有限元分析英文文獻(xiàn)

1、The Basics of FEA Procedure有限元分析程序的基本知識(shí)2.1 IntroductionThis chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique.本章討論了彈簧元件，特別是用于引入使用的有限元分析技術(shù)的各種概念的目的 A spring element is not very useful in the analysis of real engineerin

2、g structures; however, it represents a structure in an ideal form for an FEA analysis. Spring element doesnt require discretization (division into smaller elements) and follows the basic equation F = ku. 在分析實(shí)際工程結(jié)構(gòu)時(shí)彈簧元件不是很有用的;然而,它代表了一個(gè)有限元分析結(jié)構(gòu)在一個(gè)理想的形式分析。彈簧元件不需要離散化(分裂成更小的元素)只遵循的基本方程F = kuWe will use it

3、 solely for the purpose of developing an understanding of FEA concepts and procedure.我們將使用它的目的僅僅是為了對(duì)開發(fā)有限元分析的概念和過程的理解。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 behavior of a spring

4、 in which the applied force is proportional to the displacement of the spring and the relationship F = ku is satisfied.有限元分析(FEA),也稱為有限元法(FEM)，是基于一個(gè)結(jié)構(gòu)可以由一個(gè)彈簧的力學(xué)行為模擬的應(yīng)用力彈簧的位移成正比,F = ku切合的關(guān)系。 In FEA, structures are modeled by a CAD program and represented by nodes and elements. The mechanical behavior

5、 of each of these elements is similar to a mechanical spring, obeying the equation, F = ku. Generally, a structure is divided into several hundred elements, generating a very large number of equations that can only be solved with the help of a computer.在有限元分析中,結(jié)構(gòu)是由CAD建模程序通過節(jié)點(diǎn)和元素建立。每一個(gè)元素的力學(xué)行為類似于機(jī)械彈簧,

6、遵守方程,F =ku。一般來說,一個(gè)結(jié)構(gòu)分為幾百元素,生成大量的方程,只能在電腦的幫助下得到解決。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, generally used in mathematical integration).“有限元”一詞源于一個(gè)結(jié)構(gòu)分為 小而有限大小 元素的過程(而不是無限大小,通常用于數(shù)學(xué)集成) The endpoi

7、nts or corner points of the element are called nodes.元素的端點(diǎn)或角點(diǎn)稱為節(jié)點(diǎn)。Each element possesses its own geometric and elastic properties. 每個(gè)元素?fù)碛凶约旱膸缀魏蛷椥?。Spring, Truss, and Beams elements, called line elements, are usually divided into small sections with nodes at each end. The cross-section shape doesnt af

8、fect the behavior of a line element; only the cross-sectional constants are relevant 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-sectional area is the same. Plane and solid elements require more than

9、two nodes and can have over 8 nodes for a 3 dimensional element.彈簧，桁架和梁元素，稱為線元素，通常分為小節(jié)，每端有節(jié)點(diǎn)。截面形狀并不影響線元素的特性; 只有橫截面常數(shù)是相關(guān)的并用于計(jì)算。因此,一個(gè)正方形或圓形截面桁架成員將產(chǎn)生完全相同的結(jié)果,只要橫截面積是一樣的。平面和立體元素需要超過兩個(gè)節(jié)點(diǎn),可以有超過8節(jié)點(diǎn)的三維元素。A line element has an exact theoretical solution, e.g., truss and beam elements are governed by their res

10、pective theories of deflection and the equations of deflection can be found in an engineering text or handbook. However, engineering structures that have stress concentration points e.g., structures with holes and other discontinuities do not have a theoretical solution, and the exact stress distrib

11、ution can only be found by an experimental method. However, the finite element method can provide an acceptable solution more efficiently. 線元件具有精確的理論解，例如桁架和梁元件由它們各自的偏轉(zhuǎn)理論控制，并且偏轉(zhuǎn)方程可以在工程文本或手冊(cè)中找到。然而，具有應(yīng)力集中點(diǎn)的工程結(jié)構(gòu)，例如具有孔和其他不連續(xù)的結(jié)構(gòu)不具有理論解，并且精確的應(yīng)力分布只能通過實(shí)驗(yàn)方法找到。 然而，有限元方法可以更有效地提供可接受的解決方案。Problems of this type cal

