Stresses and displacements in an elastic mass due to distributed loading by discretization technique

Numerical solutions have been obtained for stresses and displacements in a linear elastic half space due to distributed loads of circular, rectangular and elliptical shapes. The technique primarily involves use of a multi-dimensional numerical integration technique to integrate point load solutions over the distributed loading after discretizing the area into a finite number of elements. Both uniform vertical and shear loads have been considered as well as vertical conical loads and inward shear loads. The technique evolved facilitates the determination of stresses and displacements by the use of mini-computers and is neither as tedious and cumbersome as the use of tables and charts nor as costly as FEM solutions. A detailed comparison has been presented between the results obtained by the numerical solutions and those of the existing analytical solutions wherever they are available. It is found that the agreement between the two is within one per cent for displacements at all depths for the different cases studied. The matching is also good in the case of stresses, except at shallow depths.     

Rock Joint Surfaces Measurement and Analysis of Aperture Distribution under Different Normal and Shear Loading using GIS

Summary The geometry of the rock joint is a governing factor for joint mechanical and hydraulic behaviour. A new method for evaluating the aperture distribution, based on measurement of joint surfaces and three dimensional characteristics of each surface, is developed. This method allows one to determine and visualize the aperture distribution under different normal stresses and shear displacements, which is difficult to observe experimentally. A new laser scanner system is designed and developed for joint surface measurements. Special attention is paid to both surfaces’ data gained by measurements and processing, such as x-y coordinate table modification, data referencing, and matching between upper and lower surfaces. The surfaces of an artificial joint in granite are measured, processed, analyzed and three dimensional approaches are carried out for surface characterization. Parameters such as “asperity’s heights”, “slope angles”, and “aspects” distribution at micro scale, local concentration of elements and their spatial localization at local scale are determined by Geographic Information System (GIS). These parameters are used for joint surfaces matching and its real behavior quantitative analysis. The upper surface is brought down to make contact with the lower surface and the distance between the two surfaces is evaluated from the joint mean experimental aperture, which is obtained from normal and shear tests. Changes of aperture distribution at different normal stresses and various shear displacements are visualized and interpreted. Increasing normal load causes negative changes in aperture frequency distribution which indicates high joint matching. However, increasing shear displacement causes a rapid increase in the aperture and positive changes in the aperture frequency distribution, which could be due to un-matching, surface anisotropy and spatial localization of contact points with proceeding shear. Author’s address: Mostafa Sharifzadeh, Department of Mining, Metallurgy and Petroleum Engineering, Amirkabir University of Technology, Hafez 424, Tehran 15875-4413, Iran     

Influence functions for stress and displacement discontinuity elements in an anisotropic medium

Influence functions, that permit us to determine stresses and displacements at an arbitrary point in an infinite, homogeneous, linear elastic, anisotropic medium due to different three-dimensional (3-D) stress or displacement discontinuities distributed on infinite, flat, band-type elements, are presented. Any straight-line segment on the band, which is perpendicular to its infinite side, has the same distribution of the discontinuities. Along with the functions, their Taylor series approximations are also provided. The last can be useful to analyse stresses and displacements at points distant from the elements. The functions allow us to avoid procedures of numerical integration in the Indirect Boundary Element Method and/or the Displacement Discontinuity Method computer codes that are able to solve complete plane-strain problems with 3-D boundary conditions for an elastic, anisotropic medium. © 1997 by John Wiley & Sons, Ltd.     

2-D deformation analysis of a half-space due to a long dip-slip fault at finite depth

Closed form analytical expressions of stresses and displacements at any field point due to a very long dip-slip fault of finite width buried in a homogeneous, isotropic elastic half-space, are presented. Airy stress function is used to derive the expressions of stresses and displacements which depend on the dip angle and depth of the upper edge of the fault. The effect of dip angle and depth of the upper edge of the fault on stresses and displacements is studied numerically and the results obtained are presented graphically. Contour maps for stresses and displacements are also presented. The results of Rani and Singh (1992b) and Freund and Barnett (1976) have been reproduced.     

