【正文】 his stage, some important conceptual points should be emphasized and will be discussed 。 Procedure for Assembling Element stiffness matrices（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)性單元剛度矩陣在有限元的概念 把物體離散為多個(gè)單元分析 每個(gè)單元的剛度矩陣 也就是單元剛度矩陣簡稱單剛）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 constant, 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。 在這個(gè)階段，剛度矩陣總是對稱的，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。 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 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。下面是這個(gè)過程的一個(gè)說明,應(yīng)用的示例問題。同樣,對于元素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。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應(yīng)于相應(yīng)的行和列。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}=[k]{u}, 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 ,組裝結(jié)構(gòu)矩陣方程簡寫為{ F } =[k]{u},或簡單地，F(xiàn) = k u,每個(gè)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)。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 second step, a numerical matrix solution scheme is used to find deflection values by using a puter. Among the most popular numerical schemes are the Gauss elimination and the GaussSedel iteration method. For further reading, refer to any numerical analysis 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 of the two numerical solution schemes mentioned above.有兩個(gè)步驟可得到的撓度值。在第二步中,數(shù)值矩陣的解決是使用電腦查找撓度值。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。 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 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。In general, any boundary condition is satisfied by deleting the rows and columns corresponding to the node that 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。In the example problem, there is only one degree of freedom at each node. 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)。 下面的數(shù)字示例將利用上面得到的推導(dǎo)和概念。k2 = 25磅/。F = 5磅。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。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 terms 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 F4 = 5 lb. The final form of the equation bees,首先,應(yīng)用邊界條件u1 = 0,第一行和第一列將被化簡。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 noted that in a real structure the size of a stiffness matrix is rather large and can only be solved with the help of a puter. 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。 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。k2 = 15磅/英寸。P = 5磅。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。因此，行1和列4將化簡，得到以下矩陣方程， Solving, we get u2= amp。 F2 = 100 。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。 最終的矩陣方程是Which gives（給出）Deflections（變形量）Spring（彈簧）1: u4–u1= 0Spring 2: u2– u1= Spring 3： u3– u2= Spring 4: u3– u2= Spring 5: u4– u2= Spring 6: u4– u3= 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。Solutions of these problems are found by going through some additional steps. As discussed earlier, after obtaining the structural global matrix equation, deflections are found by solving the equation by applying a numerical scheme in a puter solu