CQC法（完全二次合成法）
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, based on the assumption that "acceleration can be ignored because the force is applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, which is the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is not correct. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior such as plasticity or creep, constitutive law extensions are needed
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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.
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix each iteration. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Lower cost per iteration but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies the full load not all at once but in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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).
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work". If the mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling the real-world constraint conditions is actually the most important step in the entire analysis.
Software 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.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types required for the CQC method? For example, in fluids, the presence of LES support; in structures, the ability to handle contact/large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
- "How far will it be extended?": Selection considering future expansion of analysis scale (HPC support), expansion to other departments, and integration with other tools leads to long-term cost reduction.
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