土木工程专业外文翻译--剪力墙框架结构中的应力分布

英文原文
Stress Distribution In a Shear Wall – Frame Structure
Using Unstructured – Refined Finite Element Mesh
ABSTRACT
A semi-automatic algorithm for finite element analysis is presented to obtain the stress and strain distribution in shear wall-frame structures. In the study, a constant strain triangle with six degrees of freedom and mesh refinement - coarsening algorithms were used in Matlab® environment. Initially the proposed algorithm generates a coarse mesh automatically for the whole domain and the user refines this finite element mesh at required regions. These regions are mostly the regions of geometric discontinuities. Deformation, normal and shear stresses are presented for an illustrative example. Consistent displacement and stress results have been obtained from comparisons with widely used engineering software.
Key Words: Shear wall, FEM, Unstructured mesh, Refinement.
1.INTRODUCTION
In the last two decades, shear walls became an important part of our mid and high rise residential buildings in Turkey. As part of an earthquake resistant building design, these walls are placed in building plans reducing lateral displacements under earthquake loads so shear-wall frame structures are obtained. Since the 1960’s several approaches have been adopted to solve displacements and stress distribution of shear wall structures. Continuous medium approaches, and frame analogy models are the examples of these approaches [1-4]. In the past and today, numerical solution methods are the main effort area because of the accuracy of solution and the ease of usage in 2D and 3D analysis of shear walls [5-7].
Shear walls with openings, coupled shear walls and combined shear wall frame structures can be modeled as thin plates where the loading is uniformly distributed over the thickness, in the plane of the plate. This 2D domain can be subdivided into a finite number of geometrical shapes. In the finite element method (FEM), these simple shaped elements such as triangles or quadrilaterals (in 2D) are called elements. The connection of these individual elements at nodes and along interelement boundaries covering the whole problem domain is called finite element mesh or grid. In the literature meshes can be grouped into two main categories such as structured and unstructured meshes. Structured meshes are constructed with geometrically similar triangular or quadrilateral elements. They are suitable especially for problems
with simple geometry and boundary shapes (Figure 1-a). Although structured meshes can be constructed as simple-time saving routines, regarding complicated domains with complex boundaries, it is a problem to fit the boundary shape. To circumvent this difficulty, unstructured meshes are used to discretize the complicated domains with internal boundaries (Figure 1-b). While it is a time consuming procedure, unstructured meshes are also suitable for local mesh refinement and coarsening. The aim of this work is to get a good quality unstructured mesh which will have smaller elements at the geometric discontinuities and bigger elements at other regions for a shear wall frame geometry.
Figure 1. Structured (a) and unstructured mesh (b)
2. TRIANGULAR FINITE ELEMENTS
2.1. Element Formulation
The first advantage of using triangular elements is that almost any plane geometry may be discretized using triangles. These elements have six degrees of freedom, two translations at each node (Figure 2). Because of the three nodes, the element has linear shape functions that are an additional benefit because of simplified mathematics. However, these functions generate constant strain and stress throughout the element. To surmount this disadvantage, smaller elements must be employed where strain and stress vary rapidly.
Figure 2. Constant strain triangular finite element
CST element has displacement functions and shape functions as follows,
()()3
322113
32211,,v N v N v N y x v u N u N u N y x u ++=++= （1） ()()()()[]y x x x y y y x y x A y x N e 23232332121,-+---=
（2） ()()()()[]y x x x y y y x y x A y x N e 31313113221,-+---= （3）
()()()()[]y x x x y y y x y x A y x N e 12121221321,-+---= （4）
where 1 u , 2 u and 3 u nodal displacements in x direction corresponding to nodes 1, 2 and 3 respectively. 1 v , 2 v and 3 v nodal displacements in y direction and 1 N , 2 N and 3 N are linear shape functions. x and y are the coordinates of corresponding nodes and e A is area of the element. In the finite element method, nodal displacements are obtained from the solution of the linear system of equations, that is
