# Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}emissions associated with the amount of materials obtained by every heuristic technique and the original design solution were studied. Finally, a parametric study was carried out according to the span length of the pedestrian bridge.

## 1. Introduction

_{2}emissions [27], the embodied energy [35], or the lifetime reliability [36]. However, these multi-criteria methods have not been applied to composite bridges. Other researchers, such Penadés-Plà et al. [37], have done a review of multi-criteria decision-making methods to evaluate sustainable bridge designs. Nevertheless, if we focused on composite bridges there is a lack of knowledge.

_{2}emissions [38,39,40]. Many researchers, such as Yepes et al. [41] and Molina-Moreno et al. [42], have used these databases to study the difference between the cost and the CO

_{2}emission optimization for reinforced concrete (RC) structures.

_{2}emissions associated with the material amounts obtained from each heuristic were analyzed and compared with the original structural design. Finally, a parametric study, according to the span length, was performed.

## 2. Optimization Problem Definition

_{i}is the prices of every construction unit and m

_{i}the measurements obtained by the design variables. For example, for a random structure, the vector x would contain the design variables of that random structure, m

_{i}would contain the amount of materials associated with these variables, and these measurements, multiplied by their unit prices (p

_{i}), result in the total cost of the structure.

_{c}) were the volume of concrete, the amount of reinforcement steel, the amount of rolled steel, and the amount of shear-connector’s steel.

_{2}emissions for each construction unit were obtained from Molina-Moreno et al. [42]. The data of rolled steel and shear-connector steel were taken to the BEDEC ITEC database of the Institute of Construction Technology of Catalonia [40]. Table 1 contains all the data on costs and CO

_{2}emissions considered in this work.

#### 2.1. Design Variables

^{27}possible solutions, because of this, the complete evaluation of the problem was unapproachable. The problem optimization was carried out by heuristic techniques.

#### 2.2. Structural Analysis and Constraints

^{2}and the deck self-weight, including the bridge railing and asphalt (see Table 2). Note that the thermal gradient [5] and the differential settling in each support were also taken into account. The model implemented obtained the beam stresses and the transversal section tensions to assess the structural design validity.

_{u}is the ultimate response of the structure and S

_{u}the ultimate load effects. For instance, the ULS of the shear and torsion interaction reduced the shear resistance due to the effect of the torsion. The SLS covers the requirements of functionality, comfort, and aspect (Equation (4)):

_{s}is the permitted value of the serviceability limit and E

_{s}is the value obtained from the model produced by the SLS actions.

## 3. Applied Heuristic Search Methods

#### 3.1. Descent Local Search

#### 3.2. Hybrid Simulated Annealing with a Mutation Operator

_{a}, given by the expression of Glauber (5), where T is now a parameter that decreases with the time, thus reducing the probability of accepting worse solutions, from an initial value, T

_{0}:

_{0}or that during a Markov chain no better solution was found.

#### 3.3. Glow-worm Swarm Optimization (GSO)

_{s}. So, the decision range for each glow-worm is also delimited by a maximum radial value r

^{i}

_{d}, that complies with 0 < r

^{i}

_{d}≤ r

_{s}, which we will call a decision radius. One glow-worm i considers another firefly j as its neighbor if j is within its decision radius r

^{i}

_{d}and the level of luciferin j is greater than that of i.

^{i}

_{d}decision radius allows the selective interaction of neighbors and helps the disjointed formation of sub-branches. Each firefly selects, through a probabilistic mechanism, a neighbor, who has a higher value of luciferin and moves towards it. These movements, which are based solely on local information and the selective interaction of neighbors, allow the swarm of fireflies to be subdivided into disjointed subgroups, that address, and are found in multiple optimums of the given multimodal function. The process can be summarized as follows:

