极限分析有限元法讲座Ⅲ--增量加载有限元法求解地基极限承载力

利用有限元法,通过增量加载的方式来求解地基极限承载力。随着荷载的逐渐增加,地基由初始的线性弹性状态逐渐过渡到塑性流动的极限破坏状态,此时有限元的计算将不收敛。它不但可以获得地基的极限荷载的值及荷载-位移关系,而且还能得到经典极限分析法中所采用到的破坏机构。当采用关联流动法则或采用剪胀角为 φ/2 的非关联流动法则时,获得的破坏机构与 Prandtl 的破坏机构一样。对 Prandtl 解的经典算例进行了分析,结果表明:屈服准则的选用对计算结果的影响很大,选择与实际问题相匹配的屈服准则方能得到比较精确的结果。在求解平面应变问题时,在关联流动法则条件下,采用Mohr-Coulomb 内切圆屈服准则,或在非关联流动法则下采用 Mohr-Coulomb 匹配 DP 准则所得结果与 Prandtl 精确解极为接近,可望应用于实际工程分析中。     

不排水双层粘性土地基极限承载力的数值分析

采用弹塑性有限元分析了条形基础作用下不排水条件的双层粘性土地基极限承载力性状。采用修正的地基承载力系数表征,并将不同的几何与土层参数条件下的数值解与上下限解和经典的经验解进行比较。表明弹塑性位移有限元法可以很好地求解地基的极限承载力问题,其求解得到的修正地基承载力系数与基于下限原理的有限元解很接近,而上限解高估了地基的极限荷载值,传统的经验解在某些条件下却偏小。     

柔性和刚性浅基础的地基承载能力分析

为了研究刚性和柔性加载面下地基的破坏机制和极限承载力,着重对比分析了不考虑土体自重条件下柔性基础、基底完全光滑或完全粗糙的刚性基础的地基,采用关联流动的Mohr-Coulomb内切圆屈服准则,通过增量加载的有限元方法,全程模拟了地基由初始的线弹性状态逐渐过渡到塑性流动的极限破坏状态的过程。通过对这3种基础类型下地基的计算结果的对比分析,并结合国内外模型试验成果,得到如下结论：柔性和刚性浅基础地基在不考虑土体自重的条件下有相近的地基极限承载力,但基底水平面上竖向应力和位移的发展规律、临塑荷载及滑动面有着明显的区别。     

弹塑性有限元分析中几个难点问题的一揽子方案

Mohr-Coulomb准则在Mohr应力空间中具有最简形式,同时也因其非常可靠而在经典的极限分析或极限平衡法中得到了最广泛的应用。然而,应力空间中的Mohr-Coulomb屈服面是非光滑的,这给基于变形分析的弹塑性有限元法中的本构积分带来了巨大的麻烦。此外,在求解强度问题时,基于载荷控制法(LCM)的求解器很难将有限元模型带入极限平衡状态。针对这些问题,本研究给出了以下解决方案。首先,设计了适用于带非光滑屈服面的塑性本构积分算法GSPC。GSPC对任意大小的应变增量都收敛,数值特性远优于现有的返回-映射算法。还为弹塑性有限元分析定制了一个位移控制法(DCM)求解器,该DCM能将有限元模型带入极限平衡状态而不存在收敛性方面的问题,计算效率和鲁棒性都远优于现有的基于LCM的求解器。最后,结合强度折减法,建议了求解边坡安全系数的割线法,并给出了极限平衡状态下坡顶拉裂缝位置和深度的确定技术。     

基于二阶锥规划与高阶单元的自适应上限有限元研究

基于六节点三角形高阶单元的自适应上限有限元法具有计算精度高,能够直观获取结构破坏模式等优点,但若采用屈服准则线性化方法,则计算效率较低。因此,为提高上限有限元计算效率,在既有研究基础上,引入计算效率较高的二阶锥规划方法,建立基于二阶锥规划与高阶单元的自适应上限有限元计算模型。通过分析边坡稳定性及条形基础地基承载力算例,表明所提方法计算精度较高且能够获取精细化的破坏模式;同时,在相同计算精度下,所提方法计算时长显著小于屈服准则线性化方法,有效提高上限有限元的计算效率。算例分析证明所提方法的正确性及有效性,可为类似理论研究及工程实践提供参考。     

