结构非线性分析ABAQUS.doc

工程结构非线性作业学院： 土 木 工 程 学 院 专业： 结 构 工 程 姓名： 汪 洋 学号： S10011056 教师： 方志（教授） 目 录 作业31 偏压柱的跨中最大挠度的解析解32用有限元软件ABAQUS建立题中所给的弯压柱的力学模型，并计算跨中最大挠4 2.1 给出一个实例4 2.2 确定材料的本构模型4 2.3 建立有限元模型5 2.4 模拟结果分析对比123 ABAQUS有限元软件分析的理论背景（来自ABAQUS帮助文件）144 对结构几何非线性和稳定的关系进行讨论24结构非线性作业一（1） 求出荷载柱中点侧移的解析解。（2） 以具体的实例给出几何非线性效应得数值解（可用有限元程序计算），并与解析解结果对比。（3） 给出有限元程序理论背景的详细描述。（4） 对结构几何非线性和稳定的关系进行讨论。1 偏压柱的跨中最大挠度的解析解图1 计算简图1.1跨中弯矩为： （1）1.2由材料力学中梁挠曲线的近似微分方程可以得到： 将（1）式代入其中得解微分方程得： 其中1.3 求跨中侧移：当时 2 用有限元软件ABAQUS建立题中所给的弯压柱的力学模型，并计算跨中最大挠度2.1 给出一个实例： 假设题中所给弯压柱所受荷载P=10KN, 偏心距e=0.1m，柱高为L=2m，采用屈服强度为345MP的钢材，弹性模量E=206000MP, 柱的截面尺寸如所示：图1 计算简图2.2 确定材料的本构模型abcdesifu fy fp 0 ee ee1 ee2 ee3 ei图1 钢材的s -e 关系采用韩林海（2007）中的二次塑性流模型来模拟钢材， 其应力-应变关系曲线,分为弹性段(Oa)、弹塑性段(ab)、塑性段 (bc)、强化段(cd)和二次塑流(de)等五个阶段，如图1所示。图1中的点划线为钢材实际的应力-应变关系曲线，实线所示为简化的应力-应变关系曲线，模型的数学表达式如式(3-1)。其中：； fp、fy和 fu分别为钢材的比例极限、屈服极限和抗拉强度极限。 (3-1) 由该本构模型计算出材料的应力应变关系表1 计算的钢管的力学参数与应力应变曲线应变分段点0.0017296120.0013836890.0020755340.020755340.218143系数A1.48877E+14B6.18E+11C-2850400002.3 建立有限元模型2.3.1 创建部件在ABAQUS里打开而为建模截面，创建一根二维的柱模型，长度为2000mm。如下图所示：图1 创建二维柱部件2.3.2 创建材料参数创建钢材料属性 采用韩林海（2007）二次 塑流模型Ec=206000;泊松比0.3；塑性应力应变参 数见表格；同时要对材料的的塑性性能进行编辑，将事先计算好的应力塑性应变数据导入到steel的塑性编辑表格里面去就可以了。在输入材料的应力塑性应变数据组的时候要保证所输入的塑性应变是递增的，并且初始塑性应变必须为零。将塑性数据输入到软件中去。表2 钢材塑性应变应力应力塑性应变应力塑性应变27603450.0152300.8403450.0171320.160.00013450.019333.960.0001386.40.056342.240.0002427.80.09313450.0003469.20.13013450.0022510.60.16723450.00415520.20423450.00595520.22493450.00785520.24563450.00975520.26633450.01155520.2873450.01345520.3077图2 创建钢材材料属性2.3.3 创建并指派截面指派截面定义柱的截面为一个宽50mm，高100mm的矩形截面如下：图3 创建柱界面并赋予构件上2.3.4 组装配件将各个部件建立起来并赋予了材料属性与截面属性之后，开始将各个部件组装在一起。Instance part选择需组装的部件。图4 装配构件2.3.5 设置分析步与相互作用在本模型中需要研究的是跨中挠度。所以在设置分析步中的历程输出中要创建相应的输出。为了便于计算机计算模型，需要创建一个分析步。需要输出的数据为跨中挠度。图5（a） 设置分析步图5（b） 设置分析步参数2.3.6 创建荷载，约束并划分网格构件受到一个偏心的轴力的作用，在这里假设柱受到一个轴力P=10KN，和一对弯矩M=2KNM作用。图6 创建荷载划分网格图7 划分网格2.3.7 新建工作并提交运行进入到工作界面，新建工作，新建完工作之后要进行数据检查，检查完数据无误之后再点击提交开始运行。运行完之后就可以在可视化对话框中观察计算结果。2.3.8 模拟结果分析 运用Abqaus6.5有限元软件建模计算之后可以得出钢管混凝土叠合柱的各个部件在既定荷载下的受力情况。图 8（a）加载前的柱图图 8（b） 加载后的变形图图13 加载后的受力云图输出跨中挠度时间曲线3.5结果分析ABAQUS模拟计算出来跨中最大挠度为：0.52561mm利用前面建立的理论公式计算跨中挠度： 所以： 对比有限元软件结果0.52561mm与理论公式结果0.51865mm，差异度只有1.3%4 ABAQUS有限元软件分析的理论背景（来自ABAQUS帮助文件）4.1Nonlinear solution methods in Abaqus/StandardProduct: Abaqus/StandardThe finite element models generated in Abaqus are usually nonlinear and can involve from a few to thousands of variables. In terms of these variables the equilibrium equations obtained by discretizing the virtual work equation can be written symbolically as where is the force component conjugate to the variable in the problem and is the value of the variable. The basic problem is to solve Equation 2.2.11 for the throughout the history of interest. Many of the problems to which Abaqus will be applied are history-dependent, so the solution must be developed by a series of “small” increments. Two issues arise: how the discrete equilibrium statement Equation 2.2.11 is to be solved at each increment, and how the increment size is chosen.Abaqus/Standard generally uses Newtons method as a numerical technique for solving the nonlinear equilibrium equations. The motivation for this choice is primarily the convergence rate obtained by using Newtons method compared to the convergence rates exhibited by alternate methods (usually modified Newton or quasi-Newton methods) for the types of nonlinear problems most often studied with Abaqus. The basic formalism of Newtons method is as follows. Assume that, after an iteration i, an approximation , to the solution has been obtained. Let be the difference between this solution and the exact solution to the discrete equilibrium equation Equation 2.2.11. This means that Expanding the left-hand side of this equation in a Taylor series about the approximate solution then gives If is a close approximation to the solution, the magnitude of each will be small, and so all but the first two terms above can be neglected giving a linear system of equations: where is the Jacobian matrix and The next approximation to the solution is then and the iteration continues. Convergence of Newtons method is best measured by ensuring that all entries in and all entries in are sufficiently small. Both these criteria are checked