Probability analysis of rock slope using FORM based on a nonlinear strength criterion

A nonlinear strength criterion for rock-like materials based on fracture mechanics

The shear strength characteristics of rock masses containing non-persistent discontinuities are strongly affected by discontinuities and rock bridges. The linear Jennings criterion cannot reflect the nonlinear mechanical behavior during progressive failure of rock masses with non-persistent discontinuities. In this study, a new nonlinear shear strength criterion was developed. First of all, a series of shear test data about artificial rock mass samples were collected on the basis of the published literatures, and five types of samples were differentiated according to the positions of discontinuities. After that, a new nonlinear shear strength criterion was proposed by introducing two correction coefficients A and B into the basic form of the Jennings criterion, which could correct the weight of the cohesion and the internal friction coefficient of rock bridges respectively. Then, the new criterion was determined by fitting the basic form of the Jennings criterion with the laboratory data. It was found that the parameters A and B had a nonlinear exponential and negative exponential relation with the connectivity rate respectively. It indicated that both the cohesion and the internal friction coefficient estimated by the new criterion were superior to those estimated by the Jennings criterion. Compared with the linear Jennings criterion, the new nonlinear shear strength criterion had a better applicability.

Strength of a matrix with elliptic criterion reinforced by rigid inclusions with imperfect interfaces

There is a need for a unified strength criterion, which is a variable suitable for describing the different strength properties of different types of geomaterials. In this study, a unified nonlinear strength (UNS) criterion is proposed based on the Lade-Duncan criterion for the sake of versatility to capture the complex strength behaviors of geomaterials under three-dimensional (3D) stress paths. The strength curves in this theory are a series of continuous and smooth conical loci that covers wide ranges of the meridian and octahedral planes to describe the nonlinear strength behaviors of soils. The strength curves are located between von Mises and Lade-Duncan strength curves in the deviatoric plane in the principal stress space. Verifications between the proposed criterion and experimental data from triaxial tests found in the literature show that this new criterion is able to reasonably reflect the three-dimensional (3D) strength properties and strength nonlinearity of a variety of geomaterials.

This study presents a database of over 900 large-scale triaxial tests that were used to calibrate and refine the Leps strength envelopes for rockfill materials. A novel nonlinear shear strength criterion for rockfill is introduced that can replicate the Leps strength envelopes, is derived in terms of both shear/normal and principal stresses, employs a single fitting parameter with direct physical significance, and allows for smooth transitions between rockfill strength categories. An accompanying empirical model is presented to estimate rockfill strength based on void ratio and rock type, offering a practical tool for rapid application. The model simplifies to five strength envelopes and demonstrates strong predictive performance, while maintaining simplicity. The evaluation of prediction accuracy indicates that the strength model is applicable for soils with a median particle size greater than 2 mm and a fines content of less than 25%. Analysis by stress range reveals a limitation of power-law strength criteria: when calibrated using tests at very low confining stresses, power-law models can significantly over- or underestimate strength for normal stresses above 100 kPa. To address this, a 60-600 Guideline is proposed, recommending the exclusion of triaxial confining stresses below 60 kPa and inclusion of at least one test with a confining stress above 600 kPa to obtain reliable calibration of power-law strength criteria for rockfill from laboratory testing data. An improved methodology to convert triaxial test results into shear and normal stresses is also introduced, accounting for the nonlinearity of the failure envelope.

Convergence of circular tunnels in elastoplastic rock masses with non-linear failure criteria and non-associated flow laws

Currently, the strength parameters are reduced by a uniform coefficient for the traditional strength-reduction technique in the slope stability analysis. However, observations proved that the contributions of the strength parameters on slope stability are different from each other when the slope reaches the limit equilibrium (LE) state, which means that the strength parameters should be reduced with their own reduction coefficient. Thus, based on the assumptions of stresses on slip surfaces and the overall mechanical equilibrium conditions of the slope sliding body, this work establishes the LE solution for slope stability under the double-strength reduction (DSR) technique. In the LE stress-based analysis, the loop iteration strategy is given to solve the LE stability of slope with the nonlinear strength criterion. Moreover, a new method (i.e., the square root method) is proposed to calculate the slope comprehensive factor of safety (FOS) for evaluating the slope stability with application of the DSR technique. Thus, by comparing and analyzing some slope examples, the accuracy of the present method is verified. Furthermore, from the obtained stability charts of slope under the DSR technique, the study shows that if the characteristic stability number is less than 0.04 for clay soil, then the internal friction angle would have a larger contribution to the slope stability than cohesion; whereas if the characteristic stability number is more than 1.00, then the larger contributor on the slope stability would be the cohesion.

