Test Examples - The Boundary Element Method with Programming - page 266

Required are the displacements and the changed stress distribution due to excavation. To
obtain this solution we actually have to solve two problems (Figure 10.6): One (trivial)
one where no excavation exists and one where the supporting tractions t 0 computed in
the first step are released, i.e., applied in the opposite direction.
We can use equation (4.28) to solve the problem (a) i.e. to compute the tractions t 0 as

0 . 0

- ½

® ¾

(10.1)

t

0

n V

¯ ¿

y y 0

10.3.2 Boundary element discretisation and input

To solve problem (b) we use the BEM with two planes of symmetry. For the first mesh a
single parabolic element (Figure 10.7) is used. Subsequently two (Mesh 2) and four
elements (Mesh 3) are used for a quarter of the boundary. The mesh is subjected to
Neuman boundary conditions with values of t 0 computed using (10.1) and applied as
shown in Figure 10.7.

1

2 . 986

3

2 . 121

Axes of symmetry

0 . 281 2

Figure 10.7

Boundary element mesh with Neuman boundary conditions

If the element would be able to describe an exact circle then the values of traction t y0
should be exactly -3.0 at node 1 and 0.0 at node 2. However, since the element can only
describe a parabola, the y-component of the normal vector will not be exactly -1.0 at
node 1 and not exactly 0.0 at node 2. Therefore, a small geometrical error occurs due to
the coarse discretisation. Alternatively we could specify the values of traction that
correspond to an exact circle (-3.0,-2.12, 0.0) The input file for program 7.1 for this
problem is

Circular hole
2 ! 2-D
2 ! Elasticity problem
1 ! Plane strain
2 ! Finite Region