A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

Bounding solutions for the pressuremeter modulus using variational principles in elasticity

It is classically assumed that the pressuremeter modulus (determined from the initial slope of the pressuremeter expansion curve) can be used as an estimate for the soil's shear modulus, provided disturbance effects can be neglected. This approximation is only rigorous in the plane-strain case and requires that the height of the cell be large in comparison with its diameter. The present paper allows to quantify the error that results from this approximation as a function of the height over diameter ratio. The analysis is based on an application of the variational principles of elasticity. It is shown that the static and the kinematic approaches lead, respectively, to a lower- and an upper-bound estimate of the pressuremeter modulus expressed in terms of the soil's elastic coefficients. The calculations are completely carried out in the case of an infinite cavity subjected to a pressure on a finite length. It is shown that the plane-strain approximation overestimates the shear modulus. Based on the kinematic approach, a method is developed to derive the shear modulus from the pressuremeter modulus which accounts for the finite slenderness of the cell. This method can be extended to more complex geometries and material properties, including soil heterogeneity and anisotropy.     

六面体有限覆盖的三维数值流形方法的非线性分析

对三维问题的分析是数值流形方法发展的必然,在数值流形方法覆盖位移函数的基础上构造了一种六面体有限覆盖的三维流形单元,推导了相应的应变矩阵、刚度矩阵及平衡方程等表达式。同时,由于目前数值流形方法的模拟分析主要是采用线弹性模型,而对于非线性模型分析研究很少;根据数值流形方法的特点和岩土体的本构模型,给出了适用于数值流形方法进行非线性分析的算法。该方法利用中点增量法进行求解,以改变 模型和 模型中弹性模量的方式来反映非线性,其实质是用分段线性来取代非线性。通过地基沉降计算算例表明,数值流形方法在三维岩土体中进行非线性分析中是有效的。     

Settlement analysis of rectangular piles in nonhomogeneous soil using a Winkler model approach

An analytical approach using a Winkler model is investigated to provide analytical solutions of settlement of a rectangular pile subjected to vertical loads in nonhomogeneous soils. For a vertically loaded pile with a rectangular cross section, the settlement influence factor of a normal pile in nonhomogeneous soils is derived from Mindlin's solution for elastic continuum analysis. For short piles with rectangular and circular cross sections, the modified forms of settlement influence factors of normal piles are produced taking into account the load transfer parameter proposed by Randolph for short circular piles. The modulus of subgrade reaction along a rectangular pile in nonhomogeneous soils is expressed by using the settlement influence factor related to Mindlin's solution to combine the elastic continuum approach with the subgrade‐reaction approach. The relationship between settlement and vertical load for a rectangular pile in nonhomogeneous soils is available in the form of the recurrence equation. The formulation of settlement of soils surrounding a rectangular pile subjected to vertical loads in nonhomogeneous soils is proposed by taking into account Mindlin's solution and both the equivalent thickness and the equivalent elastic modulus for layers in the equivalent elastic method. The difference of settlement between square and circular piles is insignificant, and the settlement of a rectangular pile decreases as the aspect ratio of the rectangular pile cross section increases. The comparison of results calculated by the present method for a rectangular pile in nonhomogeneous soils has shown good agreement with those obtained from the analytical methods and the finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

An analytical solution for stresses induced by a post-tensioned anchor in rocks containing two perpendicular joint sets

A new analytical approach for the analysis of stress around a post-tensioned anchor in rock with two perpendicular joint sets is presented. The solution considers nonlinear shear stresses developed along the anchor bond length as well as debonding at the tendon–grout interface. The bearing plate effects are also considered in the analyses. The following assumptions are made: (1) homogeneous and orthotropic elastic rock; (2) half-space rock mass with plane strain conditions; (3) trilinear bond-slip model for the behavior of the tendon–grout interface; (4) an elastic anchor. The employed methodology is to decompose the anchor problem into two problems of simpler loadings: stresses produced under a rigid bearing plate and stresses induced by interfacial shear stresses mobilized along the bond length. Based on the proposed solution, an illustrative example is given and the results show that a large compressive region is formed around the anchor free length. Tensile stress concentrations are also observed around the bond length and near the rock surface outside the bearing plate. A parametric study is also conducted to investigate the effects of joint properties on the induced stresses. The results show that as the number of joints intersecting the anchor axis increases, the magnitude of compressive stresses in the free zone decreases, while the size of compression zone around the anchor increases. To verify the results of the analytical approach, a comparison is made with numerical results obtained by using the finite element method. The analytical solutions compare very well with the results obtained by numerical method.     

