Structure With a Resonance Inside the Peak Spectrum Region
A worked example showing how a structure with its natural frequency in the peak region of the response spectrum experiences maximum amplification, and how this drives the seismic demand.
Problem definition
Two equipment frames are assessed for seismic loading. Frame A has a natural frequency of 8.0 Hz, which coincides with the peak of the floor response spectrum. Frame B has a natural frequency of 25 Hz, well beyond the peak. Compare the seismic demand on each frame.
Inputs
| Parameter | Frame A | Frame B | Unit |
|---|---|---|---|
| Natural frequency | 8.0 | 25.0 | Hz |
| Mass | 600 | 600 | kg |
| Damping ratio | 0.03 | 0.03 | - |
| FRS Sa at 8 Hz | 0.45g | - | - |
| FRS Sa at 25 Hz | - | 0.08g | - |
Step 1: Calculate seismic force for Frame A
Sa = 0.45g = 4.42 m/s^2. F_A = m x Sa = 600 x 4.42 = 2652 N.
Step 2: Calculate seismic force for Frame B
Sa = 0.08g = 0.785 m/s^2. F_B = m x Sa = 600 x 0.785 = 471 N.
Step 3: Compare demand
Frame A force: 2652 N. Frame B force: 471 N. Ratio: 2652/471 = 5.6. Frame A, with its frequency at the spectrum peak, experiences 5.6 times the seismic force of Frame B, despite having identical mass and damping.
Results
| Frame | Frequency (Hz) | Sa (g) | Force (N) | Ratio to Frame B |
|---|---|---|---|---|
| A | 8.0 | 0.45 | 2652 | 5.6x |
| B | 25.0 | 0.08 | 471 | 1.0x |
Checks
- Frame A is at the FRS peak - this is the worst-case location for seismic demand
- Frame B is well beyond the peak - the spectral acceleration is much lower
- The 5.6x difference is entirely due to frequency content - the frames have identical mass and damping
- Frame A should be flagged for sensitivity study: small changes in frequency could move it closer to or further from the peak
Interpretation
Frame A, with its natural frequency at 8.0 Hz coinciding with the peak of the floor response spectrum, experiences maximum amplification. The spectral acceleration is 0.45g, producing a seismic force of 2652 N. Frame B, at 25 Hz, is well beyond the spectrum peak where the acceleration has dropped to 0.08g, producing only 471 N. The 5.6:1 ratio in demand demonstrates why the frequency content of the input motion and the structural natural frequency together govern seismic response - not just the acceleration level. Frame A should be assessed with particular care, including a sensitivity study on frequency and damping, because the demand is near its maximum.