Page 539 - Aircraft Stuctures for Engineering Student

520 Matrix methods of structural analysis

The element stiffness matrix is finally obtained in step six in which we replace the
internal 'stresses' {a} by a statically equivalent nodal load system {Fe}, thereby
relating nodal loads to nodal displacements (from Eq. (12.69)) and defining the
element stiffness matrix [IC]. This is achieved by employing the principle of the
stationary value of the total potential energy of the beam (see Section 4.4) which com-
prises the internal strain energy U and the potential energy V of the nodal loads. Thus
1
U + V = - {&}T{a}d(Vol) - {6"}T{F'} (12.70)
2 1.1
Substituting in Eq. (12.70) for {E} from Eq. (12.66) and {a} from Eq. (12.69) we have
1
U + I/ = - {6"}T[A-']T[~T[D][~[A-1]{~}d(~~l) {a"}T{Fe} (12.71)
-
2 Iv0l
The total potential energy of the beam has a stationary value with respect to the nodal
displacements {6"}T; hence, from Eq. (12.71)
a(u+ V)
[A-']T[CIT[D][q[A-1]{6e}d(~~1) {Fe} = 0 (12.72)
-
whence

(12.73)

or writing [C][A-'] as [B] we obtain

(12.74)

from which the element stiffness matrix is clearly

(12.75)
[K"I = [ J vol ~~lT[ol~~l~~~~~)]
From Eqs (12.62) and (12.64) we have
1 0 0
0 1 0
[B] = [q[A-'] = [O 0 2 6x1
-3/L2 -2/L 3/L2 -1/L
21~3 ip.2 -21~3 i/L2

or
L2 +z
- 6 12x
-_
4 6x
--+_
L L2
[BIT = (12.76)
6 12x
L2 L3