Analysis of piled raft with nodular pile subjected to vertical load using a Winkler model approach

An investigation is made to present analytical solutions provided by a Winkler model approach for analysis of piled rafts with nodular pile subjected to vertical loads in nonhomogeneous soils. The vertical stiffness coefficient along a piled raft with the nodular pile in nonhomogeneous soils is derived from the displacement given by the Mindlin solution for elastic continuum analysis. The vertical stiffness coefficients for the bases of the raft and the nodular part in the nodular pile in a soil are expressed by the Muki solution for the 3‐D elastic analysis. The relationship between settlement and vertical load on the pile base is presented considering the Mindlin solution and the equivalent thickness in the equivalent elastic method. The interaction factor between the shaft of the nodular pile and the soil is expressed taking into account the Mindlin solution and the equivalent elastic modulus. The relationship between settlement and vertical load for a piled raft with the nodular pile in nonhomogeneous soils is obtained by using the recurrence equation of influence factors of the pile for each layer. The percentage of each load carried by both nodular pile and raft subjected to vertical load is represented through the vertical influence factors proposed here. Comparison of the results calculated by the present method for piled rafts with nodular piles in nonhomogeneous soils has shown good agreement with those obtained from the finite element method and a field test. Copyright © 2016 John Wiley & Sons, Ltd.     

Upper-bound multi-rigid-block solutions for bearing capacity of two-layered soils

The limit analysis method has been widely used in the stability analysis of geotechnical problems including the bearing capacity of foundations. Two main approaches have been followed in the limit analysis to improve the calculation of the bearing capacity of foundations. One approach is to combine limit analysis with the finite element method and linear/nonlinear programming. The other is to use a multi-rigid-block mechanism to obtain an upper-bound solution. In this paper, the multi-rigid-block upper-bound method with a modified failure mechanism from that of Florkiewicz [Florkiewicz A. Upper bound to bearing capacity of layered soils. Can Geotech J 1989;26(4):730–6.] was employed to calculate the bearing capacity of foundations. Attention was paid particularly to the bearing capacity of strip footings over a two-layered soil. In order to verify the effectiveness of the modified mechanism, comparisons were made with other well-known solutions. The results showed improvements over the best available multi-rigid-block upper-bound solutions given by Michalowski and Shi [Michalowski RL, Shi L. Bearing capacity of footings over two-layer foundation soils. J Geotech Eng ASCE 1995;121(5):421–8.], and fair consistence with the results from the finite element limit analysis in Shiau et al. [Shiau JS, Lyamin AV, Sloan SW. Bearing capacity of a sand layer on clay by finite element limit analysis. Can Geotech J 2003;40(5):900–15.].     

基于方位离散线性化的上限原理有限元法

上限原理有限元法不仅可以得到边坡的安全系数,还可以给出临界滑动面,且具有比极限平衡法更严谨的理论基础,因此,拥有更广阔的应用前景。针对传统的上限有限元法不能考虑强度各向异性的问题,提出了一种新的摩尔-库仑屈服面线性化方法。该方法在对方位角离散化的基础上,建立了线性化的方位离散塑性流动约束方程,丰富了基于线性规划的上限法理论。两个算例结果表明：该方法可以稳定地从极限解的上方收敛;且对边坡进行稳定性分析,若忽略了边坡的强度各向异性,则会高估边坡的稳定性,得到较大的安全系数。     

Soil–pipe interaction due to tunnelling: Assessment of Winkler modulus for underground pipelines

One of the key problems in the implementation of Winkler models to analyze tunnelling effects on existing pipelines lies in the assessment of subgrade modulus under external soil displacement. In this paper, an expression of the Winkler subgrade modulus for a pipeline buried at arbitrary depth and subjected to free soil displacement with arbitrary curve shape is given. Using superposition principle and the Fourier integral, the subgrade modulus of an infinite beam resting on the surface of an elastic half space and buried infinitely are obtained respectively. Then the influence of embedment depth is estimated based on Mindlin and Kelvin solution. The validity of the proposed subgrade modulus is verified by comparison with the results from an elastic continuum solution and two centrifuge model tests for the responses of buried pipeline due to nearby tunnelling. Thereafter, parametric studies are shown to assess the accuracy of the proposed subgrade modulus by comparing with an elastic continuum solution in homogeneous and non-homogeneous soil stratum and the amount of error is estimated.     

