# 关于斜拉桥的外文翻译--高度超静定斜拉桥的非线性分析研究

PDF外文：http://www.bisheziliao.com/p-133549.html中文5008字Study on nonlinear analysis of a highly redundant cable-stayed bridge 1． Abstract A 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 iteration 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 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 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 geometry 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 cable-stayed bridge response appear only in the modes determined on basis of the initial shape found by the nonlinear computation. 2. Introduction Rapid progress in the analysis and construction of cable-stayed bridges has been made over the last three decades. The progress is mainly due to developments in the fields of computer technology, high strength steel cables, orthotropic steel decks and construction technology. Since the first modern cable-stayed bridge was built in Sweden in 1955, their popularity has rapidly been increasing all over the world. Because of its aesthetic appeal, economic grounds and ease of erection, the cable-stayed bridge is considered as the most suitable construction type for spans ranging from 200 to about 1000 m. The world’ s longest cable-stayed bridge today is the Tatara bridge across the Seto Inland Sea, linking the main islands Honshu and Shikoku in Japan. The Tatara cable-stayed bridge was opened in 1 May, 1999 and has a center span of 890m and a total length of 1480m. A cable-stayed bridge consists of three principal components, namely girders, towers and inclined cable stays. The girder is supported elastically at points along its length by inclined cable stays so that the girder can span a much longer distance without intermediate piers. The dead load and traffic load on the girders are transmitted to the towers by inclined cables. High tensile forces exist in cable-stays which induce high compression forces in towers and part of girders. The sources of nonlinearity in cable-stayed bridges mainly include the cable sag, beam-column and large deflection effects. Since high pretension force exists in inclined cables before live loads are applied, the initial geometry and the prestress of cable-stayed bridges depend on each other. They cannot be specified independently as for conventional steel or reinforced concrete bridges. Therefore the initial shape has to be determined correctly prior to analyzing the bridge. Only based on the correct initial shape a correct deflection and vibration analysis can be achieved. The purpose of this paper is to present a comparison on the nonlinear analysis of a highly redundant stiff cable-stayed bridge, in which the initial shape of the bridge will be determined iteratively by using both linear and nonlinear computation procedures. Based on the initial shapes evaluated, the vibration frequencies and modes of the bridge are examined. 3. System equations 3.1. General system equation When only nonlinearities in stiffness are taken into account, and the system mass and damping matrices are considered as constant, the general system equation of a finite element model of structures in nonlinear dynamics can be derived from the Lagrange’ s virtual work principle and written as follows: Kjbα j-∑ Sjajα = Mαβ qβ ”+ Dαβ qβ ’ 3.2. Linearized system equation In order to incrementally solve the large deflection problem, the linearized system equations has to be derived. By taking the first order terms of the Taylor’ s expansion of the general system equation, the linearized equation for a small time (or load) interval is obtained as follows: Mαβ Δ qβ ”+Δ Dαβ qβ ’ +2Kαβ Δ qβ =Δ pα - upα 3.3. Linearized system equation in statics In nonlinear statics, the linearized system equation becomes 2Kαβ Δ qβ =Δ pα - upα 4. Nonlinear analysis 4.1. Initial shape analysis The initial shape of a cable-stayed bridge provides the geometric configuration as well as the prestress distribution of the bridge 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 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 arbitrary 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 iteration). 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 very 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’ . 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 points (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., |s p a nm a in p o in t s c o n t r o la t n t d is p la c e m e v e r t ic a l| The shape iteration will be repeated until the convergence toleranceε , 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 geometry 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 procedure. 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 carried 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 nonlinear 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. Newton– Raphson method is utilized here for equilibrium iteration. 4.2. Static deflection analysis Based 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 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 ‘ full’iteration procedure (iteration performed for a single full load step) will often fail. An increment– iteration 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 shape finding procedure using a linear or nonlinear computation.