Nonlinear Hoek Brown Failure Criterion Research Articles

Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone

The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic–brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable.

Non-linear critical taper model and determination of accretionary wedge strength

The critical taper model has been widely used to evaluate the strength contrast between the wedge and the basal detachment of fold-and-thrust belts and accretionary wedges. However, determination of the strength parameters using the traditional critical taper model, which adopts the Mohr–Coulomb failure criterion, is difficult, if not impossible. In this study, we propose a modified critical taper model that incorporates the non-linear Hoek–Brown failure criterion. The parameters in the proposed critical Hoek–Brown wedge CHBW model can be directly evaluated via field investigations and laboratory tests. Meanwhile, the wedge strength is a function of the wedge thickness, which is oriented from stress non-linearity. The fold-and-thrust belt in western central Taiwan was used as an example to validate the proposed model. The determined wedge strength was 0.86 using a representative wedge thickness of 5.3km; this was close to the inferred value of 0.6 from the critical taper. Interestingly, a concave topographic relief is predicted as a result of the wedge thickness dependency of the wedge strength, even if the wedge is composed of homogeneous materials and if the strength of the detachment is uniform. This study demonstrates that the influence of wedge strength on the critical taper angle can be quantified by the spatial distribution of strength variables and by the consideration of the wedge thickness dependency of wedge strength.

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.

Energy analysis of face stability of deep rock tunnels using nonlinear Hoek-Brown failure criterion

The nonlinear Hoek-Brown failure criterion was introduced to limit analysis by applying the tangent method. Based on the failure mechanism of double-logarithmic spiral curves on the face of deep rock tunnels, the analytical solutions of collapse pressure were derived through utilizing the virtual power principle in the case of pore water, and the optimal solutions of collapse pressure were obtained by using the optimization programs of mathematical model with regard of a maximum problem. In comparison with existing research with the same parameters, the consistency of change rule shows the validity of the proposed method. Moreover, parametric study indicates that nonlinear Hoek-Brown failure criterion and pore water pressure have great influence on collapse pressure and failure shape of tunnel faces in deep rock masses, particularly when the surrounding rock is too weak or under the condition of great disturbance and abundant ground water, and in this case, supporting measures should be intensified so as to prevent the occurrence of collapse.

Numerical simulation of triaxial compression test for brittle rock sample using a modified constitutive law considering degradation and dilation behavior

The understanding of the rock deformation and failure process and the development of appropriate constitutive models are the basis for solving problems in rock engineering. In order to investigate progressive failure behavior in brittle rocks, a modified constitutive model was developed which follows the principles of the continuum damage mechanics method. It incorporates non-linear Hoek-Brown failure criterion, confining pressure-dependent strength degradation and volume dilation laws, and is able to represent the nonlinear degradation and dilation behaviors of brittle rocks in the post-failure region. A series of triaxial compression tests were carried out on Eibenstock (Germany) granite samples. Based on a lab data fitting procedure, a consistent parameter set for the modified constitutive model was deduced and implemented into the numerical code FLAC3D. The good agreement between numerical and laboratory results indicates that the modified constitutive law is well suited to represent the nonlinear mechanical behavior of brittle rock especially in the post-failure region.

Collapsed Shape of Shallow Unlined Tunnels Based on Functional Catastrophe Theory

This paper investigates the collapse mechanisms and possible collapsing block shapes of shallow unlined tunnels under conditions of plane strain. The analysis is performed following the framework from a branch of catastrophe theory, functional catastrophe theory. First, the basic principles of functional catastrophe theory are introduced. Then, an analytical solution for the shape curve of the collapsing block of a shallow unlined tunnel is derived using functional catastrophe theory based on the nonlinear Hoek-Brown failure criterion. The effects of the rock mass parameters of the proposed method on the shape and weight of the collapsing block are examined. Moreover, a critical cover depth expression to classify deep and shallow tunnels is proposed. The analytical results are consistent with those obtained by numerical simulation using the particle flow code, demonstrating the validity of the proposed analytical method. The obtained formulas can be used to predict the height and width of the collapsing block of a shallow unlined tunnel and to provide a direct estimate of the overburden on the tunnel lining. The obtained formulas can be easily used by tunnel engineers and researchers due to their simplicity.