基于二阶锥规划与高阶单元的自适应上限有限元研究

基于六节点三角形高阶单元的自适应上限有限元法具有计算精度高,能够直观获取结构破坏模式等优点,但若采用屈服准则线性化方法,则计算效率较低。因此,为提高上限有限元计算效率,在既有研究基础上,引入计算效率较高的二阶锥规划方法,建立基于二阶锥规划与高阶单元的自适应上限有限元计算模型。通过分析边坡稳定性及条形基础地基承载力算例,表明所提方法计算精度较高且能够获取精细化的破坏模式。同时,在相同计算精度下,所提方法计算时长显著小于屈服准则线性化方法,有效提高上限有限元的计算效率。算例分析证明所提方法的正确性及有效性,可为类似理论研究及工程实践提供参考。     

非线性破坏准则与岩土材料地基承载力研究

根据线性Mohr-Coulomb破坏准则,前人运用极限平衡法、滑移线理论或极限分析等方法求解地基承载力问题,但实际上岩土在剪切破坏过程中破坏准则具有非线性。因此,为了研究非线性破坏准则对地基承载力的影响,基于上限定理,通过“切线法”引进变量,根据能量耗散情况,将承载力问题转变为非线性规划问题,运用“序列二次规划算法”求出地基承载力的最优解。数值计算结果表明,当非线性破坏准则转变为线性破坏准则时,非线性参数对地基承载力有重要影响。     

水平受荷桩的弹性有限元虚拟加载上限分析

基于将塑性上限分析等效为弹性迭代计算的总量虚拟加载上限分析理论,在商业化有限元软件ABAQUS中实现了弹性有限元虚拟加载上限方法（弹性有限元T-EMSD）。应用弹性有限元T-EMSD法分析了不排水黏土中的二维水平受荷桩,其获得的荷载-位移曲线与弹塑性有限元分析结果一致,其极限承载力与塑性解相近。在极限位移加载量下弹性有限元T-EMSD法对应的上限机构从弹性始速度场开始随迭代逐渐演化,迭代收敛后的速度场和解析塑性破坏机构相似。与其他基于可变强度概念（MSD）的方法相比,弹性有限元T-EMSD法对水平受荷桩桩身的分析具有更高的精度。弹性有限元T-EMSD法最大的优势在于可在计算中自然地获得塑性机构,因而可被用于研究一些塑性机构难以构造的复杂问题,并对弹塑性数值方法进行验证。     

复合加载情况下双层地基极限承载力研究

在复合加载情况下精确求解层状非均质地基的极限承载力,具有很强的工程实用与理论参考价值。基于土体极限平衡理论与通用有限元软件ABAQUS,针对复合加载情况下上硬下软的双层不排水饱和软黏土地基的极限承载力,进行了大量的数值计算,得出了上层土临界深度Hcr的计算公式、竖向极限承载力Pv的计算公式以及复合加载情况下地基破坏时的破坏包络面方程。研究结果表明：上层土临界深度Hcr取决于土层间强度比Su1/Su2;竖向极限承载力Pv与破坏包络面取决于土层间强度比Su1/Su2、上层土深度H1与基础型式。     

