The examples that follow were solved using the analysis/redesign technique that was introduced in the previous section. For the sake of comparison, these examples were also solved using a formal optimization methodology similar to the one introduced by Lavan and Levy (2005b). This methodology uses an appropriate first order optimization scheme that requires the derivation of the gradient
of pi. The derivation of this gradient is done by first formulating the equations of motion and the performance indices in a differentiable equivalent state space formulation, i. e. as a differentiable set of first order differential equations, and then applying a variational approach for the gradient derivation.
Examples
Example 1 — linear 2 story shear frame
In order to demonstrate the proposed methodology, and the characteristics of the optimization problem the 2story shear frame introduced by Lavan and Levy (2005a) is studied.
A 5% Rayleigh damping was assumed for the first and second modes. The constraint on the maximum drift given in (2) was set to 0.009m, which is 50% of the maximum drift of the bare frame. The two periods of the structure are 0.281 s and 0.115 s. The structure was excited by the record LA02 from the “LA 10% in 50 years” ground motions ensemble (Somerville et al. 1997), which is the NS component of ElCentro 1940 scaled by a factor of 2.01 downloaded from “http://quiver. eerc. berkeley. edu:8080/studies/system/ground_motions. html”. The mass, damping and stiffness matrices (displacements DOFs) to be used in (2) are:
The contribution of the dampers to the damping matrix is:
and
e = {1 1}r. (10)
The example was solved for a single record. Thus the “active” ground motion in Stage 1 is LA02. Applying the proposed analysis redesign procedure of Stage 2, with starting values obtained from (7) as cd^ = 2795 ■ 1 kNs/m and using q = 0.5 in (6), Figure 2 shows a contour map of the constraint, maxi(pii) < 1, the objective function at the optimum value (straight line) and the iterative progress towards convergence of the analysis redesign.
The total added damping and the constraint’s error (max; (pii) — 1) versus the iteration number are shown in Figure 3.
As can be seen from that figure, the convergence to the region of the optimum is quite fast (a constraint error of 0.35% and a total damping of 1543 occurred in 5 iterations), however, full convergence took 12 iterations. The final damping is cd1 = 1300.4kNs/m and cd2 = 181.4kNs/m. There are no remaining records to apply on the design (feasibility check of Stage 3) since only a single record is considered in this example. Hence, the design achieved is the final design. The value of the total added damping for the optimal design and the value of pi are compared in the first line of Table 1 with the ones achieved using the gradient based approach. As can be seen, the gradient based optimization leads to a, somewhat, lower value of the objective function, however, the violation of the constraint is a bit larger. It should be noted that the formal optimization technique yielded the same height distribution and practically equal values of the dampers (not shown).
Fig. 2. Contour map of the constraint (curved lines), objective function at optimum (straight line) and iterative progress towards convergence (dots). 
Table 1. Optimal design values.

Example 2 — linear 8 story 3 bay by 3 bay asymmetric framed structure
In order to demonstrate the applicability of the proposed methodology to 3D structures, the 8story 3bay by 3bay asymmetric framed structure introduced by Tso and Yao (1994) is used. Inherent 5% Rayleigh damping in the first and second modes is assumed. The methodology was performed neglecting axial deformations, i. e. 3 degrees of freedom per floor were used (two horizontal displacements and torsional angle). The ground motion ensemble was chosen as the "LA 10% in 50 years” ensemble (Somerville et al. 1997), and the allowable drift at the peripheral frames was chosen as 1.0% of the story height. Stage 1 sketches the maximal displacement of a single degree of freedom system having the natural period of the frame (1.15 sec), versus the damping coefficient for each record in the whole ensemble. The record LA16 was chosen to start the process since its spectral displacement for all reasonable damping range had the largest value.
Applying the proposed analysis redesign procedure of Stage 2, with starting values obtained from
(7) as = 1000 • 1kNs/m and using q = 0.5 in (6), Figure 4 shows the total added damping and the constraint’s error (max/ (pii) — 1) versus the iteration number for the record LA16.
As can be seen, the convergence to the region of the optimum is quite fast (a constraint error of 3.9% and a total damping of 160121 kNs/m occurred in 5 iterations), however, full convergence took 20 iterations. The final damping and the components of pi for the damped frame are show in Figure 5.
Applying the remaining records in the ensemble on the design (Stage 3) indicated that there was no record that led to greater performance indices than that of the active record, LA16. Hence, the optimization process was terminated. A comparison of the total added damping and pi with these of the gradient based optimization solution is given in the second line of Table 1 with the same behavior as the previous example, i. e. the same distribution and practically equal values of the dampers where achieved using the formal optimization technique (not shown).
As can be seen (Figure 5), the optimal solution assigns damping in stories of peripheral frames that fully utilized their local performance index (stories number 25 of frame 1 and stories number
Fig. 5. (a) optimal damping of the damped structure for LA16: left edge (frame 1) and right edge (frame 4) and b) maximal drifts for LA16 (= envelope values): left edge (frame 1) and right edge (frame 4). 
24 of frame 4), and no damping is assigned elsewhere.
Example 3 — yielding 10 story shear frame
In order to demonstrate the applicability of the proposed methodology to yielding shear frames under an ensemble of ground motion records, the 10story shear frame introduced by Lavan and Levy (2005b) with inherent 2% Rayleigh damping in the first and second modes is used.
The fundamental period of the structure is 1.0 sec; Han /, which appears in (4), is taken as the elastic energy at yielding of the ith story multiplied by 0.2 x 16 = 3.2. The secondary slope ratio for all floors was chosen as 0.02. The ground motion ensemble was chosen as the "SE 10% in 50 years” ensemble (Somerville et al. 1997). The record SE19 started the process (Stage 1) since its spectral input energy for the fundamental period of the structure had the largest value for a reasonable damping range (see Lavan and Levy, 2005b). A nonlinear analysis was performed on the bare frame for this record and revealed values much larger than 1 for the components pii. Applying the proposed analysis redesign procedure of Stage 2, with starting values obtained from (7) as c^ = 12095 • 1 kNs/m and using q = 5 in (6), leads to the optimal damping and the components for the damped frame excited by SE19 are shown in Figures 6(a) and 6(b).
The total added damping and the constraint’s error (maxi (pii) — 1) versus the iteration number are shown in Figure 7.
As can be seen from that figure full convergence took 12 iterations. Applying the remaining records in the ensemble on the design indicated that the design is feasible, thus, the design achieved is the final design. A comparison of the total added damping and pi with those of the gradient based optimization solution is given in the third line of Table 1. In this case of yielding structure, the same result observed in the linear cases is repeated, i. e. here too, the formal optimization technique yielded the same distribution and practically equal values of the dampers (not shown).
Fig. 6. (a) Design supplemental damping and (b) damage indices envelope for the designed damped frame excited by the SE 10% in 50 years ensemble. 
Fig. 7. Convergence to optimum. 
A methodology for the optimal design of added viscous damping for an ensemble of realistic ground motion records with a constraint on the maximum drift for linear structures, and on the maximum energy based damage index for nonlinear shear frames, was presented.
The optimization methodology is based on an iterative procedure of the analysis/redesign type that is appropriate for engineering practice. This scheme seems to converge very fast to the region of the final design and is applicable to nonconvex problems.
The final solution coming from formal optimization is characterized by equal maximal drifts in the linear case, and equal maximal damage indices in the nonlinear case, for floors with assigned dampers, and lower maximal drifts/damage indices for floors with no assigned damping. This fully stressed result was targeted by the iterative procedure and is desired in structures due to the uniform distribution of damage (structural and nonstructural) throughout the structure.