Limit analysis of supporting pressure in tunnels with regard to surface settlement

The proliferation of Hoek-Brown nonlinear failure criterion and upper bound theorem makes it possible to evaluate the stability of circular tunnels with an original curved collapsing mechanism. The arch effect of shallow circle tunnel is not taken into consideration so that the mechanical characteristics can be easily described. Based on the mechanism, the upper bound solution of supporting pressure of tunnels under the condition of surface settlements and overloads on the ground surface is derived. The objective function is formed from virtual work equations under the variational principle, and solutions are presented by the optimum theory. Comparisons with previous works are made. The numerical results of the present method show great agreement with those of existing ones. With regard to the surface settlement and overloads, the influence of different rock parameters on the collapsing shape is analyzed.

Functional catastrophe analysis of collapse mechanisms for deep tunnels based on the Hoek-Brown failure criterion

A reliable prediction of the characteristics of roof collapse in deep tunnels is still one of the most important and challenging tasks in tunnel engineering. To investigate the collapse mechanisms and possible shapes of the collapsing blocks in deep tunnels, an analytical solution of shape curves for collapsing blocks is derived based on the nonlinear Hoek-Brown failure criterion by using the functional catastrophe theory. The obtained formulas are extremely simple. Furthermore, a judging criterion is proposed to distinguish whether the roof collapses of deep tunnels will occur or not. The effects of rock mass parameters used in the proposed method on the collapsing block shapes of deep tunnels are also discussed. The proposed analytical solution is verified by both the empirical method and the model test.

An elasto-plastic model for underwater tunnels considering seepage body forces and strain-softening behaviour

This paper proposes an analytical–numerical model for elasto-plastic analysis of underwater tunnels that considers seepage body forces and strain-softening behaviour. Using the finite difference method, a theoretical solution is presented for calculation of the pore-water pressure, stress and strain distributions around the circular tunnels in axisymmetric and plain strain conditions. In this method, strain-softening behaviour and non-linear Hoek–Brown failure criterion are used. The proposed model also takes into account the effects of elastic strain increments and dilatancy angle () of the rock mass in the plastic zone. Through developing previous seepage models, a more accurate model is proposed for calculation of pore-water pressure distribution in all directions around underwater tunnels. Taking into account strain-softening behaviour of the rock mass, the stepwise method is applied to the plastic zone where parameters of strength and dilatancy, stresses, strains and deformation vary from their elasto-plastic boundary values to the tunnel boundary values. On the other hand, the analytical equations are derived for the elastic zone. Accuracy and the practical performance of the proposed model are evaluated by a number of examples. The results present the effects of seepage body forces, elastic strain increment and variations of the dilatancy angle in the plastic zone appropriately.

Upper bound solutions for supporting pressures of shallow tunnels with nonlinear failure criterion

Linear failure criterion is widely used in calculation of earth pressure acting on shallow tunnels. However, experimental evidence shows that nonlinear failure criterion is able to represent fairly well the failure of almost all types of rocks. A nonlinear Hoek-Brown failure criterion is employed to estimate the supporting pressures of shallow tunnels in limit analysis framework. Two failure mechanisms are proposed for calculating the work rate of external force and the internal energy dissipation. A tangential line to the nonlinear failure criterion is used to formulate the supporting pressure problem as a nonlinear programming problem. The objective function formulated in this way is minimized with respect to the failure mechanism and the location of tangency point. In order to assess the validity, the supporting pressures for the proposed failure mechanisms are calculated and compared with each other, and the present results are compared with previously published solutions when the nonlinear criterion is reduced to linear criterion. The agreement supports the validity of the proposed failure mechanisms. An experiment is conducted to investigate the influences of the nonlinear criterion on collapse shape and supporting pressures of shallow tunnels.

Tunnel face stability in heavily fractured rock masses that follow the Hoek–Brown failure criterion

A tunnel face may collapse if the support pressure is lower than a limit value called the ‘critical’ or ‘collapse’ pressure. In this work, an advanced rotational failure mechanism is developed to compute, in the context of limit analysis, the collapse pressure for tunnel faces in fractured rock masses characterized by the Hoek–Brown non-linear failure criterion. The non-linearity introduces the need for additional assumptions about the distribution of normal stresses along the slip surface, which translate into new parameters in the limit analysis optimization problem. A numerical 3D finite difference code is employed to identify adequate approximations of the distribution of normal stresses along the failure surface, with results showing that linear stress distributions along the failure surface are needed to obtain improved results in the case of weaker rock masses. Test-cases are employed to validate the new mechanism with the three-dimensional numerical model. Results show that critical pressures computed with limit analysis are very similar to those obtained with the numerical model, and that the failure mechanisms obtained in the limit analysis approach are also very similar to those obtained in small scale model tests and with the numerical simulations. The limit analysis approach based on the new failure mechanism is significantly more computationally efficient than the 3D numerical approach, providing fast, yet accurate, estimates of critical pressures for tunnel face stability in weak and fractured rock masses. The methodology has been further employed to develop simple design charts that provide the face collapse pressure of tunnels within a wide variety of practical situations.

