An approximate nonlinear modified Mohr-Coulomb shear strength criterion with critical state for intact rocks

In this paper, the Mohr-Coulomb shear strength criterion is modified by mobilising the cohesion and internal friction angle with normal stress, in order to capture the nonlinearity and critical state concept for intact rocks reported in the literature. The mathematical expression for the strength is the same as the classical form, but the terms of cohesion and internal friction angle depend on the normal stress now,leading to a nonlinear relationship between the strength and normal stress. It covers both the tension and compression regions with different expressions for cohesion and internal friction angle. The strengths from the two regions join continuously at the transition of zero normal stress. The part in the compression region approximately satisfies the conditions of critical state, where the maximum shear strength is reached. Due to the nonlinearity, the classical simple relationship between the parameters of cohesion, internal friction angle and uniaxial compressive strength from the linear Mohr-Coulomb criterion does not hold anymore. The equation for determining one of the three parameters in terms of the other two is supplied. This equation is nonlinear and thus a nonlinear equation solver is needed. For simplicity, the classical linear relationship is used as a local approximation. The approximate modified Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD,and an example case of deep tunnel failure is presented to demonstrate the difference between the original and modified Mohr-Coulomb criteria. It is shown that the nonlinear modified Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion. A more comprehensive piecewise nonlinear shear strength criterion is also included in Appendix B for those readers who are interested. It covers the tensile, compressive, brittle-ductile behaviour transition and the critical state, and gives smooth transitions.

Modified Mohr-Coulomb strength criterion considering rock mass intrinsic material strength factorization

With the increase of mining depth of mineral resources,the rock mass stress state is being more and more complex.The rock mass show different features,namely,with the increase of hydrostatic pressure,rock mass failure mode turns from brittle tension failure to structure ductile failure and its limit strength also increases.The restriction of minimal principal stress on the initiation and development of microcrack and the change of micro-unit stress state by the intermediate principal stress play a decisive role in the increase of rock mass limit strength.Based on the rock mass failure behavior law under complex stress state and the σ2-dependence on the rock mass strength,we proposed a Modified Mohr-Coulomb(M-MC) strength criterion which is smooth and convex.Finally,the M-MC criterion is validated by multiaxial test data of eight kinds of rock mass.We also compared the fitting results with Mohr-Coulomb criterion(MC).It shows that the new criterion fits the test data better than the Mohr-Coulomb criterion.So the M-MC strength criterion well reveals the rock mass bearing behavior and can be widely used in the rock mass strength analysis.The results can provide theoretical foundations for stability analysis and reinforcement design of complex underground engineering.

Determination of rock resistant coefficient based on Mohr-Coulomb criterion for underwater tunnel

According to the load-structure method, the wall rock with lining can bear the load caused by the surrounding rock, and the rock resistant coefficient (RRC) is a key parameter for evaluating the capacity of this wall rock. Based on the Mohr-Coulomb yield criterion, this paper develops a formula for calculating the RRC, which has been applied to the real engi-neering project, such as Xiamen Xiang’an East Passage Underwater Tunnel Project. The fact shows that this formula is helpful for designers to determine the RRC value.

Upper bound analysis of ultimate pullout capacity of shallow 3-D circular plate anchors based on nonlinear Mohr-Coulomb failure criterion

Based on the nonlinear Mohr-Coulomb failure criterion and the associated flow rules,the three-dimensional(3-D)axisymmetric failure mechanism of shallow horizontal circular plate anchors that are subjected to the ultimate pullout capacity(UPC)is determined.A derivative function of the projection function for projecting the 3-D axisymmetric failure surface on plane is deduced using the variation theory.By using difference principle,the primitive function of failure surface satisfying boundary condition and numerical solution to its corresponding ultimate pullout capacity function are obtained.The influences of nonlinear Mohr-Coulomb parameters on UPC and failure mechanism are studied.The result shows that UPC decreases with dimensionless parameter m and uniaxial tensile strength increases but increases when depth and radius of plate anchor,surface overload,initial cohesion,geomaterial density and friction angle increase.The failure surface is similar to a symmetrical spatial funnel,and its shape is mainly determined by dimensionless parameter m;the surface damage range expands with the increase of radius and depth of the plate anchor as well as initial cohesion but decreases with the increase of dimensionless parameter m and uniaxial tensile strength as well as geomaterial density.As the dimensionless parameter m=2.0,the numerical solution of UPC based on the difference principle is proved to be feasible and effective through the comparison with the exact solution.In addition,the comparison between solutions of UPC computed by variation method and those computed by upper bound method indicate that variation method outperforms upper bound method.

