【正文】 Study on nonlinear analysis of a highly redundant cablestayed bridge1．AbstractA parison on nonlinear analysis of a highly redundant cablestayed bridge is performed in the study. The initial shapes including geometry and prestress distribution of the bridge are determined by using a twoloop iteration method, ., an equilibrium iteration loop and a shape iteration loop. For the initial shape analysis a linear and a nonlinear putation procedure are set up. In the former all nonlinearities of cablestayed 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 twoloop iteration method, a reasonable initial shape can be determined by using the linear putation procedure, and a lot of putation efforts can thus be saved. There are only small differences in geometry and prestress distribution between the results determined by linear and nonlinear putation 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 cablestayed bridge response appear only in the modes determined on basis of the initial shape found by the nonlinear putation.2. IntroductionRapid progress in the analysis and construction of cablestayed bridges has been made over the last three decades. The progress is mainly due to developments in the fields of puter technology, high strength steel cables, orthotropic steel decks and construction technology. Since the first modern cablestayed 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 cablestayed bridge is considered as the most suitable construction type for spans ranging from 200 to about 1000 m. The world’s longest cablestayed bridge today is the Tatara bridge across the Seto Inland Sea, linking the main islands Honshu and Shikoku in Japan. The Tatara cablestayed bridge was opened in 1 May, 1999 and has a center span of 890m and a total length of 1480m. A cablestayed bridge consists of three principal ponents, 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 cablestays which induce high pression forces in towers and part of girders. The sources of nonlinearity in cablestayed bridges mainly include the cable sag, beamcolumn and large deflection effects. Since high pretension force exists in inclined cables before live loads are applied, the initial geometry and the prestress of cablestayed 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 parison on the nonlinear analysis of a highly redundant stiff cablestayed bridge, in which the initial shape of the bridge will be determined iteratively by using both linear and nonlinear putation procedures. Based on the initial shapes evaluated, the vibration frequencies and modes of the bridge are examined.3. System equations. General system equationWhen 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:Kjbαj∑Sjajα= Mαβqβ”+ Dαβqβ’. Linearized system equationIn 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: MαβΔqβ”+ΔDαβqβ’ +2KαβΔqβ=Δpα upα. Linearized system equation in staticsIn nonlinear statics, the linearized system equation bees2KαβΔqβ=Δpα upα4. Nonlinear analysis. Initial shape analysisThe initial shape of a cablestayed 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 putations, 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 putation for shape finding is performed by using the twolo