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Since the failure modes of fully grouted bolts subject to displacements of rock joints are difficult to predict, the joint displacements in this study are categorised into two types: opening displacement and shear displacement. At first, a closed-form solution incorporating the nonlinear constitutive model of interfaces between bolts and grout is developed, which is able to formulate the full-range behaviour of bolts subject to the opening displacements of rock joints. Then, a novel numerical approach considering the nonlinear rock reaction force is proposed, which can analyse stress and displacement conditions on each point along a bolt. The results of the proposed numerical approach are validated by existing test results. Combining the proposed numerical approach with the improved closed-form solution, a comprehensive model for predicting yield and failure patterns of rock bolts is proposed. In this way, decoupling between the bolt and grout, the yield and failure patterns of two potential points along the bolt are also highlighted and discussed. Further discussions about the influences of bolt inclined angles to the discontinuity and the dilatancy effect of the joint face are carried out at last. This work provides a useful tool to evaluate the behaviours and estimate the load-bearing capacity of fully grouted bolts in jointed rock masses.

The kinematical approach of limit analysis has been historically employed to assess slope stability in the geotechnical community, for which most three-dimensional (3D) failure mechanisms ...

The aim of this study is to propose an effective approach to evaluate the stability of a shield tunnel face with tensile strength cutoff by using the upper-bound limit analysis method. Bas...

In this work, the stability of a reinforced tunnel face is examined by combining the homogenization method with the kinematic approach of limit analysis. A three-dimensional (3D) heterogen...

Geotechnical structures often have position-dependent soil strength parameters as a consequence of stress history, deposition process, mineralogical composition, and weathering effect. How...

On the basis of the yield criterion of Matsuoka and Nakai (1974), novel elasto-plastic solutions are obtained for the cavity expansion problem in an anisotropic medium. The solutions are formulated by employing a small-strain analysis in an elastic zone and a large-strain analysis in a plastic zone around the cavity in a soil mass. In light of the initial condition, the plastic radius and the stress in the plastic zone are derived by incorporating the large-strain analysis, the influence of stress anisotropy (K0) and the non-associated flow rule, and two fundamental solutions for the cavity expansion problem under undrained and drained conditions are presented in an anisotropic soil mass. Finally, a comparison with the previous results is performed for verification.

This paper investigates the roof collapse of deep circular cavities in jointed rock masses using adaptive finite element limit analysis. The rock mass strength is described by the generalized Hoek-Brown failure criterion. The calculated results of the stability number Ns are presented in the form of dimensionless stability charts and are compared to the transformational results of an analytical upper bound solution by former investigators. The transformations are conducted using an algorithm proposed in the appendix of this paper. Parametric studies are performed to investigate the impacts of the Hoek-Brown parameters on the stability number and failure mechanism of underground cavities. In particular, different levels of rock mass disturbance are considered by adopting various rock mass disturbance factors D in the analyses.

An elastic-plastic semi-analytical approach is proposed for predicting the cylindrical cavity expansion in K0-consolidated clay. The approach is based on the assumption of small deformation in the ...

To better assess the stability of tunnel faces in cohesion-frictional soils, tunnel face stability was investigated considering the soil arching effect in this study. An improved 3D rotational failure model based on the limit analysis method and an improved wedge-prism model based on the limit equilibrium method were proposed. Simultaneously, the lateral stress ratio K s was improved to reflect the soil arching effect. The support pressures obtained by the two proposed models were validated by the numerical results and existing works. Finally, the effects of the geometrical and geotechnical parameters and loading conditions on the tunnel face stability were assessed.

Most subways excavated in nonhomogeneous and water-bearing soils but few existing analytical models focus on the combined effect of soil nonhomogeneity and pore water pressures. To address this issue, a three-dimensional (3D) rotational collapse mechanism is improved to investigate the effect of vertical variability of soil strength on the face stability of a circular tunnel excavated in saturated nonhomogeneous soils by means of the kinematical approach of limit analysis. The pore water pressure distribution obtained numerically is interpolated on the 3D improved collapse mechanism by a linear interpolation technology. The effectiveness of the developed model is verified by comparing the required face pressures with the existing solutions, the numerical simulations and the monitoring data of the Shenzhen urban rail transit line 9 project. The effects of nonhomogeneous friction angles and cohesion and water table elevation on the face stability are investigated.