Cracking Patterns of Shear Walls in Reinforced Concrete Structure due to Strong Earthquake Based on Mohr-Coulomb Criterion

A primary goal of seismic design of buildings is to protect people's life safety during strong earthquakes. Fundamentally,to predict the survived space and estimate people escape intervals during structural collapse are very important,which requires to describe the failure process more detailedly not only for structural joints but also for slabs and shear walls. In the present paper,the details of Mohr-Coulomb failure criterion with tension-cutoff together with its expression in failure surface and local coordinate system of reinforced concrete( RC) shear wall are given firstly,and then several typical cracking patterns of RC shear wall,such as tension shear crack and compression shear crack, are analyzed based on Mohr-Coulomb failure criterion with tension-cutoff.

PBX模拟材料压剪应力状态下的幂函数M-C强度准则与失效特性

为了研究高聚物粘结炸药(PBX)在压剪复合应力状态下的强度特性,采用变角剪切加载方法进行了0°,15°,30°,45°,60°,75°以及90°的压剪实验研究;建立了PBX模拟材料的幂函数Mohr-Coulomb(M-C)准则,并对其围压强度进行了预测;利用SEM对压剪加载下的细观失效特性进行了分析。结果表明,PBX模拟材料在加载角度θ为0°,15°,30°,45°时主要表现为沿压剪面剪切的破坏模式,在θ为60°,75°,90°时表现为剪切和张裂的混合破坏模式;在剪切破坏模式下,随着压应力的增加,PBX模拟材料剪切强度呈非线性增加的趋势,细观损伤会从颗粒剪断、破碎逐步发展到次生裂纹的形成与贯通。分析发现,幂函数M-C模型适用于描述PBX模拟材料剪切强度的非线性变化规律,拟合得到的强度准则能较为准确地预测不同围压下的轴向压缩强度;压应力增加引起的内摩擦阻力增大和颗粒破碎、次生裂纹等引起的内摩擦系数降低是导致PBX模拟材料剪切破坏强度随压应力增加而非线性增加的重要原因。

Triaxial strength behaviour of rockmass satisfying Modified Mohr-Coulomb and Generalized Hoek-Brown criteria

The triaxial strength of twenty rockmass types was predicted using two non-linear triaxial strength criteria for rockmass i.e. Modified Mohr-Coulomb(MMC) criterion and Generalized Hoek-Brown(GHB)criterion. Four different rockmass classification systems were used for the calculation of MMC criterion parameters while only GSI classification system has been used for calculation of GHB parameters. The representative value of the uniaxial compressive strength and elastic modulus of rockmass have been estimated using probabilistic approach. A hypothetical case of an unsupported tunnel has been analyzed considering both MMC and GHB criteria. The analysis was done using the convergence-confinement method with two different approaches. The first approach predicts the tunnel response using GHB criterion directly. The second approach predicts the tunnel response using equivalent Mohr-Coulomb parameters obtained by linearization of triaxial data points obtained from MMC and GHB criteria. The tunnel response has been estimated in terms of radius of plastic zone, tunnel convergence and tunnel convergence strain. For very poor rockmasses the tunnel response predicted by MMC criterion is less than that predicted by GHB criterion. For poor and fair rockmass, the tunnel response estimated considering both the criteria are comparable except for few cases. Squeezing condition in rockmass has been also evaluated.

