Page 538 - Aircraft Stuctures for Engineering Student

12.8 Finite element method for continuum structures 519

where [A-'1 is obtained by inverting [A] in Eq. (12.58) and may be shown to be given
by
I o 0 0 01
r
1
1
0
[A-'I = -3/L2 -2/L 3/L2 -1/L (12.62)
1 2/L3 1/L2 -2/L3 1/Lq
In step four we relate the strain {E(.)} at any point x in the element to the displace-
ment {.(.)} and hence to the nodal displacements {g}. Since we are concerned here
with bending deformations only we may represent the strain by the curvature
a2w/dx2. Hence from Eq. (12.54)

(12.63)

or in matrix form

(12.64)

which we write as

{E> = [Cl{a) (12.65)
Substituting for {a} in Eq. (12.65) from Eq. (12.60) we have
{E} = [cI[A-'l{@} (12.66)
Step five relates the internal stresses in the element to the strain {E} and hence,
using Eq. (12.66), to the nodal displacements (6"). In our beam-element the stress
distribution at any section depends entirely on the value of the bending moment M
at that section. Thus we may represent a 'state of stress' {a} at any section by the
bending moment M, which, from simple beam theory, is given by

a2W
M=EI- ax2

(12.67)

{a} = [DI{E) (12.68)
The matrix [D] in Eq. (12.68) is the 'elasticity' matrix relating 'stress' and 'strain'. In
this case [D] consists of a single term, the flexural rigidity EI of the beam. Generally,
however, [D] is of a higher order. If we now substitute for {E} in Eq. (12.68) from Eq.
(12.66) we obtain the 'stress' in terms of the nodal displacements, i.e.
{a> = [Dl[cl[A-'l{@) (12.69)