Influence of joint spacing and rock characteristics on the toppling stability of cut rock slope through a simplified limit equilibrium method

Toppling failure of rock mass/soil slope is an important geological and environmental problem.Clarifying its failure mechanism under different conditions has great significance in engineering.The toppling failure of a cutting slope occurred in a hydropower station in Kyushu,Japan illustrates that the joint characteristic played a significant role in the occurrence of rock slope tipping failure.Thus,in order to consider the mechanical properties of jointed rock mass and the influence of geometric conditions,a simplified analytical approach based on the limit equilibrium method for modeling the flexural toppling of cut rock slopes is proposed to consider the influence of the mechanical properties and geometry condition of jointed rock mass.The theoretical solution is compared with the numerical solution taking Kyushu Hydropower Station in Japan as one case,and it is found that the theoretical solution obtained by the simplified analysis method is consistent with the numerical analytical solution,thus verifying the accuracy of the simplified method.Meanwhile,the Goodman-Bray approach conventionally used in engineering practice is improved according to the analytical results.The results show that the allowable slope angle may be obtained by the improved Goodman-Bray approach considering the joint spacing,the joint frictional angle and the tensile strength of rock mass together.

An approach for determination of lateral limit angle in kinematic planar sliding analysis for rock slopes

Planar sliding is one of the frequently observed types of failure in rock slopes.Kinematic analysis is a classic and widely used method to examine the potential failure modes in rock masses.The accuracy of planar sliding kinematic analysis is significantly influenced by the value assigned to the lateral limit angleγlim.However,the assignment ofγlim is currently used generally based on an empirical criterion.This study aims to propose an approach for determining the value ofγlim in deterministic and probabilistic kinematic planar sliding analysis.A new perspective is presented to reveal thatγlim essentially influences the probability of forming a potential planar sliding block.The procedure to calculate this probability is introduced using the block theory method.It is found that the probability is correlated with the number of discontinuity sets presented in rock masses.Thus,different values ofγlim for rock masses with different sets of discontinuities are recommended in both probabilistic and deterministic planar sliding kinematic analyses;whereas a fixed value ofγlim is commonly assigned to different types of rock masses in traditional method.Finally,an engineering case was used to compare the proposed and traditional kinematic analysis methods.The error rates of the traditional method vary from 45%to 119%,while that of the proposed method ranges between 1%and 17%.Therefore,it is likely that the proposed method is superior to the traditional one.

Limit load and failure mechanisms of a vertical Hoek-Brown rock slope

The problem considered in this short note is the limit load determination of a vertical rock slope.The classical limit theorem is employed with the use of adaptive finite elements and nonlinear programming to determine upper and lower bound limit loads of a Hoek-Brown vertical rock slope.The objective function of the mathematical programming problem is such as to optimize a boundary load,which is known as the limit load,resembling the ultimate bearing capacity of a strip footing.While focusing on the vertical slope,parametric studies are carried out for several dimensionless ratios such as the dimensionless footing distance ratio,the dimensionless height ratio,and the dimensionless rock strength ratio.A comprehensive set of design charts is presented,and failure envelopes shown with the results explained in terms of three identified failure mechanisms,i.e.the face,the toe,and the Prandtl-type failures.These novel results can be used with great confidence in design practice,in particularly noting that the current industry-based design procedures for the presented problem are rarely found.

Effect of surface crack on the bearing capacity of strip footing placed on rock mass

