Base isolator hysteresis
Digitized force–displacement loops at four strain levels; per‑bearing storage stiffness \(K_1\), loss stiffness \(K_2\), equivalent viscous damping ratio \(\zeta_{\mathrm{eff}}\), and energy per cycle; trend plots vs strain.
CE223 - Earthquake Protective Systems
This page presents portfolio work from CE223 – Earthquake Protective Systems: linear multi‑degree‑of‑freedom (MDOF) time‑history analysis and an earlier single‑degree‑of‑freedom (SDOF) base‑isolation project built from measured hysteresis, equivalent SDOF models, and time‑ and frequency‑domain response.
Terms used below: MDOF = multi‑degree‑of‑freedom; SDOF = single‑degree‑of‑freedom; HDR = high‑damping rubber; K₁ = storage stiffness, K₂ = loss stiffness (from hysteresis loops); ζeff = equivalent viscous damping ratio; RSA = response spectrum analysis; SRSS = square‑root‑of‑sum‑of‑squares (modal combination rule); FFT = fast Fourier transform; 2‑DOF = two‑degree‑of‑freedom.
SDOF base‑isolation project
The first set of dashboards documents a base‑isolation workflow using a single‑degree‑of‑freedom (SDOF) model built from measured hysteresis of high‑damping rubber (HDR) bearings. The workflow yields equivalent SDOF parameters—storage stiffness \(K_1\), loss stiffness \(K_2\), and equivalent viscous damping ratio \(\zeta_\mathrm{eff}\)—and uses fixed‑point iteration on strain and \(\zeta_\mathrm{eff}\), plus time‑ and frequency‑domain response.
What this project shows
- From loops to parameters: extract storage stiffness \(K_1\), loss stiffness \(K_2\), and equivalent viscous damping ratio \(\zeta_\mathrm{eff}\) from measured HDR hysteresis.
- Design‑style iteration: fixed‑point update on strain and \(\zeta_\mathrm{eff}\) for a four‑bearing isolation system.
- Time vs frequency domain: compare Newmark time integration and fast Fourier transform (FFT) response for the same SDOF.
- Damping models: contrast viscous, hysteretic, and fractional Kelvin–Voigt in terms of loops and earthquake response.
Iteration and response
Fixed‑point iteration with four bearings (system stiffness \(k = 4K_1\)); convergence of strain and \(\zeta_{\mathrm{eff}}\); Newmark time integration vs fast Fourier transform (FFT) under the Kobe University record, component 090.
SDOF damping model comparisons
Models A, B, C: force–displacement loops; storage and loss stiffness \(K_1(\omega)\), \(K_2(\omega)\) vs frequency; equivalent damping coefficient (EDC); frequency‑domain earthquake response.
SDOF hysteresis loops
Force–displacement loops only for damping models A, B, C.
Linear MDOF project
A second project focuses on linear multi‑degree‑of‑freedom (MDOF) response: a Newmark time‑integration implementation in matrix form and its relationship to modal superposition and response spectrum analysis (RSA). It is illustrated on a two‑degree‑of‑freedom (2‑DOF) shear frame and then on a 2‑DOF base‑isolated building under the Kobe University record, component 090 (KBU090).
What this project shows
- Newmark MDOF implementation: constant‑average‑acceleration scheme in matrix form for the full system.
- Non‑classical damping: damping matrix not of the form \(\alpha\mathbf{M}+\beta\mathbf{K}\), so modal analysis is only approximate.
- Direct vs modal time history: when the modal approximation matches the full MDOF solution and when it drifts.
- Link to response spectrum analysis (RSA): 2‑DOF isolation example compares direct MDOF, modal time history, and RSA with square‑root‑of‑sum‑of‑squares (SRSS) modal combination.
Newmark MDOF demo with non‑classical damping
Two‑degree‑of‑freedom (2‑DOF) shear frame under a short sustained load; non‑classical damping; direct MDOF time history vs modal approximation.
2‑DOF base‑isolated building under Kobe KBU090
Direct MDOF time history via Newmark; modal time history and response spectrum analysis (RSA) with square‑root‑of‑sum‑of‑squares (SRSS) modal combination; drift, isolator displacement, base shear coefficient.
Related pages in this portfolio
Further background on dynamics, spectra, and numerical methods appears in the following course pages.