Seismic Time History Response Analysis
Seismic Time History Response Analysis: Theoretical Foundations
What is Seismic Time History Response Analysis?
Professor, how is seismic time history response analysis different from the response spectrum method?
The response spectrum method calculates only the maximum response. Time history response analysis calculates the complete history of the response over time (time histories of displacement, velocity, acceleration). Time history analysis is essential for evaluating plastic deformation and energy absorption.
Equation of Motion
Earthquake input as base excitation:
$\ddot{u}_g(t)$ is the earthquake acceleration time history (seismic waveform). $\{u\}$ is the relative displacement to the base.
So you input the seismic waveform directly.
As design seismic waveforms:
- Recorded Waves — Actual recorded waves like El Centro (1940), Hyogo-ken Nanbu Earthquake (1995), etc.
- Simulated Seismic Waves — Artificially generated to match a design response spectrum.
- Site-Specific Waves — Generated at the ground surface through ground response analysis.
Linear vs. Nonlinear
| Method | Material | Analysis Method | Position in Design Codes |
|---|---|---|---|
| Elastic Time History | Elastic | Modal Method or Direct Method | Level 1 Earthquake |
| Elastoplastic Time History | Elastoplastic | Direct Method (Newmark Method) | Level 2 Earthquake |
So elastoplastic analysis is necessary for Level 2 earthquakes (major earthquakes).
For Level 2 earthquakes (Hyogo-ken Nanbu Earthquake class), structures yield, making elastoplastic time history analysis essential to track plastic hinge formation, energy absorption, and residual deformation.
Summary
Key Points:
- Calculate the complete time history of the response by directly inputting the seismic waveform.
- $[M]\{\ddot{u}\} + [C]\{\dot{u}\} + [K]\{u\} = -[M]\ddot{u}_g$ — Base excitation.
- Elastic time history uses modal method, elastoplastic uses direct method — Newmark/HHT-α.
- Input recorded waves or simulated seismic waves — Specified by design codes.
- Elastoplastic time history is mandatory for Level 2 earthquakes — Plastic deformation and energy absorption.
Seismic time history analysis became widespread with computerization in the 1970s
Seismic time history response analysis is considered to have been practically established with calculations using the CAL16 code with the 1940 El Centro earthquake wave (accelerometer record 0.319g) in 1971. Subsequently, following the 1978 Miyagi-ken-oki earthquake, the revision of the Japanese Building Standards Act (1981, New Seismic Design Code) mandated time history analysis for super high-rise building design for structures over 60m tall. Computer centers like NTT Data began providing batch processing calculation services.
Computational Methods for Seismic Time History Response Analysis
Inputting Seismic Waveforms
How do you input seismic waveforms into FEM?
Nastran
```
SOL 109 $ Direct method time history
CEND
DLOAD = 100
BEGIN BULK
TLOAD1, 100, 200, , 0, 300
TABLED1, 300, , ,
, 0.0, 0.0, 0.01, 1.23, 0.02, -0.56, ... $ Acceleration time history
```
Define acceleration with a TABLED1 table and apply it to the base SPC point.
Abaqus
```
*AMPLITUDE, NAME=earthquake
0.0, 0.0
0.01, 1.23
0.02, -0.56
...
*STEP
*DYNAMIC
0.01, 40.0 $ dt=0.01s, 40 seconds
*BASE MOTION, DOF=1, AMPLITUDE=earthquake
*END STEP
```
Ansys
```
/SOLU
ANTYPE, TRANSIENT
DELTIM, 0.01
TIME, 40.0
ACEL, , 9.81*amp(t) ! Acceleration input
SOLVE
```
Abaqus's *BASE MOTION seems the simplest.
*BASE MOTION directly defines base excitation. You only specify the direction (DOF) and waveform (AMPLITUDE).
Elastoplastic Models
Material/element models used in seismic elastoplastic analysis:
| Structure Type | Model | Features |
|---|---|---|
| RC Column | Fiber Model (OpenSees) | Tracks section plastification. |
| Steel Beam | Plastic Hinge (Lumped Plasticity) | Hinge element at ends. |
| Seismic Isolation Device | Bilinear Spring | Yield force and secondary stiffness. |
| Viscous Damper | Maxwell Model | Viscous + Elastic. |
Summary
Japan's review criteria require averaging over 3 seismic waves
For structural review of super high-rise buildings in Japan, Notification No. 457 requires that "the average of three or more waves (notification wave, site-specific wave, recorded wave) must keep each response value within the target value." Analysis methods are direct integration (Newmark-β, HHT-α) or mode superposition. For nonlinear analysis, trilinear or slip-type restoring force characteristics are adopted. The cost for one time history analysis run is about 30 minutes to 2 hours for a super high-rise building (~100 stories, ~30k DOF).
Seismic Time History Response Analysis in Practice
Seismic Time History in Practice
Used in the Building Standards Act's "Limit Strength Calculation Method" and "Time History Response Analysis".
Seismic Waveform Selection
Seismic waves specified by design codes:
| Code | Seismic Wave Selection |
|---|---|
| Japan (Notification) | 3 or more recorded waves + site-specific waves. Extremely rare ground motions. |
| Eurocode 8 | Simulated seismic waves compatible with response spectrum. Minimum 3 waves. |
| ASCE 7 | 7 or more waves. Spectrum compatible with ground characteristics. |
| NRC (Nuclear) | SSE (Safe Shutdown Earthquake) compatible waves. Probabilistic hazard. |
So you analyze with a minimum of 3 seismic waves.
Average over 3 waves, use the maximum value from each of 7 waves (ASCE 7). The basic principle is to evaluate result variability with multiple seismic waves.