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.

Fatigue Resistance of RCA Modified Asphalt Based on Dielectric Properties

In order to study the anti-fatigue performance of RCA modified asphalt (RMA),the performance of RMA and 90#matrix asphalt with different modifier content were measured by asphalt penetration,ductility,softening point,Brookfield viscosity,rheological index,infrared spectrum and dielectric constant test.This paper discusses the changes of asphalt basic indexes,fatigue properties and asphalt components based on dielectric properties under different modifier contents,and analyzes the grey correlation degree between components and asphalt pavement performance indexes.The results show that the optimum content of RCA modifier is 16.7%of the asphalt quality according to the penetration,ductility,softening point,Brockfield viscosity,viscosity temperature curve and fatigue life.In the phase angle-strain curve,there is disorder in the latter part of the curve.According to the strain (ε_(d)) corresponding to the disorder point,a new fatigue failure criterion is proposed and proved.Based on the new asphalt fatigue failure criterion,the fatigue prediction model of asphalt mixture is improved,and the fatigue life predicted by the improved fatigue model is compared with the fatigue life obtained by four-point bending fatigue test.The results show that the proposed new asphalt fatigue failure criterion is reasonable,and the fatigue life predicted by the improved asphalt mixture fatigue prediction model is accurate.The research method of classifying asphalt components based on dielectric properties is simple and effective,and the components have a high correlation with the road performance of base asphalt and modified asphalt.

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.

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.

A modified Hoek-Brown failure criterion considering the damage to reservoir bank slope rocks under water saturation-dehydration circulation

After water is impounded in a reservoir, rock mass in the hydro-fluctuation belt of the reservoir bank slope is subject to water saturation- dehydration circulation （WSDC）. To quantify the rate of change of rock mechanical properties, samples from the Longtan dam area were measured with uniaxial compression tests after different numbers （1, 5, 10, 15, and 20） of simulated WSDC cycles. Based on the curves derived from these tests, a modified Hock- Brown failure criterion was proposed, in which a new parameter was introduced to model the cumulative damage to rocks after WSDC. A case of an engineering application was analyzed, and the results showed that the modified Hock-Brown failure criterion is useful. Under similar WSDC-influenced engineering and geological conditions, rock mass strength parameters required for analysis and evaluation of rock slope stability can be estimated according to this modified Hoek-Brown failure criterion.

Modified Burzynski criterion with non-associated flow rule for anisotropic asymmetric metals in plane stress problems

The Burzynski criterion is developed for anisotropic asymmetric metals with the non-associated flow rule （NAFR） for plane stress problems. The presented pressure depending on the yield criterion can be calibrated with ten experimental data, i.e., the tensile yield stresses at 0°, 45°, and 90°, the compressive yield stresses at 0°, 15°, 30°, 45°, 75°, and 90° from the rolling direction, and the biaxial tensile yield stress. The corresponding pressure independent plastic potential function can be calibrated with six experimental data, i.e., the tensile R-values at 0°, 15°, 45°, 75°, and 90° from the rolling direction and the tensile biaxial R-value. The downhill simplex method is used to solve these ten and six high nonlinear equations for the yield and plastic potential functions, respectively. The results show that the presented new criterion is appropriate for anisotropic asymmetric metals.

修正Mohr-Coulomb准则下倾斜基岩中桩端极限承载力研究

为计算倾斜岩面浅嵌岩桩极限承载力,本文基于修正Mohr-Coulomb强度准则提出了倾斜岩面浅嵌岩桩桩端承载力计算方法。采用特征线法构建了桩端面以下塑性发展区,考虑岩面倾斜角几何关系,推导了倾斜岩面位移滑移场;进一步通过黎曼不变量控制方程,求解了倾斜岩面浅嵌岩桩基极限承载力公式。分析了倾斜岩面坡角、岩体完整性参数以及岩体内摩擦角等因素对桩端承载力的影响。结果表明:浅嵌岩桩基嵌入深度达到极限嵌入比时,桩基达到极限承载力;当嵌入比大于极限嵌入比时,桩端承载力系数不受倾斜坡角和嵌入比影响,此时可按平地桩基进行分析计算;极限嵌入比随倾斜坡角的增大呈幂函数增大,且在岩石完整性较差时,极限嵌入比随岩体内摩擦角的增大而显著增大;当嵌入比小于极限嵌入比时,桩端承载力系数随倾斜坡角的增大近似线性减小,需考虑岩面倾角对桩基承载力的影响。

Simplifying Triangular Mesh Model by Modified Distance Criterion

Model simplification is used in a variety of disciplines, such as vir'tUal reality, CAD anddynamic simulation. The method of model simplification is an important problem. This paperpresents a new method to delete vertex from mesh model for simplification, in which the modifieddistance actS as deletion criterion. In order to improve the mesh quality, the poyer uses an optimalmethod based on swapping diagonal for retriangulating the local polygon. By the method. thesimplified model can approximate original model very well. An example to illustrate the efficiencyof the method is given.

