Seismic Response Spectrum Analysis
Seismic Response Spectrum: Theoretical Foundations
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.
Computational Methods for Seismic Response Spectrum
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.