NRC耐震解析手法
Theory and Physics
NRC Seismic Analysis Methods
Professor, what are the NRC (Nuclear Regulatory Commission) seismic analysis methods?
The U.S. Nuclear Regulatory Commission (NRC) issues detailed regulatory guidelines for seismic design of nuclear facilities. Analysis methods are specified in Reg Guides (Regulatory Guides) and the SRP (Standard Review Plan).
Major Reg Guides
| Reg Guide | Content |
|---|---|
| RG 1.60 | Design Response Spectra |
| RG 1.92 | Modal Response Combination Methods (SRSS, CQC, Grouping) |
| RG 1.122 | Broadening of Floor Response Spectra |
| RG 1.61 | Damping Ratio Values |
So there are design spectra and combination methods specific to nuclear power.
NRC Reg Guides represent the world's most conservative and systematic seismic regulations. Many countries' nuclear regulations are based on the NRC.
RG 1.92 Modal Combination
RG 1.92 specifies the Grouping Method for closely spaced modes:
1. Group modes with natural frequencies within 10% of each other
2. Within a group, use Absolute Sum
3. Between groups, use SRSS
Is the reason for using absolute sum for closely spaced modes to take the worst-case scenario?
Yes. Closely spaced modes may respond in-phase, so the absolute sum is the most conservative. CQC can also handle it, but the NRC has long used the Grouping Method + SRSS.
RG 1.61 Damping Ratios
| Structure Type | OBE (Operating Basis Earthquake) | SSE (Safe Shutdown Earthquake) |
|---|---|---|
| RC Structure | 4% | 7% |
| Steel Structure (Welded) | 2% | 4% |
| Steel Structure (Bolted) | 4% | 7% |
| Piping | 2% | 3% |
| Equipment | 2% | 3% |
Damping is larger for SSE (Safe Shutdown Earthquake)?
In SSE, the structure may experience some damage (micro-cracks, etc.), increasing energy dissipation. This effect is represented equivalently by a larger damping ratio.
Summary
Key Points:
- NRC Reg Guides are the world standard for nuclear seismic — Most conservative and systematic
- RG 1.60 — Design Response Spectra
- RG 1.92 — Grouping Method + SRSS (for closely spaced modes)
- RG 1.61 — Damping ratios by structure type
- RG 1.122 — Broadening of floor spectra (±15%)
Three Mile Island Accident Revamped NRC Analysis Methods
Post-accident analysis of the 1979 Three Mile Island nuclear accident (TMI-2) revealed significant conservatism deficiencies in seismic analysis methods for piping systems. In response, the NRC comprehensively revised Regulatory Guides 1.61, 1.92, and 1.122 in the 1980s. Particularly, the regulatory change to shift the modal combination method from SRSS to CQC (RG 1.92 Rev.2, 1976 → Rev.3, 2006) greatly influenced Japan's nuclear regulations as well.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced being thrown forward when a car brakes suddenly? 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 enough that acceleration is negligible". 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'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 under 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 forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), the force of the tires pushing on the road 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" — 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 vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational 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 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions
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. Beware 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
FEM Implementation of NRC Methods
Seismic analysis compliant with NRC standards uses Nastran as the industry standard:
1. Eigenvalue analysis with SOL 103 (covering 90% effective mass)
2. Time history or response spectrum analysis with SOL 111/112
3. Combination using RG 1.92's Grouping Method + SRSS
4. Floor response spectrum generation (PARAM,SRS) + broadening
5. Evaluation with RG 1.61 damping ratios
So Nastran is the standard for nuclear seismic analysis too.
Nastran is overwhelmingly dominant for nuclear FEM. It has over 30 years of verification history and accumulated NRC certification.
Summary
NRC Regulatory Guide 1.92 is the Bible of Seismic Analysis
The NRC's Regulatory Guide 1.92 "Combining Modal Responses and Spatial Components in Seismic Response Analysis" is the de facto world standard, detailing application conditions, thresholds, and correlation coefficient formulas for the SRSS/CQC method. The 3rd edition (2006) fully adopted Wilson's (1981) work, explicitly stating the closely spaced mode threshold as "within 10% frequency difference". It is referenced in over 40 countries worldwide.
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 tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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 total load in small increments rather than all at once. 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 "flexible curves" — can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
NRC Methods in Practice
Nuclear seismic analysis has very strict regulatory requirements. Document that all analysis methods comply with Reg Guides.
Practical Checklist
NRC-Certified Software Requires V&V Documentation
Software used for seismic analysis of nuclear facilities must have Verification and Validation (V&V) documentation per NRC standards (NUREG/CR-6430, etc.). Ansys Mechanical has a process for obtaining NRC V&V certification (Ansys Nuclear Quality Assurance Program). In Japan, Mitsubishi Heavy Industries and Toshiba Energy Systems also conduct analysis compliant with NRC methods using CAE environments with certified software.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question — which step in cooking is most prone to failure? Actually, it's "prep work". If mesh quality is poor, results will be garbage no matter how good 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 is far from reality. Verify that results stabilize with at least three levels of mesh density — neglecting this leads to the dangerous assumption "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 truly fully fixed?" "Is this load truly uniformly distributed?" — Correctly modeling real-world constraints is often the most critical step in the entire analysis.
Software Comparison
Tools for NRC Seismic
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