Nonlinear Hoek-Brown Criterion Research Articles

Analytical Solution for Seismic Stability of 3D Rock Slope Reinforced with Prestressed Anchor Cables

Currently, the study of analytical solutions for the seismic stability of slopes under anchorage conditions is one of the hottest subjects in engineering. In this paper, an analytical solution for the seismic safety of the three-dimensional (3D) two-stage rock slope reinforced with prestressed anchor cables governed by the nonlinear Hoek–Brown criterion was deduced, in which the analyses of seismicity were performed by the latest modified pseudo-dynamic method. This method supplements the consideration of the damping effect of the rock medium on seismic waves, which is more in line with the real seismic situation. A mathematical geometric model was developed for calculating the external forces work and internal energy dissipation acting on 3D rotating rock masses reinforced with prestressed anchor cables, in which the seismic work rate was calculated using a new layer-by-layer superposition summation method. The analytical solution of the safety factor could be collated as an explicit function of several variables, and then, the optimal value was obtained by the Particle Swarm Optimization (PSO) algorithm. To corroborate the accuracy of new analytical solutions, the results were contrasted with those of the pertinent literature. The results of the two comparisons were very close. Ultimately, the sensitivity analyses and coupling effects of seismic pseudo-dynamic factors and prestressing anchorage factors were carried out. It was found that even small seismic intensities had a large effect on the stability of rock slopes with developed joints. Increasing the number of steps and prestressing anchors can effectively improve the stability of rock slopes under seismic effects. The conclusions have significant implications for the anchorage design of the 3D two-stage rock slope as seismic events occur.

Analysis of the interaction between bolt-reinforced rock and surface support in tunnels based on convergence-confinement method

To investigate the interaction of the bolt-reinforced rock and the surface support, an analytical model of the convergence-confinement type is proposed, considering the sequential installation of the fully grouted rockbolts and the surface support. The rock mass is assumed to be elastic-brittle-plastic material, obeying the linear Mohr-Coulomb criterion or the non-linear Hoek-Brown criterion. According to the strain states of the tunnel wall at bolt and surface support installation and the relative magnitude between the bolt length and the plastic depth during the whole process, six cases are categorized upon solving the problem. Each case is divided into three stages due to the different effects of the active rockbolts and the passive surface support. The fictitious pressure is introduced to quantify the three-dimensional (3D) effect of the tunnel face, and thus, the actual physical location along the tunnel axis of the analytical section can be considered. By using the bolt-rock strain compatibility and the rock-surface support displacement compatibility conditions, the solutions of longitudinal tunnel displacement and the reaction pressure of surface support along the tunnel axis are obtained. The proposed analytical solutions are validated by a series of 3D numerical simulations. Extensive parametric studies are conducted to examine the effect of the typical parameters of rockbolts and surface support on the tunnel displacement and the reaction pressure of the surface support under different rock conditions. The results show that the rockbolts are more effective in controlling the tunnel displacement than the surface support, which should be installed as soon as possible with a suitable length. For tunnels excavated in weak rocks or with restricted displacement control requirements, the surface support should also be installed or closed timely with a certain stiffness. The proposed method provides a convenient alternative approach for the optimization of rockbolts and surface support at the preliminary stage of tunnel design.

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

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

Fracture angle of Hoek–Brown materials

The fracture angle (defined as the angle between the normal vector of the fracture plane and the first principal stress) of Mohr–Coulomb is 45°+ ϕ/2, which plays a crucial role in soil mechanics. However, for the non-linear Hoek–Brown criterion, the fracture angle cannot be derived as easily as Mohr–Coulomb. Based on the physical evidence that there is no parallel plastic strain in the shear zone, the fracture angle of Hoek–Brown materials, is derived as tan−11+m2s+mσ3/σc. The experimental data of marble and granite and numerical results demonstrate that the obtained theoretical fracture angles agree well with the experimental and numerical fracture angles.

