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

Víctor Yepes; Manuel Dasí-Gil; David Martínez-Muñoz; Vicente  J. López-Desfilis; Jose  V. Martí

doi:10.3390/app9163253

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

Yepes, Víctor;Dasí-Gil, Manuel;Martínez-Muñoz, David;López-Desfilis, Vicente J.;Martí, Jose V. 2019-08-09 00:00:00 applied sciences Article Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges 1 1 1 2 Víctor Yepes , Manuel Dasí-Gil , David Martínez-Muñoz , Vicente J. López-Desﬁlis and 1 , Jose V. Martí * Institute of Concrete Science and Technology (ICITECH), Universitat Politècnica de València, 46022 Valencia, Spain Department. of Continuous Medium Mechanics and Theory of Structures, Universitat Politècnica de València, 46022 Valencia, Spain * Correspondence: jvmartia@cst.upv.es; Tel.: +34-96-387-7000; Fax: +34-96-387-7569 Received: 5 June 2019; Accepted: 4 August 2019; Published: 9 August 2019 Abstract: The objective of this work was to apply heuristic optimization techniques to a steel-concrete composite pedestrian bridge, modeled like a beam on two supports. A program has been developed in Fortran programming language, capable of generating pedestrian bridges, checking them, and evaluating their cost. The following algorithms were implemented: descent local search (DLS), a hybrid simulated annealing with a mutation operator (SAMO2), and a glow-worms swarm optimization (GSO) in two variants. The ﬁrst one only considers the GSO and the second combines GSO and DLS, applying the DSL heuristic to the best solutions obtained by the GSO. The results were compared according to the lowest cost. The GSO and DLS algorithms combined obtained the best results in terms of cost. Furthermore, a comparison between the CO 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. Keywords: pedestrian bridge; composite structures; optimization; metaheuristics; structural design 1. Introduction Nowadays, society’s concern about the impact of activities is rising, not only their economic inﬂuence, but also the environmental impact. The construction sector is one of the most carbon intensive industries [1] due to the need for large amounts of materials and, henceforth, large amounts of natural resources. Therefore, researchers are investigating how to achieve cost ecient and environmentally sustainable processes for the construction industry. The term sustainable development was introduced for the ﬁrst time by the Brundtland Commission, deﬁning it as, “development that meets the needs of the present without compromising the ability of future generations to meet their own needs” [2]. Since then, countries have been raising awareness about the compromise to the future generations, modifying their policies and demanding cheaper, ecofriendly constructions, without forgetting their safety and durability. In essence, the demands of the governments are to reach solutions that reduce their impact on the three main pillars: the economy, the environment, and society. These demands translate into restrictions for constructors and designer. The former need to carry out the constructions with new strategies to improve sustainability, while the designers have to conceive their projects in a cheaper, ecofriendly way. This means proﬁting the materials and taking maximum advantage of their characteristics, and maintaining durability and safety. The traditional recommendation for the designers is to take a starting point for their designs. Furthermore, there are lots of strict codes and regulations to ensure the safe and reliability of Appl. Sci. 2019, 9, 3253; doi:10.3390/app9163253 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3253 2 of 18 constructions, mostly on the structure ﬁeld. There are many codes, which deﬁne bridge loads [3] and concrete, steel, and composite bridge design [4–6]. In the same way, the European Committee for Standardization (CEN) has advertised regulations related to structural bridge design [7–9] as has the American Association of State Highway and Transportation Ocials (AASHTO) [10]. In addition, authors have been working to provide proper box-girder bridge designs [11]. This search for optimal structures has led researchers to search for new forms of design, minimizing structural weight [12] and economic cost [13]. Since the ﬁrst relevant works carried out in the ﬁeld of structural optimization [14,15], the interest in the application of these techniques has grown a great deal, due to the dierent structural typologies studied (steel structures [16], reinforced or pre-stressed concrete structures [17,18], or composite structures [19–24]), as well as for the methods and algorithms used [25,26]. However, researchers have focused, to a large extent, on the cost optimization of these types of structures. Some researchers have stated that there is a relationship between the cost and environmental optimization, and cost optimization is a good approach for the environmental one [27]. Optimization looks to ﬁnd the values of the parameters that deﬁne the problem that allow us to ﬁnd the optimal solution. In the structural ﬁeld, the problems tend to have too many variables to analyze all the possible solutions. Therefore, the use of approximate methods that allow us to reach optimal solutions have been studied. Research about heuristics and metaheuristics has been performed recently [28], such as neural networks [29], Hybrid Harmony Search [30], genetic algorithms [31,32], or simulated annealing [25,26]. Other authors have applied accelerated optimization methods, like kriging [33], allowing to simplify the main structural problems. Some researchers have already applied the multi-criteria optimization for bridge design [34], considering other factors, besides the cost, like the security of the infrastructure and the CO 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. In recent years, databases measuring the environmental impact of materials have been elaborated, because of the importance of incorporating the design criteria to consider the impact on CO emissions [38–40]. Many researchers, such as Yepes et al. [41] and Molina-Moreno et al. [42], have used these databases to study the dierence between the cost and the CO emission optimization for reinforced concrete (RC) structures. The objective of this research was to study the dierences between three heuristic optimization techniques applied to a steel-concrete composite pedestrian bridge. Furthermore, the dierences between the CO 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 The problem proposed in this study is a single-objective optimization of a composite pedestrian bridge. To reach this purpose, a program that determined the optimum values of the variables was created. The objective was to minimize the objective function associated with the cost (1), satisfying the constraints imposed on the problem, represented by the Equation (2): ! ! C x = p m x , (1) i i i=1 G x 0, (2) j Appl. Sci. 2019, 9, 3253 3 of 18 where vector x contains the design variables. To adjust the variables to real cases, these are discrete. The total cost objective function is given in Equation (1), where p is the prices of every construction unit and m 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 would contain the amount of materials associated with these variables, and these measurements, multiplied by their unit prices (p ), result in the total cost of the structure. The values of cost for each variable contains the materials, labor and machinery. Because of the characteristics of the construction materials, the maintenance was not taken into account in this study. The construction place assumed in this paper was Valencia. The construction units included in this study (r ) were the volume of concrete, the amount of reinforcement steel, the amount of rolled steel, and the amount of shear-connector ’s steel. The unit prices of the materials were taken from the price table of the College of Civil Engineers of the Valencian Community for the year 2012. Furthermore, the CO 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 emissions considered in this work. Table 1. Prices and CO emissions. Unit Measurements Cost (¿) Emissions (kg CO ) 93.71 224.34 m of concrete C25/30 m of concrete C30/37 102.41 224.94 105.56 265.28 m of concrete C35/45 m 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 2.1. Design Variables The structural solution was deﬁned by the parameters and variables, the ﬁxed and variable data, respectively. In this work, the objective was to obtain an optimum steel-concrete composite pedestrian bridge with a box-girder cross section of 38 m of span length, modeled like a beam on two supports. The parameters are deﬁned in Table 2. These values were considered ﬁxed for the optimization. The construction was carried out on the ground and then large tonnage cranes lifted the structure. This constructive process made it possible to leverage the materials, because the steel and concrete deﬂection beginnings were the same when the structure was put into service. Figure 1 shows the deck cross section geometrical variables and the reinforcement. Table 3 shows the limits deﬁned for all the variables considered in this problem. The section is formed by two main elements, one the one hand, the steel beam conformed to steel sheets, welded and bolstered with longitudinal and transversal stieners. On the other hand, a reinforced concrete slab placed in the top of the steel beam and connected to this element by steel shear-connectors. Optimization variables were discrete, to bring the problem into line with reality. It was noted that some dimensions used may not have been practical, but allowed the algorithm to visit feasible intermediate solutions to ﬁnd better optimal solutions. The total dimension of the problem was 1.67 10 possible solutions, because of this, the complete evaluation of the problem was unapproachable. The problem optimization was carried out by heuristic techniques. Appl. Sci. 2019, 9, 3253 4 of 18 Table 2. Parameters considered in the analysis. Geometrical Parameters Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 20 Pedestrian bridge width B = 2.5 m Number of spans 1 Termical variation between steel and concrete ± 18 °C Span length 38 m Exposure related parameters Material Parameters External ambient conditions IIb Maximum aggregate size 20 mm Code related parameters Reinforcing steel B-500-S Code regulations EHE-08/IAP-11/RPX-95 Loading Parameters Service working life 100 years Reinforced concrete speciﬁc weight 25 kN/m 116 Figure 1 shows the deck Auxiliary assembly cross sect triangulations ion geomet self-weight rical variables and the reinforcement 0.14 kN . Table 3 shows /m 117 the limits defi Live n Load ed for all the variables considered in this problem. The section is formed by two main 5 kN/m 118 elements, one the one hand, the steel beam conformed to steel sheets, welded and bolstered with Dead load 1.15 kN/m 119 longitudinal and transversal stiffeners. On the other hand, a reinforced concrete slab placed in the top Temperature variation between steel and concrete 18 C 120 of the steel beam and connected to this element by steel shear-connectors. Optimization variables are Exposure Related Parameters 121 discrete to bring the problem into line with reality. It is noted that some dimensions used may not be External ambient conditions IIb 122 practical, but allow the algorithm to visit feasible intermediate solutions to find better optimal solutions. 123 The total dimension of the problem is 1.67·10 possible solutions, because of this, the complete Code Related Parameters 124 evaluation of the problem is unapproachable. The problem optimization has been carried out by Code regulations EHE-08/IAP-11/RPX-95 125 heuristic techniques. Service working life 100 years 129 Figure 1. Box-girder geometrical variables and reinforcement Figure 1. Box-girder geometrical variables and reinforcement. 130 Table 3. Box-girder geometrical variables description and values range Table 3. Box-girder geometrical variables description and values range. Concrete slab variables Range values Step size CL Slab depth 15 to 30 cm 1 cm Concrete Slab Variables Range Values Step Size CL CB Slab Slab ed depth ge depth 10 to 30 cm 15 to 30 cm 1 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 VL Lateral slab cantilever 0.5 to 0.625 m 5 mm DT Transversal reinforcement diameter 6, 8, 10, 12 and 16 mm - DT Transversal reinforcement diameter 6, 8, 10, 12, and 16 mm - DLS Top longitudinal 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 - 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 SBTC Tranversal reinforcement separation in span center 10 to 30 cm 1 cm SBTA Transversal reinforcement separation in supports 5 to 25 cm 1 cm SBTA Tranversal reinforcement separation in supports 5 to 25 cm 1 cm NLS Number of top longitudinal reinforcement bars 10 to 40 1 NLI NLS Number Number of of tbo op l ttom ongitud longitudinal inal reinforcem reinfor ent bars cement bars 10 to 40 10 to 40 1 1 Metal NLBeam I Nu V m ariables ber of botom longitudinal reinforcement bars 10 to 40 1 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 stiener spacing 0, 0.16, 0.26, 0.36, and 0.46 - SRT Transversal stiener spacing 1, 2, 3.8, 7.6, 38 m - Appl. Sci. 2019, 9, 3253 5 of 18 Table 3. Cont. 