岩石应变软化本构模型建立及NR-AL法求解研究

针对岩土工程材料应变软化问题及有限元对其数值计算时切线刚度矩阵负定造成求解困难的问题进行研究。建立了基于Drucker-Prager（D-P）强度准则的岩石弹塑性应变软化本构模型,本构积分算法采用一种完全隐式返回映射算法,它具有无条件稳定和精确的特点,详细论述了如何进行本构模型的程序化求解;考虑弧长法在判断切线刚度矩阵正定性导致效率低的缺点,在弹塑性增量有限元方程的迭代计算中尝试采用Newton-Raphson法和arc-length法（NR-AL法）联合迭代求解的思路,即在结构未达到极限荷载前采用NR迭代法,而当结构接近极限荷载时转换为AL法控制迭代,从而使结构越过峰值点进入软化区直至破坏,NR-AL法汲取了2者迭代求解中具有的优势;利用C++语言对所建应变软化模型的本构求解和弹塑性增量有限元方程迭代求解过程给予程序实现,应用所编程序进行数值计算,分析了D-P理想弹塑性模型、应变软化模型、应变硬化模型计算的应力-应变曲线的区别,同时将应变软化模型计算结果与试验数据进行了对比。研究结果表明：所建应变软化本构模型可以较好地模拟岩石材料的峰后软化特性,能够揭示峰后应变软化特性和破坏机制,同时NR-AL法能够求解由于应变软化造成的负刚度问题,也克服了单独使用弧长法时判断切线刚度矩阵正定性效率低的缺点。     

一种二阶锥线性化方法在上限有限元中的应用研究

对于平面应变条件下岩土稳定性分析,基于线性规划的上限有限元需对常用的摩尔-库仑屈服准则形成的二阶锥约束进行线性化,直接地处理方法是以外接多边形替代锥体投影形成的圆域。为了提高线性化精度往往需直接增加外接多边形边数,从而造成线性规划模型中决策变量包含大量的塑性乘子变量,使计算难度大为增加甚至变得不可行。为此,引入Ben-Tal和Nemirovsky提出的一种二阶锥线性化方法,并将其嵌入到自编的上限有限元程序。经算例分析发现,该法与外接多边形线性化方法所获计算结果相互印证,且其通过适量的增加决策变量和等式约束数目,能保证摩尔-库仑屈服准则线性化精度,同时形成的线性规划规模更小,可望应用于基于线性规划模型的上限有限元中。     

Method of deformation analysis for composite structures of soils and masonry stones

The coupled numerical manifold method (NMM) and discontinuous deformation analysis (DDA) are enhanced to simulate deformations of continuous soil and discontinuous masonry structures. An elasto-plastic NMM-DDA is formulated that incorporates elasto-plastic constitutive laws into incremental forms of the equation of motion. A node-based uniform strain element is applied to avoid volumetric locking, which often occurs in conventional NMM-DDA. The proposed method is applied to three fundamental boundary value problems: a beam bending problem, a bearing capacity problem of a footing, and a bearing capacity problem of a masonry structure. The method is verified through comparisons with conventional solutions.     

Reliability bearing capacity analysis of footings on cohesive soil slopes using RFEM

Reliability analysis of bearing capacity of a strip footing at the crest of a simple slope with cohesive soil was carried out using the random finite element method (RFEM). Analyses showed that the coefficient of variation and the spatial correlation length of soil cohesion can have a large influence on footing bearing capacity, particularly for slopes with large height to footing width ratios. The paper demonstrates cases where a footing satisfies a deterministic design factor of safety of 3 but the probability of design failure is unacceptably high. Isotropic and anisotropic spatial variability of the soil strength was also considered.     

Upper bound finite element analysis of slope stability using a nonlinear failure criterion

In this study, upper bound finite element (FE) limit analysis is applied to stability problems of slopes using a nonlinear criterion. After formulating the upper bound analysis as the dual form of a second-order cone programming (SOCP) problem, the stress field and corresponding shear strength parameters can be determined iteratively. Thus, the nonlinear failure criterion is represented by the shear strength parameters associated with stress so that the analysis of slope stability using a nonlinear failure criterion can be transformed into the traditional upper bound method with a linear Mohr–Coulomb failure criterion. Comparison with published solutions illustrates the accuracy and feasibility of the proposed method for a simple homogeneous slope stability problem. The proposed approach is also applied to a seismic stability problem for a rockfill dam to study the influence of different failure criterions on the upper bound solutions. The results show that the seismic stability coefficients obtained using two different nonlinear failure criteria are similar but that the convergence differs significantly.     