12、l for use of elements other than the line elements mentioned earlier, and the real power of the finite element is manifested. 這種類型的問題要求使用前面提到的行元素以外的元素。有限元法能真正的來體現(xiàn)證明。In order to develop an understanding of the FEA procedure, we will first deal with the spring element. 為了能深刻理解有限元分析過程,我們將首先處理彈簧元件。In th

13、is chapter, spring structures will be used as building blocks for developing an understanding of the finite element analysis procedure. 在這一章,彈簧結(jié)構(gòu)將被用作構(gòu)建塊來使用有利于有限元分析過程的理解。Both spring and truss elements give an easier modeling overview of the finite element analysis procedure, due to the fact that each

14、 spring and truss element, regardless of length, is an ideally sized element and does not need any further division.彈簧和桁架元件給出一個(gè)簡單的建模概述了有限元分析過程,由于每個(gè)彈簧和桁架元件,不計(jì)長度,是一種理想的元素不需要任何進(jìn)一步的細(xì)化。2.3 Understanding Computer and FEA software interaction -Using the Spring Element as an example2.3理解計(jì)算機(jī)和有限元分析軟件交互，使用彈性元件

15、作為一個(gè)例子In the following example, a two-element structure is analyzed by finite element method. 在接下來的例子中,對(duì)一個(gè)雙元素結(jié)構(gòu)有限元方法進(jìn)行了分析。The analysis procedure presented here will be exactly the same as that used for a complex structural problem, except, in the following example, all calculations will be carried o

16、ut 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 computer. The finite element equations are derived using Direct Equilibrium method.本文提供的分析過

17、程將一模一樣,用于復(fù)雜的結(jié)構(gòu)性問題,除了在以下示例中,所有的計(jì)算將手算進(jìn)行,這樣可以清楚地理解每一步的分析。所有方程的推導(dǎo)都是由計(jì)算機(jī)處理的形式編寫的，因?yàn)樗械挠邢拊治龆际窃谟?jì)算機(jī)上完成的。有限元方程導(dǎo)出可直接使用平衡方法。Two springs are connected in series with spring constant k1, and k2 (lb./in) and a force F(lb.) is applied. Find the deflection at nodes 2, and 3.兩個(gè)串聯(lián)鏈接的彈簧其彈簧常數(shù)為k1和k2(磅/)以及一個(gè)力F(磅)。求在節(jié)點(diǎn)的撓

18、度。Solution: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 element equations into a common equation, knows as the globalor Master equation.Step 3: Solve the global equation for deflectio

19、n at nodes 1 through 3解:這種結(jié)構(gòu)的有限元分析,以下步驟是必要的:步驟1:為每個(gè)彈簧元件方程推導(dǎo)出元素。步驟2:組裝元素到一個(gè)共同的方程,知道整體的或者主方程。步驟3:求出在節(jié)點(diǎn)1到3全局撓曲方程Detailed description of these steps follows.詳細(xì)描述這些步驟。Step 1: Derive the element equation for each spring element.步驟1:為每個(gè)彈簧元件方程推導(dǎo)。First, a general equation is derived for an element e that can

20、 be used for any springelement and expressed in terms of its own forces, spring constant, and node deflections,as illustrated in figure 2.2.首先,一般方程導(dǎo)出為一個(gè)元素,可用于任何彈簧元件和表達(dá)自己的組合,彈簧常數(shù),和節(jié)點(diǎn)變位,如圖2.2所示。Element e can be thought of as any element in the structure with nodes i and j, forces fi and fj, deflection

21、s ui and uj, and the spring constant ke. Node forces fi and fj are internal orces and are generated by the deflections ui and uj at nodes i and j, respectively.元素“e”可以被認(rèn)為是結(jié)構(gòu)中的任何元素節(jié)點(diǎn)i和j,組合fi和fj,變位ui和uj,彈簧常數(shù)k e。節(jié)點(diǎn)fi和fj和由變位生成ui和uj節(jié)點(diǎn)i和j。For a linear spring f = ku, and對(duì)于一個(gè)線性彈簧f = ku,fi = k e(uj ui) = - k

