The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 572

FIGURE 9.8
A hollow cylinder with internal pressure.

kept below prescribed values. These conditions can then be added to the constraint condi-
tions of the minimax formulation.

EXAMPLE 9.5 Boundary Element Solution for a Thick Cylinder under Internal Pressure
C ompute the displacements of the thick cylinder of Fig. 9.8 under an internal pressure of
p 1 GN/m 2 200 GN/m 2 , and Poisson's = 0 . . The modulus of elasticity of the material is E =

ratio is 0.25.
This problem can be treated as a plane strain problem (Section 1.3.1). The displacements
on a point

ξ inside the body or on the boundary are expressed as [Eq. (9.88)]

u j p ij dS p j u ij dS p Vj u ij dA

c i u i (ξ ) + = + (1)

S S A

or

p ∗ u dS u ∗ p dS u ∗ p V dA

cu (ξ ) + = + (2)

S S A

The terms u ij and
p ij are given in Eq. (9.74) and (9.78). The displacement form of the governing differential
equation is [Eq. (9.53)]

where A is the area of the cross-section of the cylinder, and i, j = 1 , 2 .

1 p Vi

G

2 u i +

∇ u k,ki + = 0 i, k = 1 , 2 (3)

1 − 2 ν

Express the displacement in terms of the Galerkin vector [Eq. (9.58)]

1

u i = g i,kk − g k,ik i, k = 1 , 2 (4)

2 ( 1 − ν)

where g i is the component of the Galerkin vector. Substitute (4) into (3) to obtain

p Vi
G

4 g i +

∇ = 0 (5)

Let

p Vi = δ(ξ ,x ) a i (6)