Improved time integration for pseudodynamic tests

Abstract

Converting the second-order differential equation to a first-order equation by integrating it with respect to time once as the governing equation of motion for a structural system can be very promising in the pseudodynamic testing. This was originally found and developed by Chang.\ The application of this time-integration technique to the Newmark explicit method is implimented and investigated in this paper. The main advantages of using the integral form of Newmark explicit method instead of the commonly used Newmark explicit method in a pseudodynamic test are: a less-error propagation effect, a better capability in capturing the rapid changes of dynamic loading and in eliminating the adverse linearization errors. All these improvements have been verified by theoretical studies and experimental tests. Consequently, for a same time step this time-integration technique may result in less-error propagation and achieve more accurate test results than applying the original form of Newmark explicit method in a pseudodynamic test due to these significant improvements. Thus, the incorporation of this proposed time-integration technique into the direct integration method for pseudodynamic testings is strongly recommended.

An overview of seismic hybrid testing of engineering structures

“…In explicit/implicit time integration methods the fundamental second order differential equation of motion is solved, however an approach solving the integrated first order form of the equation of motion has been implemented by [79] & [80]. Chang et al, [79] first proposed the scheme that is derived from Chen and Robinson [81], in which an incremental form of the explicit Newmark method is integrated.…”

“…Chang et al, [79] first proposed the scheme that is derived from Chen and Robinson [81], in which an incremental form of the explicit Newmark method is integrated. The equation of motion is now solved in terms of change in velocity rather than acceleration [80].…”

“…The equation of motion is now solved in terms of change in velocity rather than acceleration [80]. Using direct integration for PsD testing, cumulative errors may become large if the rapid variation of resistance of the test system is not captured accurately due to non-linear behaviour [82]. However, using the first order solution method errors resulting from the assumption that the structural properties remaining constant throughout a time step (linearisation error) are eliminated by the integration of the restoring force, while the sharp characteristics of the seismic loading are smoothed out by the integration of the applied force [82].…”

“…Using direct integration for PsD testing, cumulative errors may become large if the rapid variation of resistance of the test system is not captured accurately due to non-linear behaviour [82]. However, using the first order solution method errors resulting from the assumption that the structural properties remaining constant throughout a time step (linearisation error) are eliminated by the integration of the restoring force, while the sharp characteristics of the seismic loading are smoothed out by the integration of the applied force [82]. The result is an algorithm which is capable of picking up the effects of rapidly varying excitation forces and stiffness changes, while at the same time displaying improved error propagation characteristics over the standard Newmark explicit representation.…”

Nonlinear evaluations of unconditionally stable explicit algorithms

“…To accomplish this, two explicit pseudodynamic algorithms with unconditional stability for linear elastic systems have been developed (Chang, 2002 and2007). These two algorithms can be implemented as a general explicit pseudodynamic algorithm (Newmark, 1959;Shing and Mahin, 1987a, b;Chang, 1996;1997;2000;2001;and 2002;Chang et al 1998). There is no need to solve any implicit systems or involve any iterative procedures in each time step, which is often required in an implicit pseudodynamic algorithms (Nakashima et al, 1990;Shing et al, 1991;Chang and Mahin, 1993;Thewalt and Mahin, 1995).…”