Amongst the different methods for designing the joints for segmental linings there are a few different approaches based on the assumption of a triangular stress distribution at the joint. In these models the rotational stiffness of the joint is derived by assuming that the concrete to concrete contact can be approximate to linear elastic springs. The stiffness of the springs is defined by the stiffness of the concrete and by a depth of concrete beneath the joint over which the strains in the concrete distribute.
It should be fairly obvious that these triangular stress distributions are not strictly correct because of the assumption of linear elasticity, however it is a model used by many designers and appears to be a reasonable approach for the design of joints. The application of the method is generally relatively simple, however there is one unknown, the depth of concrete to use when defining the stiffness of the springs. I have seen a number of different values used for this length from the thickness of the segmental lining through to half the circumferential length of the segment. All of these approaches have some justification and are probably more or less appropriate in different circumstances. In this post I am not going to go into why these different lengths are used, but what I would like to consider instead is how these different approaches perform when applied to the design as a whole. In particular I’m going to consider a few different design cases and look at how the bearing and bursting stress vary at the joints when different numbers of segments are adopted for each approach.
In this post I’m going to consider three basic development lengths
- half the circumferential length of a segment
- the thickness of the segment
- twice the thickness of the segment capped at half the circumferential length of a segment
I have seen all three of these values applied to reasonable designs for segmental linings before and so I would like to see how the results of the different approaches compare.
To do the comparison I’ve run a typical design for an 11.65m ID tunnel in soft ground. This is indicative of a typical twin lane road tunnel crossing a river. All the other parameters used for the assessment are at the end of this post1.
The following chart illustrates how the mobilised bearing capacity at the joints varies as the number of segments increases. There are a few few obvious points in this chart. The use of half the circumferential length of a segment is clearly the least cautious approach. This is actually a very consistent result with most design checks exhibiting similar behaviour which often leads to it being a preferred approach where the design loads on a lining are significant. The problem with this approach is that the curve of mobilised capacity with the number of joints in the ring shows a slight upward trend i.e. as the number of joints increases in a ring, the bearing stresses at the joint also increase. This is not consistent with observed behaviour of tunnels and essentially invalidates a large amount of the benefit of having joints in a segmental lining.
Considering the other two models there is a clear reduction in the total stress at the joints as the number of joints increases which is the behaviour that would be expected. The use of a longer development length of twice the segment thickness rather than the segment thickness gives a clear reduction in the bearing stress at the joint.
A smilar chart can be produced for bursting stress rather than bearining stress but which gives a similar behaviour to the bearing stress calculation.
At this point it is clear that the case for using the development length based on a proportion of the thickness of the segment is clear to me. The exact dimension remains to be addressed, however the behaviour of the model using the circumferential length is certainly flawed in its behaviour and appears to be far from ideal.
In a future post I will consider the development length in more detail to try and understand what proportion of the segment thickness may be correct.