify[Factor[ D[StrainEnergyDensity, epyy, kapxy] ] ] ];

Print[" "]; Print[" "];

Print["Bstiffener66 = ", FullSimplify[Factor[ D[StrainEnergyDensity, gamxy, kapxy] ] ] ];

Print[" "]; Print[" "]; Print[" "];

(* COMPUTE THE D-MATRIX OF FIRST-ORDER TRANSVERSE-SHEAR DEFORMATION PLATE THEORY; see equation (43c) *)

Print["Dstiffener11 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapxx, kapxx] ] ] ];

Print[" "]; Print[" "];

Print["Dstiffener12 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapxx, kapyy] ] ] ];

Print[" "]; Print[" "];

Print["Dstiffener16 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapxx, kapxy] ] ] ];

Print[" "]; Print[" "];

Print["Dstiffener22 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapyy, kapyy] ] ] ];

Print[" "]; Print[" "];

Print["Dstiffener26 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapyy, kapxy] ] ] ];

Print[" "]; Print[" "];

Print["Dstiffener66 = ", FullSimplify[Factor[ D[StrainEnergyDensity, kapxy, kapxy] ] ] ];

Print[" "]; Print[" "]; Print[" "];

(* COMPUTE THE TRANSVERSE-SHEAR STIFFNESS MATRIX OF FIRST-ORDER TRANSVERSE-SHEAR DEFORMATION PLATE

 THEORY; see equation (43d) *)

Print["Astiffener44 = ", FullSimplify[Factor[ D[StrainEnergyDensity, gamyz, gamyz] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener45 = ", FullSimplify[Factor[ D[StrainEnergyDensity, gamyz, gamxz] ] ] ];

Print[" "]; Print[" "];

Print["Astiffener55 = ", FullSimplify[Factor[ D[StrainEnergyDensity, gamxz, gamxz] ] ] ];

Print[" "]; Print[" "]; Print[" "];

(* RE-INITIALIZE MATHEMATICA *)

 Remove["Global`*"]; 

(* CHANGE INPUT AND SELECT "EVALUATE NOTEBOOK" UNDER THE EVALUATION MENU ON THE MATHEMATICA WINDOW HEADER *)
125

Appendix F

Equivalent-Plate Stiffnesses for a Plate Reinforced with a Hexagon-Shaped

Stiffener Arrangement

The expressions presented in this appendix are for a general laminated-composite plate that is 

stiffened with ribs and two nonidentical families of diagonal braces, as shown in figure 21. The 

ribs and diagonals are eccentric with respect to the plate midplane and the pockets formed by the 

skin and stiffeners shown in figure 21 are isosceles hexagons; that is, the two pairs of diagonal 

members have the same length, which is generally different from the length of the two identical 

ribs. The notation used for the material and section properties and the orientation angle of each 

stiffener family is given in Table 8. In this table and the expressions that follow, the subscript and 

superscript "r" refers to the ribs and the subscripts and superscripts "d1" and "d2" refer to the two 

nonidentical families of diagonals. The stiffness expressions presented subsequently are obtained 

from equations (23) and (25)-(28) by applying equations (25)-(28) to each family of stiffeners 

with the attributes given in Table 8. In the expressions that follow, the stiffener extensional 

modulus, shear modulus, eccentricity, moment of inertia, torsion constant, and the transverseshear correction factors refer to the corresponding effective quantities defined in Appendix A for 

a nonhomogeneous, specially orthotropic beam. 

 (F1)

 (F2)

 (F3)

 (F4)

 (F5)

A11 = A11

plate

+

a

3

2 b + c L3 Ed1Ad1 1 + τY

d1 b

2

a

2 + Ed2Ad2 1 + τY

d2 b

2

a

2

A12 = A12

plate

+ ab2

2 b + c L3 Ed1Ad1 1 − τY

d1

+ Ed2Ad2 1 − τY

d2

A16 = A16

plate

+

a

2

b

2 b + c L3 Ed1Ad1 1 +

τZ

d1

2

b

2

a

2 − 1 − Ed2Ad2 1 +

τY

d2

2

b

2

a

2 − 1

A22 = A22

plate

+

ErArc

2a b + c

+ b

4

2a b + c L3 Ed1Ad1 1 + τY

d1 a

2

b

2 + Ed2Ad2 1 + τY

d2 a

2

b

2

A26 = A26

plate

+ b

3

2 b + c L3 Ed1Ad1 1 +

τY

d1

2

a

2

b

2 − 1 − Ed2Ad2 1 +

τY

d2

2

a

2

b

2 − 1
126

 (F6)

 (F7)

 (F8)

 (F9)

 (F10)

 (F11)

 (F12)

 (F13)

