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SUMMARYA methodology for the optimal design of supplemental viscous dampers for regular as well as irregular yielding shear-frames is presented. It addresses the problem of minimizing the added damping subject to a constraint on an energy-based global damage index (GDI) for an ensemble of realistic ground motion records. The applicability of the methodology for irregular structures is achieved by choosing an appropriate GDI. For a particular choice of the parameters comprising the GDI, a design for the elastic behavior of the frame or equal damage for all stories is achieved. The use of a gradient-based optimization algorithm for the solution of the optimization problem is enabled by ÿrst deriving an expression for the gradient of the constraint. The optimization process is started for one 'active' ground motion record which is e ciently selected from the given ensemble. If the resulting optimal design fails to satisfy the constraints for other records from the original ensemble, additional ground motions (loading conditions) are added one by one to the 'active' set until the optimum is reached. Two examples for the optimal designs of supplemental dampers are given: a 2-story shear frame with varying strength distribution and a 10-story shear frame. The 2-story shear frame is designed for one given ground motion whereas the 10-story frame is designed for an ensemble of twenty ground motions.