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 stiener thickness 8 to 40 mm 2 mm ERT Transversal stiener 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 2.2. Structural Analysis and Constraints The structure was analyzed like a linear element. The model of the structure considered the shear deformation and the eective ﬂange width [6]. To obtain the stresses of the structure to check regulations and recommendations constraints, a Fortran language program was implemented. This program calculated the stresses in two sections: mid span and supports. In this structure, the highest stresses occurred in these areas. Prior to the verifying limit states, the program needed to calculate stress envelopes due to the loads. This program evaluated the stress envelopes due to a uniform load of 5 kN/m and the deck self-weight, including the bridge railing and asphalt (see Table 2). Note that the thermal gradient [5] and the dierential 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. Once the stresses were obtained, a structural integrity analysis was performed. The ultimate limit states (ULS) assessed the capacity of the structure against the ﬂexure, shear, torsion, and the combination of the stresses. It further considered the minimum reinforcements to resist the stresses and the examination of the geometrical conditions. To evaluate the structural capacity, the regulations employed to obtain the equations that allow the veriﬁcation of the pedestrian bridge have been the Spanish code on structural concrete [4], the Spanish recommendations for composite road bridge project [6], and the code on the actions for the design of road bridges [3]. The serviceability limit states (SLS) assessed the capacity of the structure to continue its service. The ULS considered in this study were: bending, shear, torsion, shear–torsion interaction, and the stieners veriﬁcation. It must be taken into account that once the variables that deﬁne a frame solution have been chosen, then geometry, the materials, and the passive reinforcement are deﬁned. It should be noted that no attempt has been made to calculate the reinforcement in such a way as to comply with common design rules. Such common design procedures follow a conventional order for obtaining reinforcement bars from ﬂexural ULS, and then checking the SLS and redeﬁning if required. While this order is eective, it ignores other possibilities that heuristic search algorithms do not oversee. The seismic veriﬁcations are not necessary due to the small value of the calculus acceleration for the location of the constructions. On the other side, the SLS assessed were: deﬂections, vibrations, cracking, and web deformation. The vibration limit state was veriﬁed in accordance with the restrictions for footbridges [4]. The SLS of cracking included compliance with limitations of the crack width for existing durability conditions. With respect to deﬂection, the instantaneous and time-dependent deﬂection was limited to 1/500th of the main span length for the characteristic combination [4], and the frequent value for the live loads was limited to 1/1200th of the main span length [5]. Concrete and steel fatigue were not considered, as this ultimate limit state is rarely checked in pedestrian bridges. In addition, the recommendations indicated in the specialized bibliography [43–45] were considered. The modulus implemented compared the structure model values with the values obtained from the regulations equations. This modulus veriﬁed the demands of the safety, as well as those relating to the aptitude for service requirement. Therefore, the limit states and the geometrical and constructability Appl. Sci. 2019, 9, 3253 6 of 18 requirements must be guaranteed. The ULS checked that the ultimate load eects were lower than the resistance of the structure, as seen in Equation (3): R S , (3) U u where R is the ultimate response of the structure and S the ultimate load eects. For instance, the u u ULS of the shear and torsion interaction reduced the shear resistance due to the eect of the torsion. The SLS covers the requirements of functionality, comfort, and aspect (Equation (4)): Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 18 C E , (4) s s where Ru is the ultimate response of the structure and Su the ultimate load effects. For instance, the ULS of the shear and torsion interaction reduced the shear resistance due to the effect of the where C is the permitted value of the serviceability limit and E is the value obtained from the model s s torsion. The SLS covers the requirements of functionality, comfort, and aspect (Equation (4)): produced by the SLS actions. (4) 𝐶 ≥𝐸 , 3. Applied Heuristic Search Methods where Cs is the permitted value of the serviceability limit and Es is the value obtained from the model produced by the SLS actions. 3.1. Descent Local Search 3. Applied Heuristic Search Methods This algorithm (Figure 2) begins by obtaining a random initial solution. Then, a small movement 3.1. Descent Local Search is produced in randomly chosen variables, increasing or decreasing them by a unit step. The algorithm This algorithm (Figure 2) begins by obtaining a random initial solution. Then, a small movement obtains the cost and the evaluation modulus check if the alternative fulﬁls the constraints. If the cost of is produced in randomly chosen variables, increasing or decreasing them by a unit step. The the working solution is lower than the ﬁrst and the new solution fulﬁlls the restrictions, then it replaces algorithm obtains the cost and the evaluation modulus check if the alternative fulfils the constraints. the previous one. This process is continued until no better solutions are found, after a certain number If the cost of the working solution is lower than the first and the new solution fulfills the restrictions, of iterations. then it replaces the previous one. This process is continued until no better solutions are found, after a certain number of iterations. Figure 2. Flowchart of the descent local search (DLS) process. Figure 2. Flowchart of the descent local search (DLS) process. The movements set out in the study modified 1 to 25 variables. The number of iterations used with no improvement were 100, 500, 1000, 2000, 5000, 10,000, 100,000, 500,000, 1 million, and 10 Appl. Sci. 2019, 9, 3253 7 of 18 The movements set out in the study modiﬁed 1 to 25 variables. The number of iterations used with no improvement were 100, 500, 1000, 2000, 5000, 10,000, 100,000, 500,000, 1 million, and 10 million. Figure 3 shows the average cost values according to the number of iterations stop criteria. With movements of one, two, three, four, and ﬁve variables, the algorithm converged very quickly in such a way that improvements were no longer achieved from 500, 5,000, 100,000, and 500,000 iterations, respectively. Movements of more than ﬁve variables always improved the results, as the number of iterations increased, starting to converge from 10,000 iterations. The most balanced evolution was given to a movement of ﬁve or six variables; given that it converged from 10,000 iterations. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 20 197 Figure 3. Average costs and number of iterations stop criterion Figure 3. Average costs and number of iterations stop criteria. 198 3.2. Hybrid Simulated Annealing with a Mutation Operator 3.2. Hybrid Simulated Annealing with a Mutation Operator 199 The hybrid Simulated Annealing (SA) with a mutation operator (SAMO2) is an algorithm The hybrid simulated annealing (SA) with a mutation operator (SAMO2) is an algorithm developed 200 developed by Martí et al. [26] (Figure 4). This technique is used to combine the advantages of good by Martí et al. [26] (Figure 4). This technique is used to combine the advantages of good convergence of 201 convergence of SA and the promotion of the diversity of the genetic strategy. SA, developed by SA and the promotion of the diversity of the genetic strategy. SA, developed by Kirkpatrick et al. [46], 202 Kirkpatrick et al. [46], is based on the analogy of the thermodynamic behavior of a set of atoms to is based on the analogy of the thermodynamic behavior of a set of atoms to form a crystal. “Annealing” 203 form a crystal. “Annealing” is the chemical process of heating and cooling a material in a controlled is the chemical process of heating and cooling a material in a controlled fashion. Genetic algorithms seek 204 fashion. Genetic algorithms seek the best solution through operators such as selection, crossover and the best solution through operators such as selection, crossover, and mutation. Soke and Bingul [32] 205 mutation. Soke and Bingul [32] have combined effectively both algorithms. SAMO2 introduces the 206 eectively probabilcombined istic acceptboth ance o algorithms. f poorer quality so SAMO2 lution introduces s duringthe thepr pr obabilistic ocess, allowing acceptance it to esc ofap poor e from er quality 207 local optimums and to finally find highest quality solutions. To do this, it accepts worse solutions solutions during the process, allowing it to escape from local optimums and to ﬁnally ﬁnd the highest 208 with a probability Pa, given by the expression of Glauber (5), where T is now a parameter that quality solutions. To do this, it accepts worse solutions with a probability P , given by the expression 209 decreases with the time and, thus, reducing the probability of accepting worse solutions, from an of Glauber (5), where T is now a parameter that decreases with the time, thus reducing the probability 210 initial value, T0. of accepting worse solutions, from an initial value, T : 𝑃 = P = . (5) a (5) DE 1+ 1 +𝑒 e This method was applied using the following ﬁxed variable movements: 2, 3, 4, 5, 6, and 12. 212 This method has been applied using the following fixed variable movements: 2, 3, 4, 5, 6 and 12. The initial temperature was set by the method proposed by Medina [47]. Markov chain lengths of 5,000, 213 The initial temperature was set by the method proposed by Medina [47]. Markov chain lengths of: 10,000, 20,000, and 30,000 were tested. In this work, a geometrical cooling of the type T = kT was i+1 i 214 5,000, 10,000, 20,000 and 30,000 have been tested. In this work, a geometrical cooling of the type adopted, considering k < 1, which has the advantage of prolonging the ﬁnal phase of the search when 215 𝑇 =𝑘 · 𝑇 has been adopted, considering k < 1, which has the advantage to prolong the final phase the temperature is low. The coecients k used were 0.80, 0.85, 0.90, and 0.95. For the stop criteria, two 216 of the search when the temperature is low. The coefficients k used have been 0.80, 0.85, 0.90 and 0.95. 217 werFor the stop cri e set in this study: teria, two a that rthe e set i temperatur n this study: tha e was t the tempera less than 0.001 ture is l T eor ss tha that n 0.00 during 1·T0 or that a Markov durin chain g no 218 a Markov chain no better solution has been found. better solution was found. 219 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 18 Appl. Sci. 2019, 9, 3253 8 of 18 Figure 4. Flowchart of the hybrid simulated annealing with a mutation operator (SAMO2) process. Figure 4. Flowchart of the hybrid simulated annealing with a mutation operator (SAMO2) process. From the application of the SAMO2 algorithm, graphs of trajectories of the cost, according to From the application of the SAMO2 algorithm, graphs of trajectories of the cost, according to the the number of iterations or the time, were obtained, as in Figure 5. This ﬁgure shows the trajectory number of iterations or the time, were obtained, as in Figure 5. This figure shows the trajectory of one of one of the rehearsed processes, where a correct operation of the algorithm is appreciated, initially of the rehearsed processes, where a correct operation of the algorithm is appreciated, initially accepting high worsening, which decreases as the process progresses, focusing the search on solutions accepting high worsening, which decreases as the process progresses, focusing the search on with similar or lower costs, which divides the process into an initial diversiﬁcation phase and a ﬁnal solutions with similar or lower costs, which divides the process into an initial diversification phase intensiﬁcation phase. and a final intensification phase. Appl. Sci. 2019, 9, 3253 9 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 18 Figure 5. Trajectory cost and temperature-iterations. Figure 5. Trajectory cost and temperature-iterations. 