有限元分析英文文獻(編輯修改稿)

　

【文章內容簡介】 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代表外部力量在節點2和3,在一般條款,而不是實際的已知值的力量。在示例問題,F1 = = 0 F2和F3 = f .實際力矩陣 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 ,組裝結構矩陣方程簡寫為{ F } =[k]{u},或簡單地，F = k u,每個術語的理解是一個m n矩陣m和n的行數的列數。Step 3: Solve the global equation for deflections at nodes.步驟3:解決全局方程在節點變位。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 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.有兩個步驟可得到的撓度值。在第一步中,所有的應用邊界條件,這將導致減少整體結構性矩陣的大小。在第二步中,數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數值方案GaussSedel算法。為進一步閱讀,指的是任何數值分析有關此主題的書。下面的例子和章節,所有的矩陣計算解決方案將是有限的手雖然實際矩陣在有限元的解決方案總是使用上面提到的兩個數值解方案之一。 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,節點1是固定的,因此u1 = 0。在不進入數學證明的情況下，可以說，該條件通過刪除結構矩陣的行1和列1來實現，從而將矩陣的大小從33減小到22。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，通過刪除對應于具有零偏轉的節點的行和列，滿足任何邊界條件。節點具有六個自由度（DOF），其包括在x，y和z方向上的三個平移和三個旋轉。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，在每個節點處只有一個自由度，即節點僅沿著彈簧的軸線偏轉。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.在本節中，已經建立了用于彈簧結構的有限元分析程序。 下面的數字示例將利用上面得到的推導和概念。Example In 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.在給定的彈簧結構,k1 = 20磅/。k2 = 25磅/。,k3 = 30磅/。F = 5磅。在所有節點確定撓度。Solution（解）We would apply the three steps discussed earlier.我們將使用前面討論的三個步驟。Step 1: Derive the Element Equations步驟1:方程推導出元素。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 terms according to their row and column position, we get根據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,全局結構方程為, 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 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,這是應用所有邊界條件的最終結構矩陣。 由于最終矩陣的尺寸小，可以手算偏轉。 應當注意，在實際結構中，剛度矩陣的大小相當大，并且只能借助于計算機來求解。 用手算求解上述矩陣方程， Example In the spring structure shown k1 = 10 lb./in., k2 = 15 lb./in., k3 = 20 lb./in., P= 5 lb. Determine the deflection at nodes 2 and 3. = 10磅/英寸。k2 = 15磅/英寸。,k3 = 20磅/英寸。P = 5磅。確定撓度在節點2和3。Solution:Again apply the three steps outlined previously.Step 1: Find the Element Stiffness Equations解決方案:再次應用前面所述的三個步驟。第一步: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現在全局結構方程可以寫成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 matrix equation,已知的邊界條件是：u1 = u4 = 0，F3 = P = 3lb。因此，行1和列4將化簡，得到以下矩陣方程， Solving, we get u2= amp。 u3=求解，我們得到u2 = ＆u3 = Example （）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 in all springs.在所示的彈簧結構中，k1 = 10N / mm，k2 = 15N / mm，k3 = 20N / mm，k4= 25N / mm，k5 = 30N / mm，k6 = 35N / mm。 F2 = 100 。Solution（解）Here again, we follow the threestep approach described earlier, without specificallymentioning at each step.在這里,我們遵