# Multi-Objective Optimization Applied to the Design of Sustainable Pedestrian Bridges

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. For both scenarios, the optimal ratio for the web height and total span (L

_{e}) lies between L

_{e}/20 and L

_{e}/16. The web height, the concrete strength, and the slab thickness were the design variables with more influence on the value of the vertical acceleration. The Pareto-optimal solutions were considerably sensitive to the parameters varied in each scenario, changing concrete consumption and dimensions of the welded steel I-beam, evidencing the importance of carrying out a sensitivity analysis in optimization problems.

## 1. Introduction

_{2}emissions. The optimization considering tuned mass dampers in pedestrian bridges is a topic of interest in different publications [24,25,26]. Although several studies have been developed regarding the optimization of costs and environmental impacts or the dynamic response of the pedestrian bridge, there is a lack of publications that consider the three objectives simultaneously. This justifies the research on the assessment of the interaction between the sustainability of the pedestrian bridge and its vertical acceleration generated by human-induced vibrations in a multiobjective optimization problem. Furthermore, the topic of optimization of steel-concrete composite pedestrian bridges is barely explored, as also pointed out by Yepes et al. [10].

_{2}unit emission of the concrete slab are studied, with different values obtained from a life cycle assessment presented by Santoro and Kripka [27]. This sensitivity analysis aims to verify the influence of this parameter in the Pareto-optimal solutions.

## 2. Materials and Methods

#### 2.1. Multiobjective Harmony Search

_{min}and PAR

_{max}), maximum and minimum bandwidth (bw

_{min}and bw

_{max}) and the maximum improvisation number (MI).

_{2}) with the same number of harmonies as the HMS value. Each harmony can be generated in three different ways: random selection, memory consideration, or pitch adjustment. If a random number rand

_{1}is equal to or bigger than HMCR, the value of the variable is obtained by random generation; otherwise, it is obtained by memory consideration. In this case, a new random number rand

_{2}is compared to PAR, and if the rand

_{2}value is equal or bigger, the value of the variable is randomly selected from harmony in the HM. If rand

_{2}is smaller than PAR, then the new value is submitted to pitch adjustment, considering a value of the variable stored in HM and performing a step based on the bw parameter.

_{2}in a matrix H

_{u}, with 2*HMS harmonies. The solutions are ranked by a non-dominated sorting method proposed by Fonseca and Fleming [37]. The ranking value of each solution is equal to the number of solutions that dominate the solution evaluated plus one, so non-dominated solutions have a rank equal to 1. Then, the solutions are transferred to HM by rank order until the next rank has more solutions than the left space in HM. At this stage, the crowding distance of the solutions is evaluated using the algorithm presented by Deb et al. [36]. The harmonies with the greatest crowding distance are moved to HM until it is filled.

#### 2.2. Problem Formulation and Implementation

_{t}is the concrete slab thickness, both defined by the optimization process. The longitudinal section, as well as the span of the structure, are illustrated in Figure 3.

#### 2.2.1. Design Variables

_{i}) are the characteristic strength of concrete (f

_{ck}), the concrete slab thickness, the steel beam dimensions, and the interaction degree (α) of the composite beam, as shown in Figure 4. In the illustration, h

_{w}is the web height, t

_{w}, t

_{fs,}and t

_{fi}are, respectively, the thickness of the web, of the superior and inferior flanges, and b

_{fs}and b

_{fi}are the width of the superior and inferior flanges. In total, 9 design variables are considered, and their values are discrete to ensure that the solutions are feasible and represent a real design. The possible intervals for each variable are shown in the following equations, which form the lateral constraints of the optimization problem. These discrete intervals also limit the minimum and maximum dimensions of the structural elements, as well as consider the manufacturing standard of Brazilian metallurgists.

#### 2.2.2. Objective Functions and Emissions Scenarios

_{2}emissions associated with the construction phase of the pedestrian bridge. Equation (7) is used to evaluate the monetary cost of the structure (C), in the function of the unit costs (c

_{i}) and the respective quantity of material consumed (q

_{i}), the elements ($ec$) considered and their unit is the volume of concrete, the mass of the slab’s reinforcement steel and the steel beam mass, which also considers the mass of the headed studs. Unit costs were obtained from the SINAPI report [38], made available by the Brazilian Federal Savings Bank. These values are given in Table 2, where the monetary costs are expressed in real (R$), the Brazilian currency. As a reference, on 22 January 2023, the exchange rate against the US dollar is R$ 1.00 for $ 0.19 (or $ 1.00 for R$ 5.21).

_{2}emissions (E) and evaluated similarly to the cost, as can be seen in Equation (8). The elements (ee) are the same as in the cost evaluation but consider a unit emission of CO

_{2}(e

_{i}). For the steel I-beam, the value of unit emission adopted was provided by Worldsteel [39]. On the other hand, the unit emissions of each concrete strength and the steel reinforcement rebars are those published by Santoro and Kripka [27] and are summarized in Table 2.

