Dynamic Analysis Of Structures Based On Spectral Element Method

Numerical simulation software, which has been widely used in Civil Engineering, is based on all kinds of discrete numerical methods, such as the finite element method (FEM), the finite difference method (FDM), the meshless Method, etc. All these numerical methods have a common characteristic. That is the structure must be meshed before analysis. Elements or nodes meshed form structures are necessary in the analysis process of these numerical methods. The calculation accuracy and efficiency will decrease greatly because of the discrete mesh in these numerical methods, especially for structural dynamic analysis. Sometimes, these discrete numerical methods may not converge or may be failure because mesh generation is not simple under certain circumstances. The spectral element method (SEM), which is based on the theory of continuum mechanics, is a high precise method in structural dynamics. Only one spectral element need be placed between any two joints for structures with uniform area. Conventional SEM can be applied only to structures subjected to concentrated loads applied on nodes. Furthemore, there are few literatures about dynamic analysis for structures under distributed loads and other complex loads. In this dissertation, the following aspects are studied by theoretical and numerical analysis, some important results and conclusions have been achieved as follows:(1) The spectral stiffness matrices of axial vibration rod, torsional vibration rod and flexural vibration Bernoulli-Euler beam were derived from the vibration differential equations of these members. Both internal viscoelastic damping and external viscous damping were considered in the vibration equations. The derived results indicate that the damping effect in SEM can be easily considered by modifying the wave number in the spectral stiffness matrix. The relationship of damping coefficients in SEM and classic Rayleigh damping coefficients in FEM was obtained in the derivation process. The space truss, multi-span continuous beam and space frame structures were analyzed. The SEM has been proved to be an efficient method to analyze the dynamic responses of structures while the number of elements can greatly decrease.(2) The dynamic analysis of structures under distributed loads using Laplace-based SEM was proposed. The distributed loads applied on structures were equivalent to concentrated nodal forces using two different methods. The first method is based on the principle of linear superposition and the other is based on the principle of virtual work. The computational expressions were derived in both two methods. While the equivalent forces were obtained, the dynamic responses of structures can be calculated without any difficulty by use of traditional SEM which solved the vibration problem of structures under concentrated forces. The analytical expressions of equivalent nodal forces under uniform distributed dynamic loads, triangle distributed dynamic loads and trapezium distributed dynamic loads were derived. The equivalent nodal forces of rod subjected to uniform axial dynamic loads and torsional dynamic loads were also presented. By comparison of the numerical results obtained from SEM and those from FEM, the ability and effectiveness of SEM under distributed dynamic loads have been proved.(3) The seismic load applied on structures could be considered as inertia force of structures. Therefore, for homogeneous structures, the seismic loads were uniform distributed loads acted on structures. In other words, the seismic loads can also be equivalent to concentrated forces applied on nodes based on the principle of virtual work, which is the same way as the uniform distributed loads. The equivalent nodal forces got form adjacent elements were superimposed at the same node. The total seismic equivalent nodal forces were the sum of the equivalent nodal forces obtained form adjacent elements. The calculated total seismic equivalent forces are then used to analyze seismic responses of structures. To evaluate the accuracy of SEM, the dynamic responses of space truss and frame structures under seismic load are analyzed. It has been found that the SEM provides good dynamic results under seismic load by using the equivalent forces.(4) The traditional SEM is limited to structures subjected to loads with fixed positions. An extended SEM for dynamic analysis of bridges subjected to moving loads was proposed here. The Dirac function was introduced to simulate the moving loads applied on beam bridge. The moving loads in time domain were transformed into frequency domain by use of Laplcace tranform. The moving loads in frequency were then equivalent to fixed concentrated nodal forces by integrating the shape function in SEM based on the principle of virtual work. These equivalent nodal forces were used to calculate the dynamic responses of bridges in SEM. No matter how many moving loads and their directions acted on the continuous beam bridge, their equivalent nodal forces could be simply superimposed using linear superposition principle at the end of bridges. Only one spectral element need to be placed between any two joints in SEM and this avoid the element meshing in other discrete numerical methods. The numerical results calculated by SEM were compared with those obtained from FEM and the precise time integration method. It has been shown that the present extended SEM provides very high precision results while high efficiency could be obtained.