Rock masses

In this chapter, we will concentrate on extending the ideas discussed in the previous chapter on discontinuities to provide a predictive model for the deformability and strength of rock masses. In Chapter 12, we will be discussing rock mass classification (which is a method of combining selected geometrical and mechanical parameters) to semi-quantitatively provide an overall characterization, mainly for assessing excavation support requirements.

DeformabilityConsider first, as an initial step in the overall development of a deformability model, the deformation of a set of parallel discontinuities under the action of a normal stress, assuming linear elastic discontinuity stiffnesses. This circumstance is illustrated in Fig. 8.1. To calculate the overall modulus of deformation, the applied stress is divided by the total deformation. We will assume that the thickness of the discontinuities is negligible in comparison to the overall length under consideration, L. Additionally, we will assume that the deformation is made up of two components: one due to deformation of the intact rock; the other due to the deformability of the discontinuities.The contribution made by the intact rock to the deformation, 61, is oL/E (i.e. strain multiplied by length). The contribution made by a single discontinuity to the deformation, &, is o/ED (remembering that ED relates stress to displacement directly). Assuming a discontinuity frequency of A, there will be AL discontinuities in the rock mass and the total contribution made by these to the deformation will be L& which is equal to oAL/ED. Hence, the total displacement, &, is()with the overall strain being gven by(€)Finally, the overall modulus, EMASS, is given by()

A suite of curves illustrating this relation is given in Fig. 8.2 for varying discontinuity frequencies and stiffnesses. It is simple to extend this formula for multiple intact rock strata with differing properties, discontinuity frequencies and discontinuity stiffnesses, and hence model stratified rock with discontinuities parallel to the bedding planes.

The case illustrated via the mathematics above and shown in Fig. 8.2 only involves loading parallel to the discontinuity normals. Clearly, even in these idealized circumstances, we need to extend the ideas to loading at any angle and the possibility of any number of non-parallel sets. An argument similar to that given above can be invoked in the derivation of shear loading parallel to the discontinuities, as succinctly described by Goodman (1989), to give()The mathematics associated with further extensions to account for discontinuity geometry rapidly becomes complex. A complete solution has been provided by Wei (1988), which can incorporate the four stiffnesses of a discontinuity (normal, shear and the two cross terms), any number of sets and can approximate the effect of impersistent discontinuities.In the stress transformations presented in Chapter 3, the resolution of the stress components involves only powers of two in the trigonometrical terms, because theforce is being resolved and the urea is also being resolved. However, for the calculation of the deformability modulus, powers of four are necessary because of the additional resolution of the discontinuity frequency (explained in Chapter 7) and the displacements. An example equation from Wei's theory, the roots of which provide the directions of the extreme values of the modulus, is()where A, B, C, D, E and F are constants formed by various combinations of the discontinuity stiffnesses and a is the angle between the applied stress and one of the global Cartesian axes. The reader is referred to Wei's work for a complete explanation.The utility of this type of analysis is illustrated by the polar diagrams in Fig. 8.3 representing the moduli variations for two discontinuity sets in two dimensions. (It is emphasized that this figure is one example of a general theory.) When k is high, as in the left-hand diagram, the lowest moduli are in a direction at 45" to the discontinuity sets, and the highest moduli are perpendicular to the sets. Conversely, when k is low, as in the right-hand diagram, the minimum moduli are in a direction perpendicular to the sets, and the maximum moduli are at a direction of 45" to the sets. Like the discontinuity frequency, the directions of maximum and minimum moduli are not perpendicular.A most interesting case occurs when k = 1, i.e. the normal and shear stiffnesses are equal, and the modulus is isotropic. The significance of

even this very simple case of rock mass deformability for in situ testing and numerical modelling is apparent.