Interlaminar stress predictions for laminated composites under bending, torsion and their combined effect

Composires Engineering, Printed in Great Britain.

Vol. I, No. 5. pp. 277-291,

0961-9526/91 $3.00+ .GU 0 1991 Pergamon Press plc

1991.

INTERLAMINAR STRESS PREDICTIONS FOR LAMINATED COMPOSITES UNDER BENDING, TORSION AND THEIR COMBINED EFFECT+ ERIAN A. ARMANIOSand JIAN LI School of Aerospace Engineering, Georgia Institute of Technoloty, Atlanta, GA 30332-0150, U.S.A. (Received 24 May 1991; accepted 10 July 1991) Abstract-A simple analytical formulation is used to predict the interlaminar stresses in a symmetric laminate under extension, bending, torsion and their combined effect. The method is based on a shear deformation theory and a sublaminate approach. The solution is obtained in closed form and the controlling parameters are isolated. Extensive comparisons with a finite element solution are made to verify the model for the case of laminates subjected to torsion loading. An assessment of the induced curvature predicted from classical lamination theory and the present approach is provided. The influence of the induced curvature on the interlaminar stress prediction is investigated for torsion loading. The interlaminar stresses in laminates subjected to combined bending and torsion are obtained by superposition.

1. INTRODUCTION

Composite materials have played an important role in achieving overall performance improvements in advanced structures. However, these are often limited by delamination resulting from interlaminar stresses. An example is the flapping flexure region of the Bell 680 rotor hub. In addition to axial centrifugal force, the composite rotor flexure is also subjected to bending and torsion loading. The interlaminar stresses induced by these loads can cause free edge delamination. It is well documented (Puppo and Evensen, 1970; Pipes and Pagano, 1970; Pagano, 1978; O’Brien, 1982, 1984; Whitney and Knight, 1985; Whitney, 1986; Armanios and Rehfield, 1989a) that interlaminar stresses arise at the free edges of laminates under uniform extension, and delamination caused by interlaminar stresses can initiate at the free edges in these structures. However, there is limited work on bending, torsion and combined loadings. Salamon (1975, 1978) predicted the interlaminar stresses in four-layer [+45], and [0,0], laminates under uniform bending using a finite difference approach to solve the exact elasticity equations. He found that the interlaminar shear and normal stresses rise sharply near the free edges. Armanios and Rehfield (1988) studied bending and combined bending and extension using a transverse shear deformation theory and a sublaminate approach for laminate layups where Mode III is negligible, such as [0,,/90,], and [0,,/(*45),], laminates. Interlaminar stresses and energy release rates were obtained in closed form. They concluded that the energy release rate in a combined bending-extension loading may be more critical than extension loading only. Ye and Yang (1986) developed a quasi-three-dimensional finite element procedure to investiage the free edge effects in symmetric composite laminates of finite width under bending. Results were presented for angle-ply [*45], and symmetric cross-ply laminates. Since the twisting effect induced by bending was not considered, their solution can only be used for those symmetric laminates where twisting and bending coupling effects are negligible. Chan and Ochoa (1987) calculated the interlaminar stresses and energy release rates in symmetric laminates with various layups subjected to bending. They found that the total and Mode I strain energy release rate decrease as delamination size increases and reach a lower bound. They also obtained (Chan and Ochoa, 1987b) the interlaminar stress distributions and energy release rates for laminates with [O],, , [45,, -45,],, and [45,, -452, 0, , go,], layups under torsion ‘A partial summary is published in the proceedings of ICCM VII, Honolulu, pp. 27-A-l to 27-A-l 1. 277