f Ku = （5） where, K is stiffness matrix, u is nodal displacement vector, and f is nodal load vector. Stiffness matrix may be calculated as
()()SN C SN t A K T e = （6） where t is thickness of the element,
⎥⎦
⎤⎢⎣⎡=321321000000N N N N N N N （7） and differential operator is,
⎥⎥⎥⎥⎦
⎤⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂=x y y x S 00 （8） and the elasticity matris is defined by
⎥⎥
⎥⎥⎦
⎤⎢⎢⎢⎢⎣⎡--=2100010112v v v E C ν （9） where E is modulus of elasticity and v is the Poisson’s ratio.
2.2. Mesh Generation and Refinement
Unstructured mesh generation procedure consists of some basic steps. These are the generations of boundary and interior nodes and connection of these nodes which has a specific name as triangulation for triangular finite elements. In this work, a random point generation scheme is used to form interior nodes that were explained by Fukuda and Suhura in detail [8]. This procedure uses subsquares in which only one node is generated randomly. Although, here, node generation is a random process, we can define some restriction on this generation. For example, we can say that the distance between any two nodes must not be less than a minimum value which can be taken as the width of a subsquare. In fact this random procedure is a time consuming procedure, but this is not too long for today’s desktop computers for initial node generation. Other algorithms can be used in large scale finite element problems. For example, instead of generating interior nodes by random trials, a rectangular grid could be used creating one node in each rectangle. The width and height of the rectangles are in the ratio of 2/√3 and they are placed in a zigzag manner and nodes are placed at the center of the rectangles. The ratio helps ensure equilateral triangles [9]. Another node generation scheme in which imaginary horizontal lines cut the domain in an even number of points could also be used.Interior nodes are generated on the horizontal line between the cuts according to a prescribed spacing parameter [10].
For the triangulation step, a number of algorithms have been suggested by various authors. These techniques involve simple automatic triangulation methods [8,9],advancing front methods [10,11], domain decomposition methods [12,13], and coordinate transformation methods [14]. In the present work,triangular elements in the problem domain are obtained using a condition known as Delaunay or empty circle criterion [15-18]. According to this rule, in a circumcircle of a triangle no node must exist in the problem domain (Figure 3). The nodes in the problem domain are scanned from the first node to the last node selecting three candidate nodes which will obey the empty circle criterion. This criterion is very useful to eliminate intersection check of the interelement boundaries.
Figure 3. A triangulation which obeys the empty circle criterion (a),
triangulation which does not obey the criterion (b).
Initially we could not know the necessary degree of smallness of finite elements to represent the stress distribution accurately. For that reason, a few numbers of initial nodes are used for triangulation giving an initial triangulation. Around geometrical discontinuities such as interior holes at shear walls and shear wall frame connection points, a local mesh refinement must be done by inserting new nodes and repeating the triangulation procedure. At this stage, a number of alternative algorithms could be employed to get a fine mesh where it is necessary. Those are,
• One point mesh refinement
• Line or polygon mesh refinement
• Central point mesh refinement
• and, Delaunay point mesh refinement
In FEM, all generated nodes and elements have some special id denoted by numbers. When this number is known for a node, it is easy to search surrounding elements of this node. For the first algorithm mentioned above, once these elements are determined, they aregrouped forming first kind of triangles which are affected by refining. Some second kinds of triangles exist, which are the neighboring triangles to the first kind. According to the one point local mesh refinement algorithm, four new triangles are generated in the first kind of triangles and two new triangles are generated in the second kind of triangles and old ones are deleted (Figure 4-a and b). Line and polygon local mesh refinement algorithms also use the same logic which differs from the first one in such a way that, selected nodes make a line or a closed polygon. In the third refinement algorithm, a point is added into a triangle, using the geometrical center of apex nodes. The last algorithm, named the Delaunay point mesh refinement,adds a node into the triangulation at the center of the circumcircle of a selected triangle.
