Nonlinear Failure Criterion Research Articles

Reliability analysis of three-dimensional rock slope

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

Energy analysis of rock plug thickness in karst tunnels based on non-associated flow rule and nonlinear failure criterion

The geological hazards, such as water inrush and mud outburst, are easily induced by the high water pressure caverns ahead of a karst tunnel face. Therefore, it is a pivotal issue to determine the reserved thickness of rock plug during the construction of tunnels. The limit analysis principle is employed to analyze the safe thickness from the point of energy dissipation, and the nonlinear and non-associated characteristics of geotechnical materials are both considered. On the basis of a plane failure pattern of rock plug, the expressions of detaching curve and rock plug thickness are derived. The effect of each parameter on the safe thickness of rock plug is discussed in detail, which interprets the corresponding failure scope of rock plug. The obtained results indicate that the thickness of rock plug is highly influenced by the nonlinear dilatancy coefficient and the nonlinear coefficient. The proposed method is validated by a comparison of the calculated results with those of the engineering project of the “526 karst cavern” of Yunwushan tunnel. This proposed method can provide reference basis for the design and excavation of karst tunnels in the future.

Unified Flexural Design Method for Deep and Shallow Beams Using Non-Linear Grid Model

The fundamental design theory of reinforced concrete deep beams is not consistent with that of shallow beams due to non-compliance with the plane section assumption. This paper proposes a grid model that is valid for the flexural capacity prediction and reinforcement design of both shallow and deep beams, which unifies the design process and extends the sectional analysis to deep beams. The grid model is actually based on structural analysis such that stress distribution can be calculated automatically without the plane section assumption. In order to apply sectional design principles, it assumes that design of the critical section is independent of the status of other sections, which means that other sections will remain elastic when the critical section fails. As a result, by inserting the concerned section into the grid model with non-linear material constitutive relationship and failure criteria, the capacity prediction and reinforcement design on that section can be achieved. The axial grid in the grid model guides the design of the flexural reinforcement in two kinds of layouts. Two series of experiments are listed to validate the model. The results of the proposed method are also compared with other wellestablished methods, for example traditional sectional method, layered section approach and strut-and-tie model (STM) method.

Active Earth Pressure with Linear and Nonlinear Failure Criteria

Most conventional calculation methods of active earth pressure behind a retaining wall are mainly based on the linear Mohr–Coulomb failure criterion. However, numerous experimental data indicate that various kinds of soils obey the nonlinear failure criterion. Assuming a Coulomb failure plane, an analytical expression following linear and nonlinear failure criteria using the limit equilibrium method is presented in this paper for estimating active earth pressure with a translating retaining wall. Meanwhile, both the adhesion and friction angle of the wall as well as cohesion and internal friction angle of the soil backfill are considered. Special cases show that the derived formula for the linear failure criterion can be reduced to expressions proposed by previous literature under different field conditions of soil backfill. Numerical studies show that the results of the presented method are consistent with previous solutions, which indicates that the method of this paper is feasible for evaluating active earth pressures. At last, the effects of the the nonlinear failure criterion on active earth pressure are further examined.

Three-dimensional limit analysis of seismic stability of tunnel faces with quasi-static method

Based on the existing research results, a three-dimensional failure mechanism of tunnel face was constructed. The dynamic seismic effect was taken into account on the basis of quasi-static method, and the nonlinear Mohr-Coulomb failure criterion was introduced into the limit analysis by using the tangent technique. The collapse pressure along with the failure scope of tunnel face was obtained through nonlinear limit analysis. Results show that nonlinear coefficient and initial cohesion have a significant impact on the collapse pressure and failure zone. However, horizontal seismic coefficient and vertical seismic proportional coefficient merely affect the collapse pressure and the location of failure surface. And their influences on the volume and height of failure mechanism are not obvious. By virtue of reliability theory, the influences of horizontal and vertical seismic forces on supporting pressure were discussed. Meanwhile, safety factors and supporting pressures with respect to 3 different safety levels are also obtained, which may provide references to seismic design of tunnels.