Hawaii, 15-19 July 1991,

218

E. A. ARMANIOS

and JIAN LI

loading. Unlike the bending case, they found that the Mode III strain energy release rate increases steadily as a function of crack length and eventually reaches a plateau. Kurtz and Whitney (1988) developed an exact elasticity solution for the torsion of cross-ply laminates. Comparison of this solution with the existing elasticity theory for the torsion of homogeneous orthotropic bars (Lekhnitskii, 1963) showed that the homogeneous solution is sufficiently rigorous for most practical applications. Murthy and Chamis (1989) used a three-dimensional finite element analysis to investigate the width and loading conditions effects on the free-edge stress fields in composite laminates. The analysis included a special free-edge region refinement or superelement with progressive substructuring. Various loading conditions were considered including out-of-plane twisting moment and in-plane bending. The three-dimensional free-edge stress was determined using a cantilever geometry. They found that axial extension produces the smallest magnitude of interlaminar free-edge stress compared to other loading conditions. Daniel and Vizzini (1989) calculated the interlaminar stresses in a [O/90], and [+15], laminate under torsion using MSC/NASTRAN anisotropic solid elements. In contrast to the results of Kurtz and Whitney (1988), they reported a non-zero interlaminar normal stress in a [O/90], laminate. Armanios and Li (1990) developed a simple shear deformation model for the analysis of laminated composites subjected to torsion loading. The interlaminar stresses were obtained in closed form and were in good agreement with the exact elasticity solution (Lekhnitskii, 1963) for a [O],, laminate. Comparisons of the induced bending curvature predicted by the classical laminate plate theory adopted in Chan and Ochoa (1987) and the shear deformation model for [45,, -45&, and [452, -452, 0,) 90,], layups were also provided. Li and Armanios (1990) obtained a closed-form solution for the interlaminar stress and strain energy release rate for unidirectional and cross-ply laminates under torsion. The results showed excellent agreement with the exact elasticity solution for the interlaminar stress distributions. The Mode III strain energy release rate in a [0/9014, layup with a delamination at the mid-surface was found to increase with crack growth. A constant value was reached for crack lengths larger than 10 ply thicknesses. This behavior is in agreement with the results of Chan and Ochoa (1987b). Whitney (1991) obtained a closed-form solution for the deformation of a laminated anisotropic plate subjected to torsional loading. Both classical laminated plate theory and a modified laminated plate shear deformation theory were considered. A comparison of solutions indicated that the classical theory is not capable of representing the response of a laminate subjected to torsional loading. Yin (1991) used a variational method based on Lekhnitskii’s stress functions to determine the free-edge interlaminar stresses in a multilayered laminate subjected to combined axial extension, bending and twisting loads. Solutions were calculated for symmetric four-layer cross-ply and angle-ply laminates. It is seen that closed-form-type models used for parametric studies are not available for generally symmetric laminates subjected to torsion and bending. The finite element method and finite difference method are computationally intensive and can be expensive to use for preliminary design analysis. For example, in Murthy and Chamis (1989) the finite element model for a four-layer cantilever has a total of 22,683 degrees of freedom. On the other hand, exact elasticity solutions are limited to orthotropic laminates, The objective of this work is to develop a simple analytical model that provides accurate estimates of the interlaminar stresses in composite laminates under combined loading. Improvements in the delamination strength of these structures are achieved by identifying the critical interfaces and using design concepts that minimize the interlaminar stresses. 2. MATHEMATICAL

MODEL

In this analysis, the interlaminar stresses are estimated based upon a displacement field that includes shear deformation and a sublaminate approach. Shear deformation is significant in composites because of their high extensional-to-shear-modulus ratios. The sublaminate approach provides the flexibility of treating groups of plies as single laminated units and thus avoiding the complexity of a ply-by-ply modeling. It has been

Stress predictions for laminated composites

FJ

219

‘T

Fig. 1. Laminate configuration

and loading.

validated by comparison of predictions with a finite element solution in Armanios and Rehfield (1989) for laminates under uniform extension. Further simplification is achieved in the analysis by neglecting thickness strain. In this case, the interlaminar peel stress is estimated by enforcing the equilibrium and boundary conditions on the transverse shear stress resultant, as described in Armanios and Badir (1990). The generic laminate shown in Fig. 1 is subjected to combined extension F, torsion T and bending moment M. For a homogeneous anisotropic elastic body bounded by a cylindrical surface, a generalized plane deformation (Lekhnitskii, 1963) exists when the stress tensor does not vary along the generator, x. The displacement field in the absence of rigid body modes can be written as u(x,y,z)

= &o-X+

u(x, y, z) = -+K, w(x,y,z)

= -*K-X2

K’X’Z

+ KI’X’Y

+ U(y,z)

* x2 + c * x * z + V(y, z) - c-x*y

(la)

+ W(y,z)

where u, u and w denote displacements relative to the x, y and z axes, respectively. Axial extension is denoted by E,-,, while K and K~ are the bending curvatures in the x-z plane and x-y plane, respectively. The relative angle of rotation about the x axis is denoted by C. The laminate is considered as being made of sublaminates or groups of plies. Each sublaminate is treated as a homogeneous anisotropic body bounded by a cylindrical surface and subjected to interfacial stresses, as shown in Fig. 2. The thickness-to-width

Fig. 2. Sublaminate notation and sign convention.

