土体稳定性弹塑性大变形有限元分析
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摘要
土体稳定包括土坡稳定和地基稳定,是土力学的基本课题之一,是岩土工程的一个重要研究内容。传统的土体稳定分析方法极限平衡法、极限分析法虽经验丰富,但理论上缺点明显。有限单元法是一种理论体系更为严密的方法,近期基于弹塑性小变形理论的有限元强度折减系数法进行土体稳定分析在国内外受到关注,已取得了一些可观的研究成果。已有研究表明,土体失稳是局部化变形形成与发展的结果,是弹塑性大变形问题,尤其软土更表现出大位移、大应变,因此,在分析中不仅要考虑土体的材料非线性,还要引入大变形问题的几何非线性分析方法。本文对大变形研究成果进行分析总结的基础上,对土体失稳过程中弹塑性大变形的问题,采用Updated Lagrangian描述方法,建立了土体稳定分析的弹塑性大变形有限元模型,提出了坐标更新和非线性方程组线性化的方法;在土体本构模型中采用了适合进行土体稳定分析的Mohr-Coulomb准则,同时考虑了土体拉伸屈服及拉伸屈服后强度软化对土体稳定性的影响,并用Fortran语言编制了相应的计算程序,分析了复杂荷载作用下的边坡稳定、地基整体稳定和建筑物地基表层土体抗滑稳定。本文具体工作内容如下:
1.简要叙述了大变形有限元分析中用到的连续介质力学有关理论,包括物体的构形和物体运动的两种描述方法、变形梯度、应变的两种度量、变形率、三种应力张量及其相互关系、客观应力张量变化率、三种应力应变能量共轭对、虚功方程和虚功率方程。
2.推导了应力空间和应变空间表述的弹塑性小变形本构方程,明确了应变空间表述本构方程对强化材料、理想材料以及软化材料都能普遍适用。采用变形率的和分解,将应变空间表述的弹塑性小变形本构方程扩展到欧拉描述的大变形条件下率型弹塑性本构方程,为方便地利用现成的塑性理论和采用修正的拉格朗曰法(U.L)建立大变形条件下的有限元列式提供了理论依据。
3.在土体本构模型中,选用了与最大拉应力结合的Mohr-Coulomb准则屈服准则,并将土体拉伸屈服后的抗拉强度弱化为零,这对建筑物地基表层
The subject of soil mass stability to include slope stability and foundation stability is the one of basic problem of soil mechanics ,it is a important researchcontent of geotechnical engineering.Though traditional analysis methods-----limitequilibrium method and limit analysis method of soil mass stability have rich experience, they have obvious shortcoming theoretically .The Finite Element Method (FEM) is a kind of more strict method in theoretical system. Based on the theory of small deformation elastic-plastic, the strength reduction method with FEM for soil mass stability analysis is recently regarded at home and overseas, some considerable research findings has been gotten.To study has shown that the soil mass unstability is the results for local deformation to occur and develop, it is the problem of large elastic-plastic deformation, especially soft soil more shows large deformation and large strain, therefore to analyse soil mass stability has to consider the material nonlinearity of soil mass and the geometry nonlinearity of large deformation problem.The research findings of large deformation having been analysesed and summarized, based on the updated Lagrangian description of large deformation and the large elastic-plastic deformation of the soil unstability process, the model of large deformation elastic-plastic finite element for soil stability is constructed. The methods of updating coordinates and nonlinear equation group linearization are propounded. In a constitutive model the Mohr-Coulomb criterions of most fit for soil steady analysis were adopted.Soil tension yield and softening of tensile strength after tension yield were also considered.The calculation program of soil mass stability was compiled with Fortran. Slope stability , general foundation stability and soil local stability of against sliding under the action of complicated loads were calculated.These works are as follows:1. The theory of continuum mechanics to be used in Large deformation analysis with finite element method has briefly been related, it include two methods
describing object configuration and movement, deformation gradient, two kinds of strain metrics, deformation rate , 3 kinds of stress tensor and its correlation , objective stress tensor rate of change,3 conjugates of stress and strain energy, the virtual work equation and the fietitious power equation.2. Increment constitution equation of small elastic-plastic deformation in the stress space and the strain space was derived. It was clarified that constitution equation in the strain space is universal for strain hardening,elastic-perfectly plastic and strain softening.Used sum decom- pose of deformation rate, increment constitution equation of small elastic-plastic deformation in the strain space was transformed into the elastic-plastic rate constitution equation of large deformation with Eulerian deseription.That provide theoretical basis to use readytouse plastic theory and to constructe finite element formula with the updated Lagrangian description of large deformation.3. In a constitutive model of soil mass,the Mohr-Coulomb yield criterions with the maximum tension stress yield criterion was chosen.The tensile strength were weakened to be zero after soil tension yield,that is highly important for analysesing soil local stability of against sliding.Normal flow rule was chosen as plastic flow rule, so plasticity stiffness tensor was symmetrical, in order to use readytouse continuum mechanics theory. The singular points are replace by pertinent surface of Drucker—Prager circular cone ,these singular points are on the Mohr-Coulomb yield surface or the maximum tension stress yield surface and are uncertain on derivative.4. In the U.L system, according to the rate constitution equation of large deformation with Eulerian deseription,increment constitution equation with Lagrangian description was derived, i.e. the relation of Kirchhoff stress increment AStJ and Green strain increment AEtj was derived. The finite element formula of large deformation was derivedY(Aa) = ￡ B'ASdV + ￡ B'SdV -R=0Where Aa is the node displacement increment, .Bis the strain- displacement
