Abstract
Form function matrix is created by introducing high order displacement interpolation function in the node. Based on the virtual work principle and dynamic finite element theory, the spatial element stiffness matrix, mass matrix and earthquake mass matrix of a thin-walled box girder having 9 freedom degrees at each node are deduced. The D’Alembert vibration equation is also established. Newmark-β method is used through MATLAB to solve the seismic response of a long-span continuous curved box girder bridge under El-centro seismic waves. Meanwhile the spatial finite element model of the whole bridge is established by ANSYS. The results indicate that the dynamic responses of pier columns exhibit spatiality. The dynamic response of a bridge structure under 2D coupling horizontal seismic excitation is much bigger than that under 1D horizontal seismic excitation. The critical angle of seismic waves is 50° for radial displacement response. Theoretical calculation results are in agreement with the finite element analysis results. The deduced element matrix not only can be used to calculate the seismic response of long-span curved beam bridge structures but also can provide significant references for the structures in vibration response caused by moving traffic.
Highlights
Thin-walled box girder is widely used in bridge structures
Mass matrix and earthquake mass matrix of curved box girder bridge are deduced in present work, vibration characteristics and seismic waves of a long-span curved box girder bridge are solved by eigenvalue function and Newmark-β method, the following conclusions can be draw: 1) The theoretical results present a good agreement with the finite element analysis, which can verify the accuracy and reliability of the deducing element matrix
Satisfied results can be acquired when the curved box girder bridge is meshed with several elements, which shows the high efficiency of the proposed method in the present work
Summary
Thin-walled box girder is widely used in bridge structures. Spatial mechanical behavior of thin-walled box girder is complex, as the fundamental deformation include vertical bending and shear lag [1, 2] under symmetrical loads. For thin-walled curved box girder, coupling deformation prevails in terms of bending, torsion, warping, distortion and shear lag, regardless of whether the load is symmetric or not. Based on the stiffness method, spatial element stiffness matrix and stiffness equation of a straight box girder having 10 freedoms at each node were deduced [5], in which the restricted torsion, distortion and shear lag effect were considered. According to the finite element theory, spatial element stiffness matrix of a thin-walled curved box girder having 14 freedoms at each node was proposed [6], the compression-tension, bending, torsion, warping, distortion and shear lag effect were considered. Based on the generalized coordinate method and energy principle, Chan et al [8] deduced the vibration governed differential equation of curved bridges and provided the explicit stiffness matrix and mass matrix, whilst the distortion and shear lag effect were neglected. Eigenvalue function and Newmark-β method are used through MATLAB to solve the characteristic equation and seismic response of box girder bridges
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