【正文】 are used. Both the shape iteration and the equilibrium iteration are carried out in the nonlinear putation. Newton– Raphson method is utilized here for equilibrium iteration. . Static deflection analysis Based on the determined initial shape, the nonlinear static deflection analysis of cablestayed 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 putation and a desired convergence tolerance can be achieved. Newton– Raphson iteration procedure is employed. For nonlinear analysis of large or plex structural systems, a ‘ full’iteration procedure (iteration performed for a single full load step) will often fail. An increment– iteration procedure is highly remended, 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 putation. The algorithm of the static deflection analysis of cablestayed bridges is summarized in Section . . Linearized vibration analysis When a structural system is stiff enough and the external excitation is not too intensive, the system 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 vibration is different from the linear vibration, where the system vibrates with small amplitude around a linear static state. The nonlinear static state qα a can be statically determined by nonlinear deflection analysis. After determining qα a , the system matrices may be established with respect to such a nonlinear static state, and the linearized system equation has the form as follows: Mαβ Aqβ ”+ Dαβ Aqβ ’+ 2Kαβ Aqβ =pα (t) Tα A where the superscript ‘ A’ denotes the quantity calculated at the nonlinear static state qα a . This equation represents a set of linear ordinary differential equations of second order with constant coefficient matrices Mαβ A, Dαβ A and 2Kαβ A. 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 bees Mαβ Aqβ ” + 2Kαβ Aqβ =0 This equation represents the natural vibrations of an undamped system based on the nonlinear static state qα a The natural vibration frequencies and modes can be obtained from the above equation by using eigensolution procedures, ., subspace iteration methods. For the cablestayed bridge, its initial shape is the nonlinear static state qα a . When the cablestayed bridge vibrates with small amplitude based on the initial shape, the natural frequencies and modes can be found by solving the above equation. . Computation algorithms of cablestayed bridge analysis The algorithms for shape finding putation, static deflection analysis and vibration analysis of cablestayed bridges are briefly summarized in the following. . Initial shape analysis 1. 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 position (i) Linear procedure ? Linear cable and beamcolumn stiffness elements are used. ? Linear constant coordinate transformation coefficients ajα are used. ? Establish the linear system stiffness matrix Kαβ by assembling element stiffness matrices. ? Solve the linear system equation for qα (equilibrium position). ? No equilibrium iteration is carried out. (ii) Nonlinear procedure ? Nonlinear cables with sag effect and beamcolumn elements are used. ? Nonlinear coordinate transformation coeffi cients ajα 。 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 cab