*4.4. Seismic Performance*

Now, the seismic performance of the building is evaluated under the action of the four seismic records used in the optimization process. It is worth noting that the scope of this investigation is limited to the linear behavior of the structural system. Thus, the numerical model does not take into account any type of damage or yielding process affecting the structure. Furthermore, the computed displacements are presumable within the elastic behavior of the system.

Figures 5 and 6 show the history of displacements at the 32nd story of the building, for *μ* = 0.02 and 0.05, respectively. Even though both TMD options (*μ* = 0.02, and 0.05) can effectively reduce the dynamic response of displacements, it is visibly clear that TMDs

with *μ* = 0.05 reach greater reductions in lateral displacements at the upper level of the structural system, regardless of the objective function.

**Figure 5.** Displacement of the 32nd story of the structure equipped with TMDs with *μ* = 0.02.

**Figure 6.** Displacement of the 32nd story of the structure equipped with TMDs with *μ* = 0.05.

Figures 7 and 8 present the comparison of the peak displacement for *μ* = 0.02 and 0.05, taking into account the six proposed objective functions. It is shown that the greatest reductions in the peak displacement at each story-level are achieved using the devices with the highest mass ratio, as reported in the literature [30–50]. Moreover, the greatest reductions are observed at the higher story-levels of the structure. This behavior can be better understood by analyzing the proposed objective functions, which are mainly focused on diminishing the dynamic response at the 32nd story of the building. Hence, the optimization process influences indirectly the response parameters of the floors near the upper level.

**Figure 7.** Maximum floor displacement at each story level of the structure equipped with TMDs with *μ* = 0.02.

The numerical results show that, using *μ* = 0.02 TMD, the reduction of the maximum displacements of each floor of the structure is more effective using Kobe and Loma Prieta accelerograms than El Centro and Northridge cases. Moreover, it may be affirmed that the greatest reduction in displacement is approximately 32.13%, reached at the 25th story-level of the structure when it is controlled by the TMD designed by the J4 approach and using the Loma Prieta earthquake excitation. This reduction shows consistency, since the J4 approach gives the greatest possible weight to the reduction of the displacement peak over the RMS displacement values.

**Figure 8.** Maximum floor displacement at each story level of the structure equipped with TMDs with *μ* = 0.05.

The greatest reductions in the horizontal peak displacement at the top-story level are attained using TMDs designed through the objective functions OA1 and J4. Using the El Centro earthquake, the horizontal peak displacement is reduced from 0.336 m to 0.280 m (16.60% reduction) using the function OA1 and J4 indistinctly; similar results are observed using the Kobe earthquake, where the peak displacement decreases from 1.243 m to 1.032 m (17% reduction). Now, under the Loma Prieta accelerograms, the maximum displacement reduces from 0.222 m to 0.162 m (27% reduction) using the objective function J4, while, for the Northridge excitation, the maximum displacement is decreased from 1.051 m to 0.980 m (7% reduction) with the OA1 approach.

The controlled response of the structure using optimally designed TMDs with *μ* = 0.05 is remarkable. The best performances are achieved with devices optimized through OA1 and J4 approaches, which are mainly focused on reducing the peak displacements, showing reductions of up to 43 and 42%, respectively. Nevertheless, the reductions in the maximum displacements at each floor achieved by the devices designed with the remaining objective functions are notable as well, exhibiting reductions up to 39, 38, 38, and 40% for the cases of optimization OA2, J1, J2, and J3, respectively. Once more, using the Northridge earthquake, the smallest reductions obtained ranged from 12 to 17% at the higher story-levels of the structure. Besides, for the three remaining accelerograms, the reductions in the maximum floor displacement oscillate between 20 and 43%.

