Optimal design of viscous dampers and their supporting members for the seismic retrofitting of 3D irregular frame structures

This paper presents a formal optimization methodology for the design of seismic retrofitting of three-dimensional irregular buildings. The damping coefficients of viscous dampers potentially allocated in given feasible locations as well as the stiffness of their supporting braces are adopted as design variables. The objective function minimizes a cost function of the dampers while constraints are added to limit various responses of interest to allowable values under a filtered white noise excitation (e.g., interstory drift at each location separately, total acceleration at each location separately, force of each damper, stress in each supporting brace, force/forces of each structural member, base shear, overturning moment at the base, side constraints on dampers' forces and braces' cross sections, etc.). A first-order optimization method is adopted for that purpose. The constraints on various normalized responses are condensed to a single constraint on their maximum value and the gradient required is efficiently derived analytically using the adjoint analytical method. Thus, a computational effort at the order of a single additional analysis is required for the evaluation of the gradient of the constraint regardless of the number of design variables considered or responses to be constrained. This efficient scheme enables a study on the effect of limiting the brace size with and without a limit on its stresses.