This post has the derivation of a function that I have derived in the last couple of weeks. if you have an existing segmental lining and then fill the lining with a liquid under pressure the simple design approach is to reduce the hoop force in the lining the internal pressure of the fluid multiple by the tunnel radius. This is a simple and cautious approach to addressing the problem but at times it may not be sufficient. If the tunnel is well grouted into a stiff ground the ground reaction could result in a significant reduction in the axial force that need to be applied due to the internal pressure. This derivation considers a relatively simple model for this lining where a bolted segmental lining is considered in elastic ground.

Considering the effective pressures on the lining at the point of balance the internal pressure is equal to the external ground reaction plus the tensile force in the lining converted to an equivalent pressure:

where:

* p _{i}* = Internal pressure in the tunnel

*p*= Equivalent pressure due to generated axial force in the lining

_{l}*p*= Ground reaction pressure

_{g}Initially considering the ground reaction, this can be defined using the sub-grade reaction approach. Muir-Wood ^{1} gave a spring stiffness for tunnel lining design of:

where:

* ν * = Poisson's ratio of the ground* E * = Elastic modulus of the ground* r _{o} * = Radius of the excavation

Given a radial deformation of the tunnel lining and the ground of δ_{radial} the reaction pressure of the ground is given by:

Considering the flexibility of the tunnel lining both the axial stiffness of the tunnel lining and the stiffness of the bolted joints will need to be considered. The hoop force in the lining is given by:

Given the hoop force in the lining this can be converted into axial extension of the lining due to the hoop force. Initially consider the tunnel lining along. The circumfrential increase in the length of the lining by treating it simple as a column:

where:* E _{l} * = Elastic modulus of the tunnel lining.

*A*= Cross section area fo the tunnel lining.

_{l}A similar function can be derived for the circumfrential expansion of the lining due to the expansion of the bolted joints:

where:* n * = the number of segmental lining joints in a ring* l _{bolt} * = the effective length of the bolts at the segmental lining joints

*E*= the elastic modulus of the bolts at the segmental lining joints

_{bolt}*A*= the cross sectional area of the bolts at the segmental lining joints

_{bolt}The circumfrential displacement can be converted to a radial displacement by dividing by 2π. The total radial displacement is then given by:

Substitute ③ and ⑦ into ①:

⑧ can be rearranged to get a value for δ _{radial} _ from known values:

Having derived a value for δ_{radial} the equivalent ground reaction can be determinned from Equation ③. Equation ① can then be used to determine the hoop force in the lining. The expansion of the lining and the opening of the bolts can then be determinned from Equations ⑤ and ⑥.

^{1. A. M. Wood, “Tunnelling: Management by Design,” pp. 1–320, Nov. 2002.↩}