I have previously looked at the equation that relates the internal pressure in a segmental lining to the forces in the linng and the opening at the joints. In this post I want to look at the implications of this equation in terms of the ground stiffness. Is there a typical minimum ground stiffness that is required for a benefical effect in terms of the internal pressure?
To asses this I've run two sumple check cases through the equation for variations in ground stiffness. The equation is a strong function of the tunnel radius and so I'vve looked at two concrete linings, one 9m in diameter and one 3m in diameter. I've selected typical lining properties and joint configuratins for each case and had a look at how the ground stiffness affects the results.
The followng two plots show the varaition in axial forces for the two lining types:
The results of the two cases are very similar when considered as normalised results. This indicates that a strong rule of thumb could be applied to this problem for tpical segmental linings.
Up to around 100MPa ground sitffness (≈stiff clay ) there is very little beneficial effect in considering the ground reaction to an internal pressure. Above 100MPa there is a gradual beneficial effect, increasing with increasing ground stiffness. At around 1000MPa (≈chalk) the load is shared approximately equally. At around 10,000MPa stiffness the vast majority of the load is taken by the ground reaction.
A similar chart can be produced for hte raidal displacement and jint opening of the lining under the action of internal pressure:
The displacement charts give very similar results to the foop force results in terms of the impact of the ground stiffness. A crude rule of thumb therefore appears to be appropriate for the design of segmental linings under the action of internal pressures.
Under 100MPa stiffness there is no signficiant benefit from ground support.
Above 100MPa there is some benefit increasing to about 50% of the unsupported condition at 1000MPa.
At around 10,000MPa ground support takes the majority of the internal pressure.