一种新的节理裂隙岩体弹塑性模型

提出一种节理裂隙岩体弹塑性模型,将结构划分为块体单元和缝单元,其中缝单元可以是实际节理裂隙,也可以是人为缝单元。以块体单元形心的刚体位移和块体的平均应变作为基本未知量。该模型能充分考虑节理裂隙材料和块体材料的本构关系,计算量不大,是一种位移不协调单元。该模型特别适合节理裂隙岩体的数值分析。     

Constitutive modelling of aggregate interlock in concrete

A model for formed cracks in concrete is presented. The model can be used in isolation for existing cracks or linked with other damage or plasticity models and applied once a crack has fully formed. It can be applied directly to interface finite elements or used to control the behaviour of crack planes in more general constitutive models that are applied to 2D and 3D continuum elements. The focus of the present development is on aggregate interlock and crack closing behaviour, which is examined from available experimental data. A contact function is derived and is used to differentiate between three contact states. These states are named open, where there is no contact, interlock, for which the stresses depend upon the nearest distance to the contact surface and closed, for which the stresses depend upon the relative displacements directly. The model is developed within an elasto‐plastic framework using effective stresses, which are related to the total stresses via a contact proportion function. The relationship between the effective normal and shear yield stresses is found to be parabolic and the yield shear stress to be dependent upon the opening and embedment displacements. The performance of the model is assessed against experimental data from shear‐normal tests and it is concluded that the model is able to represent the key characteristics of the behaviour of formed cracks in concrete. Copyright © 2002 John Wiley & Sons, Ltd.     

Displacement and stress distributions under a uniform inclined rectangular load on a cross‐anisotropic geomaterial

In practical engineering, an applied rectangular area load is not often horizontally or vertically distributed but is frequently inclined at a certain angle with respect to the horizontal and vertical axes. Thus, the solutions of displacements and stresses due to such a load are essential to the design of foundations. This article yields the analytical solutions of displacements and stresses subjected to a uniform rectangular load that inclines with respect to the horizontal and vertical axes, resting on the surface of a cross‐anisotropic geomaterial. The planes of cross‐anisotropy are assumed to be parallel to the horizontal ground surface. The procedures to derive the solutions can be integrated the modified point load solutions, which are represented by several displacement and stresses elementary functions. Then, upon integrations, the displacement and stress integral functions resulting from a uniform inclined rectangular load for (1) the displacements at any depth, (2) the surface displacements, (3) the average displacements in a given layer, (4) the stresses at any depth, and (5) the average stresses in a given layer are yielded. The proposed solutions are clear and concise, and they can be employed to construct a series of calculation charts. In addition, the present solutions clarify the load inclinations, the dimensions of a loaded rectangle, and the analyzed depths, and the type and degree of geomaterial anisotropy profoundly affect the displacements and stresses in a cross‐anisotropic medium. Parametric results show that the load inclination factor should be considered when an inclined rectangular load uniformly distributed on the cross‐anisotropic material. Copyright © 2008 John Wiley & Sons, Ltd.     

A theoretical study of reinforcement influence on the stability of a tunnel face

Summary The problem of tunnel face stability is studied and the analysis of the stabilizing effect achieved by pre-reinforcement of the core which has to be excavated is dealt with. In Italy, the insertion of longitudinal fibreglass pipes in the ground has proved to be efficient in solving face instability problems. A thorough review of other applications of the technology has been carried out.The results of a series of parametric, three-dimensional, elasto-plastic finite element analyses are presented. The tunnel excavation with or without support and face reinforcement is simulated in the models. The stresses, plastic zones and displacements in the ground, pipes and lining have been studied. The support effect of the pipes is demonstrated by the results. Face reinforcement significantly reduces the displacements of the face. The stresses in the material at the face are also charged from tension to compression, thus enhancing face stability.     

基坑开挖与支护模拟的位移迭代法

将基坑支护体系分解为彼此独立又相互作用的3个模型,即支护结构外侧主动土体、支护结构本身和支护结构内侧被动土体。按平面问题进行有限元分析时,土体采用面单元,支护桩采用梁单元,若桩上有内支撑或锚杆,则将梁单元与杆单元结合。各模型可分别求解,但共享边界条件。以开挖后桩体侧向位移为零、两侧都为静止土压力作为初始条件,然后按位移迭代求解,前后两次计算位移值基本相等时迭代结束,此时的位移和相应荷载就是基坑支护结构稳定后的变形与受力。该方法能客观反映土与结构相互作用的动态过程,实例计算结果与实测结果很吻合。     

Displacement control of geologic structures

Structural geologists routinely undertake geological analyses, particularly studies of faulting, by assuming that applied stresses are the controlling parameters. An alternative view is the assumption that material velocities, incremental displacements, or total displacements are imposed on the system, with stresses then part of the material response to these imposed boundary conditions. In our view, taking velocities and displacements as independent variables in deformation and stresses as dependent variables requires fewer assumptions and is more consistent with the observed geology.     

