Analytical approach for the design and optimal allocation of shape memory alloy dampers in three-dimensional nonlinear structures

The purpose of this paper is to present a simplified nonlinear equations-model that is intended to design shape memory alloy (SMA) dampers in three-dimensional structures of unsymmetrical-plan and in yielding shear-frames. The equations-model is based on a schematic idealization of the SMA damper's hysteretic behavior and the structure's transient tangent stiffness matrix, which corresponds to yielding shear-frames, and is commensurate with structural design methodologies. As of today, there are very few seismic design methodologies that use SMA dampers. In this paper, the analytical equations-model is implemented in a seismic retrofit search algorithm modified for SMA dampers. The algorithm sequentially places the SMA dampers at the story with the maximum interstory drift gain magnitude, calculated using the transfer function matrix between the ground acceleration components and the frames' interstory drifts. The transfer function matrix stems from a linear state-space formulation, but the tangent stiffness matrix is nonlinear. In order to cater to this issue, a new iterative approach suggests updating the tangent stiffness matrix in accordance with the transfer function matrix's peak interstory drift magnitudes – until the tangent stiffness matrix converges. A numerical example shows that the iterative process takes no more than two iterations.