Stage 1: Selection of the “active” ground motion

In general, the active ground motion is chosen by first sketching a certain response quantity of a single degree of freedom system, having the natural frequency of the structure, versus the damping coefficient, for all records in the ground motion ensemble. For the linear problem, the response quantity sketched is the maximal displacement whereas, for the nonlinear case it is the input energy (according to Uang and Bertero 1990). The record for which the response quantity takes the largest values, for a reasonable range of damping, is chosen as the active record.

Stage 2: Analysis redesign

Following the discussion on fully stressed design an analysis/redesign approach using (6) is adopted.

As will be seen from the examples, the choice of a starting point does not have a large effect on the methodology since the methodology converges very fast to the region of the final solution. In the examples to follow a uniform distributed damping contributing a predetermined percentage of critical damping to the first mode shape is used, hence:

where fd1 = predetermined damping ratio of the first mode; rn1 = circular frequency of the first mode, and ф(1) = first mode shape.

The analysis redesign stage is stopped when the constraint error, max; (pii) -1 takes a small value and the changes in the objective function or in the damping vector for two subsequent iterations is small.

Stage 3: Feasibility check and stopping criteria

Once a design for the “active” ground motions is achieved in Stage 2, the performance of the damped structure for each of the remaining ground motions separately in the ensemble is evaluated using a time history analysis. If the design achieved in Stage 2 violates constraints of other records in the ensemble, i. e. maxi(pii) > 1, the ground motion for which maxi(pii) receives the largest value is added to the active set.

In Examples 2 and 3 only one record is active. This record is easily tracked by the algorithm, and it is expected that the optimization scheme is likely to use, in general, only a few of the records and not whole ensembles. Therefore, the scheme becomes practical in the sense of the computational effort.

The methodology is terminated when no additional ground motion is added to the active set at this stage. If an additional ground motion is needed, then Stage 2 is repeated with the new active set.