Modal Combination in
Response Spectrum Analysis
A complete student guide to the CQC method, SRSS method, and the special treatment of closely-spaced modes — with interactive tools for hands-on practice.
Key Definitions (IS 1893 Clause 3)
Before diving into modal combination, it is essential to understand the foundational vocabulary that IS 1893 (Part 1) : 2016 establishes. These terms appear repeatedly in dynamic analysis.
| Term | Symbol | Definition (simplified) | Clause |
|---|---|---|---|
| Closely-Spaced Modes | — |
Natural modes whose frequencies differ by ≤ 10% of the lower frequency. These need special combination rules because they are correlated. | Cl 3.1 |
| Modal Mass | Mₖ |
The effective portion of total seismic mass that participates in mode k. It determines how much a mode contributes to the response. | Cl 3.14 |
| Modal Participation Factor | Pₖ |
How much structural mode k contributes to the overall oscillation during earthquake. Depends on mode shape scaling. | Cl 3.15 |
| Mode Shape Coefficient | φᵢₖ |
The deformation at degree-of-freedom i when the structure vibrates in mode k. Forms the mode shape matrix [Φ]. | Cl 3.17 |
| Natural Period | Tₖ |
Time (in seconds) for one complete cycle in mode k. The fundamental period T₁ is the longest period. | Cl 3.18 |
| Response Spectrum | Sₐ/g |
A graph of maximum acceleration for SDOF systems vs. natural period, at a given damping ratio. The design spectrum is a smoothed version. | Cl 3.22 |
| Cross-Modal Correlation | ρᵢⱼ |
A coefficient (between 0 and 1) that measures how statistically correlated two modes i and j are. ρ = 1 for identical modes; ρ ≈ 0 for widely separated modes. | Cl 7.7.5.3 |
| Peak Response Quantity | λ |
The combined peak response — displacement, shear, moment, etc. — after combining all modal responses via CQC or SRSS rules. | Cl 5 |
"Closely-spaced modes of a structure are those of the natural modes of oscillation whose natural frequencies differ from each other by 10 percent or less of the lower frequency." This is the exact threshold you must check before choosing your combination method.
Why Can't We Just Add Modal Responses Directly?
In a Response Spectrum Analysis, we compute the peak response in each mode separately. The critical problem is: these peaks do not occur at the same instant in time. So simple algebraic summation would be grossly unconservative or overly conservative.
❌ Direct Sum
λ = λ₁ + λ₂ + λ₃
This assumes all modes peak simultaneously — physically impossible for random ground motion. Severely overestimates response.
✅ SRSS
λ = √(λ₁² + λ₂² + λ₃²)
Assumes modes are statistically independent. Works well when frequencies are well-separated. Misses correlation effects.
✅✅ CQC
λ = √(Σᵢ Σⱼ λᵢ ρᵢⱼ λⱼ)
Accounts for correlation between all mode pairs. Preferred method per IS 1893. Generalises SRSS.
Response Spectrum Analysis — IS 1893 Step-by-Step
Clause 7.7.5 of IS 1893 (Part 1) : 2016 lays out the Response Spectrum Method. Here is the full process presented as a logical flow:
Free Vibration Analysis Cl 7.7.5.1
Perform undamped free vibration analysis of the entire building using the mass matrix [M] and stiffness matrix [K] to extract all Nₘ natural periods Tₖ and mode shape vectors {φₖ}. This is typically done by an eigenvalue solver.
Select Number of Modes Cl 7.7.5.2
Include enough modes so that the total modal mass accounts for ≥ 90% of the total seismic mass. For modes above 33 Hz, apply the missing mass correction. Designers may use a different cutoff if justified by rigorous analysis.
Compute Modal Mass Mₖ and Participation Factor Pₖ Cl 7.7.5.4a,b
For each mode k, calculate how much seismic mass is effectively mobilised and how strongly the mode participates in the response. These are needed to distribute the spectral acceleration to floor forces.
