Numerical Experiments on Time Step Selection in Linear Dynamic Analysis of Structures

ABSTRACT

The complex nature of the exciting force in earthquake engineering applications mandates employing numerical methods to obtain the response of structures to ground motions. Several numerical methods are in use in the field of earthquake engineering and structural dynamics to solve the ordinary differential equation of motion. Previous studies assumed that the appropriate time step for numerical schemes can be selected as a constant fraction of natural period of the system disregarding the period range. This study is concerned with assessing, through a comprehensive numerical experiment; the numerical error of the most commonly used schemes in structural dynamics applications and assessing the appropriateness of each method for different excitations and damping conditions using different natural period ranges. The current investigation involved testing the central difference method, Newmark, HHT, and HHT1 methods. The four methods were tested for linear single degree of freedom (SDOF) oscillator. Three different exciting forces were used to test the schemes; half-sine pulse wave, harmonic force, and actual ground motion record. Three natural periods were used to conduct the experiment, representing short, medium, and long period systems including the resonant condition. Three structural damping ratios, representing lightly, moderately, and heavily damped systems, were used to assess the structural damping effect on the accuracy associated with selected time steps. A new displacement error norm was introduced to define the accuracy of numerical schemes. The results of this investigation indicated that the commonly used assumption of time step as a constant fraction of natural period of the system, even for SDOF linear oscillator, disregarding the period range could result in significant numerical errors. The study also confirmed the significant effect of structural damping ratio on the accuracy of the numerical schemes. The paper also presents a recommendation matrix for the most appropriate time step for each scheme for different applications and structural conditions. The results of the current investigation are strictly limited to linear systems.

KEYWORDS:

• Numerical methods
• Seismic Analysis
• HHT
• Newmark Method
• Equation of Motion
• Dynamic Analysis

Acknowledgments

The invaluable revision and contribution of Prof. Jon Wilkening, Professor of Applied Mathematics and Vice Chair of Department of Mathematics, University of California, Berkeley in developing this study are much appreciated.