A66 = A66

plate

+

ErArτY

r

c

8a b + c

+

ab2

2 b + c L3 Ed1Ad1 1 +

τY

d1

4

a

2

− b

2

ab

2

+ Ed2Ad2 1 +

τY

d2

4

a

2

− b

2

ab

2

B11 = B11

plate

+

a

3

2 b + c L3 Ed1Ad1 zd1 + τY

d1

zd1

b

2

a

2 + Ed2Ad2 zd2 + τY

d2

zd2

b

2

a

2

B12 = B12

plate

+ ab2

2 b + c L3 Ed1Ad1zd1 1 − τY

d1

zd1 + Ed2Ad2zd2 1 − τY

d2

zd2

B16 = B16

plate

+

a

2

b

2 b + c L3 Ed1Ad1 zd1 +

τY

d1

zd1

2

b

2

a

2 − 1 − Ed2Ad2 zd2 +

τY

d2

zd2

2

b

2

a

2 − 1

B22 = B22

plate

+

ErArczr

2a b + c

+

b

4

2a b + c L3 Ed1Ad1 zd1 + τY

d1

zd1

a

2

b

2 + Ed2Ad2 zd2 + τY

d2

zd2

a

2

b

2

B26 = B26

plate

+ b

3

2 b + c L3 Ed1Ad1 zd1 +

τY

d1

zd1

2

a

2

b

2 − 1 − Ed2Ad2 zd2 +

τY

d2

zd2

2

a

2

b

2 − 1

B66 = B66

plate

+

ErArc

8a b + c

τY

r

zr +

ab2

2 b + c L3 Ed1Ad1 zd1 +

τY

d1

zd1

4

a

2

− b

2

ab

2

+ Ed2Ad2 zd2 +

τY

d2

zd2

4

a

2

− b

2

ab

2

D11 = D11

plate

+

a

3

2 b + c L

3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

b

2

a

2
127

 (F14)

 (F15)

 (F16)

 (F17)

 (F18)

 (F19)

 (F20)

 (F21)

where the inplane-shear-deformation parameters are defined as

 (F22)

 (F23)

 (F24)

D12 = D12

plate

+ ab2

2 b + c L3 Ed1Id1 + Ed2Id2 − Gd1Jd1 − Gd2Jd2

D16 = D16

plate

+

a

2

b

2 b + c L

3 Ed1Id1 − Ed2Id2 +

1

2 Gd1Jd1 − Gd2Jd2

b

2

a

2 − 1

D22 = D22

plate

+

ErIrc

2a b + c

+ b

4

2a b + c L3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

a

2

b

2

D26 = D26

plate

+ b

3

2 b + c L

3 Ed1Id1 − Ed2Id2 +

1

2 Gd1Jd1 − Gd2Jd2

a

2

b

2 − 1

D66 = D66

plate

+

cGrJr

8a b + c

+ ab2

2 b + c L3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

a

2

− b

2

2ab

2

A44 = A44

plate

+

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2 b

2

+ Er

ArτY

r

cL

2a b + c L

A45 = A45

plate

+

Ed1Ad1τZ

d1

− Ed2Ad2τZ

d2 b

2 b + c L

A55 = A55

plate

+

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

a

2 b + c L

τY

r

≡ k Y

r

Gr

Er

τY

d1

≡ k Y

d1

Gd1

Ed1

τY

d2

≡ k Y

d2

Gd2

Ed2
128

and the transverse-shear-deformation parameters are defined as

 (F25)

 (F26)

 (F27)

Setting these shear-deformation parameters equal to zero eliminates the effects of stiffener shear 

deformation.

 For the special case in which all stiffeners have the same length, , b = L/2, and c = L. 

For this simplification and for identical families of diagonals, the stiffnesses reduce to 

 (F22)

 (F23)

 (F24)

 (F25)

 (F26)

 (F27)

 (F28)

 (F29)

τZ

r

≡ k Z

r

Gr

Er

τZ

d1

≡ k Z

d1

Gd1

Ed1

τZ

d2

≡ k Z

d2

Gd2

Ed2

a = 3L

2

A11 = A11

plate

+

3EdAd

4L

1 + τY

d

3

A12 = A12

plate

+

3EdAd

12L 1 − τY

d

A16 = A16

plate

A22 = A22

plate

+

3

36L

8ErAr + EdAd 1 + 3τY

d

A26 = A26

plate

A66 = A66

plate

+

3

12L

2Er

ArτY

r

3 + EdAd 1 + τY

d

3

B11 = B11

plate

+

3EdAd

4L zd +

τY

d

zd

3

B12 = B12

plate

+

3EdAd

12L

zd − τY

d

zd
129

 (F30)

 (F31)

 (F32)

 (F33)

 (F34)

 (F35)

 (F36)

 (F37)

 (F38)

 (F39)

 (F40)

 (F41)

 (F42)

where the subscript and superscript "d" indicates the properties of the diagonal stiffeners.

B16 = B16

plate

B22 = B22

plate

+

3

36L 8Er

Arzr + EdAdzd zd + 3τY

d

zd

B26 = B26

plate

B66 = B66

plate

+

3

12L

2ErArτY

r

zr

3 + EdAd zd +

τY

d

zd

3

D11 = D11

plate

+ 3

4L EdId +

GdJd

3

D12 = D12

plate

+

3

12L

EdId − GdJd

D16 = D16

plate

D22 = D22

plate

+

3

36L

8ErIr + EdId + 3GdJd

D26 = D26

plate

D66 = D66

plate

+

3

36L

3EdId + 2GrJr + GdJd

A44 = A44

plate

+ 3

9L EdAdτZ

d

+ 2Er

ArτZ

r

A45 = A45

plate

A55 = A55

plate

+

3EdAdτZ

d

3L
130

Appendix G

Equivalent-Plate Stiffnesses for a Plate Reinforced with a Star-Cell-Shaped

Stiffener Arrangement

The expressions presented in this appendix are for a general laminated-composite plate that is 

stiffened with stringers and two nonidentical families of diagonal braces, as shown in figure 23. 

