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Road bridge designs are based on technical standards, which, to date, consider dynamic loading as equivalent static loads. Additionally, the few engineers who perform a dynamic analysis typically do not consider the effects of bridge-vehicle interaction and also simplify the road’s irregularity profile. Moreover, often, even when a simplified dynamic analysis is carried out and shows that there will be a high dynamic amplification factor (DAF), designers prefer to solve this problem by adopting high safety factors and thereby oversizing the bridge, rather than using energy dissipation devices that would allow reducing the amplitude of vibration. In this context, the present work proposes a complete methodology to minimize the dynamic response of road bridges by optimizing multiple tuned mass dampers (MTMD), taking into account the bridge-vehicle interaction, the random profile of pavement irregularities, and the uncertainties present in the coupled system and in the excitation. For illustrative purposes, the coupled vibration problem of a regular truck traveling on a random road profile over a typical Brazilian bridge is analyzed. Three different scenarios for the MTMD are considered. The proposed optimization problem is solved by employing the Whale Optimization Algorithm (WOA). The results showed the excellent ability of the proposed methodology, reducing the bridge’s DAF to acceptable values for all analyzed cases, considering or not the uncertainties present in the system. Furthermore, the results obtained by the proposed methodology are compared with results obtained using classical tuned mass damper (TMD) design methods, showing the best performance of the proposed optimization method. Thus, the proposed method can be employed to optimize MTMD, improving bridge design.

The design of road bridges is based on technical standards, which generally consider the dynamic loads caused by vehicle traffic as equivalent static loads. This assumption simplifies the design, however, does not adequately represent reality. Thus, many discussions have taken place on how to improve road bridge designs. In this context, some studies that consider a basic dynamic analysis, adopting simplified models of beams and/or vehicles, have emerged, as, for instance, Inbanathan and Wieland [

Initially, the bridge-vehicle interaction problem was considered only in railway bridges, in which Willis [

However, even when a dynamic analysis is performed, as described in the above papers, and it shows that the dynamic amplification factor (DAF) is high and consequently large vibration amplitudes will occur, even so, most engineers prefer to use high safety factors, often oversizing the design, instead of opting for solutions that increase the energy dissipation capacity of the system, thereby reducing vibration amplitudes. Thus, studies on energy dissipation devices are very important and can help to improve the design of bridges.

Among passive energy dissipation devices, one of the most widely used is the tuned mass damper (TMD). The use of vibration absorbers began in 1909 when Hermann Frahm proposed a kind of TMD. After that, especially nowadays, a rapid increase in the development and application of passive energy dissipation devices, such as viscous fluid dampers, viscoelastic dampers, friction dampers, and metallic yield dampers, has occurred (Soong and Dargush [

Concerning bridges and footbridges, Pakrashi et al. [

However, in addition to the use of energy dissipation devices, as described in the last paragraph, it is also important to optimize these devices to achieve the best possible performance with minimum cost. In this context, the most recent works, besides presenting the implementation of vibration control devices, also present the optimization of their parameters. Regarding footbridges and bridges, Miguel et al. [

Thus, the present work aims to connect all the topics discussed above, proposing a complete methodology for dynamic analysis of road bridges, which takes into account the interaction among vehicle-pavement-bridge-TMD, optimizing the parameters of a single TMD and MTMD, in order to minimize the dynamic response of the bridge, thus ensuring safety and comfort to users. Uncertainties present in the bridge, vehicle, and pavement parameters are also taken into account. So, this work aims to contribute with a complete methodology of dynamic analysis and optimization of TMD and MTMD to improve the design of road bridges.

The present work is organized as follows: after this introduction, Section

This section presents the theoretical background and the essential equations for understanding and formulating the problem.

In order to model the stochastic pavement irregularity profile, the ISO 8608 [^{3},

The values of the reference vertical displacement PSD,

Geometric mean of the reference PSD for each road class.

