Test Examples - The Boundary Element Method with Programming - page 283

10.5.3 Results

According to the highlighted value in the output from 7.1, the maximum displacement
for mesh 1at the top is 0.72 mm compared to a theoretical value of 0.9. To obtain the
correct stress results of the excavation problem we must add the virgin stress to the
values highlighted in the output from 9.1. According to the highlighted value we obtain a
maximum compressive stress in the z-direction at the boundary of -1.47 MPa, compared
to a theoretical value of -2.0 MPa.

2,0

1,5

linear elements
parabo lic elements
exact

1,0

0,5

0,0

1,0 1,5 2,0 2,5 3,0

R/a

Figure 10.18 Distribution of vertical stress

The results are obviously improved for Mesh 2. For the mesh with parabolic boundary
elements the maximum displacement at the crown is computed as 0.88 mm. The
maximum value of stress at the meridian of the sphere is computed as 2.02 MPa
compared with the theoretical solution of 2.0. The results for the internal stresses are
summarized in Figure 10.18 where the vertical stress is plotted along a horizontal line
originating from the meridian. It can be seen that even a relatively coarse mesh with 3
quadratic elements gives a reasonable accuracy.

10.5.4 Comparison with FEM

To be able to model this problem with the FEM we have to truncate the mesh, as with
the 2-D example. A fairly coarse mesh is shown in Figure 10.19. The mesh has 135
degrees of freedom, as compared with 16 degrees of freedom for the boundary element
mesh 2. The maximum displacement obtained from this analysis is 0.084 which
represents a poor agreement with the theory. A much finer mesh would be required.