有限元分析报告报告材料英文文献

1、The Basics of FEA Procedure有限元分析程序的基本知识2.1 IntroductionThis chapter discusses the spri ng eleme nt, especially for the purpose of in troduc ing various con cepts invo Ived in use of the FEA tech niq ue.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spri ng eleme nt is not very useful in the an alysis of real

2、engin eeri ng structures;however, it represents a structure in an ideal form for an FEA analysis. Spring element doesn ' t require discretization (division into smaller elements) and follows the basic equatio n F = ku.在分析实际工程结构时弹簧元件不是很有用的；然而,它代表了一个有限元分析结构在一 个理想的形式分析。弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F

3、 = ku We will use it solely for the purpose of developingan understandingof FEAcon cepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。2.2 Overview 概述Finite Element Analysis (FEA), also known as finite element method (FEM) is based on the concept that a structure can be simulated by the mechanical b

4、ehavior of a spring in which the applied force is proportional to the displaceme nt of the spri ng and the relati on ship F = ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模 拟的应用力弹簧的位移成正比,F = ku切合的关系。In FEA, structures are modeled by a CAD program and represe nted by no desand eleme nts

5、. The mecha ni cal behavior of each of these eleme nts is similar to a mecha ni cal spri ng, obey ing the equati on, F = ku. Gen erally, a structure is divided into several hun dred eleme nts, gen erati ng a very large nu mber of equati ons that can only be solved with the help of a computer.在有限元分析中

6、，结构是由CAD建模程序通过节点和元素建立。每一个元素的力学行为类 似于机械弹簧，遵守方程,F =ku。一般来说，一个结构分为几百元素，生成大量的方程，只能在 电脑的帮助下得到解决。The term finite element ' stems from the procedure in which a structure is divided into small but finite size elements (as opposed to an infinite size, gen erally used in mathematical in tegrati on).“有限元”一

7、词源于一个结构分为小而有限大小 元素的过程(而不是无限大小，通常用于数学集成)The en dpo ints or corner points of the eleme nt are called no des.元素的端点或角点称为节点。Each eleme nt possesses its own geometric and elastic properties.每个元素拥有自己的几何和弹性。Spring, Truss, and Beams eleme nts, called line eleme nts, are usually divided intosmall secti ons wi

8、th no des at each end. The cross-secti on shape does n' t affectthe behavior of a line eleme nt; only the cross-sect ional con sta nts are releva nt and used in calculations. Thus, a square or a circular cross-section of a truss member will yield exactly the same results as long as the cross-sec

9、tional area is the same. Pla ne and solid eleme nts require more tha n two no des and can have over 8 no des for a 3 dime nsional eleme nt.弹簧，桁架和梁元素，称为线元素，通常分为小节，每端有节点。截面形状并不影响线元素的特性；只有横截面常数是相关的并用于计算。因此一个正方形或圆形截面桁架成员将产生完全相同的结果，只要横截面积是一样的。平面和立体元素需要超过两个节点，可以有超过8节点的三维元素。A line element has an exact theo

10、retical solution, e.g., truss and beam elementsare gover ned by their respective theories of deflect ion and the equati ons of deflect ion can be found in an engin eeri ng text or han dbook. However, engin eeri ng structures that have stress concen trati on poi nts e.g., structures with holes and ot

11、her disc ontin uities do not have a theoretical soluti on, and the exact stress distribution can only be found by an experimental method. However, the fin ite eleme nt method can provide an acceptable soluti on more efficie ntly.线元件具有精确的理论解，例如桁架和梁元件由它们各自的偏转理论控制，并且偏转方程可以在工程文本或手册中找到。然而，具有应力集中点的工程结构，例如

12、具有孔和其 他不连续的结构不具有理论解，并且精确的应力分布只能通过实验方法找到。然而，有限元方法可以更有效地提供可接受的解决方案。Problems of this type call for use of elements other than the line elements mentioned earlier, and the real power of the finite element is manifested.这种类型的问题要求使用前面提到的行元素以外的元素。有限元法能真正的来体现证明。In order to develop an understanding of the FE

13、A procedure, we will first deal withthe spri ng eleme nt.为了能深刻理解有限元分析过程，我们将首先处理弹簧元件。In this chapter, spring structures will be used as building blocks for developing anunderstanding of the finite element analysis procedure.在这一章，弹簧结构将被用作构建块来使用有利于有限元分析过程的理解。Both spring and truss elements give an easie

