On the effect of segment curvature on joint rotations in segmental linings
Previously I wrote about a few approaches to modelling segment joints for the design of segmental tunnel linings. The model that I discussed is typically applied in a particular way where it is assumed that all of the deformation of the ring occurs due to rotation at the segment joints. This model is relatively simple to derive and importantly the outputs can be clearly understood. The simplicity does however hide the complexity of the problem that it is trying to address.
The approaches I previously discussed for calculating the bearing stress at segment joints typically result in eccentric loads being applied at the joints. If we consider a simple case of a single segment with large joint rotations at each end, if the rotations are in the same direction and of the same magnitude then there will be constant bending moment along the length of the segment due to the eccentricity of the hoop force in the segment. This constant bending moment will result in a constant curvature along the entire length of the segment in the same direction as the rotation of the joints. Under the right conditions bending in the segments caused by the eccentricity of the force at the joints can result in a reduced joint rotation… all the deformation of the ring is no longer assumed to be due to joint rotation but can include curvature of the segment as well.
Obviously having a high rotation and hence high bending moment across the entire length of a segment is unlikely in most linings where there is a change in radial deformation and and bending moment around the tunnel profile. The simple case considered above with a uniform bending moment is however instructive in considering this problem and how a typical assessment of joint rotation may overestimate the rotation and hence bearing and bursting stresses.
So how much of an impact might the segment rotation have on the results? Unfortunately building an analytical model of the problem is not trivial because of the interaction between the joint rotations and segment rotations. There are also a lot of feasible combinations of results so you need to define additional criterion to be used when finding the solution you are looking for.
Because of this I’ve just looked at a relatively simple model to estimate the impact of including segment curvature in the rotation calculation. I’ve assumed that the curvature in the segment is defined by a simple bending moment diagram with full eccentricity at one end and no eccentricity at the other which gives half the rotation of the constant curvature case considered above. The following results are taken from the lining I detailed in the last post about joint stress models at joints.
From these results you can clearly see that considering curvature of the segment does result in a reduced bearing and bursting stress at the joints, the reduction being very significant where relatively few joints are considered in a large diameter tunnel. The behaviour of the jointed lining is still consistent with the expected result that increasing the number of joints decreases the stresses at the joint but the effect is much less pronounced. The results, whilst still worse than the case when you consider half the segment length, are now more consistent with this model. Note that I have not applied the segment curvature model to the half a segment length model as I believe that this would be double counting the curvature of the segment.
So where does this leave us? Over the years I have encountered many situations where some adjustments to the joint bursting calculation are applied because it is felt that the design is too conservative such as doing the check at SLS rather than ULS. Joint bursting of course is typically the critical design case and therefore whilst these modifications may be beneficial at times they could be considered ill advised. If the model considered above is considered reasonable then there is a clear source of a potential reduction in the stresses at the joint that that should remain code compliment and not require any adjustments.
Of course this is not the only source of complexity that you might want to consider. In the coming weeks I’ll consider the design of a bolted joint and how this can impact the design of a ring.