【正文】 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 cablestayed bridges is started by use of arbitrary small initial cable forces suggested in the papers of Wang et al. Therefore, to estimate a suitable trial initial cable forces in order to get a convergent solution bees important for the shape finding analysis. In the following, several methods to estimate trial initial cable forces will be discussed. . Balance of vertical loads . Zero moment control . Zero displacement control . Concept of cable equivalent modulus ratio . Consideration of the unsymmetry If the estimated initial cable forces are determined independently for each cable stay by the methods mentioned above, there may exist unbalanced horizontal forces on the tower in unsymmetric cablestayed bridges. Forsymmetric arrangements of the cablestays on the central (main) span and the side span with respect to the tower, the resultant of the horizontal ponents of the cablestays acting on the tower is zero, ., no unbalanced horizontal forces exist on the tower. For unsymmetric cablestayed bridges, in which the arrangement of cablestays on the central (main) span and the side span is unsymmetric, and if the forces of cable stays on the central span and the side span are determined independently, evidently unbalanced horizontal forces will exist on the tower and will induce large bending moments and deflections therein. Therefore, for unsymmetric cablestayed bridges, this problem can be overe as follows. The force of cable stays on the central (main) span Tim can be determined by the methods mentioned above independently, where the superscript m denotes the main span, the subscript I denotes the ith cable stay. Then the force of cable stays on the side span is found by taking the equilibrium of horizontal force ponents at the node on the tower attached with the cable stays, ., Tim cosα i= Tis cosβ i, and Tis = Tim cosα i/ cosβ i, where α i is the angle between the ith cable stay and the girder on the main span, andβ i, angle between the ith cable stay and the girder on the side span. 6. Examples In this study, two different types of small cablestayed bridges are taken from literature, and their initial shapes will be determined by the previously described shape finding method using linear and nonlinear procedures. Finally, a highly redundant stiff cablestayed bridge will be examined. A convergence tolerance e =104 is used for both the equilibrium iteration and the shape iteration. The maximum number of iteration cycles is set as 20. The putation is considered as not convergent, if the number of the iteration cycles exceeds 20. The initial shapes of the following two small cable stayed bridges in Sections and are first determined by using arbitrary trial initial cable forces. The iteration converges monotonously in these two examples. Their convergent initial shapes can be obtained easily without difficulties. There are only small differences between the initial shapes determined by the linear and the nonlinear putation. Convergent solutions offer similar results, and they are independent of the trial initial cable forces. 7. Conclusion The twoloop iteration with linear and nonlinear putation is established for finding the initial shapes of cablestayed bridges. This method can achieve the architecturally designed form having uniform prestress distribution, and satisfies all equilibrium and boundary conditions. The determination of the initial shape is the most important work in the analysis of cablestayed bridges. Only with a correct initial shape, a meaningful and accurate deflection and/or vibration analysis can be achieved. 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 cablestayed bridges, where arbitrary initial trial cable forces can be used to start the putation. However for large scale cablestayed bridges, serious difficulties occurred in convergence of iterations. (2). Difficulties often occur in convergence of the shape finding putation of large cablestayed bridge, when trial initial cable forces are given by the methods of