Robert Levy and Oren Lavan

Faculty of Civil and Environmental Engineering,
Technion – Israel Institute of Technology, Haifa 32000, Israel
E-mail: cvrlevy@tx. technion. ac. il


This paper presents an efficient and practical procedure for the optimal design of added damping in framed structures. The total added damping is minimized while inter-story performance indices for linear and nonlinear structures are chosen and restricted to allowable values under the excitation of an ensemble of realistic ground motion records. Optimality criteria are formulated based on fully stressed characteristics of the optimal solution and a simple analysis/redesign procedure is proposed for attaining optimal designs. Results of three examples presented compare well to those obtained using formal gradient based optimization.


In the modern design of buildings to withstand strong earthquakes life safety is no longer the only concern but rather, performance whereby a prescribed level of damage is designed for. Hence, ret­rofitting of structures for a higher level of seismic protection may be needed. One of the means for achieving this enhancement to seismic performance is using supplemental damping which is the concern of this paper.

Various procedures for the design of added viscous damping, for linear behavior of damped 2D structures, were proposed by several researchers (for example Constantinou and Tadjbakhsh 1983; Zhang and Soong 1992; Fu and Kasai 1998; Inaudi et al. 1993; Gluck et al. 1996; Takewaki 1997; Lavan and Levy 2005a). Some of these procedures were extended to linear 3D structures (for example Wu et al. 1997; Takewaki et al. 1999). These methodologies, will usually require mathematics of stochastic processes, optimization methods, and/or variational mathematics – tools that are not that familiar to the practicing engineer. Garcia (2001) simplified the Sequential Search Algorithm, originally proposed by Zhang and Soong (1992) for stochastic models of the excitation, and made it appropriate for practical use. However, his method is restricted to linear structures under the excitation of a single deterministic record that does not guarantee optimal damping distribution of the dampers.

Procedures were proposed for the design of viscous dampers for yielding structures as well (Kim et al. 2003; Shen and Soong 1996; Lavan and Levy 2005b. The primary concern of the methodologies proposed by Kim et al. (2003), and by Shen and Soong (1996), was to estimate the total added damp­ing needed rather than its distribution. The procedure proposed by Lavan and Levy (2005b) requires some nonlinear programming background and variational mathematics and may not be that easy to implement in the practicing design office. Allowable stress algorithms of the analysis/redesign procedures may be more suitable.

Allowable stress algorithms go back to the classical design of trusses, whereby the weight is minimized for a given allowable stress. Design problems of this type may be achieved iteratively using a two step algorithm in each iteration cycle. In the first step an analysis is performed for a given preliminary design, whereas in the second step the design is changed using a recurrence


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 303-315.

© 2006 Springer. Printed in the Netherlands.

relationship which, for the truss problem is the ratio between the current stress and the allowable stress for each member. The algorithm possesses a fixed point (Levy 1991), i. e. fully stressedness and exhibits monotonic convergence properties. Convergence yields a statically determinate fully stressed design, with members out of the design having strains smaller than the allowable. This result appeared in the literature as early as 1900 (Cilley 1900). It was later shown (Levy 1985) that this design is a Karush-Kuhn-Tucker point and therefore, an optimal design. Algorithms closely related are the optimality criteria based algorithms as described by Khot et al. (1976), Venkayya (1978) and Rozvany (1989) to name only a few.

This paper presents a fully stressed design algorithm of the analysis/redesign type for the design of added viscous damping for linear as well as yielding structures for a given ensemble of realistic earthquake records and a specified target performance index. An algorithm of this kind is well liked by design engineers because its process is transparent and uses available and familiar dynamic time-history analysis programs that are common in civil engineering practice rather than unfamiliar mathematical optimization tools that need problem specific tailoring. Results were found to be in good agreement with optimal designs achieved using gradient based optimization. It should be noted in passing, that the fully stressed design described herein for viscous dampers has not been rigorously proven to be optimal yet.