In hilly regions,the existence of surface cracks in rock mass induces a potential threat to structural stability.Thus,the present research aims to explore the impact of surface cracks on the loadbearing capacity of strip footing placed on the rock mass.By taking into account the various boundary constraints across the surface of crack edges,the study investigates the presence of two categories of surface cracks,namely(1)separated crack,and(2)fine crack.The lower bound limit analysis is employed in conjunction with the finite element method(LBFELA)to conduct the numerical analysis.In order to evaluate rock mass yielding,the power conic programming(PCP)method is utilized to implement the generalized Hoek-Brown(GHB)failure criterion.The stability of the strip footing is analyzed by determining the bearing capacity factor(Nσγ),which is presented in the form of design charts by varying the strength parameters of rock,including the Geological Strength Index(GSI),Hoek-Brown material parameter(mi),Disturbance factor(D),and Normalised Uniaxial Compressive Strength(σci/γB),whereγis the unit weight of rock mass,and B is the width of strip footing.The study also investigates the impact of cracks on strip footings,considering different positions of the crack(LC)and depths of the crack(DC).The results demonstrate that the influence of the fine crack is only noticeable until the LC/B ratio reaches 6.However,for the separated crack,its impact remains significant even when the LC/B ratio exceeds 16.The appearance of fine crack at the edge of the footing results in a decrease in the magnitude Nσγof up to 45%,indicating a substantial reduction in the stability of the footing.The failure patterns are presented and discussed in detail for various cases in this study to examine the effect of surface cracks on the strip footing and to address the extent of the plastic collapse.

Overhanging rock slope by design:An integrated approach using rock mass strength characterisation,large-scale numerical modelling and limit equilibrium methods

Overhanging rock slopes(steeper than 90°) are typically avoided in rock engineering design, particularly where the scale of the slope exceeds the scale of fracturing present in the rock mass. This paper highlights an integrated approach of designing overhanging rock slopes where the relative dimensions of the slope exceed the scale of fracturing and the rock mass failure needs to be considered rather than kinematic release of individual blocks. The key to the method is a simplified limit equilibrium(LE) tool that was used for the support design and analysis of a multi-faceted overhanging rock slope. The overhanging slopes required complex geometries with constantly changing orientations. The overhanging rock varied in height from 30 m to 66 m. Geomechanical modelling combined with discrete fracture network(DFN)representation of the rock mass was used to validate the rock mass strength assumptions and the failure mechanism assumed in the LE model. The advantage of the simplified LE method is that buttress and support design iterations(along with sensitivity analysis of design parameters) can be completed for various cross-sections along the proposed overhanging rock sections in an efficient manner, compared to the more time-intensive, sophisticated methods that were used for the initial validation. The method described presents the development of this design tool and assumptions made for a specific overhanging rock slope design. Other locations will have different geological conditions that can control the potential behaviour of rock slopes, however, the approach presented can be applied as a general guiding design principle for overhanging rock cut slope.

Force analysis of pile foundation in rock slope based on upper-bound theorem of limit

Based on the characteristic that the potential sliding surfaces of rock slope are commonly in the shape of either line or fold line,analysis thought of conventional pile foundation in the flat ground under complex load condition was applied and the upper-bound theorem of limit analysis was used to compute thrust of rock layers with all possible distribution shapes. The interaction of slope and pile was considered design load in terms of slope thrust,and the finite difference method was derived to calculate inner-force and displacement of bridge pile foundation in rock slope under complex load condition. The result of example shows that the distribution model of slope thrust has certain impact on displacement and inner-force of bridge pile foundation. The maximum displacement growth rate reaches 54% and the maximum moment and shear growth rates reach only 15% and 20%,respectively,but the trends of inner-force and displacement of bridge pile foundation are basically the same as those of the conventional pile foundation in the flat ground. When the piles bear the same level lateral thrust,the distribution shapes of slope thrust have different influence on inner-force of pile foundation,especially the rectangle distribution,and the triangle thrust has the smallest displacement and inner-force of pile foundation.

Mechanism of energy limit equilibrium of rock burst in coal mine

With the increase of mining depth,the effect of rock burst on coal mining is becoming more and more obvious and the rock burst mechanism becomes more and more complicated.Scholars from many countries had put forward different mechanisms,but no one gave a reasonable explanation to the mechanism of rock burst.In this paper,based on the energy theories,we studied the energy limit equilibrium（ELE） of coal mine rock burst The coal seam with rock burst is divided into energy limit equilibrium zone（ELEZ）（A） and elastic zone（B）;we also determined the position where the rock burst occurs,including the roof and floor of coal seams;in addition,we derived the limit width of ELEZ and the mathematic relationship between the limit width and occurrence mechanism of rock burst：the energy difference function（EDF）,w（x） = w_j - w_p,because first-order derivative w＇（x）,is less than 0.So EDF is a monotonically decreasing function.The graph of the energy difference function was also determined, through which we analysed the occurrence mechanism of rock burst.