E. A. ARMANIOSand JIAN LI

280

ratio of each sublaminate is much smaller than unity. Therefore, the bending curvature K, is negligible. Further simplification is achieved by considering transverse shear deformation only in the expression for the displacement components U(y, z), V(y, z) and W(y, z) associated with the reference surface. The displacement field in eqn (la) takes the form u(x,y,z)

= EO’X+

K’X’(Z

+ 6) + U(y) + z-P,(y)

u(x, y, z) = V(y) + z * Py(y) + c * x * (z + 6) W(X,y,Z)

=

-+K’X’

-

c*X*(y

+

p)

+

(lb)

w(y).

The arbitrary constants 6 and p are to be determined from continuity of displacements between sublaminates and from the overall boundary conditions. Shear deformation is recognized through the rotations /3, and j$ . The corresponding strains are &xx =

&&

+

ZK,

&YY =

&;y

+

ZKy

0

y v = )‘$ + ZK,

YYZ =

E,, Yxz

YYZ

=

0

(2)

0

=

Yxz*

The strain components associated with the reference surface are denoted by superscript ‘. These are defined as 0 &XX= Eo + K - 6 K,

$z

=

K

KY =

&;y = yy Kg

fiY,Y

= Py + yy

y;=qy+c+ =

Y,9; = Px -

&,

+

(3)

c

c * (Y + PI

where partial differentiation is denoted by a comma. The constitutive relationship can be written in terms of resultant forces and moments and strains and curvatures as follows:

‘&O xx\

All

An

A16

&I

42

46

A,2

-422

A26

42

822

B26

A,6

A26

A66

B16

B26

B66

0 &vu 0 y,

4,

42

46

D,,

012

46

KX

42

B22

B26

D12

D22

O26

KY

(4)

bKW

For a sublaminate

of thickness h, the stiffness coefficients are defined as h/2 (Aij,B,,Dij)

=

-h,2

&ij(ls

Zs Z2)

dZ

(6)

s reduced stiffnesses as defined in Vinson and Sierakowski

where oij are the transformed (1986). The equilibrium equations can be written as

N xy,Y + f2x - t1, = 0 Ny ,y + hy - tly = 0

Qy,y +

~2

-

PI

= 0 (7)

M xy.y

-

Qx +

5 * (tz,

+ fd

= 0

M y,y

-

Qy +

; - (by

+ fly)

= 0

Stress predictions for laminated composites

281

where the interlaminar shear and peel stresses at the sublaminate upper and lower surfaces are denoted by tzx, tD, p2 and t,, , t, , pl, respectively, as shown in Fig. 2. 3. INTERLAMINAR

STRESSES

A solution for a laminate subject to torsion loading is sought first. With minor modifications, it can be applied to a laminate under bending. A complete solution for combined loadings can be obtained by superposition. Considering the antisymmetrical nature of the loading, one half of the cross-section is modeled by using two sublaminates referred to as sublaminate 0 and sublaminate 1, as shown in Fig. 3. The interlaminar stresses at the interface between sublaminate 0 and 1 are determined by enforcing the continuities of displacements at the interface between the sublaminates. These are

u,(x,Y, $) =U’(X.y. -2) .(x.,.$) =Vl(X& -2) ,(x.,.2) =WI(X. y.-:) where hI and h,, denote the thicknesses of sublaminates laminate central plane we have

(8)

1 and 0, respectively.

At the

(9)

To suppress the rigid body modes, it is assumed that the center of the laminate That is, at x = y = z = 0 we have

is fixed.

u=v=w=o

(10) V ,x =

Substituting we have

w,x

=

v,,

-

w,y

=

0.

eqn (lb) into eqns (8) and (9), and considering

6, = h, + 2

6, = 2

Pl

=

the conditions

PO= 0

and

2b

Fig. 3. Sublaminate model and coordinate system.

in eqn (lo), (11)

E. A. ARMANIOSand

282

JIAN

LI

The subscripts in eqns (ll), (12) and the equations to follow refer to the respective sublaminate. The response associated with sublaminates 0 and 1 is coupled through the interface continuity conditions. The upper surface of sublaminate 1 is stress free. The shear and peel stresses at the bottom surface will be denoted by t,, tY and p, respectively. From reciprocity of stresses at the interface between sublaminates 1 and 0, the stresses at the upper surface of sublaminate 0 are tx , tY and p. From the antisymmetric condition at the sublaminate bottom surface, the peel stress is zero. The interlaminar shear stresses at the bottom surface are denoted by t,, and t,,, for sublaminate 0. There are five boundary conditions at each sublaminate free edge, namely N.iIy=*b

