结构非线性分析ABAQUS

《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……………………………………………………………………………………………3 1 偏压柱的跨中最大挠度的解析解………………………………………………………………3 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…………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…………………………………………………………………………12 3 ABAQUS有限元软件分析的理论背景（来自ABAQUS帮助文件）………………………14 4 对结构几何非线性和稳定的关系进行讨论…………………………………………………24

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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： M?P(e?y) （1）

d2yM??1.2由材料力学中梁挠曲线的近似微分方程可以得到： 2dxEI将（1）式代入其中得

P(e?y)PPe y''???y?EIEIEI解微分方程得：

l?x?x???y?e?csc?lsin?(l?x)??ecsc?lsin?x??l?l????? ?e?sin?(l?x)?sin?x?csc?l?e其中??P EI1.3 求跨中侧移：当x? ymax

l时 2?l?l?2esincsc?l?e?e(sec?1)

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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

采用韩林海（2007）中的二次塑性流模型来模拟钢材， ? i 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：

a fu fy fp b c d e ?e?0.8fy/Es,?e1?1.5?e,?e2?10?e1,?e3?100?e1； 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。

0 ?e ?e1 ?e2 ?e3 ?i 图1 钢材的? -? 关系 ??Es?s??2?A??B?s?Cs???s??fy??f?1?0.6?s??e2??y??e3??e2????1.6fy???s??e?e??s??e1?e1??s??e2?e2??s??e3?s??e3 (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

应力应 ?e 变 ?e1 分 ?e2 段 点 ?e3 系 数 A B ?0 0.001729612 0.001383689 0.002075534 0.02075534 0.218143 1.48877E+14 6.18E+11 钢管应力应变图700600500400系列1300200100000000.10.010.010.010.020.020.040.070.140.170.2 4

C -285040000

2.3 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

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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 276 300.84 320.16 333.96 342.24 345 345 345 345 345 345 345 345

塑性应变 0 0 0.0001 0.0001 0.0002 0.0003 0.0022 0.0041 0.0059 0.0078 0.0097 0.0115 0.0134 应力 345 345 345 386.4 427.8 469.2 510.6 552 552 552 552 552 552 塑性应变 0.0152 0.0171 0.019 0.056 0.0931 0.1301 0.1672 0.2042 0.2249 0.2456 0.2663 0.287 0.3077

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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

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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

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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

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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

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where K is the elastic stiffness matrix and i stands for the iteration number. Because the

elastic stiffness serves as the Jacobian matrix throughout the analysis, the equation system is solved only once. Therefore, the direct cyclic algorithm is likely to be less expensive to use than the full Newton approach to the solution of the nonlinear equations, especially when the problem is large. We also expand the residual vector in a truncated Fourier series in the same form as the displacement solution:

where each residual vector coefficient displacement coefficient. The conversion of element-by-element basis:

, and

in the Fourier series corresponds to a

into Fourier terms is done incrementally on an

At the end of each loading cycle, we solve for the corrections to the displacement Fourier coefficients— and . The next displacement coefficients are then

The updated displacement coefficients are used in the next iteration to obtain displacements at each instant in time. This process is repeated until convergence is obtained. Each pass through the

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complete load cycle can therefore be thought of as a single iteration of the solution to the nonlinear problem.

Convergence of the direct cyclic method is best measured by ensuring that all the entries in and are sufficiently small. By default, both these criteria are checked in an Abaqus/Standard solution.

There are two accuracy aspects to this algorithm: the number of Fourier terms and the number of iterations to obtain convergence. The number of Fourier terms needed to obtain a solution

depends on the time variation of the cyclic load as well as the variation of the structure response. In determining the number of terms, keep in mind that the objective of this kind of analysis is to make low-cycle fatigue life predictions. Hence, the goal is to obtain a good approximation of the plastic strain cycle at each point; local inaccuracies in the stress are less important. More Fourier terms usually provide a more accurate solution but at the expense of additional data storage and computational time. Abaqus/Standard uses an adaptive algorithm to determine the number of

Fourier terms during the analysis. Both “automatic” time incrementation and direct user control of the time incrementation can be used in the direct cyclic method.

Since the direct cyclic algorithm uses the modified Newton method, in which a constant elastic stiffness matrix serves as the Jacobian throughout the analysis, interface nonlinearities such as contact and friction are not taken into account. These nonlinearities are severe and would probably lead to convergence difficulties if they were included in the direct cyclic algorithm.

By default, the periodicity condition, in which the solution of an iteration starts with the solution at the end of the previous iteration, is always imposed from the beginning of an analysis. However, in cases where the periodic solution is not easily found (for example, when the loading is close to causing ratchetting), the state around which the periodic solution is obtained may show considerably more “drift” than would be obtained in a transient analysis. In such cases the user may wish to delay the application of the periodicity condition as an artificial method to reduce this drift. Abaqus/Standard allows the user to choose when to impose the periodicity condition. By delaying the application of the periodicity condition, the user can influence the mean stress and strain level, without affecting the shape of the stress-strain curves or the amount of energy dissipated during the cycle. Therefore, this is rarely necessary since the average stress and strain levels are usually not needed for low-cycle fatigue life predictions.

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4 对结构几何非线性和稳定的关系进行讨论

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