by default in an Abaqus/Standard solution. Abaqus/Standard also prints peak values in the force residuals, incremental displacements, and corrections to the incremental displacements at each iteration so that the user can check for these contingencies himself.Newtons method is usually avoided in large finite element codes, apparently for two reasons. First, the complete Jacobian matrix is sometimes difficult to formulate; and for some problems it can be impossible to obtain this matrix in closed form, so it must be calculated numericallyan expensive (and not always reliable) process. Secondly, the method is expensive per iteration, because the Jacobian must be formed and solved at each iteration. The most commonly used alternative to Newton is the modified Newton method, in which the Jacobian in Equation 2.2.12 is recalculated only occasionally (or not at all, as in the initial strain method of simple contained plasticity problems). This method is attractive for mildly nonlinear problems involving softening behavior (such as contained plasticity with monotonic straining) but is not suitable for severely nonlinear cases. (In some cases Abaqus/Standard uses an approximate Newton method if it is either not able to compute the exact Jacobian matrix or if an approximation would result in a quicker total solution time. For example, several of the models in Abaqus/Standard result in a nonsymmetric Jacobian matrix, but the user is allowed to choose a symmetric approximation to the Jacobian on the grounds that the resulting modified Newton method converges quite well and that the extra cost of solving the full nonsymmetric system does not justify the savings in iteration achieved by the quadratic convergence of the full Newton method. In other cases the user is allowed to drop interfield coupling terms in coupled procedures for similar reasons.)Another alternative is the quasi-Newton method, in which Equation 2.2.12 is symbolically rewritten and the inverse Jacobian is obtained by an iteration process. There are a wide range of quasi-Newton methods. The more appropriate methods for structural applications appear to be reasonably well behaved in all but the most extremely nonlinear casesthe trade-off is that more iterations are required to converge, compared to Newton. While the savings in forming and solving the Jacobian might seem large, the savings might be offset by the additional arithmetic involved in the residual evaluations (that is, in calculating the ), and in the cascading vector transformations associated with the quasi-Newton iterations. Thus, for some practical cases quasi-Newton methods are more economic than full Newton, but in other cases they are more expensive. Abaqus/Standard offers the “BFGS” quasi-Newton method: it is described in “Quasi-Newton solution technique,” Section 2.2.2.When any iterative algorithm is applied to a history-dependent problem, the intermediate, nonconverged solutions obtained during the iteration process are usually not on the actual solution path; thus, the integration of history-dependent variables must be performed completely over the increment at each iteration and not obtained as the sum of integrations associated with each Newton iteration, . In Abaqus/Standard this is done by assuming that the basic nodal variables, , vary linearly over the increment, so that where represents “time” during the increment. Then, for any history-dependent variable, , we compute at each iteration. The issue of choosing suitable time steps is a difficult problem to resolve. First of all, the considerations are quite different in static, dynamic, or diffusion cases. It is always necessary to model the response as a function of time to some acceptable level of accuracy. In the case of dynamic or diffusion problems time is a physical dimension for the problem and the time stepping scheme must provide suitable steps to allow accurate modeling in this dimension. Even if the problem is linear, this accuracy requirement imposes restrictions on the choice of time step. In contrast, most static problems have no imposed time scale, and the only criterion involved in time step choice is accuracy in modeling nonlinear effects. In dynamic and diffusion