=

O,

W-yiIy=*b

=

O9

Nyi

Iy = ib

=

0,

(13) Myi Iy = *b = 0,

QyiIy=*b

= 0

where i refers to the respective sublaminate: 1 or 0. The laminate semi-width is denoted by b. By using the kinematic relations provided in eqn (12), these boundary conditions reduce to

Q~I + Qyoly=fb = o. By applying eqn (7) to sublaminates 1 and 0, and prescribing the boundary conditions eqn (14) at the sublaminate-free edges, the interlaminar stresses are given as

in

where h, = B; + ?A;. The superscript in eqns (15) and (16) refers to sublaminate given by

(16)

1. The rotations in eqn (15) are

WI = Ml W siWv9) - WI[RI MITbW~

(17)

where WI = Ub ,Pxo,Pyl,PyolT, PI = k#4 M2h k4, M411, [H sinh(sy)] = (H, sinh(s, y), H2 sinh(s,y), H3 sinh(s,y), I-I4 sinh(s,y))T, and la1

=

A45 -7A45 44

o -

A&,

A FA:, 44

T A45 - A:,, -Aj4 A 44

- A&

(18)

(19) and Si and (+i) (i = l-4) are the eigenvalues and eigenvectors of the following system of linear homogeneous equations:

s2Flld = m(yll.

(20)

Stress predictions for laminated composites

283

The matrices [F] in eqn (20) and [K] in eqns (19) and (20) are defined as

h, 46 + -2 h6,

SYM

ho&

ho + -2 c66

e66

iFI

(214 ho h,, k&i

e26

ho + - 2 c26

62 + 2 h22

+

hiA:6

e,,

h&2

+

ho -c22 2

+ h2A’ 0 22

@lb)

rA;5 (2d2

SYM

44

-~A!& WI =

At5 ($I2 44

Ao;; 44

-~A:,&

45

A 44

A 44

A44

--A:&

A,,&

A 44

(22)

&A:, --A:‘&, A:k, ~ A 44

A44

A44

_

The superscripts in eqns (21) and (22) refer to the respective sublaminate. The integration constants (H) in eqn (17) are determined from the solution of the following system of four linear equations: (23)

where (d,,+

(h,++b++(d,,+

e66 + 366 (

Id =

(d,,+

(ho+~)h,)~

+ ho(h,, + hoAA6) C + e16 + $k,6 > (

(ho++bj)C+

(d,,+

+ ho(h16 + hoA;,)

K >

(24)

(ho++,+

I

! + p,,,

+ ho(h12 + hoA;,)

)J

K

As a result of neglecting the thickness strain, the interlaminar peel stress distribution predicted from eqn (7) does not satisfy the shear force boundary condition at the sublaminate free edge. In order to enforce this condition a boundary function expressed in terms of the characteristic roots is added to the transverse shear force expression as outlined in Armanios and Badir (1990). The modified peel stress can be written as

284

E. A. ARMANIOS

and JIAN LI

where Q,,r = Q,,r + cl sinh(s,y) + c2 sinh(s,y)

(26)

and ~3

‘r = (.s4tanh(s,b)

Qy,(@

- s3 tanh(s,b)) * cosh(s36) (27)

Qyl(b)

~4

” = - (s4 tanh(s, b) - s3 tanh(s, 6)) - cosh(.s, b) * where Q,,r denotes the resultant shear force in sublaminate 1 as provided from the constitutive relationships in eqn (5) or the equilibrium equation in eqn (7). The eigenvalues in eqns (26) and (27) are arranged in increasing order of magnitude s1 s s2 I s3 I s4 . For a laminate subjected to pure bending the analysis under torsion is modified to reflect the end boundary condition in order to determine the induced twist, as outlined in the Appendix. 4. VERIFICATION

OF PREDICTIONS

The present theory is applied to the prediction of interlaminar stresses in unidirectional and angle-ply laminates with [45],r and [45, -451, layups, respectively. The laminates are subjected to torsion. A comparison of these predictions with the finite element method (FEM) of Chan and Ochoa (1987a) is presented in Figs 4-14. The laminate geometry and material properties used in the FEM model are given in Table 1 for convenience. The applied twist C = 0.5 rad in-’ (0.02 rad mm-‘). The through-thethickness distribution in Figs 7, 8, 12 and 13 are plotted for Y/H = 50.89 or 0.11 ply 1.5 -

-.-- *---

1.0 -

PlaseM pEM

E b

0.5 -

i 0.0 $ E: 4.5 P

-

-1.0 -

-1.51 -60

.

