Linear Frames – A Particular Case

A class of structures, usually regular, can be brought to behave elastically under an earthquake excitation, by the addition of a reasonable amount of added damping. In this case nonlinear analysis methodologies, which are computationally expensive, are not essential and some of the nonlinear performance indices, such as hysteretic energy, become meaningless. Since the optimization problem of linear structures is a particular case the same procedure presented earlier holds. Minor changes ease the computations. First, the equations of motion are reduced to their linear form by substituting

Ka = K and fh (t) = 0 (or, equivalently, fh (t) = 0 ; fh (to )= 0 ). Substituting these relations into Eqs. 25 and 26 leads to simpler equations for the gradient computation as well. Then, the constraint on the hysteretic energy will be omitted since no hysteretic energy dissipates in the elastic range. This leads to simpler equations for the gradient computation as well since Eh (tf )= 0 . Inter-story drifts remain

the only constraints. In 3D structures the inter-story drifts of the peripheral frames are used. Fundamental Results

• The optimal design of added damping in 2D frames, assuming linear behavior of the damped structure, and in 2D yielding frames is characterized by assigning damping only in stories that reached the allowable drift.

• The optimal design of added damping in 2D yielding shear frames is characterized by assigning damping only in stories that reached the allowable normalized hysteretic energy.

• The optimal design of added damping of 3D framed structures is characterized by assigning damping at the peripheral frames only, where the peripheral drift has reached the allowable.

These observations are strikingly analogous to the classical “fully-stressed-design” behavior of optimal trusses reported in the sixties.


A gradient based methodology for the optimal design of added viscous damping for an ensemble of realistic ground motion records with constraints on the maximum inter-story drifts for linear frames, and additional constraints on maximum energy based local damage indices for nonlinear frames, was presented. This methodology is appropriate for use in linear, as well as nonlinear, frames.

The computational effort is appreciably reduced by first using one “active” ground motion record since experience shows that one or two records dominate the design.

The gradients of the constraints were derived so as to enable the use of an efficient first order optimization scheme for the solution of the optimization problem. The approach for the gradient derivation has several advantages over other approaches. It is appropriate for use when the equations of motion assume nonlinear plastic behavior as well, and it requires a relatively small computational effort, in the form of a single additional solution of a set of differential equations (that is, the equations for the Lagrange multipliers).


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