Mathematical Modelling of Peak and Residual Shear Strength of Rough Rock Discontinuities Using Continued Fractions

Abstract

AbstractEstimating the shear strength of large in situ rock discontinuities is often required to assess the stability of rock masses. This estimation is, however, complicated by the well-known scale effect and the fact that the discontinuity surfaces are only partially accessible through traces. A new approach, referred to as the stochastic approach for discontinuity shear strength (StADSS), was recently presented to address these two points. This approach relies on a random field model and a semi-analytical shear strength model, the latter of which is referred to as the NDSS (Newcastle discontinuity shear strength) model. The NDSS model has to be implemented as a numerical code, and because the StADSS model is a Monte Carlo approach with hundreds if not thousands of simulations, the computational time to obtain a shear strength distribution is not negligible. The objective of this study is to find an efficient alternative to the NDSS model in the form of a continued fraction model that can predict the sheared area within a rough discontinuity subjected to direct shearing under constant normal stress as a function of the material strength, effective normal stress applied to the discontinuity and the standard deviation of asperity gradients (defined as the difference in elevation of two points of the surface over the horizontal distance between these points) of the surface. Using a 10/90 training/testing split of the dataset, a memetic algorithm-based truncated continued fraction regression (CFR) model was formulated. The distribution of CFR predictions was found to be very close to that of the dataset used for training. Then, the CFR model was tested against experimental data of the sheared area and shear strength (peak and residual) obtained from small (90 mm per 90 mm) and large (2 m per 2 m) specimens. It was found that 75% of the predictions fall within 20% of the experimental values. The continued fraction regression model can be used as an efficient alternative to the semi-analytical NDSS model, provided that it is used within the bounds of variables used to establish it.