Furthermore, the dynamic response of displacements observed in Figures 7 and 8 denotes an S-shape trend. The curve shows a linear behavior between 1st and 20th story-levels; then, there is a setback in the curve between 27th and 32nd story-levels, where it becomes linear again, although with a different inclination from the first segmen<sup>t</sup> representing the greatest response reductions. This behavior is closely related to the whipping lash effect that has been previously observed in high rise buildings [66]. This behavior is largely caused by changes in the compressive strength in the concrete used in the column sections of the building, which causes changes in stiffness and the dynamic behavior of the structural system.

On the other hand, Figures 9 and 10 present the behavior of the RMS values of the displacements at each level of the building controlled by TMDs with mass ratios *μ* = 0.02 and *μ* = 0.05, respectively.

**Figure 9.** RMS displacement at each story level of the structure equipped with TMDs with *μ* = 0.02.

**Figure 10.** RMS displacement at each story level of the structure equipped with TMDs with *μ* = 0.05.

Figure 9 shows reductions fluctuating from 19% (Northridge excitation at 23rd storylevel) and 42% (Kobe excitation at 23rd story-level). Besides, the reductions in the RMS response of displacement of the roof story-level are 12, 31, 24, and 17% for the El Centro, Kobe, Loma Prieta, and Northridge accelerograms, respectively. The analysis of the RMS response of displacement curves at each floor allows the establishment that the control devices designed through the OA2, J1, and J2 approaches present the best performances when the RMS of displacements is analyzed; accordingly, TMDs designed with objective functions OA1 and J4 achieve lower reductions.

The best reductions in the RMS values of the displacements at each level of the structure are obtained when the building is equipped with TMDs with *μ* = 0.05. These reductions are attained using the OA2 and J1 approaches: 20, 42, 33, and 26% for the El Centro, Kobe, Loma Prieta, and Northridge accelerograms, respectively. Among these reductions, the most outstanding is attained under Kobe excitation, since the reductions between 15th to 32nd story-levels ranged between 40% (0.286 m to 0.170 m at 15th storylevel) and 52% (0.334 m to 0.159 m at the 22nd story-level).

It can be noticed that the reductions in the RMS response of displacement at each floor of the structure exhibit a much more uniform and efficient behavior than the reductions of the maximum horizontal peak floor displacements, especially with *μ* = 0.05 TMDs, and objective functions OA2, which focus on reducing the RMS displacements, and J1, which gives the greatest weight to this parameter. Using the Kobe and Northridge records, the results report the highest response magnitudes and, at the same time, the TMDs exhibit the greatest performance. This behavior suggests that the objective functions for the optimal design of TMDs should be focused on minimizing the RMS values of displacements, or, instead, a linear combination in which the greatest weight is given to this parameter should be used. To verify this statement, Figures 11 and 12 present a comparative analysis of the six objective functions and the attained PI.

**Figure 11.** PI values for every optimization approach using TMDs with *μ* = 0.02.

**Figure 12.** PI values for every optimization approach using TMDs with *μ* = 0.05.

Figures 11 and 12 demonstrate that objective functions OA2, J1, and J2 lead to the most efficient design parameters for TMDs. Consequently, TMDs designed through these objective functions are the ones that best improve the seismic performance of the case-study by given major importance to reducing the RMS response of displacements. Furthermore, the PI values of the TMDs designed through OA2, J1, and J2 demonstrates that the control of the RMS values of displacements in the structure is the best possible within both evaluated parameters, which implies significant reductions in the dynamic response over time. Moreover, the horizontal peak response of displacements is markedly reduced as well. Hence, according to these results, it may be claimed that the objective function J1 is the most balanced and effective among the six objective functions analyzed in this paper, despite the small differences in performance with OA2 and J2.