22、 e(ui-uj) = - k eui + k euj平衡方程：fj = -fi = k e(ui-uj) = k eui - k euj或 -fi = k eui - k euj - fj = - k eui + k eujWriting these equations in a matrix form, we get寫出這些方程的矩陣形式,我們得到：Element （元素）1:力矩陣上的上標(biāo)表示相應(yīng)的元素因此f1 = -k1(u1 u2) f2 = k1(u1-u2)f2 = -k2(u2 u3) f3 = k2(u2-u3)這就完成第一步的過程。Note that f3 = F (lb.

23、). This will be substituted in step 2. The above equations representindividual elements only and not the entire structure.請(qǐng)注意,f3 = F(磅)。這將是在步驟2中代替。上面的方程表示僅單個(gè)元素,而不是整個(gè)結(jié)構(gòu)。Step 2 : Assemble the element equations into a global equation.步驟2:組裝元素方程為全局方程。The basis for combining or assembling the element equ

24、ation into a global equation is the equilibrium condition at each node.結(jié)合或組裝元素的基礎(chǔ)方程為全局方程是每個(gè)節(jié)點(diǎn)的平衡條件。 When the equilibrium condition is satisfied by summing all forces at each node, a set of linear equations is created which links each element force, spring constant, and deflections. In general, let t

25、he external forces at each node be F1, F2, and F3, as shown in figure 2.3. Using the equilibrium equation, we can find the element equations, as follows.滿足平衡條件時(shí),通過總結(jié)所有部隊(duì)在每個(gè)節(jié)點(diǎn),創(chuàng)建一組線性方程聯(lián)系每個(gè)元素力,彈簧常數(shù),變形量。一般來說,讓每個(gè)節(jié)點(diǎn)的外部力量F1,F2,F3,如圖2.3所示。使用平衡方程,我們可以找到方程的元素,如下所示。The superscript “e” in force fn(e) indicates

26、 the contribution made by the element numbere, and the subscript “n” indicates the node “n” at which forces are summed.力fn（e）中的上標(biāo)“e”表示元素號(hào)e，下標(biāo)“n”表示力相加的節(jié)點(diǎn)“n”。Rewriting the equations, we get,重寫方程,我們得到,k1 u1 k1 u2 = F1- k1 u1 + k1 u2 + k2 u2 k2 u3 = F2 (2.1)- k2 u2 + k2 u3 = F3These equations can now be

27、 written in a matrix form, givingk1 -這些方程可以寫成矩陣形式,代入k1 -This completes step 2 for assembling the element equations into a global equation. At this stage, some important conceptual points should be emphasized and will be discussed below.這將完成組裝的步驟2元素方程為全局方程。在這個(gè)階段,一些重要的概念點(diǎn)應(yīng)該強(qiáng)調(diào),將在下面討論。2.3.1 Procedure fo

28、r Assembling Element stiffness matrices元素剛度矩陣的步驟（就是把剛度變到了多維，比考慮了在多維的情況下 各個(gè)維度的相關(guān)性單元?jiǎng)偠染仃囋谟邢拊母拍?把物體離散為多個(gè)單元分析 每個(gè)單元的剛度矩陣 也就是單元?jiǎng)偠染仃嚭喎Q單剛）The first term on the left hand side in the above equation represents the stiffness constant for the entire structure and can be thought of as an equivalent stiffness co

29、nstant, given as a single spring element with a value Keq will have an identical mechanical property as the structural stiffness in the above example.第一項(xiàng)左邊在上面的方程代表了整個(gè)結(jié)構(gòu)的剛度常數(shù)和可以被認(rèn)為是一個(gè)等效剛度常數(shù), 給定為具有值為Keq的單個(gè)彈簧元件將具有與上述示例中的結(jié)構(gòu)剛度相同的機(jī)械特性,結(jié)構(gòu)剛度在上面的例子中。 The assembled matrix equation represents the deflection eq