Road class | Degree of roughness, ^{−6} m^{3}) |
---|---|

A | 16 |

B | 64 |

C | 256 |

D | 1024 |

E | 4096 |

F | 16384 |

G | 65536 |

H | 262144 |

After obtaining

This random pavement roughness model was already adopted by the authors in previous works, as, for instance, in Fossati et al. [

The vehicle used in this work is a regular Brazilian truck, which is modeled as a five DOFs system, as shown in Figure _{s}), including the mass of the main body of the truck, and three unsprung masses (_{ns}), which include the masses of the suspension, wheels, and tires. _{s} is the mass moment of inertia of the main body. The three unsprung masses are connected to the sprung mass through springs and dampers, which represent the dynamic properties of the suspensions (_{s} and _{s}). In turn, these three unsprung masses are linked to the ground again through springs and dampers, which represent the dynamic properties of the tires (_{t} and _{t}). _{1}, _{2}, and _{3} are the distances from the center of mass to the rear/front axles and the 5 DOFs are the vertical displacement of the sprung mass at the center of mass (_{s}), the pitch angle of the sprung mass at the center of mass (_{s}), and the three vertical displacements of the 3 unsprung masses (_{ns1}, _{ns2}, and _{ns3}).

Vehicle model.

The bridge used in this work is a typical RC girder bridge without balances, regularly found in Brazil, which is modeled as an Euler-Bernoulli beam, discretizing the deck through the Finite Element Method. Initially, the bridge’s mechanical properties are assumed to be constant throughout its length; however, in the last analysis, to take into account the uncertainties present in the system, Young’s modulus, the density, and the damping ratio of the bridge are considered as independent Gaussian random variables with known mean and coefficient of variation. Each node of the beam has two DOFs, being the vertical displacement and the rotation.

Each one of the tuned mass dampers is modeled as a 1 DOF mass-spring-damper system which is coupled to the bridge central nodes, as shown in Figure _{tmd}) is assumed to be a percentage fixed value of the total mass of the bridge, while stiffness and damping constants (_{tmd} and _{tmd}) are design variables that are optimized to minimize the bridge dynamic response.

TMD model coupled to the bridge.

Three different scenarios are evaluated, with 1, 2, and 3 TMDs, installed on the bridge central nodes, as shown in Figure

(a) Scenario 1: 1 TMD with the total TMD mass, (b) Scenario 2: 2 TMDs with half of the total mass each, and (c) Scenario 3: 3 TMDs with one-third of the total mass each.

The model of Figure

Coupled vehicle-pavement-bridge model.

The equations of motion of the coupled bridge-vehicle problem can be expressed as

To solve this complex coupled problem, the coupled mass, damping, and stiffness matrices should be assembled, as explained, for instance, in a previous paper (Pagnoncelli and Miguel [

Thus, the interaction force between bridge and vehicle for the ^{th} tire is given by^{th} tire, ^{th} tire, ^{th} unsprung mass, ^{th} tire, ^{th} tire, and a dot over a symbol indicates differentiation with respect to time.

Additionally, it is important to take into account the weight force given by^{th} unsprung mass, and ^{2}.

As explained previously, the TMDs are installed on the bridge central nodes, adding the TMD constants (_{tmd} and _{tmd}) in the corresponding DOFs of the bridge, while the masses of the TMDs (_{tmd}) are added to the mass matrix of the bridge-vehicle coupled system, as depicted in Figure

After that, the dynamic analysis of the coupled problem is carried out through the Newmark integration method, emphasizing the maximum vertical displacement at the bridge central node.

After assembly of the coupled problem, the optimization procedure may be performed. The proposed optimization process has as objective function the minimization of the maximum vertical displacement at the center of the bridge span (_{max}), having as design variables the stiffness (_{tmd}) and damping (_{tmd}) constants of the TMDs, while the mass of the MTMD (_{tmd}) is considered a percentage fixed value of the total mass of the bridge (_{b}). The constraints are the lower (

This optimization problem may be solved through the WOA summarized in the next section.

As described earlier, the optimization problem discussed in this work is complex. Such problems may be nonconvex and, therefore, must be solved through optimization methods capable of dealing with such problems. Metaheuristic algorithms are well suited for solving these optimization problems (Miguel and Fadel Miguel [

In this context, among the metaheuristic algorithms, the nature-inspired metaheuristic optimization algorithm, called WOA, recently proposed by Mirjalili and Lewis [

The WOA is a nature-inspired metaheuristic optimization algorithm that imitates the social behavior of humpback whales. It is based on the strategy of bubble-net hunting. According to Mirjalili and Lewis [

The WOA included three operators to simulate: the search for prey, encircling prey, and bubble-net foraging behavior of humpback whales. Each step of the algorithm is briefly explained in the next subsections.