’

-40

.

’

.

-20

’

0

’

’

20

.

’

40

I

’

60

Y/H

Fig. 4. Interlaminar shear stress distribution

across the width of a [4Q2. laminate.

100 60 5

60-

4

40-

-40 .BO

-40

-20

Present

0

20

40

4 60

Y/H

Fig. 5. Interlaminar normal stress distribution across the width of a [45],, laminate.

Stress predictions for laminated composites 1.5

-

---_ r)---

1.0 E i ab

285

Pmamt FEM

Z.h/2 0.5 -

0.0 8

II r"

-0.5 -

-1.0 -

.1.51

* -60

’

-40

’

’

.

-20

’

.

0

’

20

.

’

40

.

’

60

Y/H

Fig. 6. Interlaminar shear stress distribution across the width of a [45],, laminate.

Fig. 7. Interlaminar normal stress distribution across the thickness of a [45],, laminate.

-

Presml

----*---

FEM

Fig. 8. Interlaminar shear stress distribution across the thickness of a [45],, laminate.

thicknesses from the free edge. The comparisons in Figs 4-13 show that the interlaminar stresses are in good agreement with the FEM solution. The induced bending curvature due to twist was estimated in Chan and Ochoa (1987b) from classical lamination theory (CLT). In CLT however, all four edges of the laminate are assumed to be subjected to torsion and consequently the free-edge boundary conditions along the unloaded edges is not enforced. In the present approach the induced bending curvature is calculated based upon satisfying the stress-free boundary condition along the unloaded edge, as outlined in the Appendix. A comparison between the induced

E. A.

286

and

ARMANIOS

JIAN

LI

Fig. 9. Interlaminar shear stress distribution across the width of a [45, -451, laminate.

0.6 -

E L k

0.4 -

8

0.2 -

-

Present

-------.

FEM

b” 0.0 -

-0.2

&

-40

-60

-20

0

20

40

60

Y/H

Fig. 10. Interlaminar normal stress distribution across the width of a [45, -451, laminate.

.*-- *---

Present FEM

i!.h/2

-2 -60

.

’ -40

’ .20

’ 0

.

’ 20

.

’ 40

.

’ 60

Y/H

Fig. 11. Interlaminar

shear stress distribution across the width of a [45, -451, laminate.

bending curvature predicted by the present approach and the CLT method of Chan and Ochoa (1987b) is provided in Table 2 for four laminates. The CLT prediction is denoted by K, and the twist per unit length by C. The CLT predictions are in good agreement with the present approach for laminates with small thickness-to-width ratio (h/b). This is because the free-edge boundary zone is small and consequently CLT prevails in a larger portion of the laminate. However, CLT overestimates the induced bending curvature by 23.5% relative to the present approach for a [45,, -45,],, laminate and by 22.6% for a [45*, -45,) 0, ,90,], laminate. Although the difference in the induced bending curvature

Stress predictions for laminated composites 1

E L 8

-

Prsrni

---*--.

PEY

:

P

. ..--.

:

281

’

Y/H=SO.BB

-2

-1

0

I

2

Z/H

Fig. 12. Interlaminar normal stress distribution

across the thickness of a 145, -451, laminate.

YIH.50.99

-3 1 -2

-1

, 2

1

0 Z/H

Fig. 13. Interlaminar

-15

shear stress distribution across the thickness of a [45, -451, laminate.

-6

.

’ -6

.

’ -4

.

I -2

.

’ 0

’ 2

.

’ 4

.

’ 6

a’

Z/H

Fig. 14. The influence of induced bending curvature on the present approach for a [45,, -45,],, laminate. Table 1. Material properties of Graphite/epoxy E,, = 19.3 Msi (133.07 GPa) = 1.62 Msi (11 .I7 GPa) G,, = G,, = G,3 = 1.02 Msi (7.03 GPa)

E,, = E,,

v = 0.228

Ply thickness H = 0.0052 in. (0.132 mm) Semi-width b = 51H

E. A. ARMANIOS

288

and JIAN LI

Table 2. Induced bending curvature Stacking sequence

K,/2c

[451,, 145% -451, [45,, -45,1,,

]45,. -45*,% 990,1,

K/2c

(K -

K,)/K

h/b

(VO)

