Seismic Response Spectrum Analysis
Theory and Physics
What is the Response Spectrum Method?
Professor, how is the "Response Spectrum Method" different from time history analysis?
Time history analysis calculates the "entire time history" of the response, whereas the response spectrum method finds only the maximum response of each mode and combines them statistically. The computation is orders of magnitude faster.
Definition of Response Spectrum
Response Spectrum $S_a(T, \zeta)$ is the "maximum acceleration response of a single-degree-of-freedom system with natural period $T$ and damping ratio $\zeta$ subjected to seismic wave input":
It's a graph listing the maximum response at each natural period.
Design response spectra are specified in design codes. Examples include the Building Standard Law's $S_a$, Eurocode 8's elastic response spectrum, and ASCE 7's MCER spectrum.
Mode Superposition Method (RSA: Response Spectrum Analysis)
Procedure:
1. Eigenvalue Analysis — $N$ modes (frequency, mode shape, effective mass)
2. Maximum Response per Mode — Read the maximum acceleration $S_{a,i}$ for mode $i$ from the response spectrum
3. Maximum Displacement per Mode — $u_{max,i} = \Gamma_i S_{d,i} \{\phi_i\}$
4. Combination of Modal Responses — Combine using SRSS or CQC
Since the "maximum values for each mode" do not occur simultaneously, they are combined statistically, right?
Exactly. SRSS (Square Root of Sum of Squares) combines uncorrelated modes, CQC (Complete Quadratic Combination) combines correlated modes.
SRSS vs. CQC
| Combination Method | Formula | Application |
|---|---|---|
| SRSS | $R = \sqrt{\sum R_i^2}$ | When modes are sufficiently separated |
| CQC | $R = \sqrt{\sum \sum \rho_{ij} R_i R_j}$ | When closely spaced modes exist |
$\rho_{ij}$ is the modal correlation coefficient (Der Kiureghian, 1981).
If there are closely spaced modes, CQC is the only choice then.
Current design codes (Eurocode 8, ASCE 7) recommend CQC. SRSS can be non-conservative for closely spaced modes.
Summary
Key Points:
- Response Spectrum = Graph of maximum response per period — Specified by design codes
- Mode Superposition Method (RSA) — Eigenvalue analysis → Spectrum reading → Combination
- SRSS (uncorrelated), CQC (correlated) — CQC is currently recommended
- Orders of magnitude faster than time history analysis — Mainstay in design practice
- Number of modes covering 90% effective mass — Requirement of Building Standard Law/Eurocode 8
Housner made the response spectrum practical in 1952
The concept of the seismic response spectrum was conceived by K.A. Terzaghi (1943) and established as a practical calculation method by George W. Housner (Caltech) in 1952. Housner calculated spectra from four actual records including the 1940 El Centro earthquake record and proposed their application to seismic design. This method was incorporated into the building codes of the Western US by 1959 and became the starting point for seismic design methods worldwide.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body 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, which assumes "forces are applied slowly enough that acceleration can be ignored". 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. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. 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"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) 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), 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 it becomes "compression"—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. 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 deliberately 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, 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, considers 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. 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 deformations) |
| 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
Nastran
```
SOL 103 $ Eigenvalue analysis
CEND
METHOD = 10
BEGIN BULK
EIGRL, 10, , , 50
```
+ Post-processing for spectrum combination. Or SOL 111 + TABRND for response spectrum input.
Abaqus
```
*STEP
*FREQUENCY
50, ,
*END STEP
*STEP
*RESPONSE SPECTRUM
0.01, 10.0, 0.05 $ Period range, damping ratio
*SPECTRUM, NAME=design_spectrum, TYPE=ACCELERATION
0.0, 9.81
0.5, 24.5
1.0, 9.81
3.0, 3.27
*END STEP
```
Ansys
```
/SOLU
ANTYPE, SPECTR
SPOPT, SPRS ! Response spectrum analysis
SVTYPE, 2 ! Acceleration spectrum
SV, 1, freq1, Sa1, freq2, Sa2, ... ! Spectrum data
SOLVE
```
Abaqus's *RESPONSE SPECTRUM seems the most intuitive.
Just input the spectrum data (period or frequency vs. acceleration) directly and select the CQC/SRSS combination method.
Design Response Spectra
| Code | Spectrum Definition |
|---|---|
| Building Standard Law (Japan) | $S_a = C_0 \cdot Z \cdot R_t \cdot A_i$ frequency dependence |
| Eurocode 8 | Elastic response spectrum Type 1/2. Ground types A–E |
| ASCE 7 | MCER (Maximum Considered Earthquake) spectrum. Defined by $S_{DS}, S_{D1}$ |
Summary
Why the 5% Damped Spectrum Became the World Standard
The 5% damping ratio (ζ=0.05) for seismic design spectra became the world standard after the ATC-3-06 project (1978) following the 1971 San Fernando earthquake established 5% as the representative value. Measured damping in reinforced concrete structures primarily fell within the 3–7% range, and 5% was judged to be a conservative yet realistic central value. Japan's Building Standard Law and Road Bridge Specifications also adopted 5% as the standard for the same reason.
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. Recommendation: 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 tangent stiffness matrix each 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 not all at once, but in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy for Direct vs. Iterative Methods
Direct methods are like "solving simultaneous equations accurately by hand calculation"—reliable but too time-consuming for large-scale problems. Iterative methods are like "repeatedly making educated guesses to approach the correct answer"—initially rough
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