Page 217 - Mechanics Analysis Composite Materials

202 Mechanics and analysis of composite materials

(4.153)

(4.154)

where elastic constants of an individual ply are specified by Eqs. (4.76). Strain-
displacement equations, Eqs. (2.22), for the problem under study are

(4.155)

Integration of the first equation yields for the +q5 and -4 plies
u;4 =z.x+u(y), U,-b =&.X-u(y) , (4.156)

where u(y) is the displacement shown in Fig. 4.73. This displacement results in the
following transverse shear deformation and transverse shear stress

(4.157)

where G, is the transverse shear modulus of the ply specified by Eqs. (4.76).
Consider the equilibrium state of +4 ply element shown in Fig. 4.74. Equilibrium
equations can be written as

(4.158)

The first of these equations shows that z,,, does not depend on x. Because the axial
stress, a,, in the middle part of a long specimen also does not depend on x,
Eqs. (4.153) and (4.155) allow us to conclude that zY and hence do not depend on
x. As a result, the last equation of Eqs. (4.155) yields in conjunction with the first
equation of Eqs. (4.156):

Fig. 4.74. Forces acting on the infinitesimal element of a ply.