LIMIT EQUILIBRIUM STATE OF ISOTROPIC ROCKS AROUND THE TUNNEL OUTLINE

The assumed presupposition of limit equilibrium theory is recounted. Based on this, the expressive formula of radius of the non-elastic zone and rock displacement of tunnel outline was given.

Limit variation analysis of shallow rectangular tunnels collapsing with double-layer rock mass based on a three-dimensional failure mechanism

In the framework of upper bound theorem of limit analysis, the progressive collapse of shallow rectangular tunnels with double-layer rock mass has been theoretically analyzed based on the three-dimensional (3D) velocity discontinuity surfaces. According to the virtual work principle, the difference theorem and the variation method, the collapse surface of double-layer rock mass is determined based on the Hoek-Brown failure criterion. The formula can be degenerated to a single-layer rock collapsing problem when the rock mass is homogeneous. To estimate the validity of the result, the numerical simulation software PLAXIS 3D is used to simulate the collapse of shallow tunnels with double-layer rock mass, and the comparative analysis shows that numerical results are in good agreement with upper-bound solutions. According to the results of parametric analysis, the potential range of collapse of a double-layer rock mass above a shallow cavity decreases with a decrease in A1/A2,σci1/σci2 and σtm1/σtm2 and an increase in B1/B2,γ1/γ2. The range will decrease with a decrease in support pressure q and increase with a decrease in surface overload σs. Therefore, reinforced supporting is beneficial to improve the stability of the cavity during actual construction.

The influence of inter-band rock on rib spalling in longwall panel with large mining height

In order to improve rib stability,failure criteria and instability mode of a thick coal seam with inter-band rock layer are analysed in this study.A three-dimensional mechanical model is established for the rib by considering the rock layer.A safety factor is defined foy the rib,and it is observed that the safety factor exhibits a positive correlation with the thickness and strength of the inter-band rock.A calculation method for determining critical parameters of the rock layer is presented to ensure the rib stability.It is revealed that incomplete propagation of the fracture at the hard rock constitutes a fundamental prerequisite for ensuring the rib stability.The influence of the position of the inter-band rock in the coal seam on failure mechanism of the rib was thoroughly investigated by developing a series of physical models for the rib at the face area.The best position for the inter-band rock in the coal seam is at a height of 1.5 m away from the roof line,which tends to provide a good stability state for the rib.For different inter-band rock positions,two ways of controlling rib by increasing supports stiffness and flexible grouting reinforcement are proposed.

Effect of eccentric and inclined loading on the bearing capacity of strip footing placed on rock mass

This paper deals with the bearing capacity determination of strip footing on a rock mass in hilly area by considering the influence of inclined and eccentric loading. Applying the generalized HoekBrown failure criterion, the failure behavior of the rock mass is modeled with the help of the power cone programming in the lower bound finite element limit analysis framework. Using bearing capacity factor(Ns), the change in bearing capacity of the strip footing due to the occurrence of eccentrically inclined loading is presented. The variations of the magnitude of Ns are obtained by examining the effects of the Hoek-Brown rock mass strength parameters(uniaxial compressive strength(sci), disturbance factor(D), rock parameter(mi), and Geological Strength Index(GSI)) in the presence of different magnitudes of eccentricity(e) and inclination angle(λ) with respect to the vertical plane, and presented as design charts. Both the inclined loading modes, i.e., inclination towards the center of strip footing(+λ) and inclination away from the center of strip footing(-λ), are adopted to perform the investigation. In addition, the correlation between the input parameters and the corresponding output is developed by utilizing the artificial neural network(ANN). Additionally, from sensitivity analysis, it is observed that inclination angle(λ) is the most sensitive parameter. For practicing engineers, the obtained design equation and design charts can be beneficial to understand the bearing capacity variation in the existence of eccentrically inclined loading in mountain areas.

