What is being done and why

Base isolators are devices that reduce seismic demand on a structure by supporting it on flexible, energy-dissipating elements. Their force–displacement response under cyclic loading is hysteretic: the load path on loading differs from that on unloading, and the area enclosed by the loop represents energy dissipated per cycle. Quantifying this behavior—peak force \(P_0\) and displacement \(U_0\), secant stiffness \(K_0 = P_0/U_0\), and equivalent viscous damping \(\zeta_\mathrm{{eff}}\)—is standard in earthquake engineering for design and assessment.

This page starts from scanned images of force–displacement loops (from reports or papers) rather than raw data. Each image is digitized to obtain a cloud of \((u, F)\) points; the same mechanical definitions are applied to compute \(U_0\), \(P_0\), \(K_0\), \(W_D\), \(K_1\), \(K_2\), and \(\zeta_\mathrm{{eff}}\). The result is a consistent set of parameters across four strain levels (approximately 9.8%, 74%, 124%, and 180%) for comparison and potential use in models. All values reported here are per bearing; the problem’s isolation system uses 4 HDR bearings in parallel, so the system storage stiffness for SDOF analysis is \(4\times K_1\).

Digitization workflow and quantity definitions

The following steps are applied to each scanned PNG:

1. The image is converted to grayscale and the plotting window is detected from dark-pixel density.
2. The outer frame and axis lines are removed in image space.
3. Remaining dark pixels are mapped to physical coordinates \((u, F)\) using strain-specific axis limits taken from the original plot labels (displacement in inches, force in kips).
4. Small connected components (annotation digits and specks) are filtered out.
5. Physical axis limits (displacement and force) are set from the plot labels (or from a per-image sidecar basename_limits.json), so \(U_0\) and \(P_0\) are obtained directly from the digitized curve.

From the calibrated point cloud the following quantities are computed:

• \(U_0\) and \(P_0\): maximum absolute displacement and force.
• \(K_0 = P_0/U_0\): secant stiffness to the loop tip.
• \(W_D\): area enclosed by the loop (envelope integration over displacement bins).
• \(K_2 = W_D/(\pi U_0^2)\): loss stiffness.
• \(K_1\): storage stiffness from \(K_0^2 = K_1^2 + K_2^2\).
• \(\zeta_\mathrm{{eff}} = W_D/\bigl(2\pi (\omega/\omega_n) K_1 U_0^2\bigr)\), evaluated at \(\omega/\omega_n = 1\).

On each figure, the green marker indicates the point corresponding to \(U_0\) and the red marker the point corresponding to \(P_0\).

Trends with shear strain

The following plots summarize how \(K_1\), \(K_2\), and \(\zeta_\mathrm{eff}\) vary with shear strain, based on the values computed from each digitized hysteresis loop.