Equivalent Mohr Coulomb Strength Parameters Research Articles

Modified minimum principal stress estimation formula based on Hoek–Brown criterion and equivalent Mohr–Coulomb strength parameters

The most critical parameter for determining equivalent values for the Mohr–Coulomb friction angle and cohesion from the nonlinear Hoek–Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress (sigma_{3,max }^{prime }) on the potential failure surface. The existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine sigma_{3,max }^{prime } of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on sigma_{3,max }^{prime } while the influence of intact rock strength and the material constant m_{i} are relatively small. According to the variation of sigma_{3,max }^{prime } with different factors, two new formulas for estimating sigma_{3,max }^{prime } are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.

Bearing capacity of foundations resting on rock masses subjected to Rayleigh waves

This study investigates the ultimate bearing capacity of shallow foundations resting on rock masses under earthquakes using upper-bound limit analysis. A modified pseudodynamic method (MPD) that considers the influence of Rayleigh waves is initially used to characterize the spatiotemporal variation in earthquake acceleration. The strength of rock masses is governed by the generalized Hoek‒Brown criterion and the plastic flow at failure is assumed to respect the associated flow rule. The generalized tangential technique is introduced to provide a linear approximation of the strength envelope for capturing the equivalent Mohr‒Coulomb strength parameters. The classic nonsymmetrical multi-block translational mechanism, which is capable of assessing a rock-foundation system under asymmetrical load, is reconstructed to portray the velocity discontinuity of the plastic failure region. Based upon the energy equilibrium theorem of the limit analysis, an analytical solution of seismic bearing capacity is derived and then formulated as a multivariate optimization problem. Comparisons show good agreement with the literature and validate the correctness and rationality of this work. Parametric study indicates that the seismic foundation capacity obtained by MPD tends to be conservative compared to the pseudostatic method with the same seismic coefficients, and that the period of a Rayleigh wave has only a slight effect on the bearing capacity. The failure region is more sensitive to the disturbance coefficient than it is to the other rock strength parameters.

A Return Mapping Algorithm for Nonlinear Yield Criteria with the Equivalent Mohr–Coulomb Strength Parameters

This paper proposes a modified return mapping algorithm for a series of nonlinear yield criteria. The algorithm is established in the principal stress space and ignores the effect of the intermediate principal stress. Three stress return schemes are derived in this paper: return to the yield surface, return to the curve, and return to the apex point. The conditions used for determining the correct stress return type are also constructed. After the proposed algorithm is programmed in the finite element software, we merely need the equivalent Mohr–Coulomb (M-C) strength parameters, the derivatives of their functions, and the tensile strength of these nonlinear yield criteria. In addition, the Hoek–Brown (H-B) yield criterion is taken as an example to validate the proposed method. The results show that the updated stresses and the final principal stresses obtained by the proposed method are in good agreement with those obtained by other methods. Furthermore, the proposed method is more suitable for the associated plastic-flow rule.

Rock Slope Analysis with Nonlinear Hoek–Brown Criterion Incorporating Equivalent Mohr–Coulomb Parameters

In the present work, the results of rock slope analyses incorporating nonlinear Hoek–Brown yield criterion with equivalent Mohr–Coulomb strength parameters to simulate rock mass behaviour are presented. The factor of safety (FOS) of rock slopes are calculated using Morgenstern–Price and Spencer’s method. A simplified method of generating non-circular segmented failure surface is proposed. The search of minimum FOS and associated critical non-circular failure mass for rock slopes is performed by Particle swarm optimization (PSO) technique. A MATLAB code is developed incorporating nonlinear Hoek–Brown criteria with equivalent Mohr–Coulomb strength parameters and coupled with PSO to analyze stability of rock slopes. The efficiency and accuracy of the developed MATLAB code is established by comparing the obtained results with those from existing literatures. Furthermore, the analyses of slopes are performed with varying height and slope angle for an intact rock slope with varying material parameters such as Hoek–Brown constant $$m_{i}$$ and uniaxial compressive strength $$\sigma_{ci}$$ for rock material. It is found that the FOS value follows a decreasing pattern if height and the slope angles are increased.

Upper Bound Solution for Required Supporting Pressure Applied on a Deep Shield Tunnel Face Under Different Groundwater Levels

The construction of shield tunnels is greatly influenced by the presence of groundwater. Based on the upper bound theorem of limit analysis, this paper presents an analytical investigation on the stability of shield tunnel face subjected to the impact of pore water pressure. Considering the nonlinear strength feature of soil mass, the nonlinear failure criterion of power-law type is employed, and the tangential technique is utilized to determine its equivalent Mohr–Coulomb strength parameters. The log-spiral failure mechanism for shield tunnel face is constructed, and three different cases of the relation between groundwater level and this mechanism are summarized. After that, rates of work of external forces and rate of internal energy dissipation are computed and analytical expression for the supporting pressure is deduced. Numerical solution with respect to specified parameters is calculated through optimization. Parametric analysis is performed and the variation of required supporting pressure with increasing height of groundwater level is presented.

