Seismic bearing capacity of shallow foundations near rock slopes using the generalized Hoek–Brown criterion

The seismic bearing capacity of shallow foundations resting on a modified Hoek–Brown rock mass is investigated within the framework of the kinematic approach of limit analysis theory. The analysis focuses on evaluating the reduction in bearing capacity induced by seismic loading and by the proximity of a rock slope. A pseudo‐static approach is adopted to account for the earthquake effects for the seismic bearing capacity evaluations. At the rock material level, the closed‐form expressions previously obtained for the support functions of the rock failure criterion allow the implementation of different failure mechanisms families, and thus to derive rigorous upper bounds estimates of the load‐bearing capacity in both static and seismic conditions. The effects of geometrical, strength and loading parameters are assessed through a large number of parametric computations. Finally, design tables are presented for practical use in rock engineering. Copyright © 2010 John Wiley & Sons, Ltd.     

An anisotropic strength criterion for jointed rock masses and its application in wellbore stability analyses

In this paper, an anisotropic strength criterion is established for jointed rock masses. An orientation distribution function (ODF) of joint connectivity, is introduced to characterize the anisotropic strength of jointed rock masses related to directional distributed joint sets. Coulomb failure condition is formulated for each plane of jointed rock masses by joint connectivity, where the friction coefficient and cohesion of the jointed rock mass are related to those of the intact rock and joint and become orientation dependent. When approximating joint connectivity by its second‐order fabric tensor, an anisotropic strength criterion is derived through an approximate analytical solution to the critical plane problem. To demonstrate the effects of joint distribution on the anisotropic strength of jointed rock masses, the failure envelopes are worked out for different relative orientations of material anisotropy and principal stress axes. The anisotropic strength criterion is also applied to wellbore stability analyses. It is shown that a borehole drilled in the direction of the maximum principal in situ stress is not always the safest due to the anisotropic strength of the jointed rock mass. Copyright © 2007 John Wiley & Sons, Ltd.     

土岩组合岩体中抗拔桩极限承载力的确定

为全面探究土岩组合岩体中抗拔桩的极限承载力,结合工程岩土参数及试验数据,运用Flac3D数值分析软件对其进行数值模拟分析,即可得到土岩组合岩体中抗拔桩的极限承载力。基于被动状态下的Kotter极限平衡方程式求解土层提供的抗拔力,根据岩石强度,基于Hoek-Brown破坏准则求解抗拔桩嵌岩端岩体的抗拉强度,从而可计算得到嵌岩端岩体的抗拔力;由静力平衡原理,叠加土层及嵌岩端岩体提供的抗拔力及破坏锥体重量,即可得到土岩组合岩体中嵌岩抗拔桩的极限承载力理论解析式。在嵌岩深度较小的情况下,该解析式的理论计算值与数值模拟分析值相接近,但随着嵌岩深度的增加,理论计算值会偏离数值计算值。故结合数值模拟试验值,对提出的极限承载力理论解析式作进一步的修正,得到修正后的极限承载力解析式能反映嵌岩端岩石风化程度、嵌岩深度、土层厚度、桩长对极限承载力的影响。运用修正后的解析式对该地质条件下不同抗拔桩的极限承载力计算表明：数值模拟结果与理论计算结果相吻合,说明所建立的抗拔桩极限承载力解析式的方法是可行的。同时,运用该方法可确定类似工程中嵌岩抗拔桩的极限承载力。     

基于统计岩体力学的岩体强度特征分析

结构面的存在改变了岩体力学性质,影响了岩体强度特征。基于统计岩体力学强度判据,结合摩尔-库仑准则,得到了含单组结构面岩体破坏的4种不同方式和相应的结构面倾角范围,推导了岩体强度由结构面控制转化为应力控制的临界围压表达式。在此基础上,根据岩石和结构面参数之间的关系,将含单组结构面岩体分为4类,并探讨了可能破坏方式和发生破坏的条件。最后,举例分析了含单组结构面闪长岩的强度特征,结果表明,该岩体属于第I类岩体,在垂向受压时先沿结构面后沿岩块破坏,临界围压为9.12 MPa;从全空间分析,该岩体强度各向异性显著,围压增大可使岩体在一些方向上受力时强度由结构控制转化为应力控制。     