14、r modeling overview of the finite eleme nt an alysis procedure, due to the fact that each spri ng and truss eleme nt, regardless of len gth, is an ideally sized eleme nt and does not n eed any further divisio n.弹簧和桁架元件给出一个简单的建模概述了有限元分析过程，由于每个弹簧和桁架元件不计长度，是一种理想的元素不需要任何进一步的细化。2.3 Understanding Computer

15、 and FEA softwareinteraction -Using the Spring Element as an example2.3理解计算机和有限元分析软件交互，使用弹性元件作为一个例子In the following example, a two-element structure is analyzed by finite element method.在接下来的例子中，对一个双元素结构有限元方法进行了分析。The an alysis procedure prese nted here will be exactly the same as that used for acom

16、plex structural problem, except, in the following example, all calculations willbe carried out by hand so that each step of the analysis can be clearly understood. All derivations and equations are written in a form, which can be handled by a computer, since all finite element analyses are done on a

17、 computer.The finite element equations are derived using Direct Equilibrium method.本文提供的分析过程将一模一样，用于复杂的结构性问题，除了在以下示例中，所有的计算 将手算进行，这样可以清楚地理解每一步的分析。所有方程的推导都是由计算机处理的形 式编写的，因为所有的有限元分析都是在计算机上完成的。有限元方程导出可直接使用 平衡方法。Two springs are connected in series with spring constant k1, and k 2 (lb./in) and aforce F（l

18、b.） is applied. Find the deflecti on at no des 2, and 3.两个串联链接的弹簧其弹簧常数为 k1和k2（磅/）以及一个力F（磅）。求在节点的挠度。彳123Solutio n:For finite element analysis of this structure, the following steps are necessary:Step 1: Derive the element equation for each spring element.Step 2: Assemble the eleme nt equati ons into

19、a com mon equati on, knows as the global or Master equati on.Step 3: Solve the global equati on for deflect ion at no des 1 through 3解:这种结构的有限元分析，以下步骤是必要的：步骤1:为每个弹簧元件方程推导出元素。步骤2:组装元素到一个共同的方程，知道整体的或者主方程。步骤3:求出在节点1到3全局挠曲方程Detailed description of these steps follows.详细描述这些步骤。Step 1: Derive the element

20、equation for each spring element.步骤1:为每个弹簧元件方程推导。First, a gen eral equati on is derived for an eleme nt e that can be used for anyspri ngeleme nt and expressed in terms of its own forces, spri ng con sta nt, and nodedeflect ions,as illustrated in figure 2.2.首先，一般方程导出为一个元素，可用于任何弹簧元件和表达自己的组合，弹簧常数，和节点变

21、位，如图2.2所示。U,k出fi/ AfjvVVv-eEleme nt e' can be thought of as any eleme nt in the structure with no des i andj, forces f i and f j, deflect ions u i and u j, and the spri ng con sta nt k e. Node forces f i and f j are internal orces and are gen erated by the deflecti ons ui and u j at no des iand

22、j, respectively.元素“ e”可以被认为是结构中的任何元素节点i和j,组合fi和fj,变位ui和uj,弹簧常数 k?,节点fi和fj和由变位生成ui和uj节点i和j。For a linear spring f = ku, and 对于一个线性弹簧 f = ku,fi = k ?攵5 ui) = - k ?攵ui-uj) = - k ?ui + k ?uj平衡方程：fj = -fi =k ?(ui-uj) = k ?ui - k ?jj或-fi = k?ji - k?jj-fj = - k ?ji + k ?jjWriting these equations in a matrix

23、 form, we get写出这些方程的矩阵形式，我们得到：Eleme nt(元素)1:因此f1 = -k 1(u1 U2)f2 = k 1(U1-U2) f2 = -k 2(u2 U3) f3 = k 2(u2-u 3) 这就完成第一步的过程。Note that f 3 = F (lb.). This will be substituted in step 2. The above equations represe ntin dividual eleme nts only and not the en tire structure.请注意,f3 = F（磅）。这将是在步骤2中代替。上面的方

24、程表示仅单个元素，而不是整个结Step 2 : Assemble the eleme nt equatio ns into a global equati on步骤2:组装元素方程为全局方程。The basis for combiningor assembling the element equation into a globalequati on is the equilibrium con diti on at each no de.结合或组装元素的基础方程为全局方程是每个节点的平衡条件。Whe n the equilibrium con diti on is satisfied by

