Continuum Grain-Based Modelling (CGBM) has emerged as an efficient alternative to Bonded Block Modelling (BBM) for simulating grain-scale rock behaviour; however, its application to dynamic rock failure problems remains relatively limited, and the associated micromechanical failure processes, which involves strong rate-dependent and inertial effects, have not yet been extensively investigated. The major novelty is to develop CGBM as a mechanistic framework to uncover the micro-physical processes governing dynamic rock failure. CGBM models are developed for two Split Hopkinson Pressure Bar (SHPB) experimental campaigns involving different rock types (model material and marble), specimen geometries (intact, jointed, grouted, and cavity-containing), and loading modes (dynamic compression and indirect tension). After calibration using intact specimens, the models reveal how grain-scale stress redistribution, and localized yielding govern strength, anisotropy, and failure patterns under dynamic loading. The results demonstrate that strain-rate effects and U-shaped anisotropy arise from rate-dependent contact yielding. Cavity-induced strength reduction and size dependency are shown to originate from amplified grain-scale stress concentrations and premature damage localization. The simulations reproduce complex failure modes and reveal how grouting alters stress transfer pathways and suppresses fracture coalescence. Importantly, matching macroscopic stress–strain behaviour alone is shown to be insufficient for physically consistent calibration, as multiple micro-parameter combinations can produce similar global responses but fundamentally different failure mechanisms, reflecting the non-uniqueness of inverse calibration. By linking grain- and contact-scale damage evolution, and stress heterogeneity, to macroscopic behaviour, this study establishes a mechanism-based validation framework for dynamic rock modelling, supported by comparative UDEC–BBM analyses and parametric assessment.

Bonded Block Model (BBM) is gaining popularity in understanding the grain-scale micromechanics along with the macroscopic response of rocks to a variety of loading. However, the understanding of the application of BBM and the effect of different parameters on the estimated dynamic behaviour of rocks is lacking. This study aims to investigate the effect of different sample, experimental and modelling parameters on the dynamic response of Kishangarh marble represented via BBM in UDEC. The Voronoi tessellation is used to create the polygonal/ polyhedral grain structures along the samples. The modelling micro-parameters of the numerical model are initially calibrated using the experimental observations on intact samples for Split Hopkinson Pressure Bar (SHPB) tests. The numerical model and calibrated parameters were then fully validated with experimental results of jointed and grouted samples. The effect of different flaws (orientations, shape and persistence), experimental (strain rate), grain (heterogeneity and lamina orientation) and modelling (micro-parameters and modelling type) parameters are investigated. The orientation (anisotropy), shape and arrangement (persistence) of flaw affect the dynamic response of jointed samples significantly. The effect was suppressed in grouted samples due to possible strengthening of flaw tips, impeding stress concentration and fracture propagation. The effect of lamina orientation and strain rate was more dominant in grouted samples due to the transition in the behaviour of grouted samples towards intact rocks. The effect of grain/contact heterogeneity was observed to be negligible on micro/ macro level responses of samples. The initiation, progression and coalescence of different types of micro-level cracks and failure mechanisms were explored and explained with the BBM model of the rock specimens.

Slope instability is a complex geological phenomenon triggered by heavy rainfall, earthquakes, tectonic forces and anthropogenic activities. Unplanned excavation of rock slopes for the development and maintenance of infrastructure such as highways, railways, and buildings in hilly regions plays a significant role in causing slope instability. The present work conducts a comprehensive stability assessment of rock slopes along the national highway (NH-44) section of the Ramban district of Jammu and Kashmir. The kinematic investigation was carried out to determine the different modes of failure of road-cut slopes, and two main modes were found: planar and wedge. Various empirical classifications have been applied for the assessment of slope stability, like geological strength index (GSI), slope mass rating (SMR), Chinese slope mass rating (CSMR), and continuous slope mass rating (CoSMR). Numerical analysis through universal distinct element code (UDEC) has been conducted to evaluate the stability of four critical rock slopes under static and dynamic loading conditions. Modelling results have provided insights into the failure mechanism, and based on these findings, remedial measures have been proposed. Three (L-1, L-3, and L-13) out of four slopes were determined to be unstable under static conditions, with factors of safety (FoS) < 1.2 and under dynamic conditions, two slopes were unstable with FoS < 1.0. However, one (L-8) slope was found to be stable under both static and dynamic conditions, with FoS 1.29 and 1.12, respectively. Installing rock bolts reduced displacement for slopes L-1 and L-13, with reductions of 11.86

