Nonlinear dynamic analysis of two-dimensional solid elements by modal superposition

A method is proposed for the analysis of nonlinear 2D solid elements subjected to earthquake excitation. The method is based on the transformation of the original equation of motion to a linear equation that governs the dynamic behavior of these elements and the solution of this linear equation by through a modal decomposition. To this end, the nonlinear terms that appear in the original equation of motion are moved to the right-hand side of the equation and considered as equivalent or pseudo forces. As the magnitude of these equivalent or pseudo forces depends on the solution of the equation of motion, an iterative approach is employed to define these forces and solve the linear equations by the modal superposition technique. The resulting modal equations are then solved by means of an integration procedure that is exact for the analysis of systems subject to piecewise linear excitations in order to minimize the solution time. With the intention of minimizing the solution too, use is made of the modal acceleration method to allow for a reduction in the number of modes considered in the analysis but approximately take into account the effect of the truncated modes. The use of load-dependent Lanczos-Ritz vectors as a means to improve the efficiency and accuracy of the method is also investigated. In addition, a scheme based on the properties of the modal participation factors is introduced to decide how many modes need to be included in the analysis to achieve acceptable and reliable results. Along the same lines, an energy balance formulation is introduced to detect accumulated errors during an analysis and avoid thus unnecessary runs, and to serve as a criterion to assess the global accuracy of the solution. Only systems with material nonlinearities and classical damping are considered. A comparative study with three 2D systems is also conducted to verify the accuracy and efficiency of the proposed method. A plate, a dam, and a shear wall with openings under three different earthquake excitations are the systems considered. From the results of this comparative study, it is concluded that the proposed method is reliable and efficient and represents thus a convenient alternative for the analysis of large and complex structure that have a large number of degrees of freedom, undergo extensive nonlinear deformations, and are subjected to strong and long earthquake ground motions.*; *This dissertation is a compound document (contains both a paper copy and a CD as part of the dissertation).