The Earthquake Engineering Online Archive

Analysis and design of numerical integration methods in structural dynamics

Hilber, Hans M.

UCB/EERC-76/29, Earthquake Engineering Research Center, University of California, Berkeley, 1976-11, 102 pages (555/H521/1976)

The objective of the present work is to develop one-step methods for the integration of the equations of structural dynamics which (a) are unconditionally stable, (b) have an order of accuracy not less than two, and (c) possess numerical dissipation which can be controlled by a parameter other than the time-step size. In particular, no numerical dissipation must be included. Four new families of algorithms are discussed from this point of view, and compared with algorithms, such as the Newmark, Wilson and Houbolt methods, which are commonly used in structural dynamics and which do not achieve these requirements. The algorithms introduced first are derived from the Newmark formulas by adding step-size-dependent viscous damping terms in the discrete equations of motion. The analysis reveals that the artificial viscosity of these schemes does not effectively filter out undesirable high-frequency modes. In fact, it is shown that the qualitative behavior of artificial and real viscous damping terms is essentially the same, inasmuch as both are practically ineffective in high-frequency modes. Based on this experience a one-parameter family of algorithms is developed with properties (a) to (c). The third family of algorithms discussed in this work is based on equilibrium collocation and can be viewed as a generalization of the Wilson methods. Finally, a set of higher-order algorithms is analyzed.

Available online: http://nisee.berkeley.edu/documents/EERC/EERC-76-29.pdf (2 MB)