Combination of modal responses: A closed-form formulation for rigid response coefficient

Abstract

Recent studies conducted by US Nuclear Regulatory Commission (USNRC) for combining modal responses in a response spectrum method of seismic analysis and design have emphasized that each modal response quantity should be separated into damped-periodic and rigid parts before combining the contributions from different modes. The damped-periodic parts of modal responses are combined using the double-sum equation whereas the rigid parts are combined algebraically. A particular modal response quantity is separated into damped-periodic and rigid parts using the “rigid response coefficient”. The USNRC sponsored study recommends the calculation of rigid response coefficient by either the Lindley–Yow approach or Gupta method. While Lindley–Yow's method has a heuristic basis and gives incorrect results in low frequency region, Gupta's method is based on numerical studies of free-field earthquake motions and works well in the frequency regions of interest for a free-field ground motion. A closed-form solution was developed by Hahn and Valenti in 1997 using a frequency domain approach. With appropriate simplifications, their work can be shown to result in an expression which is very similar to that given by Gupta. It must be noted that the earthquake input to the secondary systems such as piping and equipment is defined by a floor motion and not a free-field ground motion. The frequency characteristics of a floor motion are very different from those of a free-field ground motion. In this paper, we study the validity of existing formulations for the case of floor motions and develop a closed-form solution based on a time domain approach to explain the behavior of rigid response coefficient. The formulation is then used to explain the nature of variation in rigid response coefficient for ground as well as floor motions. It is shown that the proposed formulation and its simplified form gives results that are identical to those evaluated numerically in the complete frequency region of interest.