_{2}emissions, considering the acquisition of raw materials, transport to the concrete plant, production in the concrete batch, and the concrete transport to the building site. More details are presented in the original publication.

- Scenario (A)—unit emissions obtained from the on-site survey for the study region. In Table 2, emission A shows the values considered in this scenario.
- Scenario (B)—unit emissions calculated using the SimaPro software, with Ecoinvent 3.5 database and ReCiPe 2016 method, with adjustments in the processes and quantities to make the values compatible with the same region. These values are displayed as emission B in Table 2.

_{a,}

_{95%}, C, K

_{1}, K

_{2}e K

_{f}are tabled constant values from the same code, d is the pedestrian density (with a value adopted of 1.0 pedestrian/m

^{2}), L is the structure length, b is the width of the pedestrian bridge, and M

_{i}is the modal mass, associated to mode i.

#### 2.2.3. Verifications and Constraints

^{2}), and railing (1 kN/m). As variable actions, there is the constructive live load (1 kN/m

^{2}), the pedestrian live load (5 kN/m

^{2}) [41], as well as horizontal actions, such as wind [42] and a 100 kN punctual load in the most unfavorable situation to the structure.

^{2}[41].

^{4}in this research.

## 3. Results and Discussion

^{2}, it is necessary to invest R$ 200.00 ($ 38.40) per meter of the structure, which corresponds to an increase of 15% in the cost. To reduce the acceleration from 2.5 to 1.5 m/s

^{2}, it is necessary an increase of only 7% in the structure cost.

_{2}emissions due to the unit emissions being considerably lower in comparison to Scenario (B). It is also possible to notice that for Scenario (A), solutions are distributed almost linearly, with the emissions growing as the cost increases. On the other hand, in Scenario (B), the solutions are more dispersed, without a clear tendency between the two objectives. However, the cost and the CO

_{2}emissions of the pedestrian bridge are not conflicting objectives, with solutions that are at the same time efficient in terms of costs and environmental impacts.

_{2}emissions. In contrast, other solutions adopted smaller web heights, bigger slab thickness, and higher concrete strengths, reducing the costs but trading off a poorer performance in terms of environmental impacts.

_{2}emissions. A possible explanation for this is that to minimize costs, the optimization algorithm favors solutions that reduce material consumption, which also reduces the environmental impact of the structure. Therefore, minimizing costs and emissions are not conflicting objectives, as the literature also points out [10,46,47], for different types of structures and materials.

_{w}) is a design variable that notoriously influences the cost, the environmental impact, and the vertical acceleration generated by human-induced vibrations. This occurs due to the influence of the variable in the structure’s resistance and stiffness. For Scenario (A), Pareto-optimal solutions with smaller h

_{w}have lower cost and environmental impact but perform poorer in comfort for the pedestrians, as Figure 7 shows. For the problem studied in this paper, the optimal values to h

_{w}, for Scenario (A) are in the range of 900 mm and 1100 mm. The figure also shows that increasing this variable is an effective way to reduce the vertical accelerations from 2.5 m/s

^{2}to 1.0 m/s

^{2}. However, to reduce accelerations further, other variables should be considered, such as the superior and inferior flange width, slab thickness, and concrete strength. The change in the behavior occurs as an attempt to satisfy the constraints regarding slenderness and the bending moment about the weak axis of the I-beam.

_{w}is very similar. Two groups of Pareto-optimal solutions stand out in Figure 7. The first remains practically constant around the value of 850 mm, while the second is grouped near the value of 1050 mm. For this scenario, variables such as slab thickness, concrete strength, and superior and inferior flange width played a bigger role in obtaining solutions with lower vertical accelerations. The solutions with h

_{w}close to 850 mm have, in general, a slab thickness bigger than 14 cm and a preference for concrete strength of 50 MPa. The other configurations, with bigger h

_{w}, have the minimum slab thickness and concrete strength. For both cases, the optimal ratio for h

_{w}and total span (L

_{e}) lies between L

_{e}/20 and L

_{e}/16.

_{w}that complies with the maximum slenderness, given that this is an effective way to increase the beam’s stiffness. However, solutions that increase only the slenderness ratio can have a lower resisting capacity in the weak axis of the I-beam profile, and because of that, the constraint related to the verification of horizontal loads is also important in the sizing of the beam. For this restriction to be satisfied, solutions must also increase the flanges width and thickness of the I beam cross-section.

_{2}released in its construction is due to the structural element. In Scenario (B), the concrete slab contribution of emissions increases to almost 46%. For this reason, the results show a certain preference for solutions with the minimum slab thickness, reducing the volume of concrete consumed.