Limit equilibrium analysis for rock slope stability using basic Hoek–Brown strength criterion

Hoek–Brown(HB)strength criterion can reflect rock’s inherent failure nature,so it is more suitable for analyzing the stability of rock slopes.However,the traditional limit equilibrium methods are at present only suitable for analyzing the rock slope stability using the linear equivalent Mohr–Coulomb(EMC)strength parameters instead of the nonlinear HB strength criterion.Therefore,a new method derived to analyze directly the rock slope stability using the nonlinear HB strength criterion for arbitrary curve slip surface was described in the limit equilibrium framework.The current method was established based on certain assumptions concerning the stresses on the slip surface through amending the initial normal stressσ0 obtained without considering the effect of inter-slice forces,and it can satisfy all static equilibrium conditions of the sliding body,so the current method can obtain the reasonable and strict factor of safety(FOS)solutions.Compared with the results of other methods in some examples,the feasibility of the current method was verified.Meanwhile,the parametric analysis shows that the slope angleβhas an important influence on the difference of the results obtained using the nonlinear HB strength criterion and its linear EMC strength parameters.Forβ≤45°,both of the results are similar,showing the traditional limit equilibrium methods using the linear EMC strength parameters and the current method are all suitable to analyze rock slope stability,but forβ>60°,the differences of both the results are obvious,showing the actual slope stability state can not be reflected in the traditional limit equilibrium methods,and then the current method should be used.

Validation of Hoek-Brown failure criterion charts for rock slopes

Although stability charts suggested by Hoek and Bray on the basis of Mohr-Coulomb criterion are used for rock slopes,but complete and precise recognition is required for distinguishing cohesive strength and Mohr-Coulomb equivalent internal friction angle for rock mass.The paper by Lia et al.[6]is the only one that introduced rock slope charts according to Hoek-Brown failure criterion.In this paper,at first,this type of charts is introduced.Then,Mohr-Coulomb failure criterion charts[2]are compared and validated with Hoek-Brown failure criterion ones[6].Next,Bishop method utilizing Slide software is compared with Hoek-Brown failure criterion stability charts.Average standard deviation(ASD),root mean square error(RMSE)and variance account for(VAF)were used for the comparison.According to the results,because of high distribution and very low correlation among the comparisons,Hoek-Brown failure criterion charts are not efficient.

Mohr-Coulomb准则模型的能量描述

以Mohr-Coulomb(M-C)准则的力学模型为基础,建立了材料压缩破坏面上一点的能量函数。该函数的极值条件给出了破坏面的位向,其最大值确定了真实破坏能。真实破坏能和裂纹表面能的能量守恒临界条件导出了一个新的材料破坏强度关系。两向压缩时,这个关系与M-C准则形式上完全一致。在体积变形饱和近似条件下,它与侧限压缩时的M-C准则存在确定性联系。体积应力的引入使这个强度关系克服了经典M-C准则无法考虑到中间主应力对破坏强度影响的不足。理论与试验结果的对比分析表明,M-C准则模型的能量描述具有更好的材料压缩破坏强度预测能力。

Elastoplastic homogenization of particulate composites complying with the Mohr–Coulomb criterion and undergoing isotropic loading

This work aims at determining the overall response of a two-phase elastoplastic composite to isotropic loading. The composite under investigation consists of elastic particles embedded in an elastic perfectly plastic matrix governed by the Mohr-Coulomb yield criterion and a non-associated plastic flow rule. The composite sphere assemblage model is adopted, and closed-form estimates are derived for the effective elastoplastic properties of the composite either under tensile or compressive isotropic loading. In the case when elastic particles reduce to voids, the composite in question degenerates into a porous elastoplastic material. The results obtained in the present work are of interest, in particular, for soil mechanics.

理想M-C材料强度评估判据与等效塑性应变

基于理想Mohr-Coulomb(M-C)屈服准则,该文提出了拉伸、压缩和剪切等效应力的概念及其公式,并给出了三个相应的强度评估条件.根据塑性功等效原则,分别导出了与上述等效应力共轭的拉伸、压缩等效塑性应变和等效塑性剪应变,探论了不同的摩擦因数下等效应变的变化特征.与Mises等效应变不同,所得到的M-C等效应变能够反映静水压力的影响,也可退化为简单应力状态.这些等效应力和等效应变概念都具有明确的物理意义,将能够应用于更准确、有效地评估拉、压性能不同材料的强度,对于用简单拉伸、压缩或剪切试验标定复杂应力状态下本构模型参数也具有直接应用价值.