An analytical method for predicting load acting on geosynthetic overlying voids

Soil arching often occurs in geosynthetic-reinforced structures, where the underlying soil has voids, resulting in load transmission from the subsided area to surrounding less deformed area. A new method is proposed to predict load acting on gensynthetic overlying voids. The shape of soil arch and stress states of all the points at the soil arch can be obtained by combining nonlinear M-C yielding criterion, non-associated flow rule with static equilibrium of segmental arch through a dilatancy coefficient. The load applied to the geosynthetic can be determined by load transmission from the overlying soil, to the soil arch, and onto the collapsed soil. The model is verified using a model test conducted by Zhu et al. (2012), the soil pressure acting on the deflected geosynthetic is reasonably predicted. Due to the inherent nonlinear behaviour of soil, nonlinear failure criterion can better describe the stresses and deformations of the soil and geosynthetic. Soil nonlinearity has significant influence on the evaluation of arching effect. Ignoring the nonlinear behaviour of soil tends to underestimate the soil pressure acting on the geosynthetic. There exists an optimal subsidence width for which the soil pressure acting on the geosynthetic is minimal. The method used in this study is more appropriate where a large deflection occurs in the geosynthetic and provides a novel approach to evaluating soil arching under these conditions.

Combined influence of nonlinearity and dilation on slope stability evaluated by upper-bound limit analysis

The combined influence of nonlinearity and dilation on slope stability was evaluated using the upper-bound limit analysis theorem. The mechanism of slope collapse was analyzed by dividing it into arbitrary discrete soil blocks with the nonlinear Mohr–Coulomb failure criterion and nonassociated flow rule. The multipoint tangent (multi-tangent) technique was used to analyze the slope stability by linearizing the nonlinear failure criterion. A general expression for the slope safety factor was derived based on the virtual work principle and the strength reduction technique, and the global slope safety factor can be obtained by the optimization method of nonlinear sequential quadratic programming. The results show better agreement with previous research result when the nonlinear failure criterion reduces to a linear failure criterion or the non-associated flow rule reduces to an associated flow rule, which demonstrates the rationality of the presented method. Slope safety factors calculated by the multi-tangent inclined-slices technique were smaller than those obtained by the traditional single-tangent inclined-slices technique. The results show that the multi-tangent inclined-slices technique is a safe and effective method of slope stability limit analysis. The combined effect of nonlinearity and dilation on slope stability was analyzed, and the parameter analysis indicates that nonlinearity and dilation have significant influence on the result of slope stability analysis.

An iterative method for the linearization of nonlinear failure criteria

Linearization of a nonlinear failure criterion may be required if a software incorporating the nonlinear failure criterion is not readily available or if an additional computational difficulty is encountered due to the nonlinearity of the failure criterion. In this paper, a novel method is proposed for the linearization of nonlinear failure criteria. The proposed method is based on an iterative procedure and the least-squares regression is used for the linearization. The material is assumed to be elastic-perfectly plastic. By conducting analytical and finite element analyses of stresses and displacements around underground openings in rock mass governed by the generalized Hoek–Brown failure criterion, it is shown that the proposed method is convergent, efficient and effective.

A generalized nonlinear failure criterion for frictional materials

A generalized nonlinear failure criterion formulated in terms of stress invariants is proposed for describing the failure characteristics of different frictional materials. This failure criterion combines a power function and a versatile function in the meridian and deviatoric plane, respectively, which is a generalization of several classic criteria, including the Tresca, Drucker–Prager, Mohr–Coulomb, Lade–Duncan and Matsuoka–Nakai failure criterion. The procedure for determination of the strength parameters was demonstrated in detail. Comparisons between the failure criterion and experimental results were presented for uncemented/cemented Monterey sand, normally consolidated Fujinomori clay, rockfill, concrete, Mu-San sandstone and granite, which reveal that the proposed failure criterion captures experimental trend quite well.

