【正文】 工程文本或手冊中找到。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。截面形狀并不影響線元素的特性。Spring, Truss, and Beams elements, called line elements, are usually divided into small sections with nodes at each end. The crosssection shape doesn’t affect the behavior of a line element。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 endpoints 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)。每一個(gè)元素的力學(xué)行為類似于機(jī)械彈簧,遵守方程,F =ku。 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 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。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。The Basics of FEA Procedure有限元分析程序的基本知識(shí) 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 engineering 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 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。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 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。 In FEA, structures are modeled by a CAD program and represented by nodes and elements. The mechanical behavior 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 puter.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。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。Each element possesses its own geometric and elastic properties. 每個(gè)元素?fù)碛凶约旱膸缀魏蛷椥浴?only the crosssectional constants are relevant and used in calculations. Thus, a square or a circular crosssection of a truss member will yield exactly the same results as long as the crosssectional area is the same. Plane and solid elements require more than 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)。 只有橫截面常數(shù)是相關(guān)的并用于計(jì)算。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。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。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. 這種類型的問題要求使用前面提到的行元素以外的元素。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。Both spring and truss elements give an easier modeling overview of the finite element analysis procedure, due to the fact that each 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。The analysis procedure presented here will be exactly the same as that used for a plex structural problem, except, in the following example, all calculations will be 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 puter, since all finite element analyses are done on a puter. The finite element equations are derived using Direct Equilibrium method.本文提供的分析過程將一模一樣,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。有限元方程導(dǎo)出可直接使用平衡方法。求在節(jié)點(diǎn)的撓度。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。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)。Element ‘e’ can be thought of as any element in the structure with nodes i and j, forces fi and fj, deflections 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。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 e(uiuj) = k eui + k euj平衡方程：fj = fi = k e(uiuj) = 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(u1u2)f2 = k2(u2 – u3) f3 = k2(u2u3)這就完成第一步的過程。這將是在步驟2中代替。Step 2 : Assemble the element equations into a global equation.步驟2:組裝元素方程為全局方程。 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 the external forces at each node be F1, F2, and F3, as shown in figure . 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。使用平衡方程,我們可以找到方程的元素,如下所示。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 () k2 u2 + k2 u3 = F3These equations can now be written in a matrix form, givingk1 這些方程可以寫成矩陣形式,代入k1 This pletes step 2 for assembling the element equations into a global equation. At t