Effects of nonlinear strength parameters on stability of 3D soil slopes

Actual slope stability problems have three-dimensional (3D) characteristics and the soils of slopes have curved failure envelopes. This incorporates a power-law nonlinear failure criterion into the kinematic approach of limit analysis to conduct the evaluation of the stability of 3D slopes. A tangential technique is adopted to simplify the nonlinear failure criterion in the form of equivalent Mohr-Coulomb strength parameters. A class of 3D admissible rotational failure mechanisms is selected for soil slopes including three types of failure mechanisms: face failure, base failure, and toe failure. The upper-bound solutions and corresponding critical slip surfaces can be obtained by an efficient optimization method. The results indicate that the nonlinear parameters have significant influences on the assessment of slope stability, especially on the type of failure mechanism. The effects of nonlinear parameters appear to be pronounced for gentle slopes constrained to a narrow width. Compared with the solutions derived from plane-strain analysis, the 3D solutions are more sensitive to the values of nonlinear parameters.

Limit analysis of 3D rock slope stability with non-linear failure criterion

The non-linear Hoek-Brown failure criterion has been widely accepted and applied to evaluate the stability of rock slopes under plane-strain conditions. This paper presents a kinematic approach of limit analysis to assessing the static and seismic stability of three-dimensional (3D) rock slopes using the generalized Hoek.Brown failure criterion. A tangential technique is employed to obtain the equivalent Mohr-Coulomb strength parameters of rock material from the generalized Hoek-Brown criterion. The least upper bounds to the stability number are obtained in an optimization procedure and presented in the form of graphs and tables for a wide range of parameters. The calculated results demonstrate the influences of 3D geometrical constraint, non-linear strength parameters and seismic acceleration on the stability number and equivalent strength parameters. The presented upper-bound solutions can be used for preliminary assessment on the 3D rock slope stability in design and assessing other solutions from the developing methods in the stability analysis of 3D rock slopes.

비선형 암석 파괴조건식의 접선 마찰각과 점착력의 중간주응력 의존성

Although Mohr-Coulomb failure criterion has limitations in that it is a linear criterion and the effect of the intermediate principal stress on failure is ignored, this criterion has been widely accepted in rock mechanics design. In order to overcome these shortcomings, the Hoek-Brown failure criterion was introduced and recently a number of 3-D failure criteria incorporating the effect of the intermediate principal stress on failure have been proposed. However, in many rock mechanics designs, the possible failure of rock mass is still evaluated based on Mohr-Coulomb criterion and most of practitioners are accustomed to understanding the strength of rock mass in terms of the internal friction angle and cohesion. Therefore, if the equivalent Mohr-Coulomb strength parameters of the advanced failure criteria are calculated, it is possible to take advantage of the advanced failure criteria in the framework of the Mohr-Coulomb criterion. In this study, a method expressing the tangential Mohr-Coulomb strength parameters in terms of the stress invariant is proposed and it is applied to the generalized Hoek-Brown criterion and the HB-WW criterion. In addition, a new approach describing the geometric meaning of the -dependency of failure criteria in 3-D principal stress space is proposed. Implementation examples of the proposed method show that the influence of the intermediate principal stress on the tangential friction angle and cohesion of the HB-WW criterion is considerable, which is not the case for the 2-D failure criterion.

The feasibility analysis of underground gas storage caverns

This paper presents the fuzzy concept for the feasibility analysis of Underground Gas Storage (UGS) designed from Lined Rock Caverns (LRC). For this purpose, it is necessary to carry out a number of steps. The first is the determination of the geological model of the UGS region. The next is transferring of the geological model into the geomechanical model, which could be achieved by applying the system of Hoek and Brown based on the geological strength index (GSI). The equivalent Mohr–Coulomb strength parameters should be determined. The geomechanical analysis for risk during construction and later during operation of UGS using Finite Element Method (FEM) is carried out. The series of FEM analyses should be calculated for the determination of changes in the response functions due to changes in the design variables. The results of FEM analyses are given in table form. The cost of UGS is calculated with developed equations. The construction costs comprise the investment and operational costs of the UGS system. The total construction costs per unit of gas include the sum of fixed costs and variable costs. Once the results of FEM and cost analysis are obtained, we use those solutions to develop ANFIS models. In this paper the ANFIS model for cost prediction is described in detail and the similar procedure is used to develop geomechanical constrain functions. These models are the basis for the feasibility analysis of underground gas storage caverns.