- Initially a swarm of n feasible glow-worms is generated and distributed in the search space. Each glow-worm has assigned the initial luciferin value l
_{0}and the initial sensitivity radius r_{s}; - Depending on the previous luciferin l
_{i}and the objective function value, the luciferin is updated as is shown on Equation (6). The luciferin value decays constant $\rho (0\rho 1)$ simulates the decrease in luciferin level over time, and the luciferin enhancement constant γ (0 < γ < 1) is the proportion of the improvement in the objective that glow-worm adds to its luciferin. J(x_{i}(t)) is the value of the objective function of the glow-worm i at iteration j:$${l}_{i}\left(t\right)=\left(1-\rho \right)\xb7{l}_{i}\left(t-1\right)+\gamma \xb7J\left({x}_{i}\left(t\right)\right);$$ - Each glow-worm uses a probability sampling mechanism to move towards a neighbor with a higher luciferin value. For each glow-worm i, the probability of moving to a neighbor j is given by Equation (7), where N
_{i}(t) is the set of neighbors of the glow-worm i in the iteration t, d_{ij}represents the Euclidean distance between glow-worms i and j in iteration t. r^{i}_{d}(t) is the decision ratio of glow-worm i in iteration j:$${p}_{ij}\left(t\right)=\frac{{l}_{j}\left(t\right)-{l}_{i}\left(t\right)}{{{\displaystyle \sum}}_{k\in {N}_{i}\left(t\right)}{l}_{k}\left(t\right)-{l}_{i}\left(t\right)};j\in {N}_{i}\left(t\right),{N}_{i}\left(t\right)=\left\{j:{d}_{ij}\left(t\right){r}_{d}^{i}\left(t\right);{l}_{i}\left(t\right){l}_{j}\left(t\right)\right\};$$ - During the movement phase, the glow-worm i moves to glow-worm j. Equation (8) describes the model of the movement of a glow-worm at any given moment, where x
_{i}(t) is the location of the glow-worm i at iteration t and s is the step factor constant:$${x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+s\xb7\left(\frac{{x}_{j}\left(t\right)-{x}_{i}\left(t\right)}{\Vert {x}_{j}\left(t\right)-{x}_{i}\left(t\right)\Vert}\right);$$ - Once the movement is finished, the update of the radial sensor range is carried out by the expression of Equation (9), where β is a constant parameter and n
_{t}is another parameter that controls the number of neighbors:$${r}_{d}^{i}\left(t+1\right)=min\left\{{r}_{s},max\left\{0,\text{}{r}_{d}^{i}\left(t\right)+\beta \xb7\left({n}_{t}-\left|{N}_{i}\left(t\right)\right|\right)\right\}\right\}.$$

_{t}, s, and l

_{0}, respectively. The maximum number of iterations was fixed at 4000. The values of n and r

_{0}were taken as different values; the values adopted in the study are shown in Table 4.

## 4. Discussion

#### 4.1. Comparison of the Heuristic Techniques

#### 4.2. Sustainability Study

_{2}emissions associated with the amount of the materials obtained from every cost heuristic optimization was carried out. In addition, a comparison with the original project of this steel-concrete composite pedestrian bridge was performed. In Table 6, the values of cost and CO

_{2}emissions of every solution are compared.

_{2}emissions compared with the reference. This means that an improvement of 1 €/m

^{2}produced a reduction of the 1.74 kg CO

_{2}/m

^{2}.

#### 4.3. Parametric Study

^{2}− 8.0873L + 325.96 with a regression coefficient of R

^{2}= 0.9994. The cost rising is produced by the need for larger amounts of materials to satisfy the deflection requirements. Note that the R

^{2}regression coefficient in Figure 8 is almost 1, this indicates a very good correlation. The variations between the minimum and the mean cost of the pedestrian bridge produced by the GSO and DLS combination are 0.42%.

^{2}= 0.9927. Again, the good correlation factor represents a functional relationship.

^{2}− 0.0067L + 0.2568 with R

^{2}= 0.9849 when the span length is larger than 38 m. Related to the compressive strength of the concrete of the slab, Figure 11 shows the relationship of the concrete compressive characteristic strength and the span length. This relationship adjusts well to a quadratic function. The concrete compressive strength adjusts to FCK = 0.0292L

^{2}− 1.6959L + 49.714 with R

^{2}= 0.9987. Note that the highest concrete compressive strength considered for this study was 35 MPa.

_{s}) and the surface of the slab (S

_{s}) Figure 12 illustrates the increase in the amount of rolled steel needed to resist the flexural requirements. The slope of the curve tends to increase as the span length increases. The mean amount of rolled steel in relation to the surface of slab adjusts to R

_{s}/S

_{s}= 0.01913L

^{2}− 3.5553L + 154.96 with R

^{2}= 0.9996. However, the ratio of the volume of concrete (V

_{c}) and the surface slab fits a second order equation that increases with the span length in the same way as rolled steel amount, as seen in Figure 13. The mean ratio of the volume of concrete in relation to the surface of slab adjusts to V

_{c}/S

_{s}= 0.0001L

^{2}− 0.0067L + 0.245. Moreover, the ratio of reinforcing steel (RF

_{s}) measured per square meter of slab shows the same tendency as rolled steel amount and concrete volume. The ratio of reinforced steel in relation to the surface of slab adjusts to RF

_{s}/S

_{s}= 0.0246L

^{2}− 1.2009L + 32.516 with R

^{2}= 0.9955, as shown in Figure 14.