A new rock mass failure criterion for biaxial loading conditions

To simulate brittle rocks, a mixture of glastone, sand and water was used as a model material. Thin galvanized sheets of thickness 0.254 mm were used to create joints in blocks made out of the model material. To investigate the failure modes and strength, both the intact material blocks as well as jointed model material blocks of size 35.6 × 17.8 × 2.5 cm having different joint geometry configurations were subjected to uniaxial and biaxial compressive loadings. A new intact rock failure criterion is proposed at the 3-D level. This criterion is validated for biaxial loading through laboratory experimental results obtained on intact model material blocks. Results obtained from both the intact and jointed model material blocks are used to develop a strongly non-linear new rock mass failure criterion for biaxial loading. In this failure criterion, the fracture tensor component is used to incorporate the directional effect of fracture geometry system on jointed block strength. The failure criterion shows the important role, the intermediate principal stress plays on rock mass strength.     

On the role of joint roughness on the micromechanics of rock fracturing process: a numerical study

The deformation process and failure mechanism of rock mass with increased joint roughness subjected to unconfined compression are investigated in this study using discrete element method. A numerical model is developed using soft-bonded particle and validated to realistically replicate the mechanical response of the rock mass. The micro-parameters of the rock material are first determined, and the effects of the joint roughness on the macromechanical response and fracture growth mechanism are then investigated. Analyses are also performed to examine the tensile and shear crack distributions, acoustic emission (AE) characteristics, coordination number, and crack anisotropy to advance the current understanding of the role of joint roughness on the mechanical behavior and deformability of rock mass. The results show that strength and deformability of the jointed rocks are highly dependent on the joint orientation and roughness. Joint roughness is found to restrain the propagation and coalescence of microcracks and AE events from the interlocking of asperities. In addition, the spatial distribution of the contact forces allows for better understanding of the effect of joint inclination angle on the response of the investigated rock samples.

     

Numerical investigation of the effect of joint geometrical parameters on the mechanical properties of a non-persistent jointed rock mass under uniaxial compression

The strength and deformability of rock masses transected by non-persistent joints are controlled by complex interactions of joints and intact rock bridges. The emergence of synthetic rock mass (SRM) numerical modelling offers a promising approach to the analysis of rock masses, but has not been rigorously compared with actual physical experiments. In this work, SRM modelling by the discrete element software PFC3D is used to investigate the effect of geometric parameters of joints on the rock mass failure mechanism, unconfined compressive strength and deformation modulus. Firstly, a validation study is undertaken to investigate the ability of SRM modelling to reproduce rock mass failure modes and strength as determined by uniaxial and biaxial compression testing in the laboratory. The numerical analyses agree well with physical experimentation at low confining pressure. A sensitivity study is then undertaken of the effect of joint configuration parameters on the failure mode, unconfined compressive strength and deformation modulus of the rock mass. Five failure modes are predicted to occur: intact rock, planar, block rotation, step-path and semi-block generation. It is found that the failure mode is determined principally by joint orientation and step angle and the joint orientation with respect to principal stress direction is the parameter with the greatest influence on rock mass properties.     

Analysis of strength and moduli of jointed rocks

This paper deals with two aspects of jointed rock mass behavior, first the finite element modeling of a jointed rock mass as an equivalent continuum, second the comparison of empirical strength criteria of a jointed rock mass. In finite element modeling the jointed rock properties are represented by a set of empirical relationships, which express the properties of the jointed medium as a function of joint factor and the properties of the intact rock. These relationships have been derived from a large set of experimental data of tangent elastic modulus. It is concluded that equivalent continuum analysis gives the best results for both single and multiple jointed rock. The reliability of the analysis depends on the estimation of joint factor, which is a function of the joint orientation, joint frequency and joint strength.Empirical strength criteria for jointed rocks, namely Hoek and Brown, Yudhbir et al., Ramamurthy and Arora, Mohr–Coulomb have been incorporated in a nonlinear finite element analysis of jointed rock using the equivalent continuum approach, to determine the failure stress. The major principal stress at failure, obtained using Ramamurthy's criteria, compares very well with experimental results.     

