土建外文翻譯----剪力墻框架結(jié)構(gòu)中的應(yīng)力分布

1、附錄英文原文及翻譯stress distribution in a shear wall frame structureusing unstructured refined finite element meshabstracta 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

2、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

3、 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.introducti

4、onin 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

5、are obtained. since the 1960s 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 methodsare the main e

6、ffort 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 pl

7、ane 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

8、 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 ar

9、e 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, uns

10、tructured 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 ha

11、ve 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 elements2.1. element formulationthe first advantage of using triangular elements is that almost any plane g

12、eometry 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 con

13、stant 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 elementcst element has displacement functions and shape functions as follows,（1）（2）（3）（4）where 1 u , 2 u a

14、nd 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 me

15、thod, nodal displacements are obtained from the solution of the linear system of equations, that is （5）where, k is stiffness matrix, u is nodal displacement vector, and f is nodal load vector. stiffness matrix may be calculated as （6）where t is thickness of the element, （7）and differential operator

16、is, （8）and the elasticity matris is defined by （9）where e is modulus of elasticity and v is the poissons ratio.2.2. mesh generation and refinementunstructured mesh generation procedure consists of some basic steps. these are the generations of boundary and interior nodes and connection of these node

17、s 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

18、, 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,

19、 but this is not too long for todays 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

20、 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 po

21、ints 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,a

22、dvancing 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 t

23、riangle 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

24、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 init

25、ial 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,

26、 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 refinementin fem, all generated nodes and elements have some special id denoted

27、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,

28、 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 an

29、d 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,

30、 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 improv

31、e 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 bo

32、undary 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 m

33、oved 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

34、 coordinates defined as,（10）（11）（12）where n represents the number of surrounding nodes and zero indices represent initial values.3. the shear wall - frame problema shear wall model with small window openings connected to a beam column system was considered to perform a fem analysis using unstructure

35、d 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 a

36、s 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 refinementalgorithm was used around sharp corners of window openings and beam shear

37、-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 aft

38、er analysis.4. conclusionsin 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 domai

39、n 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 cir

40、cle 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

41、=100n.table 1. lateral displacement of shear wall structure with openings.location,lateral disp. (m), lateral disp. (m)ansys (plane2)error (%)4.82.8: nodes at 8 m height.: nodes at 4 m height.table 2. shear stress, , values at some points given in figure 5.locationstress (pa)stress (pa)ansys (plane2

42、)error (%)p1205.6222.157.4p2189.25175.068.1p336.7540.138.4p4180.35164.389.83) 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,

43、 the work presented here provided us with a springboard for further developments to get an adaptive finite element procedure for shear wall frame structures.剪力墻框架結(jié)構(gòu)中的應(yīng)力分布 非結(jié)構(gòu)細(xì)化有限單元網(wǎng)格法的計(jì)算摘要：本文通過有限元分析中的一個(gè)半自動(dòng)運(yùn)算法則來獲得剪力墻中的應(yīng)變和應(yīng)力分布。在研究中， 用matlab軟件采用了六自由度的三角形常應(yīng)變單元和網(wǎng)格的精細(xì)粗化運(yùn)算法則計(jì)算。最初，被采用的運(yùn)算法則在整個(gè)范圍內(nèi)自動(dòng)生成粗化網(wǎng)格，然后用

44、戶在需要精細(xì)網(wǎng)格的區(qū)域?qū)@種有限單元網(wǎng)格進(jìn)行細(xì)化，這些區(qū)域往往在幾何不連續(xù)處。在一個(gè)例子中，分別列出了變形、正應(yīng)力和剪應(yīng)力，通過和用廣泛運(yùn)用的工程軟件運(yùn)算的結(jié)果的比較，其位移和應(yīng)力結(jié)果大致相同。關(guān)鍵詞：剪力墻；有限元分析方法；非結(jié)構(gòu)網(wǎng)格；細(xì)化1. 引言在最近二十年，剪力墻成為了土耳其中高層民用建筑中的一個(gè)重要組成部分。作為抗震設(shè)計(jì)中的一部分，這些剪力墻被設(shè)置在建筑中以減少地震荷載作用下的橫向位移，于是就有了剪力墻框架結(jié)構(gòu)。自從19世紀(jì)60年代以來，許多方法被用來求解剪力墻框架結(jié)構(gòu)的位移和應(yīng)力分布，如連續(xù)介質(zhì)法和框架分析模型法1-4。迄今為止，由于數(shù)值分析法的精確度以及在對(duì)剪力墻進(jìn)行2d和3d分

45、析中的簡(jiǎn)單使用，已成為卓有成就的一塊領(lǐng)域5-7。開口剪力墻、雙肢剪力墻和組合剪力墻均可用荷載沿厚度方向均勻分布的薄板模型來模擬，在薄板平面內(nèi)對(duì)其進(jìn)行分析。這種2d區(qū)域可細(xì)分為有限個(gè)幾何形狀。在有限元分析方法中，劃分的簡(jiǎn)單形狀如三角形、四邊形（2d中）等被稱為單元，單元之間通過節(jié)點(diǎn)和公共邊進(jìn)行連接。在研究問題的區(qū)域內(nèi)，這種單元間的連接被稱為有限元網(wǎng)格。在不同文獻(xiàn)中，網(wǎng)格可分為兩大類，即結(jié)構(gòu)網(wǎng)格和非結(jié)構(gòu)網(wǎng)格。結(jié)構(gòu)網(wǎng)格由幾何類似的三角形和矩形單元組成，尤其適用于具有簡(jiǎn)單形狀和邊界的問題，如圖1-a。盡管結(jié)構(gòu)網(wǎng)格可大大減少程序的運(yùn)行時(shí)間，但是在有著復(fù)雜邊界的復(fù)雜區(qū)域上，這種網(wǎng)格往往很難適應(yīng)復(fù)雜的邊界形