In this step, the WOA defines which is the best search agent and tries to update the positions of the other agents in relation to this one, simulating the behavior of humpback whales, which can identify the position of the prey and encircle them. Mathematically, it is given by

In this step, the exploitation phase is carried out, modeling the bubble-net behavior of humpback whales. For this, two approaches are applied:

Shrinking encircling mechanism

The value of

Spiral updating position

This approach initially calculates the distance between the humpback whale and the prey. Thus, in order to imitate the movement of humpback whales, a spiral equation is created between the position of whale and prey. Additionally, humpback whales are known to swim around their prey within a shrinking circle and along a spiral-shaped path. To model this simultaneous behavior, it is assumed that there is a 50% probability of choosing between the shrinking encircling mechanism or the spiral model to update the whale position. Mathematically, this behavior is given by

where ^{th} whale to the prey,

In this step, the exploration phase is carried out, adopting the same approach based on the variation of the vector

The pseudocode of the WOA is summarized in Figure

Pseudocode of the WOA, adapted from Mirjalili and Lewis [

In order to illustrate the proposed method for optimal MTMD design aiming to minimize the dynamic response of road bridges taking into account the bridge-vehicle interaction and random pavement irregularity, a typical truck traveling on a common bridge in Brazil is simulated in this section.

The next subsections present simulations of the bridge, vehicle, and pavement irregularities, as well as the scenarios considered for the MTMD and the results of the coupled problem. Initially, uncertainties are not considered; however, after that, a robust optimization is proposed, taking into account the uncertainties present in the bridge parameters, in the vehicle velocity, and also in the pavement roughness. All simulations are performed in Matlab software, using subroutines developed by the authors.

A typical RC girder bridge without balances, regularly found in Brazil, is simulated. The bridge is modeled as a 2D simply supported beam, discretized into 34 finite elements of 50 cm each, totalizing 17 meters long (35 nodes). The bridge has a “double T” cross section, as shown in Figure ^{2} and a moment of inertia equal to 1.068 m^{4}. The RC has Young’s modulus of 30 GPa and a density of 2450 kg/m^{3}.

Bridge cross section (dimensions in meters).

Thus, the first three natural frequencies of the bridge, obtained by solving the eigenvalue problem, are 10.359, 41.438, and 93.235 Hz. The damping matrix is supposed to be proportional to the stiffness matrix. A damping ratio of 3.0% is assumed for the first mode.

A regular nonsymmetrical three-axle truck in Brazil is simulated (Figure _{s} = 10000 kg and _{s} = 35000 kgm^{2}, respectively; unsprung masses equal to _{ns1} = 530 kg, _{ns2} = 530 kg, and _{ns3} = 320 kg; suspension stiffness and damping coefficients equal to _{s1} = 585 kN/m, _{s2} = 585 kN/m, _{s3} = 432 kN/m, _{s1} = 6 kNs/m, _{s2} = 6 kNs/m, and _{s3} = 3 kNs/m; and tire stiffness and damping coefficients equal to _{t1} = 1680 kN/m, _{t2} = 1680 kN/m, _{t3} = 840 kN/m, _{t1} = 1 kNs/m, _{t2} = 1 kNs/m, and _{t3} = 1 kNs/m. The axle positions are not symmetrical, being _{1} = 2.5 m, _{2} = 1.0 m, and _{3} = 4.0 m. Additionally, the axles support different portions of the sprung mass, being _{s1} = 0.73/2 _{s}, _{s2} = 0.73/2 _{s}, and _{s3} = 0.27 _{s}.

Thus, the natural frequencies of the truck are 1.671, 2.354, 10.138, 10.409, and 10.482 Hz. Initially, it is assumed that the truck crosses the bridge with a constant velocity of 90 km/h (25 m/s), which is the maximum permitted velocity for trucks on most Brazilian roads.

Following the procedure described in Section

Sample of irregularity amplitude for road class C.

To improve the bridge design by reducing the DAF, an optimization process is proposed, in which the objective function is to minimize the maximum vertical displacement of the central node of the bridge, while the design variables are the properties of the TMDs, that is, their stiffness and damping constants.

For this purpose, three different scenarios for MTMD installation are proposed. The first scenario is to consider a single TMD installed on the bridge central node (node 18), as shown in Figure

The second scenario is considering two TMDs installed on the bridge central nodes, with a distance of 3 m between the two TMDs (nodes 15 and 21), as shown in Figure

Finally, the third scenario is considering three TMDs installed on the bridge central, with a distance of 1.5 m between each of the TMDs (nodes 15, 18, and 21), as shown in Figure

To solve the dynamic coupled vehicle-pavement-bridge-TMD problem, the authors implemented the Newmark method, with a time step equal to 2

To solve the optimization problem (equation (

Initially, to compare results, a static analysis is also carried out. For this purpose, it is assumed that the vehicle crosses the bridge considering only its own weight, distributed on each axle as specified previously in Section

Thus, the black curve of Figure

Maximum vertical displacement for each node of the bridge, for static case (black curve), dynamic without TMD (red curve), with 1 TMD (magenta curve), with 2 TMDs (green curve), and with 3 TMDs (blue curve).

