Civil Engineering Reference
In-Depth Information
18.3
LINEAR INCLUSIONS
18.3.1
Theory
The treatment of linear inclusions follows a similar approach as with heterogeneities
1
.
Figure 18.5 shows an example of a tunnel with a rock bolt which will be treated as an
inclusion with different material properties. The rock bolt is assumed to be fully grouted
i.e. in contact to the rock mass along it's length.
Figure 18.5
Example of a tunnel with a rock bolt
The cross-section of the rock bolt is assumed to be small compared to its length and
therefore the variation of the stress across the section can be assumed to be constant. The
approach is very similar to the previous one, i.e. first an analysis is carried out without
the rock bolt and then a correction made due to the presence of the rock bolt. It is
explained on a plane problem but the extension to 3-D is straightforward.
The first system of equations to be solved is for the
Neumann
example in Fig. 18.5
> @
^` ^`
7
uF
(18.9)
After the first analysis the strain in the direction of the rock bolt is computed. Because of
the difference in moduli between the rock and the bolt the stress is different at a point
depending if the point lies in the bolt or in the rock (Fig. 18.6). The difference in the
stress (in the direction of the bolt
x
) is computed by:
VV V
(
E
E
)
H
(18.10)
x
xRock
xBolt
Rock
Bolt
x
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