岩体结构面强度的可靠度分析

基于岩体结构面破坏的Mohr-Coulomb准则和可靠度理论,视结构面和岩块强度参数以及结构面倾角为随机变量,分析了结构面强度参数的变异性对结构面破坏概率的影响,探讨了结构面破坏概率随结构面倾角的变化规律,提出了结构面和岩块破坏概率的计算公式,找出了影响结构面破坏概率的主要因素。结果表明,岩体结构面强度的可靠度分析比传统的定值分析方法更为客观合理。     

Application of the Christensen Failure Criterion to Intact Rock

The Christensen criterion, originally introduced in materials science, has a simple mathematical form and uniaxial tensile and compressive strength as the only parameters, making it an attractive candidate for rock engineering purposes. In this study, the applicability of the criterion to rock materials is examined. Explicit equations for application of the criterion under biaxial, triaxial compression, triaxial extension, and polyaxial states of stresses are derived. A comprehensive strength data set including the results of tests on synthetic rock, chert dyke, Carrara marble and Westerly granite is utilized to examine the accuracy of the Christensen criterion to the failure of rock material. The two surprising findings about the Christensen criterion are the zero values of tensile strength and the very low slopes of the failure envelope obtained from fitting analyses for chert dyke and Westerly granite. It is shown that the two problems are interrelated and the values of tensile strength tend to zero to produce higher slopes. It is then mathematically proven that the maximum initial slope of the Christensen failure envelope is limited to 4 in triaxial compression and 2.5 in triaxial extension which is considerably lower than the slope of experimental data. The accuracy of the Christensen criterion was found to be significantly lower than the well-established Hoek–Brown criterion. The circular π-plane representations and brittle-to-ductile transition limits from the Christensen criterion are also inconsistent with the observed behavior of rocks.     

基于统一强度理论分析不连续面对岩体强度的影响

基于统一强度理论,对含有结构面的岩体,在平面形式下引入材料统一强度参数ct,φt,推导了含单个节理裂隙岩体材料的统一强度公式,即考虑中间主应力的岩体破坏强度公式,并分析了其有效范围。运用硅藻质软岩和石膏的三轴试验结果,验证了所建立的公式的正确性,并分析了具有不连续面岩体的强度随中间主剪应力作用系数b的变化。结果表明,岩体强度随b值的增大而增大。根据岩土材料统一强度参数的确定方法,计算并分析了b值对强度参数的影响,从而说明强度参数随着b值的变化而变化。     

Anisotropic Deformation of a Circular Tunnel Excavated in a Rock Mass Containing Sets of Ubiquitous Joints: Theory Analysis and Numerical Modeling

Determining anisotropic deformation surrounding underground excavations for tunnels is an intuitional task that involves many difficulties due to the inherent anisotropies in the strength and deformability of natural rocks. This study investigates joint-induced anisotropic deformation surrounding a tunnel via a numerical simulation that accounts for the mechanical behavior of intact rock, the orientations of joint sets, and the mechanical behavior of joint planes; this numerical simulation can model the complete stress–strain relationship with anisotropic rock mass characteristics. Simulation results demonstrate that the well-known excavation-induced stress variation–decrease in the radial component and increase in the tangential component–decrease shear strength and increase shear stress for the joint plane tangential to the tunnel wall, resulting in joint sliding failure and considerable shear deformation. This joint sliding failure and significant shear deformation account for the joint-induced anisotropic deformation surrounding a tunnel. When a rock mass has two joint sets with unfavorable joint orientations, the area with joint sliding failure can deteriorate mutually, resulting in large anisotropic deformation. Additionally, for a rock mass containing three joint sets with well-distributed orientations, joint sliding in various joint sets and associated stress variations can counter balance each other, resulting in less anisotropic deformation than those of rock masses containing one or two joint sets.     