This research aims to study the deformation and fracture behavior of shale by a novel anisotropic regular lattice spring model (ARLSM). The novel ARLSM applies the normal and tangential coupling spring to release the Poisson's ratio limitation in the traditional regular lattice spring model. Meanwhile, a nonlinear strength criterion is introduced into ARLSM to simulate the fracture failure of shale. Two benchmark problems are tested to implement the research. The study shows that ARLSM has larger range of Poisson's ratio and better effects comparing with the existing anisotropic lattice spring model. Moreover, ARLSM can accurately predict the deformation and fracture behavior of shale under different conditions.

The stability of rocky slope is determined mainly by discontinuities. The discrete element calculation method can be used to analyze the geometric features of rock structures and to deal with the nonlinear deformation and destruction of the rock mass that may be affected by the discontinuities. In this paper, the Barton–Bandis (B-B) nonlinear strength criterion was introduced to improve the Mohr–Coulomb (M-C) slip model of joint slope. The modified model could reflect the real-time shear strength in a changing stress state. Using the numerical calculation of the shear test of the structural plane, we found that the corrected slip curve fits well with the process before the failure of the shear test. Furthermore, the modified model can track the disintegration between blocks while sliding failure of joint slope. With an increase in the number of structural planes and the complexity of the relative location of blocks, the interaction force between blocks and the sliding failure mode of the joint slope would be more complex. Changing the nonlinear parameters with the stress state of the structure plane could effectively solve this problem.

Rainfall infiltration is one of the major causes of slope instability. While shear failures were commonly adopted, unsaturated soil slopes are in fact highly susceptible to composite tensile-and-shear failure. To address this problem, a nonlinear Mohr–Coulomb (MC) criterion was modified to capture the time dependent tensile and shear strengths for various soils during rainfall, where the coupled effect between capillary and adsorption pressures in the soils was considered. The results showed that the proportion of adsorption stress in the total suction stress varied greatly across different soil types, with the transitional and saturated gravimetric water contents being the two most sensitive parameters affecting the proportion of adsorption stress. On this basis, energy balance equation was established through a two-stage composite tensile–shear failure mechanism of three-dimensional (3D ) slopes following the modified MC criterion. Based on the upper bound theorem of limit analysis, the factor of safety (FoS) for the slopes was calculated by incorporating the strength reduction technique to systematically investigate the effects of rainfall patterns including intensity and duration and soil types on slope stability. Comparisons by analytical and numerical approaches were conducted to verify the present work. Parametric analysis revealed that a 3D analysis could better reflect the slope stability, and that the falling range of slope FoS was strongly correlated with the rainfall patterns and soil permeability. The present study provides a reference and method for assess the stability of various unsaturated soil slopes under different rainfall conditions.

The conventional factor of safety as used in geotechnical engineering does not reflect the degree of uncertainty of the relevant parameters. Then in the geotechnical engineering, there have been efforts to reflect the uncertainties of the geotechnical properties through probabilistic analysis. In this study, a practical method for probabilistic analysis using the Hasofer-Lind reliability index is introduced. The method is based on the perspective of an ellipsoid that just touches the failure surface in the original space of the variables. The method is applied to prefabricated vertical drains (PVD) and compared with the result of Monte Carlo Simulation method.