Response surfaces are commonly adopted as a surrogate for complex performance functions due to their ability to solve rock slope reliability problems with low computational costs. Most widely used methods employ a static scheme that uses input (rock properties) and output (factor of safety) samples generated using some experimental design to obtain the best fit parameters of the response surface. However, since the selection of input samples is not optimized, the increase in accuracy of the response surface often comes at the cost of an increased number of performance function evaluations, particularly, for slopes having a low probability of failure ( P_f ). This is addressed by an active learning scheme that iteratively selects input samples that improve the prediction of the response surface around the failure region. In this paper, an active learning scheme with support vector machine (SVM) is adopted for estimating the P_f of a rock slope along the Rishikesh–Badrinath highway against planar failure. The analytical expression for the factor of safety is utilized for conducting Monte Carlo simulation to estimate P_f , which is treated as a benchmark for determining the accuracy of the proposed method. Comparison with static scheme SVM illustrates the advantages of active learning scheme in increasing the accuracy in estimating the P_f for a similar number of performance function evaluations.

The simulation of field conditions for seismically induced slope failures incorporates model uncertainties, which account for the difference between simulated and observed slope behaviour. The quantification of this uncertainty is mandatory to understand the field response of the geotechnical system and make decisions for geotechnical systems. Previous studies have partially studied uncertainty for slope systems under seismic loading. To this aim, this study proposes a methodology based on probabilistic back analysis to estimate uncertainties in soil parameters considering the observed slope response under seismic loading. The proposed method involves support vector regression (SVR) model to map the relationship between soil parameters and seismically induced slope displacement. The SVR model is generated using the data from the numerical simulation of slope system under seismic loading using FLAC 2D. Further, the developed SVR model is used for probabilistic back analysis using Markov Chain Monte Carlo (MCMC) simulation. The Noto Hanto earthquake in 2007 and the subsequent slope failure along Noto Yuryo Road, Japan, are considered as a case study to validate the proposed methodology. The results of the case study show that the updated or inferred soil parameters have less variability than the prior distribution. Further, the uncertainties in the slope system influence the inferred soil parameters. Hence, a parametric study is conducted to investigate the effect of model uncertainty on the posterior statistics of soil parameters. The study results facilitate a better understanding of the slope deformation mechanism and the effect of model uncertainty on the updated statistics of soil parameters.

This paper presents the stability analysis of three real large rock slopes prone to failure with different mechanisms to investigate the applicability of the response surface methodology (RSM) for reliability-based rock slope problems. Initially, a detailed review of studies was performed to identify the gaps and common types of reliability-based rock slope problems. The applicability of RSMs based reliability methods was then investigated for three types of identified rock slope reliability problems via three case studies–type (1) Chenab rock slope prone to stress-controlled failures neglecting spatial variability, type (2) Deccan gold mine slope prone to stress-controlled failures considering spatial variability, and type (3) Rishikesh-Badrinath Highway slope prone to structurally controlled failures neglecting spatial variability. Analysis was performed by coupling MATLAB coded RSMs with advanced numerical tools and results were compared with those of direct Monte-Carlo Simulations (MCSs) based reliability method. Further, a detailed comparative study was carried out to evaluate the effect of the use of different response surfaces, i.e., polynomial based, Radial Basis Functions (RBFs) based, Support Vector Machine (SVM) based, kriging, Moving Least Square (MLS) and Gaussian Process Regression (GPR) on the accuracy and efficiency of RSMs based reliability analysis. Accuracy was evaluated using the Nash–Sutcliffe Efficiency (NSE) and Relative Difference in Reliability Index (RDRI), while computational efficiency was evaluated via the computational time required to evaluate the reliability with acceptable accuracy. RSMs based methods are observed to be highly accurate and efficient for the reliability analysis of large rock slopes. Further, the accuracy and efficiency are observed to be dependent upon the employed response surfaces and type of the problem. Suggestive guidelines are provided for selecting most suitable response surfaces for different problem types– a) for type 1 problem, Kriging and Least Square (LS)-SVM, b) for type 2 problem, some RBFs based and LS-SVM, and c) for type 3 problem, some RBFs based and LS-SVM are the most suitable RSMs.

