Step 3: Modal Response Analysis Summary
This page summarizes the modal response analysis of the 3-story MDOF building subjected to the 100% Loma Prieta at Palo Alto ground motion. All responses are computed using modal superposition with mass-orthonormal mode shapes and Newmark's method for numerical integration.
System Properties
The following system properties were used for the modal response analysis:
Mass Matrix:
\[ \mathbf{m} = \begin{bmatrix} 1180 & 0 & 0 \\ 0 & 1180 & 0 \\ 0 & 0 & 910 \\ \end{bmatrix} \quad \text{[kg]} \]Mass-Orthonormal Mode Shape Matrix:
\[ \boldsymbol{\Phi} = \begin{bmatrix} 0.771 & -1.918 & 2.051 \\ 1.755 & -1.334 & -1.892 \\ 2.495 & 1.978 & 0.944 \\ \end{bmatrix} \times 10^{-2} \]| Mode | $f_n$ [Hz] | $T_n$ [s] | $\omega_n$ [rad/s] | $\zeta_n$ [%] | $\Gamma_n$ [-] | $M_n^*$ [kg] | $h_n^*$ [m] |
|---|---|---|---|---|---|---|---|
| Mode 1 | 2.00 | 0.500 | 12.57 | 1.13 | 52.5081 | 2757.1 | 4.71 |
| Mode 2 | 7.20 | 0.139 | 45.24 | 1.57 | -20.3723 | 415.0 | 0.01 |
| Mode 3 | 13.75 | 0.073 | 86.39 | 0.93 | 10.4710 | 109.6 | 1.06 |
Ground Acceleration Input
The applied ground motion (from the input file) and the measured table acceleration are shown below.
Key Values
- Applied ground accel |üg|max: 340.57 in/s²
- Measured table accel |üg|max: 340.57 in/s²
(a) Modal Displacement Responses qn(t)
The displacement response for each mode is computed using:
Governing Equation:
\[ \ddot{D}_n(t) + 2\zeta_n\omega_n\dot{D}_n(t) + \omega_n^2 D_n(t) = -\ddot{u}_g(t) \]where $D_n(t)$ is the modal displacement, $\zeta_n$ is the damping ratio, $\omega_n$ is the natural frequency, and $\ddot{u}_g(t)$ is the ground acceleration.
Modal Coordinate:
\[ q_n(t) = \Gamma_n D_n(t) \quad \text{[in]} \]where $\Gamma_n = L_n / M_n$ is the participation factor, with $L_n = \boldsymbol{\phi}_n^{T} \mathbf{m} \boldsymbol{\iota}$ and $M_n = \boldsymbol{\phi}_n^{T} \mathbf{m} \boldsymbol{\phi}_n = 1$ (mass-orthonormal).
Note: The modal equations were integrated using Newmark's method for SDOF systems.
Response Statistics
| Response | Maximum [in] | Minimum [in] | Time of Max [s] |
|---|---|---|---|
| Mode 1 (q1) | 244.033 | -244.033 | 20.64 |
| Mode 2 (q2) | 6.199 | -5.735 | 20.64 |
| Mode 3 (q3) | 1.099 | -1.061 | 19.32 |
(b) Floor Displacement Responses uj(t) [inches]
The displacement response of each floor is computed using modal superposition:
Governing Equation:
\[ u_j(t) = \sum_{n=1}^{3} \Gamma_n \phi_{jn} D_n(t) = \sum_{n=1}^{3} \phi_{jn} q_n(t) \quad \text{[in]} \]where $u_j(t)$ is the displacement at floor $j$ and $\phi_{jn}$ is the mode shape component at floor $j$ for mode $n$.
Response Statistics
| Response | Maximum [in] | Minimum [in] | Time of Max [s] |
|---|---|---|---|
| Floor 1 (u1) | 1.996 | -1.996 | 20.64 |
| Floor 2 (u2) | 4.365 | -4.365 | 20.64 |
| Floor 3 (u3) | 5.974 | -5.974 | 20.65 |
Note: The left column shows the displacement time history for each floor. The right column shows the scaled mode shape contributions, where each mode's contribution is scaled to match the maximum displacement magnitude for visual comparison.
(c) Floor Acceleration Responses üj(t) [inches/s²]
The acceleration response of each floor includes both relative and ground acceleration:
Governing Equation:
\[ \ddot{u}_j(t) = \sum_{n=1}^{3} \Gamma_n \phi_{jn} \ddot{D}_n(t) + \ddot{u}_g(t) \quad \text{[in/s²]} \]Total floor acceleration includes both relative acceleration from modal response and the ground acceleration component.
Response Statistics
| Response | Maximum [in/s²] | Minimum [in/s²] | Time of Max [s] |
|---|---|---|---|
| Floor 1 (ü1) | 566.50 | -486.08 | 20.63 |
| Floor 2 (ü2) | 853.09 | -731.33 | 20.64 |
| Floor 3 (ü3) | 1044.35 | -1031.56 | 21.65 |
Note: Comparison with measured floor accelerations (from Item (v)) can be performed by overlaying the measured data on these computed responses.
(d) Base Shear Vb(t) [kips]
The base shear is computed using modal superposition:
Governing Equation:
\[ V_b(t) = \sum_{n=1}^{3} V_{b,n}^{\text{st}} A_n(t), \quad A_n(t) = \omega_n^2 D_n(t) \quad \text{[kips]} \]where $V_{b,n}^{\text{st}}$ is the modal static base shear and $A_n(t)$ is the pseudo-acceleration for mode $n$.
Response Statistics
| Response | Maximum [kips] | Minimum [kips] | Time of Max [s] |
|---|---|---|---|
| Base Shear (Vb) | 13.01 | -13.01 | 20.64 |
(e) Base Overturning Moment Mb(t) [kip-ft]
The base overturning moment is computed using modal superposition:
Governing Equation:
\[ M_b(t) = \sum_{n=1}^{3} M_{b,n}^{\text{st}} A_n(t) \quad \text{[kip-ft]} \]where $M_{b,n}^{\text{st}}$ is the modal static base moment and $A_n(t) = \omega_n^2 D_n(t)$ is the pseudo-acceleration for mode $n$.
Response Statistics
| Response | Maximum [kip-ft] | Minimum [kip-ft] | Time of Max [s] |
|---|---|---|---|
| Base Moment (Mb) | 178.32 | -178.32 | 20.64 |
Summary of Maximum Responses
The following table summarizes the maximum absolute responses for each floor and the base:
| Floor | Max Displacement [in] | Time [s] | Max Acceleration [in/s²] | Time [s] |
|---|---|---|---|---|
| Floor 1 | 1.996 | 20.64 | 566.50 | 20.63 |
| Floor 2 | 4.365 | 20.64 | 853.09 | 20.64 |
| Floor 3 | 5.974 | 20.65 | 1044.35 | 21.65 |
Base Response:
- Maximum Base Shear: 13.01 kips (at t = 20.64 s)
- Maximum Base Moment: 178.32 kip-ft (at t = 20.64 s)