Figure 4. First kind of triangles (a), second kind of triangles (b), and after refinement (c)
•selected point for refinement.
A mesh smoothing process was made to improve the quality of triangles after triangulation. In any branch of mechanics, the shape of triangles is an important factor for finite element meshes. Especially in solid mechanics triangles are regarded as good if they are nearly equiangular [19]. In computational fluid dynamics, problems concerning boundary layers and shocks, skinny and long triangles provide a better solution [20].In order to obtain well-distributed good quality triangles, here, we used a Laplacian smoothing method which uses neighboring triangles for node repositioning (Figure 5). This is an iterative method in which a node is moved to the centroid of the nodes to which it is connected [21]. One or two iterations for moving the nodes are sufficient but iterations could be continued until each movement satisfies a convergence distance.
Figure 5. Before smoothing (a), neighboring nodes (b), after Laplacian smoothing (c).
The new coordinates defined as,
()()u y x y x ∆+=0,,
（10） ∑==∆n
i i u n u 11 （11）
()00,y y x x u i i i --=
（12） Where n represents the number of surrounding nodes and zero indices represent initial values.
3. THE SHEAR WALL - FRAME PROBLEM
A shear wall model with small window openings connected to a beam column system was considered to perform a FEM analysis using unstructured mesh generation with refinement. This shear wall frame structure was loaded as tip load but the loading was distributed to the nodes at the top of the frame. The necessary dimensions of geometry are given in Figure 6-a. Boundary nodes were generated in a direct manner using the width of a column as a spacing parameter. A few initial nodes were obtained using the random procedure given in section 2 (Figure 6-b). Line mesh refinement algorithm was used to refine the mesh at the beams and columns. One point mesh refinement algorithm was used around sharp corners of window openings and beam shear-wall connections, in order to get the final mesh (Figure 7). The deformed shape and stress distributions obtained using the final mesh is given in Figure 8.
Figure 6. Example problem a) dimensions, loading and b) initial node generation.
Figure 7. After mesh refinement.
Figure 8. Graphical results after analysis.
4. CONCLUSIONS
In order to get an accurate solution for stress distribution at beam to column and beam to shear wall connections, we developed a finite element program in MATLAB environment. Although the geometry of the problem is quite complex because of interior holes, the problem domain was discretized with an unstructured finite element
with refinement.
1) The element mesh was refined employing small triangles at the regions of geometric discontinuities. Smooth transitions from small to large triangles were obtained. Good quality equiangular elements were obtained using empty circle criterion and Laplace smoothing.
2) A good agreement exists between ANSYS (PLANE2 elements) results and the work presented here in lateral displacements and shear stresses. Comparisons are given in Table 1 and Table 2 for lateral displacements and shear stresses respectively for a load value of P=100N.
Table 1. Lateral displacement of shear wall structure with openings.
m N 8: Nodes at 8 m height.
m N 4: Nodes at 4 m height.
Table 2. Shear stress, xy σ, values at some points given in Figure 5.
3) The algorithm reviewed here falls into the semi automatic approach category. There is an automatic mesh generation for arbitrary domain with inner holes. A manual decision is required for the refinement regions.
4) In fact, the work presented here provided us with a springboard for further developments to get an adaptive finite element procedure for shear wall frame structures.
中文译文
剪力墙框架结构中的应力分布
——非结构细化有限单元网格法的计算摘要：
本文通过有限元分析中的一个半自动运算法则来获得剪力墙中的应变和应力分布。

