Summary: Foreword - Combination of modal responses -  Combination of spatial component responses - Description of the combination rules - Requirements of the applicable Codes - Conmparison of the combination rules.
 
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Revised 2022-02-26

Foreword

In seismic response analysis two types of combinations are to be considered:
  1. the modal responses combination, for any assigned seismic spatial component; and
  2. the combination of the responses to the seismic spatial components (two horizontal components, orthogonal each other,  and one vertical)
A comprehensive illustration of these combinations is provided in Chapters 3 (Combination of modal responses) and 4 (Response to multi-components of earthquake) of the book “Response Spectrum Method In Seismic Analysis and Design of Structures” by Ajaya Kumar Gupta (see ref. [16.]).
This subject is dealt with in several standards or regulatory guides among which the following are examined herein below:
  1.  U.S. NRC Regulatory Guide 1.92, Revision 3 “Combining Modal Responses and Spatial Components in Seismic Response Analysis”.
  2.  ASCE 4-98, “Seismic Analysis of Safety-Related Nuclear Structures and Commentary”.
  3.  ASCE/SEI 7-10, “Minimum Design Loads for Buildings and Other Structures”.
  4.  Eurocode 8 – Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings – UNI EN 1998-1:2005.
  5. NTC 2008 - Norme tecniche per le costruzioni – D. M. Infrastrutture 14 gennaio 2008.
 

Combination of modal responses

The combination rule to be used for individual modal responses depends upon whether the responses are periodic, rigid or both (see R.G. 1.92 Rev. 3).
Periodic responses are dominant in the region of amplified spectral displacement, amplified spectral velocity and amplified spectral acceleration.
These regions are AB, BC and CD, respectively, in Figure 1 and Figure 2.
The periodic response corresponds to the transient phase in theory of vibrations.
Rigid responses occur in the high-frequency regions of the spectrum (regions EF and FG in Figure 1 and Figure 2).
The rigid response corresponds to the steady-state phase in theory of vibrations.
Both periodic and rigid responses occur in the intermediate region between amplified spectral acceleration and high-frequency (region DE in Figure 1 and Figure 2).
Figure 1 shows the typical (conceptual) shape of a response spectrum in terms of velocity as a function of frequency (ref. to R.G. 1.92).
Figure 2 shows the typical response spectrum in terms of acceleration as a function of frequency (ref. to R.G. 1.92). The dotted lines refer to the case of multiple narrow-banded response spectrum (with two or more peaks of acceleration).
The periodic modal responses of an earthquake are combined with the following rules:
-        SRSS, which stands for Square Root of Sum of Squares;
-        CQC, which stands for Complete Quadratic Combination.
According to the US standards and Eurocode 8 (EC8-1, para. 4.3.3.3.2), the selection of the most appropriate method depends upon the main modal frequencies separation or independence (un-correlation); while the NTC 2008 allows only the CRC method (see para. 7.3.3.1).
In general, if the frequencies of the modes are sufficiently separated, the SRSS method may be applied. The Eurocode uses the words “modes are independent” instead of “not closely spaced”, adopted by the U.S. standards,  what has an equivalent meaning.
If modes with closely spaced frequencies exist, the SRSS method is not applicable and the CQC method is to be used.
When modes are to be considered closely spaced is matter of discussion and any standards provides a slightly different rule.
According to R.G. 1.92, modes are considered closely spaced when:
  1. For critical damping ratios ≤ 2%, the frequencies are within 10% of each other, i.e. given f1  <  f2 , it shall result f2  ≤ 1.1 ∙ f1.
  2. For critical damping ratio > 2%, “the frequencies are within five times the critical damping ratio of each other”, i.e. given 4% damping and f1  <  f2, it shall result f2  ≤  1.20 ∙ f1; or, for 8% damping ratio, it shall result f2  ≤  1.40 ∙ f1.
According to EC8-1, paragraph 4.3.3.3.2, “the response in two vibration modes i and j may be taken as independent of each other, if their periods Ti and Tj satisfy (with Ti  ≤  Tj) the following condition: Ti  ≤  0.9 ∙ Tj. This condition is equivalent to the one adopted by R.G. 1.92 (being 1/1.1 ≈ 0.9), since, using frequencies in lieu of periods, the condition becomes:  fi   ≥  1.1 ∙  fj , where fi  ≡  f2 and fj  ≡  f1 being fi  >  fj . This condition is therefore equivalent to  f2  ≥  1.1  ∙  f1 stipulated by R.G. 1.92.
The EC8-1 definition is more conservative than R.G. 1.92 since does not provide any allowance for the critical damping ratio effect.
The rigid responses and rigid components of responses are combined algebraically.
 