Experimental and analytical investigations on strength and deformation behaviour of red sandstone under conventional triaxial compression

In this article, a series of conventional triaxial compression tests were carried out to investigate strength characteristics and deformation behaviour of red sandstone. According to experimental results, the initial yield stress, volume dilatation stress and peak strength of the red sandstone all increase with confining pressure. Moreover, with the increase in confining pressure, the failure modes of the specimens are changed from splitting failure to compressive-shear failure. The non-linear Hoek–Brown criterion is found suitable for fitting the strength data. An elastoplastic damage model with a unified plastic hardening/softening law is developed on the basis of the experiments. As an original contribution, analytical solutions to the constitutive equations are derived for the conventional triaxial loading path under consideration. The model is validated by comparing the analytical predictions with the experimental data.

Closed-form solutions for collapse mechanisms of tunnel crown in saturated non-uniform rock surrounds

The crown stability of a deep-buried metro tunnel or station is evaluated by means of a kinematic analysis procedure, with considerations of non-uniform rock surrounds and pore water effects. Non-uniform rock strata are discretized to several layers with finite thickness, with varying rock parameters assigned to different layers and constant parameters within each layer. Pore water effect is considered as an external loading acting on rock mass and boundaries. Employing the nonlinear Hoek-Brown criterion, a preliminary study is presented to analyse tunnel crown stability in a uniform rock medium, in the realm of upper bound theory. After having obtained general solutions of 2D and 3D collapse mechanisms, non-uniformity of rock masses is considered and investigated in crown stability analyses. 2D and 3D closed-form solution are formulated based on variational principle and work rate balance equation. Apart from the collapse shape and dimension, the weight of collapsing block is calculated, which gives a quick estimate of the average overburden pressure and hence guides the design of tunnel lining. Some examples related to circular and rectangular (square) tunnels are provided to offer insights of the implication of influential factors on the crown stability of a tunnel deeply excavated in saturated non-uniform rock strata.

Fracture evolution mechanism of hollow sandstone under conventional triaxial compression by X-ray micro-CT observations and three-dimensional numerical simulations

The knowledge of the fracture evolution mechanism of hollow rock under conventional triaxial compression is very important to evaluate the stability and safety in deep tunnel engineering. In this research, a series of X-ray micro-CT observations and three-dimensional (3D) numerical simulations were conducted on hollow sandstone specimens with various borehole diameters (d = 0, 15 and 26 mm) under different confining pressures (0, 8, 16, 24 and 35 MPa). A 3D realistic failure process analysis method (RFPA3D) was established for the simulation of the deformation failure behavior of hollow sandstone under conventional triaxial compression. First, a group of micro-parameters used in RFPA3D, which reflect the macro mechanical behavior of the tested sandstone material, was calibrated using the experimental triaxial compression results obtained for intact specimens. Subsequently, a series of 3D numerical simulations was conducted on hollow sandstone specimens with various borehole diameters under different confining pressures. The quantitative comparison of the axial deviatoric stress-strain curve and peak strength of hollow sandstone obtained from the experiments and simulations indicates that the 3D numerical results agrees very well with the experimental results. The nonlinear Hoek-Brown criterion better reflects the peak strengths of intact and hollow sandstone materials than the linear Mohr-Coulomb criterion. The effect of the borehole diameter on the peak strength of hollow sandstone depends on the confining pressure, which is interpreted by radial and tangent stress distribution around the borehole. In addition, the surface and internal fracture characteristics of the hollow sandstone obtained by X-ray micro CT observations and simulations were compared. Generally, the internal cracks of the intact and hollow sandstone specimens determined using numerical simulations are very similar to those observed using X-ray micro CT. The influence of confining pressure on the fracture evolution mechanism of hollow sandstone with various borehole diameters was analyzed in detail based on the results of numerical simulations. The conclusions drawn in this study are significant for the prediction of the instability around deep well-bores in petroleum engineering and to ensure the safety of deep excavation damage zones in tunnel engineering.

