【正文】 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 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 ’ . 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 . Linearized system equation in statics In nonlinear statics, the linearized system equation bees 2Kαβ Δ qβ =Δ pα upα 4. Nonlinear analysis . Initial shape analysis The 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 twoloop iteration method, ., 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 cablestayed 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, ., ??|s pa nm a i n po i nt s c on t rolat nt di s pl a c e m e ve rt i c a l| The shape iteration will be repeated until the convergence toleranceε , say 104, is achieved. When the convergence tolerance is reached, the putation will stop and the initial shape of the cablestayed 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 beamcolumn and large deflection effects bee insignificant. The initial analysis can be performed in two different ways: a linear and a nonlinear putation procedure. 1. Linear putation procedure: To find the equilibrium configuration of the bridge, all nonlinearities of cable stayed bridges are neglected and only the linear elastic cable, beamcolumn 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 putation efforts can be saved. 2. Nonlinear putation procedure: All nonlinearities of cablestayed bridges are taken into consideration during the whole putation process. The nonlinear cable element with sag effect and the beamcolumn element including stability coefficients and nonlinear coordinate transformation coefficients