conversion matrix, ￡ is the Kirchhoff stress at the time t, R is the equivalent node load vector. The new methods of updating coordinates are propounded, i.e. node coordinates will be updated at once after finishing each iterative calculation.The calculation step of nonlinear equation group is iteration step. Based on these , The new methods of nonlinear equation group linearization are propounded, i.e. the iterative form under certain load increment level is tF"(Aan) = (^B[DeBldV)Aan + ^ (B1"1)'SndV-R = 0where superscript n express the iteration order number in this load increment level, BL is the linear strain-displacement conversion matrix, De is the matrix form of linear elastic material tensor. BLandDe will be updated just at the first iteration in certain load increment level. B"~J is the strain-displacement conversion matrix calculated with nodal displacement increment that was result of last iterative computation. S" is the Kirchhoff stress at the start of this iteration.In this way, iteration method of the constant stiffness was realized under certain load increment level, the calculation of FEM of ideal plasticity material go on smoothly. As the correlative plasitc flow rule was used and the existing unstability criterion of soil mass was analysed,in certain load increment level,limited iterative time—permited maximum of iterative time was chosen as the unstability criterion of soil mass.Exceeded this limited value,iterate has not met the convergence criterion,that show that whole soil structure is bereaved of load carrying capacity,i.e.soil mass is unstability.5. Based on the existent program PLAST of the planar elastic- plastic small deformation,the calculation program of the planar large deformation elastic-plastic finite element for soil stability analysis is compiled with Fortran.Slope stability,general foundation stability and soil local stability of against sliding were calculated by this program.lt showed that the computation on soil stability with the large deformation elastic-plastic FEM is rational and important for the engineering safety.
1.简要叙述了大变形有限元分析中用到的连续介质力学有关理论,包括物体的构形和物体运动的两种描述方法、变形梯度、应变的两种度量、变形率、三种应力张量及其相互关系、客观应力张量变化率、三种应力应变能量共轭对、虚功方程和虚功率方程。
2.推导了应力空间和应变空间表述的弹塑性小变形本构方程,明确了应变空间表述本构方程对强化材料、理想材料以及软化材料都能普遍适用。采用变形率的和分解,将应变空间表述的弹塑性小变形本构方程扩展到欧拉描述的大变形条件下率型弹塑性本构方程,为方便地利用现成的塑性理论和采用修正的拉格朗曰法(U.L)建立大变形条件下的有限元列式提供了理论依据。
3.在土体本构模型中,选用了与最大拉应力结合的Mohr-Coulomb准则屈服准则,并将土体拉伸屈服后的抗拉强度弱化为零,这对建筑物地基表层
The subject of soil mass stability to include slope stability and foundation stability is the one of basic problem of soil mechanics ,it is a important researchcontent of geotechnical engineering.Though traditional analysis methods-----limitequilibrium method and limit analysis method of soil mass stability have rich experience, they have obvious shortcoming theoretically .The Finite Element Method (FEM) is a kind of more strict method in theoretical system. Based on the theory of small deformation elastic-plastic, the strength reduction method with FEM for soil mass stability analysis is recently regarded at home and overseas, some considerable research findings has been gotten.To study has shown that the soil mass unstability is the results for local deformation to occur and develop, it is the problem of large elastic-plastic deformation, especially soft soil more shows large deformation and large strain, therefore to analyse soil mass stability has to consider the material nonlinearity of soil mass and the geometry nonlinearity of large deformation problem.The research findings of large deformation having been analysesed and summarized, based on the updated Lagrangian description of large deformation and the large elastic-plastic deformation of the soil unstability process, the model of large deformation elastic-plastic finite element for soil stability is constructed. The methods of updating coordinates and nonlinear equation group linearization are propounded. In a constitutive model the Mohr-Coulomb criterions of most fit for soil steady analysis were adopted.Soil tension yield and softening of tensile strength after tension yield were also considered.The calculation program of soil mass stability was compiled with Fortran. Slope stability , general foundation stability and soil local stability of against sliding under the action of complicated loads were calculated.These works are as follows:1. The theory of continuum mechanics to be used in Large deformation analysis with finite element method has briefly been related, it include two methods
