Page 86 - Mechanics Analysis Composite Materials

Chapter 3. Mechanics of a unidirectional ply 71

Finite-difference equation, (3.31) can be reduced to the following form

A,T+I- 2A,,cosO + &.I = 0 , (3.33)
where

;1'
cos@=1 -~ (3.34)
2/2
As can be readily checked the solution for Eq. (3.33) is

A,, = B cos nO + C sin nO (3.35)

while Eq. (3.34) yields after some transformation
e
I = 2p sin - (3.36)
2
Substituting the solution, Eq. (3.35), into Eq. (3.30) we obtain after some
transformation

B = -Ctan(k + I)$
Thus, Eq. (3.35) can be written as

A,, = C [sinn8 - cos a@.tan(k + l)O] . (3.37)

Substituting Eq. (3.37) into Eq. (3.32) and omitting rather cumbersome trigono-
metric transformations we arrive at the following equation for 0
e
tank$= -tan- 2 . (3.38)

Z
Fig. 3.18. Geometric interpretation of Eq. (3.38) for k = 4.