Page 543 - Aircraft Stuctures for Engineering Student

524 Matrix methods of structural analysis

From Eqs (1.18) and (1.20) we see that
all dv
dU aV rxy =-+-
Ex = - ay ax (12.88)
ax?
Substituting for u and v in Eqs (12.88) from Eqs (12.82) gives
{E}= [" 0 10 0 0 o,[q
E, =a2
Ey =a6
rxy = Q3 + Q5
or in matrix form

0
0
0
1
0
(12.89)
001010
a5
a6
which is of the form
{E} = [C]{a} (see Eqs (12.64) and (12.65))
Substituting for {a}(= [A-']{6e}) we obtain

{E} = [C'l[A-']{Se} (compare with Eq. (12.66))
or
{E} = [B]{Se} (see Eq. (12.76))
where [q is defined in Eq. (12.89).
In step five we relate the internal stresses {a} to the strain {E} and hence, using step
four, to the nodal displacements {g}. For plane stress problems
{u} = {;} (12.90)

7"y
and
ux va,
E, = - - -
E E
- ay -vox (see Chapter 1)
Ey-77-E

Thus, in matrix form,
{E} = { :} [ --v 0 (12.91)
1 I( z}

0
YXY =; :v 0 2(1+v) Tx,