The Finite Difference Method - Mechanics of Structures: Variational and Computational Methods - page 529

The principle of stationary potential energy leads to the same result with a segment
matrix

2 − 42 w k − 1

w k
w k + 1

0
2 p 0 h
0

=

h 3

( EI / ) − 48 − 4

2 − 42

In this case, the force boundary conditions are satisfied exactly by the variational
approach.

8.13 Suppose a thick elastic spherical shell contains a gas exerting a pressure p on the inner
wall. The radial ( r ) displacement u is governed by the differential equation

du
dr +

d
dr

u
r

= 0

/ =− = / = = .

subject to the boundary conditions du dr p at r r inner ,du dr 0at r r outer

Discretize this problem and set up a finite difference solution.
8.14 Use the principle of virtual work to implement a finite difference for the beam of
Fig. P8.14. Employ a simple central difference expression on the grid shown.

FIGURE P8.14

w 0 =

Answer: After application of the displacement boundary conditions, w = 0 ,

0

(w − w − 1 )/ h = 0 ,

1

7 − 4 1 w 1

w 2
w 3
w 4
w

0
0
0
P
0

− 46 − 4 1

L 4

( 64 EI / ) 1 − 45 . 5 − 30 . 5 =

1 − 3 3 1

0 . 5 − 10 . 5

5

This can be solved for w

1 ,

w

2 ,

w

3 , and

w

4 .