#### Example 1: Required brace stiffness for an SDOF

To illustrate the use of the design procedure proposed in Section 2, consider an SDOF structure with floor mass of 1000 Kg, lateral stiffness of 150 kN/m and damping equal to 3*%* of critical damping. Suppose that a linear damper with coefficient *c*_{d} = 4.2 kNs/m has been selected to increase the overall structural damping up to 20*%*.

We can apply Equation (9) to calculate the stiffness required by the brace to provide a brace–damper assembly that behaves close to the infinite brace stiffness case. After substituting the numerical values, we can calculate

Notice that, here, for , the brace stiffness is high compared to the structural stiffness; however, it will be seen that it is lower than that generated by three of the four other methods considered. The dynamic response of the SDOF structure fitted with the brace–damper arrangement is first evaluated in terms of the frequency response *T*(*s*) defined in Equation (4). Figure 6(a) presents three different conditions that are considered: the structure without any additional damping device (dashed line), the structure with the added damper and infinite brace stiffness (thick red line), and the structure with the damper and its supporting brace designed using the method proposed here (thin line). The plot shows that the response of the SDOF with the flexible brace is comparable to that of the idealised model with the infinitely rigid brace.

As discussed when presenting the method, the presence of a non-infinite brace stiffness results in deleterious performance only at frequencies higher than the target frequency, which in this case is *ω*_{n}. Nonetheless, this does not affect the overall structural response of the SDOF structure in a substantial manner. Figure 6(b) shows the dynamic response of the SDOF structure when subjected to an earthquake. Again, the three aforementioned conditions are presented. Note that when the brace stiffness *k*_{b} designed by the proposed procedure is used, the system is able to track the response of the ideal case with negligible errors.

For comparison, we have calculated the brace stiffness required by considering the criteria suggested by three relevant previous works in [20, 21] and [24]. The first reference recommends *k*_{b} to be 10 times the storey stiffness. The second one considers the transformation from the Kelvin model (linear spring and dashpot in parallel) to the Maxwell model. Note that this equivalence is only strictly valid at the frequency used in the transformation. Moreover, it is necessary to include a small fictitious stiffness in parallel with the damper in order to define a corresponding Maxwell model. Furthermore, the model outcome is extremely sensitive to the value of the fictitious stiffness chosen for the Maxwell model. We used the natural frequency of the system and arbitrarily selected two fictitious stiffness of 3*%* and 10*%* of the structural stiffness *k* to compute the results reported in what follows. The third reference is a recent work that presents a simple noniterative design procedure for brace–damper assemblies. We use the Optimum Brace Stiffness criterion proposed therein to calculate the brace stiffness given in Table 1.

Table 1. Brace stiffness for the SDOF structure obtained from the different approaches.Approach | *k*_{b} (kN/m) |
---|

This paper | 253.3 |

Fu and Kasai [20] | 1500.0 |

Valles, *et al.* [21] ( 0.03*k*) | 592.5 |

Valles, *et al.* [21] ( 0.10*k*) | 191.5 |

Londoño, *et al.* [24] | 280.6 |

Figure 7 presents a comparison among the frequency responses of the SDOF system with the added brace–damper assembly when the different brace stiffness are considered. Notice that although the *k*_{b} recommended by [20] results in a near-rigid behaviour, the *k*_{b} value is six times greater than that calculated using the method proposed here. As this additional stiffness results in minimal benefit, this near-rigid brace arguably does not represent a cost-effective solution. Furthermore, unlike the procedure proposed in this paper, the transformation presented in [21] offers no indication of how the fictitious stiffness must be selected and how this relates to the final system behaviour, allowing for a very large set of broadly different solutions. Besides, the result offered by the proposed method is highly consistent with that one resulting from the simplified design rule presented in [24].

#### Example 2: Required stiffness for braces in a 10-storey structure

To show the applicability of the proposed procedure to more realistic structures, a 10-storey three-bay frame structure is considered. The structure has been extensively studied before in [13] and [25]. The condensed stiffness matrix matrices that neglects axial deformations is taken from the references as

- (17)

In addition, the mass of each storey is 50 tons, and an inherent 2% Rayleigh damping in the first and second modes is used. The cited references identified an optimal solution that comprises five damper locations at the first, third, fourth, fifth and sixth storeys with corresponding added damper coefficients: kNs/m. The flexibility of the dampers' supporting braces were not considered when obtaining these optimal values.

The performance of the braces is evaluated by simulating the dynamic response of the 10-storey structure with added brace–damper systems under seismic base motion. The structure is excited by the records NGA–1048 and NGA–173 of the Pacific Earthquake Engineering Research Center ground motion database downloaded from [26]. These records are the unscaled fault-parallel component of the Northridge Earthquake in 1994 and the unscaled fault-parallel component of Imperial Valley Earthquake in 1979, respectively (Figure 8).