3.3. Glow-worm Swarm Optimization (GSO) 3.3. Glow-worm Swarm Optimization (GSO) The glow-worm The glow-woswarm rm swarm optimization optimization algorithm algorithm mim mimics ics th the e bbehavior ehavior of of a fia re ﬁr fly eﬂy swarm swarm and w and as was proposed by Krishnanand and Ghose [48]. Glow-worms produce a natural light that is used as a proposed by Krishnanand and Ghose [48]. Glow-worms produce a natural light that is used as a signal signal to attract a partner. Each glow-worm carries an amount of luminescence, which we will call to attract a partner. Each glow-worm carries an amount of luminescence, which we will call “luciferin”. “luciferin”. It is considered that the maximum distance at which this luminescence is perceived is It is considered that the maximum distance at which this luminescence is perceived is limited by a limited by a maximum radial value, which we will call the sensitivity radius rs. So, the decision range maximum radial value, which we will call the sensitivity radius r . So, the decision range for each i i for each glow-worm is also delimited by a maximum radial value r d, that complies with 0 < r d ≤ rs, i i glow-worm is also delimited by a maximum radial value r , that complies with 0 < r r , which we d d s which we will call a decision radius. One glow-worm i considers another firefly j as its neighbor if j will call a decision radius. One glow-worm i considers another ﬁreﬂy j as its neighbor if j is within its is within its decision radius r d and the level of luciferin j is greater than that of i. decision radius i r and the level of luciferin j is greater than that of i. The r d decision radius allows the selective interaction of neighbors and helps the disjointed The r 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 formation of sub-branches. Each ﬁreﬂy selects, through a probabilistic mechanism, a neighbor, who has local information and the selective interaction of neighbors, allow the swarm of fireflies to be a higher value of luciferin and moves towards it. These movements, which are based solely on local subdivided into disjointed subgroups, that address, and are found in multiple optimums of the given information and the selective interaction of neighbors, allow the swarm of ﬁreﬂies to be subdivided multimodal function. The process can be summarized as follows: into disjointed subgroups, that address, and are found in multiple optimums of the given multimodal 1. Initially a swarm of n feasible glow-worms is generated and distributed in the search space. Each function. The process can be summarized as follows: glow-worm has assigned the initial luciferin value l0 and the initial sensitivity radius rs; 1. Initially a swarm of n feasible glow-worms is generated and distributed in the search space. Each 2. Depending on the previous luciferin li and the objective function value, the luciferin is updated glow-worm has assigned the initial luciferin value l and the initial sensitivity radius r ; as is shown on Equation 6. The luciferin value dec 0 ays constant 𝜌 (0 <𝜌 <1) simulates s the decrease in luciferin level over time, and the luciferin enhancement constant γ (0 < γ < 1) is the 2. Depending on the previous luciferin l and the objective function value, the luciferin is updated proportion of the improvement in the objective that glow-worm adds to its luciferin. J(xi(t)) is as is shown on Equation (6). The luciferin value decays constant (0 < < 1) simulates the the value of the objective function of the glow-worm i at iteration j: decrease in luciferin level over time, and the luciferin enhancement constant (0 < < 1) is the ( ) ( ) ( ) ( ) proportion of the improvement 𝑙 𝑡 = in1−𝜌 the objective ·𝑙 𝑡− 1 that +𝛾 ·glow-worm 𝐽 𝑡 ; adds to its luciferin. J(x (t)) (6) is the value of the objective function of the glow-worm i at iteration j: 3. 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 l (t) = (1 )l (t 1) + J(x (t)); (6) i i i by Equation 7, where Ni(t) is the set of neighbors of the glow-worm i in the iteration t, dij represents the Euclidean distance between glow-worms i and j in iteration t. r d(t) is the decision 3. Each glow-worm uses a probability sampling mechanism to move towards a neighbor with a ratio of glow-worm i in iteration j: higher luciferin value. For each glow-worm i, the probability of moving to a neighbor j is given by 𝑙 (𝑡 ) −𝑙 (𝑡) 𝑝 (𝑡 ) = ; 𝑗 ∈𝑁 (𝑡 ) , 𝑁 (𝑡 ) = 𝑗 :𝑑 (𝑡 ) <𝑟 (𝑡 ) ; 𝑙 (𝑡 ) <𝑙 (𝑡 ); (7) Equation (7), where N (t) is the set of neighbors of the glow-worm i in the iteration t, d represents i ij ∑ 𝑙 (𝑡 ) −𝑙 (𝑡) ∈ () the Euclidean distance between glow-worms i and j in iteration t. r (t) is the decision ratio of glow-worm i in iteration j: n o l (t) l (t) j i p (t) = P ; j 2 N (t) , N (t) = j : d (t) < r (t) ; l (t) < l (t) ; (7) i j i i i j i j ( ) ( ) l t l t k2N (t) k 𝑥 Appl. Sci. 2019, 9, 3253 10 of 18 4. During the movement phase, the glow-worm i moves to glow-worm j. Equation (8) describes the Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 18 model of the movement of a glow-worm at any given moment, where x (t) is the location of the glow-worm i at iteration t and s is the step factor constant: 4. During the movement phase, the glow-worm i moves to glow-worm j. Equation 8 describes the x (t) x (t) model of the movement of a glow-worm at any given moment, where xi(t) is the location of the j i x (t + 1) = x (t) + s ; (8) i i glow-worm i at iteration t and s is the step factor constant: kx (t) x (t)k j i 𝑥 (𝑡 ) −𝑥 (𝑡) ( ) ( ) 𝑥 𝑡+ 1 =𝑥 𝑡 +𝑠 · ; (8) 5. Once the movement is ﬁnished, the update of the radial sensor range is carried out by the ( ) 𝑡 −𝑥 (𝑡) expression of Equation (9), where is a constant parameter and n is another parameter that 5. Once the movement is finished, the update of the radial sensor range is carried out by the controls the number of neighbors: expression of Equation 9, where β is a constant parameter and nt is another parameter that n n oo i i controls the number of neighbors: r (t + 1) = min r , max 0, r (t) + n N (t) . (9) s t d d 𝑟 (𝑡+ 1 ) =𝑚𝑖𝑛 𝑟 ,𝑚𝑥𝑎0, 𝑟 (𝑡 ) +𝛽 · (𝑛 − |𝑁 (𝑡 )|). (9) In this work, the GSO algorithm was applied to reach an optimum solution for a steel-concrete composite In thipedestrian s work, the GSO al bridge. The gorivalues thm wa of s a the ppl parameters ied to reach a used n opti to apply mum this solu method tion for a steel– were 0.5,c0.1, oncret 0.5, e composite pedestrian bridge. The values of the parameters used to apply this method were 0.5, 0.1, 2, 0.25, and 4 for , , , n , s, and l , respectively. The maximum number of iterations was ﬁxed at t 0 4000. 0.5, 2,The 0.25 values , and 4 for of n and 𝜌 , γ r, β wer , nt,e s, taken and las 0, re dispectively erent values; . The m theavalues ximumadopted number o infthe iter study ations w areashown s fixed at 4000. The values of n and r0 were taken as different values; the values adopted in the study are in Table 4. shown in Table 4. Table 4. n and r adopted values. Table 4. n and r0 adopted values. n 10 20 30 40 50 60 80 100 n 10 20 30 40 50 60 80 100 r 50 100 150 r0 50 100 150 In this work, the GSO algorithm was applied in two experiments. The ﬁrst one only used the GSO In this work, the GSO algorithm was applied in two experiments. The first one only used the to reach the optimum solution, but in the second, the DLS algorithm was applied to the best solutions GSO to reach the optimum solution, but in the second, the DLS algorithm was applied to the best of the GSO to improve those solutions. Figures 6 and 7 show the results of the GSO and GSO with solutions of the GSO to improve those solutions. Figures 6 and 7 show the results of the GSO and DLS, respectively. GSO with DLS, respectively. Figure 6. Average cost to average iterations for the glow-worms swarm optimization (GSO) experiment. Figure 6. Average cost to average iterations for the glow-worms swarm optimization (GSO) experiment. 𝑥 Appl. Sci. 2019, 9, 3253 11 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 18 Figure 7. Average cost to average iterations for GSO and DLS combination experiment. Figure 7. Average cost to average iterations for GSO and DLS combination experiment. 4. Discussion 4. Discussion 4.1. Comparison of the Heuristic Techniques 4.1. Comparison of the Heuristic Techniques To compare the results of the heuristic techniques, we focused on the cost obtained by each one. To compare the results of the heuristic techniques, we focused on the cost obtained by each one. In Table 5, the results of material amounts and cost for each solution are shown. The heuristic that In Table 5, the results of material amounts and cost for each solution are shown. The heuristic that obtained the lowest cost is GSO combined with DLS. This result is related to the low amount of rolled obtained the lowest cost is GSO combined with DLS. This result is related to the low amount of rolled steel achieved, due to the importance of this material in steel-concrete composite pedestrian bridges. steel achieved, due to the importance of this material in steel–concrete composite pedestrian bridges. As is seen in Table 5, the heuristics with lower values of cost (SA and GSO with DLS) have a lower As is seen in Table 5, the heuristics with lower values of cost (SA and GSO with DLS) have a lower amount of rolled steel. amount of rolled steel. The concrete strength was the same for DSL, SA, and GSO, but when GSO and DSL were combined, The concrete strength was the same for DSL, SA, and GSO, but when GSO and DSL were the geometry variables of the slab decreased due to the increase in the concrete strength, leveraging combined, the geometry variables of the slab decreased due to the increase in the concrete strength, the material. The optimum solution for a steel beam consists of locating the area of steel in a way leveraging the material. The optimum solution for a steel beam consists of locating the area of steel that allows the mobilization of the highest possible mechanical arm. In order to reach the lowest cost, in a way that allows the mobilization of the highest possible mechanical arm. In order to reach the the solutions obtained by the optimization algorithms looked for greater depths with lower amounts lowest cost, the solutions obtained by the optimization algorithms looked for greater depths with of material, increasing the inertia and reducing the structure weight. lower amounts of material, increasing the inertia and reducing the structure weight. Table 5. Cost and material amount for the best heuristic solutions. Table 5. Cost and material amount for the best heuristic solutions. DLS SA GSO GSO and DLS DLS SA GSO GSO and DLS Rolled steel kg/m 153.99 151.26 157.49 150.66 Rolled steel kg/m 153.99 151.26 157.49 150.66 % Rolled/record % 2.21% 0.40% 4.54% 0.00% % Rolled/record % 2.21% 0.40% 4.54% 0.00% Shear-connector steel kg/m 0.90 0.90 0.79 0.90 % Shear-connector/record % 14.28% 14.28% 0.00% 14.28% Shear-connector steel kg/m 0.90 0.90 0.79 0.90 3 2 Concrete 0.15 0.15 0.14 0.14 m /m % Shear-connector/record % 14.28% 14.28% 0.00% 14.28% % Concrete/record % 7.52% 7.52% 0.83% 0.00% 3 2 Concrete m /m 0.15 0.15 0.14 0.14 Reinforcement steel kg/m 22.57 22.22 25.50 22.23 % Concrete/record % 7.52% 7.52% 0.83% 0.00% % Reinforcement/record % 1.58% 0.00% 14.76% 0.04% Reinforcement steel kg/m 22.57 22.22 25.50 22.23 Cost ¿/m 304.82 299.77 313.18 297.76 % Cost/record % 2.37% 0.67% 5.18% 0.00% % Reinforcement/record % 1.58% 0.00% 14.76% 0.04% Cost The rows with percentages express the €/ incr m ease 30 in the4. quantity 82 of29 material9.77 with31 respect3.18 to the minimum.297.76 % Cost/record % 2.37% 0.67% 5.18% 0.00% The rows with percentages express the increase in the quantity of material with respect to the minimum. Appl. Sci. 2019, 9, 3253 12 of 18 4.2. Sustainability Study An analysis of the CO 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 emissions of every solution are compared. Table 6. CO emissions and cost data comparison from the reference solution and the heuristics. Reference DLS SA GSO GSO and DLS Cost ¿/m 399.10 304.82 299.77 313.18 297.76 % Cost/reference % - 23.62% 24.89% 21.53% 25.39% CO emissions kg CO /m 700.22 536.39 527.70 552.54 523.68 % Emissions/reference % - 23.40% 24.64% 21.09% 25.21% As is seen in this table, the GSO and DLS combination heuristic obtained a reduction of 8.12% of the CO emissions compared with the reference. This means that an improvement of 1 ¿/m produced a reduction of the 1.74 kg CO /m . 