Mₖ = [Σ Wᵢ φᵢₖ]² / (g × Σ Wᵢ φᵢₖ²)
Pₖ = Σ Wᵢ φᵢₖ / Σ Wᵢ φᵢₖ²
Read Spectral Acceleration Aₖ for Each Mode Cl 6.4.2
For each mode k with period Tₖ and damping ζ = 5%, read the design acceleration spectrum Sₐ/g and compute the design horizontal acceleration Aₖ = (Z/2) × (I/R) × (Sₐ/g)ₖ.
Compute Modal Lateral Forces at Each Floor Cl 7.7.5.4c
Peak lateral force at floor i in mode k:
Qᵢₖ = Aₖ × Pₖ × Wᵢ × φᵢₖ
Combine Modes — CQC or SRSS+Closely Spaced Cl 7.7.5.3
This is the critical step. Combine the peak responses from all modes using either the CQC method (recommended) or the SRSS method with special treatment for closely-spaced modes. See the next sections for detail.
Apply Torsion Provisions Cl 7.8
Apply design forces at the displaced centre of mass to account for eccentricity between centre of mass and centre of resistance. Design eccentricity = max(1.5eₛ + 0.05b, eₛ − 0.05b).
Modal Combination Methods — IS 1893 Clause 7.7.5.3
IS 1893 (Part 1) : 2016 permits two approaches for combining modal responses:
| Method | Description | When to Use | IS 1893 Status |
|---|---|---|---|
| CQC Complete Quadratic Combination |
Uses cross-modal correlation coefficients ρᵢⱼ. Accounts for correlation between all mode pairs including closely spaced ones. | Always applicable. Especially important when modes are closely spaced. | Method (a) — Preferred |
| SRSS + CSM With Closely Spaced Mode treatment |
SRSS for well-separated modes; arithmetic sum of absolute values for closely spaced mode cluster; then SRSS combination of the cluster with remaining modes. | Alternate method. Must correctly identify and group closely-spaced modes (|Δω/ωᵢ| ≤ 10%). | Method (b) — Alternate |
CQC Method — Complete Quadratic Combination
The CQC method was developed by Wilson, Der Kiureghian and Bayo (1981) and is grounded in random vibration theory. It computes the statistically expected peak response, accounting for the correlation between any pair of modes.
The CQC Formula
λ = √[ Σᵢ Σⱼ λᵢ · ρᵢⱼ · λⱼ ]
λ = combined peak response quantity (shear, moment, displacement…)
λᵢ = response in mode i (with algebraic sign)
λⱼ = response in mode j (with algebraic sign)
ρᵢⱼ = cross-modal correlation coefficient between modes i and j
Nₘ = total number of modes considered
The Cross-Modal Correlation Coefficient ρᵢⱼ
This is the heart of the CQC method. It captures how correlated the peak responses of modes i and j are:
ρᵢⱼ = 8ζ²(1 + β)β^1.5 / [(1 − β²)² + 4ζ²β(1 + β)²]
β = ωⱼ/ωᵢ = frequency ratio (circular frequencies)
ωᵢ = circular natural frequency in mode i = 2π/Tᵢ (rad/s)
ωⱼ = circular natural frequency in mode j = 2π/Tⱼ (rad/s)
Note: ρᵢⱼ = ρⱼᵢ (symmetric). ρᵢᵢ = 1 always (a mode is perfectly correlated with itself).
Behaviour of ρᵢⱼ
λ² = λ₁²ρ₁₁ + λ₂²ρ₂₂ + ... + 2λ₁λ₂ρ₁₂ + 2λ₁λ₃ρ₁₃ + ...
The diagonal terms (ρᵢᵢ = 1) are just the squared modal responses (same as SRSS). The off-diagonal cross-terms (2λᵢλⱼρᵢⱼ) are the extra correction that CQC adds over SRSS — these are significant only when modes are closely spaced.