Selection of optimal strength criteria for the terrigenous Pashiyan horizon of the Romashkinskoe field Tashliyarskaya area

In the process of oil reserves’ development, the in-situ stresses change. Knowledge of rock failure constraints will allow prediction of behavior of rock when subject to subsurface stress change. In this study, we used the results of studies of the Pashiyan sandstone core samples recovered from the Tashliyarskaya area well No. 14403. Six sets of samples, each consisting of three samples taken from the homogeneous intervals at the same depth, were used to determine the ultimate tensile strength, uniaxial and triaxial compressive strength in the in-situ conditions. An analysis of the methods for constructing a rock strength certificate, and comparison of the strength criteria described in State Standard 21153.8-88, the Mohr-Coulomb linear strength criterion and the non-linear Hoek-Brown criterion are provided. The Hoek-Brown criterion has the advantage of describing a non-linear increase in strength with an increase in overburden pressure and more adequately reflects the properties of rock. For the first time, a comparison of applicability of strength criteria obtained by different methods and based on the laboratory core analysis was made to determine their practical applicability. Comprehensive studies of the strength characteristics have never been previously conducted, and the results obtained will serve as the basis for further analysis and application in order to improve the development of the terrigenous Devonian Romashkinskoe field.

RELIABILITY BASED DESIGN OF THE ROCK SIDE SLOPE FOR MIDDLE PLATEAU IN MOKATTAM AREA CONSIDERING OPTIMAL COST VALUE

Design solutions for the problems of the rock side slope stability, in general, are one of themost critical issues in geo-technical engineering profession, due to the fact that the values ofrock parameters are difficult to be considered reliable, because of the uncertainties existing ingeological system. This research aims to introduce reliability based design of the rock sideslope stability considering optimal cost value. The idea for obtaining the optimal cost valuedepends on two main critical parameters. These two main critical parameters are uni-axialcompressive strength (U.C.S) and geologic strength index (G.S.I). Both of them have beencalculated based on statistical calculations of mean and standard deviations. A strengthcriterion is needed to characterize the rockmass in geotechnical engineering. There are threecriteria are used to describe the strength of a material: bilinear Mohr-Coulomb criterion,nonlinear Hoek-Brown criterion and special spalling criterion must be used in special case ofmassive, brittle rocks. Nonlinear Hoek-Brown criterion has been used in this study; becauseof some limitations associated with the use of the Mohr-Coulomb criterion. RocData programhave been utilized for obtaining the Hoek-Brown parameters (m, s), Slide program has beenemployed for calculating the reliability index, factor of safety and probability of failure foreach case study for upper plateau in Mokattam area. The concept of design is based on thenecessity of using reinforcement. If the design of using rock parameters is safe and reliable,there is no need for reinforcement; otherwise design of rock bolts should be used as a supportsystem to improve reliability and factor of safety. The current simplified method has beenutilized to consider the cost of failure to find the optimal length of support and optimalreliability index. The study concluded that the suggested technique of nonlinear Hoek BrownCriterion is effective for helping the designer in finding the optimal length of support thatmeets the optimal reliability index and optimal cost value for two main critical parameters.

The Hoek–Brown failure criterion and GSI – 2018 edition

The Hoek–Brown criterion was introduced in 1980 to provide input for the design of underground excavations in rock. The criterion now incorporates both intact rock and discontinuities, such as joints, characterized by the geological strength index (GSI), into a system designed to estimate the mechanical behaviour of typical rock masses encountered in tunnels, slopes and foundations. The strength and deformation properties of intact rock, derived from laboratory tests, are reduced based on the properties of discontinuities in the rock mass. The nonlinear Hoek–Brown criterion for rock masses is widely accepted and has been applied in many projects around the world. While, in general, it has been found to provide satisfactory estimates, there are several questions on the limits of its applicability and on the inaccuracies related to the quality of the input data. This paper introduces relatively few fundamental changes, but it does discuss many of the issues of utilization and presents case histories to demonstrate practical applications of the criterion and the GSI system.

