Implicit Numerical Integration for Hybrid Simulation of Complex Structural Systems

A fully implicit iterative integration procedure is developed for local and geographically distributed hybrid simulation of the seismic response of complex structural systems with distributed nonlinear behavior. The purpose of this procedure is to allow experimental elements in simulations using existing fully implicit integration algorithms designed for pure numerical simulations. The implementation difficulties of implicit integration are addressed at the element level by introducing a safe iteration strategy and using an efficient procedure for online estimation of the experimental tangent stiffness matrix. In order to avoid physical application of iterative displacements, the required experimental restoring force at each iteration step is estimated from polynomial curve fitting of recent experimental measurements. The tangent stiffness matrix is estimated using readily available experimental measurements and by a classical diagonalization approach that reduces the number of unknowns in the matrix. Numerical and hybrid simulations are used to demonstrate that the proposed procedure provides an efficient method for implementation of fully implicit numerical integration in hybrid simulations of complex nonlinear structures. The hybrid simulations presented include distributed nonlinear behavior in both the numerical and experimental substructures. . INTRODUCTION Hybrid simulation combines numerical and experimental methods for cost-effective, large-scale laboratory testing of structures under simulated earthquake loading (Mahin et al. 1989; Shing et al. 1996; Takanashi and Nakashima 1987). The equation of motion is expressed for the combined experimental and numerical components and solved using a time-stepping integration procedure as in numerical simulations. Explicit integrators, such as the Central Difference Method, are simple to implement in a hybrid test, but their conditional stability limits their application to simple structural models. A combination of non-iterative implicit and explicit integration algorithms (Dermitzakis and Mahin 1985), including the operator-splitting method (Nakashima et al. 1990), offer improved stability and accuracy, but use the initial stiffness matrix to predict the nonlinear response of the specimen. A tangent stiffness matrix has been proposed to improve this correction (Ahmadizadeh and Mosqueda 2008b). Fully implicit integration algorithms are widely used in pure numerical simulations of the seismic response of structures for their superior stability and accuracy at larger time steps compared to explicit methods. However, the direct application of implicit integration algorithms to hybrid simulation has been partially limited by the requirement to iterate with experimental substructures and difficulties in estimating the experimental tangent stiffness matrix. Past implementations have relied on conservative iterations that ensure each iteration results in a 1 Dept. of Civil, Structural and Environmental Engineering, University at Buffalo; Buffalo, NY 2 Dept. Civil Engineering, Shiraz University, Shiraz, Iran