在研究中，用Matlab软件采用了六自由度的三角形常应变单元和网格的精细——粗化运算法则计算。

最初，被采用的运算法则在整个范围内自动生成粗化网格，然后用户在需要精细网格的区域对这种有限单元网格进行细化，这些区域往往在几何不连续处。

在一个例子中，分别列出了变形、正应力和剪应力，通过和用广泛运用的工程软件运算的结果的比较，其位移和应力结果大致相同。

关键词：剪力墙；有限元分析方法；非结构网格；细化
1. 引言
在最近二十年，剪力墙成为了土耳其中高层民用建筑中的一个重要组成部分。

作为抗震设计中的一部分，这些剪力墙被设置在建筑中以减少地震荷载作用下的横向位移，于是就有了剪力墙框架结构。

自从19世纪60年代以来，许多方法被用来求解剪力墙框架结构的位移和应力分布，如连续介质法和框架分析模型法[1-4]。

迄今为止，由于数值分析法的精确度以及在对剪力墙进行2D和3D分析中的简单使用，已成为卓有成就的一块领域[5-7]。

开口剪力墙、双肢剪力墙和组合剪力墙均可用荷载沿厚度方向均匀分布的薄板模型来模拟，在薄板平面内对其进行分析。

这种2D区域可细分为有限个几何形状。

在有限元分析方法中，划分的简单形状如三角形、四边形（2D中）等被称为单元，单元之间通过节点和公共边进行连接。

在研究问题的区域内，这种单元间的连接被称为有限元网格。

在不同文献中，网格可分为两大类，即结构网格和非结构网格。

结构网格由几何类似的三角形和矩形单元组成，尤其适用于具有简单形状和边界的问题，如图1-a。

尽管结构网格可大大减少程序的运行时间，但是在有着复杂边界的复杂区域上，这种网格往往很难适应复杂的边界形状。

为了解决这个问题，非结构网格就可用来离散有内部边界的复杂区域，如图1-b。

尽管它是一种很费时的程序，非结构网格也依然适合局部网格的细化和粗化。

这种工作的目标是为剪力墙框架几何形状得到一个优品质的非结构网格，在几何不
连续处有更小的单元，而在其他区域内有更大的单元。

图1 结构网格（a ）和非结构网格（b ）
2. 三角形有限单元 2.1 单元的公式表示
几乎所有平面集合形状均可离散成三角形有限单元，这就是使用三角形有限单元的第一个优点。

这种单元有6个自由度，每个接点处有2个方向的平移，如图2。

由于3节点的存在，这种单元有着线形形状函数，单一化的数学处理，这就是使用三角形有限单元的第二个优点。

然而，这些公式导致单元内部的应变和应力为常量，在单元边界处应变和应力有跳跃变化。

为了解决这个缺陷，在应力和应变变化迅速的区域，就必须采用更小的单元来划分区域。

图2 常应变三角有限单元
CST 单元的位移函数和形函数如下：
()()3
32211332211,,v N v N v N y x v u N u N u N y x u ++=++= （1）
()()()()[]y x x x y y y x y x A y x N e
23232332121
,-+---=
（2）
()()()()[]y x x x y y y x y x A y x N e
31313113221
,-+---= （3）
()()()()[]y x x x y y y x y x A y x N e
12121221321
,-+---=
（4）
式中：1u 、2u 、3u ——节点1、2、3在x 方向的节点位移；
1v 、2v 、3v ——节点1、2、3在y 方向的节点位移； 1N 、2N 、3N ——线性形函数；
x 、y ——相应节点的坐标；
e A ——单元的面积。

在有限元方法中，节点位移可通过解线性方程组得到，如下式。

f Ku =
（5）
式中：K ——刚度矩阵； u ——节点位移向量；
f ——节点荷载向量。

刚度矩阵可通过下式计算。

()()SN C SN t A K T
e =
（6）
式中：t ——单元厚度；
⎥⎦
⎤⎢⎣⎡=32
1
321
000N N N N N N N （7）
微分算子如下式。

⎥⎥⎥⎥⎦
⎤⎢⎢⎢
⎢⎣
⎡∂∂∂∂
∂∂∂∂
=x y
y x S 00 （8）
弹性矩阵由下式计算。

⎥⎥⎥⎥⎦
⎤⎢⎢
⎢⎢
⎣
⎡--=210
0010112
v v v
E C ν （9）
式中： E ——弹性模量；
v ——泊松比。

2.2 网格的生成和细化
非结构网格法由一些基本步骤组成，即：边界和内部节点的生成，节点的连接。

这种采用三角有限单元的方法叫三角化，Fukuda 和Suhura 详细介绍了这种方法，即选取一个随机点来生成内部节点[8]。

其中只有一个节点是随机生成的。

尽管节点生成是一个随机过程，我们依然可以对这种生成方式加以限制。

例如，我们可以将两节点的距离定义一个最小值，以作为单元的宽度。

实际上，这种随机生成节点的过程很费时间，但如今的台式电脑不会在初始节点的生成上花费很多时间。

在大尺寸有限单元问题上，可以采取其他运算法则，例如可以采用在矩形栅格的每个矩形内生成一个节点来取代随机生成内部节点。

矩形的宽高比在2：3内，并且以Z 字形排列，节点在矩形中心，以保证生成的单元为等边三角形[9]。

也可以采用另一种节点生成方式，即用假想的水平线将区域双数分数划分，内部节点在根据指定间距参数划分的水平线上生成[10]。

在三角法步骤中，不同的作者提出了许多运算法则，包括简单自动三角法[8,9]，前沿法[10,11]，区域分解法[12,13]和坐标变换法[14]。

在现今的研究中，问题区域内的三角单元是通过Delaunay 或空圆法则来得到的[15-18]。

根据这个法则，一个三角形的外接圆中不能有任何节点，如图3。

圆区域内的节点被从头到尾检查，以获得三个服从空圆法则的候选节点。

这种法则可以有效的排除单元间边界的交叉检查。

图3 服从空圆法则的三角化（a）和不服从空圆法则的三角化
最初我们不知道要对有限单元划分多大才能获得所需精度的应力分布。

因此，对少数初始节点三角化，在像剪力墙内开洞这样的几何不连续处周围和剪力墙框架连接点上，就必须插入新节点，重复三角化程序来进行局部网格细化。

在这一步骤中，当需要细化时，可以在众多运算法则中采用一种来得到精细网格，这些法则如下。

1. 单节点网格细化
2. 线性或多边形网格细化
3. 中央节点网格细化
4. Delaunay节点网格细化
在有限元分析方法中，所有生成的节点和单元都用数字专门编号。