4.3. Parametric Study A parametric study for varying span lengths is presented with the GSO and DLS combination optimization model. Five span lengths were considered: 28, 32, 38, 42 and 48 m. The characteristics that were studied are the economy, the geometry, and the amount of materials. Tables 7 and 8 compile the values of the features of the optimization solutions: Table 7 gives the main values of the geometry of the structure, and Table 8 gives the values of the measurements of the amount of materials of the structure. Table 7. GSO with DLS combination for 28, 32, 38, 42, and 48 m spans. Span CL CA CT EAS EAL EAI FCK Total (m) (m) (m) (m) (mm) (mm) (mm) (MPa) 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 Table 8. GSO with DLS combination measurements of the materials for 28, 32, 38, 42, and 48 m spans. 2 3 2 2 Span (m) Beam Rolled Steel (kg/m ) Slab Concrete (m /m ) Slab Reinforcement (kg/m ) 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 The results of the parametric study led to practical rules for the preliminary design of cost-optimized steel-concrete composite pedestrian bridges with box-girder cross sections isostatic spans. The discussion of the results was carried out together, with a regression analysis. The functions obtained were valid approximations within the range of the studied parameters. The extrapolation of these results to other span lengths should be carried out carefully. Figure 8 shows the average results of the cost of the structure per square meter of the steel-concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the horizontal span leads to a very good quadratic correlation. The average footbridge cost adjusted to C = 0.1954L 8.0873L + 325.96 with a regression coecient of R = 0.9994. The cost rising is produced by the need Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 20 Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 18 323 Figure 8 shows the average results of the cost of the structure per square meter of the steel- Figure 8 shows the average results of the cost of the structure per square meter of the steel– 324 concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the Appl. Sci. 2019, 9, 3253 13 of 18 325 horizontal span leads to a very good quadratic correlation. (The average footbridge cost adjust to C = horizontal span leads to a very good quadratic correlation. The average footbridge cost adjusted to C 2 2 326 0.1954L – 8.0873L + 325.96 with a regression coefficient of R = 0.9994.) The cost rising is produced by 2 2 = 0.1954L − 8.0873L + 325.96 with a regression coefficient of R = 0.9994. The cost rising is produced 2 2 327 for the needs of larger amounts larger ofamoun materials ts of mater to satisfy ials to sati the deﬂection sfy the deflection requirements. requirement Note thats. Note th the R regr at the R ession by the need for larger amounts of materials to satisfy the deflection requirements. Note that the R 328 regression coefficient in Figure 8 is almost 1, this indicates a very good correlation. The variations coecient in Figure 8 is almost 1, this indicates a very good correlation. The variations between the regression coefficient in Figure 8 is almost 1, this indicates a very good correlation. The variations 329 minimum between the and minimum the mean an cost d the me of thean pedestrian cost of the pe bridge destrian br produced idby ge prod the GSO uced and by the G DLS combination SO and DLS between the minimum and the mean cost of the pedestrian bridge produced by the GSO and DLS 330 combination are 0.42%. are 0.42%. combination are 0.42%. 333 Figure 8. Average cost for different span lengths. Figure 8. Average cost for different span lengths. Figure 8. Average cost for dierent span lengths. 334 Figure 9 shows the mean values of the depth of the steel beam (CA) for different span lengths. Figure 9 shows the mean values of the depth of the steel beam (CA) for different span lengths. Figure 9 shows the mean values of the depth of the steel beam (CA) for dierent span lengths. 335 The depth of the beam has a good linear variation according to the span length of the bridge. (The The depth of the beam has a good linear variation, according to the span length of the bridge. The The depth of the beam has a good linear variation, according to the span length of the bridge. 336 average depth of the beam adjusts to CA = 0.0369L – 0.2179 with R = 0.9927) Again, the good 2 2 average depth of the beam adjusts to CA = 0.0369L − 0.2179 with R = 0.9927. Again, the good The average depth of the beam adjusts to CA = 0.0369L 0.2179 with R = 0.9927. Again, the good 337 correlation factor represents a functional relation. correlation factor represents a functional relationship. correlation factor represents a functional relationship. 340 Figure 9. Mean steel beam depth for different span lengths. Figure 9. Mean steel beam depth for dierent span lengths. Figure 9. Mean steel beam depth for different span lengths. 341 As shown in Figure 10, a function is found relating the thickness of the slab with different span 342 lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of the slab As shown in Figure 10, a function was found relating the thickness of the slab with dierent span As shown in Figure 10, a function was found relating the thickness of the slab with different 343 is determined by the transverse flexion. Once the slab stresses are determined by the longitudinal lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of the slab span lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of 344 deflection, the slope of the curve increases. (The average slab thickness adjusts to CL= 0.0001L – is determined by the transverse ﬂexion. Once the slab stresses are determined by the longitudinal the slab is determined by the transverse flexion. Once the slab stresses are determined by the 345 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m). Related to the compressive deﬂection, the slope of the curve increases. The average slab thickness adjusts to CL = 0.0001L longitudinal deflection, the slope of the curve increases. The average slab thickness adjusts to CL = 346 strength of the concrete of the slab, in Figure 11 the relation to the concrete compressive characteristic 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m. Related to the compressive 2 2 0.0001L − 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m. Related to the 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. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 18 Appl. Sci. 2019, 9, 3253 14 of 18 compressive strength of the concrete of the slab, Figure 11 shows the relationship of the concrete Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 20 compressive characteristic strength and the span length. This relationship adjusts well to a quadratic 2 2 The concrete compressive strength adjusts to FCK = 0.0292L 1.6959L 2 + 49.714 with R = 0.9987. 2 function. The concrete compressive strength adjusts to FCK = 0.0292L − 1.6959L + 49.714 with R = 347 strength and the span length is shown. This relation has a good adjust to a quadratic function (The Note that the highest concrete compressive strength considered for this study was 35 MPa. 0.9987. Note that the highest concrete compressive strength considered for this study was 35 MPa. 2 2 348 concrete compressive strength adjusts to FCK = 0.0292L – 1.6959L + 49.714 with R = 0.9987). Note 349 that the highest concrete compressive strength considered for this study is 35 MPa. Figure 10. Mean slab thickness for dierent span lengths. Figure 10. Mean slab thickness for different span lengths. 351 Figure 10. Mean slab thickness for different span lengths. 353 Figure 11. Average compressive strength for different span lengths. Figure 11. Average compressive strength for dierent span lengths. Figure 11. Average compressive strength for different span lengths. 354 Regarding ratio of the amount of rolled steel (Rs) and the surface of the slab (Ss); Figure 12 Regarding the ratio of the amount of rolled steel (R ) and the surface of the slab (S ) Figure 12 s s 355 illustrates the increasing of the amount of rolled steel needed to resist the flexural requirements. The Regarding the ratio of the amount of rolled steel (Rs) and the surface of the slab (Ss) Figure 12 illustrates the increase in the amount of rolled steel needed to resist the ﬂexural requirements. The slope 356 slope of the curve tens to increase as the span length increases (The mean amount of rolled steel in illustrates the increase in the amount of rolled steel needed to resist the flexural requirements. The of the curve tends to increase as the span length increases. The mean amount of rolled steel in relation 2 2 357 relation to the surface of slab adjusts to Rs/Ss = 0.01913L – 3.5553L + 154.96 with R = 0.9996). Although, slope of the curve tends to increase as the span length increases. The mean amount of rolled steel in 2 2 to the surface of slab adjusts to R /S = 0.01913L 3.5553L + 154.96 with R = 0.9996. However, s s 2 2 358 the ratio of volume concrete (Vc) and the surface slab fits a second order equation that increases with relation to the surface of slab adjusts to Rs/Ss = 0.01913L − 3.5553L + 154.96 with R = 0.9996. However, the ratio of the volume of concrete (V ) and the surface slab ﬁts a second order equation that increases 359 the span length in the same way as rolled steel amount as it seen in Figure 13 (The mean ratio of the the ratio of the volume of concrete (Vc) 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 360 volume of concrete in relation to the surface of slab adjusts to Vc /Ss = 0.0001L – 0.0067L + 0.245). with the span length in the same way as rolled steel amount, as seen in Figure 13. The mean ratio of of the volume of concrete in relation to the surface of slab adjusts to V /S = 0.0001L 0.0067L + c s 361 Moreover, the ratio of reinforcing steel (RFs) measured per square meter of slab shows the same the volume of concrete in relation to the surface of slab adjusts to Vc/Ss = 0.0001L − 0.0067L + 0.245. 0.245. Moreover, the ratio of reinforcing steel (RF ) measured per square meter of slab shows the same 362 tendency as rolled steel amount and concrete volume (The ratio of reinforced steel in relation to the Moreover, the ratio of reinforcing steel (RFs) 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 2 2 363 surface of slab adjusts to RFs / Ss = 0.0246L – 1.2009L + 32.516 with R = 0.9955), it shows in figure 14 tendency as rolled steel amount and concrete volume. The ratio of reinforced steel in relation to the 2 2 surface of slab adjusts to RF /S = 0.0246L 1.2009L + 32.516 with R = 0.9955, as shown in Figure 14. s s 2 2 surface of slab adjusts to RFs/Ss = 0.0246L − 1.2009L + 32.516 with R = 0.9955, as shown in Figure 14. Appl. Sci. 2019, 9, 3253 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with the span. Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with the span. the span. 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 13. Variation in the ratio of the volume of concrete 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 Figure 13. Variation in the ratio of the volume of concrete in relation to the surface of the slab with the span. the span. Figure 14. Variation in the ratio of the reinforcement steel 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. Figure 14. Variation in the ratio of the reinforcement steel in relation to the surface of the slab with Figure 14. Variation in the ratio of the reinforcement steel in relation to the surface of the slab with the span. the span. Appl. Sci. 2019, 9, 3253 16 of 18 5. Conclusions Three heuristic algorithms were applied to a steel-concrete composite pedestrian bridge to ﬁnd an ecient design. DLS, SA, and GSO are used to automatically ﬁnd optimum solutions. All procedures are useful in the automated design of steel-concrete composite pedestrian bridges. Furthermore, a parametric study was carried out. The conclusions are as follows: 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 emission comparison showed that the reduction between the original structure and the GSO with DLS combination optimized structure reduced the CO emission by 21.21%. Thus, 2 2 an improvement of 1 ¿/m produced a reduction of 1.74 kg CO /m . Therefore, the solutions that are acceptable in terms of CO 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 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: This paper represents a result of teamwork. The authors jointly designed the research. D.M.-M. drafted the manuscript. V.Y., M.D.-G., J.V.M., and V.J.L.-D. edited and improved the manuscript until all authors are satisﬁed with the ﬁnal version. Acknowledgments: The authors acknowledge the ﬁnancial support of the Spanish Ministry of Economy and Business, along with FEDER funding (DIMALIFE Project: BIA2017-85098-R). Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Liu, S.; Tao, R.; Tam, C.M. Optimizing cost and CO emission for construction projects using particle swarm optimization. Habitat Int. 2013, 37, 155–162. [CrossRef] 2. World Commission on Environment and Development. Our Common Future; Oxford University Press: Oxford, UK, 1987. 3. Ministerio de Fomento. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/heuristic-techniques-for-the-design-of-steel-concrete-composite-HtRQCU00mj