25、sum ming all forces at each no de,a set of linear equations is created which links each element force, springcon sta nt, and deflect ions. In gen eral, let the exter nal forces at each node be F1,F2, and F3, as shown in figure 2.3. Using the equilibrium equation, we can find the eleme nt equati ons,

26、 as follows.满足平衡条件时，通过总结所有部队在每个节点，创建一组线性方程联系每个元素力，弹簧 常数,变形量。一般来说，让每个节点的外部力量 F1,F2,F3,如图2.3所示。使用平衡方程,我们可以找到方程的元素，如下所示。忙F 二 0二£+片or 片=_£ =上 1（旳 _些）=&气 _£nrNode 2 :or F2=-f2(n-f2a)= -k|(Ui-Un) + k：(U2-U3)=-k仙+上+阪_1朗Node3:IF=O. fp + F尸0OrFg 二3"二-k?(U2 - ui)Node 3The superscript “

27、e” in force f n(e)indicates the contribution made by the element nu mbere, and the subscript“n ” in dicates the node “n ” at which forces are summed.力fn (e)中的上标"e”表示元素号e,下标"n”表示力相加的节点"n”。Rewriting the equations, we get,重写方程，我们得至U ,k1 u1 - k1 u2 = F1-k 1 u1 + k 1 u2 + k 2 u2 - k2 u3 =

28、F2 (2.1)-k 2 u2 + k 2 u3 = F3These equations can now be written in a matrix form, givingk1 -这些方程可以写成矩阵形式，代入k1 -r 、k- ki0UiFi-kik|+ k3u2=Pj0七k2JU3J巧L JThis completes step 2 for assemblingthe element equationsinto a globalequati on. At this stage, some importa nt con ceptual points should be emphasize

29、d and will be discussed below.这将完成组装的步骤2元素方程为全局方程。在这个阶段,一些重要的概念点应该强调，将在下面讨论。元素刚度矩阵的步骤（就是把刚度变到了多维，比考虑了在多维的情况下各个维度的相关性单元刚度矩阵在有限元的概念把物体离散为多个单元分析 每个单元的刚度矩阵也就是单元刚度矩阵简称单刚）The first term on the left hand side in the above equation represents the stiffnesscon sta nt for the en tire structure and can be thou

30、ght of as an equivale nt stiff nessconstant, given as a single spring element with a value Keq will have an identicalmecha ni cal property as the structural stiff ness in the above example.第一项左边在上面的方程代表了整个结构的刚度常数和可以被认为是一个等效刚度常数，给定为具有值为Keq的单个弹簧元件将具有与上述示例中的结构刚度相同的机械特性，结构刚度在上面的例子中。kt0【7kik汁-k：0kjThe as

31、sembled matrix equati on represe nts the deflecti on equati on of a structure without any con stra in ts, and cannot be solved for deflecti ons without modifyi ng it to in corporate the boun dary con diti ons.At this stage, the stiff nessmatrix isalways symmetric with corresp onding rows and colu mn

32、s in tercha ngeable组装的矩阵方程表示没有任何约束的结构的偏转方程，并且不能解出偏转而不修改它以 并入边界条件。在这个阶段，刚度矩阵总是对称的，相应的行和列是可互换的The global equation was derived by applying equilibrium conditions at each no de. In actual fin ite eleme nt an alysis, this procedure is skipped and a much simpler procedure is used.全局方程是通过在每个节点应用平衡条件得到的。在实际

33、的有限元分析中，跳过该过程并且使用更简单的过程。The simpler procedure is based on the fact that the equilibrium condition at each node must always be satisfied, and in doing so, it leads to an orderly placeme nt of in dividual eleme nt stiffness con sta nt accord ing to the node nu mbers of that eleme nt.更简单的程序是基于每个节点处的平衡条

34、件必须始终满足的客观事实，并在这一过程中：它会导致有序放置单独的元素刚度常数根据元素的节点的数量。The procedure invoIves numberingthe rows and columns of each element,accord ing to the node nu mbers of the eleme nts, and the n, plac ing the stiff ness constant in its corresponding position in the global stiffness matrix. Following isan illustratio

