# Effect of Small Numbers of Test Results on Accuracy of Hoek–Brown Strength Parameter Estimations: A Statistical Simulation Study

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## Abstract

The Hoek–Brown empirical strength criterion for intact rock is widely used as the basis for estimating the strength of rock masses. Estimations of the intact rock H–B parameters, namely the empirical constant *m* and the uniaxial compressive strength \(\sigma_{\text{c}}\), 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 and different combinations of H–B parameters, and then investigate the uncertainty in H–B parameters estimated from these limited data sets. We show that the uncertainties depend not only on the level of variability but also on the particular combination of parameters being investigated. As particular combinations of H–B parameters can informally be considered to represent specific rock types, we discuss that as the minimum number of required samples depends on rock type it 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 ongoing implementation of reliability-based design protocols and conclude with suggestions for improvements in this respect.

## Keywords

Rock strength Uncertainty Statistical simulation Sample size Hoek–Brown strength criterion## List of symbols

*m*Hoek–Brown regression parameter

*s*Sample standard deviation of axial strength

- \(\varepsilon\)
Random error

- \(\sigma_{1}\)
Axial strength

- \(\sigma_{3}\)
Confining pressure

- \(\sigma_{\text{c}}\)
Hoek–Brown regression parameter corresponding to mean unconfined compressive strength

- \(\varsigma\)
Standard deviation (as a Hoek–Brown regression parameter)

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