A block-spring model for jointed rocks.

Abstract

A numerical method called the Block-Spring Model for analyzing heavily jointed rocks has been developed from first principles. The model simulates the jointed rock mass by an assemblage of rigid blocks interacted through contacts. The contact forces between blocks are determined in terms of the relative displacements between the blocks. By introducing the contact forces and the boundary conditions into the equilibrium equations of the blocks, a wet of stiffness equations are obtained in which the unknown variables are the displacements of the blocks. The displacements of the blocks can therefore be determined by solving the stiffness equations. The contact forces between blocks are calculated after the displacements are obtained. The discontinuities of the jointed rock mass are automatically modelled as the interfaces between the blocks. The deformation behaviour of the joints are governed by a non-tension rule in normal direction. Patton's bi-linear criterion is used to describe the shear failure behaviour of the rock joints. An interaction procedure is applied to describe the progressive failure along the joints. The positions of the blocks are continuously updated based on the displacements obtained after each step of iteration. Therefore, the large scale displacements of the blocks are modelled incrementally with the iteration procedure. During evolution of the block movement, the blocks are allowed to rearrange. The block system may become statically stable or unstable after undergoing large scale displacements. The proposed model therefore identifies unstable blocks by considering the rearrangement of the blocks. A numerical procedure for simulating the rock bolts has been proposed. The rockbolts are modelled by a series of one dimensional elements interacted with the rock blocks through nodal points. The effect of the rockbolts on the rock masses is evaluated by relating the bolt forces to the displacements of the blocks and by introducing the bolt forces into the equilibrium equations of the blocks. The model can be used to analyze both the end-anchored rock bolts and the fully grouted rock bolts either pre-tensioned or untensioned. A procedure for modelling the groundwater pressure has also been developed. It has been considered that groundwater may affect the stability of the jointed rock masses by reducing the effective stresses between the rock blocks. In simulating the groundwater effect on the rock masses, the water pressures imposed on the surfaces of the rock blocks are calculated. These water pressures are introduced into the equilibrium equations of the blocks. The contact forces between blocks determined after solving the equations are the effective forces across the rock joints. A FORTRAN program, BLOSMER, has been entirely written by the Author based on the procedures of the Block-Spring Model developed during this study. The program can be used to analyze the stability of the surrounding rock masses of either open pits or underground excavations at the shallow depth. It has been applied to several examples to demonstrate the capability of the proposed model. Two case histories have been analyzed in detail with the Block-Spring Model. The numerical results have been compared with the field instrumentation data. Good agreement has been observed between the results from the back analyses with the proposed model and the field measurements in both cases. The proposed model can be applied for analysis of rock excavations by practicing civil and mining engineers.