The solutions of stress and displacement of a circular opening excavated in brittle and strain-softening rock mass incorporating rockbolts effectiveness and seepage force are presented in this study. The evolution equation is reconstructed for the strength parameters that incorporate these factors. Based on the evolution equation, an improved numerical method and stepwise procedure are presented which are compatible with the Mohr–Coulomb (M–C) and the generalized Hoek–Brown (H–B) failure criteria, respectively. Then given three interaction mechanisms between rockbolts and surrounding rock, solutions for stress and displacement are proposed in line with the improved numerical method and numerical stepwise procedure. The proposed approach can be reduced to Fahimifar and Soroush’s (Tunn Undergr Space Technol 20:333–343, 2005) solutions for special cases. The proposed method was validated by field monitoring data and FLAC results of Yanzidong tunnel. Examples under the M–C and generalized H–B failure criteria for rock mass are generated through MATLAB programming. Moreover, parametric studies are conducted to highlight the influence of rockbolts effectiveness in combination with seepage force on the stress and displacement of very good, average, and very poor surrounding rock. Results show that in this case, stress confinement is higher and tunnel convergences are lower than the corresponding stresses and displacements obtained in non-reinforced tunnels. Displacement and plastic radius are also higher than those without considering seepage force.

A new procedure for the ground response curve (GRC) is investigated in strain-softening surrounding rock for a circular opening. The procedure started each step with a radius increment and the analytical solutions of stress and strain in each annulus were presented. The plastic region is divided into a finite number of concentric annuli, whose thickness is uniformly determined by a small radius increment. Combining the equilibrium equation and failure criterion, stress for each annulus can be obtained analytically. The displacement for each step can be calculated analytically through solving the differential equation by invoking flow rule and Hooke’s law. The strains for each annulus can be obtained by the strain-displacement relationship. In the successive manner, the distributions of stress and displacement can be found. It should be noted that the finial stress and displacement at radial direction are the internal support pressure and deformation at the excavation surface, respectively. By assuming different plastic radii (using a plastic radius increment), GRC, the evolution curve of plastic radii and internal support pressure can be obtained analytically. Some numerical and engineering examples are performed to demonstrate the validity of the proposed procedure. It is shown that the results of the proposed procedure at the tunnel crown are basically consistent with field measuring data. The influence of the annulus number, plastic radius increment and dilation on the accuracy of the proposed approach is investigated. Results show that the solutions are more accurate and the calculation efficiency is higher.

The problem of cylindrical cavity expansion incorporating deformation dependent of intermediate principal stress in rock or soil mass is investigated in the paper. Assumptions that the initial axial total strain is a nonzero constant and the axial plastic strain is not zero are defined to obtain the numerical solution of strain which incorporates deformation-dependent intermediate principal stress. The numerical solution of plastic strains are achieved by the 3-D plastic potential functions based on the M-C and generalized H-B failure criteria, respectively. The intermediate principal stress is derived with the Hook\'s law and plastic strains. Solution of limited expansion pressure, stress and strain during cylindrical cavity expanding are given and the corresponding calculation approaches are also presented, which the axial stress and strain are incorporated. Validation of the proposed approach is conducted by the published results.

This paper focuses on the quasi-plane strain-softening problem of cylindrical cavity expanding in Cam-Clay soil mass. The quasi-plane strain-softening problem is defined based on the assumptions that the initial axial total strain is a non-zero constant and the axial plastic strain is not zero. A new approach for the major, intermediate, and minor principal strains are proposed based on the associated flow rule, 3-D plastic potential function, Hooke’s law, stress equilibrium equation and Cam-Clay model. The correctness of the proposed approach is validated by the results in Cao and Teh. Parametric studies were performed to investigate the influences of the intermediate principal stress on the radial and circumferential stresses and the radial displacement, respectively.

A closed-form solution for the undrained cavity expansion problem in an anisotropic soil mass is investigated based on the spatially mobilized plane (SMP) model in this study. Stress and st...

Considering the effect of seepage force, a dimensionless approach was introduced to improve the stress and strain increment approach on the stresses and radial displacement around a circular tunnel excavated in a strain-softening generalized Hoek–Brown or Mohr–Coulomb rock mass. The circular tunnel can be simplified as axisymmetric problem, and the plastic zone was divided into a finite number of concentric rings which satisfy the equilibrium and compatibility equations. Increments of stresses and strains for each ring were obtained by solving the equilibrium and compatibility equations. Then, the stresses and displacements in softening zone can be calculated. The correctness and reliability of the proposed approach were performed by the existing solutions.

Polynomial chaos expansions (PCEs) have been extensively used to perform reliability analyses of slopes. The accuracy of a PCE metamodel is highly dependent on the experimental design samples, which are commonly selected according to their uniformity. However, the method of uniform sampling fails to put additional weight on the regions with high nonlinearity, in which more samples are required to give a good approximation. To address this issue, the Voronoi-based exploration and the local linear approximation-based exploitation (Voronoi-LOLA) are combined to determine experimental design samples for PCE constructions. A sequential sampling scheme that employs the sparse polynomial chaos expansion (SPCE) output information is further proposed to choose the most informative samples, which are crucial for building a PCE. This method not only improves computational efficiency but also enhances the accuracy of the PCE metamodel. The performance of the proposed Voronoi-LOLA-SPCE method is illustrated with four representative examples, and the results show that the proposed Voronoi-LOLA-SPCE is an effective and accurate method for slope reliability assessment.

