Technical note

Rock variability and establishing confining pressure levels for triaxial tests on rocks

The critical strain concept has been widely used in analytical or numerical approaches to evaluate the stability of underground excavations. Analytical, empirical, and numerical procedures are usually used to determine the critical strain values. This paper presents a reliability assessment procedure for evaluating excavation stability using the empirical approach based on the rock mass classification Q and the first order reliability method (FORM). In contrast to deterministic critical strain values, a probabilistic critical strain, which considers uncertainties in rock mass parameters, was incorporated in a limit state function for reliability analysis. Using the rock mass classification Q, the empirically estimated tunnel stain was included in the limit state function. The critical strain and estimated tunnel strain were probabilistically characterized based on the rock mass classification Q-derived rock mass properties. Monte Carlo simulations were also conducted for comparing the reliability analysis results with those derived from the FORM algorithm. A highway tunnel case study was used to demonstrate the reliability assessment procedure. The effects of the input ground parameter correlations, probability distributions, and coefficients of variation on tunnel reliability were investigated. Results show that uncorrelated and normally distributed input parameters (intact rock strength and elastic modulus) have generated more conservative reliability. The reliability analysis results also show that the tunnel had relatively high reliability (reliability index of 2.78 and probability of failure of 0.27%), indicating the tunnel is not expected to experience instability after excavation. The tunnel excavation stability was assessed using analytical and numerical approaches for comparison. The results were consistent with the reliability analysis using the FORM algorithm's Q-based empirical method.

The uniaxial compressive strength (UCS) and Youngs’ modulus (E) of rock are important parameters required during design and stability analysis of mining and geotechnical structures. There is correlation between UCS and E of rock, and the proper estimation of such correlation is important for reliable mining engineering analysis. However, limited quantity of UCS and E data pairs often available for most mining project sites makes it difficult to estimate reliable correlation between UCS and E. This study addresses the difficulty by developing Bayesian approach for characterizing the site-specific joint probability distribution of UCS and E that is data-driven, without the use of an empirical model. A major novelty of the proposed approach over previous studies is that it does not require selection or integration of a regression model as input to characterize the correlation between UCS and E. The likelihood function in the proposed approach is directly constructed from only limited UCS and E data pairs and their prior information as inputs. The Bayesian approach is incorporated into Markov Chain Monte Carlo (MCMC) simulation to generate samples pairs of UCS and E, which are then analysed for marginal statistics, marginal probability distribution, joint probability distribution and correlation. Real data of UCS and E obtained from uniaxial compression tests on migmatites at the Sanandaj-Sirjan zone in Iran is used to demonstrate the approach. The marginal statistics, distributions and correlation coefficient from the proposed approach is consistent with those of the measured data from the adopted site. This indicates that the approach is effective in characterizing the correlation between UCS and E, and can be used when there is need for such characterization at a site with limited data. Simulated data are also used in the approach and the results show that the quality and quantity of information available as inputs play an important role in the efficiency of the characterization by the approach. The hallmark of the proposed approach is that it is data-driven, and practitioners do not need to determine and select an appropriate site-specific regression model to evaluate the correlation between UCS and E of rock.

Triaxial compression testing can yield realistic simulations of in-situ rock properties in underground excavation. This paper presents laboratory triaxial compression tests for marble from Jinping II hydropower station project, southwestern China. The test results were used to determine shear strength parameters (i.e., angles of internal friction and cohesive strengths) of the marble. Prominent uncertainties were found among the shear strength parameters. Modelling and quantifying these uncertainties in rock properties are initial, critical steps in probabilistic analysis and design of rock engineering. However, disagreement exists among researchers about which probability distribution should be used to characterize the rock angle of internal friction and the cohesive strength. This paper proposes an alternative universal form of probability distributions for these shear strength parameters, i.e., maximum entropy probability density functions. The entropy distributions are obtained from maximum entropy method, a combination of maximum entropy principle and a second order Akaike information criterion. The maximum entropy principle is used to determine a probability density function that maximizes Shannon’s entropy subjected to constraints in terms of sample integral moments. The second order Akaike information criterion can provide strong consistent and unbiased estimate for Kullback-Leibler entropy, and thus the most unbiased and objective probability density distribution with optimum order can be determined from an available sample. Kolmogorov-Smirnov test is used to establish the maximum entropy probability distributions, which are compared with conventional normal and lognormal probability distributions to illustrate the advantages.

