Deformation of stratified rock masses: A laminated model

In view of an urgent need for a simple means of simulating the behaviour of stratified rock masses, a long-dormant linear model is revived, generalized, and discussed in some depth. The model is piecewise homogeneous, consisting of a pile of homogeneous isotropic laminae in which the interfaces between beds are parallel and free of shear stresses and cohesion. The continuity of stresses and normal displacements is assured across the bedding planes and, to maintain simplicity, the theory of thin plates is used as the kernel of the model. The paper covers the case where the rock mass consists of discrete beds in which the layers are arranged either heterogeneously or systematically. The stratification is regarded as uniform when all the layers are of the same thickness and their Young's modulus changes exponentially with depth; and as homogeneous when all the laminae have identical thickness and properties. The case of uniform stratification is also treated on the basis of an equivalent quasi-continuum derived from the basic model. Attention is focused on the derivation of influence functions. It is demonstrated that the quasi-continuous models, when they are available, can replace their distinct-layer counterparts. The homogeneous quasi-continuum, which permits the derivation of its influence functions in closed form, facilitates the determination of the roof and floor convergence. In the case of a significant class of problems, the convergence distribution is defined by a second-order differential equation in the xy plane, which simplifies to a Poisson's equation if the mining excavations are unsupported. With this result, the paper prepares for the solution of a host of practical problems. These solutions are presented in a series of papers that are in the course of publication.