problems it is exceptional to encounter discontinuities in the time history, because inertia or viscous effects provide smoothing in the solution. (One of the exceptions is impact. The technique used in Abaqus/Standard for this is discussed in “Intermittent contact/impact,” Section 2.4.2.) However, in static cases sharp discontinuities (such as bifurcations caused by buckling) are common. Softening systems, or unconstrained systems, require special consideration in static cases but are handled naturally in dynamic or diffusion cases. Thus, the considerations upon which time step choice is made are quite different for the three different problem classes.Abaqus provides both “automatic” time step choice and direct user control for all classes of problems. Direct user control can be useful in cases where the problem behavior is well understood (as might occur when the user is carrying out a series of parameter studies) or in cases where the automatic algorithms do not handle the problem well. However, the automatic schemes in Abaqus are based on extensive experience with a wide range of problems and, therefore, generally provide a reliable approach.For static problems a number of schemes have been suggested for automatic step control (see, for example, Bergan et al., 1978). Abaqus/Standard uses a scheme based predominantly on the maximum force residuals following each iteration. By comparing consecutive values of these quantities, Abaqus/Standard determines whether convergence is likely in a reasonable number of iterations. If convergence is deemed unlikely, Abaqus/Standard adjusts the load increment; if convergence is deemed likely, Abaqus/Standard continues with the iteration process. In this way excessive iteration is eliminated in cases where convergence is unlikely, and an increment that appears to be converging is not aborted because it needed a few more iterations. One other ingredient in this algorithm is that a minimum increment size is specified, which prevents excessive computation in cases where buckling, limit load, or some modeling error causes the solution to stall. This control is handled internally, with user override if needed. Several other controls are built into the algorithm; for example, it will cut back the increment size if an element inverts due to excessively large geometry changes. These detailed controls are based on empirical testing.In dynamic analysis when implicit integration is used, the automatic time stepping is based on the concept of half-step residuals (Hibbitt and Karlsson, 1979). The basic idea is that the time stepping operator defines the velocities and accelerations at the end of the step in terms of displacement at the end of the step and conditions at the beginning of the step. Equilibrium is then established at which ensures an equilibrium solution at the end of each time step and, thus, at the beginning and end of any individual time step. However, these equilibrium solutions do not guarantee equilibrium throughout the step. The time step control is based on measuring the equilibrium error (the force residuals) at some point during the time step, by using the integration operator, together with the solution obtained at , to interpolate within the time step. The evaluation is performed at the half step . If the maximum entry in this residual vectorthe maximum “half-step residual”is greater than a user-specified tolerance, the time step is considered to be too big and is reduced by an appropriate factor. If the maximum half-step residual is sufficiently below the user-specified tolerance, the time step can be increased by an appropriate factor for the next increment. Otherwise, the time step is deemed adequate. The algorithm is somewhat more complicated at traumatic events such as impact. Here, the time step can also be adjusted based on the magnitude of residuals in the first or second iteration following such events. Clearly, if these residuals are several orders of magnitude greater than those permitted, convergence is unlikely and the time step is altered immediately to avoid unproductive iteration. These algorithms are discussed in more detail in “Intermittent contact/impact,” Section 2.4.2, as well as in the