Reliability analysis of shallow tunnel with surface settlement

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

An experimental study to characterize the initiation of the seismic-induced Tsaoling rock avalanche

The Tsaoling rock avalanche, which had a sliding volume of 120millionm3 and a maximum thickness of 170m, was triggered by the Chi-Chi earthquake in 1999. However, the initiation mechanism has not been well understood because different friction models have been proposed to describe the triggering process using the Newmark sliding model and numerical simulations. In this study, a new dynamic ring-shear test with a maximum normal stress higher than 3400kPa is applied as an alternative approach to investigate the initiation of the seismic-induced rock avalanche along the wet sliding surface at the interface of the Cholan sandstone and consolidated remold shale. Experimental results verify that the seismic stresses at the wet sliding surface touch the failure surface and generate significant cumulative shear displacements. Additionally, the shear surface should have non-linear failure criteria rather than the well-known linear Mohr-Coulomb failure criteria with a constant friction angle. Furthermore, the initiation time of the Tsaoling rock avalanche obtained from the dynamic ring-shear test is strongly correlated with the available seismic signal interpretations and the interviews of witnesses who travelled with the sliding mass. This is the first experiment clarifying the co-seismic behavior of the sliding surface and the initiation of the Tsaoling landslide during the Chi-Chi earthquake. Additionally, from the development of the ring-shear test, the integrations of high normal stress, the wet sandstone/shale interface, and the dynamic test with seismic loadings greatly extend the use of ring-shear tests to rock avalanches.

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

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

Effects of nonlinear strength parameters on stability of 3D soil slopes

Actual slope stability problems have three-dimensional (3D) characteristics and the soils of slopes have curved failure envelopes. This incorporates a power-law nonlinear failure criterion into the kinematic approach of limit analysis to conduct the evaluation of the stability of 3D slopes. A tangential technique is adopted to simplify the nonlinear failure criterion in the form of equivalent Mohr-Coulomb strength parameters. A class of 3D admissible rotational failure mechanisms is selected for soil slopes including three types of failure mechanisms: face failure, base failure, and toe failure. The upper-bound solutions and corresponding critical slip surfaces can be obtained by an efficient optimization method. The results indicate that the nonlinear parameters have significant influences on the assessment of slope stability, especially on the type of failure mechanism. The effects of nonlinear parameters appear to be pronounced for gentle slopes constrained to a narrow width. Compared with the solutions derived from plane-strain analysis, the 3D solutions are more sensitive to the values of nonlinear parameters.

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.

Study on mechanism of macro failure and micro fracture of local nearly horizontal stratum in super-large section and deep buried tunnel

The stability of surrounding rock will be poor when the tunnel is excavated through nearly horizontal stratum. In this paper, the instability mechanism of local nearly horizontal stratum in super-large section and deep buried tunnel is revealed by the analysis of the macro failure and micro fracture. A structural model is proposed to explain the mechanics of surrounding rock collapse under the action of stress redistribution and shed light on the macroscopic analytical approach of the stability of surrounding rock. Then, some highly effective formulas applied in the tunnel engineering are developed according to the theory of mixed-mode micro fracture. And well-documented field case is made to demonstrate the effectiveness and accuracy of the proposed analytical methods of mixed-mode fracture. Meanwhile, in order to make the more accurate judgment about yield failure of rock mass, a series of comprehensive failure criteria are formed. In addition, the relationship between the nonlinear failure criterion and <TEX>$K_I$</TEX> and <TEX>$K_{II}$</TEX> of micro fracture is established to make the surrounding rock failure criterion more comprehensive and accurate. Further, the influence of the parameters related to the tension-shear mixed-mode fracture and compression-shear mixed-mode fracture on the propagation of rock crack is analyzed. Results show that <TEX>${\sigma}_3$</TEX> changes linearly with the change of <TEX>${\sigma}_1$</TEX>. And the change rate is related to <TEX>${\beta}$</TEX>, angle between the cracks and <TEX>${\sigma}_1$</TEX>. The proposed simple analytical approach is economical and efficient, and suitable for the analysis of local nearly horizontal stratum in super-large section and deep buried tunnel.