## 4. Conclusions

_{2}emission of the concrete. The first (A) uses unit emissions of a cradle-to-gate LCA from the on-site survey for the study region, while the second (B) are values obtained from the SimaPro software. The program developed was able to successfully obtain a Pareto front of non-dominated and feasible solutions, considering the three distinct objectives. With the analysis of the results, several conclusions are drawn which can be of aid during the decision-making in the design of sustainable pedestrian bridges of composite structure, taking into account the user’s comfort.

^{2}to 1.5 m/s

^{2}by an increase of 7% in its cost and to 1.0 m/s

^{2}by an increase of 15%. For Scenario (A), an efficient way to reduce the vertical acceleration down to 1.0 m/s

^{2}is to increase the web height of the I-beam, although further minimization must also increase superior and inferior flange width, slab thickness, and concrete strength. Scenario (B) showed a preference for changing the slab thickness, concrete strength, and flanges width, with solutions maintaining similar web heights while the acceleration is still reduced. For both cases, the optimal ratio for the web height and the total span (L

_{e}) is from L

_{e}/20 and L

_{e}/16. For Scenario (A), the web height grows from 850 mm while the vertical acceleration decreases until 1.0 m/s

^{2}, and from then on, the values stabilize at around 1050 mm. The optimal web heights for Scenario (B) are concentrated in both extremes in an almost constant behavior along vertical acceleration, with no values in the middle of the range.

_{2}emissions and vertical acceleration but with higher monetary costs. The web height is also important for the structure sizing since it is directly related to the displacements, the active constraint in most solutions. Nevertheless, the width of superior and inferior flanges is crucial to satisfy restrictions regarding the slenderness ratio and moments acting in the weak axis of the I-beam section. These results indicate that other design variables other than web height can play an important role in the vertical acceleration and sizing of the structure, evidencing the importance of them being considered in the decision-making process by engineers.

_{2}emissions and the cost of the pedestrian bridge. For the first scenario, the solutions presented a clear tendency of linear growth between the cost and emission, while in the second, the Pareto-optimal solutions were more dispersed. The optimal values of the design variables were also affected, with scenario (B) presenting a bigger consumption of concrete when compared to scenario (A). Therefore, two broader conclusions can be drawn from the variations in the results of the sensitivity analysis. First, the importance of considering different scenarios in optimization problems, as the results can be highly sensitive to changes in certain parameters. Second, the necessity of guaranteeing the reliability of the life cycle assessment is to make sure the results will be representative and applicable in practical ways, especially when raising sizing parameters for project development.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Average optimal values for design variables in each scenario * Values in the y axis divided by 10 ** Values in the y axis divided by 100 *** Values in the y axis multiplied by 10.

Parameter | Value |
---|---|

HMS | 20 |

HMCR | 0.5 |

PAR_{min} | 0.1 |

PAR_{max} | 0.9 |

bw_{min} | 0.1 |

bw_{max} | 0.5 |

MI | 500,000 |

Material | Unit | Cost (R$) | Emission A ^{1} (kgCO_{2}) | Emission B ^{2} (kgCO_{2}) |
---|---|---|---|---|

Concrete 30 MPa | m^{3} | 533.88 | 157.65 | 348.76 |

Concrete 45 MPa | m^{3} | 591.15 | 194.70 | 381.72 |

Concrete 50 MPa | m^{3} | 631.60 | 225.78 | 508.63 |

Reinforcement | kg | 9.68 | 1.05 | 2.10 |

Steel I-beam | kg | 14.56 | 1.91 | 1.91 |

^{1}Values used in scenario A optimization, where concrete and reinforcement emissions are from the LCA performed by Santoro and Kripka [27] to a city in the south of Brazil.

^{2}Values used in scenario B optimization, with emissions evaluated by Santoro and Kripka [18] using the SimaPro software, with adjustments in processes and quantities to match the same region.

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**MDPI and ACS Style**

Tres Junior, F.L.; Yepes, V.; Medeiros, G.F.d.; Kripka, M. Multi-Objective Optimization Applied to the Design of Sustainable Pedestrian Bridges. *Int. J. Environ. Res. Public Health* **2023**, *20*, 3190.
https://doi.org/10.3390/ijerph20043190

**AMA Style**

Tres Junior FL, Yepes V, Medeiros GFd, Kripka M. Multi-Objective Optimization Applied to the Design of Sustainable Pedestrian Bridges. *International Journal of Environmental Research and Public Health*. 2023; 20(4):3190.
https://doi.org/10.3390/ijerph20043190

**Chicago/Turabian Style**

Tres Junior, Fernando Luiz, Víctor Yepes, Guilherme Fleith de Medeiros, and Moacir Kripka. 2023. "Multi-Objective Optimization Applied to the Design of Sustainable Pedestrian Bridges" *International Journal of Environmental Research and Public Health* 20, no. 4: 3190.
https://doi.org/10.3390/ijerph20043190