Page 549 - Aircraft Stuctures for Engineering Student

530 Matrix methods of structural analysis

which is of the form
(0 = [Al{a>
Then
( 12.100)
The inversion of [A] is illustrated in Example 12.4 but, as in the case of the triangular
element, is most easily carried out by means of a computer. The remaining analysis
is identical to that for the triangular element except that the {E} - {a} relationship
(see Eq. (12.89)) becomes

[: o y o o 0

{E}= 0 0000 1 (12.101)
1x01 0

Example 12.4
A rectangular element used in a plane stress analysis has corners whose coordinates
(in metres), referred to an Oxy axes system, are 1(-2, -l), 2(2, -l), 3(2,1) and
4(-2,l); the displacements (also in metres) of the corners were
~1 = 0.001, ~2 = 0.003, ~3 = -0.003, ~4 = 0
211 = -0.004, 212 = -0.002, 213 = 0.001, 214 = 0.001
If Young's modulus E = 200000N/mm2 and Poisson's ratio v = 0.3, calculate the
stresses at the centre of the element.

From the first of Eqs (12.96)
u1 =a1 - 2a2 - a3 + 2a4 = 0.001
u2 = a1 + 2a2 - a3 - 2a4 = 0.003
u3 = a1 + 2a2 + a3 + 2a4 = -0.003
U4 =a1 -2a2 +a3 -2a4 = 0

Subtracting Eq. (ii) from Eq. (i)
a2 - a4 = 0.0005
Now subtracting Eq. (iv) from Eq. (iii)
CY^ + a4 = -0.00075
Then subtracting Eq. (vi) from Eq. (v)
a4 = -0.000625 (vii)