Reliability based design approaches are usually adopted to explicitly consider uncertainties in the rock mass and make decisions on selecting design parameters for tunnel-support problems. However, it is difficult to accurately characterize the input random variables due to fewer sample availability and costly observations. Thus, it is necessary to focus on the characterization and reduction of epistemic uncertainty of those input variables, which have higher relative contributions towards the variability in output. These input variables are identified using global sensitivity analysis based on Sobol indices conducted for the supported circular tunnel in Hoek–Brown rock mass. Convergent confinement analysis method is applied to obtain the output—radius of yield zone ( $$R_{pl}$$ ), tunnel convergence ( $$u_{r}^{D}$$ ) and induced load on installed support ( $$p_{s}^{D}$$ ). The analysis is conducted for both uncorrelated and correlated input parameters. For uncorrelated input, the sensitivity measures are divided into correlated and uncorrelated contributions to provide insights into the mechanism of the tunnel-support system. The results obtained by assuming input variables as uncorrelated suggest that the relative contribution of variation in uniaxial compressive strength (UCS) of rock is highest towards the variation in the outputs followed by geological strength index (GSI) of the rock mass, its Young’s modulus ( $$E_{i}$$ ) and thickness of the liner support. However, by considering the correlation between input parameters, the sensitivity measures change significantly. The ranking of input parameters in descending orders of their relative contribution also changed due to the presence of high correlated contributions. The strong correlation between UCS and $$E_{i}$$ makes them most sensitive, followed by GSI, while thickness of the liner impacts $$p_{s}^{D}$$ the most. It is found that strong correlations do not necessarily mean higher correlated contributions, rather, it depends on the physics of the tunnel-support system. The 95% confidence interval of ground response curve and longitudinal deformation profile changes considerably for correlated rock mass parameters compared to the uncorrelated case resulting in higher variance of $$u_{r}^{D}$$ and lower variances of $$R_{pl}$$ and $$p_{s}^{D}$$ . Additionally, the investigation of average error in the variance of output, induced by keeping individual parameters as deterministic, yielded high values for highly sensitive inputs and vice versa.

Characterization of properties governing the stability of rock slopes is essential for their design and analysis. Importance ranking of these properties can be obtained by the sensitivity indices that quantifies the extent to which different properties influence the stability of the slopes. This helps the designers to divert major laboratory/in situ investigation resources to evaluate highly ranked properties. Another usage is to perform the probabilistic stability analysis of slopes efficiently by treating low-ranked properties deterministically in the estimation of probability of failure ( P_f ). In this paper, the importance ranking of rock properties affecting the P_f of rock slopes prone to different failure mechanisms is performed using local/global sensitivity approaches (L/GSAs) and the accuracy of these approaches was assessed quantitatively. Four slope case studies indicating different structurally and stress-controlled failures were considered, and sensitivity analyses were performed using six different L/GSAs. Accuracy of the approaches was assessed by comparing the importance ranking of properties based on sensitivity approaches to that of the normalized errors, i.e., ε_i invoked in the P_f by neglecting the uncertainties in these properties. Results indicated the superior accuracy of GSAs as compared to LSAs. Importance ranking was dependent upon the considered slope (failure mechanisms), with some slopes showing higher sensitivities to external parameters and others to inherent rock properties. An important guideline based on the analysis is suggested to consider the properties as deterministic/random variables in the probabilistic analysis. For the slopes with the minimum interaction effects in their sensitivity (planar, wedge, and stress controlled), uncertainties in multiple properties can be neglected based on the allowable error in the P_f . Further, a dependence of ε_i and corresponding importance ranking was observed on the selected value of property of interest (assumed as deterministic) across its domain in the analysis.

Analysis of tunnel-support system stability with uncertain rock mass properties is conducted in this paper. A horseshoe shaped tunnel driven in weak rock mass is analyzed through deterministic, random variable and random field approaches. Performance of the tunnel in probabilistic analysis was assessed by defining three limit states which ensure that tunnel convergence remained below a safe threshold level, the rock bolts installed are embedded sufficiently beyond the depth of yielded rock mass and load induced on liner support does not exceed its capacity. Both unsupported and supported tunnels were analyzed with deterministic, random variable and random field approaches. Fourier series method was applied to discretize the random fields, and random finite difference analysis was conducted using Monte Carlo simulations. Scale of fluctuation (SOF) for isotropic random fields and horizontal and vertical SOF ratios for anisotropic random fields were varied to study their effect on performance of the tunnel and failure mechanisms involved. It was found that SOF significantly influences the output statistics and Pf of the limit states. It was observed that, random variable approach underestimates the performance of the tunnel-support system; however, it can be adopted as conservative option in absence of data required for random field characterization.

