Stability of large excavations in laminated hard rock masses: the voussoir analogue revisited

Abstract

The voussoir beam analogue has provided a useful stability assessment tool for more than 55 years and has seen numerous improvements and revisions over the years. In this paper, a simplified and robust iterative algorithm is presented for this model. This approach includes an improved assumption for internal compression arch geometry, simplified displacement determination, support pressure and surcharge analysis and a corrected stabilizing moment in the two dimensional case. A discrete element simulation is used to verify these enhancements and to confirm traditional assumptions inherent in the model. In the case of beam snap-through failure, dominant in hard rock excavations of moderately large span, design criteria are traditionally based on a stability limit which represents an upper bound for stable span estimates. A deflection threshold has been identified and verified through field evidence, which corresponds to the onset of non-linear deformation behaviour and therefore, of initial instability. This threshold is proposed as a more reasonable stability limit for this failure mode in rockmasses and particularly for data limited cases. Design charts, based on this linearity limit for unsupported stability of jointed rock beams, are presented here summarizing critical span–thickness–modulus relationships.

Introduction

Rockmass behaviour dominated by parallel laminations is often encountered in underground excavations in numerous geological environments. These laminations can be the result of sedimentary layering, extensile jointing, fabric created through metamorphic or igneous flow processes or through excavation-parallel stress fracturing of massive ground. This structure can be the dominant factor controlling the stability of roofs in large civil excavations, in coal or other horizontal mining stopes1, 2and can also dominate the stability of inclined open stope hangingwalls such as those encountered in hard rock mining in Canada3, 4, in Australia5, 6and elsewhere.

In rare circumstances, the lamination partings represent the only joint set present in the rockmass. Roof stability and deflection in this instance can be assessed using conventional elastic beam deflection and lateral stress calculations7, 8. It is more common, however, to encounter other joint sets cutting through the laminations. These joints reduce and, in the extreme, eliminate the ability of the rockmass to sustain boundary-parallel tensile stresses such as those assumed in conventional beam analysis. However, where these joints cut across the laminations at steep angles or where reinforcement has been installed, it is possible to assume that a compression arch can be generated within the beam which will transmit the beam loads to the abutments (Fig. 1).

It was noted by Fayol[9]that underground strata seemed to separate upon deflection such that each laminated beam transferred its own weight to the abutments rather than loading the beam beneath. Stability of an excavation in this situation, it was concluded, could be determined by analyzing the stability of a single beam deflecting under its own weight. Conventional beam analysis, however, significantly underestimated the inherent stability of such beams. Even intact laminations would crack at midspan as predicted, but would, after additional deformation, become stable again. The notion of the voussoir, while traceable back to the architecture of ancient Rome[10], was first proposed by Evans[11]specifically to explain the stability of a jointed or cracked beam. After generating a great deal of controversy when first published, the voussoir beam analogue has been generally accepted and has since been reworked and presented as a simplified tool for stability analysis of excavations in civil construction and in mining10, 12, 13, 14. Fig. 2 illustrates two example cases where the voussoir beam analogue can be invoked to explain the inherent stability of a laminated hangingwall (Fig. 2a) and cross jointed back (Fig. 2b) in hard rock environments.

The primary modes of failure assumed in the model and verified in laboratory tests by Sterling[15]are buckling or snap-through failure, lateral compressive failure (crushing) at the midspan and abutments, abutment slip and diagonal fracturing (Fig. 3). Shear failure (Fig. 3c) is observed at low span-to-thickness ratios (thick beams), while crushing (Fig. 3b) and snap-through failure (Fig. 3a) is observed at higher span-to-thickness ratios (thin beams). An examination of the model data presented by Ran et al.[16]shows that if the angle between the plane of the cross-cutting joints and the normal to the lamination plane (and the normal to the excavation surface) is less than one third to one half of the effective friction angle of these joints, then it is valid to apply the voussoir beam solution. For shallower cross-jointing, slip along these joints and premature shear failure of the beam is likely[16].

Stimpson and Ahmed[17]also showed by physical modelling that for thick beams, external loading (surcharge) can produce diagonal tensile ruptures (Fig. 3d) extending from the upper midspan to the lower abutments (parallel to the compression arch). While this failure mode is partially the result of the loading configuration, it may also be an important mechanism where weak or broken material exists above the beam or where internal rockmass damage and bulking due to stress overloads the surface beam. This paper deals with thin laminations (span-to-thickness ratio greater than 10) under the influence of self-weight or moderate surcharge loading. Therefore, only crushing and snap-through failure are hereafter considered. Sliding stability is included in the analyses but does not control limiting dimensions for thin beams.

The following is a summary of the voussoir analysis procedure based on the iterative scheme proposed by Brady and Brown[13], including a number of improvements and corrections by the authors. Most importantly, a more realistic yield threshold is introduced for snap-through failure, to replace the ultimate rupture limit originally proposed11, 12. This procedure is summarized in several design charts and is verified using a discrete element simulation.