30、uation of a structure without any constraints, and cannot be solved for deflections without modifying it to incorporate the boundary conditions. At this stage, the stiffness matrix is always symmetric with corresponding rows and columns interchangeable組裝的矩陣方程表示沒有任何約束的結(jié)構(gòu)的偏轉(zhuǎn)方程，并且不能解出偏轉(zhuǎn)而不修改它以并入邊界條件。 在這

31、個(gè)階段，剛度矩陣總是對(duì)稱的，相應(yīng)的行和列是可互換的The global equation was derived by applying equilibrium conditions at each node. In actual finite element analysis, this procedure is skipped and a much simpler procedure is used.全局方程是通過在每個(gè)節(jié)點(diǎn)應(yīng)用平衡條件得到的。 在實(shí)際的有限元分析中，跳過該過程并且使用更簡單的過程。 The simpler procedure is based on the fact th

32、at the equilibrium condition at each node must always be satisfied, and in doing so, it leads to an orderly placement of individual element stiffness constant according to the node numbers of that element.更簡單的程序是基于每個(gè)節(jié)點(diǎn)處的平衡條件必須始終滿足的客觀事實(shí),并在這一過程中,它會(huì)導(dǎo)致有序放置單獨(dú)的元素剛度常數(shù)根據(jù)元素的節(jié)點(diǎn)的數(shù)量。 The procedure involves numb

33、ering the rows and columns of each element, according to the node numbers of the elements, and then, placing the stiffness constant in its corresponding position in the global stiffness matrix. Following is an illustration of this procedure, applied to the example problem.過程包括編號(hào)每個(gè)元素的行和列,根據(jù)元素的節(jié)點(diǎn)數(shù)量,然后

34、,將剛度常數(shù)在全局剛度矩陣對(duì)應(yīng)的位置。下面是這個(gè)過程的一個(gè)說明,應(yīng)用的示例問題。Element 1:元素1Assembling it according with the above-described procedure, we get, 由上述程序組裝它得到，Note that the first constant k1 in row 1 and column 1 for element 1 occupies the row 1 and column 1 in the global matrix. Similarly, for element 2, the constant k2 in r

35、ow 2 and column 2 occupies exactly the same position (row 2 and column 2) in the global matrix, etc.注意,第一個(gè)常數(shù)k1在第一行和第一列元素1占據(jù)全局第一行和第一列矩陣。同樣,對(duì)于元素2,第2行和列2中的常數(shù)k2占據(jù)了完全相同的位置(第二行和列2)在全局矩陣,等等。In a large model, the node numbers can occur randomly, but the assembly procedure remains the same. Its important to

36、place the row and column elements from an element into the global matrix at exactly the same position corresponding to the respective row and column.在大型模型中,節(jié)點(diǎn)隨機(jī)數(shù)字可以發(fā)生,但裝配程序是相同的。重要的是要將從一個(gè)元素的行和列元素融入全局矩陣在完全相同的位置對(duì)應(yīng)于相應(yīng)的行和列。2.3.2 Force matrix力矩陣At this stage, the force matrix is represented in a general f

37、orm, with unknown forces F1,F2, and F3在這個(gè)階段,力矩陣的一般形式表示,F1與未知的力量,F2和F3Representing the external forces at nodes 1, 2, and 3, in general terms, and not in terms of the actual known value of the forces. In the example problem, F1 = F2 = 0 and F3 = F. The actual force matrix is then代表外部力量在節(jié)點(diǎn)1、2和3,在一般條款,

38、而不是實(shí)際的已知值的力量。在示例問題,F1 = = 0 F2和F3 = f .實(shí)際力矩陣 Generally, the assembled structural matrix equation is written in short as F=ku, orsimply, F = k u, with the understanding that each term is an m x n matrix where m is thenumber of rows and n is the number of columns.一般來說,組裝結(jié)構(gòu)矩陣方程簡寫為 F =ku,或簡單地，F(xiàn) = k u,每個(gè)

39、術(shù)語的理解是一個(gè)m × n矩陣m和n的行數(shù)的列數(shù)。Step 3: Solve the global equation for deflections at nodes.步驟3:解決全局方程在節(jié)點(diǎn)變位。There are two steps for obtaining the deflection values. In the first step, all the boundary conditions are applied, which will result in reducing the size of the global structural matrix. In the

