Fully Stressed Design

Optimal seismic design of added damping for linear structures, as well as for yielding shear frames, has been achieved by the authors using formal nonlinear programming optimization techniques (Lavan and Levy 2005a, 2005b). Closer to the designer’s heart, however are the analysis/redesign type techniques where an initial choice of the design variables is made (in this case damping coeffi­cients, cdi) then, based on analysis results and a pre-defined recurrence relationship, the initial values are changed. A new analysis is made and the process continues until the designer is satisfied. Usually methods of this type converge very fast (up to five iterations for reasonable convergence of between 5-10% error in the examples of this paper). Moreover, in the classical FSD of trusses, for example, monotonic convergence is exhibited by the objective function (Spillers 1975). The designer is, thus, in full control and may stop at any iteration knowing that his results are the best up to that point.

Having studied the results achieved by the optimization methodologies mentioned above an analogy to the classical fully stressed design (FSD) of trusses seems to emerge. It is observed that, for 2D frames, the optimal design will attain nonzero values of cdi in stories for which the local performance index has reached the allowable, and zero values of cdi in stories for which the local performance index is less than the allowable. In other words dampers are assigned only where the performance index is full. Similarly, for the 3D structures the optimal design will attain nonzero values of cdi in stories of peripheral frames only, for which the local performance index has reached the allowable in at least one loading condition. In stories with no dampers the performance index is less than the allowable.

The recurrence relationship that is suggested in this work thus targets “fully stressedness” of the local performance index and is written as

(6)

where c’dk = the i-th component of the damping vector at the k-th iteration, q is a convergence parameter and piik> = the actual i-th component of the performance index at the k-th iteration (using cdk> as a damping vector). Note that cІ> and and piik> refer to the same location i. e. same story of the

same peripheral frame. In case the active set which is the subset of the ensemble that is considered

ik>

at a certain stage, as will be explained later on, is comprised of more than one ground motion, pii

ik>

is taken as the envelope of pi] for the records within the active set.

The choice of q in (6) affects the efficiency of the method. For larger values of q the method is more stable, i. e. the method is more likely to converge, however the convergence is slower. For the linear problem q = 0.5 may be used whereas for the nonlinear case a value of q = 5 seems to be appropriate.