Strength and dilatancy of jointed rocks with granular fill

It is well recognised that the strength of rock masses depends upon the strain history, extent of discontinuities, orientation of plane of weakness, condition of joints, fill material in closely packed joints and extent of confinement. Several solutions are available for strength of jointed rock mass with a set of discontinuities. There is a great multiplicity in the proposed relationships for the strength of jointed rocks. In the present study, the author conceives the effect of increasing stresses to induce permanent strains. This permanent strain appears as micro crack, macro crack and fracture. A fully developed network of permanent deformations forms joint. The joint may contain deposits of hydraulic and hydrothermal origin commonly known as gouge. The joint factor numerically captures varied engineering possibilities of joints in a rock mass. The joints grow as an effect of loading. The growth of the joints is progressive in nature. It increases the joint factor, which modifies the failure stresses. The dilatancy explains the progressive failure of granular media. Hence, a mutual relationship conjoins effectively the strength of jointed rock and a dilatancy-dependent parameter known as relative dilatancy. This study provides a simple and integral solution for strength of jointed rocks, interpreted in relation to the commonly used soil, and rock parameters, used for a realistic design of structure on rock masses. It has scope for prediction of an equivalent strength for tri-axial and plane strain conditions for unconfined and confined rock masses using a simple technique.     

IV级软弱围岩相似材料的试验研究

以IV级围岩为参照对象,研制可用于隧道围岩稳定性物理模型试验的软弱围岩相似材料。采用的5种原材料（重晶石粉、石膏、细砂、洗衣液和水）,通过直剪试验、无侧限抗压强度试验及三轴压缩试验测得所配相似材料的强度特征和物理力学参数,包括黏聚力c、内摩擦角?、单轴抗压强度Rc、弹性模量E和泊松比?。首先,通过试验分析每种原材料的不同比例对所配相似材料强度特征及物理力学参数的影响,选择适合于模拟IV级围岩相似材料的原材料配比;其次,针对优选的材料配比所配制的相似材料,开展大量的三轴压缩试验,研究在不同围压下该相似材料的强度及其破坏特征。结果表明,含水率减少或石膏含量增加导致Rc、E和c明显增大,而?降低不明显;增加砂含量可增大E 和?,但Rc值减少;石膏含量对相似材料的脆性和延性影响最明显,而含水率对相似材料的强度影响最为突出。试验结果还表明,相似材料在低围压时呈现脆性,在高围压时呈延性;就破坏准则而言,试验所得结果小于Mohr-Coulomb准则,略大于Hoek-Brown准则,但更符合Hoek-Brown准则。研制软岩相似材料的试验数据可为下一步拟开展的隧道围岩稳定性模型试验和数值模拟提供重要的参考和依据。     

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

This paper presents a probabilistic analysis to compute the probability density function of the bearing capacity of a strip footing resting on a spatially varying rock mass. The rock is assumed to follow the generalised Hoek–Brown failure criterion. The uniaxial compressive strength of the intact rock (σc) was considered as a random field and the geological strength index was modelled as a random variable. The uncertainty propagation methodology employed in the analysis is the sparse polynomial chaos expansion. A global sensitivity analysis based on Sobol indices was performed. Some numerical results were presented and discussed.     

Rock slope stability assessment using finite element based modelling – examples from the Indian Himalayas

Numerical modelling of rock slides is a versatile approach to understand the failure mechanism and the dynamics of rock slopes. Finite element slope stability analysis of three rock slopes in Garhwal Himalaya, India has been carried out using a two dimensional plane strain approach. Two different modelling techniques have been attempted for this study. Firstly, the slope is represented as a continuum in which the effect of discontinuities is considered by reducing the properties and strength of intact rock to those of rock mass. The equivalent Mohr-Coulomb shear strength parameters of generalised Hoek-Brown (GHB) criterion and modified Mohr-Coulomb (MMC) criterion has been used for this continuum approach. Secondly, a combined continuum-interface numerical method has been attempted in which the discontinuities are represented as interface elements in between the rock walls. Two different joint shear strength models such as Barton-Bandis and Patton’s model are used for the interface elements. Shear strength reduction (SSR) analysis has been carried out using a finite element formulation provided in the PHASE². For blocky or very blocky rock mass structure combined continuum-interface model is found to be the most suitable one, as this model is capable of simulating the actual field scenario.     

