【正文】 s in row 2 to zero, except the term in column 2 (kij = 0, kii = k22 ≠0)步驟1：除了第2列中的項（kij = 0，kij = k22≠0），將第2行中的所有項設(shè)置為零， Step 2: Replace F2 with the term k22 U2 = (90)() = 135, (Fi =Kiiui) 步驟2：用項k22替換F2 U2 =（90）（）= 135，（Fi = Kiiui） Step 3: Subtract the value k22 U2 from all the forces, except F2 (subtract kji from the existing values of fj)步驟3：從除了F2之外的所有力中減去值K22 U2（從fj的現(xiàn)有值中減去kji）F1 → F1– (15)() = Row (行)1: kj2 = k12 = 15F3 → F3– (45)() = Row(行)1: kj2 = k32 = 45F4 → F4– (30)() =45 Row(行)1: kj2 = k42 = 30Note(注釋): F1 = F3 = F4 = 0.The new force equation now is,得到的新的力學(xué)方程是 Step 4: Set all the elements in column 2 to zero, except, row2 (all kji = 0, kii ≠ 0)步驟4：將第2列中的所有元素設(shè)置為零，除了第2行（所有kji = 0，kii≠0）Or, k12 = k32 = k42 = 0, and the new equation is, 或者，k12 = k32 = k42 = 0，并且新方程是，This is the final equation after the nodal value u2 = mm is incorporated into the structural equation.將節(jié)點(diǎn)值u2 = 。The same procedure can be followed for the boundary conditions u1 = u4 = 0. It can be stated that for zero nodal values, the procedure will always lead to elimination of rows and columns corresponding to these nodes, that is, the first and fourth rows as well as columns will drop out. The reader is encouraged to verify this statement.對于邊界條件u1 = u4 = 0可以遵循相同的過程?？梢哉f，對于零節(jié)點(diǎn)值，過程將總是導(dǎo)致消除與這些節(jié)點(diǎn)相對應(yīng)的行和列，即第一和第四行以及列將化簡消除。 鼓勵讀者核實此聲明Thus, the final equation is, 最后的方程是， Solving for u2 and u3, we get求解u2和u3，我們得到 Spring deflection is: 彈簧的變形量：彈簧Spring 1: u2 – u1 = Spring 2: u3 – u1 = Spring 3: u3 – u2 = Spring 4: u3 – u2 = Spring 5: u4 – u2 = Spring 6: u4 – u3 = Structures that can be Modeled Using a Spring Elements可以使用彈簧元素建模的結(jié)構(gòu)As mentioned earlier, almost all engineering structures (linear structures) are similar to a linear spring, satisfying the relation F = ku. Therefore, any structure that deflects only along its axial direction (with one degree of freedom) can be modeled as a spring element. The following example illustrates this concept.如前所述，幾乎所有工程結(jié)構(gòu)（線性結(jié)構(gòu)）類似于線性彈簧，滿足關(guān)系F = ku。 因此，任何僅沿其軸向方向（具有一個自由度）偏移的結(jié)構(gòu)可以被建模為彈簧元件。 以下示例說明了此概念。Example(例)A circular concrete beam structure is loaded as shown. Find the deflection of points at 8”, 16”, and the end of the beam. E = 4 x 10 6 psi如圖所示裝載圓形混凝土梁結(jié)構(gòu)。 找到在8，16和梁的端部的點(diǎn)的撓度。 E = 410 6 磅Solution（解）The beam structure looks very different from a spring. 梁結(jié)構(gòu)看起來與彈簧區(qū)別很大。However, its behavior is very similar. 但是，它的行為非常相似。Deflection occurs along the xaxis only. 僅沿x軸發(fā)生變形。The only significant difference between the beam and a spring is that the beam has a variable crosssectional area. 梁和彈簧之間唯一的顯著區(qū)別是梁具有可變的橫截面積。An exact solution can be found if the beam is divided into an infinite number of elements, then, each element can be considered as a constant crosssection spring element, obeying the relation F = ku, where k is the stiffness constant of a beam element and is given by k= AE/L.如果梁被分成無限數(shù)量的元件，則可以得到精確解，然后，每個元件可以被認(rèn)為是恒定橫截面彈簧元件，服從關(guān)系F = ku，其中k是梁的剛度常數(shù) 并且由k = AE / L給出。In order to keep size of the matrices small (for hand calculations), let us divide the beam into only three elements. For engineering accuracy, the answer obtained will be in an acceptable range. If needed, accuracy can be improved by increasing the number of elements.為了保持矩陣的大小（手算），讓我們將波束劃分為僅三個元素。 對于工程精度，獲得的答案將在可接受的范圍內(nèi)。 如果需要，可以通過增加元素的數(shù)量來提高精度。As mentioned earlier in this chapter, spring, truss, and beam elements are lineelements and the shape of the cross section of an element is irrelevant.如本章前面所述，彈簧，桁架和梁單元是線單元，與單元橫截面的形狀是不相關(guān)的。Only the crosssectional area is needed (also, moment of inertia for a beam element undergoing a bending load need to be defined). 只需要橫截面積（此外，需要確定經(jīng)受彎曲載荷的梁元件的慣性矩）。The beam elements and their puter models are shown in figure .。Here, the question of which crosssectional area to be used for each beam section arises. 這里，出現(xiàn)了用于每個梁部分的橫截面積的問題。A good approximation would be to take the diameter of the midsection and use that to approximate the area of the element.想得到更精確的近似值需要采用中間截面的直徑并且使用近似元素的面積。Beam sections（梁部分） Equivalent spring elements（等效彈簧元件）Crosssectional area橫截面面積The average diameters are: d1 = in., d2 = in., d3 = . (diameters are taken at the mid sections and the values are found from the height and length ratio of the triangles shown in figure ), which is given as平均直徑為：d1 = ，d2 = ，d3 = 。 （在中間部分取直徑，），其給出為12/L = 3/(L24), L = 32Average areas are:平均面積A1 = in2 A2 = in2 A3 = in2注：in2為平方英寸Stiffness（剛度）k1 = A1 E/L1 = ()(4 10 7 /8) = 10e7 lb./in., similarly,k2 = A2 E/L2 = 10 7 lb./in.k3 = A3 E/L3 = 10 6 lb./in.Element Stiffness Equations元素剛度方程Similarly, 類似地，Global stiffness matrix is全局剛度矩陣為Now the global structural equations can be written as, 現(xiàn)在全局結(jié)構(gòu)方程可以寫成：Applying the boundary conditions: u1 = 0, and F1 = F2 = F3 = 0, F4 = 5000 lb., results in the reduced matrix,應(yīng)用邊界條件：u1 = 0，并且F1 = F2 = F3 = 0，F(xiàn)4 = 5000磅。得出簡化的矩陣Solving we get,解得The deflections u2, u3, and u4 are only the approximate values, which can be improved by dividing the beam into more elements. As the number of elements increases, the accuracy will improve.撓度u2，u3和u4僅是近似值，其可以通過細(xì)化成更多元素來改進(jìn)。 隨著元件數(shù)量的增加，精度將提高。