Upper bound solution for supporting pressure acting on shallow tunnel based on modified tangential technique

Based on the nonlinear failure criterion and the upper bound theorem, the modified tangential technique method was proposed to derive the expression of supporting pressure acting on shallow tunnel. Instead of the same stress state, different normal stresses on element boundaries were used. In order to investigate the influence of different factors on supporting pressures, the failure mechanism was established. The solution of supporting pressure, with different parameters, was obtained by optimization theory. The corresponding failure mechanism and numerical results were presented. In comparison with the results using the single tangential technique method, it is found that the proposed method is effective, and the good agreement shows that the present solution of supporting pressure is reliable.

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.

基于上限定理确定隧道式锚碇极限抗拔承载力

隧道式锚碇作为一种新型的悬索桥锚碇结构,因具有因地制宜、经济环保的优点在多山的西南地区得到了广泛应用。已有研究表明,锚塞体围岩性质较差、埋深较浅,锚塞体与围岩接触界面结合程度较好的隧道式锚碇极易发生倒楔形冲切破坏。将破坏体视为以锚塞体中心线为轴线的旋转楔形体,假定破坏面为最小旋转曲面,利用变分法经典欧拉方程求解破坏体最小旋转曲面问题,确定旋转曲面母线方程;基于改进的Mohr-Coulomb强度准则,由主缆拉拔荷载和楔形体自重所做的外功率与楔形体沿滑移面所消耗的内功率相等的条件建立虚功方程,推导隧道式锚碇极限抗拔承载力上限解。开展隧道式锚碇室内模型试验,研究了拉拔承载过程中的坡面位移的变化规律及隧道式锚碇的破坏模式。研究结果表明,拉拔作用点的荷载-位移曲线可分为4个阶段:克服重力阶段、弹性阶段、弹塑性阶段以及破坏阶段;隧道式锚碇破坏模式为围岩倒楔形冲切破坏,破坏体可近似视为以锚塞体中心线为轴线的旋转楔形体;由推导公式得到的隧道式锚碇极限抗拔承载力计算值与实测值一致性较好,具有一定的可靠性,可为隧道式锚碇的设计提供参考。

基于修正Mohr-Coulomb准则的弹塑性本构模型及其数值实施

针对Mohr-Coulomb准则高估岩土体抗拉性能的局限性,建立考虑最大拉应力准则的修正Mohr-Coulomb模型;系统地论述隐式本构积分算法的主要内容,推导相应的一致性刚度矩阵。以ABAQUS软件为平台,采用向后欧拉隐式应力积分算法编制了UMAT本构程序,对单轴拉伸试验和三轴压缩试验进行数值模拟,对比分析ABAQUS自带模型和自编模型的优劣,结果表明编写的修正Mohr-Coulomb模型能够有效地反映岩土介质的抗拉性能,弥补了ABAQUS自带模型的不足。

修正Hoek-Brown准则的等效Mohr-Coulomb强度参数确定方法研究

合理确定岩石材料强度参数是工程设计的关键问题之一。为有效描述岩石的强度特性,众多学者提出了大量三维强度准则,其中基于两参数偏函数的修正Hoek-Brown(H-B)准则取得了良好效果,能够有效地反映岩石的静水压力效应、Lode角效应及中间主应力效应等强度特性。确定修正H-B准则的等效Mohr-Coulomb(M-C)参数有利于提高其在工程应用中的便利性,因此提出一种基于I1-√J2平面计算三维强度准则瞬时等效M-C参数的方法,应用于修正H-B准则,并对计算方法的合理性及瞬时等效M-C参数与岩石强度特性的关系进行讨论。结果表明:瞬时等效M-C参数可有效表征非线性静水压力效应,且合理反映Lode角效应和中间主应力效应的区间性特点;瞬时等效M-C参数可有效表征静水压力对Lode角效应的影响和最小主应力对中间主应力效应的影响,静水压力增大时,Lode角效应更为明显,且当最小主应力增大时,瞬时等效内聚力反映出更为显著的中间主应力效应;该方法能够合理地计算瞬时等效M-C参数,修正H-B准则具有准确预测岩石强度的能力。上述成果对修正H-B准则在实际工程中的应用具有较强的指导意义。

Stability analysis of intermittently jointed rock slopes based on the stepped failure mode