40、 second step, a numerical matrix solution scheme is used to find deflection values by using a computer. Among the most popular numerical schemes are the Gauss elimination and the Gauss-Sedel iteration method. For further reading, refer to any numerical analysis book on this topic. In the following e

41、xamples 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 of the two numerical solution schemes mentioned above.有兩個(gè)步驟可得到的撓度值。在第一步中,所有的應(yīng)用邊界條件,這將導(dǎo)致減少整體結(jié)構(gòu)性矩陣的大小。在第二步中,數(shù)值矩陣的解決是使用電腦查找撓度值。最受歡迎的是高斯消去法和

42、數(shù)值方案Gauss-Sedel算法。為進(jìn)一步閱讀,指的是任何數(shù)值分析有關(guān)此主題的書。下面的例子和章節(jié),所有的矩陣計(jì)算解決方案將是有限的手雖然實(shí)際矩陣在有限元的解決方案總是使用上面提到的兩個(gè)數(shù)值解方案之一。2.3.3 Boundary conditions邊界條件In the example problem, node 1 is fixed and therefore u1 = 0. Without going into a mathematical proof, it can be stated that this condition is effected by deleting row 1

43、 and column 1 of the structural matrix, thereby reducing the size of the matrix from 3 x 3 to 2 x 2.在問題的例子中,節(jié)點(diǎn)1是固定的,因此u1 = 0。在不進(jìn)入數(shù)學(xué)證明的情況下，可以說，該條件通過刪除結(jié)構(gòu)矩陣的行1和列1來實(shí)現(xiàn)，從而將矩陣的大小從3×3減小到2×2。In general, any boundary condition is satisfied by deleting the rows and columns corresponding to the node t

44、hat has zero deflection. In general, a node has six degrees of freedom (DOF), which include three translations and three rotations in x, y and z directions.一般來說，通過刪除對(duì)應(yīng)于具有零偏轉(zhuǎn)的節(jié)點(diǎn)的行和列，滿足任何邊界條件。節(jié)點(diǎn)具有六個(gè)自由度（DOF），其包括在x，y和z方向上的三個(gè)平移和三個(gè)旋轉(zhuǎn)。In the example problem, there is only one degree of freedom at each node

45、. The node deflects only along the axis of the spring.在示例問題中，在每個(gè)節(jié)點(diǎn)處只有一個(gè)自由度，即節(jié)點(diǎn)僅沿著彈簧的軸線偏轉(zhuǎn)。In this section, the finite element analysis procedure for a spring structure has been stablished. The following numerical example will utilize the derivation and concepts developed above.在本節(jié)中，已經(jīng)建立了用于彈簧結(jié)構(gòu)的有限元分析程

46、序。 下面的數(shù)字示例將利用上面得到的推導(dǎo)和概念。Example 2.2例2.2In the given spring structure, k1 = 20 lb./in., k2 = 25 lb./in., k3 = 30 lb./in., F = 5 lb. Determine deflection at all the nodes.在給定的彈簧結(jié)構(gòu),k1 = 20磅/。k2 = 25磅/。,k3 = 30磅/。F = 5磅。在所有節(jié)點(diǎn)確定撓度。Solution（解）We would apply the three steps discussed earlier.我們將使用前面討論的三個(gè)步驟

47、。Step 1: Derive the Element Equations步驟1:方程推導(dǎo)出元素。As derived earlier, the stiffness matrix equations for an element e is,如前所述，元素e的剛度矩陣方程是 Therefore, stiffness matrix of elements 1, 2, and 3 are, 因此，元素1,2和3的剛度矩陣為Step 2: Assemble element equations into a global equation步驟2:將子方程組裝為全局方程Assembling the ter

48、ms according to their row and column position, we get根據(jù)他們的行和列的位置裝配條件,我們得到 Or, by simplifying或者,通過簡化 The global structural equation is,全局結(jié)構(gòu)方程為, Step 3: Solve for deflections第三步:求解變形量First, applying the boundary conditions u1=0, the first row and first column will drop out. Next,F1= F2 = F3 = 0, and F