35、n of this procedure, applied to the example problem.过程包括编号每个元素的行和列，根据元素的节点数量，然后，将刚度常数在全局刚度矩阵 对应的位置。下面是这个过程的一个说明，应用的示例问题。Element 1:元素 1I 2kj-k kJ 212Assembli ng it accord ing with the above-described procedure, we get,组装它得到,/23r/kiki0KJ =2-ki-k?30kz由上述程序Note that the first con sta nt k1 in row 1 and

36、colu mn 1 for eleme nt 1 occupies therow 1 and colum n 1 in the global matrix. Similarly, for eleme nt 2, the con sta nt k2in row 2 and colu mn 2 occupies exactly the same positi on (row 2 and colu mn 2)in the global matrix, etc.注意，第一个常数k1在第一行和第一列元素1占据全局第一行和第一列矩阵。同样，对于 元素2,第2行和列2中的常数k2占据了完全相同的位置(第二行

37、和列2)在全局矩阵，等等。In a large model, the node numbers can occur randomly, but the assembly procedure remains the same. It' s important to place the row and column elements from an element into the global matrix at exactly the same position corresp onding to the respective row and colum n.在大型模型中，节点随机数字

38、可以发生，但装配程序是相同的。重要的是要将从一个元素的 行和列元素融入全局矩阵在完全相同的位置对应于相应的行和列。力矩阵At this stage, the force matrix is represented in a general form, with unknownforces F 1,F2, and F 3在这个阶段，力矩阵的一般形式表示，F1与未知的力量,F2和F3F1F2F3Represe nting the exter nal forces at no des 1,2, and 3, in gen eral terms, and not interms of the actu

39、al known value of the forces. In the example problem, F1 = F 2 = 0and F 3 = F. The actual force matrix is then代表外部力量在节点1、2和3,在一般条款，而不是实际的已知值的力量。在示例问题,F1 =0 F2和F3 = f .实际力矩阵0-0 *FLJGen erally, the assembled structural matrix equati on is writte n in short asF=ku, orsimply, F = k u, with the understan

40、ding that each term is an m x n matrix where mis thenumber of rows and n is the number of columns.般来说，组装结构矩阵方程简写为 F =ku,或简单地，F = k u ,每个术语的理解是一个 m x n矩阵m和n的行数的列 数。Step 3: Solve the global equati on for deflect ions at no des.步骤3:解决全局方程在节点变位。There are two steps for obtaining the deflection values. In

41、 the first step, all theboundary conditions are applied, which will result in reducing the size of the global structural matrix. In the second step, a numerical matrix solution scheme is used to find deflect ionvalues by using a computer. Among the most popularnumerical schemes are the Gauss elimina

42、tionand the Gauss-Sedel iterationmethod. For further read in g, refer to any nu merical an alysis book on this topic. In the following examples and chapters, all the matrix solutions will be limited to a hand calculation even though the actual matrix in a finite element solution will always use one

43、of the two nu merical soluti on schemes men ti oned above.有两个步骤可得到的挠度值。在第一步中，所有的应用边界条件，这将导致减少整体结构性 矩阵的大小。在第二步中，数值矩阵的解决是使用电脑查找挠度值。最受欢迎的是高斯消 去法和数值方案Gauss-Sedel算法。为进一步阅读，指的是任何数值分析有关此主题的书。 下面的例子和章节，所有的矩阵计算解决方案将是有限的手虽然实际矩阵在有限元的解决 方案总是使用上面提到的两个数值解方案之一。233 Bou ndary con diti ons边界条件In the example problem,

44、node 1 is fixed and therefore u1 = 0. Without going intoa mathematical proof, it can be stated that this con diti on is effected by delet ingrow 1 and column 1 of the structural matrix, thereby reducing the size of the matrix from 3 x 3 to 2 x 2.在问题的例子中，节点1是固定的，因此u1 = 0。在不进入数学证明的情况下，可以说，该条件通过删除结构矩阵的

45、行1和列1来实现，从而将矩阵的大小从 3 X3减小到2 X2oIn general, any boundary condition is satisfied by deleting the rows and columns corresp onding to the node that has zero deflecti on. In gen eral, a node has sixdegrees of freedom (DOF), which in elude three tran slati ons and three rotati ons inx, y and z directi ons