The CQC Matrix — Expanded View
The CQC formula is equivalent to computing λ = √[{λ}ᵀ [ρ] {λ}], where [ρ] is the correlation matrix:
| ρᵢⱼ | Mode 1 | Mode 2 | Mode 3 |
|---|---|---|---|
| Mode 1 | ρ₁₁ = 1 | ρ₁₂ (0 to 1) | ρ₁₃ (0 to 1) |
| Mode 2 | ρ₂₁ = ρ₁₂ | ρ₂₂ = 1 | ρ₂₃ (0 to 1) |
| Mode 3 | ρ₃₁ = ρ₁₃ | ρ₃₂ = ρ₂₃ | ρ₃₃ = 1 |
The matrix is symmetric. Off-diagonal terms capture modal correlation. For SRSS, the entire off-diagonal is assumed to be zero.
Closely Spaced Modes — Definition & Special Treatment
Why Are Closely Spaced Modes a Problem?
When two or more modes have similar frequencies, they respond to the earthquake at nearly the same time. Their peaks are no longer statistically independent — they are correlated. Using plain SRSS (which assumes independence) underestimates the combined response.
Physically, closely spaced modes often occur in:
- Buildings with nearly symmetric floor plans (similar stiffness in both directions)
- Structures with coupled translational and torsional modes
- Irregular buildings where modes 2 and 3 are very close
- 3D structures with many closely spaced frequencies due to complex geometry
Checking for Closely Spaced Modes
If: (fⱼ − fᵢ) / fᵢ ≤ 0.10 → Modes i and j are CLOSELY SPACED
Equivalently using periods: Tᵢ and Tⱼ are closely spaced if
(1/Tᵢ − 1/Tⱼ) / (1/Tᵢ) ≤ 0.10 (using lower frequency i)
SRSS Method with Closely Spaced Mode Treatment Cl 7.7.5.3(b)
IS 1893 provides this alternative procedure when some modes are closely spaced:
Identify all closely-spaced mode groups
Check all mode pairs. Group modes that are closely spaced together into "clusters". Modes in a cluster are those whose frequencies mutually differ by ≤ 10% of the lowest in the cluster.
Combine within each closely-spaced cluster: Arithmetic Sum of Absolute Values
For each cluster c, the combined peak response is simply the absolute sum of individual modal responses:
λ' = Σ |λc| (sum over closely spaced modes in cluster only)
Combine clusters and remaining well-separated modes using SRSS
Treat each cluster result (λ') as a single modal response, and combine it with the other well-separated modes using SRSS:
λ = √[ (λ')² + Σ(λₖ)² ]
SRSS Method — Pure Square Root of Sum of Squares
λ = √[ Σ (λₖ)² for k = 1 to Nₘ ]
λₖ = peak response in mode k (positive value used)
Nₘ = number of modes considered
SRSS is mathematically derived from the assumption that the response processes of different modes are statistically independent (white noise excitation, uncorrelated modes). It gives the RMS (root mean square) combination of peak modal responses.
SRSS is a special case of CQC where all off-diagonal ρᵢⱼ = 0. This simplification is valid when modes are well-separated. When modes approach each other in frequency, ρᵢⱼ grows significantly and SRSS becomes unconservative.
Worked Example 1 — 3-Mode CQC Storey Shear
Problem Statement
A 5-storey RC building has the following peak storey shear values (at ground storey) from three modes of vibration under RSA:
| Mode k | Period Tₖ (s) | Freq fₖ (Hz) | ωₖ (rad/s) | Peak Shear Vₖ (kN) |
|---|---|---|---|---|
| 1 | 0.80 | 1.25 | 7.854 | 850 kN |
| 2 | 0.28 | 3.57 | 22.44 | 260 kN |
| 3 | 0.16 | 6.25 | 39.27 | 110 kN |
Damping ζ = 0.05. Find combined peak base shear using CQC method.