当已知一个节点的编号时，就很容易查到节点周围的单元号。

以上述的第一种法则为例，一旦这些单元确定下来，它们就形成第一种三角单元，在第一种三角单元附近的是第二种三角单元。

根据单节点网格细化法则，在第一种三角单元中生成4个新三角形单元，在第二种三角单元中生成2个新三角形单元，如图4-a和4-b。

线性和多边形局部网格细化法则也是如此，不同于第一中法则的是它选择节点以形成直线或封闭的多边形。

在第三种细化法则中，在一个三角形的几何中心处添加一个节点。

Delaunay节点网格细化法则中，在一个选择的三角形的外接圆的中心处添加一个节点。

图4 第一种三角单元（a），第二种三角单元（b）和细化后的三角单元（c）
●代表选择的细化点
为了改进三角化后三角单元的性质，可以采用一种网格滤波程序。

在力学的
所有分支中，三角单元的形状是有限单元法中的一个重要因素，特别是在固体力学中，近似等边的三角形被认为是很好的一个形状[19]；在计算流体动力学问题时，关系到边界层及其震动，薄长的三角形单元可以提供更好的解答[20]。

为了获得较好性质的三角形单元，我们使用了拉普拉斯算子滤波法，这种方法利用周围的三角形来进行节点重新配置，如图5所示。

通过迭代法使节点移至与其连接的节点形状的质心[21]，通常一次或两次迭代就可足够，但迭代也可继续直到每次节点移动都满足一个收敛距离。

图5 节点移动前（a ），周围的节点（b ），节点移动后（c ）
新节点的坐标由下式确定。

()()u y x y x ∆+=0,,
（10） ∑==∆n
i i u n u 1
1
（11）
()00,y y x x u i i i --=
（12）
式中，n 代表周围节点的个数，下标0代表初始位置。

3. 剪力墙框架问题
为完成一个有限元分析，我们考虑了一个和梁柱连接的带有开窗的剪力墙模型，用非结构网格法进行网格生成和细化。

该剪力墙框架结构在顶部加载，但荷载被分配给框架顶部的几个节点上，所需的几何尺寸如图6-a 。

边界节点由直接法生成，以柱的宽度作为间距参数，少数初始节点采用随机程序生成，如图6-b 。

采用线性网格细化法则来对梁和柱处的网格细化；采用单节点网格细化法则来对开窗、梁和剪力墙连接处的尖角部位进行网格细化，得到最终的网格如图7。

用
最终网格计算出变形形状和应力分布如图8。

图6 剪力墙尺寸和荷载（a）和初始节点生成（b）
图7细化后的有限元网格
图8 有限元分析后的图表结果
4. 结论
为了得到梁和柱、梁和剪力墙连接处应力分布的精确解，我们用Matlab软件编了一个有限元程序。

由于内部开洞的存在，这个问题区域的几何形状非常复杂，尽管如此，该结构还是用非结构有限元网格细化方法进行了离散。

1）在集合不连续区域，采用了小三角单元对有限元网格进行了细化，在小三角形和大三角形处采用了平滑过渡，采用了空圆法则和拉普拉斯滤波法，获得了较好性质的等边三角形单元。

2）Ansys结果和实际的横向位移和剪应力大致吻合，表1和表2列出了在荷载P=100N下的横向位移和剪应力结果比较。

表1. 开洞剪力墙侧移计算结果
m N 8：节点在8m 高度 m N 4: 节点在4m 高度
表2. 图5所示节点的剪应力xy σ
3）本文介绍的运算法则属于半自动法，也有一种自动网格生成法可适用于任意内部开洞的区域，不过在区域网格细化时需要一本结果手册。

4）实际上，本文的工作为我们提供了一个更深发展的跳板来得到一个适合剪力墙框架结构计算的有限元程序。