46、.一般来说，通过删除对应于具有零偏转的节点的行和列，满足任何边界条件。节点具有 六个自由度(DOF)，其包括在x，y和z方向上的三个平移和三个旋转。In the example problem, there is only one degree of freedom at each no de. The node deflects only along the axis of the spri ng.在示例问题中，在每个节点处只有一个自由度，即节点仅沿着弹簧的轴线偏转。In this section, the finite element analysis procedure for a sp

47、ring structure hasbeen stablished. The following numerical example will utilize the derivation and con cepts developed above.在本节中，已经建立了用于弹簧结构的有限元分析程序。下面的数字示例将利用上面得到的推导和概念。Example 2.2 例 2.2In the given spring structure, k 1 = 20 lb./in., k 2 = 25 lb./in., k 3 = 30 lb./in., F = 5 lb. Determ ine deflec

48、t ion at all the no des.在给定的弹簧结构,k1 = 20磅/。k2 = 25磅/。，k3 = 30磅/。F = 5磅。在所有节点确 定挠度。o 4& 2Solution （解）We would apply the three steps discussed earlier.我们将使用前面讨论的三个步骤。Step 1: Derive the Eleme nt Equati ons步骤1:方程推导出元素。As derived earlier, the stiffness matrix equations for an element e is,如前所述，元素e的刚度

49、矩阵方程是K£e） =-匕1qTherefore, stiff ness matrix of eleme nts 1,2, and 3 are,因此，元素1,2和3的刚度矩阵为Element 1:K1"=Element 2;EIcnietn 3:34K(3) = f30 -3(f| 330 30 4<JStep 2: Assemble eleme nt equati ons into a global equati on步骤2:将子方程组装为全局方程Assembling the terms according to their row and column posit

50、ion, we get根据他们的行和列的位置装配条件，我们得到1Go -20301【KJ二-20204-25-2501盃0-2525+30-303<003030j1 4The global structural equatio n is,Or, by simplifyi ng或者，通过简化厂20-2000、陶二-2045-25002555-30L o03030JF 1FiGo -20 0 rir-20 45 -250叽压0 25 55 -30p4E *003030全局结构方程为,Step 3: Solve for deflecti ons第三步:求解变形量First, appl ying

51、 the boun dary con diti ons ui =0, the first row and first column willdrop out. Next,F1= F 2 = F 3 = 0, and F 4 = 5 lb. The final form of the equation becomes,首先,应用边界条件u1 = 0,第一行和第一列将被化简。接下来，F1 = F2 = F3 = 0,F4 = 5磅。方程的最终形式为rr045 -2500-25 55 -30卜5J £Lo 30 30,.AThis is the final structural matri

52、x with all the boundary conditions being applied.Since the size of the final matrices is small, deflecti ons can be calculated by hand.It should be no ted that in a real structure the size of a stiffness matrix is rather large and can only be solved with the help of a computer. Solving the above mat

53、rix equati on by hand we get,这是应用所有边界条件的最终结构矩阵。由于最终矩阵的尺寸小，可以手算偏转。应当注意，在实际结构中，刚度矩阵的大小相当大，并且只能借助于计算机来求解。用手 算求解上述矩阵方程,f rUj0.2500OrUjU4卜=0.45000.6167 <J0= 45-25 uj0 = -25 ui + 55 uj-30u45 = -3030 mExample 2.3In the spring structure shown k 1 = 10 lb./in., k 2 = 15 lb./in., k 3 = 20 lb./in., P= 5 lb.

54、Determ ine the deflect ion at no des 2 and 3.例2.3所示的弹簧结构中k1 = 10磅/英寸。k2 = 15磅/英寸。，k3 = 20磅/英寸。P = 5Again apply the three steps outlined previously.Step 1: Find the Element Stiffness Equations解决方案:再次应用前面所述的三个步骤。第一步：找到元素刚度方程Element 7/Element 2:Element 3:Step 2: Find the Global stiff ness matrix步骤二：获得整

55、体刚度矩阵FiF2Fma Now the global structural equation can be written as现在全局结构方程可以写成r 气Firio-100-1025-150T0-1535-201 FLo0-2020jStep 3: Solve for Deflectio ns步骤3 :解决变形量The known boun dary con diti ons are: u1 = u4 = 0, F3 = P = 31b. Thus, rows and columns 1 and 4 will drop out, resulting in the following matrix equation,已知的边界条件是:矩阵方程，