循环冻融条件下节理岩体损伤破坏试验研究

通过循环冻融和相似材料试验,研究了节理岩体在冻融条件下的损伤破坏机制及其相应的力学特性。通过对经历冻融循环后的试件损伤破坏模式的观察和单轴压缩试验,重点研究了节理倾角、节理贯通度、节理条数、节理充填物厚度、节理充填物类型、试件饱和度、冻融循环次数等对试件冻融损伤破坏模式、单轴抗压强度和弹性模量的影响。研究发现：节理存在及其物理力学性质对岩体的冻融损伤破坏模式及强度均有很大影响。节理倾角通过影响冻融裂纹的起裂位置进而影响其破坏模式和强度;随着贯通度的增加,试件表面裂纹由稀变密;随着节理条数增加,试件受冻融影响明显加剧;随着节理充填物厚度增加,试件冻融损伤程度先增加后减小;节理充填物类型对试件冻融损伤程度也有一定影响;随着试件饱和度的增加,试件冻融损伤程度先减小后增加;随着冻融循环次数的增加,节理试件表面因冻胀引起的裂纹逐渐增多、变宽,且其抗冻融特性较完整试件差。上述研究成果对寒区岩体工程建设及安全运营具有重要的参考价值。     

Stability analysis of tunnels driven in jointed rocks by means of a homogenized limit analysis approach

Conceived as a potential alternative to the classical design methods employed for analyzing the stability of underground structures driven in jointed rocks, the homogenization approach stems from the heuristic idea that, from a macroscopic point of view, a rock mass cut by a network of joints may be perceived as a homogenized continuum. The strength properties of the latter can be theoretically obtained from the failure conditions of its individual constituents: rock matrix and joint interfaces. At the material level, the limit analysis reasoning is used in the context of homogenization to formulate the homogenized strength criterion of a jointed rock mass in the particular situation of a single set of parallel joints. As it could be expected, the obtained closed‐form expressions show the strength anisotropy induced by joint preferential orientation. The support functions (π functions) associated with the homogenized strength criterion are also determined in both plane strain and three‐dimensional cases. This criterion is then applied to the investigation of stability analysis of a tunnel excavated in a jointed rock mass. Upper bounds estimated of the stability factor are derived from the implementation of the kinematic approach directly on the homogenized underground structure. Finally, the approach is applied to analyze and discuss the collapse of the Pinheiros subway station (São Paulo, Brazil). Copyright © 2014 John Wiley & Sons, Ltd.     

SPH procedures for modeling multiple intersecting discontinuities in geomaterial

A methodology is developed in SPH framework to analyze the behavior of preexisting multiple intersecting discontinuities or joints in rock material. The procedure does not require any additional unknowns to represent discontinuities and to capture velocity jump across them. Instead, a discontinuity is represented by a set of joint particles placed along the discontinuity plane, in which relative velocity and traction vector is evaluated, obeying the Mohr–Coulomb friction law with zero tension constrain. For failure of continuous rock material, the Drucker–Prager yield criterion with tensile cracking is employed in the elastic‐plastic constitutive model. Free‐sip, no‐slip, and symmetric boundary conditions are also implemented in SPH framework for proper representation of physical system. The paper analyzes behavior of a rock sample having a discontinuity plane under uniaxial loading and compares velocity and stress with a theoretical solution derived considering effective vertical stiffness of the joint planes. The efficacy of the proposed method is successfully demonstrated by solving another two problems of jointed rock mass under uniaxial and gravitational loading conditions.Copyright © 2014 John Wiley & Sons, Ltd.     

基于非线性统一强度理论的节理 岩体地基承载力研究

现有的节理岩体地基承载力计算方法大多是以Hoek-Brown强度准则为基础建立的,没有考虑中间主应力的效应,不能充分发挥岩体的强度潜能。综合统一强度理论和Hoek-Brown强度准则的优点,结合临界滑动场理论求解地基承载力,并考虑地震荷载的影响。计算结果表明,基于非线性统一强度理论的地基承载力有不同程度的提高,现有方法中的地基承载力偏于保守;在地震荷载作用下地基承载力有一定程度的降低。