Nonlinear dynamic response of reinforced concrete structures using sets of eigenvectors

A computational procedure is presented for predicting the nonlinear dynamic response of planer frame R/C structures. In general, the nonlinear response may be computed by direct integration of the equations of motion or by mode superposition. With a specification of the initial conditions, the motion of nonlinear systems can be integrated to produce the response time history. However, the computational effort involved in applying these techniques which requires an update of the instantaneous stiffness can be quite time consuming. The cost and the amount of effort of a nonlinear dynamic analysis using the full system of equations may be prohibitive and practically unfeasible except on powerful computers unless a procedure to reduce the number of degrees of freedom is used. Thus an efficient dynamic analysis procedure which does not require the calculation of global stiffness at each time step is proposed.; A method for speeding the integration process is developed; herein it will be called the prediction procedure. Integration is performed using the normal coordinates, i.e., eigen quantities. In conventional nonlinear normal coordinate analyses it is necessary to compute a new set of eigenvectors in each time step. The computation of eigenvectors can be exhausting so it is prudent to devise a procedure for reducing the number of times it is required. The procedure which was selected uses a limited number of sets of predetermined eigenvectors. The sets of vectors are predicted to accurately represent the normal coordinate properties of the nonlinear structures. They are obtained using stiffness states of the structure which occur when it is deformed into the nonlinear range. It is convenient to deform the structure into these nonlinear states using incremental static analysis. Therefore the dynamic time history analysis is performed with sets of eigenvectors which are obtained in a so called static phase. The set of eigenvalues and eigenvectors which can closely represent the dynamic stiffness state at each time step is chosen by comparing summed differences of stiffnesses between the static and dynamic states. This procedure will reduce the considerable amount of numerical effort when compared with the general step-by-step methods or with the classical modal superposition method. The favorable accuracy and the computational savings which accure from its use indicate that the prediction procedure is a useful method.