This paper focuses on the influence of intermediate principal stress on visco-elasto-plastic solution of surrounding rock. Assuming that the surrounding rock satisfies the spatially mobilized plane (SMP) failure criterion, the H-Kelvin model was adopted and improved by considering the effect of intermediate principal stress. For elasto-plastic surrounding rock, an analytical approach was presented. For strain-softening surrounding rock, the potential plastic zone was subdivided into a finite number of concentric annuli and a numerical model for calculating the stress and displacement of each annulus was presented. Thus, based on SMP failure criterion and the improved H-Kelvin model, the nonlinear visco-elasto-plastic models considering the effect of intermediate principal stress were proposed for elasto-plastic and strain-softening surrounding rock, respectively. The accuracy of the proposed approaches were validated with the published results. Comparisons show that the long-term creep deformation based on M-C failure criterion is 9.4 mm which is 34.3% greater than the results of SMP failure criterion. Taking the safety factor as 1.5, the proposed model is applied in predicting the long-term deformation of Shangxinzhai Tunnel, the deformation is calculated to be 8.5 cm which is 29.2% smaller than the designed value.

This paper presented solutions of displacement and stress for a circular opening which is reinforced with grouted rock bolt. It satisfies the Mohr-Coulomb (M-C) or generalized Hoek-Brown (H-B) failure criterion, and exhibits elastic-brittle-plastic or strain-softening behavior. The numerical stepwise produce for strain-softening rock mass reinforced with grouted rock bolt was developed with non-associative flow rules and two segments piecewise linear functions related to a principle strain-dependent plastic parameter, to model the transition from peak to residual strength. Three models of the interaction mechanism between grouted rock bolt and surrounding rock proposed by Fahimifar and Soroush (2005) were adopted. Based on the axial symmetrical plane strain assumption, the theoretical solution of the displacement and stress were proposed for a circular tunnel excavated in elastic-brittle-plastic and strain-softening rock mass compatible with M-C or generalized H-B failure criterion, which is reinforced with grouted rock bolt. It showed that Fahimifar and Soroush's (2005) solution is a special case of the proposed solution for n = 0.5. Further, the proposed method is validated through example comparison calculated by MATLAB programming. Meanwhile, some particular examples for M-C or generalized H-B failure criterion have been conducted, and parametric studies were carried out to highlight the influence of different parameters (e.g., the very good, average and very poor rock mass). The results showed that, stress field in plastic region of surrounding rock with considering the supporting effectiveness of the grouted rock bolt is more than that without considering the effectiveness of the grouted rock bolt, and the convergence and plastic radius are reduced.

This paper focuses on the influence of intermediate principal stress and seepage force on visco-elasto-plastic solution of surrounding rock. By assuming that the surrounding rock satisfies the 3-D ...

Based on the virtual image technique, stress function method and Boussinesq’s solutions, an approximate solution of the cylindrical cavity expansion is investigated under the non-axisymmetric displacement boundary condition on hypotenuse. The stress harmonic functions on the ground surface are improved. At first, the elastic solution without any correction is deduced, considering the combined effects of the actual and image sources. Then, stress on the ground surface is revised by the improved stress harmonic function. Stress on the slope surface boundary is corrected by integrating Boussinesq’s solution and then applying the coordinate transformation technique. Moreover, the newly induced stresses on the ground surface boundary (resulting from the correction process of the slope surface boundary) are further revised by the improved stress harmonic function. Finally, the linear elastic superposition principle is adopted to obtain the approximate solution under the non-axisymmetric and free displacement bo...

This study focuses on the stress and displacement of a circular opening that is excavated in a strain-softening rock mass incorporating the effects of hydraulic–mechanical coupling and rockbolts effectiveness. It follows the generalized Hoek–Brown failure criterion. Moreover, an improved numerical approach and stepwise procedure are proposed. This approach considers the deterioration of the strength, deformation, and dilation angle and the variation of elastic strain in the plastic region considering the effect of the hydraulic–mechanical coupling and the rockbolts effectiveness. The presented solutions were validated by FLAC results. Several examples are conducted to demonstrate the validity and accuracy of the proposed solution through MATLAB programming. Parametric studies are also conducted to highlight the influences of hydraulic–mechanical coupling and rockbolts effectiveness on stress and displacement. Results show that stress and displacement, incorporating the effects of hydraulic–mechanical coupling and rockbolts effectiveness, are between those when hydraulic–mechanical coupling or rockbolts effectiveness is considered separately. However, this theory needs more verification from practical engineering.