Step 1: Check for Closely Spaced Modes
| Pair (i,j) | fᵢ (Hz) | fⱼ (Hz) | |fⱼ−fᵢ|/fᵢ | Closely Spaced? |
|---|---|---|---|---|
| (1,2) | 1.25 | 3.57 | 1.86 = 186% | ❌ No |
| (1,3) | 1.25 | 6.25 | 4.00 = 400% | ❌ No |
| (2,3) | 3.57 | 6.25 | 0.75 = 75% | ❌ No |
All modes are well-separated → CQC will have small cross-terms (but we still apply it rigorously)
Step 2: Compute ρᵢⱼ for each pair
Using ρᵢⱼ = 8ζ²(1+β)β^1.5 / [(1−β²)² + 4ζ²β(1+β)²] with ζ = 0.05:
| Pair (i,j) | β = ωⱼ/ωᵢ | ρᵢⱼ (computed) | Note |
|---|---|---|---|
| ρ₁₁, ρ₂₂, ρ₃₃ | β = 1 | 1.000 | Diagonal always = 1 |
| ρ₁₂ = ρ₂₁ | 22.44/7.854 = 2.858 | 0.0037 | Very small — modes well separated |
| ρ₁₃ = ρ₃₁ | 39.27/7.854 = 5.000 | 0.0005 | Negligible |
| ρ₂₃ = ρ₃₂ | 39.27/22.44 = 1.750 | 0.0075 | Very small |
Step 3: Apply CQC Formula
λ² = V₁²·ρ₁₁ + V₂²·ρ₂₂ + V₃²·ρ₃₃
+ 2·V₁·V₂·ρ₁₂ + 2·V₁·V₃·ρ₁₃ + 2·V₂·V₃·ρ₂₃
= 850²×1 + 260²×1 + 110²×1
+ 2×850×260×0.0037 + 2×850×110×0.0005 + 2×260×110×0.0075
= 722500 + 67600 + 12100
+ 1632 + 94 + 429
= 802200 + 2155 = 804355 kN²
λ = √804355 = 897 kN
Worked Example 2 — Closely Spaced Modes (SRSS+CSM Method)
Problem Statement
A structure has 4 modes with the following data. Combine the storey shear at floor 3 using IS 1893 SRSS+CSM method.
| Mode k | Period Tₖ (s) | fₖ (Hz) | Peak Shear V₃ₖ (kN) |
|---|---|---|---|
| 1 | 1.00 | 1.000 | 620 kN |
| 2 | 0.38 | 2.632 | 195 kN |
| 3 | 0.35 | 2.857 | 175 kN |
| 4 | 0.18 | 5.556 | 80 kN |
Check: Modes 2 and 3 have f₂ = 2.632 Hz and f₃ = 2.857 Hz. Ratio: (2.857 − 2.632)/2.632 = 0.0855 = 8.55% < 10% → Closely Spaced!
Step 1: Identify Closely Spaced Mode Cluster
Step 2: Combine within Cluster — Absolute Sum
λ' = |V₂| + |V₃| = 195 + 175 = 370 kN
Step 3: Combine Cluster Result with Remaining Modes via SRSS
λ = √[ (λ')² + V₁² + V₄² ]
= √[ 370² + 620² + 80² ]
= √[ 136900 + 384400 + 6400 ]
= √527700
= 726 kN
Interactive CQC / SRSS+CSM Calculator
Modal Combination Calculator
Enter modal data below. Choose CQC (recommended) or SRSS+CSM. The calculator shows step-by-step working.
