This paper proposes a new cable element and a novel formation procedure of unbalanced loads for cable-stayed bridges. The proposed method couples the effects of geometric nonlinearity and concrete ...

Coupled nonlinear and time-dependent analysis for long span cable-stayed bridges

The determination of the final cable shape under the self-weight of the suspension bridge enables its safe construction and operation. Most existing studies solve the cable shape segment-by-segment in the Lagrangian coordinate system. This paper develops a novel shape finding method for the main cable of suspension bridge using nonlinear finite element approach with Eulerian description. The governing differential equations for a three-dimensional spatial main cable is developed before a one-dimensional linear shape function is introduced to solve the cable shape utilizing the Newton iteration method. The proposed method can be readily reduced to solve the two-dimensional parallel cable shape. Two iteration layers are required for the proposed method. The shape finding process has no need for the information of the cable material or cross section using the present technique. The commonly used segmental catenary method is compared with the present method using three cases study, i.e., a 1666-m-main-span earth-anchored suspension bridge with 2D parallel and 3D spatial main cables as well as a 300-m-main-span self-anchored suspension bridge with 3D spatial main cables. Numerical studies and iteration results show that the proposed shape finding technique is sufficiently accurate and operationally convenient to achieve the target configuration of the main cable.

A Novel Shape Finding Method for the Main Cable of Suspension Bridge Using Nonlinear Finite Element Approach

The investigation aims to propose a refined model to analyze the parametric resonance under multicable systems such as cable-stayed bridges. Considering the interaction between the adjacent beam portions, the shear difference is applied to modify the vibration equations derived from the multi-degree-of-freedom stiffness method. Furthermore, the difference method is adopted to make the equations more accessible for numerical analysis. The comparison results indicate that the refined model exhibits the key character of parametric resonance and also further verified the simulation methods. The consequences show that the cable will resonate at the fundamental frequency under the support excitation. In particular, when resonance occurs, most of the energy in the subsystem is transferred to the cable, resulting in the resonance amplitude of the beam portion being weakened to some certain extent. Moreover, the global resonance will have a sufficient excitation on the local resonance only when the resonance condition is satisfied.

The Numerical Analysis for Parametric Resonance of Multicable System considering Interaction between Adjacent Beam Portions

To study the complex mechanism of internal resonance in cable-stayed bridges, an integrated dynamic system comprised of four stay cables and a bridge beam was established in this study. Considering the effect of the beam’s shear force and axial force and by applying the finite difference method, the dynamic equations under non-linear boundary conditions were derived based on the D’Alembert principle and Galerkin method. By comparing the results obtained from the dynamic equations with those obtained by a finite element model (FEM), the linearization and non-linear characteristics of the dynamic model were further verified and discussed. The results of the analytical simulation based on the 4th Runge–Kutta Method show that a severe internal resonance is observed once the ratio of local mode and the undamped in-plane vertical natural mode (IVNM) meets 1:1 or 1:2. The conditions regarding the threshold of the initial excitations are also discussed. For the exceptions, the concept of the zero-point of a mode shape (ZPS), where the correlation coefficient in the vertical mode of a certain order equals zero, was proposed and applied in the analysis of the influence of the IVNMs on internal resonance. It is further illustrated that the cables located at the ZPS are not excited by the IVNM of the corresponding order. Additionally, when the excitations act on the ZPS, the IVNMs of the corresponding order are not initially observed in the structure. By using the filtering technology with a zero-phase-shift, the separated vibration signals further revealed that the IVNM becomes re-excited in the global system and thus the re-excited resonance occurs in the cables under specific conditions: an increase in time and excitation value. Three conditions regarding the modal interaction processes of the re-excited resonance were further investigated and discussed. • The dynamic model proposed in this paper is more refined by considering the effect of the beam’s axial force and shear force. • The refined model proposed in this paper can reduce a lot the time of simulation compared to other 3D FE models. • The concept of the zero-point shape of the in-plane vertical mode shape is firstly proposed in this paper. • The mechanism of the re-excited internal resonance is further discussed in this paper. 

Modelling and analysis of the influence of in-plane vertical modes on the internal resonance of cable-stayed bridges

To determine the cable sag effect and solve the rigid connection problem at cable ends of a long-span cable-stayed bridge, a two-node spatial catenary cable element with arbitrary rigid arms is developed. Using the finite rotation formula of the space vector and a differential method, the incremental relation between displacement and force at both ends of the rigid arm is given. Then, explicit expression of the tangent stiffness matrix of the element with arbitrary rigid arms is derived based on the catenary equations. Two numerical examples are provided to verify the validity of the new element. A long-span cable-stayed bridge application model is established, and the cables are simulated using three methods. The results show that the rigid end effect has influence on displacements, bending moments and rigidity and should not be ignored. The catenary cable element with arbitrary rigid arms can be used to simulate the geometric nonlinear mechanical behavior of the cables and can well solve the rigid connection problem.

Zhiqiang Wu

Papers

Nonlinear Analysis of Spatial Cable of Long-Span Cable-Stayed Bridge considering Rigid Connection