The stringers and diagonals are eccentric with respect to the plate midplane and the pockets 

formed by the skin and stiffeners shown in figure 23 are star shapes, formed by isosceles triangles, 

and isosceles hexagons. The two pairs of diagonal members have the same length, which is 

generally different from the length of the two identical stringers. The notation used for the 

material and section properties and orientation angle of each stiffener family is given in Table 9. 

The stiffness expressions presented subsequently are obtained from equations (23) and (25)-(28) 

by applying equations (25)-(28) to each family of stiffeners with the attributes given in Table 9. 

In the expressions that follow, the stiffener extensional modulus, shear modulus, eccentricity, 

moment of inertia, torsion constant, and transverse-shear correction factors refer to the 

corresponding effective quantities defined in Appendix A for a nonhomogeneous, specially 

orthotropic beam. The subscript and superscript "s" refers to the stringers and the subscripts and 

superscripts "d1" and "d2" refer to the two nonidentical families of diagonals.

 (G1)

 (G2)

 (G3)

 (G4)

 (G5)

 (G6)

A11 = A11

plate

+

EsAs

H

+ B3

16HL3 Ed1Ad1 1 + 4τY

d1 H2

B2 + Ed2Ad2 1 + 4τY

d2 H2

B2

A12 = A12

plate

+

BH

4L3 Ed1Ad1 1 − τY

d1

+ Ed2Ad2 1 − τY

d2

A16 = A16

plate

+ B2

8L3 Ed1Ad1 1 + τY

d1

2

4H2

B2 − 1 − Ed2Ad2 1 + τY

d2

2

4H2

B2 − 1

A22 = A22

plate

+ H3

BL3 Ed1Ad1 1 + τY

d1

4

B2

H2 + Ed2Ad2 1 + τY

d2

4

B2

H2

A26 = A26

plate

+ H2

2L3 Ed1Ad1 1 + τY

d1

2

B2

4H2 − 1 − Ed2Ad2 1 + τY

d2

2

B2

4H2 − 1

A66 = A66

plate

+

EsAsτY

s

4H

+

BH

4L3 Ed1Ad1 1 +τY

d1 B

4H − HB

2

+ Ed2Ad2 1 + τY

d2 B

4H − HB

2
131

 (G7)

 (G8)

 (G9)

 (G10)

 (G11)

 (G12)

 (G13)

 (G14)

 (G15)

 (G16)

B11 = B11

plate

+

EsAszs

H

+ B3

16HL3 Ed1Ad1 zd1 + 4τY

d1

zd1

H2

B2 + Ed2Ad2 zd2 + 4τY

d2

zd2

H2

B2

B12 = B12

plate

+

BH

4L3 Ed1Ad1 zd1 − τY

d1

zd1 + Ed2Ad2 zd2 − τY

d2

zd2

B16 = B16

plate

+ B2

8L3 Ed1Ad1 zd1 +

τY

d1

zd1

2

4H2

B2 − 1 − Ed2Ad2 zd2 +

τY

d2

zd2

2

4H2

B2 − 1

B22 = B22

plate

+ H3

BL3 Ed1Ad1 zd1 +

τY

d1

zd1

4

B2

H2 + Ed2Ad2 zd2 +

τY

d2

zd2

4

B2

H2

B26 = B26

plate

+ H2

2L3 Ed1Ad1 zd1 +

τY

d1

zd1

2

B2

4H2 − 1 − Ed2Ad2 zd2 +

τY

d2

zd2

2

B2

4H2 − 1

B66 = B66

plate

+

EsAsτY

s

zs

4H

+

BH

4L3 Ed1Ad1 zd1 +τY

d1

zd1

B

4H − HB

2

+ Ed2Ad2 zd2 + τY

d2

zd2

B

4H − HB

2

D11 = D11

plate

+

EsIs

H

+ B3

16HL3 Ed1Id1 + Ed2Id2 + 4H2B2 Gd1Jd1 + Gd2Jd2

D12 = D12

plate

+

BH

4L3 Ed1Id1 + Ed2Id2 − Gd1Jd1 − Gd2Jd2

D16 = D16

plate

+ B2

8L3 Ed1Id1 − Ed2Id2 + 1

2 Gd1Jd1 − Gd2Jd2

4H2

B2 − 1

D22 = D22

plate

+ H3

BL3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

B2

4H2
132

 (G17)

 (G18)

 (G19)

 (G20)

 (G21)

where the inplane-shear-deformation parameters are defined as

 (G22)

 (G23)

 (G24)

and the transverse-shear-deformation parameters are defined as

 (G25)

 (G26)

 (G27)

Setting these shear-deformation parameters equal to zero eliminates the effects of stiffener shear 

