The paper deals with mathematical modelling of interaction of hydraulic tunnel linings with the surrounding rock mass with different kinds of static loads and earthquake seismic effects, as well as design methods for linings regarded as elements of a single deformable "liningrock" unit subject to loads and other effects.


The principle of interaction treating the rock mass and the lining as elements of a single deformable "lining- rock" unit sensitive to all external loads and actions is assumed as a basis for mathematical models and design procedures for linings of underground structures.

According to the above principle the design scheme (physical problem statement) for an underground structure (hydraulic tunnel) is a pattern of contact lining-rock interaction. The scheme can serve as a basis for a mathematical model of an underground structure (analytical solution) or for a numerical experiment (using the finite element method). The scheme of lining-rock interaction can be also realized in the form of a physical model of an 'underground structure of optically active (photoelastic method) or equivalent materials. The most important feature of the contact interaction schemes is that stresses at the lining-rock contact (loads, pressure acting on the lining) are not given a priori, but are determined in the course of analysis (modelling).

Schemes of contact interaction of linings (supports) of underground structures with rock masses are studied within the scope of underground structure mechanics - a scientific discipline developed in the Soviet Union (Bulychev 1982).

The paper discusses mathematical models simulating the rock mass by a linearly deformable medium including viscoelastic one. So each mathematical model is the result of analytical solution of the corresponding plane problem of the elasticity theory. The models and design procedures for linings are realized as computer programs enabling one to perform multivariant designs of underground structures.

Reliability of the mathematical models and design procedures for linings is ensured both by qualitative and quantitative correspondence of the design values to the results of model experiments and in situ measurements. In addition, these mathematical models enable one to solve inverse problems using in situ measurements of stresses or displacements in linings.


The design scheme for tunnel linings (Fig. 1) is one of the contact interaction of an arbitrarily shaped elastic ring, with one axis of symmetry, having the characteristics EI (modulus of deformation), VJ (Polsson's ratio), simulating the Iining and supporting a hole in a linearly deformable homogeneous medium having the characteristics E, V, and simulating the rock mass The mathematical model considers rock rheologic characteristics within the scope of 269 H Figure 1.

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