This study proposes a numerical approach that considers the effect of out-of-plane stress for circular tunnels excavated in strain-softening rock. The entire plastic region is divided into a finite number of concentric rings whose thickness is determined to satisfy equilibrium and compatibility equations internally. The increments of stress and strain for each ring are obtained by successively incorporating the effect of out-of-plane stress. The initial value of the outermost ring is determined using equilibrium and compatibility equations. Numerical solutions of the stress, displacement and plastic radius are derived by considering the effect of out-of-plane stress for circular tunnels excavated in strain-softening rock obeying Mohr–Coulomb or generalised Hoek–Brown failure criteria. The proposed approach is compared with previously proposed solutions. Several examples are presented to validate the accuracy of the proposed approach and the effect of different parameters is determined. The results show that the out-of-plane stress influences stress and displacement distributions in the plastic region. The results also show that the residual and softening plastic radii increase with increasing out-of-plane stress when out-of-plane stress is the major or intermediate principal stress, and the residual and softening radii increase with decreasing out-of-plane stress when out-of-plane stress is the minor principal stress.

Seismic stability of slopes reinforced with soil nails has been traditionally investigated by two-dimensional limit equilibrium method. In this paper, the strength reduction method in combination with the kinematic approach of limit analysis is employed to assess the factor of safety (Fos) of slopes reinforced by soil nails using the three-dimensional rotational failure mechanism. The pseudo static approach is used to represent the seismic effects. Both the tensile failure and pull-out failure of soil nails are considered in the computations of internal energy dissipations. Comparisons are made to validate the proposed approach, which shows that the implemented approach is an efficient design tool for evaluating the factor of safety of slopes reinforced by soil nails. Parametric analysis is conducted to discuss the influence of model parameters, including nail length, nail density, soil shear strength and seismic forces, on slope stability. A set of stability charts is finally provided for fast assessments of slope safety.

The available theoretical models for evaluating seismic active earth pressures are limited to plane strain analyses and linear failure criterion. To fill this research gap, with a combination of the kinematical approach of limit analysis, the spatial discretization technique as well as the pseudo-static approach, this work develops a discretized three-dimensional failure mechanism of sloping backfills that is composed of an end cap and a central plane-strain insert to estimate the seismic active earth pressure coefficient under nonlinear failure criterion. Compared with the existing three-dimensional kinematically admissible failure mechanisms, the developed one has three significant merits: (1) the normality condition is strictly satisfied in the generation of the end cap; (2) the work rates can be computed by a summation of elementary work rates conveniently instead of tedious integral schemes; (3) the considerations of the nonlinear failure criterion and the sloping backfill. Comparisons with the existing upper bound solutions and experimental results in terms of active earth forces or active earth pressure coefficients are conducted to verify the proposed model. Parametric studies are performed to study the influences of model parameters, including the seismic coefficient, the sloping angle of backfills as well as the nonlinear parameters, on the active earth pressure coefficients.

This study aims to propose an effective approach to estimate the earthquake-induced displacements of a slope in heterogeneous soils considering the tensile strength cut-off. In the framework of the upper-bound limit analysis theory, an improved failure mechanism generated from the slope crest using the discretization technique is developed to determine the critical seismic acceleration coefficients with tensile strength cut-off. By using the Newmark’s method, the earthquake-induced displacements of slopes are then estimated based on the proposed failure mechanism. The proposed method is validated by comparisons with experimental results and numerical modellings in terms of the predicted displacements and failure surfaces. Finally, the displacements induced by a real earthquake are presented for two heterogeneous slopes with considering tensile strength cut-off.

The numerical solutions for strain-softening surrounding rock under 3D Hoek-Brown strength criterion are developed incorporating the influence of passive bolts and the seepage flow field. Based on 3D Hoek-Brown failure criterion, a numerical-increment-approach is reconstructed for the first time under the spatial axisymmetric conditions, in which the elastic and plastic zones are divided into a finite number of concentric annuli with constant increments of radial stress and radius between the adjacent annuli, respectively. Furthermore, the hydraulic-mechanical coupling model is improved by considering the effects of out-of-plane plastic strain and the spring-slider model is improved by considering the axial force at rockbolt head. On the basis of the reconstructed numerical-increment-approach and the improved spring-slider model, the bolt self-equilibrium conditions for five cases of passive bolts are also reconstructed (e.g., neutral point in elastic or plastic zones, neutral length in elastic, plastic or elastic-plastic zones, respectively). Ultimately, an improved numerical-iterative-approach is proposed in terms of the developed numerical-increment-approach and the reconstructed bolt self-equilibrium conditions, through which stresses and displacement of surrounding rock can be obtained by incorporating the effects of hydraulic-mechanical coupling and passive bolts. The corresponding calculation program is also presented. The reliability of the presented approach has been verified by the numerical and existed results. From a practical point of view, the proposed approach can be applied to estimate the reinforced ground response curves and optimize the length of passive bolts.