Fractional Differential Equation Bearing Models for Base-Isolated Buildings: Framework Development

Base isolation is a powerful technique to prevent damage in low- and medium-rise structures during an earthquake. Nowadays, the extensive use of high-damping viscoelastic (VE) materials in base isolators has motivated the necessity to model its mechanical behavior more accurately. Traditional approaches, such as Maxwell and Kelvin models, are often used to predict VE properties. However, these models cannot precisely represent the material’s frequency-dependent behavior. Mathematical models using fractional derivatives (FD) have been shown to have the features needed to capture and predict the salient VE characteristics. Despite the possibility of such accurate descriptions using FD models in VE material applications, their implementation has been limited because of the complex computations needed to obtain the mathematical solution. This is especially true in a base-isolated (BI) building, when the bearing-superstructure coupled system includes both fractional and integer differential equations. In this study, a novel framework is developed based on dynamic substructuring to analyze a hybrid BI system with both integer and fractional order differential equations. To demonstrate the framework, the seismic response of a multidegree of freedom building model coupled with a FD state space model governing the base isolators is obtained. The structural response of the asymmetric-plan 8-story benchmark building with 92 high-damping VE bearings, governed by the fractional derivative Zener (FDZ) model, is calculated and evaluated. Using this well-known benchmark structure, a numerical comparison is performed considering the seismic response of the building fixed to the ground, the BI building with traditional elastomeric bearings, and the BI building with FDZ bearings. The results demonstrate that the framework developed provides an effective and reliable approach to evaluate the hybrid base-isolated system equations.