Step 4: Peak Response via Response Spectrum Analysis
This summary presents the RSA of the 3-DOF frame under the 100% Loma Prieta (Palo Alto) motion. It shows (1) the design spectrum and modal ordinates obtained by period and damping interpolation; (2) the resulting spectral displacements per mode; (3) SRSS peak floor displacements; and (4) SRSS base shear. Compare these SRSS results to the direct time-history results from Step 3.
Computation steps (RSA):
- Period interpolation: For each damping curve, \(A_{n,0}\) at the modal period \(T_n\) is obtained by linear interpolation in \(T\).
- Damping interpolation: \(A_{n,0}\) is then interpolated between the bounding damping curves (0, 1, 2, 3, 5%) toward the mode damping.
- Spectral displacement: \(D_{n,0} = \dfrac{A_{n,0}}{\omega_n^2}\) (reported in inches).
- SRSS floor displacement: modal contributions \(\Gamma_n \cdot \phi_{jn} \cdot D_{n,0}\) are combined as \(u_{j,SRSS} = \sqrt{\sum (\Gamma_n \cdot \phi_{jn} \cdot D_{n,0})^2}\) (assumes statistical independence).
- SRSS base shear: modal pseudo-accelerations multiply modal static base shears \(V_{b,n}^{st}\); combined as \(V_{b,SRSS} = \sqrt{\sum \left(V_{b,n}^{st} \cdot A_{n,0}\right)^2}\), then converted to kips.
Design Spectrum and Modal Ordinates
The provided design spectrum (Item vi) includes 0, 1, 2, 3, and 5% damping curves. Each modal point is obtained by interpolating in period along the appropriate damping curves and then interpolating between damping curves to match the mode's damping.
Modal Spectral Ordinates
| Mode | T [s] | ζ (interp) | An,0 [g] | An,0 [in/s²] | Dn,0 [in] |
|---|---|---|---|---|---|
| Mode 1 | 0.500 | 1.13% (bracket 1%→2%, w=0.13) | 1.904 | 735.3 | 4.656 |
| Mode 2 | 0.139 | 1.57% (bracket 1%→2%, w=0.57) | 1.897 | 732.8 | 0.358 |
| Mode 3 | 0.073 | 0.93% (bracket 0%→1%, w=0.93) | 1.577 | 609.1 | 0.082 |
SRSS Peak Floor Displacements (in)
Peak floor displacements are combined by SRSS from modal spectral displacements (Gamma_n * phi_jn * D_n0). Values are given in inches.
| Floor | uSRSS [in] |
|---|---|
| Floor 1 | 1.890 |
| Floor 2 | 4.292 |
| Floor 3 | 6.102 |
SRSS Base Shear
The SRSS base shear combines modal pseudo-accelerations with modal static base shears \(V_{b,n}^{st}\); the result is reported in kips.
\(V_{b,SRSS} = 11.71\) kips