Rock Failure Analysis of the Broken Zone Around a Circular Opening / Analiza pęknięcia skały w strefie naruszonej wokół otworu kolistego

In this paper, considering the non-linear Hoek-Brown failure criterion, a new theoretical model is presented to predict the stress components and estimate the plastic zone radius around a circular tunnel. The tunnel is excavated in an elasto-plastic rock mass subjected to plane hydrostatic and axial in situ stresses. Effects of the axial in situ stress on the plastic zone radius and stress components are studied. Based on the combination of plane hydrostatic and axial in situ stresses with the equilibrium equation and a suitable failure criterion (Hoek & Brown failure criterion), several cases are considered. For each case, the stress components, the plastic zone radius and the necessary conditions for its occurrence are determined. The results obtained by the present method are compared with those using Mohr-Coulomb criterion and with the experimental data, illustrating the validity of the present model in predicting the failure zone radius.

Distinct element simulation of ultimate bearing capacity in jointed rock foundations

In this paper, distinct element method numerical modeling is applied to evaluate bearing capacity of strip footing rested on anisotropic discontinuous rock mass. As yet, a little work has been carried out to investigate the effect of joint set orientation on the bearing capacity of rock mass. Generally, the overall behavior of rock mass under loading is very complicated and such analysis should include deformation determination, sliding along discontinuities and failure of rock material. Failure mechanism of rock mass depended on both geometrical parameters of joint sets and strength parameters of rock mass. In this research, it is assumed that rock mass contains one joint set, and therefore the anisotropy in bearing capacity and rock behavior is only due to the existence and orientation of the joint set. In this study, by assuming constant strength parameters and using Mohr–Coulomb failure criterion for the single joint set and nonlinear Hoek–Brown failure criterion for rock material, variation of the bearing capacity values and the type of failure mechanism of rock mass with different joint set dips is investigated. The obtained results indicate that the ultimate bearing capacity of rock mass containing one joint set varies between 27 and 86 % of intact rock.

A Comparative Study of Different Approaches for Factor of Safety Calculations by Shear Strength Reduction Technique for Non-linear Hoek–Brown Failure Criterion

The shear strength reduction technique is becoming more and more popular to determine the factor-of-safety for geotechnical constructions, especially for slopes. At present, two in principal different procedures are used to apply the numerical shear strength reduction technique for materials characterised by non-linear failure envelopes, like the Hoek–Brown criterion. One procedure is based on the determination on local stress and strength values, whereas the other is based on a global linearization of the non-linear failure envelope. This article shortly describes and discusses these two different procedures and compares results for a broad spectrum of parameter constellations based on slope stability calculations. The local approach is physically more correct. The global approach can be considered as a first approximation. A comparison of both methods reveal that the global approach in comparison to the local approach, can leads to a deviation of up to 15 % in both directions. If one considers the local approach as the ‘correct’ one, depending on the parameters the results of the global approach can lie on the safe or unsafe site. The practical conclusion is that evaluation of slope stability using the global approach can result in uneconomic slope design or overestimation of safety margin. The use of the local approach instead of the global should be preferred. In case of small safety margins (e.g. 20 % or less) the use of the local approach is strictly recommended.

Upper bound solution of supporting pressure for a shallow square tunnel based on the Hoek-Brown failure criterion

To analyze the stability of a shallow square tunnel, a new curved failure mechanism, representing the mechanical characteristics and collapsing form of this type of tunnel, is constructed. Based on the upper bound theorem of limit analysis and the Hoek-Brown nonlinear failure criterion, the supporting pressure derived from the virtual work rate equation is regarded as an objective function to achieve optimal calculation. By employing variational calculation to optimize the objective function, an upper bound solution for the supporting pressure and the collapsing block shape of a shallow square tunnel are obtained. To evaluate the validity of the failure mechanism proposed in this paper, the solutions computed by the curved failure mechanism are compared with the results calculated by the linear multiple blocks failure mechanism when the Hoek-Brown nonlinear failure criterion is converted into the Mohr-Coulomb linear criterion. The influences of rock mass parameters on the supporting pressure and collapsing block shape are discussed.

