【正文】 只有基于非線性程序的初始形狀確定的模態(tài)展示非線性拉索下沉和主梁的偏轉(zhuǎn)效應。基于研究的數(shù)字實驗 ,一些結(jié)論概述如下 : (1).對于小型斜拉橋 初始形狀的確定不會出現(xiàn)現(xiàn)在的困難，任意的初始試驗拉索應力都能用來計算。在這二個例 子中收斂于一點是重復單調(diào)的。中間跨 (主要部份 )的拉索受力可以通過上面獨立介紹的方法確定，其中上標 m代表主跨，上標 i代表第 i條拉索。在非線性分析中，牛頓 瑞普生類型迭代計算能收斂 到一點 ,只有當解決的被估計的價值是在真正的價值附近時才能實現(xiàn)。 6. 對于線性偏轉(zhuǎn)分析，只有線性剛度單元和變形系數(shù)被采用且沒有平衡迭代的實行。 (ii)非線性程序 o 非線性下垂效應的拉索和主梁單元被使用。 . 斜拉橋計算運算法則分析 斜拉橋的形狀的確定計算、靜態(tài)偏轉(zhuǎn)分析和振動分析的計算法則簡短的概述如下。在決定 qα a之后，系統(tǒng)矩陣可能被建立有關(guān)于如此的一個非線性靜態(tài)系數(shù)，線性化系統(tǒng)的等式如下所示： Mαβ Aqβ ”+ Dαβ Aqβ ’+ 2Kαβ Aqβ =pα (t) Tα A 上面的上標字母‘ A39。因為非線性分析較大或復雜的結(jié)構(gòu)系統(tǒng) ,一個‘完整’的迭代程序 (重復為一個單一全部荷載運行步驟 )將會時常失敗。 (2) 非線性計算程序 :斜拉橋所有的非線性因素在整個的計算程序中被考慮。也就是 , ??|| 主跨控制點的垂直位移 形狀迭代將會重復直到應變可以達到所說的 104?；趨⒖冀Y(jié)構(gòu) (建筑設計形式 )，沒有歪斜和零的預應力在主梁和索塔中 ,斜拉 橋平衡位置在恒載作用下是由迭代首先確定的 (平衡迭代 )。這篇論文的目的要提供一個高度冗余的斜拉橋的非線性分析的比較，橋的開始形狀將會由線性和非線性計算程序迭代來決定。主梁在沿縱向方向由拉索彈性支撐以使主梁能跨越一個更長的距離而不需要中間橋墩。然而，對于自振頻率和震動模態(tài)的分析來說，基本的頻率和震動模態(tài)將會有顯著的不同，而且斜拉橋反應的非線性只出現(xiàn)在由非線性計算得到的初始形狀的基礎(chǔ)之上的模態(tài)中。 ajα，β are used. ? Establish the tangent system stiffness matrix 2Kαβ . ? Solve the incremental system equation for △ qα . ? Equilibrium iteration is performed by using the Newton– Raphson method. 4. Shape iteration 5. Output of the initial shape including geometric shape and element forces. 6. For linear static deflection analysis, only linear stiffness elements and transformation coefficients are used and no equilibrium iteration is carried out. . Vibration analysis 1. Input of the geometric and physical data of the bridge. 2. Input of the initial shape data including initial geometry and initial element forces. 3. Set up the linearized system equation of free vibrations based on the initial shape. 4. Find vibration frequencies and modes by subspace iteration methods, such as the Rutishauser Method. 5. Estimation of the trial initial cable forces In the recent study of Wang and Lin, the shape finding of small cablestayed bridges has been performed by using arbitrary small or large trial initial cable forces. There the iteration converges monotonously, and the convergent solutions have similar results, if different trial values of initial cable forces are used. However for large cablestayed bridges, shape finding putations bee more difficult to converge. In nonlinear analysis, the Newtontype iterative putation can converge, only when the estimated values of the solution is locate in the neighborhood of the true values. Difficulties in convergence may appear, when the shape finding analysis of cablestayed bridges is started by use of arbitrary small initial cable forces suggested in the papers of Wang et al. Therefore, to estimate a suitable trial initial cable forces in order to get a convergent solution bees important for the shape finding analysis. In the following, several methods to estimate trial initial cable forces will be discussed. . Balance of vertical loads . Zero moment control . Zero displacement control . Concept of cable equivalent modulus ratio . Consideration of the unsymmetry If the estimated initial cable forces are determined independently for each cable stay by the methods mentioned above, there may exist unbalanced horizontal forces on the tower in unsymmetric cablestayed bridges. Forsymmetric arrangements of the cablestays on the central (main) span and the side span with respect to the tower, the resultant of the horizontal ponents of the cablestays acting on the tower is zero, ., no unbalanced horizontal forces exist on the tower. For unsymmetric cablestayed bridges, in which the arrangement of cablestays on the central (main) span and the side span is unsymmetric, and if the forces of cable stays on the central span and the side span are determined independently, evidently unbalanced horizontal forces will exist on the tower and will induce large bending moments and deflections therein. Therefore, for unsymmetric cablestayed bridges, this problem can be overe as follows. The force of cable stays on the central (main) span Tim can be determined by the methods mentioned above independently, where the superscript m denotes the main span, the subscript I denotes the ith cable stay. Then the force of cable stays on the side span is found by taking the equilibrium of horizontal force ponents at the node on the tower attached with the cable stays, ., Tim cosα i= Tis cosβ i, and Tis = Tim cosα i/ cosβ i, where α i is the angle between the ith cable stay and the girder on the main span, andβ i, angle between the ith cable stay and the girder on the side span. 6. Examples In this study, two different types of small cablestayed bridges are taken from literature, and their initial shapes will be determined by the previously described shape finding method using linear and nonlinear procedures. Finally, a highly redundant stiff cablestayed bridge will be examined. A convergence tolerance e =104 is used for both the equilibrium iteration and the shape iteration. The maximum number of iteration cycles is set as 20. The putation is considered as not convergent, if the number of the iteration cycles exceeds 20. The initial shapes of the following two small cable stayed bridges in Sections and are first determined by using arbitrary trial initial cable forces. The iteration converges monotonously in these two examples. Their convergent initial shapes can be obtained easily without difficulties. There are only small differences between the initial shapes determined by the linear and the nonlinear putation. Convergent solutions offer similar results, and they are independent of the trial initial cable forces. 7. Conclusion The twolo