Page 275 - Mechanics Analysis Composite Materials

260 Mechanics and analysis of composite materials

Applying Eqs. (5.71) to calculate the strains in the plies principal material
coordinates and using Eqs. (5.72) to find the stresses, we get:
0 in the angle-ply layer,

P P P
a{’)= -0.26- ay) = -0.028-, 21:) = 0.023- ;
Rh’ Rh Rh
0 in the hoop layer,

P P
a?) = 0.073-, CY’ = -0.089 -, T$) = 0 ,
Rh Rh
where h = hl +hz is the total thickness of the laminate. To calculate interlaminar
stresses acting between the angle-ply and the hoop layers, we apply Eqs. (5.73).
Using Eqs. (5.4) and taking Eqs. (5.82) and (5.83) into account, we first find the
stresses in the layers referred to the global coordinate frame x, y, z, i.e.,

a!) = A(’)&O x +~(i)&o +). =A(~)&oA@),O22 I’ ,Wxy = 0 ’ (5.84)
+
21
x
12
y’
11
where, i = I, 2 and A$; are given by Eqs. (5.80) and (5.81). Because these stresses
do not depend on x and y, the first two equations in Eqs. (5.73) yield

This means that both interlaminar shear stresses do not depend on z. But on the
inner and on the outer surfaces of the cylinder shear stresses are equal to zero. So
zxz = 0 and z, = 0. The fact that zp = 0 is natural. Both layers are orthotropic and
do not tend to twist under axial compression of the cylinder. Concerning zxi = 0 a
question arises as to how compatibility of the axial deformations of the layers with
different stiffnesses can be provided without interlaminar shear stresses. The answer
follows from the model used above to describe the stress state of the cylinder.
According to this model, the transverse shear deformation yx is zero. Actually, this
condition can be met if the part of the axial force applied to the layer is proportional
to the layer stiffness, Le., as

(5.85)

Substituting strains from Eqs. (5.79) we can conclude that within the accuracy of
a small parameter hJR which was neglected in comparison with unity when we
calculated stresses PI +P2 = -P, and that the axial strains are the same even if the
layers are not bonded together. In the middle part of a long cylinder, axial forces
are automatically distributed between the layers in accordance with Eqs. (5.85).
However, in the vicinity of the cylinder ends this distribution depends on the loading