Comparison of Numerical Methods for Dynamic Response Evaluation

Abstract

Response of structure subjected to various types of arbitrary dynamic forces like Pulse, Ramp, Unit Impulse, Earthquake is obtained by employing suitable Numerical Integration Method, since closed form solution of such problems are not evident. Mostly used numerical integration methods includes Central Difference Method, Newmark's Method and Runge-Kutta Method. Apart, other numerical methods like Wilson Theta Method and Recursive Formula are also available. Analyst, having many numerical integration methods, faces issue of utilizing suitable numerical method pertaining to structural system subjected to different types of dynamic loads. The present study is an attempt to access suitability of various numerical methods. Three systems considered for the present study are-(i) Linear Single Degree of Freedom (SDOF) System, (ii) Linear Multi Degree of Freedom (MDOF) System and (iii) Nonlinear Single Degree of Freedom (SDOF) System. Dynamic forces considered are Half-Cycle Sine Pulse, Harmonic Excitation (applied at Mass and Base), Step Force, Ramp Force, Unit Impulsive Force and Earthquake Excitations. Dynamic response of all three systems subjected to above mentioned dynamic loads are evaluated using numerical methods, closed form solution are also obtained wherever possible. Convergence study is carried out for each case considered in order to derive appropriate time step (t) value. Comparison among numerical solution obtained by various numerical methods is carried out. Apart, comparison is also carried out with closed form solution, wherever possible. It has been observed that, 4th Order Runge-Kutta method is more suitable to derive dynamic response of SDOF/MDOF system subjected to Harmonic type of forcing function. However, Newmark Beta Method (Linear acceleration) and Recurrence Formula Method are more useful for deriving dynamic response of systems subjected to arbitrary excitation. It has been noted that all numerical methods are equally good for extracting dynamic response of system subjected to Earthquake excitations. An experimental study is also carried out employing linear SDOF system subjected to harmonic forcing function at base through shake table. A comparison of experimental response to one obtained through various numerical methods shows good agreement. Thus proves suitability of Numerical methods to extract dynamic structural response.