| Most of the continuum models of suspension bridges are analytical ideal models based on deflection theory.For the preliminary design of suspension bridges,continuum models can be used to identify the key dimensionless parameters that control the dynamic response of the bridge and to conduct extensive parametric analyses.Although they are not accurate enough,they have simple governing equations and do not require initial equilibrium state analysis.At present,there have been a lot of studies on the dynamic analysis models of suspension bridges,but most of them do not consider the bending stiffness of the main cables.The actual main cables have a certain bending stiffness.With the increase of the spans of the suspension bridges,the cross-sectional areas of the main cables are also increasing.The actual bending stiffness of the main cables will inevitably have a certain effect on the internal force and deformation of the structure under dynamic load.This paper embeds the bending stiffness term of the main cables into the motion equations of the suspension bridges with extensible hangers.The coupled differential equations of the free vertical vibration of the main cable and the girder are derived with dimensionless parameters.Based on the Galerkin method,equations of motion are obtained in matrix form,and the analytical solutions for the dynamic characteristics of the three-span plane main cable and three-span spatial main cable suspension bridges are obtained.According to these two instances of suspension bridges,the finite element models are established by Midas Civil software,and the analytical solutions and finite element results of the natural frequencies are calculated respectively to verify the accuracy of the analytical solutions.The verification results show that the analytical solutions are relatively close to the finite element results,and the continuum models can obtain the dynamic characteristics accurately.On this basis,parametric analyses of the analytical solutions of dynamic characteristics of the two suspension bridges are conducted in dimensionless form.The results show that the bending stiffness of the main cables have a slightly more pronounced influence on the natural frequencies of the higher-order modes and anti-symmetric modes of the suspension bridges.Within the range of dimensionless parameters of common suspension bridges selected in this paper,for the first ten symmetric and antisymmetric modes of these two types of suspension bridges,the differences of the natural frequencies between the before and after considering the bending stiffness of the main cables gradually increase with the increase of the relative bending stiffness of the main cables and the relative axial stiffness of the hangers;gradually decrease with the increase of the relative length of the side span;with the increase of the relative bending stiffness of the girder,the higher-order modes(such as the ninth and tenth order)tend to decrease first and then increase;with the increase of the relative axial stiffness of the main cables,the lower-order(such as the third and fourth order)of the symmetric modes show two trends of gradual increase and gradual decrease,but there is no effect on antisymmetric modes.For three-span plane main cable suspension bridges,the differences of natural frequencies gradually decrease with the increase of the relative mass of the girder,and the change in the relative sag of the mid-span main cable has no obvious effect on the differences of natural frequencies.For three-span spatial main cable suspension bridges,the differences of natural frequencies gradually increase with the increase of the inclination of the main cables and hangers plane and the relative sag of the mid-span main cable.Under the same dimensionless parameters,for the most values selected in this paper,the bending stiffness of the main cable has a greater influence on the natural frequencies of the plane main cable suspension bridges than on the spatial main cable suspension bridges. |
PDF Full Text Request