## 5. Conclusions

- The GSO optimization algorithm obtained worse results than DLS and SA, but if we apply the DLS to the best GSO solutions, then this combination of heuristic techniques reaches the lowest cost solution;
- The results show the potential of the application of heuristic techniques to reach automatic designs of composite pedestrian bridges. The reduction in costs exceeds 20%. It is important to note that the current approach eliminates the need for experience-based design rules;
- The CO
_{2}emission comparison showed that the reduction between the original structure and the GSO with DLS combination optimized structure reduced the CO_{2}emission by 21.21%. Thus, an improvement of 1 €/m^{2}produced a reduction of 1.74 kg CO_{2}/m^{2}. Therefore, the solutions that are acceptable in terms of CO_{2}emissions are also viable in terms of cost, and vice versa; - The results indicate that cost optimization is a good approach to environmentally friendly design, as long as cost and CO
_{2}emission criteria reduce material consumption; - The parametric study showed that there is a good correlation between span length and cost, amount of material, and geometry. This relationship could be useful for designers, to have a guide to the day-to-day design of steel-concrete composite pedestrian bridges. However, the tendencies of the thickness of the flanges and webs of the steel beam are not clear;
- The heuristic techniques look for lower amounts of materials, which allows the reduction of the self-weight of the structure. In addition, the optimization algorithms look for an increase of the depth of the section to improve their mechanical characteristics. For an optimized pedestrian bridge, the relationship between the steel beam depth and span length takes a value of 1/27.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Average cost to average iterations for the glow-worms swarm optimization (GSO) experiment.

**Figure 12.**Variation in the ratio of the rolled steel amount in relation to the surface of the slab with the span.

**Figure 13.**Variation in the ratio of the volume of concrete in relation to the surface of the slab with the span.

**Figure 14.**Variation in the ratio of the reinforcement steel in relation to the surface of the slab with the span.

Unit Measurements | Cost (€) | Emissions (kg CO_{2}) |
---|---|---|

m^{3} of concrete C25/30 | 93.71 | 224.34 |

m^{3} of concrete C30/37 | 102.41 | 224.94 |

m^{3} of concrete C35/45 | 105.56 | 265.28 |

m^{3} of concrete C40/50 | 111.64 | 265.28 |

kg of steel (B-500-S) | 1.20 | 3.02 |

kg of shear-connector steel | 2.04 | 3.63 |

kg rolled steel (S-355-W) | 1.70 | 2.8 |

Concrete classification according to EN 1992 |

Geometrical Parameters | |

Pedestrian bridge width | B = 2.5 m |

Number of spans | 1 |

Span length | 38 m |

Material Parameters | |

Maximum aggregate size | 20 mm |

Reinforcing steel | B-500-S |

Loading Parameters | |

Reinforced concrete specific weight | 25 kN/m^{3} |

Auxiliary assembly triangulations self-weight | 0.14 kN/m^{2} |

Live Load | 5 kN/m^{2} |

Dead load | 1.15 kN/m^{2} |

Temperature variation between steel and concrete | ±18 °C |

Exposure Related Parameters | |

External ambient conditions | IIb |

Code Related Parameters | |

Code regulations | EHE-08/IAP-11/RPX-95 |

Service working life | 100 years |

Concrete Slab Variables | Range Values | Step Size | |

CL | Slab depth | 15 to 30 cm | 1 cm |

CB | Slab edge depth | 10 to 30 cm | 1 cm |

VL | Lateral slab cantilever | 0.5 to 0.625 m | 5 mm |

DT | Transversal reinforcement diameter | 6, 8, 10, 12, and 16 mm | - |

DLS | Top longitudinal reinforcement diameter | 6, 8, 10, 12, and 16 mm | - |

DLI | Bottom longitudinal reinforcement diameter | 6, 8, 10, 12, 16, 20, and 32 mm | - |

SBTC | Transversal reinforcement separation in span center | 10 to 30 cm | 1 cm |

