CQC Method (Complete Quadratic Combination)
CQC Method (Complete Quadratic Combination): Theoretical Foundations
What is the CQC Method?
Professor, is the CQC method an improved version of SRSS?
CQC (Complete Quadratic Combination) is a combination method that considers the correlation between modes proposed by Der Kiureghian (1981). It is a superior alternative to SRSS.
$\rho_{ij}$ is the modal correlation coefficient.
Modal Correlation Coefficient
$\rho_{ij}$ is given by Der Kiureghian's formula:
$r = \omega_j / \omega_i$.
If $r = 1$ (same frequency) then $\rho = 1$ (perfect correlation), and if $r$ diverges then $\rho \to 0$ (uncorrelated), right?
If the modes are sufficiently separated, then $\rho_{ij} \to 0$ and CQC degenerates to SRSS. In other words, CQC encompasses SRSS.
Difference Between SRSS and CQC
When there are closely spaced modes, the CQC result can be 10-30% larger than SRSS. This is because SRSS ignores the positive correlation between modes, thus underestimating the contribution of closely spaced modes.
Summary
Key Points:
- $R = \sqrt{\sum \sum \rho_{ij} R_i R_j}$ — Complete combination including inter-modal correlation
- $\rho_{ij}$ — Der Kiureghian's formula. Depends on frequency ratio and damping
- CQC is a superior alternative to SRSS — Degenerates to SRSS if modes are well separated
- 10-30% larger than SRSS for closely spaced modes — SRSS is non-conservative
- Current design codes recommend CQC — Eurocode 8, ASCE 7
CQC emerged in 1981 as the "accurate version" of SRSS
The CQC (Complete Quadratic Combination) method was proposed in the 1981 paper "A Replacement for the SRSS Method in Seismic Analysis" by E.L. Wilson et al. (UC Berkeley). The paper demonstrated that the SRSS method ignores correlation between closely spaced natural frequencies, leading to errors, and introduced a quadratic combination formula using the correlation coefficient ρij. It provides particularly high accuracy for irregular buildings on soft ground and reactor buildings where closely spaced modes are problematic.
Computational Methods for the CQC Method (Complete Quadratic Combination)
CQC Implementation
CQC is a standard option in all FEM solvers:
- Nastran: *RESPONSE SPECTRUM, CQC
- Abaqus: *RESPONSE SPECTRUM, COMBINATION=CQC
- Ansys: Switch from SRSS to CQC
- ETABS/SAP2000: CQC is the default
If we set CQC as the default, we don't have to worry about SRSS issues, right?
Exactly. Always using CQC is the safest approach. If the modes are separated, it gives the same result as SRSS, so there is no downside to using CQC.
Summary
Calculating the correlation coefficient requires modal damping ratios
The CQC method's correlation coefficient ρij is calculated from the natural frequency ratio βij = ωj/ωi and the damping ratios ζi, ζj of both modes. For a damping ratio ζ = 5% (standard for buildings), when the frequency ratio is within 1.1 (within 10% difference), the correlation coefficient ρ > 0.1 and cannot be ignored. The maximum difference from SRSS (which assumes ρ=0) occurs when βij=1 (perfect match) and ρij reaches its maximum value of 1, resulting in a response value difference of √2 = 1.41 times.
CQC Method (Complete Quadratic Combination) in Practice
CQC in Practice
CQC is the standard combination method in current seismic design.
Practical Checklist
So if CQC results are the same as SRSS, it also confirms "no closely spaced modes", right?
Comparing CQC and SRSS can also be used as a tool to check for the presence of inter-modal correlation.
In some countries, CQC is a legal requirement for reactor buildings
The US NRC Regulatory Guide RG 1.92 (revised 2006) recommends (effectively mandates) the CQC method for structures with closely spaced modes (within 10% natural frequency difference). Japan's Nuclear Regulation Authority has similar guidelines, and CQC is standard for seismic analysis of all BWR and PWR nuclear power plants in the country. SRSS use is permitted only when it can be proven that modal spacing is sufficiently wide (>10% difference).
CQC Method (Complete Quadratic Combination): Software & Solver Comparison
CQC Tools
All FEM solvers have standard CQC support. Building design software (ETABS, SAP2000, MIDAS Gen) has CQC as the default.
Selection Guide
SAP2000 and ETABS are the de facto implementations of CQC
The software most frequently using CQC for response spectrum analysis in building/civil structures is CSI's (Computers and Structures, Inc.) SAP2000 and ETABS. Their origin is SAP (Structural Analysis Program) developed in the 1970s by UC Berkeley's Professor E.L. Wilson, and due to Wilson's involvement in founding CSI, their CQC implementation is particularly strong. In nuclear power, NASAP/NRC's PIPESYS/PRIME are used as dedicated tools.
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