As pointed out in Section 3.1, the key factor when determining the brace stiffness for an MDOF structure is to identify the target frequency *ω*_{t}. To emphasise this aspect, we consider two cases: (i) selecting the first natural frequency of the structure without dampers as the target frequency (0.4 Hz for this structure) and (ii) selecting *ω*_{t} = 3.5 Hz. The second boundary was selected in such a way that the strongest segment of the frequency content of the earthquakes is covered (dashed lines in Figure 8). After selecting the target frequency, Equation (16) can be used straightforwardly. Table 2 presents the resulting brace stiffness calculated for each storey level.

Table 2. Storey brace stiffness as a result of applying Equation (16) for every value .*ω*_{t} | Brace stiffness ( × 10^{3}kN/m) |
---|

| | | | |
---|

| 189.39 | 51.75 | 6.43 | 17.73 | 42.43 |

3.5 Hz | 1657.5 | 452.91 | 56.32 | 155.19 | 371.36 |

It is worth noticing that one may want to distribute the additional damping in the *i*-th storey into *q* damping devices each one with damping coefficient equal to . In such case, the required brace stiffness would be the value reported in the previous table divided by *q*. Without loss of generality, we assume that the additional damping will be condensed into one device at each corresponding storey – this will not affect the results presented hereafter.

Table 3 presents the resulting frequencies and damping ratios of the first five modes of vibration when considering the different scenarios. The row starting with ‘None’ refers to the case of the structure with no additional damping, while the row for *ω*_{t} = ∞ refers to the ideal case of disregarding the presence of the supporting braces. By examining these values, it is clear that the case considering the target frequency equal to 3.5 Hz gives satisfactory agreement with the ideal case of pure dashpots with regard to both frequencies and damping ratios.

Table 3. Natural frequencies and damping ratios of the first five modes.*ω*_{t} | Natural frequencies (Hz) | Damping ratio (%) |
---|

*ω*_{1}* * | *ω*_{2}* * | *ω*_{3}* * | *ω*_{4}* * | *ω*_{5}* * | *ξ*_{1}* * | *ξ*_{2}* * | *ξ*_{3}* * | *ξ*_{4}* * | *ξ*_{5}* * |
---|

None | 0.40 | 1.32 | 2.43 | 3.80 | 5.55 | 2.0 | 2.0 | 3.1 | 4.6 | 6.6 |

| 0.51 | 1.72 | 3.29 | 4.64 | 7.04 | 23.8 | 10.2 | 11.0 | 8.0 | 9.5 |

3.5 Hz | 0.50 | 1.76 | 4.22 | 4.48 | 8.03 | 26.5 | 15.9 | 31.2 | 20.9 | 15.0 |

∞ | 0.50 | 1.76 | 4.23 | 4.43 | 8.23 | 26.7 | 16.6 | 50.6 | 16.4 | 15.6 |

In addition, the frequency response of the structure in terms of the relative structural displacement at the top storey is presented in Figure 9. The response of two limit cases, the structure without dampers and the idealised structure with only dampers (infinitely rigid braces) are compared against the realistic case of including flexible supporting braces. Both plots show that for all frequencies lower than the target frequency, the structure with flexible braces closely follows the behaviour of the structure with only dampers. In view of the frequency content of the earthquakes under consideration, it is evident that the case in Figure 9(a) will yield poorer results.

To further assess the behaviour of the brace–damper assemblies with braces sized in line with the proposed procedure, we use the set of performance indices presented in Table 4 [27]. The indices *J*_{1} and *J*_{2} measure the improvements in terms of the acceleration response of the structure with added dampers compared to the system without dampers. The indices *J*_{3} and *J*_{4} concern the peak displacement of the top floor and the overall inter-storey drift. Finally, the index *J*_{5} considers the maximum damper force all over the building.

Figures 10 and 11 show the dynamic responses of the 10-storey structure at the top level when it is base-excited by the earthquakes presented earlier. We examine the response in terms of both top storey relative displacement and top storey absolute acceleration. As before, the response of the structure provided with braces of infinite stiffness is compared against the case of flexible braces for the two different values of the target frequency.

From the plots, it can be seen that the brace–damper assemblies whose braces were sized in accordance with the method in Section 3.1 behave satisfactorily and are comparable to the ideal case of rigid braces. A detailed quantitative evaluation was carried out by using the performance indices defined previously. The results are presented in Table 5 for the two earthquakes.

Table 5. Performance index assessment.*ω*_{t} | Northridge | Imperial Valley |
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*J*_{1} | *J*_{2} | *J*_{3} | *J*_{4} | *J*_{5} | *J*_{1} | *J*_{2} | *J*_{3} | *J*_{4} | *J*_{5} |
---|

| 1.54 | 2.37 | 0.28 | 0.65 | 6.43 | 1.15 | 1.31 | 0.60 | 0.97 | 9.37 |

3.5 Hz | 1.36 | 1.85 | 0.25 | 0.53 | 5.86 | 1.08 | 1.18 | 0.55 | 0.86 | 9.99 |

∞ | 1.30 | 1.70 | 0.25 | 0.51 | 5.83 | 1.04 | 1.09 | 0.55 | 0.84 | 10.1 |

The performance indices of the structure with flexible braces sized by using *ω*_{t} = 3.5 Hz show excellent agreement with the performance indices of the ideal case. This confirms that the proposed procedure can deliver fast solutions close to optimality.