A Winkler model approach for vertically and laterally loaded piles in nonhomogeneous soil

An investigation is made to present analytical solutions provided by a Winkler model approach for the analysis of single piles and pile groups subjected to vertical and lateral loads in nonhomogeneous soils. The load transfer parameter of a single pile in nonhomogeneous soils is derived from the displacement influence factor obtained from Mindlin's solution for an elastic continuum analysis, without using the conventional form of the load transfer parameter adopting the maximum radius of the influence of the pile proposed by Randolph and Wroth. The modulus of the subgrade reaction along the pile in nonhomogeneous soils is expressed by using the displacement influence factor related to Mindlin's equation for an elastic continuum analysis to combine the elastic continuum approach with the subgrade reaction approach. The relationship between settlement and vertical load for a single pile in nonhomogeneous soils is obtained by using the recurrence equation for each layer. Using the modulus of the subgrade reaction represented by the displacement influence factor related to Mindlin's solution for the lateral load, the relationship between horizontal displacement, rotation, moment, and shear force for a single pile subjected to lateral loads in nonhomogeneous soils is available in the form of the recurrence equation. The comparison of the results calculated by the present method for single piles and pile groups in nonhomogeneous soils has shown good agreement with those obtained from the more rigorous finite element and boundary element methods. It is found that the present procedure gives a good prediction on the behavior of piles in nonhomogeneous soils. Copyright © 2011 John Wiley & Sons, Ltd.     

隧道开挖条件下地埋管线的非线性响应分析

隧道开挖引起的地层不均匀沉降造成附近的地埋管线产生额外的变形,甚至破坏。被动管线与土体的相互作用的研究表明,不考虑土体的刚度衰减,较大的土体弹性模量使得管线的最大弯矩计算结果过大,偏于保守,造成不必要的浪费。在被动管线Winkler地基模型分析基础上,引入土体刚度衰减模型考虑土体非线性特性,提出了隧道开挖作用下管线响应的等效线性分析方法。基于自由土体位移场计算管周土体由于隧道开挖引起的附加应变,基于水平受荷桩的环状弹性介质模型考虑由于管土相互作用引起的管周土体应变,从而对被动Winkler地基模型的土体弹性参数进行修正,计算得到管线的最大弯矩。通过与现有的弹性理论方法、离心模型试验结果的对比,验证了针对隧道开挖引起的被动管土相互作用问题,该方法考虑土体非线性特性的合理性。     

爆破荷载应力波在无限弹性介质中传播的特征线法解

双指数爆破荷载的应力时程关系是爆破工程的数值分析常用的动力输入。其在弹性岩石中传播衰减的特征线法解是工程爆破数值分析中弹性以及弹塑性分析的重要参照,也是动力学数值分析算法开发的重要高阶校准工具。应用统一弹性波的特征线法解求解源自圆形孔洞的双指数爆破荷载在理想介质中的传播。介绍了源自圆形孔洞的双指数爆破荷载在周围弹性介质传播过程中的径向应力、周向应力、位移以及速度波的特征线法解的具体过程,在MATLAB中进行了4种波的特征线法解的数值实施。对径向应力、周向应力、位移以及速度波的特征线法解的结果进行了多方面讨论。结果表明,特征线法是求解双曲型偏微分方程的一种有效方法,双指数爆破荷载在弹性岩石中传播衰减的特征线法解可为爆破工程数值分析以及动力学数值分析算法开发提供重要的支撑。     

Beams on nonlinear winkler foundation with gaps

The analysis of beams on a Winkler foundation is very common in engineering. In many practical problems nonlinear foundation behavior should be considered. In some cases a gap may exist between the beam and the nonlinear foundation. Based on an exact solution for a finite beam on linear elastic foundation, an iterative procedure for a piecewise linear foundation is formulated. The approach is implemented in a continuous beam computer program and demonstrated in numerical examples.     

土工格栅加筋堆石坝的数值模拟研究

运用复合材料法模拟土工格栅加筋单元,对格栅加筋堆石坝坝顶堆石进行了数值模拟。计算中,土石料采用非线性弹性模型模拟,并用拟静力法模拟地震动荷载。对无地震和有地震两种情况下的加筋效果及加筋层数和格栅模量对加筋效果的影响进行了分析研究,并对格栅加筋机理加以解释。数值模拟结果表明：格栅加筋有利于增加土石坝坝顶堆石的稳定性和抗震性能,随加筋层数和格栅模量的增加,加筋效果愈加显著。     

弹性模量对岩样破坏前兆及全部变形特征的影响

利用FLAC内嵌语言编制的计算平面应变压缩岩样轴向、侧向、体积应变及泊松比的FISH函数,计算了弹性模量不同时单缺陷岩样的全部变形特征,研究了弹性模量对岩样的破坏过程及前兆的影响。在峰前及峰后,岩石的本构模型分别取为线弹性模型及莫尔-库仑剪破坏与拉破坏复合的应变软化模型。当弹性模量不是较高时,岩样自始至终仅出现一条倾斜的剪切带。当弹模较高时,最终剪切应变仅强烈集中于贯通岩样的剪切带内部。剪切带倾角在Arthur与Coulomb倾角之间,且随弹模的增加而降低,经典理论对此不能解释。随着弹模的增加,峰值应力增加,峰前的应力-轴向应变曲线变得陡峭,峰后的应力-轴向应变曲线的斜率几乎不变。随着弹模的增加,峰值强度所对应的轴向应变及侧向应变的大小均降低;应力-侧向应变曲线软化段变得陡峭。随着弹模的降低,侧向应变-轴向应变曲线、泊松比-轴向应变曲线及体积应变-轴向应变曲线在峰前发生转折（偏离线性状态）的程度越来越大;非弹性轴向应变增加;材料缺陷附近的最大剪切应变增量增加。岩样破坏的前兆随着弹性模量的降低而逐渐增强。     