Vertical displacement at the bridge central node (node 18), as the vehicle moves along the bridge, for static case (black curve), dynamic without TMD (red curve), with 1 TMD (magenta curve), with 2 TMDs (green curve), and with 3 TMDs (blue curve).

Next, a dynamic analysis is carried out, considering initially that there are no TMDs installed on the bridge. Thus, the red curve of Figure

As can be seen in the red curves of Figures

Thus, for the road bridge under consideration, the impact factor calculated through equation (

To reduce the DAF, the installation and optimization of MTMD are proposed. As described in Section

Results of the proposed optimization procedure.

Scenario | TMD mass (kg) | TMD optimized parameters | Maximum vertical displacement (mm) | Dynamic amplification factor |
---|---|---|---|---|

Static case | — | — | 0.3132 | — |

Without TMD | — | — | 0.5710 | 1.823 |

1 | _{tmd1} = 4498.2 | _{tmd1} = 24508942.28 | 0.3231 | 1.032 |

_{tmd1} = 10.0574 | ||||

2 | _{tmd1} = 2249.1 | _{tmd1} = 12210049.58 | 0.3272 | 1.045 |

_{tmd1} = 13.4143 | ||||

_{tmd2} = 2249.1 | _{tmd2} = 12196121.07 | |||

_{tmd2} = 14.9268 | ||||

3 | _{tmd1} = 1499.4 | _{tmd1} = 8138925.96 | 0.3257 | 1.040 |

_{tmd1} = 12.1952 | ||||

_{tmd2} = 1499.4 | _{tmd2} = 8163326.22 | |||

_{tmd2} = 10.5451 | ||||

_{tmd3} = 1499.4 | _{tmd3} = 8160046.61 | |||

_{tmd3} = 11.9033 |

As can be seen in Figures

In addition to the three proposed scenarios having considerably reduced the maximum vertical displacement and consequently the DAF, it is interesting to note that the three scenarios (with 1, 2, and 3 TMDs) presented very similar results, showing that any of the three solutions can be adopted by the designer. Scenario 1, with a single TMD, may be more convenient; however, in certain situations, the use of 2 or more TMDs may be necessary, especially when more than one frequency needs to be controlled or when the individual mass of each TMD needs to be reduced.

Figure

Convergence curves for (a) scenario 1, (b) scenario 2, and (c) scenario 3.

To prove again the effectiveness of the proposed method, this subsection shows a comparison of the results obtained by the proposed method with the results obtained employing a classical Genetic Algorithm (GA).

To carry out a fair comparison, the parameters employed for the GA are the same as those of the WOA, that is, a population of 100 individuals and 300 iterations, totalizing 30000 evaluations. The results obtained with the GA are shown in Table

Results obtained with the GA.

Scenario | TMD mass (kg) | TMD optimized parameters | Maximum vertical displacement (mm) | Dynamic amplification factor |
---|---|---|---|---|

1 | _{tmd1} = 4498.2 | _{tmd1} = 24461515.78 | 0.3242 | 1.035 |

_{tmd1} = 55.5349 | ||||

2 | _{tmd1} = 2249.1 | _{tmd1} = 13073406.38 | 0.3358 | 1.072 |

_{tmd1} = 31.9801 | ||||

_{tmd2} = 2249.1 | _{tmd2} = 12001929.32 | |||

_{tmd2} = 22.1855 | ||||

3 | _{tmd1} = 1499.4 | _{tmd1} = 7990745.97 | 0.3326 | 1.062 |

_{tmd1} = 16.6866 | ||||

_{tmd2} = 1499.4 | _{tmd2} = 8846387.21 | |||

_{tmd2} = 17.3647 | ||||

_{tmd3} = 1499.4 | _{tmd3} = 8109517.58 | |||

_{tmd3} = 51.1554 |

Looking at Table

To demonstrate the effectiveness of the proposed method in another way, the optimal solution is compared with solutions obtained by traditional TMD design methods, due to Den Hartog [

Den Hartog [_{tmd}/_{str}, in which _{str} is the main mass (the structure mass).