Limit equilibrium method(LEM) of slope stability and calculation of comprehensive factor of safety with double strength-reduction technique

When the slope is in critical limit equilibrium(LE) state, the strength parameters have different contribution to each other on maintaining slope stability. That is to say that the strength parameters are not simultaneously reduced. Hence, the LE stress method is established to analyze the slope stability by employing the double strengthreduction(DSR) technique in this work. For calculation model of slope stability under the DSR technique, the general nonlinear Mohr–Coulomb(M–C) criterion is used to describe the shear failure of slope. Meanwhile, the average and polar diameter methods via the DSR technique are both adopted to calculate the comprehensive factor of safety(FOS) of slope. To extend the application of the polar diameter method, the original method is improved in the proposed method. After comparison and analysis on some slope examples, the proposed method's feasibility is verified. Thereafter, the stability charts of slope suitable for engineering application are drawn. Moreover, the studies show that:(1) the average method yields similar results as that of the polardiameter method;(2) compared with the traditional uniform strength-reduction(USR) technique, the slope stability obtained using the DSR techniquetends to be more unsafe; and(3) for a slope in the critical LE state, the strength parameter φ, i.e., internal friction angle, has greater contribution on the slope stability than the strength parameters c, i.e., cohesion.

Strength behaviour of sandstone subjected to polyaxial state of stress

The paper presents an experimental investigation on the strength behaviour of natural rock subjected to polyaxial state of stress. The polyaxial tests were conducted on cubical specimens of sandstone obtained from the Shivpuri district in Madhya Pradesh state of India, The specimens having nominal dimensions of100 mm x 100 mm x 100 mm were tested using a polyaxial testing machine. Twenty-five combinations of intermediate and minor principal stresses were applied and the specimens were loaded till failure occurs. It was observed that the intermediate principal stress has a substantial effect on the strength of the Shivpuri sandstone. A database of rock strength under various combinations of σ_2 and σ_3 was obtained for the Shivpuri sandstone. The database was used to study the predictability of five most commonly used strength criterion. Root mean square error(RMSE), average absolute relative error percentage(AAREP) and coefficient of accordance(COA) were used as indices for the measure of goodness of fit. It was observed that the least error in the prediction was shown by modified Mohr Coulomb criterion followed by modified Weibols and Cook criterion. A probability analysis of the error in prediction was also done.

Nonlinear shear behavior of rock joints using a linearized implementation of the Barton-Bandis model

Experiments on rock joint behaviors have shown that joint surface roughness is mobilized under shearing,inducing dilation and resulting in nonlinear joint shear strength and shear stress vs.shear displacement behaviors.The Barton-Bandis(B-B) joint model provides the most realistic prediction for the nonlinear shear behavior of rock joints.The B-B model accounts for asperity roughness and strength through the joint roughness coefficient(JRC) and joint wall compressive strength(JCS) parameters.Nevertheless,many computer codes for rock engineering analysis still use the constant shear strength parameters from the linear Mohr-Coulomb(M-C) model,which is only appropriate for smooth and non-dilatant joints.This limitation prevents fractured rock models from capturing the nonlinearity of joint shear behavior.To bridge the B-B and the M C models,this paper aims to provide a linearized implementation of the B-B model using a tangential technique to obtain the equivalent M-C parameters that can satisfy the nonlinear shear behavior of rock joints.These equivalent parameters,namely the equivalent peak cohesion,friction angle,and dilation angle,are then converted into their mobilized forms to account for the mobilization and degradation of JRC under shearing.The conversion is done by expressing JRC in the equivalent peak parameters as functions of joint shear displacement using proposed hyperbolic and logarithmic functions at the pre-and post-peak regions of shear displacement,respectively.Likewise,the pre-and post-peak joint shear stiffnesses are derived so that a complete shear stress-shear displacement relationship can be established.Verifications of the linearized implementation of the B-B model show that the shear stress-shear displacement curves,the dilation behavior,and the shear strength envelopes of rock joints are consistent with available experimental and numerical results.