Combination of spatial component responses

The directional effects of an earthquake are combined with the following rules:
  • SRSS or SRSS3 (ref. [11.]);
  • CQC3 (two components) or GCQC3 (three components) (ref. [10.], [11.]);
  • 100%-30%-30% (ASCE 7-10);
  • 100%-40%-40% (R.G. 1.92).
 

Description of the combination rules

As mentioned by Gupta in ref. [16.], page 3-9, the Complete Quadratic Combination (CQC) was proposed by Wilson, Der Kiureghian and Bayo in 1981 with paper ref. [23.]; while the Square Root of the Sum of the Squares (SRSS) method was proposed in 1953 by Goodman, Rosenblueth and Newmark (see page 3-2 of ref. [16.] and full original paper ref. [30.] and [31.]).
The percentage rule was put forward by Newmark and Rosenblueth first. (ref. [30.] and [31.]).
Some design codes (such as UBC 97, Caltrans 90, ASCE 7-10, Eurocode 8 and NTC 2008) for building and bridges use the 100%-30%-30% rule.
Other design codes (such as ASCE 4-86, ATC-32 and RG 1.92) stipulate use of 40% rather than 30%.
 

Requirements of the applicable Codes

R.G. 1.92 Rev. 3 (ref. [1.]) with the Regulatory Position C.2 states that using the 100-40-40 percentage combination rule proposed by Newmark (see ref. [22.]) is acceptable as an alternative to the SRSS method for combination of spatial components.
In the ref. [22.]) paper, Newmark so describes the proposed method:
“Alternatively, one can use the procedure of taking the seismic forces corresponding to 100 percent of the motion in one direction, combined with 40 percent of the motions in the other two orthogonal directions, then adding the absolute values of these, to obtain the maximum resultant forces in a member or at a point in a particular direction, and computing the stresses corresponding to the combined effect. In general, this alternative method is slightly conservative for most cases and is quite adequate since its degree of conservatism is relatively small.“
For response spectrum (RS) analysis, the R.G. 1.92 Rev. 3 argues that the 100-40-40 percent rule “is the only alternative (to SRSS) method for spatial combination that has received any significant attention in the nuclear power industry”. With respect to the SRSS method, the 100-40-40 method “produces higher estimates of maximum response….by as much as 16%, while the maximum under-prediction is 1%.”
Eurocode 8 requirements.
In EN 1998-1:2004 (ref. [24.]), paragraph 4.3.3.3.2, the code requires that, when the main modal responses can be considered to be independent each other, the seismic response is obtained with the SRSS method applied to each mode; but, if the modes cannot be considered independent, the CQC is recommended.
The combination of the effects of the seismic components is discussed in paragraph 4.3.3.5. In paragraph 4.3.3.5.1 point (3) the 100-30 percentage rule is introduced as a valid alternative to the SRSS rule proposed under point (2) above. The 100-30 percentage rule is also proposed in paragraph 4.3.3.5.2 point (4) when the vertical component of the seismic action is considered.
NTC 2008 requirements. (ref. [25.])
The effects of each modes (for a given direction of the seismic action) shall be combined with the CQC rule (see para. 7.3.3.1).
The effects due to the different directions of the seismic action shall be combined with the 100-30-30 percentage rule (see para. 7.3.5).
ASCE 7-10 requirements.
Ref. [26.], para. 12.5.3.a. “Orthogonal Combination Procedure” reads as follows:
