Oren Lavan and Robert Levy

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


This paper presents a consistent approach for the optimal seismic design of added viscous damping in framed structures. The approach presented is appropriate for use in elastic as well as yielding frames. The sum of added damping is chosen as the objective function and the performance of the structure, under the excitation of an ensemble of deterministic ground motion records, is constrained. The performance of the structure is measured by the maximal inter-story drifts in both the linear and nonlinear cases. The nonlinear case however, uses an additional performance measure of the normalized hysteretic energy of the plastic hinges

Gradients of the performance measures are first derived to enable the use of an appropriate first order optimization scheme. Moreover, an efficient selection scheme enables the consideration of only a few records rather than the whole ensemble, hence making the optimization process efficient in terms of the computational effort.


The problem of seismic retrofitting of existing structures has gained much attention lately due to the new performance-based-design approach, which allows engineers to design structures for a desired level of seismic performance. Installation of viscous dampers is an effective means for this seismic retrofitting, hence, the problem of optimal design of these dampers is of paramount importance. This problem was tackled by several researchers with limited results for the particular class of regular buildings (see for example Zhang and Soong 1992; Inaudi et al. 1993; Gluck et al. 1996; Takewaki 1997 to name only few). Since most existing buildings are irregular, available methodologies remain academic.

For an efficient and computationally effective solution of the optimization problem of dynamic systems subjected to time varying loads, first order schemes that require constraints’ gradients are preferred. Zero order optimization schemes, e. g. genetic algorithms, require a large number of function and constraints evaluations, that is to say time history analyses, making them less attractive to use.

Several approaches for the gradient computation have been introduced in the literature. Hsieh and Arora (1985) derived the gradients of point-wise as well as integral type constraints for linear elastic systems by deriving the first variations of these constraints which depend on the variation on the displacements of the degrees of freedom. They further used a direct differentiation method of the equations of motion, and alternatively an adjoint variables method, to evaluate these variations on the displacements. Another approach for the gradient computation uses the finite difference method (see for example Falco et al., 2004). Here the derivative of the constraint with respect to each design variable is approximated by the forward or backward finite difference approximation. This method actually requires an additional analysis for each design variable. Conte et al. (2003) distinguished two methods for computing the response sensitivities considering plastic behavior of the structure. The first method uses the differentiation of the response equations with respect to each of the design variables, and then discretizes the resulting response sensitivity equations in time. The second method discretizes the response equations in time, and then differentiates the resulting discrete response


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 663-671.

© 2006 Springer. Printed in the Netherlands.

equations with respect to each of the design variables. It should be noted that both methods require an additional analysis for each design variable.

The present research proposes a gradient based approach for the optimal design of viscous dampers for the seismic retrofitting of existing, regular as well as irregular, structures. This approach uses a first order optimization scheme whose success lies in the ability to derive the gradients of the constraints with respect to the damping coefficients of the dampers. Thus, the main effort in this paper is the gradient derivation of constraints in linear as well as nonlinear dynamic optimization problems under earthquake excitations. The relatively small computational effort associated with their evaluation using the proposed scheme makes these gradients highly desirable.