Bounds for the maximum responses of multi-degree-of-freedom linear systems excited by earthquakes are investigated. The response(shock)-spectrum method yields an exact upper bound directly. A previously proposed semi-empirical lower bound based on simple pulse excitation is examined for a class of four-degree-of-freedom systems with earthquake excitation.
Using these bounds the root-mean-square estimator for the true response is evaluated. It is also shown that the addition of the lower bound to the response spectrum method produces a more valuable structural design tool.