重力坝沿建基面失稳破坏的真实安全度研究

采用强度储备系数法,对重力坝建基面破坏过程进行非线性有限元模拟,并分析了坝体地基弹性模量比和建基面应力分布状态对破坏过程和强度储备系数的影响。研究结果表明：破坏过程可划分为4个阶段：贯通率平稳期、加速期、短暂平稳过渡期、急剧增加失稳期。当弹性模量比一定时,应力状态越趋于均匀分布,大坝处于稳定状态时的强度储备系数越大,坝趾坝踵处的应力比值越大,大坝处于稳定时的强度储备系数越小;当建基面应力分布状态一定时,弹性模量比的变化不改变建基面的总体破坏特征,强度储备系数与弹性模量比成正比。综合考虑应力分布及弹性模量比的影响,给出了正常荷载作用下重力坝沿建基面失稳破坏的真实安全度取值范围为1.5～2.5。     

非均质软土基坑抗隆起稳定性的极限分析方法

考虑到实际工程中土体非均质以及软土地区基坑开挖过程中土体稳定性的特点,将极限分析上限方法应用于非均质软土基坑抗隆起稳定分析计算中,讨论了极限分析方法的适用性问题。选用Prandtl机构建立了非均质软土地区基坑抗隆起稳定分析的运动许可速度场,根据速度场推导了极限分析上限方法的计算公式。讨论了该方法中土体强度、基坑宽度、支护墙体入土深度、基岩埋置深度等因素对坑底抗隆起稳定的影响。通过几个工程的计算与对比,验证了该方法的适用性和优越性。     

非线性破坏准则下边坡稳定性极限分析斜条分法

考虑坡顶均布荷载和地震效应典型情况下,将边坡滑体进行任意斜条块划分,建立了具有倾斜界面的多块体破坏模型。基于极限分析上限法和非线性摩尔-库仑破坏准则,考虑岩体内正应力的不均匀性,引入多点切线法和强度折减法推导得出边坡临界破坏状态下的安全系数Fs通用计算公式。采用序列二次规划法对安全系数Fs的目标函数进行最优化计算,并与既有研究成果进行对比分析,其结果具有较好的一致性,相对误差不超过3.565%,表明了该方法的正确性。同时对比传统单点切线法计算结果,多点切线法较单点切线法获得的边坡安全系数值偏小,表明了多点切线斜条分法偏于保守,是安全的。参数分析表明坡顶均布荷载、地震效应和非线性参数均对边坡安全系数及潜在临界滑裂面有重要影响。多点切线法引入非线性摩尔-库仑破坏准则对边坡进行稳定性极限分析,为相关研究人员提供了一种新的思路与方法。     

Stability of a circular tunnel in cohesive-frictional soil subjected to surcharge loading

The stability of circular tunnels in cohesive-frictional soils subjected to surcharge loading has been investigated theoretically and numerically assuming plane strain conditions. Despite the importance of this problem, previous research on the subject is very limited. At present, no generally accepted design or analysis method is available to evaluate the stability of tunnels/openings in cohesive-frictional soils. In this study, continuous loading is applied to the ground surface, and both smooth and rough interface conditions are modelled. For a series of tunnel diameter-to-depth ratios and material properties, rigorous lower- and upper-bound solutions for the ultimate surcharge loading are obtained by applying finite element limit analysis techniques. For practical use, the results are presented in the form of dimensionless stability charts with the actual tunnel stability numbers being closely bracketed from above and below. As an additional check on the solutions, upper-bound rigid-block mechanisms have been developed and the predicted collapse loads from these are compared with those from finite element limit analysis. Finally, an expression that approximates the ultimate surcharge load has been devised which is convenient for use by practising engineers.     

Analysis of Circular Elastic Plate Resting on Pasternak Foundation by Strain Energy Approach

In this paper, an analytical expression for the deflection of a thin circular elastic plate resting on the Pasternak foundation is derived by adopting the strain energy approach. The parametric study is carried out to observe the location of the plate lift-off and the variation of the deflection profiles for comparing the variation pattern reported in the literature based on an approximate solution technique. It is found that the radial distance of the point of lift-off of the plate decreases with increase in the values of both the shear modulus and the modulus of subgrade reaction of the foundation soil; the reduction being more for their lower values. It is also observed that the variation in deflection of the plate at any radial distance due to change in soil subgrade conditions is significant for lower values of modulus of subgrade reaction.