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1、Study on nonlinear analysis of a highly redundant cable-stayed bridge1AbstractA comparison on nonlinear analysis of a highly redundant cable-stayed bridge is performed in the study. The initial shapes including geometry and prestress distribution of the bridge are determined by using a two-loop iter
2、ation method, i.e., an equilibrium iteration loop and a shape iteration loop. For the initial shape analysis a linear and a nonlinear computation procedure are set up. In the former all nonlinearities of cable-stayed bridges are disregarded, and the shape iteration is carried out without considering
3、 equilibrium. In the latter all nonlinearities of the bridges are taken into consideration and both the equilibrium and the shape iteration are carried out. Based on the convergent initial shapes determined by the different procedures, the natural frequencies and vibration modes are then examined in
4、 details. Numerical results show that a convergent initial shape can be found rapidly by the two-loop iteration method, a reasonable initial shape can be determined by using the linear computation procedure, and a lot of computation efforts can thus be saved. There are only small differences in geom
5、etry and prestress distribution between the results determined by linear and nonlinear computation procedures. However, for the analysis of natural frequency and vibration modes, significant differences in the fundamental frequencies and vibration modes will occur, and the nonlinearities of the cabl
6、e-stayed bridge response appear only in the modes determined on basis of the initial shape found by the nonlinear computation.2. Nonlinear analysis2.1. Initial shape analysisThe initial shape of a cable-stayed bridge provides the geometric configuration as well as the prestress distribution of the b
7、ridge under action of dead loads of girders and towers and under pretension force in inclined cable stays. The relations for the equilibrium conditions, the specified boundary conditions, and the requirements of architectural design should be satisfied. For shape finding computations, only the dead
8、load of girders and towers is taken into account, and the dead load of cables is neglected, but cable sag nonlinearity is included. The computation for shape finding is performed by using the two-loop iteration method, i.e., equilibrium iteration and shape iteration loop. This can start with an arbi
9、trary small tension force in inclined cables. Based on a reference configuration (the architectural designed form), having no deflection and zero prestress in girders and towers, the equilibrium position of the cable-stayed bridges under dead load is first determined iteratively (equilibrium iterati
10、on). Although this first determined configuration satisfies the equilibrium conditions and the boundary conditions, the requirements of architectural design are, in general, not fulfilled. Since the bridge span is large and no pretension forces exist in inclined cables, quite large deflections and v
11、ery large bending moments may appear in the girders and towers. Another iteration then has to be carried out in order to reduce the deflection and to smooth the bending moments in the girder and finally to find the correct initial shape. Such an iteration procedure is named here the shape iteration.
12、 For shape iteration, the element axial forces determined in the previous step will be taken as initial element forces for the next iteration, and a new equilibrium configuration under the action of dead load and such initial forces will be determined again. During shape iteration, several control p