49、4 = 5 lb. The final form of the equation becomes,首先,應(yīng)用邊界條件u1 = 0,第一行和第一列將被化簡。接下來,F1 = F2 = F3 = 0,F4 = 5磅。方程的最終形式為, This is the final structural matrix with all the boundary conditions being applied. Since the size of the final matrices is small, deflections can be calculated by hand. It should be n

50、oted 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 matrix equation by hand we get,這是應(yīng)用所有邊界條件的最終結(jié)構(gòu)矩陣。 由于最終矩陣的尺寸小，可以手算偏轉(zhuǎn)。 應(yīng)當(dāng)注意，在實(shí)際結(jié)構(gòu)中，剛度矩陣的大小相當(dāng)大，并且只能借助于計(jì)算機(jī)來求解。 用手算求解上述矩陣方程， Example 2.3In the spring structur

51、e shown k1 = 10 lb./in., k2 = 15 lb./in., k3 = 20 lb./in., P= 5 lb. Determine the deflection at nodes 2 and 3.例2.3所示的彈簧結(jié)構(gòu)中k1 = 10磅/英寸。k2 = 15磅/英寸。,k3 = 20磅/英寸。P = 5磅。確定撓度在節(jié)點(diǎn)2和3。Solution:Again apply the three steps outlined previously.Step 1: Find the Element Stiffness Equations解決方案:再次應(yīng)用前面所述的三個(gè)步驟。第一步

52、:找到元素剛度方程 Step 2: Find the Global stiffness matrix步驟二：獲得整體剛度矩陣Now the global structural equation can be written as現(xiàn)在全局結(jié)構(gòu)方程可以寫成Step 3: Solve for Deflections步驟3：解決變形量The known boundary conditions are: u1 = u4 = 0, F3 = P = 3lb. Thus, rows and columns 1 and 4 will drop out, resulting in the following m

53、atrix equation,已知的邊界條件是：u1 = u4 = 0，F(xiàn)3 = P = 3lb。因此，行1和列4將化簡，得到以下矩陣方程， Solving, we get u2= 0.0692 & u3=0.1154求解，我們得到u2 = 0.0692u3 = 0.1154Example 2.4（例2.4）In the spring structure shown, k1= 10 N/mm, k2= 15 N/mm, k3= 20 N/mm, k4= 25 N/mm, k5= 30 N/mm, k6= 35 N/mm. F2 = 100 N. Find the deflections

54、 in all springs.在所示的彈簧結(jié)構(gòu)中，k1 = 10N / mm，k2 = 15N / mm，k3 = 20N / mm，k4= 25N / mm，k5 = 30N / mm，k6 = 35N / mm。 F2 = 100 N.求所有彈簧的撓度。Solution（解）Here again, we follow the three-step approach described earlier, without specifically mentioning at each step.在這里,我們遵循前面描述的三步方法,沒有特別提及每一步。 The global stif

55、fness matrix is,整體剛度矩陣為： And simplifying, we get（簡化后得到）And the structural equation is,(結(jié)構(gòu)方程為)Now, apply the boundary conditions, u1 = u4 = 0, F2 = 100 N. This is carried out by deleting the rows 1 and 4, columns 1 and 4, and replacing F2by 100N. The final matrix equation is,現(xiàn)在，應(yīng)用邊界條件u1 = u4 = 0，F(xiàn)2 =

56、 100 N.這通過刪除行1和4，列1和4，以及令100N替換F2來執(zhí)行。 最終的矩陣方程是Which gives（給出）Deflections（變形量）Spring（彈簧）1: u4u1= 0Spring 2: u2 u1= 1.54590Spring 3： u3 u2= -0.6763Spring 4: u3 u2= -0.6763Spring 5: u4 u2= -1.5459Spring 6: u4 u3= -0.86962.3.4 Boundary Conditions with Known Values具有已知值的邊界條件Up to now we have considered problems that have known applied forces, and no known values of deflection.到目前為止，我們已經(jīng)考慮了已知施加的力的問題，并且沒有已知的變形量。Now we will consider the procedure for applying the boundary conditions where, deflections on some nodes are known. 現(xiàn)在我們將考慮應(yīng)用邊界條件的過程，其中已知某些節(jié)點(diǎn)