- 0.3923 - 0.2942

- 0.3829

2.5

2/51

-0.2832

3.9

2/51

-0.1471 -0.0816

-0.1191 - 0.0665

23.5 22.6

l-V51 8151

is significant in these laminates, its influence on the interlaminar stress is small due to the fact that the induced bending curvature is small relative to the applied twist. This is shown in Fig. 14, where the through-the-thickness shear stress r,, for a [452, -45& laminate is plotted at a distance of 0.5 ply thicknesses from the free edge. The predictions based on K and K, are in close agreement. The interlaminar stress distribution in laminates subjected to extension, bending and torsion is obtained by superposition. As an example, Figs 15-17 show the interlaminar stress distributions for a [02, *45], laminate under bending, torsion and combined bending and torsion. The applied bending curvature K = 1 rad in:’ (0.04 rad mm-t) and the twist C = 0.5 rad in:’ (0.02 rad mm-‘). The shear stress distribution appears in Figs 15 and 16 while the peel stress distribution is provided in Fig. 17. It is worth noting that the peel stress at the free edge of the O/45 interface is compressive when the laminate is subjected to uniform extension. Figures 15-17 show the interlaminar stress distribution in the upper half of the laminate. The interlaminar shear stress distribution is symmetric about the mid-plane of the laminate while the peel stress distribution is antisymmetric. 40 ? ? ii

Ztl0 1 m I

-20 ----m--E:

-40 -4o-

-.-.-

e

&“(j,np

l

.-.-

TOWon

-60

mlehg

+

Tw6ion

-

Z/H=2 t

-64 -

i -1OOL 0

.

’ 10

.

r 20

.

’ 30

I 40

’ 50

.

1 60

Y/H

Fig. 15. Interlaminar

N

shear stress distribution across the width at the O/45 interface of a [Oz, 45, -451, laminate.

-100

-

q w

Z/H.0

-ZOO f 0

10

20

30

40

50

60

Y/H

Fig. 16. Interlaminar shear stress distribution across the width at the mid-plane of a [O,, 45, -451, laminate.

Stress predictions for laminated composites

a

-20 -

I -30 -

----o---

&“(f,n(l

_._.-

Tomon

l .-.-

-

Bending

289

+ TomIon f

-701

. 0

’ 10

.

’ 20

.

’ 30

.

’ 40

.

’ 50

.

t 60

V/H

Fig. 17. Interlaminar

normal stress distribution across the width at the O/45 interface of a [0,, 45, -451, laminate.

Consequently, the peel stress distribution in Fig. 17 is tensile in the lower half of the laminate. This can precipitate delamination in the graphite/epoxy material considered. Moreover, the initial symmetry of the laminate will be altered. 5. CONCLUSION

A simple analytical formulation is developed for the analysis of laminated composites subjected to bending, torsion and their combined effect. Closed-form expressions for the interlaminar stresses are obtained. The analysis is applied to unidirectional and angle-ply laminates subjected to torsion loading. The interlaminar stress predictions are in good agreement with a finite element solution. Comparisons between the induced curvature predicted from classical lamination theory and the present approach show significant differences for laminates with thickness up to 16% of the width. However, its effect on the interlaminar stress prediction is negligible for the laminates considered in this study. Superposition is used to predict the interlaminar stresses in laminates subjected to combined loading. Acknowledgements-This work was sponsored by the Army Research Office under Grant DAAL 03-08-C-0003 (the Center of Excellence for Rotary Wing Aircraft Technology). This support is gratefully acknowledged. The authors would also like to thank Dr Wen Chan of the University of Texas at Arlington for providing the FEM results used in the comparisons. REFERENCES Armanios, E. A. and Badir, A. M. (1990). Hygrothermal influence on mode I edge delamination in composites. Compos.

Struct.

H(4),

323-342.

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PREDICTION

OF INDUCED EXTENSION

AND CURVATURE

The relation between cO, K and C can be found by considering the entire laminate as one sublaminate and imposing the end boundary condition as follows: ‘b

MU. . I -b

dy = T

(Al)

where F, M and T are the applied extension force, bending and twisting moments, respectively. For a laminate subjected to torsion, we have

L42)

where (A3)

(A4) and (A5) The integration constant I is associated with the twisting curvature and is defined by Ko = 2C + Iscosh(sy).

W)

Stress predictions for laminated composites

291

Similarly, for a laminate subjected to bending we have

(.47)