The Hoek-Brown empirical strength criterion for intact rock is widely used as the basis for estimating the strength of rock masses. Estimates of the H-B parameters, the empirical constant m and the uniaxial compressive strength sc are commonly obtained by fitting the criterion to triaxial strength data sets of small sample size. This paper investigates how such small sample sizes affect the uncertainty associated with the H-B parameter estimations. We use Monte Carlo (MC) simulation to generate data sets of different sizes that represent various rock types, and then investigate the uncertainty associated with parameter estimations for these synthetic data. We show that the uncertainties depend not only on the level of variability, but also on the rock type being investigated. We discuss that the minimum number of required samples depends on the rock type, and should correspond to some acceptable level of uncertainty in the estimations. Also, a comparison of the results from our analysis with actual rock strength data shows that the probability of obtaining reliable strength parameter estimations using small samples may be very low. We further discuss the impact of this on on-going implementation of reliability based design protocols, and conclude with suggestions for improvements in this respect.
In rock engineering, the determination of rock strength is a significant step in the early stages of a project in order to develop economic and safe designs. The well-known Hoek-Brown strength criterion is commonly used to characterize the strength of both intact rock and rock masses. At its simplest, for intact rock this criterion may be written as
where s1 and s3 are the major and minor principal stresses at peak strength, sc is the intact rock uniaxial compressive strength, and m is an empirical constant.
This criterion is usually calibrated, in terms of obtaining estimates of sc and m, by fitting it to the results of triaxial compressive tests conducted on intact rock specimens (e.g. Carter et al., 1991; Eberhardt, 2012; Hoek & Brown, 1988; Kovári et al., 1983). This is most regularly done using non-linear regression (Hoek & Brown, 1980; Hoek et al. 2002; Pariseau, 1997). The regression model used in such work is usually