“The requirement of Section 12.5.1 is deemed satisfied if members and their foundations are designed for 100 percent of the forces for one direction plus 30 percent of the forces for the perpendicular direction.”. The 100-30 percentage rule is therefore adopted.
Regarding the number of modes, in para. 12.9.1 the standard requires that “The analysis shall include a sufficient number of modes to obtain a combined modal mass participation of at least 90 percent of the actual mass in each of the orthogonal horizontal directions..”
For what regard the mode combination, paragraph 12.9.3 requires the use of SRSS, or the CQC or the CQC-4 (the CQC as modified by ASCE 4), or an approved equivalent approach. The CQC or CQC-4 shall be used when the closely spaced modes are important.
No other rules are specified for non-building structures dealt with in Chapter 15.
ASCE 4-98 requirements (rif. [27.])
Paragraphs 3.2.7.1.2 “Combination of spatial components” and 3.2.7.2 “Combination of Spatial Components for Time History Analysis” adopt the 100-40-40 Percent Rule as an alternative to the SRSS with the (Eq. 3.2-26) and (Eq. 3.2-30) that are identical even though the former referred to the Response Spectrum Analysis and the latter to the Time History Analysis. The equations are written as follows:
R = [± R1  ± 0,4 ∙ R2  ± 0,4 ∙ R3]
or,
R = [± 0,4 ∙ R1  ± R2  ± 0,4 ∙ R3]
or,
R = [± 0,4 ∙ R1  ± 0,4 ∙ R2  ± R3]
[A total of  3 x 23 = 24 combinations are to be considered; my equation has the first sign ± inside the squared brackets and not outside as given in ASCE 4-98].
Paragraph C3.2.7.1.2 “Combination of components” provides a short illustration of the background of the 100-40-40 percent rule adopted in equation 3.3-26 of the standard. “…It is based on the observation that the maximum increase in the resultant for two orthogonal forces occurs when these forces are equal. The maximum value is 1.4 [= √(2)] times one component. As a consequence, it can be shown that 100-40-40 Percent Rule is, in general, more conservative than the SRSS Rule and is a reasonable procedure to use given the basic uncertainties involved…”
[For two components earthquake providing the same response R1 = R2, the combined response is √(2) ∙ R1  ≈ 1.4 ∙ R1; for three seismic components with  R1 = R2 = R3, the combined reponse is √(3) ∙ R1  ≈ 1.73 ∙ R1 < R1 + 0.4 ∙ R2 + 0.4 ∙ R3  = 1.8 ∙ R1].
ASCE 43-05 requirements (rif. [28.])
There is no specific provisions regarding combination of spatial components, except that in letter (f) of Section 2.4 is stated that:
“To be considered statistically independent, the directional correlation coefficients between pairs of records shall not exceed a value of 0.30 (see Definitions in this Standard). Simply shifting the starting time of a given accelerogram does not constitute the establishment of a different accelerogram. If uncoupled response of the structure is expected, then only one time history is required. Then, the seismic analysis for each direction can be performed separately and then combined by the square root of the sum of the squares (SRSS).”
UBC-97 requirements
Section 1633.1 reads as follows:
“The requirement that orthogonal effects be considered may be satisfied by designing such elements for 100 percent of the prescribed design seismic forces in one direction plus 30 percent of  the prescribed design seismic forces in the perpendicular direction.”
As an alternative the SRSS rule can be used.
IBC-2015 requirements
There is no specific requirement.
 