Bearing Capacity of Embedded Strip Footings on Cohesionless Soil Under Vertical and Horizontal Loads

In the last decades a few attention was given to the evaluation of the bearing capacity of embedded footing under inclined loads on a frictional soil. This paper focuses on a numerical study using the finite-difference code Fast Lagrangian Analysis of Continua (FLAC), to evaluate the bearing capacity of embedded strip footing on a frictional soil. The soil is modeled by an elasto-plastic model with a Mohr–Coulomb yield criterion and associative flow rule; the effect of non-associativity of the soil on the bearing capacity is also investigated. The effect of the embedment is estimated though a depth factor, defined as a ratio of the bearing capacity of a strip footing at a depth D to that of a strip footing at the ground surface. The inclination effect is estimated by inclination factors, defined as the ratio of the limit vertical load for a footing under inclined loading to that of the vertically loaded footing. Both swipe and probe analyses were carried out to identify the vertical force–horizontal force (V–H) failure envelope. The results have been compared with those available in the literature.     

Incremental plasticity analysis of frictional soils

Three constitutive models of soil are used in finite element analyses of lateral earth pressure and bearing capacity. The three models are an elasto-plastic formulation derived from the Mohr-Coulomb law, a similar model with the plastic dilatancy removed, and a strain hardening model with a capped yield criterion. Stiffness formulations are described; the non-dilatant model has a non-symmetric stiffness. The results for the retaining walls are in close agreement with classical soil mechanics, but the bearing capacity analyses greatly overestimate the bearing capacity. The patterns of motion are, however, reasonable. Reasons for the discripancies in the bearing capacity case include: (a) the elements are too stiff and do not permit sliding on discrete failure planes; (b) the bearing capacity problem is itself not well settled theoretically; (c) very fine element divisions are necessary in areas of strong stress gradients and (d) rotation of principal stresses is significant.     

基于混合常应力−光滑应变单元的极限分析方法

极限分析是岩土工程稳定性评价的重要方法之一。传统的有限元极限分析方法,采用低阶三角形单元时需要引入速度间断面并采用特殊网格布局,或者采用高阶三角形单元等措施来克服体积锁定问题和提高数值精度。在光滑有限元法（smoothed finite element method,简称SFEM）的基础上,提出了一种基于新型混合常应力−光滑应变单元的极限分析方法（mixed constant stress-smoothed strain element limit analysis,简称MCSE-LA方法）。在服从关联流动法则和Mohr-Coulomb屈服准则的基础上,MCSE-LA方法最终将数值极限分析转化为以应力和极限荷载乘子为基本未知量的二阶锥规划（second order cone programming,简称SOCP）问题。MCSE-LA方法具有形式简单、优化变量相对较少和无需显式的写出塑性内能耗散函数的优点,并且根据凸锥优化的对偶理论,可以从对偶问题中获得速度场和塑性乘子等信息。此外,还采用基于最大塑性剪应变率的网格自适应加密算法,该算法在塑性区细化网格,显著提高了新数值极限分析方法的计算效率和精度。最后通过边坡稳定分析的结果对比,验证了MCSE-LA方法的计算精度和效率均高于传统的有限元极限分析方法。     

Estimation of bearing capacity for multiple footings in sand

In the present study, the effects of multiple-footing configurations in sand on bearing capacity were investigated using field plate load tests and finite element analyses. Both strip and spread footings were considered in the finite element analyses. In each case, different footing distances were applied for the purposes of comparison among all of the results. From these results, it was observed that the load responses of multiple footings are similar to those of the single footing at distances greater than three times the footing width. Design equation and correlation parameters, necessary for quantifying the values of the bearing capacity ratio for the different multiple-footing configuration, were derived. Experimental test results from the literature were selected and used in verifying the proposed method.