Comparison of Modal Superposition Methods for the Analytical Solution to Moving Load Problems

Report No: 95-R6

Published in 1994

About the report:

The response of bridge structures to moving loads is investigated using modal superposition methods. Two distinct modal superposition methods are available: the mode displacement method and the mode-acceleration method. While the mode-displacement method is widely used in the analysis of both continuous and discrete systems, the mode acceleration method is typically given for discrete systems only. This report introduces general operator notation as a way to extend the mode acceleration method to the analysis of arbitrary continuous or discrete structural models and shows that the formulation is available for arbitrary self-adjoint systems. The problem of a damped and undamped beam subjected to a concentrated moving load is considered. The displacement, shear force, and bending moment solutions are formulated to accommodate any time-dependent moving force, including randomly varying vehicular loads. While both mode-displacement and mode-acceleration methods provide reasonable displacement responses, the mode-acceleration method provides superior moment and shear estimates in the vicinity of the moving load at a given level of approximation because of its ability to directly incorporate the pseudo-static response of the system into the solution. This insight, which is readily applicable to the study of transportation structures, suggests that mode-acceleration solution techniques can significantly reduce computational time and labor when modeling highway bridge behavior.

Disclaimer Statement:The contents of this report reflect the views of the author(s), who is responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the Virginia Department of Transportation, the Commonwealth Transportation Board, or the Federal Highway Administration. This report does not constitute a standard, specification, or regulation. Any inclusion of manufacturer names, trade names, or trademarks is for identification purposes only and is not to be considered an endorsement.


  • Thomas Thaxton Baber, Peter J. Massarelli 

Last updated: December 18, 2023

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