Study of Post‐Peak Strain Softening Mechanical Behaviour of Rock Material Based on Hoek–Brown Criterion

In order to build the post‐peak strain softening model of rock, the evolution laws of rock parameters (m, s) were obtained by using the evolutionary mode of piecewise linear function regarding the maximum principle stress. Based on the nonlinear Hoek–Brown criterion, the analytical relationship of the rock strength parameters (m, s), cohesion (c), and friction angle (φ) has been developed by theoretical derivation. According to the analysis on the four different types of rock, it is found that, within the range from 0 to σ3min, the peak hardness of the rock becomes smaller as the confining pressure increases and the degree of rock fragmentation decreases as well. The post‐peak stress‐strain curves obtained from the developed softening model are in good agreement with the laboratory test results under different confining pressures. In conclusion, the analytical method is reasonable, and it can predict the post‐peak mechanical behaviour of rock well, which provides a new thought for the rock‐softening simulation.

Experimental study on deformation, peak strength and crack damage behavior of hollow sandstone under conventional triaxial compression

Based on a series of triaxial compression experiments using hollow sandstone specimens with various hole diameters (d=0, 11, 15 and 26mm), the deformation, peak strength and crack damage behavior of hollow sandstone specimens under different confining pressures are investigated. The experimental results show that the Young's modulus of hollow sandstone only depends on the confining pressure and is not affected by hole diameter in the tested range of d=0–26mm. Two types of methods used to confirm the elastic modulus and Poisson's ratio of rock material are proposed to evaluate the triaxial deformation characteristics of hollow sandstone. The effects of confining pressure and hole diameter on the Poisson's ratio and peak strain of hollow sandstone are analyzed. The peak strength and crack damage parameters of hollow sandstone depend on not only the confining pressure (σ3) but also the hole diameter. Under uniaxial compression, the peak strength and crack damage threshold (σcd) of hollow sandstone are independent of hole diameter, whereas the triaxial compressive strength of hollow sandstone decreases linearly with increasing hole diameter. The peak strength and crack damage threshold of hollow sandstone increase with increasing confining pressure, which can be better described by the nonlinear Hoek-Brown criterion than by the linear Mohr-Coulomb criterion. Furthermore, the sensitivity of the crack damage threshold of hollow sandstone to hole diameter is lower than that of peak strength on hole diameter. The concluding remarks can be used to improve the stability and safety of deep underground engineering.

Seismic and static 3D stability of two-stage rock slope based on Hoek–Brown failure criterion

Two-stage slope is beneficial to improve slope stability in comparison with single-stage slope. Based on nonlinear Hoek–Brown criterion, a three-dimensional failure mechanism is employed to estimate the stability of two-stage rock slope, with the effect of seismic inertia force being taken into account. A generalized tangential technique is used to formulate the stability factor problem as a classical optimization problem corresponding to the dissipated energy. The upper-bound solutions are obtained by minimizing the objective function with respect to the location of sliding body center and the location of tangency point. The seismic inertia force is considered and incorporated into the objective function. In comparison with previously published solutions using the linear Mohr–Coulomb criterion, the validity of the present solutions is shown. The analytical expressions for two-stage slope are derived to estimate the seismic stability of slopes. Numerical results for different types of rocks are presented for practical use in engineering, and the effects of different parameters on slope stability are discussed.

Three dimensional seismic and static stability of rock slopes

The kinematical approach of limit analysis is used to estimate the three dimensional stability analysis of rock slopes with nonlinear Hoek-Brown criterion under earthquake forces. The generalized tangential technique is introduced, which makes limit analysis apply to rock slope problem possible. This technique formulates the three dimensional stability problem as a classical nonlinear programming problem. A nonlinear programming algorithm is coded to search for the least upper bound solution. To prove the validity of the present approach, static stability factors are compared with the previous solutions, using a linear failure criterion. Three dimensional seismic and static stability factors are calculated for rock slopes. Numerical results of indicate that the factors increase with the ratio of slope width and height, and are presented for practical use in rock engineering.