The inhomogeneous variability of natural rock always leads to the uncertainty of the parameters gained from the compressive testing data of a limited number of specimens. In this technical note, a systematic comparison for two presented classical approaches, i.e. the Yamaguchi's “decision of the sample number approach (DSNA)” and the “confidence interval approach (CIA)”, is firstly made to identify the cause of discrepancy in the number of required specimens evaluated with different methods. Based on the advantages of both methods, the confidence interval dynamic process approach (CIDPA) is proposed to dynamically determine the number of required specimens during the test process. This presented technical method can effectively avoid the waste of rock specimens while ensuring the accuracy of parameters under a given confidence level. In addition, all of the methods mainly used for the uniaxial comprehensive strength with the Gaussian distribution are extended to obtain a reliable elastic modulus that usually obeys lognormal distribution. Finally, uniaxial compression tests of oblique porphyritic basalt (OP-basalt) and amygdaloidal basalt (Am-basalt) are analyzed to verify the contrast between the results obtained with different approaches and the rationality of CIDPA.

ISO2394:2015 contains a new informative Annex D on “Reliability of Geotechnical Structures”. The emphasis in Annex D is to identify and characterize critical elements of the geotechnical reliability-based design (RBD) process, while respecting the diversity of geotechnical engineering practice arising from the influence of site-specific conditions. Annex D should pave the way for future code revisions to inject greater realism into geotechnical RBD. The most important element is the characterization of geotechnical variability. There are some noteworthy features: (1) the coefficient of variation (COV) of a geotechnical design parameter does not fall within a narrow range of values, (2) the multivariate nature of geotechnical data can be exploited to reduce the COV, (3) spatial variability affects the limit state beyond reduction in COV due to spatial averaging, and (4) statistical uncertainty may be significant, because the volume of soil sampled is a minute fraction of the volume of interest. Another important element is the characterization of model uncertainty. Geotechnical design codes, be it reliability-based or otherwise, must cater to diverse local site conditions and diverse local practices that grew and evolved over the years to suit these conditions. One obvious example is that the COVs of geotechnical parameters can vary over a wide range, because diverse property evaluation methodologies exist to cater to these diverse practice and site conditions. Another example is that deep foundations are typically installed in layered soil profiles that vary from site to site. These diverse site specific design scenarios do not surface in structural engineering. Section D.5.2 notes that reliability calibration for geotechnical RBD is more challenging for this reason.

When there is no possibility of direct compression test, geotechnical engineers and practitioners may utilize regression models (i.e., equations) to estimate the uniaxial compressive strength, UCS, of rock from point load index, $\mathrm{I}{s}_{(50)}$, since there are established relationships between the two properties. Different studies have shown, however, that there is no unique equation relating $\mathrm{I}{s}_{(50)}$ to UCS for all rock types. This leads to the problem of how to select the most appropriate model for a particular rock deposit out of the numerous models available, since UCS of rock like other geomechanical properties are inherently variable as a result of different geological processes that rocks are subjected to. This study develops a method that rationally compares different regression models and selects the most appropriate model for a specific site or deposit. The most appropriate model is the model with the highest occurrence probability for the given set of observation data, and it is selected using only a limited number of $\mathrm{I}{s}_{(50)}$ data obtained from the specific deposit or site. The methodology starts with the formulation of a general likelihood model to determine the occurrence probability of each model based on the limited number of $\mathrm{I}{s}_{(50)}$ data available from a specific site. This is different from previous works that need both UCS and $\mathrm{I}{s}_{(50)}$ data to draw comparison. Note that UCS data are generally not available when the use or selection of regression model is needed. The selected model is subsequently used in Bayesian framework to integrate the prior knowledge about UCS with the limited number of site-specific $\mathrm{I}{s}_{(50)}$ data available for probabilistic characterization of UCS, such as obtaining its mean, standard deviation and full probabilistic distribution.