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Publisher: Multidisciplinary Digital Publishing Institute
Copyright: © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
ISSN: 2076-3417
DOI: 10.3390/app9163253
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Abstract

applied sciences Article Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges 1 1 1 2 Víctor Yepes , Manuel Dasí-Gil , David Martínez-Muñoz , Vicente J. López-Desﬁlis and 1 , Jose V. Martí * Institute of Concrete Science and Technology (ICITECH), Universitat Politècnica de València, 46022 Valencia, Spain Department. of Continuous Medium Mechanics and Theory of Structures, Universitat Politècnica de València, 46022 Valencia, Spain * Correspondence: jvmartia@cst.upv.es; Tel.: +34-96-387-7000; Fax: +34-96-387-7569 Received: 5 June 2019; Accepted: 4 August 2019; Published: 9 August 2019 Abstract: The objective of this work was to apply heuristic optimization techniques to a steel-concrete composite pedestrian bridge, modeled like a beam on two supports. A program has been developed in Fortran programming language, capable of generating pedestrian bridges, checking them, and evaluating their cost. The following algorithms were implemented: descent local search (DLS), a hybrid simulated annealing with a mutation operator (SAMO2), and a glow-worms swarm optimization (GSO) in two variants. The ﬁrst one only considers the GSO and the second combines GSO and DLS, applying the DSL heuristic to the best solutions obtained by the GSO. The results were compared according to the lowest cost. The GSO and DLS algorithms combined obtained the best results in terms of cost. Furthermore, a comparison between the CO 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. Keywords: pedestrian bridge; composite structures; optimization; metaheuristics; structural design 1. Introduction Nowadays, society’s concern about the impact of activities is rising, not only their economic inﬂuence, but also the environmental impact. The construction sector is one of the most carbon intensive industries [1] due to the need for large amounts of materials and, henceforth, large amounts of natural resources. Therefore, researchers are investigating how to achieve cost ecient and environmentally sustainable processes for the construction industry. The term sustainable development was introduced for the ﬁrst time by the Brundtland Commission, deﬁning it as, “development that meets the needs of the present without compromising the ability of future generations to meet their own needs” [2]. Since then, countries have been raising awareness about the compromise to the future generations, modifying their policies and demanding cheaper, ecofriendly constructions, without forgetting their safety and durability. In essence, the demands of the governments are to reach solutions that reduce their impact on the three main pillars: the economy, the environment, and society. These demands translate into restrictions for constructors and designer. The former need to carry out the constructions with new strategies to improve sustainability, while the designers have to conceive their projects in a cheaper, ecofriendly way. This means proﬁting the materials and taking maximum advantage of their characteristics, and maintaining durability and safety. The traditional recommendation for the designers is to take a starting point for their designs. Furthermore, there are lots of strict codes and regulations to ensure the safe and reliability of Appl. Sci. 2019, 9, 3253; doi:10.3390/app9163253 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3253 2 of 18 constructions, mostly on the structure ﬁeld. There are many codes, which deﬁne bridge loads [3] and concrete, steel, and composite bridge design [4–6]. In the same way, the European Committee for Standardization (CEN) has advertised regulations related to structural bridge design [7–9] as has the American Association of State Highway and Transportation Ocials (AASHTO) [10]. In addition, authors have been working to provide proper box-girder bridge designs [11]. This search for optimal structures has led researchers to search for new forms of design, minimizing structural weight [12] and economic cost [13]. Since the ﬁrst relevant works carried out in the ﬁeld of structural optimization [14,15], the interest in the application of these techniques has grown a great deal, due to the dierent structural typologies studied (steel structures [16], reinforced or pre-stressed concrete structures [17,18], or composite structures [19–24]), as well as for the methods and algorithms used [25,26]. However, researchers have focused, to a large extent, on the cost optimization of these types of structures. Some researchers have stated that there is a relationship between the cost and environmental optimization, and cost optimization is a good approach for the environmental one [27]. Optimization looks to ﬁnd the values of the parameters that deﬁne the problem that allow us to ﬁnd the optimal solution. In the structural ﬁeld, the problems tend to have too many variables to analyze all the possible solutions. Therefore, the use of approximate methods that allow us to reach optimal solutions have been studied. Research about heuristics and metaheuristics has been performed recently [28], such as neural networks [29], Hybrid Harmony Search [30], genetic algorithms [31,32], or simulated annealing [25,26]. Other authors have applied accelerated optimization methods, like kriging [33], allowing to simplify the main structural problems. Some researchers have already applied the multi-criteria optimization for bridge design [34], considering other factors, besides the cost, like the security of the infrastructure and the CO 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. In recent years, databases measuring the environmental impact of materials have been elaborated, because of the importance of incorporating the design criteria to consider the impact on CO emissions [38–40]. Many researchers, such as Yepes et al. [41] and Molina-Moreno et al. [42], have used these databases to study the dierence between the cost and the CO emission optimization for reinforced concrete (RC) structures. The objective of this research was to study the dierences between three heuristic optimization techniques applied to a steel-concrete composite pedestrian bridge. Furthermore, the dierences between the CO 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 The problem proposed in this study is a single-objective optimization of a composite pedestrian bridge. To reach this purpose, a program that determined the optimum values of the variables was created. The objective was to minimize the objective function associated with the cost (1), satisfying the constraints imposed on the problem, represented by the Equation (2): ! ! C x = p m x , (1) i i i=1 G x 0, (2) j Appl. Sci. 2019, 9, 3253 3 of 18 where vector x contains the design variables. To adjust the variables to real cases, these are discrete. The total cost objective function is given in Equation (1), where p is the prices of every construction unit and m 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 would contain the amount of materials associated with these variables, and these measurements, multiplied by their unit prices (p ), result in the total cost of the structure. The values of cost for each variable contains the materials, labor and machinery. Because of the characteristics of the construction materials, the maintenance was not taken into account in this study. The construction place assumed in this paper was Valencia. The construction units included in this study (r ) were the volume of concrete, the amount of reinforcement steel, the amount of rolled steel, and the amount of shear-connector ’s steel. The unit prices of the materials were taken from the price table of the College of Civil Engineers of the Valencian Community for the year 2012. Furthermore, the CO 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 emissions considered in this work. Table 1. Prices and CO emissions. Unit Measurements Cost (¿) Emissions (kg CO ) 93.71 224.34 m of concrete C25/30 m of concrete C30/37 102.41 224.94 105.56 265.28 m of concrete C35/45 m 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 2.1. Design Variables The structural solution was deﬁned by the parameters and variables, the ﬁxed and variable data, respectively. In this work, the objective was to obtain an optimum steel-concrete composite pedestrian bridge with a box-girder cross section of 38 m of span length, modeled like a beam on two supports. The parameters are deﬁned in Table 2. These values were considered ﬁxed for the optimization. The construction was carried out on the ground and then large tonnage cranes lifted the structure. This constructive process made it possible to leverage the materials, because the steel and concrete deﬂection beginnings were the same when the structure was put into service. Figure 1 shows the deck cross section geometrical variables and the reinforcement. Table 3 shows the limits deﬁned for all the variables considered in this problem. The section is formed by two main elements, one the one hand, the steel beam conformed to steel sheets, welded and bolstered with longitudinal and transversal stieners. On the other hand, a reinforced concrete slab placed in the top of the steel beam and connected to this element by steel shear-connectors. Optimization variables were discrete, to bring the problem into line with reality. It was noted that some dimensions used may not have been practical, but allowed the algorithm to visit feasible intermediate solutions to ﬁnd better optimal solutions. The total dimension of the problem was 1.67 10 possible solutions, because of this, the complete evaluation of the problem was unapproachable. The problem optimization was carried out by heuristic techniques. Appl. Sci. 2019, 9, 3253 4 of 18 Table 2. Parameters considered in the analysis. Geometrical Parameters Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 20 Pedestrian bridge width B = 2.5 m Number of spans 1 Termical variation between steel and concrete ± 18 °C Span length 38 m Exposure related parameters Material Parameters External ambient conditions IIb Maximum aggregate size 20 mm Code related parameters Reinforcing steel B-500-S Code regulations EHE-08/IAP-11/RPX-95 Loading Parameters Service working life 100 years Reinforced concrete speciﬁc weight 25 kN/m 116 Figure 1 shows the deck Auxiliary assembly cross sect triangulations ion geomet self-weight rical variables and the reinforcement 0.14 kN . Table 3 shows /m 117 the limits defi Live n Load ed for all the variables considered in this problem. The section is formed by two main 5 kN/m 118 elements, one the one hand, the steel beam conformed to steel sheets, welded and bolstered with Dead load 1.15 kN/m 119 longitudinal and transversal stiffeners. On the other hand, a reinforced concrete slab placed in the top Temperature variation between steel and concrete 18 C 120 of the steel beam and connected to this element by steel shear-connectors. Optimization variables are Exposure Related Parameters 121 discrete to bring the problem into line with reality. It is noted that some dimensions used may not be External ambient conditions IIb 122 practical, but allow the algorithm to visit feasible intermediate solutions to find better optimal solutions. 123 The total dimension of the problem is 1.67·10 possible solutions, because of this, the complete Code Related Parameters 124 evaluation of the problem is unapproachable. The problem optimization has been carried out by Code regulations EHE-08/IAP-11/RPX-95 125 heuristic techniques. Service working life 100 years 129 Figure 1. Box-girder geometrical variables and reinforcement Figure 1. Box-girder geometrical variables and reinforcement. 130 Table 3. Box-girder geometrical variables description and values range Table 3. Box-girder geometrical variables description and values range. Concrete slab variables Range values Step size CL Slab depth 15 to 30 cm 1 cm Concrete Slab Variables Range Values Step Size CL CB Slab Slab ed depth ge depth 10 to 30 cm 15 to 30 cm 1 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 VL Lateral slab cantilever 0.5 to 0.625 m 5 mm DT Transversal reinforcement diameter 6, 8, 10, 12 and 16 mm - DT Transversal reinforcement diameter 6, 8, 10, 12, and 16 mm - DLS Top longitudinal 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 - 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 SBTC Tranversal reinforcement separation in span center 10 to 30 cm 1 cm SBTA Transversal reinforcement separation in supports 5 to 25 cm 1 cm SBTA Tranversal reinforcement separation in supports 5 to 25 cm 1 cm NLS Number of top longitudinal reinforcement bars 10 to 40 1 NLI NLS Number Number of of tbo op l ttom ongitud longitudinal inal reinforcem reinfor ent bars cement bars 10 to 40 10 to 40 1 1 Metal NLBeam I Nu V m ariables ber of botom longitudinal reinforcement bars 10 to 40 1 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 stiener spacing 0, 0.16, 0.26, 0.36, and 0.46 - SRT Transversal stiener spacing 1, 2, 3.8, 7.6, 38 m - Appl. Sci. 2019, 9, 3253 5 of 18 Table 3. Cont. 