SRSS法（二乗和平方根法）
Theory and Physics
What is the SRSS Method?
Professor, what is SRSS?
SRSS (Square Root of Sum of Squares) is a method that combines the maximum response of each mode by taking the square root of the sum of squares.
Why not just add them together?
The maximum responses of each mode do not occur simultaneously. When mode 1 is at its peak, mode 2 might be at zero. The combination of statistically uncorrelated (independent) random variables is performed using SRSS.
SRSS Assumptions and Limitations
SRSS Assumption: Modes are statistically uncorrelated. This holds true when the natural frequencies of the modes are sufficiently separated.
For closely spaced modes where $f_{i+1} / f_i < 1.1$, there is correlation between modes, and SRSS can sometimes yield non-conservative results. In such cases, the CQC method is used.
Summary
Key Points:
- $R = \sqrt{\sum R_i^2}$ — Combination of uncorrelated modes
- Accurate when modes are sufficiently separated — $f_{i+1}/f_i > 1.2$ is a guideline
- Non-conservative for closely spaced modes → Use CQC method
- Historically widely used — Currently, CQC is often recommended
SRSS Spread from Nuclear Power to General Seismic Design
The SRSS (Square Root of the Sum of the Squares) method originated from a proposal by E.L. Rosenblueth in 1951 that "the worst case probabilistically is the square root of the sum of squares of each mode." It was adopted in the 1960s for multi-mode vibration analysis of NASA space equipment, and in the 1970s, through seismic design standards for nuclear power facilities (AEC standards, later NRC regulatory guides), it became a world standard. The single term SRSS became so widespread that it symbolized "common sense in seismic analysis."
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Haven't you experienced your body being thrown forward during sudden braking? 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, assuming "forces are applied slowly so acceleration can be ignored." It absolutely cannot be omitted in impact load 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," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure" — different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). 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 typical mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems rotate 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. That's because 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 for a smoother ride. What if damping were zero? Buildings would keep 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 equilibrium between external and internal forces.
- Non-Applicable Cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like 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 system 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
SRSS Calculation
Obtain the maximum response $R_i$ (displacement, stress, reaction force, etc.) for each mode and combine using SRSS. FEM solvers calculate this automatically.
Solver Settings
- Nastran: PARAM, SRSS (post-processing for SOL 103)
- Abaqus: *RESPONSE SPECTRUM, COMBINATION=SRSS
- Ansys: SRSS (default option for SPECTR analysis)
Summary
Difference Between Absolute Sum (ABS) and SRSS Can Be Up to 40%
The Absolute Sum (ABS) method, the most conservative mode response combination, assumes all modes reach their maximum simultaneously, leading to overestimation. Compared to ABS, SRSS often gives statistically 30-40% smaller estimates. ASCE 7-22 stipulates that "SRSS may be used when the number of modes ≥ 3," while ABS is required for 2 modes or fewer, being on the conservative side. Many field cases still overdesign using ABS without knowing this difference.
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 critical.
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. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
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 total load not all at once, but in small increments. The arc-length method (Riks method) can track 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: opening to an estimated page and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
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
SRSS in Practice
Current design codes recommend CQC, but SRSS continues to be used.
Practical Checklist
SRSS Adopted in Seismic Design of Tokyo Bay Aqua-Line
For the seismic design of the submerged tunnel section (maximum depth 60m below seabed) of the Tokyo Bay Crossings (opened 1997), the SRSS method was adopted for the RC section structure with numerous modes. The design seismic motion used the L2 (50-year exceedance probability 2%) spectrum, and SRSS was performed for up to 10 modes. This was the largest-scale seismic analysis for an underwater structure in Japan at the time, and the design team established a Nastran SOL 103→101 combined workflow.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question — which step in cooking is most prone to failure? Actually, it's the "prep work." If mesh quality is poor, 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 "calculation ran = results are 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 can 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 exam question." If the question 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 real-world constraints is often the most critical step in the entire analysis.
Software Comparison
SRSS Tools
All FEM solvers support SRSS as standard. No difference. Switching to CQC is also possible in all solvers.
Selection Guide
SAP2000 RSA Module Features Automatic SRSS/CQC Selection
CSI's SAP2000 v22 and later Response Spectrum Analysis (RSA) module includes an "Auto Combo" feature that automatically calculates mode spacing ratios and dynamically switches between SRSS and CQC. Users set a threshold (default 10%), and it automatically evaluates all mode pairs and selects combinations compliant with NRC RG 1.92. A similar automatic selection feature was implemented in Ansys Mechanical 2022R1.
The Three Most Important Questions for Selection
- "What are you solving?": Does the tool support the physical models/element types needed for the SRSS method? For example, in fluids, the presence of LES support; in structures, the capability for contact/large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) cars.
- "How far will it expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced SRSS Topics
Cases Exist Where SRSS Becomes Non-Conservative
Cases theoretically exist where the SRSS method becomes more non-conservative (dangerous side) than CQC. When components of closely spaced modes respond in-phase (positive correlation), SRSS underestimates. Menun (2004) showed this non-conservatism can cause up to +15% underestimation for specific structure-input combinations. In practice, switching to CQC if any part has mode spacing less than 10% has become established as a safe-side practice.
Troubleshooting
SRSS Troubles
Related Topics
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