Page 544 - Aircraft Stuctures for Engineering Student

12.8 Finite element method for continuum structures 525

It may be shown that

which has the form of Eq. (12.68), i.e.

(4 [Dl{&>
=
Substituting for {E} in terms of the nodal displacements {$} we obtain
{o} = [D][B]{a"} (see Eq. (12.69))
In the case of plane strain the elasticity matrix [D] takes a different form to that
defined in Eq. (12.92). For this type of problem

ffx uffy Uff;
Ex=----- E E
E

Eliminating a, and solving for ff.y, cy and T~~ gives
- U -
1 - 0
1-u { ;}
= U 1 0 (12.93)
(1 - 24
0 0 - /xy
- 2(1 - u)-
which again takes the form
{ff} = [DliE}
Step six, in which the internal stresses {a} are replaced by the statically equivalent
nodal forces {F'} proceeds, in an identical manner to that described for the beam-ele-
ment. Thus

as in Eq. (12.74), whence

In this expression [B] = [q[A-'] where [A] is defined in Eq. (12.85) and [C] in Eq.
(12.89). The elasticity matrix [D] is defined in Eq. (12.92) for plane stress problems
or in Eq. (12.93) for plane strain problems. We note that the [q, [A] (therefore [B])
and [D] matrices contain only constant terms and may therefore be taken outside