Similarly, Warburton [

Thus, through the values of _{tmd} and _{tmd}) can be calculated as

Finally, it is important to note that the expressions proposed by Den Hartog [

Thus, using the methodology proposed by Rana and Soong [_{tmd_DH} = 1.6961 × 10^{4} kN/m and _{tmd_DH} = 80.485 kNs/m (equation (_{tmd_W} = 1.6452 × 10^{4} kN/m and _{tmd_W} = 65.221 kNs/m (equation (

Vertical displacement at the bridge central node (node 18), as the vehicle moves along the bridge, for dynamic cases: without TMD (red curve), with 1 TMD optimized by the proposed method (magenta curve), with 1 TMD designed by Den Hartog’s method (gray curve), and with 1 TMD designed by Warburton’s method (cyan curve).

As can be seen in Figure _{max} = 0.4894 mm) and with Warburton’s method (_{max} = 0.5034 mm) is more than 50% greater than the maximum vertical displacement obtained with the proposed optimization methodology (_{max} = 0.3231 mm), highlighting the superior performance of the proposed method.

The DAF obtained employing the proposed optimization method is 1.032, that is, less than the limit recommended by the ABNT NBR 7187 [

Initially, in the previous subsections, it was assumed that the truck crosses the bridge with a constant velocity of 90 km/h, which is the maximum permitted velocity for trucks on most Brazilian roads. Now, to assess the influence of the truck velocity, in this subsection, this velocity is reduced to 50 km/h, keeping all other parameters unchanged.

However, as the truck velocity is lower, the time for the vehicle to cross the bridge is higher and, consequently, the total analysis time, as well as the computational time, is also higher. For these analyses, the computational time was approximately 11.5 minutes.

Thus, applying the proposed methodology with the WOA, the results shown in Figures

Maximum vertical displacement for each node of the bridge, for static case (black curve), dynamic without TMD (red curve), with 1 TMD (magenta curve), with 2 TMDs (green curve), and with 3 TMDs (blue curve), for a truck velocity of 50 km/h.

Vertical displacement at the bridge central node (node 18), as the vehicle moves along the bridge, for static case (black curve), dynamic without TMD (red curve), with 1 TMD (magenta curve), with 2 TMDs (green curve), and with 3 TMDs (blue curve), for a truck velocity of 50 km/h.

Results of the proposed optimization procedure for 50 km/h.

Scenario | TMD mass (kg) | TMD optimized parameters | Maximum vertical displacement (mm) | Dynamic amplification factor |
---|---|---|---|---|

Static case | — | — | 0.3132 | — |

Without TMD | — | — | 0.6026 | 1.924 |

1 | _{tmd1} = 4498.2 | _{tmd1} = 21198202.49 | 0.4477 | 1.429 |

_{tmd1} = 56064.90 | ||||

2 | _{tmd1} = 2249.1 | _{tmd1} = 6730864.37 | 0.3867 | 1.235 |

_{tmd1} = 1516.19 | ||||

_{tmd2} = 2249.1 | _{tmd2} = 11187204.86 | |||

_{tmd2} = 202.916 | ||||

3 | _{tmd1} = 1499.4 | _{tmd1} = 4598993.93 | 0.3867 | 1.235 |

_{tmd1} = 1122.01 | ||||

_{tmd2} = 1499.4 | _{tmd2} = 6925434.17 | |||

_{tmd2} = 2501.31 | ||||

_{tmd3} = 1499.4 | _{tmd3} = 7598713.09 | |||

_{tmd3} = 168.309 |

As can be seen in Figures

After installing the optimized MTMD, with a total mass of 3% of the structure’s mass, the DAF was reduced from 1.924 to 1.429 (for scenario 1) and 1.235 (for scenarios 2 and 3). Even scenario 1 leading to a reduction of 25.7% in the DAF, this scenario has not yet been able to lead the DAF below the limit recommended by the ABNT NBR 7187 [

Thus, in this case, a single TMD with a mass of 3% of the bridge’s mass is not enough to meet the standard criteria. Therefore, the designer must choose scenario 2 (with 2 TMDs with a mass of 1.5% of the bridge’s mass each) or scenario 3 (with 3 TMDs with a mass of 1% of the bridge’s mass each). Alternatively, the designer can increase the mass of a single TMD (scenario 1) and assess whether the DAF would be reduced enough.