Mining at greater depths can lead to stress-induced failure, especially in areas of high horizontal in situ stress. The induced stresses around the opening are known to be in a poly-axial stress state where, σ1 ≠ σ2 ≠ σ3 with special case of σ3 = 0 and σ1, σ2 ≠ 0 at its boundary, where σ1, σ2, σ3 are major, intermediate, minor principal stress, respectively. The conventional triaxial testing does not represent the actual in situ strength of the rock in regions of high horizontal stress, as it ignores the influence of intermediate principal stress (σ2). The typical poly-axial testing (biaxial and true-triaxial tests) of intact rock mostly requires sophisticated and expensive loading systems. This study investigated the mechanical behavior of intact rock under a poly-axial stress state using a simple and cost-effective design. The apparatus consists of a biaxial frame and a confining device. The biaxial frame has two platens that apply equal stress in both directions (σ1 = σ2) on a 50.8 mm cubical specimen when placed inside the uniaxial loading device. The confining device performed separate biaxial tests under constant intermediate principal stress (σ2 = constant) and true-triaxial tests when used along with the biaxial frame. This study then compared the failure modes and peak strength of Berea sandstone specimens with other biaxial–triaxial devices to validate the design of the poly-axial apparatus. We also performed uniaxial tests on both standard cylindrical samples and prismatic specimens of different slenderness ratios. These tests provided a complete understanding of the failure mode transition from standard uniaxial compressive tests to triaxial stress conditions on cubical specimens. Additionally, this study determined best-fitted strength envelopes for biaxial and triaxial stress state. Based on regression analysis, we found a quadratic polynomial to be a good fit to biaxial strength envelope. For the true-triaxial strength envelope, we found the three-dimensional (3D) failure criterion to be a good fit with R2 of 0.964.

Fracturing in rocks results in the formation of an inelastic region surrounding the crack tip called the fracture process zone (FPZ), which is often characterized using the Linear Cohesive Zone Model (LCZM). Various numerical studies have shown that the prediction of the FPZ characteristics is significantly influenced by variability in the input parameters of LCZM, such as crack tip opening displacement and tensile strength. In this study, an integrated approach was used for evaluating the LCZM performance in predicting fracture processes of Barre granite specimens, as a representative rock, under Mode I loading. The approach involved experimental testing, numerical simulation, uncertainty quantification of overall fracture behavior, and global sensitivity analysis. First, parameters of the LCZM were estimated from three-point bending tests on center notch Barre granite specimen using the two-dimensional digital image correlation (2D-DIC) technique. This was followed by the implementation of the LCZM in XFEM-based numerical model to simulate the evolution of the FPZ in tested geometry. The results from the deterministic numerical simulation showed that while LCZM can predict all stages of FPZ evolution, the variability in the experimental results, such as the FPZ size, cannot be accounted. The variability of the material response was quantified using a random variable analysis, which involved treating the LCZM’s parameters as random variables. This was followed by the global sensitivity analysis that revealed the most sensitive input parameters is the tensile strength for accurate prediction of the global response of rock specimens under Mode I loading.

ABSTRACT Vertical anchor plates are often provided to increase the performance of various geotechnical engineering structures such as sheet pile walls, bulkheads, bridge abutments and offshore structures. Hence, the safe design of such structures needs a better understanding of the 3D behaviour of the anchor plate. This paper presents and discusses the results obtained from a series of 3D finite-difference analyses of vertical square anchor plate embedded in cohesionless soil. The 3D model is found to closely predict experimental pullout load–displacement relationship. The failure mechanism observed in the numerical model is found to be very similar to the failure reported in experimental studies. For a given embedment depth, the stiffness of the breakout factor–displacement response substantially reduces with increase in anchor plate size. However, the ultimate reduction in anchor capacity is found to approximately 8% with an increase in anchor size from 0.1 to 1 m. Numerical analysis reveals that at deeper embedment depth, the friction angle of sand is the critical parameter in enhancing the performance of anchor plate. The obtained 3D model results are then compared with the published results and are found to be reasonably in good agreement with each other.

The paper investigates the uplift performance of horizontal anchor plate in geocell reinforced sand through a series of model tests. It is noted that the unreinforced anchor plate undergoes a clear failure at a displacement of about 3% of its width, whereas with the provision of geocell and a layer of geotextile right below the geocell mattress significantly increases the uplift capacity by about 4.5 times higher than that of unreinforced sand and could sustain anchor displacement of more than 60%. Results indicates that the geocell mattress by virtue of its rigidity distributes the uplift load in the lateral directions to a larger area, thereby reducing the stress in the overlying soil mass and hence increases the performance of anchor plate system. The provision of the additional geotextile layer right below the geocell mattress is found to be very effective in increasing the stiffness as well as load carrying capacity of anchor plate system. The optimum size (i.e., width and length) of geocell mattress giving adequate load carrying capacity of anchor plate is found to be 5.4 times of anchor width (5.4B). The comparison of model tests results with 3D numerical analysis shows good agreement, indicating that the proposed model is able to capture the uplift load-displacement behaviour of geocell reinforced anchor plate system.