Experimental Study of Mechanical Behavior and X-Ray Micro CT Observations of Sandstone Under Conventional Triaxial Compression

This paper reports a series of triaxial compression experiments and X-ray observations carried out to explore the mechanical behavior and internal damage mechanism of sandstone material. The results show that the Young’s modulus of sandstone increased nonlinearly with increasing confining pressure, but the Poisson’s ratio remained unaffected. The nonlinear Hoek–Brown criterion can better reflect the peak strength properties than the linear Mohr–Coulomb criterion. However, the residual strength of sandstone exhibits a clear linear relationship with the confining pressure, which can be best described by the linear Mohr–Coulomb criterion. The sensitivity of the crack damage threshold on the confining pressure was clearly lower than that for the peak strength. After unaxial and triaxial compression failure, the sandstone specimens were analyzed using a 3D X-ray micro CT scanning system. Based on the horizontal and vertical cross-sections of sandstone specimens, we found that under uniaxial compression and lower confining pressure, the sandstone specimen is dominated mainly by axial splitting tensile cracks; however, under higher confining pressure, the sandstone specimen is mainly dominated by a single shear crack. To quantitatively evaluate the internal damage of sandstone material, crack area and aperture extent for each horizontal cross-section were calculated by analyzing the binarized pictures. The system of crack planes under uniaxial compression is much more complicated than that under triaxial compression, which is also testified by the evolution behavior of crack area and aperture extent. Finally, the fracture mechanism of sandstone during uniaxial and triaxial compression is discussed in detail and simplified models are proposed.

Influences of pore pressure on short-term and creep mechanical behavior of red sandstone

Short-term and creep tests for saturated red sandstone under different pore pressures were carried out by a rock servo-controlled triaxial equipment. Based on the experimental results, the influences of confining pressure and pore pressure on short-term mechanical behavior of red sandstone are firstly analyzed. The results show that the peak strength of dry red sandstone increases with the confining pressure, which can be better expressed by the nonlinear Hoek–Brown criterion than the linear Mohr–Coulomb criterion. But with the increase of pore pressure, the peak strength and elastic modulus of saturated red sandstone all decrease step by step. The short-term failure mode of red sandstone is dependent of the confining pressure, but independent of the pore pressure. And then, the influence of pore pressure and axial deviatoric stress on the creep mechanical behavior of saturated red sandstone is analyzed quantitatively. Creep contribution to rock deformation increases with the pore pressure, and the specimens show significant time-dependent effect at higher deviatoric stresses. However, the steady-state creep rate of saturated red sandstone increases nonlinearly but the viscosity coefficient decreases gradually with increasing axial deviatoric stress and pore pressure. An exponential function is suggested to characterize the relationship between the creep parameters (including creep strain, steady-state creep rate and viscosity coefficient) and the axial deviatoric stress, and pore pressure. The calculated curves using the proposed function are compared with the experimental results, which is in a good agreement with the experimental data. In the end, the mechanism that the pore pressure and deviatoric stress affect the creep deformation of saturated red sandstone is discussed in detail. The presented experimental results on short-term and creep mechanical behavior of red sandstone under different pore pressures are very significant for evaluating the long-term stability and safety of deep underground rock mass engineering.

Elasto-plastic coupling analysis of circular openings in elasto-brittle-plastic rock mass

This paper deals with elasto-plastic coupling solutions for the prediction of displacements around circular openings in elasto-brittle-plastic rock mass subject to hydrostatic stress. Both linear Mohr–Coulomb criterion and nonlinear Hoek–Brown criterion on the basis of a non-associated flow rule are considered. The restrictive condition between strength parameters and Young’s modulus in post-failure region for rock materials is theoretically established. Meanwhile, the proposed restrictive condition is validated by FLAC3D code that the advanced elasto-plastic coupling model (EPCM) based on Mohr–Coulomb criterion is merged in. Finally, the elasto-plastic coupling dimensionless displacements of circular opening in elasto-brittle-plastic rock mass are analyzed with different deterioration degree of Young’s modulus using two kinds of rocks. The results show that the deformation of rock mass increases obviously with the decrease of Young’s modulus in the plastic region, but Young’s modulus has little influence on the plastic region radius and stresses distribution form.

