Geometrical Nonlinearity And Cable Force Optimization Of Multi-pylon Cable-stayed Bridges

The cable-stayed bridge is a kind of highly statically indeterminate flexible structure, which is composed of girder, tower and cable. The cable work under huge tension, the girder and tower work under the combination of pressure and bend. As the bridge span increases, the structure presents obvious geometrical nonlinearity. So, nonlinear influence should be fully considered in the calculation and analysis of cable-stayed bridges. Besides that, the cable force of cable-stayed bridges can be artificially tensioned or adjusted, which causes the diversity of internal force state of finished bridge. So, it is necessary to study the optimal cable force in order to work in a more reasonable internal force state for cable-stayed bridges. This paper focuses on geometric nonlinearity and cable force optimization of cable-stayed bridges. The core content mainly includes three parts.(1) The geometric nonlinearity analysis theory of TL, UL and CR formulation were introduced in detail. The specific expressions of tangent stiffness matrix of common elements such as truss, beam, cable and plate were given. Under the help of MATLAB software programme, theories in this paper were validated through some numerical examples. The numerical results of beam element show that, CR-TL formulation is far better than CR-UL formulation. Compared with CR-UL formulation, CR-TL formulation has high precision and efficiency which can avoid accumulative error and remove the restriction of small rotations by using CR coordinate to eliminate the rigid body motion accurately and computing the internal forces of element based on the total equilibrium condition rather than the incremuntal equilibrium condition. The numerical results of plate element show that, fourth order of Gaussian integration is appropriate in the finite element, high order is unuseful to improve the final calculation accuracy.(2) According to four analysis models of ChiShi Bridge such as the completely linear model, the nonlinear model in which equivalent elastic modulus method was used to consider sag effect only, the nonlinear model in which both large displacement and second-order effect of tower and girder were considered and the nonlinear model in which all nonlinear factors were considered, the nonlinear influence was analysed for deformation and internal force of girder and tower in each of special construction stage. In addition, by using MATLAB software programme, the calculation error of equivalent elastic modulus method was studied. In the end, based on linearity theory, deflection theory and nonlinearity theory, the response of live load was analyzed in completed bridge stage. It is concluded that the error of equivalent elastic modulus method may be very high for cable whose stress is low and length is long, its computation precision relies heavily on the number of load steps. Cable sag has a great effect on the deformation of girder but a small impact on the internal force of it. The influence to girder’s bending moment caused by cable sag stays within5percent limit in each of construction stage. It may reach over20percent under combined action of large displacement and second-order effect of tower and girder. As for the effect of live load, it can be calculated approximatively using linearity theory due to that it is small in completed bridge phase.(3) Setting up the influence matrix of ChiShi Bridge between construction cable force and variables in finished bridge state, the third tensioning of cable was optimized by taking the girder’s bending energy revised for the live load effect as objective function and utilizing forward iteration method to consider the influence of shrinkage, creep and nonlinearity. The optimization result in significant decrease in the peak of girder’s negative bending moment due to dead load of completed bridge and make bending moment more closer to the median line of feasible region. The optimized bending energy of girder is only38.7percent of that before optimization. After optimization, the maximum compressive stress of girder happens at the largest cantilever stage, its value is9.44MPa at the upper edge and15.5MPa at the lower edge. The maximum compressive stress of tower happens at the largest cantilever stage and completed bridge stage, its value is10.2MPa occorred on the outside lower pylon.