D26 = D26

plate

+ H2

2L3 Ed1Id1 − Ed2Id2 + 1

2 Gd1Jd1 − Gd2Jd2

B2

4H2 − 1

D66 = D66

plate

+

GsJs

4H

+

BH

4L3 Ed1Id1 + Ed2Id2 + Gd1Jd1 + Gd2Jd2

B

4H − HB

2

A44 = A44

plate

+

H

BL

Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2

A45 = A45

plate

+

1

2L

Ed1Ad1τZ

d1

− Ed2Ad2τZ

d2

A55 = A55

plate

+

4EsAsτZ

s

L + Ed1Ad1τZ

d1

+ Ed2Ad2τZ

d2 B

4HL

τY

s

≡ k Y

s

Gs

Es

τY

d1

≡ k Y

d1

Gd1

Ed1

τY

d2

≡ k Y

d2

Gd2

Ed2

τZ

s

≡ k Z

s

Gs

Es

τZ

d1

≡ k Z

d1

Gd1

Ed1

τZ

d2

≡ k Z

d2

Gd2

Ed2
133

deformation.

 For the special case of equilateral-star cells with identical diagonals B = L and 

and the stiffnesses reduce to

 (G28)

 (G29)

 (G30)

 (G31)

 (G32)

 (G33)

 (G34)

 (G35)

 (G36)

 (G37)

 (G38)

 (G39)

H = 3

2 L

A11 = A11

plate

+

2 3EsAs

3L +

3EdAd

12L

1 + 3τY

d

A12 = A12

plate

+

3EdAd

4L

1 − τY

d

A16 = A16

plate

A22 = A22

plate

+

3 3EdAd

4L

1 + τY

d

3

A26 = A26

plate

A66 = A66

plate

+

3EsAsτY

s

6L +

3EdAd

4L

1 + τY

d

3

B11 = B11

plate

+

2 3EsAszs

3L +

3EdAd

12L

zd + 3τY

d

zd

B12 = B12

plate

+

3EdAd

4L

zd − τY

d

zd

B16 = B16

plate

B22 = B22

plate

+

3 3EdAd

4L

zd +

τY

d

zd

3

B26 = B26

plate

B66 = B66

plate

+

3EsAsτY

s

zs

6L +

3EdAd

4L

zd +

τY

d

zd

3
134

 (G40)

 (G41)

 (G42)

 (G43)

 (G44)

 (G45)

 (G46)

 (G47)

 (G48)

where the subscript and superscript "d" indicates the properties of the diagonal stiffeners.

D11 = D11

plate

+

2 3EsIs

3L + 3

12L EdId + 3GdJd

D12 = D12

plate

+ 3

4L EdId − GdJd

D16 = D16

plate

D22 = D22

plate

+ 3 3

4L EdId +

GdJd

3

D26 = D26

plate

D66 = D66

plate

+

3GsJs

6L + 3

4L EdId +

GdJd

3

A44 = A44

plate

+

3EdAdτZ

d

L

A45 = A45

plate

A55 = A55

plate

+ 3

3L

2EsAsτZ

s

+ EdAdτZ

d
135

Appendix H

Equivalent-Plate Stiffnesses for a Hexagon-Cell-Core Sandwich Plate

 The equivalent-plate stiffnesses for a sandwich plate composed of two different, anisotropic, 

laminated-composite face plates and the hexagon-cell core, made of a homogeneous orthotropic 

material, shown in figure 25 are presented in this appendix. The thicknesses of the bottom face 

plate (plate no. 1), the top face plate (plate no. 2), and the core are denoted by h1

, h2, and hc, 

respectively. The plate reference plane is located at the midplane of the core and the eccentricities 

of face plates no. 1 and no. 2 are given by and , respectively. 

The core is composed of two member types. One member type is aligned with the y-axis of the 

plate, as shown in figures 21 and 25, and is referred to herein as a rib. The other member type 

makes an angle Φ with the x-axis and is referred to herein as a diagonal. Member attributes 

associated with the ribs and diagonals are indicated subsequently with the subscripts or 

superscripts "r" and "d," respectively.

 For a hexagon-core stiffener arrangement with each member made of a homogeneous 

orthotropic material, the effective moduli given by equations (A13a)-(A13c) reduce to the 

corresponding principal moduli of the orthotropic material, where the principal orthotropicmaterial axes are coincident with the X, Y, and Z beam coordinate axes shown in figure 6. Thus, 

the symbols Er

 and Ed denote the extensional modulus EX and the symbols Gr and Gd denote 

the mean principal shear moduli of the ribs and diagonals, respectively. Similarly, the 

stiffness-weighted eccentricities , , , and vanish for homogeneous stiffeners centered on 

the plate reference plane. Moreover, the stiffness-weighted moments of inertia reduce to the 

corresponding second moments of area and the stiffness-weighted product of inertia vanishes for 

the rectangular stiffener cross sections. The cross-sectional areas of the core members are given 

by Ar

 = hctr and Ad = hctd, where tr is the rib thickness and td is the diagonal thickness. The 

moments of inertia associated with out-of-plane bending are given by for the ribs and 

 for the diagonals. The planar geometric dimensions of the core, as shown in figure 21 

and 25, are given by the symbols a, b, c, and L. The symbols Gr

Jr and GdJd are the torsional 

stiffnesses of the ribs and diagonals, respectively. For a rib with a rectangular cross-section made 

of a homogeneous orthotropic material, equation (A16) yields 

 (H1)

and for a diagonal

e1 = − 1

2 h1 + hc e2 = 12 h2 + hc

GXZGXY

zr zd zr zd

Ir = hc

3

tr

12

I d = hc

3

td

12

GrJr = GXZ

tr

3

h c

3 1 − 96

π5

tr

h c

GXZ

GXY

1

2 1 − −1 p

p

5 tanh pπh c

2tr

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞
136

 (H2)

Likewise, the symbols and denote the stiffness-weighted transverse-shear correction 

factor of the ribs and diagonals, respectively, and and denote the stiffnessweighted inplane-shear correction factor of the ribs and diagonals, respectively. 

