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Extensive studies have been carried out on the behaviour of structures under

multicomponent earthquake excitations. These studies use either elastic modal response

analysis [Wilson and Button (1981), Smeby and Der Kiureghian (1985), Hisada et al (1987),

Ger and Cheng (1990), Wilson et al (1995), López and Torres (1997), Menun and Der

Kiureghian (1998), López et al (2000) and (2004)] or have probabilistic bases [Rosenblueth

unavailability of an inelastic transfer function. Contemporary codes and software follow these

studies and evaluate the structural combined response, RC, by combining the orthogonal

structural responses. These structural responses are evaluated by applying the same code

spectrum or equivalent static forces separately to each of the two principal directions of the

structure. On using the assumption of independence of earthquake components; the two

orthogonal responses of the structure are uncorrelated, thus the combined response is computed

either by using the SRSS of the two responses:

RC = (R12 + R22)



or by using the γ-percent rule:

R = R + C 1 γR2 (3.3)

where R1 is the peak structural response to the seismic excitation along the structure’s major

principal axis, R2 is the response in the orthogonal direction to R1 and always less or equal to R1 ,

R 2 ≤ R1, and γ is the orthogonal combination factor. The orthogonal combination factor, γ,

generally has a value of 30-percent as in many codes, such as the ISO 3010 (1988) (E), UBC

(1997), FEMA 350 (2000), Eurocode 8 (2002) and NZS 1170.5 (2004). However, fewer codes

use percentages for γ as high as 40-percent such ASCE (1986) and ATC (1996). Figure 3-5 shows the values of (1+γ) for 30-, 40-percent and SRSS methods, where equation (3.2) is reformulated using R2 as some ratio of R1 (b):

= bR 0 ≤ b ≤ 1 (3.4)

R2 1

The spectrum density ratio (b) depends on the transfer functions between the two

horizontal excitations and the particular response under consideration zi(t), i = x, y, z. The ratio b

is defined as:

b = Sa(T ) / Sn2 a(T ) n1 (3.5)

where Sa(Tn2) is the spectral acceleration at the natural period of principal direction 2 of the

considered structure and likewise Sa(T )n1 is for direction 1. Hence, equation (3.2) is reformulated as:

Thus, the 30- and 40-percent rules can be considered as a further simplification of the

SRSS rule, which was originated by Newmark (1975) with γ = 40-percent that was later reduced by Veletsos and Nair (1975) to γ = 30-percent. Rosenblueth and Contreras (1977) studied the problem stochastically and found out that when using γ = 30-percent, the maximum errors on both safe and unsafe sides become minimal. They reported that an exception to using γ = 30-percent is chimneys and slender tall tower structures where the more appropriate value of γ is 50-percent. This 50-percent value is unjustifiable as it exceeds the resultant value however it may account for

anticipated high torsional effects. Moreover, Wilson et al (1995) reported that for axisymmetric

structures or components, where the stresses are not collinear, or when both orthogonal responses

are similar, b ≈ 1; the 40-percent rule or preferably the SRSS should be used.

Clough and Penzien (1993, page 658) examined the difference between the SRSS and 30-

percent rules and showed that for b = 0.85, the 30-percent rule is a conservative linear

approximation to the SRSS rule with a maximum difference of about 5%. This can be seen on

inspecting Figure 3-5 where the SRSS curve intersects with the 30-percent line at b = 0.83. This

favourable comparison holds only when the orthogonal excitations are directed along the

structure’s principal axes and are uncorrelated, with the weaker component having 85% or less

of the intensity of the stronger one. In general, this assumption may not be satisfied and the error

associated with the 30-percent rule can be substantially higher. Likewise and on inspecting

Figure 3-5, when b = 0.98 the SRSS curve intersects with the 40-percent line. Thus, the 40-

percent rule is inherently more conservative than the 30-percent rule, however, associated errors

in 30-percent rule can be prohibitive in case of correlated responses.

On considering that earthquake components are correlated, in particular the two

horizontal ones, the SRSS and γ-percent rules become invalidated. Hence, other methods were developed that incorporates component correlations in their formulations such as CQC, Wilson

et al (1981). Many methods followed and developed to counteract shortcomings of previous

To tackle the apparent problem of combining multicomponent response and identifying

critical incidence angles (θcr), Wilson et al (1995) developed a method that is an extension to the

study of Wilson and Button (1982). It incorporates the two orthogonal responses to yield the

maximum orthogonal response of the considered member at some structural incidence angle θ. Further, it employs the response spectrum method as its analysis tool and assumes a full 1:1

correlation between the two orthogonal seismic inputs. It pre-assumes that the minor spectrum

S2 is some fraction of the major input spectrum S1, S = bS2 1; where 0 < b < 1. Also, Wilson’s

method assumes the stability of the incidence angle of the earthquake resultant over a finite

period of time, as per Penzien and Watabe (1975), around the time of maximum ground

acceleration. However, Wilson et al (1995) formulations were incorrect due to their lacking of a

term that represents the cross-correlation between the contributions of responses R1 and R2, (R12),

as reported by López and Torres (1997).

Menun and Der Kiureghian (1998) redeveloped the CQC3 combination method, firstly

introduced by Smeby and Der Kiureghian (1985), incorporating Wilson et al (1995) but

accounting for component correlation in its cross terms, which provided an efficient simple

procedure to combine responses and identify the critical incidence angle (θcr). Recent

reformulation of CQC3 method incorporates again Penzien and Watabe (1975) to evaluate the

maximum combined response out of two modal spectrum analyses, each run in one of the

principal horizontal axes of the studied structure. This method was further refined by López et al

(2000) and (2004). Nonetheless, all available formulations would integrate a maximum of 40-

odd-percent of the conjugate orthogonal response, γmax ≈ 0.414; albeit they predominantly

provide lesser combination ratios. Therefore, verification of the anticipated combination ratios