Key Symbols — IS 1893 Clause 5
| Symbol | Meaning | Clause |
|---|---|---|
Aₖ | Design horizontal acceleration spectrum value for mode k | Cl 5, 7.7.5.4c |
C | Index for closely-spaced modes | Cl 5 |
Mₖ | Modal mass of mode k | Cl 5, 7.7.5.4a |
Nₘ | Number of modes to be considered | Cl 5, 7.7.5.2 |
Pₖ | Modal participation factor of mode k | Cl 5, 7.7.5.4b |
Qᵢₖ | Design lateral force at floor i in mode k | Cl 7.7.5.4c |
Tₖ | Undamped natural period of mode k (seconds) | Cl 5 |
Vᵢₖ | Shear force in storey i in mode k | Cl 5 |
Wᵢ | Seismic weight of floor i | Cl 5 |
φᵢₖ | Mode shape coefficient at floor i in mode k | Cl 5, 3.17 |
λₖ | Peak response in mode k (with sign) | Cl 7.7.5.3 |
λ' | Peak response due to closely-spaced modes only | Cl 7.7.5.3 |
ρᵢⱼ | CQC cross-modal correlation coefficient for modes i and j | Cl 7.7.5.3 |
ωᵢ | Circular natural frequency in mode i (rad/s) | Cl 7.7.5.3 |
β | Frequency ratio = ωⱼ/ωᵢ (used in ρᵢⱼ formula) | Cl 7.7.5.3 |
ζ | Modal damping ratio = 0.05 (5%) per IS 1893 | Cl 7.7.5.3 |
IS 1893 Clause Reference Summary
Cl 3.1 Closely-Spaced Modes
Natural modes whose frequencies differ by ≤ 10% of the lower frequency are defined as closely-spaced.
Cl 7.7.5 Response Spectrum Method
Overall framework for RSA: use design acceleration spectrum from Cl 6.4.2, or site-specific spectrum.
Cl 7.7.5.1 Natural Modes
Undamped free vibration analysis to obtain all Nₘ natural periods and mode shapes.
Cl 7.7.5.2 Number of Modes
Sum of modal masses ≥ 90% of total seismic mass. Missing mass correction for modes > 33 Hz.
Cl 7.7.5.3 Combination of Modes ⭐
Method (a): CQC — preferred. Method (b): SRSS for well-separated + absolute sum for closely-spaced clusters, then SRSS of all. Damping ζ = 0.05.
Cl 7.7.5.4 Simplified Dynamic Analysis
Lumped mass model for regular buildings. Modal mass, participation factor, lateral forces, storey shears — all computed floor-by-floor.
Self-Test Quiz
Test your understanding of modal combination concepts from IS 1893 : 2016.
Q1. Two modes of a structure have natural frequencies of 2.0 Hz and 2.15 Hz. Are they closely spaced per IS 1893?
Q2. For CQC, what is the value of cross-modal correlation coefficient ρᵢᵢ (i.e., when i = j)?
Q3. Per IS 1893 Clause 7.7.5.2, how much of the total seismic mass must be captured by the selected modes?
Q4. When using the SRSS+CSM method for closely spaced modes, how are the responses within a closely-spaced cluster combined?
Q5. Which of the following is the preferred modal combination method per IS 1893 (Part 1) : 2016?
Key Takeaways
- CQC is the preferred method (IS 1893 Clause 7.7.5.3 Method a). It correctly handles all levels of modal correlation through the coefficient ρᵢⱼ.
- Closely spaced modes are defined as those whose frequencies differ by ≤ 10% of the lower frequency (Clause 3.1). These require special treatment.
- ρᵢᵢ = 1 always; ρᵢⱼ → 0 for well-separated modes; ρᵢⱼ → 1 for identical modes. The formula from IS 1893 uses ζ = 0.05 and β = ωⱼ/ωᵢ.
- CQC = SRSS when all modes are well-separated (all off-diagonal ρᵢⱼ ≈ 0). The difference grows with closely spaced modes.
- At least 90% of total seismic mass must be captured by selected modes (Clause 7.7.5.2). Missing mass correction applies above 33 Hz.
- The SRSS+CSM alternate method uses absolute sum within a closely-spaced cluster, then SRSS to combine with remaining well-separated modes.
- Damping ζ = 0.05 (5%) is specified for use in the CQC formula and design spectrum, irrespective of structural material (Clause 7.2.4).