Interaction mechanism of the interface between a deep buried sand and a paleo-weathered rock mass using a high normal stress direct shear apparatus

In order to evaluate the feasibility of safe mining close to the contact zone under reduced security coal pillar conditions at a coal mine in eastern China, the interaction mechanism of the interface between deep buried sand and a paleo-weathered rock mass was investigated in the laboratory by direct shear testing. A DRS-1 high pressure soil shear testing machine and orthogonal design method were used in the direct shear tests. Variance and range methods were applied to analyze the sensitivity of each factor that has an influence on the mechanical characters of the interface. The test results show that the normal pressure is the main influencing factor for mechanical characteristics of the interface, while the lithological characters and roughness are minor factors; the shear stress against shear displacement curve for the interface shows an overall hyperbola relationship, no obvious peak stress and dilatancy was observed.When the normal pressure is 6 MPa, the shear strengths of interfaces with different roughness are basically the same, and when the normal pressure is more than 8 MPa, the larger the roughness of the interface, the larger will be the shear strength; the shear strength has a better linear relationship with the normal pressure, which can be described by a linear Mohr–Coulomb criterion.

Constitutive models in stability analysis of rock slope

Equivalent Mohr-Coulomb yield criterion was established,and the relationship between different constitutive models was studied.The application of equivalent Mohr-Coulomb yield criterion in Ansys was achieved by means of transforming material parameters.The stability research aiming at the most common rock slope without conspicuous slide surface was accomplished,the methods of measurably assessing the stability of rock slope without conspicuous slide surface were explored,and the disadvantages of method of minimum slide-resisted reserve as dangerous slide path were pointed out.The results show that through the calculation and analysis of cases,the conception that measurable assessment of the stability of rock slope without conspicuous slide surface can be achieved under condition that equivalent Mohr-Coulomb yield criterion is validated.Its safety parameter formula is explicit in theory and credible in results.The results obtained are approximate to those obtained by using finite element intensity reducing method.

Establishment of tensile failure induced sanding onset prediction model for cased-perforated gas wells

Sand production is a challenging issue in upstream oil and gas industry,causing operational and safety problems.Therefore,before drilling the wells,it is essential to predict and evaluate sanding onset of the wells.In this paper,new poroelastoplastic stress solutions around the perforation tunnel and tip based on the Mohr-Coulomb criterion are presented firstly.Based on the stress models,a tensile failure induced sanding onset prediction model for cased-perforated gas wells is derived.Then the analytical model is applied to field data to verify its applicability.The results from the perforation tip tensile failure induced sanding model are very close to field data.Therefore,this model is recommended for forecasting the critical conditions of sand production analysis.Such predictions are necessary for providing technical support for sand control decision-making and predicting the production condition at which sanding onset occurs.

Non-linear Analysis of Bridge“Guglie”in Venice:Sensitivity to Designed and Realized Shape

In this paper,numerical non-linear analyses of the “Guglie” bridge,located in the historical city of Venice(Italy),are proposed.The focus is twofold:on the sensitivity of the realized shape of the bridge by reference to originally designed shape;and on sensitivity to mechanical parameters of constituent materials.The history of this bridge is very interesting,and the bridge actually built is different from the Marchesini’s project(1580).In fact,in the original design drawing,the shape of the arch is a perfect circle arch,while the shape of the actually built arch is not perfectly circular.Hence,in the aim to evaluate sensitivity of bridge behaviour to designed and realized shape,non-linear analyses are carried on by means of FEMs(Finite Element Models)under in-plane state hypothesis.Furthermore,parametric tests are also performed for evaluating the influence of masonry mechanical parameters on non-linear bridge behaviour.