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 stiener thickness 8 to 40 mm 2 mm ERT Transversal stiener 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 2.2. Structural Analysis and Constraints The structure was analyzed like a linear element. The model of the structure considered the shear deformation and the eective ﬂange width [6]. To obtain the stresses of the structure to check regulations and recommendations constraints, a Fortran language program was implemented. This program calculated the stresses in two sections: mid span and supports. In this structure, the highest stresses occurred in these areas. Prior to the verifying limit states, the program needed to calculate stress envelopes due to the loads. This program evaluated the stress envelopes due to a uniform load of 5 kN/m and the deck self-weight, including the bridge railing and asphalt (see Table 2). Note that the thermal gradient [5] and the dierential 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. Once the stresses were obtained, a structural integrity analysis was performed. The ultimate limit states (ULS) assessed the capacity of the structure against the ﬂexure, shear, torsion, and the combination of the stresses. It further considered the minimum reinforcements to resist the stresses and the examination of the geometrical conditions. To evaluate the structural capacity, the regulations employed to obtain the equations that allow the veriﬁcation of the pedestrian bridge have been the Spanish code on structural concrete [4], the Spanish recommendations for composite road bridge project [6], and the code on the actions for the design of road bridges [3]. The serviceability limit states (SLS) assessed the capacity of the structure to continue its service. The ULS considered in this study were: bending, shear, torsion, shear–torsion interaction, and the stieners veriﬁcation. It must be taken into account that once the variables that deﬁne a frame solution have been chosen, then geometry, the materials, and the passive reinforcement are deﬁned. It should be noted that no attempt has been made to calculate the reinforcement in such a way as to comply with common design rules. Such common design procedures follow a conventional order for obtaining reinforcement bars from ﬂexural ULS, and then checking the SLS and redeﬁning if required. While this order is eective, it ignores other possibilities that heuristic search algorithms do not oversee. The seismic veriﬁcations are not necessary due to the small value of the calculus acceleration for the location of the constructions. On the other side, the SLS assessed were: deﬂections, vibrations, cracking, and web deformation. The vibration limit state was veriﬁed in accordance with the restrictions for footbridges [4]. The SLS of cracking included compliance with limitations of the crack width for existing durability conditions. With respect to deﬂection, the instantaneous and time-dependent deﬂection was limited to 1/500th of the main span length for the characteristic combination [4], and the frequent value for the live loads was limited to 1/1200th of the main span length [5]. Concrete and steel fatigue were not considered, as this ultimate limit state is rarely checked in pedestrian bridges. In addition, the recommendations indicated in the specialized bibliography [43–45] were considered. The modulus implemented compared the structure model values with the values obtained from the regulations equations. This modulus veriﬁed the demands of the safety, as well as those relating to the aptitude for service requirement. Therefore, the limit states and the geometrical and constructability Appl. Sci. 2019, 9, 3253 6 of 18 requirements must be guaranteed. The ULS checked that the ultimate load eects were lower than the resistance of the structure, as seen in Equation (3): R S , (3) U u where R is the ultimate response of the structure and S the ultimate load eects. For instance, the u u ULS of the shear and torsion interaction reduced the shear resistance due to the eect of the torsion. The SLS covers the requirements of functionality, comfort, and aspect (Equation (4)): Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 18 C E , (4) s s where Ru is the ultimate response of the structure and Su the ultimate load effects. For instance, the ULS of the shear and torsion interaction reduced the shear resistance due to the effect of the where C is the permitted value of the serviceability limit and E is the value obtained from the model s s torsion. The SLS covers the requirements of functionality, comfort, and aspect (Equation (4)): produced by the SLS actions. (4) 𝐶 ≥𝐸 , 3. Applied Heuristic Search Methods where Cs is the permitted value of the serviceability limit and Es is the value obtained from the model produced by the SLS actions. 3.1. Descent Local Search 3. Applied Heuristic Search Methods This algorithm (Figure 2) begins by obtaining a random initial solution. Then, a small movement 3.1. Descent Local Search is produced in randomly chosen variables, increasing or decreasing them by a unit step. The algorithm This algorithm (Figure 2) begins by obtaining a random initial solution. Then, a small movement obtains the cost and the evaluation modulus check if the alternative fulﬁls the constraints. If the cost of is produced in randomly chosen variables, increasing or decreasing them by a unit step. The the working solution is lower than the ﬁrst and the new solution fulﬁlls the restrictions, then it replaces algorithm obtains the cost and the evaluation modulus check if the alternative fulfils the constraints. the previous one. This process is continued until no better solutions are found, after a certain number If the cost of the working solution is lower than the first and the new solution fulfills the restrictions, of iterations. then it replaces the previous one. This process is continued until no better solutions are found, after a certain number of iterations. Figure 2. Flowchart of the descent local search (DLS) process. Figure 2. Flowchart of the descent local search (DLS) process. The movements set out in the study modified 1 to 25 variables. The number of iterations used with no improvement were 100, 500, 1000, 2000, 5000, 10,000, 100,000, 500,000, 1 million, and 10 Appl. Sci. 2019, 9, 3253 7 of 18 The movements set out in the study modiﬁed 1 to 25 variables. The number of iterations used with no improvement were 100, 500, 1000, 2000, 5000, 10,000, 100,000, 500,000, 1 million, and 10 million. Figure 3 shows the average cost values according to the number of iterations stop criteria. With movements of one, two, three, four, and ﬁve variables, the algorithm converged very quickly in such a way that improvements were no longer achieved from 500, 5,000, 100,000, and 500,000 iterations, respectively. Movements of more than ﬁve variables always improved the results, as the number of iterations increased, starting to converge from 10,000 iterations. The most balanced evolution was given to a movement of ﬁve or six variables; given that it converged from 10,000 iterations. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 20 197 Figure 3. Average costs and number of iterations stop criterion Figure 3. Average costs and number of iterations stop criteria. 198 3.2. Hybrid Simulated Annealing with a Mutation Operator 3.2. Hybrid Simulated Annealing with a Mutation Operator 199 The hybrid Simulated Annealing (SA) with a mutation operator (SAMO2) is an algorithm The hybrid simulated annealing (SA) with a mutation operator (SAMO2) is an algorithm developed 200 developed by Martí et al. [26] (Figure 4). This technique is used to combine the advantages of good by Martí et al. [26] (Figure 4). This technique is used to combine the advantages of good convergence of 201 convergence of SA and the promotion of the diversity of the genetic strategy. SA, developed by SA and the promotion of the diversity of the genetic strategy. SA, developed by Kirkpatrick et al. [46], 202 Kirkpatrick et al. [46], is based on the analogy of the thermodynamic behavior of a set of atoms to is based on the analogy of the thermodynamic behavior of a set of atoms to form a crystal. “Annealing” 203 form a crystal. “Annealing” is the chemical process of heating and cooling a material in a controlled is the chemical process of heating and cooling a material in a controlled fashion. Genetic algorithms seek 204 fashion. Genetic algorithms seek the best solution through operators such as selection, crossover and the best solution through operators such as selection, crossover, and mutation. Soke and Bingul [32] 205 mutation. Soke and Bingul [32] have combined effectively both algorithms. SAMO2 introduces the 206 eectively probabilcombined istic acceptboth ance o algorithms. f poorer quality so SAMO2 lution introduces s duringthe thepr pr obabilistic ocess, allowing acceptance it to esc ofap poor e from er quality 207 local optimums and to finally find highest quality solutions. To do this, it accepts worse solutions solutions during the process, allowing it to escape from local optimums and to ﬁnally ﬁnd the highest 208 with a probability Pa, given by the expression of Glauber (5), where T is now a parameter that quality solutions. To do this, it accepts worse solutions with a probability P , given by the expression 209 decreases with the time and, thus, reducing the probability of accepting worse solutions, from an of Glauber (5), where T is now a parameter that decreases with the time, thus reducing the probability 210 initial value, T0. of accepting worse solutions, from an initial value, T : 𝑃 = P = . (5) a (5) DE 1+ 1 +𝑒 e This method was applied using the following ﬁxed variable movements: 2, 3, 4, 5, 6, and 12. 212 This method has been applied using the following fixed variable movements: 2, 3, 4, 5, 6 and 12. The initial temperature was set by the method proposed by Medina [47]. Markov chain lengths of 5,000, 213 The initial temperature was set by the method proposed by Medina [47]. Markov chain lengths of: 10,000, 20,000, and 30,000 were tested. In this work, a geometrical cooling of the type T = kT was i+1 i 214 5,000, 10,000, 20,000 and 30,000 have been tested. In this work, a geometrical cooling of the type adopted, considering k < 1, which has the advantage of prolonging the ﬁnal phase of the search when 215 𝑇 =𝑘 · 𝑇 has been adopted, considering k < 1, which has the advantage to prolong the final phase the temperature is low. The coecients k used were 0.80, 0.85, 0.90, and 0.95. For the stop criteria, two 216 of the search when the temperature is low. The coefficients k used have been 0.80, 0.85, 0.90 and 0.95. 217 werFor the stop cri e set in this study: teria, two a that rthe e set i temperatur n this study: tha e was t the tempera less than 0.001 ture is l T eor ss tha that n 0.00 during 1·T0 or that a Markov durin chain g no 218 a Markov chain no better solution has been found. better solution was found. 219 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 18 Appl. Sci. 2019, 9, 3253 8 of 18 Figure 4. Flowchart of the hybrid simulated annealing with a mutation operator (SAMO2) process. Figure 4. Flowchart of the hybrid simulated annealing with a mutation operator (SAMO2) process. From the application of the SAMO2 algorithm, graphs of trajectories of the cost, according to From the application of the SAMO2 algorithm, graphs of trajectories of the cost, according to the the number of iterations or the time, were obtained, as in Figure 5. This ﬁgure shows the trajectory number of iterations or the time, were obtained, as in Figure 5. This figure shows the trajectory of one of one of the rehearsed processes, where a correct operation of the algorithm is appreciated, initially of the rehearsed processes, where a correct operation of the algorithm is appreciated, initially accepting high worsening, which decreases as the process progresses, focusing the search on solutions accepting high worsening, which decreases as the process progresses, focusing the search on with similar or lower costs, which divides the process into an initial diversiﬁcation phase and a ﬁnal solutions with similar or lower costs, which divides the process into an initial diversification phase intensiﬁcation phase. and a final intensification phase. Appl. Sci. 2019, 9, 3253 9 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 18 Figure 5. Trajectory cost and temperature-iterations. Figure 5. Trajectory cost and temperature-iterations. 