Nonlinear Hoek Brown Failure Criterion Research Articles

Analysis and Experimental Study on the Stability of Large-Span Caverns’ Surrounding Rock Based on the Progressive Collapse Mechanism

The collapse failure of rock surrounding caverns involves a progressive collapse process. Based on the nonlinear Hoek–Brown failure criterion and the upper limit theorem, the whole process curve of the progressive collapse of the surrounding rock of a large-span cavern is outlined in this paper. The progressive collapse process of the surrounding rock of the large-span cavern is experimentally studied using an independently developed visualized large-span-cavern geomechanical model test device with variable angles. The results show that, through theoretical calculation and model tests, the surrounding rock at the top of the large-span cavern undergoes three collapses. Under the condition of rock mass and the shape of the cavern, the larger the span of the cavern, the more times the surrounding rock collapses; with the increase in surrounding rock pressure, the first collapse occurs in the middle part of the arch roof. When the overlying load reaches a certain level, the arch foot becomes the weakest part, and the rock undergoes shear failure along the arch foot, gradually extending upwards, accompanied by multiple collapses, forming a progressive collapse process. The theoretical calculation results of this paper are basically consistent with the scope of the model test, and the research results can provide a basis for the construction and support design of the large-span cavern.

An Improved Incremental Procedure for the Ground Reaction Based on Hoek-Brown Failure Criterion in the Tunnel Convergence-Confinement Method

The purpose of this study is to introduce the Hoek–Brown nonlinear failure criterion into the convergence–confinement method (CCM), which is based on the linear failure criterion, and implant it into the ground reaction curve (GRC) in this theory. Based on consistent and rigorous theoretical analysis and according to the mechanical relationships, the principle of equilibrium, the material constitutive law, and the strain/displacement compatibility equation are used to deduce the closed-form analytical solution of the stress/displacement of the rock mass in the plastic region and to complete the simulation and interpretation of the nonlinear behavior of surrounding rock due to the advancing excavation of a circular tunnel. In this paper, confinement loss is regarded as the concept of increment in numerical analysis, and the calculation process and analysis steps of the explicit analysis method (EAM) are proposed. This method can not only realize the calculation of analytical solutions but can also be performed using simple spreadsheets. In addition, to verify the feasibility of this calculation method, this study uses published data to verify the validity of the analytical solution implemented by the incremental procedure and to study the influence of nonlinear failure criteria on the GRC, especially the mechanical behavior at the intrados of the tunnel and the distributions of tress/displacement on the cross-section of the tunnel. A comparison between the published results and the proposed solution in this study reveals consistent and favorable trends.

Influence of Anisotropic Stress Conditions on Tunnel Deformation and Sequential Excavation Performance in Rock Mass

Abstract This study looks at how anisotropic stress conditions affect the mechanical response of circular tunnels in rock mass using a finite element plane strain model. The deformation response and evolution of the yield zone are evaluated using a non-linear Hoek-Brown failure criterion. The results show that predictions made under the widely employed assumption of isotropic stress conditions significantly underestimate the deformations for in-situ stress ratios greater than 1.0 and overestimate the values for in-situ stress ratios less than 1 under full-face excavation. The pattern of the yield zone changes noticeably as the in-situ stress ratio changes. With stress anisotropy, the yield zone becomes asymmetric. When the in-situ stress ratio is less than 1.0, the yield zone concentrates on the sidewalls and transitions to the crown when the in-situ stress ratio assumes values greater than 1.0. Also, the effect of anisotropic stress conditions on the performance of two commonly used excavation support schemes is studied. The results emphasize the importance of consideration of the in-situ stress ratio for prediction of tunnel deformation and selection of sequential excavation schemes.

Stability Analysis of the Horseshoe Tunnel Face in Rock Masses.