13、oints (nodes intersected by the girder and the cable) will be chosen for checking the convergence tolerance. In each shape iteration the ratio of the vertical displacement at control points to the main span length will be checked, i.e., The shape iteration will be repeated until the convergence tole
14、rance, say 10-4, is achieved. When the convergence tolerance is reached, the computation will stop and the initial shape of the cable-stayed bridges is found. Numerical experiments show that the iteration converges monotonously and that all three nonlinearities have less influence on the final geome
15、try of the initial shape. Only the cable sag effect is significant for cable forces determined in the initial shape analysis, and the beam-column and large deflection effects become insignificant.The initial analysis can be performed in two different ways: a linear and a nonlinear computation proced
16、ure. 1. Linear computation procedure: To find the equilibrium configuration of the bridge, all nonlinearities of cable stayed bridges are neglected and only the linear elastic cable, beam-column elements and linear constant coordinate transformation coefficients are used. The shape iteration is carr
17、ied out without considering the equilibrium iteration. A reasonable convergent initial shape is found, and a lot of computation efforts can be saved.2. Nonlinear computation procedure: All nonlinearities of cable-stayed bridges are taken into consideration during the whole computation process. The n
18、onlinear cable element with sag effect and the beam-column element including stability coefficients and nonlinear coordinate transformation coefficients are used. Both the shape iteration and the equilibrium iteration are carried out in the nonlinear computation. NewtonRaphson method is utilized her
19、e for equilibrium iteration.2.2. Static deflection analysisBased on the determined initial shape, the nonlinear static deflection analysis of cable-stayed bridges under live load can be performed incrementwise or iterationwise. It is well known that the load increment method leads to large numerical
20、 errors. The iteration method would be preferred for the nonlinear computation and a desired convergence tolerance can be achieved. Newton Raphson iteration procedure is employed. For nonlinear analysis of large or complex structural systems, a fulliteration procedure (iteration performed for a sing
21、le full load step) will often fail. An incrementiteration procedure is highly recommended, in which the load will be incremented, and the iteration will be carried out in each load step. The static deflection analysis of the cable stayed bridge will start from the initial shape determined by the sha
22、pe finding procedure using a linear or nonlinear computation. The algorithm of the static deflection analysis of cable-stayed bridges is summarized in Section 4.4.2.2.3. Linearized vibration analysisWhen a structural system is stiff enough and the external excitation is not too intensive, the system
23、 may vibrate with small amplitude around a certain nonlinear static state, where the change of the nonlinear static state induced by the vibration is very small and negligible. Such vibration with small amplitude around a certain nonlinear static state is termed linearized vibration. The linearized
24、vibration is different from the linear vibration, where the system vibrates with small amplitude around a linear static state. The nonlinear static state qa can be statically determined by nonlinear deflection analysis. After determining qa , the system matrices may be established with respect to su