In practical engineering,due to the noncontinuity characteristics of joints in rock slopes,in addition to plane failure,stepped sliding failure may occur for intermittently jointed rock slopes.Especially for intermittently bedding jointed rock slopes,the correlation and difference in strength parameters between joints and rock bridges,along with the various failure modes and intermittency of rock bridges,contribute to the complexity of stepped failure modes and the unpredictability of failure regions.Based on the upper-bound limit analysis method and multi-sliders step-path failure mode,considering the shear and tensile failure of rock bridges and the weakened relationship between the strength parameters of rock bridges and jointed surfaces,by introducing the modified M-C failure criterion and the formula for calculating the energy consumption of tensile failure of rock bridges,two failure mechanisms are constructed to obtain the safety factor(F_(s))of intermittently jointed rock slopes.The sequential quadratic programming method is used to obtain the optimal upper-bound solution for F_(s).The influence of multiple key parameters(slope height H,horizontal distance L,Slope angleβ,shear strength parameters of the rock bridgeφr and cr,Dimensionless parameter u,weakening coefficients of the internal friction angle and cohesion between the rock bridges and joint surfaces Kφand Kc)on the stability analysis of intermittently jointed rock slopes under the shear failure mode of rock bridges as well as under the tensile failure mode is also explored.The reliability of the failure mechanisms is verified by comparative analysis with theoretical results,numerical results,and landslide cases,and the variation rules of F_(s)with each key parameter are obtained.The results show that F_(s) varies linearly withφr and cr of the rock bridge and with K_(φ)and K_(c),whereas F_(s)changes nonlinearly with H and L.In particular,with the increase in Kφand Kc,Fs increases by approximately 52.78%and 171.02%on average,respectively.For rock bridge tensile failure,F_(s) shows a nonlinearly positive correlation withφr,cr,Kφand Kc.In particular,with the increase in Kφand Kc,Fs increases by approximately 13%and 61.69%on average,respectively.Fs decreases rapidly with increasing slope gradientβand decreasing dimensionless parameterμ.When Kφand Kc are both less than 1.0,the stepped sliding surface occurs more easily than the plane failure surface,especially in the case of tensile failure of the rock bridge.In addition,rock slopes with higher strength parameters,taller heights,and greater weakening coefficients are prone to rock bridge tension failure with lower Fs,and more attention should be given to the occurrence of such accidents in actual engineering.

基于修正拟动力法的岩质边坡三维稳定性分析方法及应用

岩质边坡的抗震设计是岩土工程的重要课题,其稳定性直接关系到工程安全和社会经济效益。由于岩石的非线性强度特性,岩质边坡的破坏形式和机理与土质边坡有很大的不同,需要采用不同的分析方法和设计原则。此外,地震作用是导致岩质边坡破坏的主要因素之一,其对岩质边坡的稳定性有着显著的影响。为更好地确定基于Hoek-Brown强度准则下的岩质边坡抗震设计方法,在极限分析法上限定理的框架下,基于离散化技术构建了一种更加符合实际岩石边坡失稳破坏机制的三维对数螺旋破坏预测模型。通过广义切线技术将Hoek-Brown准则的参数转化为容易计算的Mohr-Coulomb准则参数再将其嵌入到预测模型中,提出一种更加准确的岩质边坡地震稳定性分析模型。该模型可以考虑岩石的非线性强度特性,并通过修正拟动力法引入地震加速度和阻尼比参数,提高了地震作用力的预测精度。然后,将重力增加法和多元宇宙优化算法相结合,获得一种边坡临界安全系数的确定方法,实现了岩质边坡在地震作用下的三维稳定性评估。与现有研究成果对比和工程案例应用表明,此方法可以较为准确地评估岩质边坡的地震稳定性,同时也能够反映岩质边坡的失稳破坏形态和范围,在岩质边坡地震稳定性分析中具有较好的有效性和可靠性。研究成果为进一步优化岩石边坡的抗震设计方法提供了新的思路和参考。

Size-dependent thermomechanical vibration characteristics of rotating pre-twisted functionally graded shear deformable microbeams

A three-dimensional(3D)thermomechanical vibration model is developed for rotating pre-twisted functionally graded(FG)microbeams according to the refined shear deformation theory(RSDT)and the modified couple stress theory(MCST).The material properties are assumed to follow a power-law distribution along the chordwise direction.The model introduces one axial stretching variable and four transverse deflection variables including two pure bending components and two pure shear ones.The complex modal analysis and assumed mode methods are used to solve the governing equations of motion under different boundary conditions(BCs).Several examples are presented to verify the effectiveness of the developed model.By coupling the slenderness ratio,gradient index,rotation speed,and size effect with the pre-twisted angle,the effects of these factors on the thermomechanical vibration of the microbeam with different BCs are investigated.It is found that with the increase in the pre-twisted angle,the critical slenderness ratio and gradient index corresponding to the thermal instability of the microbeam increase,while the critical material length scale parameter(MLSP)and rotation speed decrease.The sensitivity of the fundamental frequency to temperature increases with the increasing slenderness ratio and gradient index,and decreases with the other increasing parameters.Moreover,the size effect can suppress the dynamic stiffening effect and enhance the Coriolis effect.Finally,the mode transition is quantitatively demonstrated by a modal assurance criterion(MAC).

二元逻辑回归模型中的修正Liu估计

在二元逻辑回归模型中,基于修正岭型估计和Liu估计的思想,提出了一类新的有偏估计作为极大似然估计的替代估计,以期克服多重共线性的影响。在均方误差矩阵意义下,探讨得到了修正Liu估计优于Liu估计、修正岭型估计、极大似然估计等的条件,通过数值模拟和实证分析对部分理论结果在均方误差准则下进行了验证。结果表明,在一定条件下,当模型存在多重共线性时,修正Liu估计优于Liu估计、修正岭型估计以及极大似然估计。

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.