A new closed-form solution for circular openings modeled by the Unified Strength Theory and radius-dependent Young’s modulus

A new closed-form solution is presented for the stress and displacement distribution surrounding circular openings with finite external radii that are subject to uniform internal and external pressures under plane strain conditions. The specific solution for a deep circular tunnel in an infinite rock mass is also provided. It is assumed that the rock mass is elastic–brittle–plastic and governed by the Unified Strength Theory (UST). In the plastic zone, the radius-dependent Young’s modulus (RDM) model and a non-associated linear flow rule were adopted to establish the radial displacement solution. The new closed-form solution obtained in this paper is a series of results rather than one specific solution; hence, it is suitable for a wide range of rock masses and engineering structures. The traditional solutions, which are based on the Mohr–Coulomb failure criterion and the Generalized Twin Shear Stress yield criterion, can be categorized as special cases of this proposed solution. This new solution agrees reasonably well with the results of a borehole collapse test, a secondary development numerical simulation and an additional closed-form solution using the generalized non-linear Hoek–Brown failure criterion. Parametric studies were conducted to investigate the effects of intermediate principal stress, RDM and dilatancy on the results. It is shown herein that the effects of intermediate principal stress and dilatancy are significant; the RDM model is recommended as the optimum approach for calculating radial displacement and support pressure.

Bearing Capacity Calculation of Rock Foundation based on Nonlinear Failure Criterion

Conventional foundation bearing capacity calculation is based on Mohr-Coulomb linear failure criterion. But it is verified that almost all kinds of rock's strength envelope is nonlinear with normal stress through tests and is in compliance with modified Hoek-Brown nonlinear failure criterion. Therefore, program is composed by using Matlab software and nonlinear Sequential Quadratic Programming method to calculate bearing capacity and analyze its affect factors according to the upper limit theory of limit analysis adopting Hoek-Brown nonlinear failure criterion and multi-tangential method. The result shows that the main affect factors of rock foundation's bearing capacity are GSI and mi of the rock, however, the dead weight γ,over load q and excavation disturbance coefficient D affect the bearing capacity largely when GSI is small; after comparison with formers’ research, it is found that the bearing capacity is overestimated and having greater risk by using “single-tangential method” while the “mufti-tangential method is more rigorous in theory and whose result is more close to the actual value and more applicable.

A Stability Analysis of Two Horizontal CBM Wells in the Deep Mannville Coals, Alberta Basin, Canada

This study describes the main technical challenges encountered while drilling the first horizontal coalbed methane well ever attempted in Canada. The well was drilled in the deep (>1,300 m), low-permeability (1–3 md) coals of the Lower Cretaceous Mannville Formation in the central Alberta Plains. The study also discusses the investigation that followed the occurrence of a few unexpected events during the drilling phase of the well, as well as the stability analysis of the horizontal borehole conducted. Suggestions are made on how to best mitigate the risk of getting the drill pipe stuck and potentially losing the hole along with drilling equipment. The findings are very instructive and should have an application in geological settings similar to those found in the Alberta Basin. Any modifications on how to drill horizontal wells in coal seams present at great depths and under high in-situ stresses must take into account the geomechanical properties of the target seam through triaxial testing, which should include obtaining the non-linear Hoek-Brown failure criterion and testing the coal core for very low confining stress. The lab-derived strength data may also have to be downgraded in order to predict, more realistically, the radius of the yielded zone around the wellbore and the horizontal borehole stability using geomechanical software, such as STABView® and FLAC.

Hoek–Brown criterion applied to circular tunnel using elastoplasticity and in situ axial stress

Based on the nonlinear Hoek–Brown failure criterion, elastoplastic analytical solutions are developed for the elastoplastic stresses, strains and plastic zones around a circular tunnel subjected to different value of the axial in situ stress. Effects of the transverse in situ stress, the axial in situ stress and the strength parameters of rock masses on the elastoplastic stresses, strains and plastic zones in the surrounding rock masses are investigated. It is found from the numerical results that the stresses, strains, and plastic zones in the surrounding rock depend not only on the transverse in situ stress but also on the axial in situ stress as well as the mechanical parameters of rock masses.

Collapse mechanism of shallow tunnel based on nonlinear Hoek–Brown failure criterion

Collapse shape of shallow circular tunnel is derived using a new curved failure mechanism within the framework of upper bound theorem. Nonlinear Hoek–Brown failure criterion is adopted in the present analysis. With the consideration of supporting pressure, the energy dissipations of the new failure mechanism are calculated by employing integral technique. Equating the rate of energy dissipation to the external rate of work, the objective function is formulated, and is optimized with the variational approach. Numerical results are presented, and collapse mechanisms of shallow tunnel taking into account supporting pressure are obtained. The present results are compared with the previous solutions, and the agreement shows that the present method is valid. The effects of different parameters on the failure mechanisms are discussed, and a critical depth expression is proposed for classifying shallow and deep tunnels.