2‑DOF Base‑Isolated Building — Kobe KBU090 (Part b)

This report evaluates a linear 2‑DOF base‑isolated building model under the 090 component of the 1995 Kobe University record (RSN1108_KOBE_KBU090.AT2). The objective is to quantify three engineering demands—story drift, isolation displacement, and base shear coefficient—and compare them against the corresponding fixed‑base response.

Three complementary procedures are shown: (i) direct multi‑degree‑of‑freedom (MDOF) time‑history integration (Newmark), (ii) modal superposition time‑history (2 modes), and (iii) response spectrum analysis (RSA) using the square‑root‑of‑sum‑of‑squares (SRSS) rule.

Open Newmark MDOF demo (non‑classical damping) Back to CE223 main page

1) Model definition and reported demands

Using the coordinate convention from the CEE223 lecture material, the generalized displacement vector is $\mathbf{u}(t) = [u_b(t),\;u_s(t)]^T$ where: (i) $u_b(t)$ is the isolation deformation (base displacement relative to the ground), and (ii) $u_s(t)$ is the superstructure deformation (superstructure displacement relative to the isolation level).

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = -\mathbf{M}\boldsymbol{\iota}\,\ddot{u}_g(t),\qquad \boldsymbol{\iota}=[1,0]^T$$
$$\mathbf{M}=\begin{bmatrix}M & m\\ m & m\end{bmatrix},\quad \mathbf{C}=\begin{bmatrix}c_b & 0\\ 0 & c_s\end{bmatrix},\quad \mathbf{K}=\begin{bmatrix}k_b & 0\\ 0 & k_s\end{bmatrix},\quad M=m_b+m$$

The right‑hand side is the effective inertia load induced by the ground acceleration $\ddot{u}_g(t)$. Because $u_s$ is already defined relative to the isolation level, the ground motion influences the system through the isolation coordinate only; this is why the influence vector is $\boldsymbol{\iota}=[1,0]^T$.

Engineering demands reported:

  • Story drift: $\Delta(t)=u_s(t)$ (superstructure deformation relative to the isolation level)
  • Isolation displacement: $u_b(t)$
  • Base shear coefficient: $C_V = \max\,|V_b|/(Mg)$ with $V_b=k_b u_b + c_b \dot{u}_b$

Coordinate meaning (CEE223 lecture convention): $u_b$ is the deformation of the isolation layer, and $u_s$ is the deformation of the superstructure. The corresponding absolute displacements are $u_b^{\mathrm{abs}}=u_g+u_b$ and $u_s^{\mathrm{abs}}=u_g+u_b+u_s$. Because ground motion enters directly in the absolute base coordinate but not in the relative drift coordinate, the influence vector is $\boldsymbol{\iota}=[1,0]^T$ (not $[1,1]^T$).

Given properties

Mass ratio: $m_s = 3m_b/2$. Target periods: $T_s=0.5$ s (superstructure), $T_b=2.0$ s (isolation). Damping ratios: $\zeta_s=0.02$ and $\zeta_b=0.15$.

Note on scaling: the absolute mass scale cancels from these results—peak displacements and the coefficient $C_V$ are invariant to uniformly scaling all masses.

2) Ground motion input

The input is the Kobe University record RSN1108_KOBE_KBU090.AT2 (090 component). The file is a PEER-format .AT2 record with acceleration values in units of $g$, converted here to SI units for computation. The plot below shows $\ddot{u}_g(t)$ in $g$ so amplitudes are immediately interpretable.

All response spectra on this page are generated directly from this same record.

3) Modal properties and damping used for RSA

The undamped modes are obtained from the generalized eigenproblem $\mathbf{K}\phi_n = \omega_n^2\mathbf{M}\phi_n$ and then scaled so that the base displacement component of each mode is one, $\phi_{b,n} = 1$. This matches the coordinate convention used in the CEE223 lecture material and makes modal contributions to base and story deformations directly readable from the components of $\phi_n$.

Mode shapes

We begin by reporting the eigenvectors (mode shapes) because they are the building blocks for participation factors, effective modal mass, and the modal time-history/RSA calculations that follow. The mode shape matrix $\Phi=[\phi_1\;\phi_2]$ is listed below; each column is a mode (base-normalized, $\phi_{b,n}=1$), and each row corresponds to a DOF in the CEE223 lecture coordinate convention.

DOF Mode 1: φ1 Mode 2: φ2
u_b (isolation deformation) 1.000000 1.000000
u_s (superstructure deformation / drift) 0.064042-1.626542

For base excitation, the key scalar is the participation factor $\Gamma_n$, which maps ground acceleration into modal forcing through the influence vector $\boldsymbol{\iota}=[1,0]^T$ defined in Section 1. To write independent modal equations (one scalar ODE per mode), we make the common (approximately) classical damping assumption: in modal coordinates, the damping matrix is nearly diagonal, i.e., $\Phi^T\mathbf{C}\Phi \approx \mathrm{diag}(c_1,c_2)$. Real structures are generally not exactly classical, so this step should be understood as an approximation adopted for tractability.

$$M_n = \phi_n^T\mathbf{M}\phi_n,\qquad \Gamma_n = \dfrac{\phi_n^T \mathbf{M}\boldsymbol{\iota}}{M_n},\qquad \zeta_n \approx \dfrac{\phi_n^T\mathbf{C}\phi_n}{2\omega_n\,M_n},\qquad M_n^* = \Gamma_n^2 M_n$$

$M_n^*/M$ (effective modal mass ratio) quantifies how much of the base-excited response is carried by each mode, where $M = \boldsymbol{\iota}^T\mathbf{M}\boldsymbol{\iota}$ is the effective total mass associated with the influence vector. In an isolation regime, Mode 1 should dominate.