describing object configuration and movement, deformation gradient, two kinds of strain metrics, deformation rate , 3 kinds of stress tensor and its correlation , objective stress tensor rate of change,3 conjugates of stress and strain energy, the virtual work equation and the fietitious power equation.2. Increment constitution equation of small elastic-plastic deformation in the stress space and the strain space was derived. It was clarified that constitution equation in the strain space is universal for strain hardening,elastic-perfectly plastic and strain softening.Used sum decom- pose of deformation rate, increment constitution equation of small elastic-plastic deformation in the strain space was transformed into the elastic-plastic rate constitution equation of large deformation with Eulerian deseription.That provide theoretical basis to use readytouse plastic theory and to constructe finite element formula with the updated Lagrangian description of large deformation.3. In a constitutive model of soil mass,the Mohr-Coulomb yield criterions with the maximum tension stress yield criterion was chosen.The tensile strength were weakened to be zero after soil tension yield,that is highly important for analysesing soil local stability of against sliding.Normal flow rule was chosen as plastic flow rule, so plasticity stiffness tensor was symmetrical, in order to use readytouse continuum mechanics theory. The singular points are replace by pertinent surface of Drucker—Prager circular cone ,these singular points are on the Mohr-Coulomb yield surface or the maximum tension stress yield surface and are uncertain on derivative.4. In the U.L system, according to the rate constitution equation of large deformation with Eulerian deseription,increment constitution equation with Lagrangian description was derived, i.e. the relation of Kirchhoff stress increment AStJ and Green strain increment AEtj was derived. The finite element formula of large deformation was derivedY(Aa) = ￡ B'ASdV + ￡ B'SdV -R=0Where Aa is the node displacement increment, .Bis the strain- displacement
conversion matrix, ￡ is the Kirchhoff stress at the time t, R is the equivalent node load vector. The new methods of updating coordinates are propounded, i.e. node coordinates will be updated at once after finishing each iterative calculation.The calculation step of nonlinear equation group is iteration step. Based on these , The new methods of nonlinear equation group linearization are propounded, i.e. the iterative form under certain load increment level is tF"(Aan) = (^B[DeBldV)Aan + ^ (B1"1)'SndV-R = 0where superscript n express the iteration order number in this load increment level, BL is the linear strain-displacement conversion matrix, De is the matrix form of linear elastic material tensor. BLandDe will be updated just at the first iteration in certain load increment level. B"~J is the strain-displacement conversion matrix calculated with nodal displacement increment that was result of last iterative computation. S" is the Kirchhoff stress at the start of this iteration.In this way, iteration method of the constant stiffness was realized under certain load increment level, the calculation of FEM of ideal plasticity material go on smoothly. As the correlative plasitc flow rule was used and the existing unstability criterion of soil mass was analysed,in certain load increment level,limited iterative time—permited maximum of iterative time was chosen as the unstability criterion of soil mass.Exceeded this limited value,iterate has not met the convergence criterion,that show that whole soil structure is bereaved of load carrying capacity,i.e.soil mass is unstability.5. Based on the existent program PLAST of the planar elastic- plastic small deformation,the calculation program of the planar large deformation elastic-plastic finite element for soil stability analysis is compiled with Fortran.Slope stability,general foundation stability and soil local stability of against sliding were calculated by this program.lt showed that the computation on soil stability with the large deformation elastic-plastic FEM is rational and important for the engineering safety.
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