Abaqus Analysis Users Manual. They are products of experience and many numerical experiments and have been shown to be effective in several problem areas of interest.4.2 Quasi-Newton solution techniqueProduct: Abaqus/StandardA major contribution to the computational effort involved in nonlinear analysis is the solution of the nonlinear equations (Equation 2.2.11). In most cases Abaqus/Standard uses Newtons method to solve these equations, as described in “Nonlinear solution methods in Abaqus/Standard,” Section 2.2.1. The principal advantage of Newtons method is its quadratic convergence rate when the approximation at iteration i is within the “radius of convergence”that is, when the gradients defined by provide an improvement to the solution. The method has two major disadvantages: the Jacobian matrix has to be calculated, and this same matrix has to be solved. The calculation of the Jacobian matrix is a problem because, in many important cases, it is difficult to derive the form of the matrix algebraically. The solution of the Jacobian is a problem because of the computational effort involved: as the problem size increases, the direct solution of the linear equations can dominate the entire computational effort.There are a number of important nonlinear applications where the Jacobian is symmetric, is fairly well conditioned, and does not change greatly from one iteration to the next. Examples are implicit dynamic time integration with small time increments relative to the periods of the natural vibrations that participate in the response or small-displacement elastic-plastic analysis where the yielding is confined (such as occurs in many practical fracture mechanics applications). In such cases, especially when the problem is large, it can be less expensive to use an alternative to the Newton approach to the solution of the nonlinear equations. The “quasi-Newton” methods are such an approach; and Matthies and Strang (1979) have shown that, for systems of equations with a symmetric Jacobian matrix, the BFGS (Broyden, Fletcher, Goldfarb, Shanno) method can be written in a simple form that is especially effective on the computer and is successful in such applications. This method is implemented in Abaqus/Standard and is described in this section. The user must select this method explicitly: by default, Abaqus/Standard uses the standard Newton method.The basis of quasi-Newton methods is to obtain a series of improved approximations to the Jacobian matrix, , that satisfy the secant condition: so that approaches as the iterations proceed. Equation 2.2.21 is the basic quasi-Newton equation. For convenience we define the change in the residual from one iteration to the next as so that Equation 2.2.21 can be written where is the correction to the solution from the previous iteration, defined in “Nonlinear solution methods in Abaqus/Standard,” Section 2.2.1. Matthies and Strangs implementation of the BFGS method is a computationally inexpensive method of creating a series of approximations to that satisfy Equation 2.2.21 and retain the symmetry and positive definiteness of . They accomplish this by updating to using a “product plus increment” form: where In the actual implementation of this version of the BFGS method, each is not stored: rather, a “kernel” matrix, , is used (as the decomposition of ), and the update is accomplished by premultiplication of the kernel matrix by the terms and postmultiplication of the kernel matrix by the terms for . Because of the form of these terms, the premultiplication and postmultiplication operations result in inner products of vectors and the scaling of vectors by constants: it is this organization that makes the method computationally attractive. However, too many such products ( being bigger than, say, 510) are not attractive, so usually a new kernel matrix is formed and stored after some iterations. In the Abaqus/Standard implementation the kernel is the actual Jacobian matrix . It is formed whenever a specified number of iterations have been done without obtaining a convergent solution; the default number of iterations is 8. Abaqus/Standard does not reform the kernel unless this value is exceeded, so the same kernel can be used fo