46、狀。為了解決這個(gè)問題，非結(jié)構(gòu)網(wǎng)格就可用來離散有內(nèi)部邊界的復(fù)雜區(qū)域，如圖1-b。盡管它是一種很費(fèi)時(shí)的程序，非結(jié)構(gòu)網(wǎng)格也依然適合局部網(wǎng)格的細(xì)化和粗化。這種工作的目標(biāo)是為剪力墻框架幾何形狀得到一個(gè)優(yōu)品質(zhì)的非結(jié)構(gòu)網(wǎng)格，在幾何不連續(xù)處有更小的單元，而在其他區(qū)域內(nèi)有更大的單元。圖1 結(jié)構(gòu)網(wǎng)格（a）和非結(jié)構(gòu)網(wǎng)格（b）2. 三角形有限單元2.1 單元的公式表示幾乎所有平面集合形狀均可離散成三角形有限單元，這就是使用三角形有限單元的第一個(gè)優(yōu)點(diǎn)。這種單元有6個(gè)自由度，每個(gè)接點(diǎn)處有2個(gè)方向的平移，如圖2。由于3節(jié)點(diǎn)的存在，這種單元有著線形形狀函數(shù)，單一化的數(shù)學(xué)處理，這就是使用三角形有限單元的第二個(gè)優(yōu)點(diǎn)。然而，這些公

47、式導(dǎo)致單元內(nèi)部的應(yīng)變和應(yīng)力為常量，在單元邊界處應(yīng)變和應(yīng)力有跳躍變化。為了解決這個(gè)缺陷，在應(yīng)力和應(yīng)變變化迅速的區(qū)域，就必須采用更小的單元來劃分區(qū)域。圖2 常應(yīng)變?nèi)怯邢迒卧猚st單元的位移函數(shù)和形函數(shù)如下：（1）（2）（3）（4）式中：、節(jié)點(diǎn)1、2、3在x方向的節(jié)點(diǎn)位移；、節(jié)點(diǎn)1、2、3在y方向的節(jié)點(diǎn)位移；、線性形函數(shù)；、相應(yīng)節(jié)點(diǎn)的坐標(biāo)；單元的面積。在有限元方法中，節(jié)點(diǎn)位移可通過解線性方程組得到，如下式。（5）式中：剛度矩陣；節(jié)點(diǎn)位移向量；節(jié)點(diǎn)荷載向量。剛度矩陣可通過下式計(jì)算。 （6）式中：?jiǎn)卧穸?；?）微分算子如下式。（8）彈性矩陣由下式計(jì)算。（9）式中： e彈性模量；泊松比。2.2 網(wǎng)格的

48、生成和細(xì)化非結(jié)構(gòu)網(wǎng)格法由一些基本步驟組成，即：邊界和內(nèi)部節(jié)點(diǎn)的生成，節(jié)點(diǎn)的連接。這種采用三角有限單元的方法叫三角化，fukuda和suhura詳細(xì)介紹了這種方法，即選取一個(gè)隨機(jī)點(diǎn)來生成內(nèi)部節(jié)點(diǎn)8。其中只有一個(gè)節(jié)點(diǎn)是隨機(jī)生成的。盡管節(jié)點(diǎn)生成是一個(gè)隨機(jī)過程，我們依然可以對(duì)這種生成方式加以限制。例如，我們可以將兩節(jié)點(diǎn)的距離定義一個(gè)最小值，以作為單元的寬度。實(shí)際上，這種隨機(jī)生成節(jié)點(diǎn)的過程很費(fèi)時(shí)間，但如今的臺(tái)式電腦不會(huì)在初始節(jié)點(diǎn)的生成上花費(fèi)很多時(shí)間。在大尺寸有限單元問題上，可以采取其他運(yùn)算法則，例如可以采用在矩形柵格的每個(gè)矩形內(nèi)生成一個(gè)節(jié)點(diǎn)來取代隨機(jī)生成內(nèi)部節(jié)點(diǎn)。矩形的寬高比在2：內(nèi)，并且以z字形排列，

49、節(jié)點(diǎn)在矩形中心，以保證生成的單元為等邊三角形9。也可以采用另一種節(jié)點(diǎn)生成方式，即用假想的水平線將區(qū)域雙數(shù)分?jǐn)?shù)劃分，內(nèi)部節(jié)點(diǎn)在根據(jù)指定間距參數(shù)劃分的水平線上生成10。在三角法步驟中，不同的作者提出了許多運(yùn)算法則，包括簡(jiǎn)單自動(dòng)三角法8,9，前沿法10,11，區(qū)域分解法12,13和坐標(biāo)變換法14。在現(xiàn)今的研究中，問題區(qū)域內(nèi)的三角單元是通過delaunay或空?qǐng)A法則來得到的15-18。根據(jù)這個(gè)法則，一個(gè)三角形的外接圓中不能有任何節(jié)點(diǎn)，如圖3。圓區(qū)域內(nèi)的節(jié)點(diǎn)被從頭到尾檢查，以獲得三個(gè)服從空?qǐng)A法則的候選節(jié)點(diǎn)。這種法則可以有效的排除單元間邊界的交叉檢查。圖3 服從空?qǐng)A法則的三角化（a）和不服從空?qǐng)A法則的三角化最初我們不知道要對(duì)有限單元?jiǎng)澐侄啻蟛拍塬@得所需精度的應(yīng)力分布。因此，對(duì)少數(shù)初始節(jié)點(diǎn)