Boundary element analysis of transversely isotropic bi-material halfspaces with inclined planes of isotropy and interfaces

In this paper, a single-region boundary element method (BEM) is presented for the analysis of transversely isotropic bi-material halfspaces with arbitrarily inclined planes of isotropy and material interfaces. The proposed BEM uses the fundamental solution of a transversely isotropic bi-material fullspace and five boundary element techniques. Infinite boundary elements are introduced to consider the far-fields of a transversely isotropic bi-material halfspace. The effective integration methods are proposed for dealing with various integrals in the discretized boundary integral equation. The stresses at internal points are obtained using the coordinate transformation of kernel functions, and the stresses on the boundary surface are calculated using an improved traction recovered method. Numerical verifications of displacements and stresses for a benchmark problem are conducted, and excellent agreement with previously published results is obtained. Numerical examples are presented to illustrate the influence of non-horizontal or horizontal planes of isotropy in bi-material halfspaces on the displacements and stresses induced by the tractions on the horizontal boundary surface. Results reveal that the elastic fields vary clearly with the dip angle of the isotropic plane and the stresses across the bi-material interface are closely related to the ratios of the elastic parameters of the bi-material.     

Numerical modeling of combined effects of upward and downward propagation of shear bands on stability of slopes with sensitive clay

Modeling of progressive development of zones of large inelastic shear deformation (shear band) that results from strain‐softening behavior of sensitive clays could explain the failure mechanisms of large landslides. Because of toe erosion, a shear band can be initiated and propagated upward (inward) from the river bank. On the other hand, upslope surcharge loading could generate shear bands that might propagate down towards the river bank. In the present study, upward and downward propagation of shear bands and failure of sensitive clay slopes are modeled using the Coupled Eulerian Lagrangian approach in Abaqus finite element (FE) software. It is shown that the formation and propagation of shear bands are significantly influenced by kinematic constraints that change with displacements of the soil masses, and therefore the propagation of an existing shear band might be stopped and new shear bands could be formed. The main advantages of the present FE modeling are: (i) extremely large strains in the shear bands can be successfully simulated without numerical issues, (ii) a priori definition of shearing zones is not required to tackle severe strains; instead, the FE program automatically identifies the critical locations for shear band formation and propagation. Toe erosion could significantly increase the slope failure potential because of upslope surcharge loading. FE analyses with a thick and thin sensitive clay layers show that the global failure could occur at lower surcharge loads in the former as compared to the latter cases. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical solutions for stresses and displacements around tunnels driven in cross-anisotropic rocks

In this paper, closed-form solutions for the stresses and displacements around unlined circular tunnels excavated in an elastic, orthotropic (cross-anisotropic) medium are developed. The effect of both the elastic parameters characterizing the behaviour of the medium and the anisotropy of the initial stress system on the stresses and displacements induced are evaluated. An example of utilizing the theoretical solution for design analysis is given. For convenience of application, design charts are prepared for the determination of stresses and displacements for given values of initial stresses and the elastic parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

A novel finite element double porosity model for multiphase flow through deformable fractured porous media

Based on the theory of double-porosity, a novel mathematical model for multiphase fluid flow in a deforming fractured reservoir is developed. The present formulation, consisting of both the equilibrium and continuity equations, accounts for the significant influence of coupling between fluid flow and solid deformation, usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretize the governing equations both in the space and time domain. Throughout the derived set of equations the solid displacements as well as the fluid pressure values are considered as the primary unknowns and may be used to determine other reservoir parameters such as stresses, saturations, etc. The final set of equations represents a highly non-linear system as the elements of the coefficient matrices are updated during each iteration in terms of the independent variables. The model is employed to solve a field scale example where the results are compared to those of ten other uncoupled models. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered. © 1997 by John Wiley & Sons, Ltd.     