Accurately estimating the stability of horseshoe tunnel faces remains a challenge, especially when excavating in rock masses. This study aims to propose an analytical model to analyze the stability of the horseshoe tunnel face in rock masses. Based on discretization and “point-by-point” techniques, a rotational failure model for horseshoe tunnel faces is first proposed. Based on the proposed failure model, the upper-bound limit analysis method is then adopted to determine the limit support pressure of the tunnel face under the nonlinear Hoek–Brown failure criterion, and the calculated results are validated by comparisons with the numerical results. Finally, the effects of the rock properties on the limit support pressure and the 3D failure surface are discussed. The results show that (1) compared with the numerical simulation method, the proposed method is an efficient and accurate approach to evaluating the face stability of the horseshoe tunnel; (2) from the parametric analysis, it can be seen that the normalized limit support pressure of the tunnel face decreases with the increasing of geological strength index, GSI, Hoek–Brown coefficient, mi, and uniaxial compressive strength, σci, and with the decreasing of the disturbance coefficient of rock, Di; and (3) a larger 3D failure surface is associated with a high value of the normalized limit support pressure.

Stability Assessment of Rock Mass System under Multiple Adjacent Structures

Numerical modeling is important for exploring the fundamental processes occurring in rock and for evaluating the real performance of structures built on and in rock mass system, and thus for supporting the design of rock engineering problems. Estimating the stability of rock mass foundation systems entirely based on a theoretical approach is a complicated task if there exists overlapping of their potential collapse modes. This paper applies finite element limit analysis to evaluate the bearing capacity of equally spaced multiple strip footings resting on rock mass obeying the modified non-linear Hoek–Brown failure criterion. Numerical solutions are expressed in terms of the efficiency factor that is dependent on the spacing between footings, as well as the rock mass properties. In addition, the effects of surface surcharge and footing roughness are quantified. The maximum spacing at which the interfering effect of adjacent footings becomes disappeared is evaluated and an algebraic expression for approximating the maximum spacing is proposed. Failure mechanisms for a few cases of rock mass under multiple strip footings are examined.

Estimation of the Critical Seismic Acceleration for Three-Dimensional Rock Slopes

An analytical approach for the estimating of critical seismic acceleration of rock slopes was proposed in this study. Based on the 3D horn failure model, the critical seismic acceleration coefficient of rock slopes was conducted with the modified Hoek–Brown (MHB) failure criterion in the framework of upper-bound theory for the first time. The nonlinear Hoek–Brown failure criterion is incorporated into the three-dimensional rotational failure mechanism, and a generalized tangent technique is introduced and employed to convert the nonlinear Hoek–Brown failure criterion into a linear criterion. The critical seismic acceleration coefficients obtained from this study were validated by the numerical simulation results based on finite element limit analysis. The agreement showed that the proposed method is effective. Finally, design charts were provided for exceptional cases for practical use in rock engineering.

Upper bound solution for the bearing capacity of two adjacent footings on rock masses

In this paper, the effect of the adjacency of two strip footings on bearing capacity of rock masses was investigated using the upper bound analysis and finite element method. The modified Hoek-Brown nonlinear failure criterion was used for the rock mass. The effect of various parameters such as the distance between the two footings, Hoek-Brown parameters, surcharge pressure and unit weight of the rock mass on the ultimate bearing capacity of rock masses were also investigated. The results show that if the distance between the two footings is significant, the failure lines occur beneath each footing will not interfere with the failure lines beneath the other adjacent footing and each footing behaves as a single one. The maximum distance for which the two footings affect each other can be obtained from the charts presented in this paper. By reducing the distance between the two footings, the bearing capacity will increase firstly and then decrease. Finally, when the two footings touch each other, the ultimate bearing capacity of the two footings will be equal to the ultimate bearing capacity of a single isolated footing with a width twice the width of each footing.

A New Procedure for Calculation of Ground Response Curve of a Circular Tunnel Considering the Influence of Young’s Modulus Variation and the Plastic Weight Loading

In this paper, an analytical–numerical solution is presented for a circular tunnel in the hydrostatic condition using a generalized nonlinear Hoek–Brown failure criterion. To a deep insight into the problem, the influence of the plastic zone weight, and the Young's modulus on the ground response curve of a deep tunnel in a strain-softening media is highlighted. Experimental results show that the magnitude of the confined stress can have a direct effect on the critical softening parameter. Thus, a variable critical softening parameter is also assumed. That is, as the radial stress magnitude increases, the critical softening parameter is enhanced. The equations are derived for different directions (e.g. crown, floor, and sidewall) around the circular tunnel. The accuracy of the proposed method is validated by some examples. The results show that assuming a constant Young's modulus in the plastic zone imposes a significant error in the predicting of the tunnel convergence.