Probabilistic methods are the most efficient methods to account for different types of uncertainties encountered in the estimated rock properties required for the stability analysis of rock slopes and tunnels. These methods require estimation of various parameters of probability distributions like mean, standard deviation (SD) and distributions types of rock properties, which requires large amount of data from laboratory and field investigations. However, in rock mechanics, the data available on rock properties for a project are often limited since the extents of projects are usually large and the test data are minimal due to cost constraints. Due to the unavailability of adequate test data, parameters (mean and SD) of probability distributions of rock properties themselves contain uncertainties. Since traditional reliability analysis uses these uncertain parameters (mean and SD) of probability distributions of rock properties, they may give incorrect estimation of the reliability of rock slope stability. This paper presents a method to overcome this limitation of traditional reliability analysis and outlines a new approach of rock mass characterization for the cases with limited data. This approach uses Sobol's global sensitivity analysis and bootstrap method coupled with augmented radial basis function based response surface. This method is capable of handling the uncertainties in the parameters (mean and SD) of probability distributions of rock properties and can include their effect in the stability estimates of rock slopes. The proposed method is more practical and efficient, since it considers uncertainty in the statistical parameters of most commonly and easily available rock properties, i.e. uniaxial compressive strength and Geological Strength Index. Further, computational effort involved in the reliability analysis of rock slopes of large dimensions is comparatively smaller in this method. Present study also demonstrates this method through reliability analysis of a large rock slope of an open pit gold mine in Karnataka region of India. Results are compared with the results from traditional reliability analysis to highlight the advantages of the proposed method. It is observed that uncertainties in probability distribution type and its parameters (mean and SD) of rock properties have considerable effect on the estimated reliability index of the rock slope and hence traditional reliability methods based on the parameters of probability distributions estimated using limited data can make incorrect estimation of rock slope stability. Further, stability of the rock slope determined from proposed approach based on bootstrap method is represented by confidence interval of reliability index instead of a fixed value of reliability index as in traditional methods, providing more realistic estimates of rock slope stability.

The paper investigates the behaviour of groups of horizontal square anchor plates in geogrid-reinforced sand using laboratory model tests. The optimum spacing for two anchor plates in unreinforced sand is 3.4 times the anchor width. The unreinforced groups of anchor plates show a clear failure at a displacement of about 5% of the anchor width, whereas this value increases to more than 45% for reinforced groups along with a two-fold increase in uplift capacity. The optimum width and length of the geogrid reinforcement for groups of two anchors is found to be 5 and 9.4 times the width of the anchor plate, respectively. The performance improvement for isolated anchor plates is found to be maximum and gradually reduces with an increase in the number of plates; however, this reduction is much less in groups of two to four. Therefore, the results obtained from the model tests for reinforced groups of two to four anchor plates can be conveniently used to extrapolate the uplift capacity of multiple anchor plate systems in a reinforced soil mass. The model test results show a reasonably good agreement with the three-dimensional (3D) numerical analysis results.

The support design for rock tunnels involves a complex problem because of the various types of uncertainties present in rock mass properties; therefore, probabilistic approaches are used to consider these uncertainties systematically in the presented analysis. In earlier studies, uncertainties in peak strength parameters and deformation modulus were considered while uncertainty in residual strength parameters was neglected. Another important factor is the assumption regarding the postpeak behavior of the rock mass, which depends on the rock mass quality in the field. An average quality rock mass generally shows strain-softening behavior in a range from peak to residual strength, which generally has been neglected because of the assumption that the rock mass is elastic-perfectly plastic. However, it was observed that the yield zone depth and displacements around the tunnel are highly sensitive to residual strength parameters, and residual strength parameters can considerably influence the estimation of the support requirements of a tunnel. In this paper, a new computational approach, based on the geological strength index (GSI) and the use of deterministic and probabilistic methods, is described for the reinforcement design of a tunnel in an average quality rock mass. This approach considers the variability in residual strength parameters along with peak strength parameters and the deformation modulus. Moreover, the strength drop in the stress-strain behavior of the rock mass is also considered in the analysis. An underground powerhouse cavern from the Himalayan region of India is taken as the case study to show the methodology used in the research. The study brings out the advantages of the probabilistic over the deterministic approach for the support design of a tunnel. Another important conclusion from the study is that the support requirements for the tunnel are greatly influenced by residual strength parameters, and therefore, uncertainty and strength drop in the residual strength parameters should be properly considered when designing support for tunnels.