Modelling the effect of climate change induced soil settling on jointed drinking water distribution pipes

Soil settlements related to groundwater lowering are expected to be accelerated by climate change and may damage underground infrastructure networks. A 1D mechanical model, previously developed for continuous pipelines, has been extended towards jointed pipelines to calculate the stresses and joint rotations induced by the soil settlements. From the mechanical model, curve fits were acquired that can be used to estimate the bending moments and joint rotation. The curve fits differ per soil type, joint stiffness, joint position and joint distance. The stresses calculated by the 1D mechanical model and curve fits were validated by means of 3D finite element modelling. Using the curve fits, a probabilistic approach was followed by means of a Monte Carlo method to calculate the probability of failure of the pipeline system. The effect of joints is that the pipe stresses are reduced as the joints absorb a part of the soil displacement. For the probability of failure, the pipe stresses have a larger contribution than the joint rotation, as the joint rotation remains small compared to the maximum allowable joint rotation.     

Coupled boundary element/finite element analysis in geomechanics including body forces

A coupling scheme for boundary and finite elements using joint elements is proposed which includes the consideration of body forces. In this scheme the boundary and joint elements are formulated in a similar way as finite elements (i.e., the equivalent FE procedure). These joint elements are efficiently used to combine different BE regions. For the evaluation of a body forces, two methods are compared on computational efficiency and it is found that the method using Galerkin tensor is more efficient than the method dividing the problem domain into several internal cells. Two main geotechnical problems considering self weight are numerically examined using this coupling procedure.     

拉力型锚索锚固段周边岩体的应力分布

采用弹性空间体内部一点受集中力的Kelvin解,推导了拉力型锚索锚固段的附加应力和位移分布的理论公式,并依此推出了锚固段周边的剪应力分布公式。在此基础上,讨论了锚固体、岩土体的力学参数对附加应力、位移和剪应力的影响。结果表明,锚固段周边岩体中的正应力场和位移为锥形,在锚固段的起始点上部形成了压应力场,下部为拉应力场,并以力的作用点为中心快速衰减。锚固段周边剪应力形成了一个锥面,并随远离作用点快速衰减。     

Study of wave attenuation across parallel fractures using propagator matrix method

Stress wave attenuation across fractured rock masses is a great concern of underground structure safety. This paper presents an analytical study on wave attenuation across parallel fractures at arbitrary incidence angles, where multiple reflections occurring between fractures are taken into account. Combined with displacement discontinuous model, plane wave analysis and propagator matrix method are applied to develop relations between the first layer and the nth layer with respect to potential amplitudes or displacements and stresses in matrix form. With initial and boundary conditions for different scenarios, potential amplitudes in any layer or displacements and stresses at any point can be obtained by solving corresponding matrixes. After parametric studies, it is found that parameters including incidence angle, normalized fracture stiffness, number of fractures, and fracture spacing have obvious effects on wave attenuation across parallel fractures. Copyright © 2011 John Wiley & Sons, Ltd.     

岗曲河混凝土面板堆石坝三维静力应力变形分析

采用邓肯模型对岗曲河面板堆石坝进行了三维有限元分析,研究软岩填筑层对坝体工作性能的影响,计算混凝土面板与岩石填筑层的位移和应力及周边缝变形。结果表明,由于受到下游堆石3D软岩填筑区的影响,坝体最大横断面最大沉降略偏向下游,总沉降量约为坝高的0.2 %,面板周边缝位移的绝对值一般都小于2 cm,周边缝的止水设计需注意选择合理的止水形式和填缝材料,由于坝址河谷狭窄,受岸坡约束,三维效应对坝体的应力变形影响较明显,为设计施工提供了参考。     

爆炸荷载作用下深埋圆形隧洞的饱和黏弹性土-衬砌系统动力响应

假定土骨架服从标准线性固体黏弹性本构关系,研究了深埋圆形隧洞的饱和黏弹性土-弹性衬砌耦合系统在轴对称爆炸作用下的瞬态动力响应。首先,基于饱和土的Biot模型和衬砌的弹性理论,通过引入势函数和Laplace变换,利用弹性衬砌和饱和黏弹性土界面处的连续性条件以及边界条件,得到饱和黏弹性土体和弹性衬砌位移、应力和孔隙水压力等在Laplace变换域中的解析解。其次,利用Laplace数值Crump逆变换得到耦合系统在时间域的动力响应,数值分析了不同土体模型下土体-衬砌耦合系统的径向位移和环向应力以及土体孔隙水压力等。结果表明：对不同土体模型的土体-衬砌耦合系统,其在爆炸载荷作用下的动力响应性态基本一致,但动力响应的振动周期和幅值等具有明显的差异。同时,对于饱和黏弹性土-弹性衬砌系统,土体黏性参数对土体径向位移和孔隙水压力有明显的影响,但对土体环向应力影响较小。