Youn-Kyou Lee

Instantaneous Friction Angle and Cohesion of 2-D and 3-D Hoek–Brown Rock Failure Criteria in Terms of Stress Invariants

The Mohr–Coulomb (M–C) failure criterion is one of the most widely used failure criteria in rock mechanics, although it has a number of shortcomings such as neglecting the nonlinear strength observed in rock or the effect of the intermediate principal stress σ2. Other failure criteria have been proposed to effectively include in the predictions of failure the non-linear response of rock to confinement or the effects of the intermediate principal stress. The M–C criterion is still widely used, and it is arguably the criterion most used in practice. For example, stability evaluations of shallow rock structures such as slopes and foundations are routinely carried out by estimating a friction angle and a cohesion of the rock mass. To include the dependency of cohesion and friction angle on stresses, efforts are being made to estimate equivalent values of the M–C parameters for the range of stresses applicable to a particular design. The paper suggests a new and convenient approach to find the equivalent friction angle and cohesion from any failure criterion that can be expressed in terms of the Nayak and Zienkiewicz’s stress invariants. To demonstrate the capabilities and application of the methodology, the new approach is applied to two failure criteria: the Hoek–Brown (H–B) criterion and the Hoek–Brown and Willam–Warnke (HB–WW) criterion, 2-D and 3-D failure criteria, respectively. Results from the new method, in terms of equivalent friction and cohesion for the H–B criterion, are exactly the same as the results obtained from Balmer’s theory, which confirms the validity of the new method. The predicted equivalent friction and cohesion for the HB–WW criterion show a dependency on σ2, which does not occur for a 2-D failure criterion.

Derivation of Mohr Envelope of Hoek-Brown Failure Criterion Using Non-Dimensional Stress Transformation

In the course of performing the stability analysis of rock structures, there are times when the strength of the Hoek-Brown rock mass needs to be understood in terms of the internal friction angle and cohesion. In this case, the original Hoek-Brown criteion, giving the relationship between  and  at failure, have to be transformed to the corresponding Mohr envelope. A new approach to derive the Mohr envelope of the Hoek-Brown criterion is suggested in this study. The new method is based on the Londes transformation making the stress components dimensionless. The correctness of the derivation leading to the new   relationship is confirmed by comparing the calculation results with the Brays solution through a verification example.

Equivalent Friction Angle and Cohesion of the Generalized Hoek-Brown Failure Criterion in terms of Stress Invariants

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.

Comparative Study on the Rock Failure Criteria Taking Account of the Intermediate Principal Stress

Although the Mohr-Coulomb and Hoek-Brown failure criteria have been adopted widely in rock mechanics, they neglect the σ2 effect. The result of true triaxial tests on rock samples, however, reveals that the σ2 effect on strength of rocks is considerable, so that rock failure criteria taking into account the influence of σ2 are necessary for the precise stability evaluation of rock structures. In this study, a new nonlinear 3-D failure criterion has been suggested by combining the Hoek-Brown criterion with the smooth octahedral shape function taken from Jiang & Pietruszczak (1988). The performance of the new criterion was assessed by comparing the strength predictions from both the suggested criterion and the corresponding linear 3-D criterion. The resulting fit of the new criterion to the true triaxial test data for six rock types taken from the literature shows that the criterion fits the experimental data very well. Furthermore, for the data sets having data taken in the low σ3 range, the nonlinear failure criterion works better than the linear criterion.

Simulation of polyaxial tests using a failure condition for transversely isotropic rocks

Considering that most sedimentary and metamorphic rocks display some degree of anisotropy in failure strength, the conventional isotropic failure criteria including the Mohr-Coulomb (M-C) and the Hoek-Brown have their limitations in characterizing the strength of these anisotropic rocks. An improved understanding of the orientation dependency of strength in anisotropic rocks can be achieved if a systematic way of establishing a condition at failure in anisotropic rock is provided. In this research, a general procedure to extend the M-C failure criterion to its anisotropic version is presented in which the friction angle and cohesion are not constant but defined as functions of the relative orientation of physical plane to the principal material triad. In the formulation of the anisotropic failure condition, the directional variation of the strength parameters is described by incorporating a traceless symmetric second-rank fabric tensor that defines the orientation bias in their spatial distribution. Subsequently, the critical plane approach is employed to implement the formulated transversely isotropic M-C failure function in the numerical polyaxial tests on the transversely isotropic rock samples, whereby the polyaxial stress at failure and the corresponding direction of the failure plane are specified by maximizing the failure function with respect to the orientation. A series of numerical polyaxial tests is conducted to investigate how the orientation of weak planes affects the failure behavior of transversely isotropic rocks. The simulation result strongly suggests that the microstructures embedded in the samples and their attitude with respect to the loading direction have much influence on the variation of the polyaxial strength and the orientation of the corresponding failure plane.