Mode $T_n$ [s] $\omega_n$ [rad/s] $\zeta_n$ $\Gamma_n$ $M_n^*/M$
12.03813.082914.18%0.962199.9%
20.310320.24736.76%0.03790.1%

Classical damping note: writing independent modal equations, $q_n'' + 2\zeta_n\omega_n q_n' + \omega_n^2 q_n = -\Gamma_n\,\ddot{u}_g(t)$, implicitly assumes that the damping matrix is diagonal in modal coordinates (so-called classical damping). Real structures are generally not exactly classical, but for lightly coupled low-order systems this approximation is standard in practice.

The spectrum plot shows pseudo-acceleration $S_a(T)$ for the modal damping ratios and the fixed-base 2% curve, with markers at $T_1$ and $T_2$.

4) Direct vs modal time-history response (sanity check)

The benchmark is the direct physical time-history solution of the coupled 2‑DOF equations using Newmark’s method: at each time step we solve a 2×2 coupled linear system for $\mathbf{u}(t)$. In parallel, a modal superposition time history is computed by integrating each modal coordinate as an independent single‑degree‑of‑freedom (SDOF) equation and reconstructing $\mathbf{u}(t)$.

$$q_n'' + 2\zeta_n\omega_n q_n' + \omega_n^2 q_n = -\Gamma_n\,\ddot{u}_g(t),\qquad \mathbf{u}(t)=\sum_{n=1}^{2}\phi_n\,q_n(t)$$

The plots below show isolation displacement $u_b(t)$ and story drift $\Delta(t)=u_s(t)$ for the direct and modal solutions. This comparison isolates two modeling choices: (a) truncating to the first two modes (here, the full system has only two), and (b) treating damping as modal (classical) to obtain independent modal ODEs. Good agreement indicates these choices are adequate for this record.

5) Peak demand summary and comparison to fixed-base

The table below reports peak values for the isolated building from: (i) direct 2‑DOF time history, (ii) modal time history, and (iii) response spectrum analysis (RSA) with SRSS modal combination. The fixed‑base entry is the corresponding SDOF response with $T_s=0.5$ s and $\zeta_s=2\%$ under the same record.

Fixed-base is modeled as an SDOF because setting $u_b\equiv 0$ removes the isolation deformation coordinate, leaving only the superstructure deformation relative to the ground. This is equivalent to constraining the base DOF in the 2‑DOF model. For RSA, fixed-base has a single mode, so SRSS combination is trivial.

Why (ii) and (iii) differ: both are built on the same modal properties $(T_n,\zeta_n,\Gamma_n)$, but they answer different questions. Modal time history integrates $q_n(t)$ under the actual record and reconstructs $\mathbf{u}(t)$, so peak response depends on modal phasing in time. RSA replaces the record with spectral ordinates $S_d(T_n,\zeta_n)$ (or $S_a$) and then combines modal peaks statistically (SRSS), discarding time sequencing and assuming peaks do not occur simultaneously.

Units: displacements in cm. The base shear coefficient $C_V$ is dimensionless.

Case $\max\,|\Delta|$ [cm] $\max\,|u_b|$ [cm] $\max\,C_V$
Isolated (direct time history, MDOF)0.8911.740.125
Isolated (modal time history, 2 modes)0.8011.730.126
Isolated (response spectrum analysis, SRSS)0.7611.750.118
Fixed-base (time history, SDOF)3.430.000.553
Fixed-base (response spectrum analysis)3.450.000.556

Interpretation

Isolation reduces drift and base shear primarily by shifting the dominant response to the longer first-mode period ($T_1\approx 2$ s) and by concentrating damping in the isolation layer (here $\zeta_b=15\%$). The main trade-off is increased base displacement $|u_b|$, which becomes the governing design check for moat clearance and isolator deformation capacity.

RSA note: SRSS combines modal peak contributions assuming statistical independence. For closely spaced modes, CQC would be preferred; here the two periods are well separated and Mode 1 dominates the effective modal mass.

References

Citations: Chopra, A.K. (2014). Dynamics of Structures (4th ed.). Pearson. Konstantinidis, D. (2025). Lectures for CEE223: Earthquake Protective Systems, UC Berkeley. DeJong, M. (2025). Lectures for CEE225: Structural Dynamics, UC Berkeley.

PDF attachment (placeholder)

A PDF version of this analysis will be embedded here in the final submission. For now, this section is intentionally left as a placeholder.

PDF placeholder
Drop the exported PDF here when ready.