Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 702

FIGURE 11.36
Longitudinal forces N , with second order effects after the 2nd iteration.

For the evaluation of the system matrix, the approach demonstrated in Example 11.5 can
be employed. For the fundamental state, we use either the results of the linear analysis or the
results of the second order analysis. Figure 11.36 contains the distribution of longitudinal
forces with second order effects as given in the last column in Table 11.7.
The element matrices are formed in the same manner as in Example 11.5: the linear part
of the stiffness is the same, but the geometric matrices are slightly different due to the choice
of the fundamental state. In this case, the second order analysis solution after the second
iteration was chosen.
To form the system stiffnesses, the element matrices have to be assembled separately for
the linear and the second order parts. As noted in Example 11.5, Eq. (9), the first row in the
assembled matrices gives the equilibrium of the forces in the horizontal direction at node
2, while the second describes the equilibrium of the moments at the same node.
Assembly of the elastic stiffness matrix:

U 2 2

. .

2109 38 8437 50 Element 1

Element 2
Element 3
Element 1
Element 2
Element 3

F X 2

(1)

8437 . 50 45 000 . 00

M Y 2 52 200 . 00

2109

. 38 8437 . 50

K lin =

8437 . 50 97 200 . 00

Assembly of the geometric stiffness matrix:

U 2

2

− 172 . 58 − 115 . 05

Element 1
Element 2
Element 3
Element 1
Element 2
Element 3

F X 2

− 514 . 87

(2)

− 115 . 05 − 1227 . 24

M Y 2 − 158 . 79

−

687 . 45 − 115 . 05

K geo =

− 115 . 05 − 1386 . 03