#### *4.5. Practical Design Recommendations*

Once the optimal design parameters have been determined for each benchmark record, an attempt is made to establish a single set of optimal design parameters to work properly under any acceleration record. In that sense, the methodology proposed by Fallah and Zamiri in [60] is implemented for that purpose. This methodology establishes average design parameters according to the weighted response reduction percentages obtained previously. Therefore, the design parameters obtained from the objective function J1, which was the most effective to control the dynamic response of displacement, are replaced in Equations (23) and (24):

$$\zeta\_{d\text{ }avg} = \frac{\sum\_{i=1}^{4} \left( \zeta\_{opt} \ast R\_{\text{mfd}} \ast R\_{\text{RMS}} \right)}{\sum\_{i=1}^{4} \left( R\_{\text{mfd}} \ast R\_{\text{RMS}} \right)} \tag{23}$$

$$f\_{\text{avg}} = \frac{\sum\_{i=1}^{4} \left(f\_{\text{opt}} \ast R\_{\text{mfd}} \ast R\_{\text{RMS}}\right)}{\sum\_{i=1}^{4} \left(R\_{\text{mfd}} \ast R\_{\text{RMS}}\right)}\tag{24}$$

where *R*mfd and *R*RMS denote, respectively, the reduction of the maximum floor displacement and the reduction of the maximum RMS displacement defined according to Equations (25) and (26). In addition, the counter *i* in the sum numbers the accelerograms used in the tuning process. More records in the optimization process will enhance the accuracy of the average design variables; however, for this example, only the numerical results computed from the records in Table 1 are considered. The computed set of average design parameters *ζd avg* and *f avg* are reported in Table 5.

$$R\_{\rm mfd} = \left(1 - \frac{\max\left(|x\_n|\right)}{\max\left(|x\_n^\*|\right)}\right) \tag{25}$$

$$R\_{\rm RMS} = \left(1 - \frac{\max(\rm RMS(x\_{ll}))}{\max\left(\rm RMS(x\_{ll}^\*)\right)}\right) \tag{26}$$

**Table 5.** Average design parameters for TMDs with *μ* = 0.02 and *μ* = 0.05.


Thereafter, the case-study is subjected to the action of the Petrolia and San Fernando acceleration records, whose details are presented in Table 6. The main purpose of using acceleration records different from the benchmark records used in the optimization process is to verify the performance of the structure and the robustness of the WOA methodology under any type of random excitation.


**Table 6.** Details of the new ground-motion records used.

Figures 13 and 14 illustrate the response of the structure with TMDs designed with the average design parameters subjected to Petrolia and San Fernando earthquakes, respectively. Similarly, Tables 7 and 8 show the response reduction on the 32nd floor of the building. As it can be seen, the behavior of the structure controlled via the TMDs with the average design parameters exhibits notable reductions in the horizontal peak displacements and the RMS response of displacements. The decrease in the response is especially significant in the RMS values of displacement, which means that the use of the weighted procedure could be potentially applied for the design of TMDs in structures subjected to earthquake loads. Furthermore, future works should consider a large number of seismic records in order to obtain a more reliable set of design parameters that best fit a wide range of possible ground motions.

 **13.** Performance of the structure equipped with TMDs with average design parameters subjected to

**Figure**Petrolia earthquake.

**Figure 14.** Performance of the structure equipped with TMDs with average design parameters subjected to San Fernando earthquake.



**Table 8.** Response parameters for the 32nd story of the structure subjected to San Fernando earthquake.

Reductions up to 6 and 30% are observed in horizontal peak displacements and RMS displacements, respectively, using the TMD designed with a unique set of parameters derived from the methodology described in [60]. Although the methodology was originally proposed for base isolation systems, it can be adapted for the tuning of linear mass dampers since, for both base isolation systems and TMDs, optimal design variables must be determined. At first glance, the reductions attained with the average design values are not substantial. However, in earthquake engineering practice, it is impossible to know the ground motion excitation that will affect the structural systems, and, accordingly, the Petrolia and San Fernando Earthquake present dynamic properties different from the records for which the TMD was originally tuned. Thus, although small, these results validate the robustness of the optimization based on WOA and demonstrated the efficiency of the algorithm over other metaheuristics, like DEM and conventional tuning methodologies based on closed-form expressions. Finally, to achieve a realistic tuning, other critical variables should be considered, and a more comprehensive analysis is required (e.g., analysis on the stochastic nature of the seismic site conditions).