Finally, in order to take into account the uncertainties present in the coupled bridge-vehicle system and also in the pavement roughness and consequently increase the robustness of the MTMD control, some input parameters are considered as random variables.

For the bridge, the random variables are Young’s modulus, the density, and the damping ratio, supposed to have a normal distribution with the mean values given in Section ^{3}, and 3.0%) and coefficients of variation of 10%, 10%, and 20%, respectively. For the vehicle, the random variable is the velocity, supposed to have a uniform distribution between 50 and 90 km/h, and for the pavement, in addition to the random phase angle with a uniform distribution between 0 and 2^{−6 }m^{3}) and coefficient of variation equal to 20%.

Therefore, in each run of the computational routine, the bridge, the vehicle velocity, and the pavement present different parameters. Since the response of the coupled system depends on these random variables, it becomes random itself. Thus, the objective function of the robust optimization problem is to minimize the mean of the maximum vertical displacement at the center of the bridge span (mean(_{max})).

The WOA is used to perform this robust optimization problem, considering 100 search agents and 200 iterations, and the sample size is 50. For this analysis, the computational time is approximately 6 hours. In this subsection, for optimization under uncertainties, in addition to the static case and the dynamic case without TMD, only scenario 3 is simulated. The results are shown in Table

Results of the proposed optimization procedure taking uncertainties into account.

Scenario | TMD mass (kg) | TMD optimized parameters | Mean maximum vertical displacement (mm) | Mean dynamic amplification factor |
---|---|---|---|---|

Static case | — | — | 0.3172 | — |

Without TMD | — | — | 0.6234 | 1.965 |

3 | _{tmd1} = 4498.2 | _{tmd1} = 31019373.55 | 0.3967 | 1.251 |

_{tmd1} = 24816.94 | ||||

_{tmd2} = 4498.2 | _{tmd2} = 16894861.26 | |||

_{tmd2} = 5887.50 | ||||

_{tmd3} = 4498.2 | _{tmd3} = 22234302.01 | |||

_{tmd3} = 2579.55 |

As can be seen in Table

The convergence curve for this robust optimization problem is shown in Figure

Convergence curve for the robust optimization.

The design of road bridges is based on technical standards, which, until today, consider the dynamic loads caused by vehicle traffic as equivalent static loads. However, it is already agreed in the academic community that this design procedure should be improved. Thus, preliminary works began to consider simplified dynamic analyses, disregarding the interaction between structure and vehicles and/or not considering the pavement randomness and/or the system uncertainties, for example. Moreover, even in the cases in which dynamic analyses show that there will be a large DAF, many designers still choose to oversize the structure rather than using energy dissipation devices.

In this context, the present work developed a complete methodology for dynamic analysis of road bridges, including a proposal for optimization of an energy dissipation system. The proposed methodology takes into account bridge-vehicle interaction and pavement randomness and may also consider uncertainties present in the bridge, vehicle, and pavement parameters, as well as proposing a method for optimizing MTMD. For this, the WOA was employed, which is characterized by being a simple structure algorithm and easily adaptable to complex optimization problems, even when dealing with multimodal and/or nonconvex problems.

For illustrative purposes, the complex coupled vibration problem of a regular truck traveling on a random road profile over a typical Brazilian bridge was analyzed. Three different scenarios for the MTMD were considered, aiming to minimize the dynamic response of the bridge. Initially, uncertainties were not taken into account; nevertheless, two different truck velocities were assessed. The results showed the excellent ability of the proposed method, reducing the DAF of the bridge to values below the limit recommended by the standards.

The comparison of the results obtained using the proposed method with the results obtained using the classical GA showed that the proposed method presented slightly superior performance, both in terms of DAFs and in terms of computational time.

To demonstrate the effectiveness of the proposed method in another way, the optimal solution for 1 TMD was compared with solutions obtained by traditional TMD design methods. The results showed that the maximum vertical displacement obtained with the Den Hartog and Warburton methods was about 50% greater than the maximum vertical displacement obtained with the proposed optimization methodology, highlighting the superior performance of the proposed method.

Finally, to evaluate the performance of the proposed methodology in the presence of uncertainties, a robust optimization was carried out, which proved the robustness and effectiveness of the proposed methodology also in optimization problems under uncertainty. Again, the DAF was reduced to acceptable values.

Thus, the methodology proposed in this paper to perform a complete dynamic analysis of bridges, including the optimization of MTMD, can be employed to improve bridge design, ensuring safety and comfort to users.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors acknowledge the financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).