 For the notation just given, the equivalent-plate-stiffness contributions of the core to equations 

(47) are given by

 (H3)

 (H4)

 (H5)

 (H6)

 (H7)

 (H8)

 (H9)

 (H10)

GdJd = GXZ

tr

3

h c

3 1 − 96

π5

t d

h c

GXZ

GXY

1

2 1 − −1 p

p

5 tanh pπh c

2t d

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞

k Z

r

k Z

d

k Z

GXZ

GXY

k Y

r

k Y

d

k Y

GXY

GXZ

A11

core

= Edh ct da

3

b + c L

3 1 + τY

d b

2

a

2

A12

core

= Edh ct d ab2

b + c L

3 1 − τY

d

Gd

A16

core = 0

A22

core

= h c

a b + c

Ertrc

2

+

Edt db

4

L3 1 + τY

d a

2

b

2

A26

core = 0

A66

core

= h c

a b + c

ErtrτY

r

c

8

+

Edt d a

2

b

2

L3 1 +

τY

d

4

a

2

− b

2

ab

2

D11

core

= a

3

b + c L

3

Edh c

3

t d

12 +

GdJdb

2

a

2

D12

core

= ab2

b + c L

3

Edh c

3

t d

12 − GdJd
137

 (H11)

 (H12)

 (H13)

 (H14)

 (H15)

 (H16)

 (H17)

where the inplane-shear-deformation parameters are defined as

 (H18)

 (H19)

and the transverse-shear-deformation parameters are defined as

 (H20)

 (H21)

Setting these shear-deformation parameters equal to zero eliminates the effects of stiffener shear 

deformation. The overall stiffnesses of the sandwich plate are obtained by substituting equations 

(H3)-(H17) into equations (47). 

D16

core = 0

D22

core

= c

a b + c

Erh c

3

tr

24 + b

4

cL

3

Edh c

3

t d

12 + GdJd

a

2

b

2

D26

core = 0

D66

core

= ab2

b + c L3

Edh c

3

t d

12 +

GrJrL3c

8a2

b

2 +

GdJd

4a2b

2 a

2

− b

2 2

A44

core

= h c

2a b + c L

2Edt dτZ

d

b

2

+ Er

trτZ

r

cL

A45

core = 0

A55

core

= Edh ct dτZ

d

a

b + c L

τY

r

≡ k Y

r

Gr

Er

τY

d

≡ k Y

d

Gd

Ed

τZ

r

≡ k Z

r

Gr

Er

τZ

d

≡ k Z

d

Gd

Ed
138

 For the special case of a homogeneous, isotropic core comprised of hexagons with all sides 

having length L, the angle Φ shown in figure 21b is 30 degrees, , b = L/2, and c = L. 

In addition, Er

 = Ed = E and Gr = Gd = G, where E and G are the extensional and shear moduli 

of an isotropic material, and . For this case, equations (H1) and (H2) are 

expressed as 

 (H22)

 (H23)

where

 (H24)

 (H25)

The equivalent-plate core stiffnesses reduce to 

 (H26)

 (H27)

 (H28)

a = 3L

2

k Y

r

= k Y

d

= k Z

r

= k Z

d

= k

GrJr = Gtr

3

h c

3 1 − Kr

GdJd = Gtd

3

h c

3 1 − Kd

Kr = 96

π5

tr

h c

1 − −1 p

p5 tanh pπh c

2t p = 1, 2, 3, ...

Σ r

∞

Kd = 96

π5

td

h c

1 − −1 p

p

5 tanh pπh c

2t p = 1, 2, 3, ...

Σ d

∞

A11

core A12core A16core

A12

core A22core A26core

A16

core A26core A66core

= 3h ctdE

12L

3 + kG

E 1 − kGE 0

1 − kG

E

1

3

1 + 8 tr

td

+ 3kG

E 0

0 0 1 + kG

3E 1 + 2 tr

td

D11

core D12core D16core

D12

core D22core D26core

D16

core D26core D66core

= 3h c

3

tdE

144L

3 + 4G

E

td

2

h c

2 1 − Kd 1 − 4G

E

td

2

h c

2 1 − Kd 0

1 − 4G

E

td

2

h c

2 1 − Kd

1

3

1 + 8 tr

td

+ 12G

E

td

2

h c

2 1 − Kd 0

0 0 1 + 4G

3E

td

2

h c

2

2tr

3

td

3 1 − Kr + 1 − Kd

A44

core A45core

A45

core A55core = 3kGh ctd

9L

1 + 2 tr

td

0

0 3
139

Appendix I

Equivalent-Plate Stiffnesses for an Orthogrid-Core Sandwich Plate

 The equivalent-plate stiffnesses for a sandwich plate composed of two different, anisotropic, 

laminated-composite face plates and the orthogrid core with homogeneous, orthotropic members 

shown in figure 26 are presented in this appendix. The thicknesses of the bottom face plate (plate 

no. 1), the top face plate (plate no. 2), and the core are denoted by h1

, h2, and hc, respectively. The 

plate reference plane is located at the midplane of the core and the eccentricities of face plates 

no. 1 and no. 2 are given by and , respectively. The stringer 

and rib elements of the core are aligned with the x- and y-axes of the plate shown in figure 26, 

respectively. Member attributes associated with the ribs and stringers are indicated subsequently 

with the subscripts or superscripts "r" and "s," respectively.