응력불변량으로 표현한 일반화된 Hoek-Brown 파괴조건식의 등가 마찰각 및 점착력

Implementing the generalized Hoek-Brown failure criterion in the framework of the Mohr-Coulomb criterion requires the calculation of the equivalent friction angle and cohesion. In the conventional method based on the Balmer (1952)'s theory, the tangential instantaneous friction angle and cohesion are expressed in terms of the minimum principal stress   , which does not provide the information about the dependency of the equivalent parameters on the hydrostatic pressure and the stress path. In this study, this defect of the conventional method has been overcome by representing the equivalent parameters in terms of stress invariants. Through the example implementation of the new method, the influence of the magnitude of the hydrostatic pressure and the Lode angle on the tangential instantaneous friction angle and cohesion is investigated. It turns out that the tangential instantaneous friction angle is maximum when the stress condition is triaxial extension, while the tangential cohesion is maximum when the stress condition is triaxial compression. The dependency of the equivalent Mohr-Coulomb strength parameters on the hydrostatic pressure and the Lode angle tends to be more substantial for the favorable rockmass of larger GSI value.

Seismic rock slope stability charts based on limit analysis methods

Earthquake effects are commonly considered in the stability analysis of rock slopes and other earth structures. The standard approach is often based on the conventional limit equilibrium method using equivalent Mohr–Coulomb strength parameters ( c and ϕ) in a slip circle slope stability analysis. The purpose of this paper is to apply the finite element upper and lower bound techniques to this problem with the aim of providing seismic stability charts for rock slopes. Within the limit analysis framework, the pseudo-static method is employed by assuming a range of the seismic coefficients. Based on the latest version of Hoek–Brown failure criterion, seismic rock slope stability charts have been produced. These chart solutions bound the true stability numbers within ±9% or better and are suited to isotropic and homogeneous intact rock or heavily jointed rock masses. A comparison of the stability numbers obtained by bounding methods and the limit equilibrium method has been performed where the later was found to predict unconservative factors of safety for steeper slopes. It was also observed that the stability numbers may increase depending on the material parameters in the Hoek–Brown model. This phenomenon has been further investigated in the paper.

Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto-plastic rock masses

We present a novel methodology for estimation of equivalent Mohr–Coulomb strength parameters that can be used for design of supported tunnels in elasto-plastic rock masses satisfying the non-linear empirical Hoek–Brown failure criterion. We work with a general adimensional formulation of the Hoek–Brown failure criterion in the space of normalized Lambe's variables for plane stress, and we perform linearization considering the stress field in the plastic region around the tunnel. The procedure is validated using analytical solutions to a series of benchmark test cases. Numerical solutions are also employed to validate the procedure in cases for which analytical solutions are not available. Results indicate that the stress field in the plastic region around the tunnel, as well as the linearization method employed and the quality of the rock mass, has a significant impact on computed estimates of equivalent Mohr–Coulomb strength parameters. Results of numerical analyses also show that our proposed linearization method can be employed to estimate loads and moments on the tunnel support system. We recommend the equating model responses (EMR) method to compute equivalent Mohr–Coulomb strength parameters when the tunnel support pressure is accurately known, and we further show that our newly introduced linearization method can be employed as an alternative to the best fitting in the existing stress range (BFe) and best fitting in an artificial stress range (BFa) methods, providing performance estimates that are generally better than estimates of the BFe and BFa methods when differences with the response of the Hoek–Brown rock mass are of engineering significance (say more than 10%).

Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system

Abstract Rock mass characterization is required for many applications in rock engineering practice including excavation design, support design, stope design, amongst others. For these purposes, it is necessary to obtain design input parameters such as deformation moduli and strength parameters for numerical modeling. Although such parameters can ultimately be determined from in situ tests, at the preliminary design stage, where access to underground is limited, the practical way to obtain these parameters is to apply a rock mass classification system to characterize the rock mass and estimate the rock mass properties. Over the years, many classification systems, such as RQD, Rock Mass Rating, Q and Geological Strength Index (GSI) systems, have been developed. Amongst them, the Q system is widely used for rock support system design and the GSI system is used for estimating design parameters. The GSI system is the only rock mass classification system that is directly linked to engineering parameters such as Mohr–Coulomb, Hoek–Brown strength parameters or rock mass modulus. However, the application of the existing GSI system is hindered by the facts that the use of the system is to some extent subjective and requires long-term experience. In the present study, a quantitative approach to assist in the use of the GSI system is presented. It employs the block volume and a joint condition factor as quantitative characterization factors. The approach is built on the linkage between descriptive geological terms and measurable field parameters such as joint spacing and joint roughness. The newly developed approach adds quantitative means to facilitate use of the system, especially by inexperienced engineers. The GSI system is applied to characterize the jointed rock masses at two underground powerhouse cavern sites in Japan. GSI values are obtained from block volume and joint condition factor, which in turn are determined from site construction documents and field mapping data. Based on GSI values and intact rock strength properties, equivalent Mohr–Coulomb strength parameters and elastic modulus of the jointed rock mass are calculated and compared to in situ test results. The point estimate method is implemented to approximate the mean and variance of the mechanical properties of the jointed rock masses. It is seen that both the means and variances of strength and deformation parameters predicted from the GSI system are in good agreement with field test data.