Comparison of combination rules

According to ref. [11.]:
“SRSS3 method is the simplified form of CQC3 method. SRSS3 method would have a rather large error if the natural periods of the structure are close to each other, namely modal correlation coefficients of structure are large....The SRSS method is convenient  to the analysis of structures; … many seismic code adopt it as normal combination method for multiple mode seismic responses. … the results of percentage rules are usually conservative, and in most cases area reasonable.”
Ref. [2.], paper “Multi-Component Demands from Instrumental Data: Assessment of Seismic Provisions”  (by Dionisio Bernal, Lester Silfa and Anshuman Kunwar – Civil and Environmental Engineering Department, Center of Digital Signal Processing, Northeastern University, Boston, MA) deals with SRSS and 30% (or 40%) rules of combination.
The authors argument that “the simultaneous action of the bidirectional moments … required by the SRSS combination is in this case (a column in a structure with orthogonal frames) unreasonably conservative.”
This statement is supported referring to a detailed discussion on this issue developed by Menun and Der Kiureghian in 1998 and 2000.
The percentage 100%-30%-30% rule is also mentioned at page 190 of ref. [13.].
In ref. [16.], Gupta discusses in depth the combination of modal responses (Chapter 3) and the response to multi-components of earthquake (Chapter 4) presenting the SRSS method (published in 1953 as Goodman-Rosenblueth-Newmark rule, see ref. [30.] and [31.]), the CQC method (provided in Der Kiureghian’s equation, see ref. [23.]), and the absolute sum method.
The percentage method is never explicitly cited, but the theoretical bases for it are provided in Chapter 4, where the superposition of the i-modes is provided as linear combination with coefficients Ci that have different values for different modes but can be approximated by the value 0.41 (page 4-15) that, if applied to the second mode, leads to a maximum error of 8%. This discussion refers anyhow to the combination of modes and not to combination of directional actions, even though  chapter 4 deals with “Response to multi-components of earthquake”.
In ref. [17.], page 42, only the SRSS and CQC methods are considered.
In ref. [18.] page 241 the 100%-30% rule is introduced as admitted by EC8-1/2004 [My note: paragraph 5.7.3.4 is mentioned by the author, but really EC8-1 discusses this subject in paragraphs 4.3.3.5.1 and 4.3.3.5.2), accepted also by BSSC 2003 (BSSC = Building Seismic Safety Council) and SEAOC 1999 (SEAOC = Structural Engineers Association of California). According to this book, this approach introduces a maximum error of 4.4% on the safe side and 8.1% on the unsafe side.
The authors point out that this rule for the simple case of a column would lead to 256 combinations because of three element forces (1 axial load and two moments) to be considered with both + or – sign and because of two horizontal seismic actions.
In ref. [19.] Chapter 15, pages 15-12, Wilson discusses the orthogonal effects in spectral analysis if the percentage 100%-30% rule is introduced. According to Wilson “… For structures that are rectangular and have clearly defined principal directions, these percentage rules yield approximately the same results as the SRSS method …”.
It also offers a discussion on the CQC and a comparison of the SRSS method with the 100/30 Rule (pages 15-16).
In ref. [20.] the author points out (see page 14) that in response spectrum analysis the percentage rule is applied for the directional combination only, while SRSS and CQC3 apply to modal combination, too.
He provides the following recommendations (page 30):
-        If the structure is regular, follow 100+30 rule as usual;
-        If the structure is irregular, 100-30 may be conservative for max-rotated RS.
But at page 36 the author states that “… 100+30 Procedure may be unconservative for regular structures…is almost guaranteed to be unconservative for: irregular structures, un-orthogonal lateral systems, ….”
This paper offers also (page 49) a short discussion of the directional combination procedure: 100-30.
Wallace in ref. [21.], slide 59, provides the UBC-97 requirements that as per 1633.1 states that “For some structures (irregular), combination must consider orthogonal effects: 100% of seismic forces in one direction, 30% in the perpendicular direction.” Really, as discussed above, the UBC 97 is formulated in a different way.