3.3. Glow-worm Swarm Optimization (GSO) 3.3. Glow-worm Swarm Optimization (GSO) The glow-worm The glow-woswarm rm swarm optimization optimization algorithm algorithm mim mimics ics th the e bbehavior ehavior of of a fia re ﬁr fly eﬂy swarm swarm and w and as was proposed by Krishnanand and Ghose [48]. Glow-worms produce a natural light that is used as a proposed by Krishnanand and Ghose [48]. Glow-worms produce a natural light that is used as a signal signal to attract a partner. Each glow-worm carries an amount of luminescence, which we will call to attract a partner. Each glow-worm carries an amount of luminescence, which we will call “luciferin”. “luciferin”. It is considered that the maximum distance at which this luminescence is perceived is It is considered that the maximum distance at which this luminescence is perceived is limited by a limited by a maximum radial value, which we will call the sensitivity radius rs. So, the decision range maximum radial value, which we will call the sensitivity radius r . So, the decision range for each i i for each glow-worm is also delimited by a maximum radial value r d, that complies with 0 < r d ≤ rs, i i glow-worm is also delimited by a maximum radial value r , that complies with 0 < r r , which we d d s which we will call a decision radius. One glow-worm i considers another firefly j as its neighbor if j will call a decision radius. One glow-worm i considers another ﬁreﬂy j as its neighbor if j is within its is within its decision radius r d and the level of luciferin j is greater than that of i. decision radius i r and the level of luciferin j is greater than that of i. The r d decision radius allows the selective interaction of neighbors and helps the disjointed The r 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 formation of sub-branches. Each ﬁreﬂy selects, through a probabilistic mechanism, a neighbor, who has local information and the selective interaction of neighbors, allow the swarm of fireflies to be a higher value of luciferin and moves towards it. These movements, which are based solely on local subdivided into disjointed subgroups, that address, and are found in multiple optimums of the given information and the selective interaction of neighbors, allow the swarm of ﬁreﬂies to be subdivided multimodal function. The process can be summarized as follows: into disjointed subgroups, that address, and are found in multiple optimums of the given multimodal 1. Initially a swarm of n feasible glow-worms is generated and distributed in the search space. Each function. The process can be summarized as follows: glow-worm has assigned the initial luciferin value l0 and the initial sensitivity radius rs; 1. Initially a swarm of n feasible glow-worms is generated and distributed in the search space. Each 2. Depending on the previous luciferin li and the objective function value, the luciferin is updated glow-worm has assigned the initial luciferin value l and the initial sensitivity radius r ; as is shown on Equation 6. The luciferin value dec 0 ays constant 𝜌 (0 <𝜌 <1) simulates s the decrease in luciferin level over time, and the luciferin enhancement constant γ (0 < γ < 1) is the 2. Depending on the previous luciferin l and the objective function value, the luciferin is updated proportion of the improvement in the objective that glow-worm adds to its luciferin. J(xi(t)) is as is shown on Equation (6). The luciferin value decays constant (0 < < 1) simulates the the value of the objective function of the glow-worm i at iteration j: decrease in luciferin level over time, and the luciferin enhancement constant (0 < < 1) is the ( ) ( ) ( ) ( ) proportion of the improvement 𝑙 𝑡 = in1−𝜌 the objective ·𝑙 𝑡− 1 that +𝛾 ·glow-worm 𝐽 𝑡 ; adds to its luciferin. J(x (t)) (6) is the value of the objective function of the glow-worm i at iteration j: 3. 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 l (t) = (1 )l (t 1) + J(x (t)); (6) i i i by Equation 7, where Ni(t) is the set of neighbors of the glow-worm i in the iteration t, dij represents the Euclidean distance between glow-worms i and j in iteration t. r d(t) is the decision 3. Each glow-worm uses a probability sampling mechanism to move towards a neighbor with a ratio of glow-worm i in iteration j: higher luciferin value. For each glow-worm i, the probability of moving to a neighbor j is given by 𝑙 (𝑡 ) −𝑙 (𝑡) 𝑝 (𝑡 ) = ; 𝑗 ∈𝑁 (𝑡 ) , 𝑁 (𝑡 ) = 𝑗 :𝑑 (𝑡 ) <𝑟 (𝑡 ) ; 𝑙 (𝑡 ) <𝑙 (𝑡 ); (7) Equation (7), where N (t) is the set of neighbors of the glow-worm i in the iteration t, d represents i ij ∑ 𝑙 (𝑡 ) −𝑙 (𝑡) ∈ () the Euclidean distance between glow-worms i and j in iteration t. r (t) is the decision ratio of glow-worm i in iteration j: n o l (t) l (t) j i p (t) = P ; j 2 N (t) , N (t) = j : d (t) < r (t) ; l (t) < l (t) ; (7) i j i i i j i j ( ) ( ) l t l t k2N (t) k 𝑥 Appl. Sci. 2019, 9, 3253 10 of 18 4. During the movement phase, the glow-worm i moves to glow-worm j. Equation (8) describes the Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 18 model of the movement of a glow-worm at any given moment, where x (t) is the location of the glow-worm i at iteration t and s is the step factor constant: 4. During the movement phase, the glow-worm i moves to glow-worm j. Equation 8 describes the x (t) x (t) model of the movement of a glow-worm at any given moment, where xi(t) is the location of the j i x (t + 1) = x (t) + s ; (8) i i glow-worm i at iteration t and s is the step factor constant: kx (t) x (t)k j i 𝑥 (𝑡 ) −𝑥 (𝑡) ( ) ( ) 𝑥 𝑡+ 1 =𝑥 𝑡 +𝑠 · ; (8) 5. Once the movement is ﬁnished, the update of the radial sensor range is carried out by the ( ) 𝑡 −𝑥 (𝑡) expression of Equation (9), where is a constant parameter and n is another parameter that 5. Once the movement is finished, the update of the radial sensor range is carried out by the controls the number of neighbors: expression of Equation 9, where β is a constant parameter and nt is another parameter that n n oo i i controls the number of neighbors: r (t + 1) = min r , max 0, r (t) + n N (t) . (9) s t d d 𝑟 (𝑡+ 1 ) =𝑚𝑖𝑛 𝑟 ,𝑚𝑥𝑎0, 𝑟 (𝑡 ) +𝛽 · (𝑛 − |𝑁 (𝑡 )|). (9) In this work, the GSO algorithm was applied to reach an optimum solution for a steel-concrete composite In thipedestrian s work, the GSO al bridge. The gorivalues thm wa of s a the ppl parameters ied to reach a used n opti to apply mum this solu method tion for a steel– were 0.5,c0.1, oncret 0.5, e composite pedestrian bridge. The values of the parameters used to apply this method were 0.5, 0.1, 2, 0.25, and 4 for , , , n , s, and l , respectively. The maximum number of iterations was ﬁxed at t 0 4000. 0.5, 2,The 0.25 values , and 4 for of n and 𝜌 , γ r, β wer , nt,e s, taken and las 0, re dispectively erent values; . The m theavalues ximumadopted number o infthe iter study ations w areashown s fixed at 4000. The values of n and r0 were taken as different values; the values adopted in the study are in Table 4. shown in Table 4. Table 4. n and r adopted values. Table 4. n and r0 adopted values. n 10 20 30 40 50 60 80 100 n 10 20 30 40 50 60 80 100 r 50 100 150 r0 50 100 150 In this work, the GSO algorithm was applied in two experiments. The ﬁrst one only used the GSO In this work, the GSO algorithm was applied in two experiments. The first one only used the to reach the optimum solution, but in the second, the DLS algorithm was applied to the best solutions GSO to reach the optimum solution, but in the second, the DLS algorithm was applied to the best of the GSO to improve those solutions. Figures 6 and 7 show the results of the GSO and GSO with solutions of the GSO to improve those solutions. Figures 6 and 7 show the results of the GSO and DLS, respectively. GSO with DLS, respectively. Figure 6. Average cost to average iterations for the glow-worms swarm optimization (GSO) experiment. Figure 6. Average cost to average iterations for the glow-worms swarm optimization (GSO) experiment. 𝑥 Appl. Sci. 2019, 9, 3253 11 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 18 Figure 7. Average cost to average iterations for GSO and DLS combination experiment. Figure 7. Average cost to average iterations for GSO and DLS combination experiment. 4. Discussion 4. Discussion 4.1. Comparison of the Heuristic Techniques 4.1. Comparison of the Heuristic Techniques To compare the results of the heuristic techniques, we focused on the cost obtained by each one. To compare the results of the heuristic techniques, we focused on the cost obtained by each one. In Table 5, the results of material amounts and cost for each solution are shown. The heuristic that In Table 5, the results of material amounts and cost for each solution are shown. The heuristic that obtained the lowest cost is GSO combined with DLS. This result is related to the low amount of rolled obtained the lowest cost is GSO combined with DLS. This result is related to the low amount of rolled steel achieved, due to the importance of this material in steel-concrete composite pedestrian bridges. steel achieved, due to the importance of this material in steel–concrete composite pedestrian bridges. As is seen in Table 5, the heuristics with lower values of cost (SA and GSO with DLS) have a lower As is seen in Table 5, the heuristics with lower values of cost (SA and GSO with DLS) have a lower amount of rolled steel. amount of rolled steel. The concrete strength was the same for DSL, SA, and GSO, but when GSO and DSL were combined, The concrete strength was the same for DSL, SA, and GSO, but when GSO and DSL were the geometry variables of the slab decreased due to the increase in the concrete strength, leveraging combined, the geometry variables of the slab decreased due to the increase in the concrete strength, the material. The optimum solution for a steel beam consists of locating the area of steel in a way leveraging the material. The optimum solution for a steel beam consists of locating the area of steel that allows the mobilization of the highest possible mechanical arm. In order to reach the lowest cost, in a way that allows the mobilization of the highest possible mechanical arm. In order to reach the the solutions obtained by the optimization algorithms looked for greater depths with lower amounts lowest cost, the solutions obtained by the optimization algorithms looked for greater depths with of material, increasing the inertia and reducing the structure weight. lower amounts of material, increasing the inertia and reducing the structure weight. Table 5. Cost and material amount for the best heuristic solutions. Table 5. Cost and material amount for the best heuristic solutions. DLS SA GSO GSO and DLS DLS SA GSO GSO and DLS Rolled steel kg/m 153.99 151.26 157.49 150.66 Rolled steel kg/m 153.99 151.26 157.49 150.66 % Rolled/record % 2.21% 0.40% 4.54% 0.00% % Rolled/record % 2.21% 0.40% 4.54% 0.00% Shear-connector steel kg/m 0.90 0.90 0.79 0.90 % Shear-connector/record % 14.28% 14.28% 0.00% 14.28% Shear-connector steel kg/m 0.90 0.90 0.79 0.90 3 2 Concrete 0.15 0.15 0.14 0.14 m /m % Shear-connector/record % 14.28% 14.28% 0.00% 14.28% % Concrete/record % 7.52% 7.52% 0.83% 0.00% 3 2 Concrete m /m 0.15 0.15 0.14 0.14 Reinforcement steel kg/m 22.57 22.22 25.50 22.23 % Concrete/record % 7.52% 7.52% 0.83% 0.00% % Reinforcement/record % 1.58% 0.00% 14.76% 0.04% Reinforcement steel kg/m 22.57 22.22 25.50 22.23 Cost ¿/m 304.82 299.77 313.18 297.76 % Cost/record % 2.37% 0.67% 5.18% 0.00% % Reinforcement/record % 1.58% 0.00% 14.76% 0.04% Cost The rows with percentages express the €/ incr m ease 30 in the4. quantity 82 of29 material9.77 with31 respect3.18 to the minimum.297.76 % Cost/record % 2.37% 0.67% 5.18% 0.00% The rows with percentages express the increase in the quantity of material with respect to the minimum. Appl. Sci. 2019, 9, 3253 12 of 18 4.2. Sustainability Study An analysis of the CO 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 emissions of every solution are compared. Table 6. CO emissions and cost data comparison from the reference solution and the heuristics. Reference DLS SA GSO GSO and DLS Cost ¿/m 399.10 304.82 299.77 313.18 297.76 % Cost/reference % - 23.62% 24.89% 21.53% 25.39% CO emissions kg CO /m 700.22 536.39 527.70 552.54 523.68 % Emissions/reference % - 23.40% 24.64% 21.09% 25.21% As is seen in this table, the GSO and DLS combination heuristic obtained a reduction of 8.12% of the CO emissions compared with the reference. This means that an improvement of 1 ¿/m produced a reduction of the 1.74 kg CO /m . 