SBTA | Transversal reinforcement separation in supports | 5 to 25 cm | 1 cm |

NLS | Number of top longitudinal reinforcement bars | 10 to 40 | 1 |

NLI | Number of bottom longitudinal reinforcement bars | 10 to 40 | 1 |

Metal Beam Variables | |||

CA | Metal beam depth | 1.086 to 2.375 m | 1 cm |

AAI | Bottom flange width | 1 to 1.5 m | 1 cm |

SRL | Longitudinal stiffener spacing | 0, 0.16, 0.26, 0.36, and 0.46 | - |

SRT | Transversal stiffener spacing | 1, 2, 3.8, 7.6, 38 m | - |

EAS | Top flange thickness | 8 to 40 mm | 2 mm |

EAL | Web thickness | 8 to 40 mm | 2 mm |

EAI | Bottom flange thickness | 8 to 40 mm | 2 mm |

ERL | Longitudinal stiffener thickness | 8 to 40 mm | 2 mm |

ERT | Transversal stiffener thickness | 8 to 40 mm | 2 mm |

DP | Shear-connectors diameter | 16, 19, and 22 mm | - |

LP | Shear-connectors length | 100, 150, 175, and 200 | - |

STP | Transversal shear-connectors spacing | 5 to 25 cm | 5 cm |

SLPC | Longitudinal shear-connectors spacing in mid span | 30 to 50 cm | 5 cm |

SLPA | Transversal shear-connectors spacing in supports | 10 to 30 cm | 5 cm |

Mechanical Variables | |||

FCK | Concrete characteristic strength | 20 to 35 MPa | 5 MPa |

n | 10 | 20 | 30 | 40 | 50 | 60 | 80 | 100 |

r_{0} | 50 | 100 | 150 |

DLS | SA | GSO | GSO and DLS | ||
---|---|---|---|---|---|

Rolled steel | kg/m^{2} | 153.99 | 151.26 | 157.49 | 150.66 |

% Rolled/record | % | 2.21% | 0.40% | 4.54% | 0.00% |

Shear-connector steel | kg/m^{2} | 0.90 | 0.90 | 0.79 | 0.90 |

% Shear-connector/record | % | 14.28% | 14.28% | 0.00% | 14.28% |

Concrete | m^{3}/m^{2} | 0.15 | 0.15 | 0.14 | 0.14 |

% Concrete/record | % | 7.52% | 7.52% | 0.83% | 0.00% |

Reinforcement steel | kg/m^{2} | 22.57 | 22.22 | 25.50 | 22.23 |

% Reinforcement/record | % | 1.58% | 0.00% | 14.76% | 0.04% |

Cost | €/m^{2} | 304.82 | 299.77 | 313.18 | 297.76 |

% Cost/record | % | 2.37% | 0.67% | 5.18% | 0.00% |

Reference | DLS | SA | GSO | GSO and DLS | ||
---|---|---|---|---|---|---|

Cost | €/m^{2} | 399.10 | 304.82 | 299.77 | 313.18 | 297.76 |

% Cost/reference | % | - | −23.62% | −24.89% | −21.53% | −25.39% |

CO_{2} emissions | kg CO_{2}/m^{2} | 700.22 | 536.39 | 527.70 | 552.54 | 523.68 |

% Emissions/reference | % | - | −23.40% | −24.64% | −21.09% | −25.21% |

Span (m) | CL (m) | CA (m) | CT (m) | EAS (mm) | EAL (mm) | EAI (mm) | FCK (MPa) | Total Depth/L |
---|---|---|---|---|---|---|---|---|

28 | 0.17 | 1.53 | 1.70 | 18 | 8 | 12 | 35 | 0.035 |

32 | 0.16 | 1.37 | 1.53 | 18 | 8 | 10 | 30 | 0.036 |

38 | 0.15 | 1.21 | 1.36 | 18 | 8 | 10 | 30 | 0.036 |

42 | 0.15 | 0.96 | 1.11 | 18 | 8 | 10 | 25 | 0.035 |

48 | 0.15 | 0.80 | 0.95 | 18 | 8 | 10 | 25 | 0.034 |

Span (m) | Beam Rolled Steel (kg/m^{2}) | Slab Concrete (m^{3}/m^{2}) | Slab Reinforcement (kg/m^{2}) |
---|---|---|---|

28 | 194.10 | 0.17 | 30.87 |

32 | 165.64 | 0.15 | 24.63 |

38 | 150.66 | 0.14 | 22.23 |

42 | 135.37 | 0.14 | 19.80 |

48 | 126.56 | 0.14 | 17.84 |

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**MDPI and ACS Style**

Yepes, V.; Dasí-Gil, M.; Martínez-Muñoz, D.; López-Desfilis, V.J.; Martí, J.V.
Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges. *Appl. Sci.* **2019**, *9*, 3253.
https://doi.org/10.3390/app9163253

**AMA Style**

Yepes V, Dasí-Gil M, Martínez-Muñoz D, López-Desfilis VJ, Martí JV.
Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges. *Applied Sciences*. 2019; 9(16):3253.
https://doi.org/10.3390/app9163253

**Chicago/Turabian Style**

Yepes, Víctor, Manuel Dasí-Gil, David Martínez-Muñoz, Vicente J. López-Desfilis, and Jose V. Martí.
2019. "Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges" *Applied Sciences* 9, no. 16: 3253.
https://doi.org/10.3390/app9163253