Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 690

TABLE 11.7
Comparison of the Results
First Order 1. Iteration 2. Iteration

(Linear) Analytical Approximate Analytical Approximate

− . − . − . − . − .

U 2 0 07831 0 15477 0 15482 0 15417 0 15423

2 . .

0.00553 0.01213 0.01214 0 01210 0 01211

M 1

584.2 1060.4 1060.2 1055.5 1055.3

M 2 − 363.8 − 708.4 − 709.0 − 704 . 8 − 705 . 4

V 1 − 154 . 5 − 235 . 5 − 235 . 5 − 233 . 9 − 233 . 9

V 2 − 82 . 5 − 163 . 5 − 163 . 5 − 161 . 9 − 161 . 9

V 2 66.4 100.8 100.9 100.5 100.5

V 3 6.4 40.8 40.9 40.5 40.5

V 3 0.0 81.0 81.0 79.4 79.4

N 1 − 1116 . 4 1150.8 − 1150 . 9 − 1150 . 5 − 1150 . 5

N 2

− . − . − . − .

0.0 81 0 81 0 79 4 79 4

N 3 − 2093 . 6 − 2059 . 2 − 2059 . 1 − 2059 . 5 − 2059 . 5

in a parameter ε = 0 . 90466 . This compares with the new result N =− 1150 . 9 kN and

ε = The differences appear to be small, depending on the desired accuracy and

the stiffness of the system. For small differences, a second iteration is not necessary. This
may not be the case for systems in which larger deflections are caused by low stiffness, for
example, for a guyed mast. Table 11.7 also shows the approximate solution resulting from a
second iteration with updated normal forces. These results confirm that, for this example,
only small differences are obtained.
In design, usually the moments are of primary importance, and it can be seen from the
results that for M , second order effects may have a rather large influence. It follows that,
for systems with highly compressed members, a stability analysis is essential from the
standpoint of safety.
In Table 11.7, the columns labeled “Approximate” correspond to the use of geometric
matrices as explained in this example. The solutions based on the exact matrices, such as
those of Table 4.4, are provided in the columns labeled “Analytical.” The differences are
quite small, which is typical for the case when the parameter

0 . 90466 .

ε

is not larger than 1. This
criterion can always be satisfied by reducing the length of the elements in the sense of the
finite element method.

11.5 Bifurcation of Equilibrium

11.5.1 Conditions for Bifurcation
The critical loads associated with a bifurcation of the equilibrium can be obtained from a
special case of the general nonlinear approach. The homogeneous part of the corresponding
system of algebraic equations KV

0 representing equilibrium of the system can be solved
only for specific solutions, the eigenvalues of the system. The eigenvalues can be determined
from the roots of

=

=

det K 0 (11.66)