Objectives: Quantification of uncertainties and its implications on stability assessment of geotechnical problems can very well be captured through probabilistic analysis in which parameters are considered to be random variables. Thus, randomness of the response of the system is also captured and safety assessment is reported in terms of a safety index, commonly called as “reliability index”. In this study, probabilistic approach in conjunction with conventional deterministic approach is used for the analysis of a deep excavation problem and advantages of probabilistic approach are highlighted. Methods/Statistical Analysis: Probabilistic analysis requires establishment of connection between input parameters and output responses, for which response surface method (RSM) concepts can effectively be utilized. RSM helps developing regression model between input and output which can be used for the probabilistic analysis through First Order Reliability Method (FORM). Advanced finite element numerical tool, i.e., PLAXIS 3D is utilized for the purpose of the analysis. Findings: Considering this typical case of deep excavation problem, it is highlighted that probabilistic analysis when used in conjunction with the deterministic approach, it brings rationality in decision making while dealing with uncertainty and provides mathematical basis for treating uncertainty. Application/Improvements: Conventionally, stability analyses of a deep excavation system are carried out through limit equilibrium based approaches and the parameter used to define extent of stability is factor of safety. With advent of computers and finite element or finite difference based computing facilities, strength reduction technique is adopted to estimate the factor of safety. These numerical tools also help in predicting the deformation pattern and judging the serviceability requirements. In either case, the soil parameters used as input in the numerical tools are taken a constant value uniformly. Choice of appropriate value of design input parameter is based on past experience, good engineering judgment and confidence in the adopted analysis methodology. Due to various uncertain sources involved in the estimation of the input properties of a natural material like soil, such as, natural variability, testing errors, model transformation, it is impractical to assign a single value to design parameter. Deterministic approach, which involves single value of design parameters for stability assessment is considered to be simple and straight forward. It is demonstrated that quantification of uncertainties and its influence on stability assessment of deep excavation problem can be captured in probabilistic framework.

Plastic systems reliability by LP and FORM

To establish a strength criterion that can consider the composite properties and internal non-homogeneity of frozen soil, a load sharing model is first defined based on homogenization theory and the binary-medium structure model, and then the strength functions of frozen soil on the p-q plane and the π plane are established, respectively. Finally, a three-dimensional nonlinear unified strength criterion for frozen soil is proposed. The applicability of the proposed strength criterion is verified based on the multi-group triaxial compression test results and the directional shear test results at different stress Lode angles.

This study presents a parallelized framework for the grid-and-radius-based Morgenstern–Price method to enable an efficient reliability analysis of unsaturated finite soil slopes under correlated and non-Gaussian conditions. To address the limitations of deterministic techniques and the significant computational demands associated with traditional Monte Carlo simulation, the framework incorporates surrogate modeling approaches, namely polynomial chaos expansion–enhanced Monte Carlo simulation (PCE-MCS) and Kriging-based Monte Carlo simulation (K-MCS). This research conducts a comprehensive evaluation of these surrogate-assisted methods with the gradient-based approaches for estimating the probability of failure (pf) under correlated and non-Gaussian distributions. The surrogate models exhibit exceptional accuracy (R2 > 99.7%, mean absolute error <0.6%) and drastically reduce computational cost compared with traditional methods. Parametric and variability studies show that increasing the mean stability number (0.005–0.065) and friction angle (20°–30°) reduces the pf from 19.19% to 0.005% and 14.9% to 0.01%, respectively, while higher variability increases pf. Similarly, increasing the air-entry value reduces pf from 29.8% to 0.01%, whereas pf rises to 15.1% and 21.8% with increases in soil–water characteristic curve parameters nf (0.1–3.25) and mf (0.25–1.65). A detailed comparison against gradient-based techniques such as the first-order reliability method (FORM) and the second-order reliability method (SORM) underscores the superiority of surrogate-based approaches. For instance, the FORM shows errors up to 92.59%, while SORM errors reach 62.75%, particularly in highly nonlinear or variable scenarios. In contrast, PCE-MCS and K-MCS errors remain generally below 10%. Furthermore, reliability-based design contour charts are developed using K-MCS. These charts align with the USACE reliability classification, enabling rapid and reliable assessment of unsaturated slope stability without the need for large-scale simulations. Therefore, by effectively bridging traditional geotechnical engineering and advanced surrogate techniques, the proposed approach enhances practical decision-making in unsaturated slope design and supports the advancement of reliability-based design optimization in geotechnical practice.