Excavation and kinematic analysis of a shallow large-span tunnel in an up-soft/low-hard rock stratum

The excavation of a shallow large-span metro tunnel in an up-soft/low-hard rock stratum restricted by a rock mass requires knowledge of the stratum kinematic mechanism to adopt an appropriate construction method to control the surface settlement and ensure structural safety. The purpose of a kinematic analysis is to predict the state of stratum collapse based on the nonlinear Hoek-Brown failure criterion and the upper bound theorem. The rock mass deformation and the supporting structural mechanics characteristics of the tunnel under different excavation methods are studied using numerical simulations. The Xinggongjie Station tunnel on the No. 1 Line of the Dalian Metro is used as an example. Furthermore, the ground surface settlement, the internal displacement of the surrounding rock mass and the stress on the supporting structure are monitored during the construction period. The accuracy of the theoretical kinematic model and the rationality of the selected excavation method are verified by comparing the theoretical results with the field monitoring results. The study shows that the slip failure surface and ground settlement curve can be accurately determined using the kinematic model. Different excavation sequences result in differences in the surface subsidence, and the most favourable sequence for restricting the subsidence is to excavate the arch-side pilot first and the middle pilot last. The rock mass deformation and the surface subsidence are acceptable according to field monitoring under this excavation sequence. The load ratios of the surrounding rock mass borne by the initial support, which is the primary bearing structure of the tunnel, at different monitoring points are greater than 80%. The proportion of the secondary lining is approximately 20%, which is the safe reserve of the tunnel. The tunnel has been demonstrated to be safe; it has been placed into service and has operated well.

Ultimate Bearing Capacity of Rock Mass Foundations Subjected to Seepage Forces Using Modified Hoek–Brown Criterion

The experimental data shows that most rocks behave nonlinearly in nature. The modified nonlinear Hoek–Brown failure criterion was considered to investigate the bearing capacity problem of shallow rigid foundations on rock masses subjected to horizontal seepage forces. Two multi-wedge translational failure mechanisms, including symmetrical and non-symmetrical mechanisms were used in the closed-form of the upper bound method of the limit analysis theory. The symmetrical failure mechanism was used in the case of no seepage, while the seepage effect was considered in the non-symmetrical mechanism. The variation of seepage forces was obtained as a function of gradient ratio i(γw/γsub) in the developed formulation. The bearing capacity coefficients Nγ, Nq and Nσ are introduced for the case of seepage flow condition. The results show that the magnitude of the bearing capacity coefficients reduces continuously with an increase in the value of gradient ratio i(γw/γsub). The obtained results were compared and offered for functional use in foundation engineering.

Upper Bound Solution of the Safety Factor for a Shield Tunnel Face Subjected to the Hoek–Brown Failure Criterion

Assessment of the stability of a tunnel face under a setting support pressure is a key challenge in shield tunneling engineering, especially when a tunnel is excavated in a heavily fractured rock mass. In this paper, a factor of safety is introduced for investigating the face stability of a tunnel using the upper bound theorem in combination with the Hoek–Brown nonlinear failure criterion. The factor of safety for the tunnel face is defined as the ratio of the rate of the energy dissipation and the external rate of work in the kinematically admissible velocity field. The upper bound solution of the factor of safety is obtained from the optimal computation. The solutions provided by the presented approach are compared with those derived from a numerical simulation. The values of the factor of safety derived from these two methods are found to be in agreement, indicating that the proposed method is valid. Furthermore, based on the measured data of the chamber pressure and the parameters provided by the geological survey report, the proposed method is used to evaluate the stability of the tunnel face for an actual project. The effects of different parameters on the factor of safety are also discussed.