25、ch a nonlinear static state, and the linearized system equation has the form as follows:MAq”+ DAq+ 2KAq=p(t)- TAwhere the superscript A denotes the quantity calculated at the nonlinear static state qa . This equation represents a set of linear ordinary differential equations of second order with con
26、stant coefficient matrices MA, DA and 2KA. The equation can be solved by the modal superposition method, the integral transformation methods or the direct integration methods.When damping effect and load terms are neglected, the system equation becomes：MAq” + 2KAq=0This equation represents the natur
27、al vibrations of an undamped system based on the nonlinear static state qa The natural vibration frequencies and modes can be obtained from the above equation by using eigensolution procedures, e.g., subspace iteration methods. For the cable-stayed bridge, its initial shape is the nonlinear static s
28、tate qa . When the cable-stayed bridge vibrates with small amplitude based on the initial shape, the natural frequencies and modes can be found by solving the above equation.2.4. Computation algorithms of cable-stayed bridge analysisThe algorithms for shape finding computation, static deflection ana
29、lysis and vibration analysis of cable-stayed bridges are briefly summarized in the following.2.4.1. Initial shape analysis1. Input of the geometric and physical data of the bridge.2. Input of the dead load of girders and towers and suitably estimated initial forces in cable stays.3. Find equilibrium
30、 position(i) Linear procedure Linear cable and beam-column stiffness elements are used. Linear constant coordinate transformation coefficients ajare used. Establish the linear system stiffness matrix K by assembling element stiffness matrices. Solve the linear system equation for q (equilibrium posi
31、tion). No equilibrium iteration is carried out.(ii) Nonlinear procedure Nonlinear cables with sag effect and beam-column elements are used. Nonlinear coordinate transformation coeffi- cients aj; aj， are used. Establish the tangent system stiffness matrix 2K. Solve the incremental system equation for
32、 q. Equilibrium iteration is performed by using the NewtonRaphson method.4. Shape iteration5. Output of the initial shape including geometric shape and element forces.6. For linear static deflection analysis, only linear stiff-ness elements and transformation coefficients are used and no equilibrium
33、 iteration is carried out.2.4.2. Vibration analysis1. 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 vibrat
34、ion frequencies and modes by sub-space iteration methods, such as the Rutishauser Method.3. ConclusionThe two-loop iteration with linear and nonlinear computation is established for finding the initial shapes of cable-stayed bridges. This method can achieve the architecturally designed form having u
35、niform prestress distribution, and satisfies all equilibrium and boundary conditions. The determination of the initial shape is the most important work in the analysis of cable-stayed bridges. Only with a correct initial shape, a meaningful and accurate deflection and/or vibration analysis can be ac
36、hieved. Based on numerical experiments in the study, some conclusions are summarized as follows:(1). No great difficulties appear in convergence of the shape finding of small cable-stayed bridges, where arbitrary initial trial cable forces can be used to start the computation. However for large scal
37、e cable-stayed bridges, serious difficulties occurred in convergence of iterations.(2). Difficulties often occur in convergence of the shape finding computation of large cable-stayed bridge, when trial initial cable forces are given by the methods of balance of vertical loads, zero moment control an