Three Dimensional Limit Analysis of Slope Stability with Nonlinear Criterion: Comparison the Effect of Volumetric Strain

In view of the complexity of three-dimensional analysis, traditional slope stability analysis was mainly conducted in 2-dimensional scope, namely in the case of plane strain condition. Based on the 3-dimensional failure mechanism proposed by Michalowski, three-dimensional upper bound analysis of the soil slope was presented in this paper. The two different conditions of both with and without volumetric strain taken into consideration are analyzed. The nonlinear failure criterion was introduced into analysis by virtue of the generalized tangential technique method, and the earthquake action was analyzed on the basis of pseudo-static method. The formulas of external work rate and interior energy dissipation were deduced, and the stability factors of slope with respect to diverse parameters were calculated through optimization. The results show that, with the increase of B/H, the difference between the stability factors of the two conditions decrease. Moreover, the change law of the stability factors of the different conditions is similar.

Reliability analysis of supporting pressure in tunnels based on three-dimensional failure mechanism

Based on nonlinear failure criterion, a three-dimensional failure mechanism of the possible collapse of deep tunnel is presented with limit analysis theory. Support pressure is taken into consideration in the virtual work equation performed under the upper bound theorem. It is necessary to point out that the properties of surrounding rock mass plays a vital role in the shape of collapsing rock mass. The first order reliability method and Monte Carlo simulation method are then employed to analyze the stability of presented mechanism. Different rock parameters are considered random variables to value the corresponding reliability index with an increasing applied support pressure. The reliability indexes calculated by two methods are in good agreement. Sensitivity analysis was performed and the influence of coefficient variation of rock parameters was discussed. It is shown that the tensile strength plays a much more important role in reliability index than dimensionless parameter, and that small changes occurring in the coefficient of variation would make great influence of reliability index. Thus, significant attention should be paid to the properties of surrounding rock mass and the applied support pressure to maintain the stability of tunnel can be determined for a given reliability index.

Safe retaining pressures for pressurized tunnel face using nonlinear failure criterion and reliability theory

Based on the active failure mechanism and passive failure mechanism for a pressurized tunnel face, the analytical solutions of the minimum collapse pressure and maximum blowout pressure that could maintain the stability of pressurized tunnel faces were deduced using limit analysis in conjunction with nonlinear failure criterion under the condition of pore water pressure. Due to the objective existence of the parameter randomness of soil, the statistical properties of random variables were determined by the maximum entropy principle, and the Monte Carlo method was employed to calculate the failure probability of a pressurized tunnel. The results show that the randomness of soil parameters exerts great influence on the stability of a pressurized tunnel, which indicates that the research should be done on the topic of determination of statistical distribution for geotechnical parameters and the level of variability. For the failure probability of a pressurized tunnel under multiple failure modes, the corresponding safe retaining pressures and optimal range of safe retaining pressures are calculated by introducing allowable failure probability and minimum allowable failure probability. The results can provide practical use in the pressurized tunnel engineering.

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.

Upper Bound Solution of Safety Factor for Shallow Tunnels Face Using a Nonlinear Failure Criterion and Shear Strength Reduction Technique

A method to evaluate the stability of tunnel face is proposed in the framework of upper bound theorem. The safety factor which is widely applied in slope stability analysis is introduced to estimate the stability of tunnel face using the upper bound theorem of limit analysis in conjunction with a strength reduction technique. Considering almost all geomaterials following a nonlinear failure criterion, a generalized tangential technique is used to calculate the external work and internal energy dissipation in the kinematically admissible velocity field. The upper bound solution of safety factor is obtained by optimization calculation. To evaluate the validity of the method proposed in this paper, the safety factor is compared with those calculated by limit equilibrium method. The comparison shows the solutions derived from these two methods match each other well, which shows the method proposed in this paper can be considered as effective.