Influence Zone Division and Risk Assessment of Underwater Tunnel Adjacent Constructions

Compared with ordinary tunnels, the influence analysis of underwater tunnel adjacent constructions is more complicated. At present, the empirical method used to divide the influence zone of the tunnel adjacent constructions has great uncertainty. So it is of great significance for actual construction and design to determine the influence zone accurately according to theoretical calculations. In this paper, based on the Hoek–Brown nonlinear failure criterion of rock mass and taking seepage factor into account, the stress state of rock mass around the underwater tunnel adjacent constructions can be deduced by elastoplastic theory. Then combined with the concept of “loose zone-bearing zone”, the influence zone division method of underwater tunnel adjacent constructions is proposed, and it is applied to the analysis of engineering examples. Through the deduced theoretical formulas, the influence zone of underwater tunnel adjacent construction can be divided into extensively strong, strong, relatively strong, weak, and noninfluence zones. Corresponding the influence zones with the risk levels in the code, different control measures are adopted for different risk levels, which can provide certain guidance for the design and construction of tunnel in practical engineering.

Upper Bound Solution of Surrounding Rock Pressure for Deep Cavity Using Nonlinear Hoek–Brown Failure Criterion

The linear failure criterion is generally adopted in the stability problem of geotechnical engineering, whereas the experiments have indicated that there is a nonlinear relationship between the maximum and minimum principal stresses in weak surrounding rock. According to the characteristics of weak rock, the failure mechanism of deep cavity is constructed by combining the nonlinear Hoek–Brown failure criterion and the upper bound theorem of limit analysis. The upper bound solution of the surrounding rock pressure was deduced using the tangent method. The results show that the surrounding rock pressure of deep cavity is affected by surrounding rock grade, cavity depth, and section size. Especially, the influence of the disturbance factor is quite obvious. Upper bound solution based on nonlinear Hoek–Brown failure criterion can fully take the influence of parameters on surrounding rock pressure. Therefore, this method is more scientific than the linear failure criterion to calculate the surrounding rock pressure.

Application of Flattened Brazilian Test to Investigate Rocks Under Confined Extension

A failure envelope that ignores the intermediate principal stress is typically adequate for the design of rock structures. To obtain the complete envelope, the rock cylinders should be tested using confined compression as well as confined extension tests. While confined compression tests are common, confined extension tests are rarely carried out. Current techniques available to test rocks under confined extension conditions require shaping the cylindrical specimens to a dog-bone geometry. The limited data available in the literature indicates that the results from these dog-boned shaped tests produces strengths that are considerably greater than the confined triaxial tests carried out on traditional cylindrical specimens. Whether this increased strength is real or simply an artifact of the stress path for triaxial confined extension tests (intermediate principal stress equal to the maximum principal stress) versus confined compression tests (intermediate principal stress equal to the minimum principal stress) is unknown. We propose here a new methodology to test rocks under confined extension condition with zero intermediate principal stress. We used the flattened Brazilian test and calculated the major and minor principal stresses from the strain on the surface of the specimens. To test the rock under increased confined conditions, we tested specimens with increasing depth of flattening. We derived equations to calculate the major and minor principal stresses in a flattened Brazilian specimen considering the bi-modularity in the stress–strain equations. We tested Lac du Bonnet granite specimens for major principal stress range 15–37% of its UCS and observed that the minor principal stresses remains in the range of Brazilian strength of the material (11.6 MPa). The results suggest that the nonlinear Hoek–Brown failure criterion based on only compressive triaxial results provides a reasonable estimate of the tensile strength.

Kinematic analysis of shallow tunnel in layered strata considering joined effects of settlement and seepage

The purpose of this work is to predict the state of collapse in shallow tunnel in layered strata by using a new curved failure mechanism within the framework of upper bound theorem. Particular emphasis is first given to consider the effects of seepage forces and surface settlement. Furthermore, the Hoek-Brown nonlinear failure criterion is adopted to analyze the influence of different factors on the collapsing shape. Two different curve functions which describe two different rock layers are obtained by virtual work equations under the variational principle. According to the numerical results, the parameter B in Hoek-Brown failure criterion and the unit weights in different rock layers have a positive relationship with the size of collapsing block while pore pressure coefficient and the parameter A in Hoek-Brown failure criterion present a reverse tend.