Relationship between Tangential Cohesion and Friction Angle Implied in the Generalized Hoek-Brown Failure Criterion

The generalized Hoek-Brown (H-B) function provides a unique failure condition for a jointed rock mass, in which the strength parameters of rock mass are deduced from the intact values by use of the GSI value. Since it is actually the only failure criterion which accounts for the rock mass conditions in a systematic manner, the generalized H-B criterion finds many applications to the various rock engineering projects. Its nonlinear character, however, limits more active usage of this criterion. Accordingly, many attempts have been made to understand the generalized H-B condition in the framework of the M-C function. This study presents the closed-form expression relating the tangential cohesion to the tangential friction angle, which is derived by the non-dimensional stress transformation of the generalized H-B criterion. By use of the derived equation, it is investigated how the relationship between the tangential cohesion and friction angle of the generalized H-B criterion varies with the quality of rock masses. When only the variation of GSI value is considered, it is found that the tangential friction angle decreases with the increase of GSI, while the tangential cohesion increases with GSI value.

Dependency of Tangential Friction Angle and Cohesion of Non-linear Failure Criteria on the Intermediate Principal Stress

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.

Spatial Distribution Functions of Strength Parameters for Simulation of Strength Anisotropy in Transversely Isotropic Rock

Abstract This study suggests three spatial distribution functions of strength parameters, which can be adopted in the derivation of failure conditions for transversely isotropic rocks. All three proposed functions, which are the oblate spheroidal function, the exponential function, and the function based on the directional projection of the strength parameter tensor, consist of two model parameters. With assumption that the cohesion and friction angle can be described by the proposed distribution functions, the transversely isotropic Mohr-Coulomb criterion is formulated and used as a failure condition in the simulation of the conventional triaxial tests. The simulation results confirm that the failure criteria incorporating the proposed distribution functions could reproduce the general trend in the variations of the axial stress at failure and the directions of failure planes with varying inclination of the weankness planes and confining pressure. Among three distribution functions, the function based on the directional projection of the strength parameter tensor yields the highest axial strength, while the axial strength estimated by the oblate spheroidal distribution function is the lowest.Key words Transversely isotropy, Failure criteria, Cohesion, Friction angle, Direction of failure plane초 록 이 연구에서는 횡등방성 암석파괴함수의 개발에 활용할 수 있는 3가지 강도정수 공간분포함수를 제안하였다. 제안된 분포함수는 편구(oblate spheroid)분포함수, 지수분포함수, 강도정수텐서 방향투영함수이며 모두 2개의 모델파라미터로 정의된다 . 제안된 분포함수들을 점착력과 마찰각의 공간분포함수로 활용하여 횡등방성 Mohr-Coulomb 파괴함수를 유도한 후 이를 활용하여 수치삼축시험을 모사하였다 . 연약면의 경사각과 구속압의 변화에 따른 파괴축응력 변화 및 파괴면 방향 변화를 계산한 결과 3개의 분포함수을 적용한 경우 모두 실제 실험에서 관찰되는 이방성 파괴특성을 재현하고 있음을 확인하였다 . 3개의 분포함수 중 강도정수텐서 방향투영함수를 채용한 경우가 가장 큰 파괴축강도를 계산하였으며 지수분포함수 , 편구분포함수 순으로 낮은 파괴축강도 값을 예측하였다.핵심어 횡등방성, 파괴조건식, 점착력, 마찰각, 파괴면의 방향

Estimation of Equivalent Friction Angle and Cohesion of Near-Surface Rock Mass Using the Upper-Bound Solution for Bearing Capacity of Strip Footing

The generalized Hoek-Brown failure criterion, the strength parameters of which are determined by using the GSI index, is an empirical nonlinear failure criterion of rock mass and has been widely employed in various rock engineering practices. Many rock engineering practitioners, however, are still familiar with the description of the strength of rock mass in terms of friction angle and cohesion. In addition, almost all rock mechanics softwares incorporate the simple linear Mohr-Coulomb function. Therefore, it is necessary to provide a tool to implement the Hoek-Brown function in the framework of the Mohr-Coulomb criterion. In this study, the use of upper-bound solution of limit analysis for bearing capacity of a strip footing resting on the ground surface is proposed for the estimation of the equivalent friction angle and cohesion of rock mass incorporating the generalized Hoek-Brown failure criterion. The upper-bound bearing capacity is expressed in terms of friction angle by use of the relationship between tangential friction angle and tangential cohesion implied in the generalized Hoek-Brown function. The friction angle minimizing the upper-bound bearing capacity is taken as the equivalent friction angle. Through the illustrative implementations of the proposed method, the influences of GSI,  and D on the equivalent friction angle and cohesion are investigated.

Intermediate Principal Stress Dependency in Strength of Transversely Isotropic Mohr-Coulomb Rock

A number of true triaxial tests on rock samples have been conducted since the late 1960 and their results strongly suggest that the intermediate principal stress has a considerable effect on rock strength. Based on these experimental evidence, various 3-D rock failure criteria accounting for the effect of the intermediate principal stress have been proposed. Most of the 3-D failure criteria, however, are focused on the phenomenological description of the rock strength from the true triaxial tests, so that the associated strength parameters have little physical meaning. In order to confirm the likelihood that the intermediate principal stress dependency of rock strength is related to the presence of weak planes and their distribution to the preferred orientation, true triaxial tests are simulated with the transversely isotropic rock model. The conventional Mohr-Coulomb criterion is extended to its anisotropic version by incorporating the concept of microstructure tensor. With the anisotropic Mohr-Coulomb criterion, the critical plane approach is applied to calculate the strength of the transversely isotropic rock model and the orientation of the fracture plane. This investigation hints that the spatial distribution of microstructural planes with respect to the principal stress triad is closely related to the intermediate principal stress dependency of rock strength.