Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 714

The governing differential equation for a beam with axial force and with p z = 0is

[Eq. (11.31)]

i v 2

w =

2

w + α 0 , α = P / EI (1)

i v

w

From Chapter 8, Table 8.1, the central difference expressions for w and are

1

h 2

w k = (w k − 1 − 2 w k + w k + 1 ) (2a)

1

h 4

i k

w = (w − 4 w + 6 w − 4 w + w ) (2b)

k − 2 k − 1 k k + 1 k + 2

where h is length of interval.
The boundary conditions of w 0 = w x = L =

0 and 0 applied to the central difference

1
2 h

w w [ w 0 = (w − w − 1 ) w = w − 1 and w =− w

expressions for and and (2a)] give

1 , 1 1 k + 1 k −

respectively (see Fig. 11.43).

Two Segments ( M = 2, Fig. 11.44)

In this case, h = length / M = L / 2, and at node 1, the substitution of (2) into (1) gives

2
h 2 (w

1
h 4 (w − 1

) + α

− 4 w + 6 w − 4 w + w − 2 w + w ) = 0 (3)

0 1 2 3 0 1 2

With the relations w − 1 = w

1 ,

w =− w

1 , and the boundary conditions

w = w = 0

3 0 2

0or 3

2

2
h 2 2

1
h 4 6

− α w w =

h 2 − α

w = 0 . (4)

1 1 1

2

h 2

This gives α = 3 / = P / EI . Finally, we conclude that

3 EI
h 2

12 EI
L 2

= =

P cr (5)

FIGURE 11.43
Finite difference boundary conditions for the beam of Fig. 11.28.

FIGURE 11.44
Central differences with two segments.