3D limit analysis of rock slopes based on equivalent linear failure criterion with tension cut-off

Hoek-Brown (HB) failure criterion is widely used to predict the strength of intact or heavily jointed rock mass. For stability analysis of rock slopes governed by the HB failure criterion, the equivalent linearity to Mohr-Coulomb (MC) criterion is often adopted, leading to the well-known equivalent Mohr-Coulomb method (EMCM). Existing studies on EMCM analysis mainly consider the shear strength of rock material, while consideration of the tensile strength is rare. This contradicts the fact that the underlying tensile strength of rock mass has considerable impact on the rock slope stability in real world. In this regard, this paper proposes a limit analysis-based approach that can account for tension in the three-dimensional (3D) stability analysis of HB rock slope. This approach is established on the equivalent linearity of the HB criterion with consideration of tensile strength, known as the equivalent tension cut-off MC method (ETMCM), and using a horn-like 3D mechanism of limit analysis. The safety factor solutions given by the proposed approach are validated by previous studies and numerical results. Parametric studies are conducted to investigate the effect of rock tensile strength on slope stability. Results show that the consideration of tension leads to a more conservative safety factor and a sharper curvature of the failure surface, and these impacts tend to be more obvious with the increases in slope inclination and slope width. Finally, the stability of the HB rock slope under seepage conditions is studied using the proposed approach. The results indicate that the effect of tensile strength is highly remarkable in seepage circumstances.

Comprehensive analysis of multiple machine learning techniques for rock slope failure prediction

In this study,twelve machine learning(ML)techniques are used to accurately estimate the safety factor of rock slopes(SFRS).The dataset used for developing these models consists of 344 rock slopes from various open-pit mines around Iran,evenly distributed between the training(80%)and testing(20%)datasets.The models are evaluated for accuracy using Janbu's limit equilibrium method(LEM)and commercial tool GeoStudio methods.Statistical assessment metrics show that the random forest model is the most accurate in estimating the SFRS(MSE=0.0182,R2=0.8319)and shows high agreement with the results from the LEM method.The results from the long-short-term memory(LSTM)model are the least accurate(MSE=0.037,R2=0.6618)of all the models tested.However,only the null space support vector regression(NuSVR)model performs accurately compared to the practice mode by altering the value of one parameter while maintaining the other parameters constant.It is suggested that this model would be the best one to use to calculate the SFRS.A graphical user interface for the proposed models is developed to further assist in the calculation of the SFRS for engineering difficulties.In this study,we attempt to bridge the gap between modern slope stability evaluation techniques and more conventional analysis methods.

考虑周边岩体约束的危岩体稳定性分析极限平衡法

现行规范在复杂形态危岩体稳定性分析方面存在缺乏同时考虑多个结构面影响的局限性。基于复杂形态危岩体的特点及三维离散元计算的剪应力分布情况,对现有极限平衡法进行了改进,对于规范中由上缘结构面控制的坠落式与由下缘面结构控制的滑移式危岩,均添加了对危岩体后缘结构面约束作用的考虑,对双结构面滑移式危岩体则增加了后缘结构面裂隙水压影响。采用改进的极限平衡法对上述三类危岩进行了稳定性分析,通过与三维离散法的模拟结果进行对比,结果表明:改进后的公式在计算多结构面影响下的不同复杂形态危岩体稳定性系数时,相对误差仅为3.06%~9.10%,相对于规范推荐公式13.8%~14.7%的相对误差,计算精度明显提高。改进后的公式充分考虑到了危岩体的周边约束的特点,为极限平衡法解决复杂形态危岩体问题提供了新的途径。

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.

Failure mechanism and stability control of a large section of very soft roadway surrounding rock shear slip

The measured data and simulation test phenomenon of surrounding rock deformation and failure at the project site indicate that shear failure which firstly occurs in surrounding rock, block slip and second shear failure are the root cause of deformation and damage of supporting structure of the surrounding rock at a large scale. We derived limit load of surrounding rock shear slip failure and reasonable support resistance of given load by means of shear slip line field theory, discussed the main factors which influence the limit load of surrounding rock. Shear slip line field and limit load of circular tunnel surrounding rock were obtained by means of physical simulation test, which agreed well with the theoretical analysis results. Based on the theoretical analysis and physical simulation test, the cause deformation and failure at large scale of Xinshanghai No. 1 coal mine big section ingate was analyzed, and the shear failure resistance and block slip in surrounding rock were proposed as the core technical supporting ideas. Proper range of supporting resistance which came from calculation was suggested. The support scheme which is mainly composed of large grouting anchor, sprayed anchor net support technique and full-face grille concrete finally ended the dilemma of repeated failure and mending of ingate and created critical conditions for smooth production in the coal mine.

Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion

The stability analysis of passive bolt-reinforced rock slopes under seismic loads is investigated within the framework of the kinematic approach of limit analysis theory.A pseudo-static method is adopted to account for the inertial forces induced in the rock mass by seismic events.The strength properties of the rock material are described by a modified Hoek-Brown strength criterion,whereas the passive bolts are modeled as bar-like inclusions that exhibit only resistance to tensile-compressive forces.Taking advantage of the ability to compute closed-form expressions for the support functions associated with the modified Hoek-Brown strength criterion,a rotational failure mechanism is implemented to derive rigorous lower bound estimates for the amount of reinforcement strength to prevent slope failure.The approach is then applied to investigating the effects of relevant geometry,strength and loading parameters in light of a preliminary parametric study.The accuracy of the approach is assessed by comparison of the lower bound estimates with finite element limit analysis solutions,thus emphasizing the ability of the approach to properly predict the stability conditions and to capture the essential features of deformation localization pattern.Finally,the extension of the approach to account for slipping at the interface between reinforcements and surrounding rock mass is outlined.

Upper bound solution of surrounding rock pressure of shallow tunnel under nonlinear failure criterion

The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspective of energy balance. The work rates of external forces and internal energy dissipation are calculated. An analytical solution of necessary face pressures is derived. The optimal upper-bound solution of the face pressures is obtained by optimization. The results show that the three dimensionless parameters A, T, n of nonlinear Baker failure criterion have different effects on the necessary face pressures and the pattern failure mechanisms ahead of tunnel face. A is the most important one;n takes the second place, and T is the least one. The computed necessary face pressures are nonlinearly increasing when A is reduced. Combined with the actual monitoring data of Taxia tunnel, the calculation results in this paper is verified. It is suggested that the tunnel face supports should be strengthened timely in soft rocks to prevent the occurrence of face collapse.

Comprehensive analysis of slope stability and determination of stable slopes in the Chador-Malu iron ore mine using numerical and limit equilibrium methods

One of the critical aspects in mine design is slope stability analysis and the determination of stable slopes. In the Chador- Malu iron ore mine, one of the most important iron ore mines in central Iran, it was considered vital to perform a comprehensive slope stability analysis. At first, we divided the existing rock hosting pit into six zones and a geotechnical map was prepared. Then, the value of MRMR （Mining Rock Mass Rating） was determined for each zone. Owing to the fact that the Chador-Malu iron ore mine is located in a highly tectonic area and the rock mass completely crushed, the Hoek-Brown failure criterion was found suitable to estimate geo-mechanical parameters. After that, the value of cohesion （c） and friction angle （tp） were calculated for different geotechnical zones and relative graphs and equations were derived as a function of slope height. The stability analyses using numerical and limit equilibrium methods showed that some instability problems might occur by increasing the slope height. Therefore, stable slopes for each geotechnical zone and prepared sections were calculated and presented as a function of slope height.

Three-dimensional upper bound limit analysis of underground cavities using nonlinear Baker failure criterion

A generalized nonlinear Baker failure criterion is employed with the upper bound limit analysis to study the surrounding rock stability of underground cavities. A three-dimensional(3D) failure mode is established by extending the two-dimensional(2D) failure mode, which offers an upper bound expression of the surrounding rock pressure. This method is validated with a series of examples before the influence of four parameters of scale parameter, curvature parameter, shift parameter and lateral pressure coefficient, on the surrounding rock pressure is analyzed. According to these results, failure ranges of the underground cavities are determined. The following conclusions are reached:(1) the proposed approach is more accurate to predict surrounding rock pressure than the Mohr-Coulomb failure criterion;(2) the surrounding rock with large scale parameter, curvature parameter, shift parameter, and lateral pressure coefficient can lead to a more stable underground cavity;(3) the failure range in 3D mode can be predicted according to the upper bound solutions.