38、d zero displacement control.(3). A converged initial shape can be found rapidly by the two-loop iteration method, if the cable stress corresponding to about 80% of Eeq=E value is used for the trial initial force of each cable stay in the main span, and the trial force of the cables in side spans is
39、determined by taking horizontal equilibrium of the cable forces acting on the tower.(4). There are only small differences in geometry and prestress distribution between the results of initial shapes determined by linear and nonlinear procedures.(5). The shape finding using linear computation offers
40、a reasonable initial shape and saves a lot of computation efforts, so that it is highly recommended from the point of view of engineering practices.(6). In small cable-stayed bridges, there are only small difference in the natural frequencies based on initial shapes determined by linear and nonlinea
41、r computation procedures, and the mode shapes are the same in both cases.(7). Significant differences in the fundamental frequency and in the mode shapes of highly redundant stiff cable stayed bridges is shown in the study. Only the vibration modes determined by the initial shape based on nonlinear
42、procedures exhibit the nonlinear cable sag and beam-column effects of cable-stayed bridges, e.g., the first and third modes of the bridge are dominated by the transversal motion of the tower, not of the girder. The difference of the fundamental frequency in both cases is about 12%. Hence a correct a
43、nalysis of vibration frequencies and modes of cable-stayed bridges can be obtained only when the correct initial shape is determined by nonlinear computation, not by the linear computation.高度超静定斜拉桥的非线性分析研究1.摘要一个拉索高度超静定的斜拉桥的非线性分析比较在研究中被实行。 包括桥的几何学和预应力分配的初始形状是使用双重迭代的方法决定的,也就是,一个平衡迭代和一个形状迭代。对于开始的形状分析，一
44、个线性和一个非线性计算程序被建立。以前斜拉桥所有非线性被忽视，而且形状迭代是不考虑平衡而实行的。后来桥的所有非线性被考虑到，而且平衡和形状的重复都实行了。基于收敛于一点的起始形状由不同的程序决定,自振频率和震动模态也被详细地研究。数字的结果表明收敛于一点的起始形状能由二个环的重复方法快速地得到,合理的起始形状能由线性的计算程序决定，而且那样许多计算工作将被节省。在由线性的和非线性计算程序决定的结果之间的几何学和预应力分配中只有很小的不同。然而，对于自振频率和震动模态的分析来说，基本的频率和震动模态将会有显著的不同，而且斜拉桥反应的非线性只出现在由非线性计算得到的初始形状的基础之上的模态中。2非
45、线性分析2.1. 起始形状分析 斜拉桥的初始形状提供了几何学的结构和桥在主梁和索塔的恒载、斜拉索的拉力作用下的预应力分配。作用的平衡条件，指定的边界条件和建筑的设计需求应该被满足。因为计算的形状,主梁和索塔的永久荷载必须被考虑，拉索的自重被疏忽，而且拉索下垂的非线性应包括在内。形状的计算通过使用二重迭代的方法运行,也就是,平衡重复和形状重复循环。这能用拉索中的任意小的张力开始。基于参考结构 (建筑设计形式)，没有歪斜和零的预应力在主梁和索塔中,斜拉桥平衡位置在恒载作用下是由迭代首先确定的(平衡迭代)。虽然首先决定结构的是使平衡情况和边界情况得到满足，但是建筑的设计需求大体上没有得到实现。因为桥
46、的跨径是很大的而没有预应力存在斜拉索中,相当大的偏转和非常大的弯矩可能在主梁和索塔中出现。那么另外的一个迭代有必要执行来减少偏转和使主梁的弯矩平滑并最后找出正确的初始形状。如此的一个迭代程序在这里命名为形状迭代。对于形状迭代,在先前步骤中确定的基本的轴线力将会被作为下个重复采取的初始基本力,这样一个新的平衡结构在恒载和这个初始力下再次被确定。在形状迭代的时候，一些控制点(主梁和拉索连接的点)将会被选择检验应力集中。在每次形状迭代过程中，主跨的控制点的垂直位移比率将会被检验。也就是, 形状迭代将会重复直到应变可以达到所说的10-4。当应变达到的时候, 计算将会停止而斜拉桥的初始形状就找到了。数字
47、的实验表明重复收敛于一点是没什么作用的，并且所有的三个非线性对最后的几何初始形状有比较少的影响。只有拉索下垂作用在确定初始形状分析中有显著作用，而偏压柱和大的偏转效应变则无关重要。 开始的分析能以二种不同的方式被实行: 一个线性和一个非线性计算程序。(1) 线性的计算程序:为了要找到桥的平衡结构,斜拉桥的所有非线性因素被疏忽，而只是线性的弹性拉索、梁单元、同等的线形的变形系数被使用。形状迭代是不考虑平衡迭代而实行的。合理的收敛于一点的起始形状被得到，而且许多计算的工作能被节省。(2) 非线性计算程序:斜拉桥所有的非线性因素在整个的计算程序中被考虑。非线性拉索元素的下沉作用、主梁元素的稳定系数和
48、非线性变形调整系数被应用。形状的迭代和平衡迭代都在非线性计算中实行。 牛顿-瑞普生方法在这里被用于平衡迭代。2.2 静态偏转分析基于确定的起始形状,斜拉桥在活载作用下的非线性静态偏转分析可通过模数或迭代运行。 荷载模数方法导致很大的数字错误是广为人知的。迭代方法比较适于非线性计算，而且需要的应变应能被达到。牛顿- 瑞普生的迭代程序将被使用。因为非线性分析较大或复杂的结构系统,一个完整的迭代程序(重复为一个单一全部荷载运行步骤)将会时常失败。一个模数-迭代程序高度地被推荐,荷载将会被增加，而且重复将会在每个荷载步骤中实行。斜拉桥的静态偏转分析将会从使用线性或非线性计算程序决定的初始形状开始。斜拉
49、桥静态的偏转分析的运算法则在第4.4.2节中被概述。2.3. 线性振动分析当一个结构系统是足够稳固而且外部的刺激不是太强烈,系统可能以一个确定的非线性的静态系数作一个小振幅振动，由振动引起的非线性静态系数的变化是很小的和可以忽略的。这种以一个非线性静态系数以一个小振幅的振动被称作线性化振动。线性化振动不同于线性振动，系统用很小的振幅以一个线性静态系数振动。非线性静态系数qa 能由非线性偏转分析决定。在决定qa之后，系统矩阵可能被建立有关于如此的一个非线性静态系数，线性化系统的等式如下所示：MAq”+ DAq+ 2KAq=p(t)- TA上面的上标字母A 代表在非线性静态系数qa 被计算的数量。这个等式用恒定系数矩阵MA、