Reliability analysis of three-dimensional rock slope

Reliability analysis is generally regarded as the most appropriate method when uncertainties are taken into account in slope designs. With the help of limit analysis, probability evaluation for three-dimensional rock slope stability was conducted based upon the Mote Carlo method. The nonlinear Hoek-Brown failure criterion was employed to reflect the practical strength characteristics of rock mass. A form of stability factor is used to perform reliability analysis for rock slopes. Results show that the variation of strength uncertainties has significant influence on probability of failure for rock slopes, as well as strength constants. It is found that the relationship between probability of failure and mean safety factor is independent of the magnitudes of input parameters but relative to the variability of variables. Due to the phenomenon, curves displaying this relationship can provide guidance for designers to obtain factor of safety according to required failure probability.

Stability Charts for Rock Slopes Subjected to Water Drawdown Based on the Modified Nonlinear Hoek-Brown Failure Criterion

Stability Charts for Rock Slopes Subjected to Water Drawdown Based on the Modified Nonlinear Hoek-Brown Failure Criterion

Reliability analysis of shallow tunnel with surface settlement

Based on the reliability theory and limit analysis method, the roof stability of a shallow tunnel is investigated under the condition of surface settlement. Nonlinear Hoek-Brown failure criterion is adopted in the present analysis. With the consideration of surface settlement, the internal energy and external work are calculated. Equating the rate of energy dissipation to the external rate of work, the expression of support pressure is derived. With the help of variational approach, a performance function is proposed to reliability analysis. Improved response surface method is used to calculate the Hasofer-Lind reliability index and the failure probability. In order to assess the validity of the present results, Monte-Carlo simulation is performed to examine the correctness. Sensitivity analysis is used to estimate the influence of different variables on reliability index. Among random variables, the unit weight significantly affects the reliability index. It is found that the greater coefficient of variation of variables lead to the higher failure probability. On the basis of the discussions, the reliability-based design is achieved to calculate the required tunnel support pressure under different situations when the target reliability index is obtained.

Roof collapse of shallow tunnel in layered Hoek-Brown rock media

Collapse shape of tunnel roof in layered Hoek-Brown rock media is investigated within the framework of upper bound theorem. The traditional collapse mechanism for homogeneous stratum is no longer suitable for the present analysis of roof stability, and it would be necessary to propose a curve failure mode to describe the velocity discontinuity surface in layered media. What is discussed in the paper is that the failure mechanism of tunnel roofs, consisting of two different functions, is proposed for layered rock media. Then it is employed to investigate the impending roof failure. Based on the nonlinear Hoek-Brown failure criterion, the collapse volume of roof blocks are derived with the upper bound theorem and variational principle. Numerical calculations and parametric analysis are carried out to illustrate the effects of different parameters on the shape of failure mechanism, which is of overriding significance to the stability analysis of tunnel roof in layered rock media.

Minimum safe thickness of rock plug in karst tunnel according to upper bound theorem

High pressure and water-bearing caverns ahead of a karst tunnel face tend to cause geological disasters, such as water and mud bursts. So, the determination of safe thickness of the reserved rock plug is a key technical problem to be solved for karst tunnel construction. Based on the Hoek-Brown nonlinear failure criterion, the minimum safe thickness of rock plug was investigated in the light of the limit analysis theory. On the basis of the proposed failure mode, the expression of the minimum thickness for rock plug was obtained by means of upper bound theorem in combination with variational principle. The calculation results show the influence of each parameter on safe thickness and reveal the damage range of rock plug. The proposed method is verified by comparing the results with those of the drain cavern of Maluqing Tunnel. The research shows that with the increase of compressive strength and tensile strength as well as constant A of Hoek-Brown criterion, the safe thickness decreases, whereas with the increase of cavern pressure, tunnel diameter, and constant B from Hoek-Brown criterion, the safe thickness increases. Besides, the tensile strength, or constants A and B affect the shear failure angle of rock plug structure, but other parameters do not. In conclusion, the proposed method can predict the minimum safe thickness of rock plug, and is useful for water burst study and prevention measures of tunnels constructed in high-risk karst regions.