 For an orthogrid core with each member made of a homogeneous orthotropic material, the 

effective moduli given by equations (A13a)-(A13c) reduce to the corresponding principal moduli 

of the orthotropic material, where the principal orthotropic-material axes are coincident with the 

X, Y, and Z beam coordinate axes shown in figure 6. Thus, the symbols Er

 and Es denote the 

extensional modulus EX and the symbols Gr

 and Gs denote the mean principal shear moduli 

 of the ribs and stringers, respectively. Similarly, the stiffness-weighted first moments 

of area vanish for homogeneous stiffeners centered on the plate reference plane. 

Moreover, the stiffness-weighted moments of inertia reduce to the corresponding second 

moments of area and the stiffness-weighted product of inertia vanishes for the rectangular 

stiffener cross sections. The cross-sectional areas of the stringers and ribs are given by As

 = hct

s

 

and Ar

 = hct

r

, respectively, where ts is the stringer thickness and tr is the rib thickness. The 

moments of inertia associated with out-of-plane bending are given by for the stringers 

and for the ribs. The symbols Ly and Lx denote the spacing of the stringers and ribs, 

respectively, as shown in figure 26. The symbols Gs

Js and GrJr are the torsional stiffnesses of the 

stringers and ribs, respectively. For a rib with a rectangular cross-section made of a homogeneous 

orthotropic material, equation (A16) yields 

 (I1)

and for a stringer

e1 = − 1

2 h1 + hc e2 = 12 h2 + hc

GXZGXY

zr, zs, zr, and zs

Is = hc

3

ts

12

Ir = hc

3

tr

12

GrJr = GXZ

tr

3

h c

3 1 − 96

π5

tr

h c

GXZ

GXY

1

2 1 − −1 p

p5 tanh pπh c

2tr

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞
140

 (I2)

Likewise, the symbols and denote the stiffness-weighted transverse-shear correction 

factor of the stringers and ribs, respectively, and and denote the stiffnessweighted inplane-shear correction factor of the stringers and ribs, respectively. 

 Based on the notation just given, the equivalent-plate-stiffness contributions of the core are

 (I3)

 (I4)

 (I5)

 As a special case, consider a sandwich plate with face sheets and a core are made from three 

different homogeneous, specially orthotropic materials. The core is made from a monolithic piece 

of orthotropic material. The face plates are relatively thin and, as a result, the transverse-shearing 

deformations of the face plates are presumed to be negligible. For this case, the principal material 

coordinate frames are taken to be identical to the structural coordinate frame shown in figure 26. 

The orthotropic-material properties are denoted by the extensional moduli Ex

 and Ey; the shear 

GsJs = GXZ

ts

3

h c

3 1 − 96

π5

ts

h c

GXZ

GXY

1

2 1 − −1 p

p5 tanh pπh c

2ts

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞

k Z

s

k Z

r

k Z

GXZ

GXY

k Y

s

k Y

r

k Y

GXY

GXZ

A11

core A12core A16core

A12

core A22core A26core

A16

core A26core A66core

=

Esh cts

Ly

0 0

0 Er

h ctr

Lx

0

0 0 k Y

s

Gsh cts

4Ly

+

k Y

r

Grh ctr

4Lx

D11

core D12core D16core

D12

core D22core D26core

D16

core D26core D66core

=

Esh c

3

ts

12Ly

0 0

0 Er

h c

3

tr

12Lx

0

0 0 1

4

GsJs

Ly

+

GrJr

Lx

A44

core A45core

A45

core A55core =

k Z

r

Grh ctr

Lx

0

0

k Z

s

Gsh cts

Ly
141

moduli Gxy, Gxz, and Gyz; and the Poisson’s ratios νxy, and νyx. In addition, superscripts are applied 

to these material-property designations to indicate their association with a given face plate or the 

core. For the stringer elements of the core, which are aligned with the x-axis; , 

, and . For the rib elements of the core, which are aligned with 

the y-axis; , , and . For a stiffener with a rectangular 

cross-section made of a homogeneous orthotropic material, equations (I1) and (I2) give 

 (I6)

 (I7)

where

 (I8)

 (I9)

Using this information with equations (I3) - (I5) and (47) gives

 (I10)

Es → Ex

core

k Y

s

Gs → ky

core

Gxy

core k Z

s

Gs → kz

core

Gxz

core

Er → Ey

core k Y

r

Gr → kx

core

Gxy

core k Z

r

Gr → kz

core

Gyz

core

GsJs = Gxz

core ts

3

h c

3 1 − Ks

GrJr = Gyz

core tr

3

h c

3 1 − Kr

Ks = 96

π5

ts

h c

Gxz

core

Gxy

core

1

2

1 − −1 p

p5 tanh pπh c

2ts

Gxy

core

Gxz

core

1

2

p = 1, 2, 3, ...