Strength and deformation behavior of red sandstone under multi-stage triaxial compression

Based on multi-stage triaxial experimental results of red sandstone with circumferential deformation control, the influence of confining pressure on strength and deformation behavior of red sandstone under multi-stage triaxial compression is investigated. The results show that the confining pressure has an obvious effect on the deformation parameters of red sandstone under multi-stage triaxial compression. A multi-stage triaxial compression experiment with only one specimen can be used to confirm the peak strength of rock under different confining pressures. Under single-stage and multi-stage triaxial compression, the peak strength behavior of red sandstone agrees better with the nonlinear Hoek–Brown criterion than the linear Mohr–Coulomb criterion. However, the difference between single-stage and multi-stage triaxial strength changes with different post-peak deformation values, and the reason for the difference, is discussed. To predict single-stage triaxial strength using the obtained multi-stage triaxial strength with only one specimen, a new method is put forward to revise multi-stage triaxial strength of red sandstone, which testifies to being reasonable. The concluding remarks are very useful and significant for deep geotechnical and underground structural engineering.

Experimental Investigation on the Strength, Deformability, Failure Behavior and Acoustic Emission Locations of Red Sandstone Under Triaxial Compression

Conventional triaxial compression and “reducing confining pressure” experiments were carried out for red sandstone by an MTS815 Flex Test GT rock mechanics experimental system. Our results show that the post-peak axial deformation characteristics of red sandstone changed as the confining pressure was increased from 5 to 65 MPa. Young’s modulus of red sandstone increased nonlinearly with increasing confining pressure, but Poisson’s ratio remained unaffected. Using our new data, the compactive and dilatant behavior, strength and failure characteristics of sandstone under triaxial compression are further discussed. For our data, the nonlinear Hoek-Brown criterion better reflects the peak strength properties than the linear Mohr-Coulomb criterion. However, the residual strength shows a clear linear relationship with confining pressure, which can be best described using the linear Mohr-Coulomb criterion. The peak and residual strengths were not directly related to the two different loading paths. The onset of dilatancy (C′), the switch from compaction-dominated to dilatant-dominated behavior (D′) and the stress at zero volumetric strain all increased linearly with the confining pressure. In our conventional triaxial compression experiments, the failure mode changed from mixed tension and shear fracture (single shear fracture) to shear fracture with double slippage planes with increasing confining pressure. However, the failure mode in our “reducing confining pressure” experiments was more complicated and results mainly from the unstable failure characteristics of the rock during the reduction in confining pressure. Finally, based on our acoustic emission (AE) locations, at a confining pressure of 35 MPa, a detailed analysis of the evolutionary process of internal cracks is presented for the entire loading process.

Numerical Analysis of Damage for Y-Shape Tunnel

There are many effective researches about tunnel at home and abroad, because the complexity of design and construction for Y-shape tunnel, in public there is no research about it yet, with the background of nanliang-tunnel which merge two single-beam into a two-lane tunnel as Y-shape. This paper obtains the rock mass mechanics parameters on the basis of nonlinear Hoek-Brown criterion first, and has a numerical simulation according the tunnel construction with FLAC-3D. we arrange many monitor sections in this model and discuss the law of deformation and failure in different section, at last have a comprehensive analysis of displacement, stress, plastic zone of different sites which caused by tunnel construction and discover that: with the distance of two single tunnels decreased, the interaction caused by the merging increase together with the compressive stress, tensile stress. The displacements of surrounding rock increase corresponding, the amplitude of variation is up to 44.8%, After the two-lane tunnel is 15m long, the stress and displacements redistribution of surrounding rock become stable.