Time-Domain Response of Simple Elastic Systems to Seismic Motion: Error Assessment and Engineering Implications

Abstract eng:
Time-stepping methods and/or direct evaluation of Duhamel’s integral are being increasingly used in earthquake engineering to solve the equations of motion for discrete and continuous systems, due to affordable computational costs, applicability to linear and non linear oscillators and permanent growth of the world data base of earthquake acelerogramas. Yet, when applying the above methods, one must be aware that all integration schemes, regardless of their stability and convergence characteristics, act as digital recursive filters that distort unevenly the frequency content of the input, and this may result in a misleading response of the system. Furthermore, in practice, the initial conditions for the computation of the response, in the case of analog excitations that have lost the initial portion of motion, are unknown and assumed to be zero for calculation purposes. This assumption, rigorously speaking, is only valid for high-frequency systems, (e.g., rigid structures), and leads to unconservative designs for long-period systems if the initial condition effects are not properly taken into account. In this paper a rational criterion is presented to rank the accuracy of various commonly used integration procedures for the dynamics of soil and/or structural systems. A few examples are shown to compare the performance of the methods and their sensitivity to the factors controlling the integration error, namely, the amplitude of the time step, the variation of the motion values between sampling points, and the dynamic properties of the vibrating system. The above question is of primary interest for short-to-intermediate period elastic systems, which are governed by the transient phase of the seismic response. For long period systems, which are governed mainly by the free-vibration part of the response, the study clearly shows large magnifications of the computed “non resting” response spectra with respect to the conventional calculations.

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