Mathematical bases of both modal response analysis in the time and
frequency domains, and the method of computing upper bounds of seismic
responses by using incomplete eigenvalue solutions, are reviewed.
Theoretical justifications are given for the application of those
upper bounds to piping analysis. The so-called left-out mass method,
total mass method, residual load method, modal acceleration method,
left-out force method, rigid response method, and the equivalent
static analysis method, are identified. Computing costs, accuracy
of solutions, and validity of certain methods of modal response combination
associated with a complete modal analysis and modified incomplete
modal representation of large systems, are discussed. A method of
computing an upper bound of spatial response combination is introduced.
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