Page 203 - Plastics Engineering

186 Mechanical Behaviour of Composites
Now, in order to determine the stiffness in the global (n, y) directions for a
lamina in which the fibres are aligned at an angle 8 to the global x direction,
it is necessary to go through the following three steps:
1. Determine the strains in the local (1,2) directions by transforming the
applied strains through 8" from the global (x,y) directions.
2. Calculate the stresses in the local directions using the on-axis stiffness
matrix [Q] derived earlier.
3. Transform the stresses back to the global directions through an angle
of -eo.

Using matrix notation, these three steps are carried out as follows:

step 1
2 s2 sc
{ ;;2} = [ s2 c2
-2sc 2sc (c2 -2)
Note the modification to [To] to give the Strain Wansformation matrix T, so
that we can use yl2 instead of iy12.

step 2
0 ] {I}
Qii Q12 0
0
(2) = [ Q21 Q22 Qss
0
Step 3
sc -sc (2 2sc -2) 1 E: 1
-2sc
c2
s2
(Z}= [2 c2
Linking these steps together to obtain the global stresses in terms of the
global strains

c2 s2 -2s~ Qll Q22 0" ]
Ql2
s2 c2 2sc ] [ Q21
sc -SC (c' -s2) 0 0 QM
c2 s2 sc
s2 c2
-2sc 2sc (2 -2)

which provides an overall stiffness matrix @]

(3.24)