SUMMARYThis paper aims to develop an improved understanding of the critical response of structures to multicomponent seismic motion characterized by three uncorrelated components that are deÿned along its principal axes: two horizontal and the vertical component. An explicit formula, convenient for code applications, has been derived to calculate the critical value of structural response to the two principal horizontal components acting along any incident angle with respect to the structural axes, and the vertical component of ground motion. The critical response is deÿned as the largest value of response for all possible incident angles. The ratio rcr=rsrss between the critical value of response and the SRSS response-corresponding to the principal components of ground acceleration applied along the structure axes-is shown to depend on three dimensionless parameters: the spectrum intensity ratio between the two principal components of horizontal ground motion characterized by design spectra A(Tn) and A(Tn); the correlation coe cient of responses rx and ry due to design spectrum A(Tn) applied in the x-and y-directions, respectively; and ÿ = ry=rx. It is demonstrated that the ratio rcr=rsrss is bounded by 1 and (2=1 + 2 ). Thus the largest value of the ratio is √ 2, 1.26, 1.13 and 1.08 for = 0, 0.5, 0.75 and 0.85, respectively. This implies that the critical response never exceeds √ 2 times the result of the SRSS analysis, and this ratio is about 1.13 for typical values of , say 0.75. The correlation coe cient depends on the structural properties but is always bounded between −1 and 1. For a ÿxed value of , the ratio rcr=rsrss is largest if ÿ = 1 and = ± 1. The parametric variations presented for one-storey buildings indicate that this condition can be satisÿed by axial forces in columns of symmetric-plan buildings or can be approximated by lateral displacements in resisting elements of unsymmetrical-plan buildings.

The paper in discussion [1] presents an interesting comparison of various combination rules for maximum response calculation under two-component horizontal earthquake motions. The authors should be commended for their contribution, in particular for the error bounds they have presented for each of the combination rules examined.The purpose of this discussion is to bring to the authors' attention some past pertinent work by the writer [2], who addressed not the same but a similar problem 20 years ago, and also to suggest one way for making the critical response approach compatible with current codes and design practices. Apparently that publication has escaped the attention of the authors.In that work, eight modal-spatial combination rules were examined, including those by the authors, with the exception of the CQC rule, which had just appeared in a 1979 Berkeley report. However, the double summation rule, very similar to the CQC, had been considered, but gave results very close to the SRSS rule, since the structures examined had no close lower modes.While the present paper addresses the problem of maximum value of any response parameter under two-component horizontal motions acting along the worst possible direction for the parameter, Reference [2] deals with the more traditional problem of peak response under three component earthquake motions acting along three principal structural axes. Although maximum member response, considering all possible earthquake incident angles is certainly of practical interest, a structure should not be designed with each of its members sized to such maximum forces, as this would lead to an over-designed but not necessarily safer structure. The structure would be safer if its response remained elastic. However, under design level earthquakes the structure will respond inelastically and its safety margins will be determined by capacity design procedures requiring consistent sets of member forces. Thus, before the useful concept of critical response is introduced into practice, it should be made compatible with capacity design procedures.The comparisons in Reference [2] were carried out using results from time history analyses with 30 real, three-component earthquake motions, acting on three di erent structures. In addition, the problem of maximum stresses, which requires proper combination of maximum member forces, thus introducing a third level of uncertainty beyond the modal and spatial * Correspondence to:

This paper aims to investigate the response spectra characteristics of the principal components of seismic motion, which are required for accurate multicomponent structural analysis. Mean spectra were determined for the three principal uncorrelated acceleration components for an ensemble of 97 earthquake records. The average inclination of the quasi-vertical component from the vertical axis is found to be 11.4°, with a standard deviation of 9.9°. The ratio of the minor and the major quasi-horizontal spectra varies between 0.63 and 0.81, which are lower than the value of 1 commonly used in design codes. Greater differences are found for near-fault motions in the intermediate-period range. The ratio of the quasi-vertical and the major quasi-horizontal spectra varies between 0.34 and 0.69 for far-fault and between 0.3 and 1.33 for near-fault motions, depending on vibration period. Smoothed spectra of the three principal components that can be used in modern multicomponent structural analysis methods are presented.

SUMMARYThis paper presents a response spectrum analysis procedure for the calculation of the maximum structural response to three translational seismic components that may act at any inclination relative to the reference axes of the structure. The formula GCQC3, a generalization of the known CQC3-rule, incorporates the correlation between the seismic components along the axes of the structure and the intensity disparities between them. Contrary to the CQC3-rule where a principal seismic component must be vertical, in the GCQC3-rule all components can have any direction. Besides, the GCQC3-rule is applicable if we impose restrictions to the maximum inclination and=or intensity of a principal seismic component; in this case two components may be quasi-horizontal and the third may be quasi-vertical. This paper demonstrates that the critical responses of the structure, deÿned as the maximum and minimum responses considering all possible directions of incidence of one seismic component, are given by the square root of the maximum and minimum eigenvalues of the response matrix R, of order 3 × 3, deÿned in this paper; the elements of R are established on the basis of the modal responses used in the well-known CQC-rule. The critical responses to the three principal seismic components with arbitrary directions in space are easily calculated by combining the eigenvalues of R and the intensities of those components. The ratio rmax=r SRSS between the maximum response and the SRSS response, the latter being the most unfavourable response to the principal seismic components acting along the axes of the structure, is bounded between 1 and 3 2 a =( 2 a + 2 b + 2 c ), where a¿ b ¿ c are the relative intensities of the three seismic components with identical spectral shape.

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