4.3. Parametric Study A parametric study for varying span lengths is presented with the GSO and DLS combination optimization model. Five span lengths were considered: 28, 32, 38, 42 and 48 m. The characteristics that were studied are the economy, the geometry, and the amount of materials. Tables 7 and 8 compile the values of the features of the optimization solutions: Table 7 gives the main values of the geometry of the structure, and Table 8 gives the values of the measurements of the amount of materials of the structure. Table 7. GSO with DLS combination for 28, 32, 38, 42, and 48 m spans. Span CL CA CT EAS EAL EAI FCK Total (m) (m) (m) (m) (mm) (mm) (mm) (MPa) 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 Table 8. GSO with DLS combination measurements of the materials for 28, 32, 38, 42, and 48 m spans. 2 3 2 2 Span (m) Beam Rolled Steel (kg/m ) Slab Concrete (m /m ) Slab Reinforcement (kg/m ) 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 The results of the parametric study led to practical rules for the preliminary design of cost-optimized steel-concrete composite pedestrian bridges with box-girder cross sections isostatic spans. The discussion of the results was carried out together, with a regression analysis. The functions obtained were valid approximations within the range of the studied parameters. The extrapolation of these results to other span lengths should be carried out carefully. Figure 8 shows the average results of the cost of the structure per square meter of the steel-concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the horizontal span leads to a very good quadratic correlation. The average footbridge cost adjusted to C = 0.1954L 8.0873L + 325.96 with a regression coecient of R = 0.9994. The cost rising is produced by the need Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 20 Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 18 323 Figure 8 shows the average results of the cost of the structure per square meter of the steel- Figure 8 shows the average results of the cost of the structure per square meter of the steel– 324 concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the concrete composite pedestrian bridge for distinct span lengths. The cost evolution as a function of the Appl. Sci. 2019, 9, 3253 13 of 18 325 horizontal span leads to a very good quadratic correlation. (The average footbridge cost adjust to C = horizontal span leads to a very good quadratic correlation. The average footbridge cost adjusted to C 2 2 326 0.1954L – 8.0873L + 325.96 with a regression coefficient of R = 0.9994.) The cost rising is produced by 2 2 = 0.1954L − 8.0873L + 325.96 with a regression coefficient of R = 0.9994. The cost rising is produced 2 2 327 for the needs of larger amounts larger ofamoun materials ts of mater to satisfy ials to sati the deﬂection sfy the deflection requirements. requirement Note thats. Note th the R regr at the R ession by the need for larger amounts of materials to satisfy the deflection requirements. Note that the R 328 regression coefficient in Figure 8 is almost 1, this indicates a very good correlation. The variations coecient in Figure 8 is almost 1, this indicates a very good correlation. The variations between the regression coefficient in Figure 8 is almost 1, this indicates a very good correlation. The variations 329 minimum between the and minimum the mean an cost d the me of thean pedestrian cost of the pe bridge destrian br produced idby ge prod the GSO uced and by the G DLS combination SO and DLS between the minimum and the mean cost of the pedestrian bridge produced by the GSO and DLS 330 combination are 0.42%. are 0.42%. combination are 0.42%. 333 Figure 8. Average cost for different span lengths. Figure 8. Average cost for different span lengths. Figure 8. Average cost for dierent span lengths. 334 Figure 9 shows the mean values of the depth of the steel beam (CA) for different span lengths. Figure 9 shows the mean values of the depth of the steel beam (CA) for different span lengths. Figure 9 shows the mean values of the depth of the steel beam (CA) for dierent span lengths. 335 The depth of the beam has a good linear variation according to the span length of the bridge. (The The depth of the beam has a good linear variation, according to the span length of the bridge. The The depth of the beam has a good linear variation, according to the span length of the bridge. 336 average depth of the beam adjusts to CA = 0.0369L – 0.2179 with R = 0.9927) Again, the good 2 2 average depth of the beam adjusts to CA = 0.0369L − 0.2179 with R = 0.9927. Again, the good The average depth of the beam adjusts to CA = 0.0369L 0.2179 with R = 0.9927. Again, the good 337 correlation factor represents a functional relation. correlation factor represents a functional relationship. correlation factor represents a functional relationship. 340 Figure 9. Mean steel beam depth for different span lengths. Figure 9. Mean steel beam depth for dierent span lengths. Figure 9. Mean steel beam depth for different span lengths. 341 As shown in Figure 10, a function is found relating the thickness of the slab with different span 342 lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of the slab As shown in Figure 10, a function was found relating the thickness of the slab with dierent span As shown in Figure 10, a function was found relating the thickness of the slab with different 343 is determined by the transverse flexion. Once the slab stresses are determined by the longitudinal lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of the slab span lengths. Up to a certain span length, the slope of the parabola is smaller because the inertia of 344 deflection, the slope of the curve increases. (The average slab thickness adjusts to CL= 0.0001L – is determined by the transverse ﬂexion. Once the slab stresses are determined by the longitudinal the slab is determined by the transverse flexion. Once the slab stresses are determined by the 345 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m). Related to the compressive deﬂection, the slope of the curve increases. The average slab thickness adjusts to CL = 0.0001L longitudinal deflection, the slope of the curve increases. The average slab thickness adjusts to CL = 346 strength of the concrete of the slab, in Figure 11 the relation to the concrete compressive characteristic 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m. Related to the compressive 2 2 0.0001L − 0.0067L + 0.2568 with R = 0.9849 when the span length is larger than 38 m. Related to the 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. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 18 Appl. Sci. 2019, 9, 3253 14 of 18 compressive strength of the concrete of the slab, Figure 11 shows the relationship of the concrete Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 20 compressive characteristic strength and the span length. This relationship adjusts well to a quadratic 2 2 The concrete compressive strength adjusts to FCK = 0.0292L 1.6959L 2 + 49.714 with R = 0.9987. 2 function. The concrete compressive strength adjusts to FCK = 0.0292L − 1.6959L + 49.714 with R = 347 strength and the span length is shown. This relation has a good adjust to a quadratic function (The Note that the highest concrete compressive strength considered for this study was 35 MPa. 0.9987. Note that the highest concrete compressive strength considered for this study was 35 MPa. 2 2 348 concrete compressive strength adjusts to FCK = 0.0292L – 1.6959L + 49.714 with R = 0.9987). Note 349 that the highest concrete compressive strength considered for this study is 35 MPa. Figure 10. Mean slab thickness for dierent span lengths. Figure 10. Mean slab thickness for different span lengths. 351 Figure 10. Mean slab thickness for different span lengths. 353 Figure 11. Average compressive strength for different span lengths. Figure 11. Average compressive strength for dierent span lengths. Figure 11. Average compressive strength for different span lengths. 354 Regarding ratio of the amount of rolled steel (Rs) and the surface of the slab (Ss); Figure 12 Regarding the ratio of the amount of rolled steel (R ) and the surface of the slab (S ) Figure 12 s s 355 illustrates the increasing of the amount of rolled steel needed to resist the flexural requirements. The Regarding the ratio of the amount of rolled steel (Rs) and the surface of the slab (Ss) Figure 12 illustrates the increase in the amount of rolled steel needed to resist the ﬂexural requirements. The slope 356 slope of the curve tens to increase as the span length increases (The mean amount of rolled steel in illustrates the increase in the amount of rolled steel needed to resist the flexural requirements. The of the curve tends to increase as the span length increases. The mean amount of rolled steel in relation 2 2 357 relation to the surface of slab adjusts to Rs/Ss = 0.01913L – 3.5553L + 154.96 with R = 0.9996). Although, slope of the curve tends to increase as the span length increases. The mean amount of rolled steel in 2 2 to the surface of slab adjusts to R /S = 0.01913L 3.5553L + 154.96 with R = 0.9996. However, s s 2 2 358 the ratio of volume concrete (Vc) and the surface slab fits a second order equation that increases with relation to the surface of slab adjusts to Rs/Ss = 0.01913L − 3.5553L + 154.96 with R = 0.9996. However, the ratio of the volume of concrete (V ) and the surface slab ﬁts a second order equation that increases 359 the span length in the same way as rolled steel amount as it seen in Figure 13 (The mean ratio of the the ratio of the volume of concrete (Vc) 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 360 volume of concrete in relation to the surface of slab adjusts to Vc /Ss = 0.0001L – 0.0067L + 0.245). with the span length in the same way as rolled steel amount, as seen in Figure 13. The mean ratio of of the volume of concrete in relation to the surface of slab adjusts to V /S = 0.0001L 0.0067L + c s 361 Moreover, the ratio of reinforcing steel (RFs) measured per square meter of slab shows the same the volume of concrete in relation to the surface of slab adjusts to Vc/Ss = 0.0001L − 0.0067L + 0.245. 0.245. Moreover, the ratio of reinforcing steel (RF ) measured per square meter of slab shows the same 362 tendency as rolled steel amount and concrete volume (The ratio of reinforced steel in relation to the Moreover, the ratio of reinforcing steel (RFs) 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 2 2 363 surface of slab adjusts to RFs / Ss = 0.0246L – 1.2009L + 32.516 with R = 0.9955), it shows in figure 14 tendency as rolled steel amount and concrete volume. The ratio of reinforced steel in relation to the 2 2 surface of slab adjusts to RF /S = 0.0246L 1.2009L + 32.516 with R = 0.9955, as shown in Figure 14. s s 2 2 surface of slab adjusts to RFs/Ss = 0.0246L − 1.2009L + 32.516 with R = 0.9955, as shown in Figure 14. Appl. Sci. 2019, 9, 3253 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 18 Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with the span. Figure 12. Variation in the ratio of the rolled steel amount in relation to the surface of the slab with the span. the span. 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 13. Variation in the ratio of the volume of concrete 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 Figure 13. Variation in the ratio of the volume of concrete in relation to the surface of the slab with the span. the span. Figure 14. Variation in the ratio of the reinforcement steel 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. Figure 14. Variation in the ratio of the reinforcement steel in relation to the surface of the slab with Figure 14. Variation in the ratio of the reinforcement steel in relation to the surface of the slab with the span. the span. Appl. Sci. 2019, 9, 3253 16 of 18 5. Conclusions Three heuristic algorithms were applied to a steel-concrete composite pedestrian bridge to ﬁnd an ecient design. DLS, SA, and GSO are used to automatically ﬁnd optimum solutions. All procedures are useful in the automated design of steel-concrete composite pedestrian bridges. Furthermore, a parametric study was carried out. The conclusions are as follows: 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 emission comparison showed that the reduction between the original structure and the GSO with DLS combination optimized structure reduced the CO emission by 21.21%. Thus, 2 2 an improvement of 1 ¿/m produced a reduction of 1.74 kg CO /m . Therefore, the solutions that are acceptable in terms of CO 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 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. 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Heuristic Techniques for the Design of Steel-Concrete Composite Pedestrian Bridges

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

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

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

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