Σ

∞

Kr = 96

π5

tr

h c

Gyz

core

Gxy

core

1

2 1 − −1 p

p5 tanh pπh c

2tr

Gxy

core

Gyz

core

1

2

p = 1, 2, 3, ...

Σ

∞

A11 A12 A16

A12 A22 A26

A16 A26 A66

= h 1

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 1

+ h 2

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 2

+

Ex

coreh cts

Ly

0 0

0 Ey

coreh ctr

Lx

0

0 0

k y

coreGxycoreh cts

4Ly

+

k x

coreGxycoreh ctr

4Lx
142

 (I11)

 (I12)

 (I13)

B11 B12 B16

B12 B22 B26

B16 B26 B66

= − 1

2 h 1 + h c h 1

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 1

+ 1

2 h 2 + h c h 2

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 2

D11 D12 D16

D12 D22 D26

D16 D26 D66

= 1

12 4h 1

2

+ 6h 1h c + 3h c

2

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 1

+ 1

12 4h 2

2

+ 6h 2h c + 3h c

2

Ex

1 − νxyνyx

νxyEy

1 − νxyνyx

0

νyxEx

1 − νxyνyx

Ey

1 − νxyνyx

0

0 0 Gxy

plate no. 2

+

Ex

coreh c

3

ts

12Ly

0 0

0

Ey

coreh c

3

tr

12Lx

0

0 0 h c

12

Gxz

corets

3 1 − Ks

Ly

+

Gyz

coretr

3 1 − Kr

Lx

A44 A45

A45 A55

=

kz

coreGyz

core

h ctr

Lx

0

0 kz

coreGxz

core

h cts

Ly
143

Appendix J

Equivalent-Plate Stiffnesses for a Star-Cell-Core Sandwich Plate

 The equivalent-plate stiffnesses for a sandwich plate composed of two nonidentical, 

anisotropic, laminated-composite face plates and the isosceles-star-cell core made of a 

homogeneous orthotropic material shown in figure 27 are presented in this appendix. The 

thicknesses of the bottom face plate (plate no. 1), the top face plate (plate no. 2), and the core are 

denoted by h1, h2, and hc

, respectively. The plate reference plane is located at the midplane of the 

core and the eccentricities of face plates no. 1 and no. 2 are given by and 

, respectively. The core is composed of two member types. One member type is 

aligned with the x-axis of the plate, as shown in figures 23 and 27, and is referred to herein as a 

stringer member. The other member type makes an angle Φ with the x-axis and is referred to 

herein as a diagonal. Member attributes associated with the stringers and diagonals are indicated 

subsequently with the subscripts or superscripts "s" and "d," respectively. 

 For an isosceles-star-cell core with each member made of a homogeneous orthotropic material, 

the effective moduli given by equations (A13a)-(A13c) reduce to the corresponding principal 

moduli of the orthotropic material, where the principal orthotropic-material axes are coincident 

with the X, Y, and Z beam coordinate axes shown in figure 6. Thus, the symbols Es

 and Ed 

denote the extensional modulus EX and the symbols Gs and Gd denote the mean principal shear 

moduli of the stringers and diagonals, respectively. Similarly, the stiffness-weighted 

first moments of area , , , and vanish for homogeneous stiffeners centered on the plate 

reference plane. Moreover, the stiffness-weighted moments of inertia reduce to the corresponding 

second moments of area and the stiffness-weighted product of inertia vanishes for the rectangular 

stiffener cross sections. The cross-sectional areas of the core members are given by As

 = hcts and 

Ad = hctd, where ts is the stringer thickness and td is the diagonal thickness. The moments of 

inertia associated with out-of-plane bending are given by for the stringers and 

for the diagonals. The planar geometric dimensions of the core, as shown in figure 23 and 27, are 

given by the symbols B and H, where B and H are the base and height, respectively, of either 

isosceles triangle forming the star shape. The symbols Gs

Js and GdJd are the torsional stiffnesses 

of the stringers and diagonals, respectively. For a stringer with a rectangular cross-section made 

of a homogeneous orthotropic material, equation (A16) yields 

 (J1)

and for a diagonal

e1 = − 1

2 h1 + hc

e2 = 1

2 h2 + hc

GXZGXY

zs zd zs zd

Is = hc

3

ts

12 I d = hc

3

td

12

GsJs = GXZ

ts

3

h c

3 1 − 96

π5

ts

h c

GXZ

GXY

1

2 1 − −1 p

p

5 tanh pπh c

2ts

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞
144

 (J2)

Likewise, the symbols and denote the stiffness-weighted transverse-shear correction 

factor of the stringers and diagonals, respectively, and and denote the 

stiffness-weighted inplane-shear correction factor of the stringers and diagonals, 

respectively. 

 Based on the notation just given, the equivalent-plate-stiffnesses contributions of the core are

 (J3)

 (J4)

 (J5)

 (J6)

 (J7)

 (J8)

 (J9)

 (J10)

 (J11)

GdJd = GXZ

tr

3

h c

3 1 − 96

π5

t d

h c

GXZ

GXY

1

2 1 − −1 p

p5 tanh pπh c

2t d

GXY

GXZ

1

2

p = 1, 2, 3, ...

Σ

∞

k Z

s

k Z

d

k Z

GXZ

GXY

k Y

s

k Y

d

k Y

GXY

GXZ

A11

core

= Esh cts

H

+

Edh ct dB3

8HL3 1 + 4τY

d H2

B2

A12

core

= Edh ct d BH

2L3 1 − τY

d

A16

core = 0

A22

core

= 2Edh ct dH3

BL3 1 + τY

d

B2

4H2

A26

core = 0

A66

core

= EsτY

s

h cts

4H

+

Edh ct d BH

2L3 1 + τY

d B

4H − HB

2

D11

core

= h c

3

ts

12H Es +

Ed

8

t d

ts

B3

L

3 1 +

16Gd

Ed

t d

2

h c

2

H2

B2 1 − Kd

D12

core

= Edh c

3

t dBH

24L3 1 − 4Gd

Ed

t d

2

h c

2 1 − Kd

D16

core = 0
145

 (J12)

 (J13)

 (J14)

 (J15)

 (J16)

 (J17)

where the inplane-shear-deformation parameters are defined as

 (J18)

 (J19)

and the transverse-shear-deformation parameters are defined as

 (J20)

 (J21)

The overall stiffnesses of the sandwich plate are obtained by substituting equations (J3)-(J17) into 

equations (47).

 For the special case of an isotropic star-cell core comprised of triangles with all sides having 

length L and thicknesses ts

 = td = t, the angle Φ shown in figure 23b is 60 degrees, , and 

B = L. In addition Es

 = Ed = E and Gs = Gd = G, where E and G are the extensional and shear 

D22

core

= Edh c

3

t dH3

6BL3 1 +

Gd

Ed

t d

2

h c

2

B2

H2 1 − Kd

D26

core = 0

D66

core

= Gsts

3

h c

12H 1 − Ks +

Edh c

3

t dBH

24L3 1 +

4Gd

Ed

t d

2

h c

2

B

4H − HB

2

1 − Kd

A44

core

= 2τZ

d

Edh ct d H

BL

A45

core = 0

A55

core

= h cts

2H

2τZ

s

Es + τZ

d

Ed

t d

ts

B

L

τY

s

≡ k Y

s

Gs

Es

τY

d

≡ k Y

d

Gd

Ed

τZ

s

≡ k Z

s

Gs

Es

τZ

d

≡ k Z

d

Gd

Ed

H = 3L

2
146

moduli of an isotropic material, and . For this case, equations (J1) and (J2) 

are expressed as 

 (J22)

where

 (J23)

Thus, the equivalent-plate core-stiffness contributions become 

 (J24)

 (J25)

 (J26)

k Y

s

= k Y

d

= k Z

s

= k Z

d

= k

GsJs = GdJd = Gt3

h c

3 1 − K

K = 96

π5

t

h c

1 − −1 p

p5 tanh pπh c

Σ 2t p = 1, 2, 3, ...

∞

A11

core A12core A16core

A12

core A22core A26core

A16

core A26core A66core

=

3h ct E

4L

3 + kG

E 1 − kGE 0

1 − kG

E 3 + kGE 0

0 0 1 + kG

E

D11

core D12core D16core

D12

core D22core D26core

D16

core D26core D66core

=

3h c

3

t E

48L

3 + 4G

E

t

2

h c

2 1 − K 1 − 4G

E

t

2

h c

2 1 − K 0

1 − 4G

E

t

2

h c

2 1 − K 3 + 4G

E

t

2

h c

2 1 − K 0

0 0 1 + 4G

E

t

2

h c

2 1 − K

A44

core A45core

A45

core A55core = 3kGh ct

L

1 0

0 1
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 4. TITLE AND SUBTITLE

A Treatise on Equivalent-Plate Stiffnesses for Stiffened 

Laminated-Composite Plates and Plate-Like Lattices

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 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 survey of studies conducted since 1914 on the use of equivalent-plate stiffnesses in modeling the overall, stiffness-critical response of 

stiffened plates and shells is presented. Two detailed, comprehensive derivations of first-approximation equivalent-plate stiffnesses are also 

presented that are based on the Reissner-Mindlin-type, first-order transverse-shear deformation theory for anisotropic plates. 

Equivalent-plate stiffness expressions, and a corresponding symbolic manipulation computer program, are also presented for several 

different stiffener configurations. These expressions are very general and exhibit the full range of anisotropies permitted by the 

Reissner-Mindlin-type, first-order transverse-shear deformation theory for anisotropic plates. The expressions presented in the present study 

were also compared with available, previously published results. For the most part, the previously published results are for special cases of 

the general expressions presented herein and are almost in complete agreement. Analysis is also presented that extends the use of the 

equivalent-plate stiffness expressions to sandwich plates.

15. SUBJECT TERMS

Homogenization; Equivalent-plate stiffnesses; Structural design; Single-layer grids; Lattices

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NASA/TP-2011-216882

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and 

Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person 

shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 01 - 2011