Optimization of steel beams with external pretension, considering the environmental and financial impact

With the advancement technology for reinforced concrete structures, it becomes increasingly feasible to use this technology for steel structures. The objective of this work is to present the formulation of the optimization problem of steel beams with external pretension with straight or polygonal tracing cables, considering the environmental and economic impacts. For the objective function formulation, the minimization of CO2 emission and cost in the design of the structure were considered. As constraints were established the states limits imposed by ABNT NBR 8800:2008. The program was developed within the MATLAB Platform (MATLAB®. Guia do usuário R2016a (2016) The Math Works Inc) and the optimization problem solution was obtained through the native genetic algorithms method. Routine validation was performed using examples found in the literature and an analysis of the predominant collapse modes was performed. The results indicate that monosymmetric profiles have gains when it comes to reducing CO2 emissions and cost when compared to doubly symmetrical profiles, in addition it was observed that straight cables generate better values of CO2 emission and cost when compared to polygonal cables.


Introduction
One of the great goals of engineering is to develop projects with maximum safety and minimum costs.However, given the current scenario, the environmental impact has become an important factor in the structures design, so that studies have sought more sustainable systems for civil construction.
Thus, a tool used to obtain optimal solutions of financial cost and environmental impact in the structures design is optimization through algorithms that use meta-heuristics.
These algorithms have proven relevant in the minimization of costs and environmental impacts.
Among the existing metaheuristics, the genetic algorithm proposed by Holland (1962) stands out for being the best known.Several works using GA have been published in the last decades in general to minimize the material and cost of the structure, such as the works proposed by Yassami and Ashtari (2015), Qin et al. (2020), Ramos and Alves (2021), Hashmi et al. (2023) and Akbari and Ayubirad (2017).
However, as stated by Santoro and Kripka (2020), Tormen et al. (2020), Payá-Saforteza et al., (2009), Camp and Huq (2013), Park et al. (2014), Yepes et al. (2015) and Yu et al. (2020), optimizations focused only on financial cost may not be enough to determine an optimal solution to the problem.Studies for the life-cycle of materials and their impact on the environment become an important tool in the search of the optimum design.Oliveira et al. (2014) show in their study that most CO 2 emission of civil construction if from concrete industry and the cement production is main responsible for that.Azam et al. (2016) analyze the impact of CO 2 emissions on economic growth taking as variables, the energy use, trade, and human capital.The authors conclude that the ultimate impact of shrinking pollution will help in supporting sustainable economic growth and maturation as well as largely improve society's welfare.Kaveh and Ardalani (2016) presented the analysis the cost and CO 2 emissions of reinforced concrete frames with non-prismatic members.Authors concluded that in CO 2 based design, the amount of CO 2 is reduced by 3.8% in RC frames with a 6% increase in cost, and in the RC industrial frames with 1.3% increase in cost, the amount of CO 2 can be reduced by 7%.Kaveh et al. (2020) proposed the study to reduce CO 2 emissions of RC frames with four, eight and twelve stories, are modeled and optimized via enhanced colliding bodies optimization (ECBO), enhanced vibrating particles system (EVPS) and particle swarm optimization (PSO).The comparison of the results showed the superiority of the ECBO in the final solution.Kaveh et al. (2022) analyzed the optimization of columns and bent caps of RC bridges for cost and CO 2 emission.The optimization is performed to minimize the cost and CO 2 emissions using the enhanced colliding bodies optimization (ECBO) algorithm.The trade-off between cost and CO 2 emissions shows that in the design for minimizing CO 2 emissions compared to the design based on the cost minimization, increasing 1.4% in cost can decrease CO 2 emissions by 6.1%.Durgam et al. (2022) present in their study a methodology for reducing greenhouse gas emissions based on a systematic analysis during the three phases of construction, preconstruction, construction, and post-construction, to define practical measures to reduce greenhouse gases from civil construction.
Santoro and Kripka (2020) point out the advantages of using high-strength concrete in reducing CO 2 emissions for reinforced concrete columns and at the same time point out that for the beams these concretes do not have much influence.Tormen et al. (2020) present a study to minimize CO 2 emissions for composite steel and concrete beams.The authors analyze the influence of the characteristic strength to concrete compression as well as the influence of the degree of interaction between the slab and the steel beam in the reduction of CO 2 .Arpini et al. (2022) present the formulation of the optimization problem for composite floor systems to reduce CO 2 emissions using genetic algorithm.The authors point out in the study that the best solution from an economic point of view is not always the best solution from an environmental point of view.
As in recent years, environmental problems have proved to be an alarming factor in society, so the optimization of structural systems considering the reduction of CO 2 and not only the costs involved in construction is very relevant.
Steel beams with external pretension are structural elements that have a high degree of strength, supporting high loads and not using concrete, tend to emit less CO 2 in their manufacture.The steel cables used in the external pretension can be positioned outside the section or inside the beam.The cables used are the same used in the structures of prestressed concrete, with some differential accessories to anchor and divert the cables.
According to Lou and Karavasilis (2019) external pretension is more effective in reducing tension at the bottom base of the steel beam in the middle of the span than in the center of the support.And according to Lou et al. (2016) the external pretension in composite beams significantly improves the behavior of the structure, as it leads to a substantial increase in ultimate loads (moments) and an obvious decrease in deformation.However, when analyzing the time-dependent responses, the characteristics of the prestressed beams seem to be almost identical to those of the beams without external prestressing, that is, there is no improvement in behavior in relation to the creep and shrinkage effects of the concrete, both beams, respond similarly to these effects.
Recent studies such as the work by Zhou et al. (2017), Zhou et al. (2020), Luo (2022) point out the importance of analyzing prestressed steel structures subjected to high temperatures and environment temperature.Cucuzza et al. (2021) presents in their study the optimal solution of prestressed lattice arch sections.The authors present the study for two arch-shaped steel trusses placed alongside the lateral faces of the beam to be consolidated.The arches develop longitudinally along the entire span of the beam and in elevation using the available height of the prestressed reinforced concrete cross-section and point out the gains when the solution was compared with traditional solutions.Abbas et al. (2018) developed a study in which the optimization of steel beams with pretension and without pretension was compared.Through the results, it can be observed that prestressed steel beams require a lower cross section than steel beams without pretension.
Aydin (2022) did a study aimed at optimizing the costs of prestressed steel trusses using cables positioned below the lower flange and molded with deviators.The optimization variables defined were the sections of the elements and layout of the truss, cable profile, dimension of the deviators.The optimization algorithm used was Jaya.Through the results it was observed that the pretension provided some savings in the costs of the steel truss project.
In the case of works on composite beams of steel and concrete with external pretension, it is worth mentioning the works of Nie et al. (2007), Chen et al. (2009), El-Sisi et al. (2021), Hassanin et al. (2021) and Turini and Calenzani (2022), who made comparative studies between composite beams with pretension and without pretension.The results show that the pretension improves the overall performance of composite beams.
This study aims to propose the optimization problem formulation of CO 2 emission and the cost of steel beams with external pretension using the straight and polygonal tracing of the tendons for double symmetrical and monosymmetric profiles.The optimization problem solution was obtained by Genetic Algorithm native in platform MATLAB.

Optimization problem formulation
The optimization aims to reduce the emission of CO 2 or the financial cost using a mono objective function for each problem of a simply supported steel beam using external pretension, with requests for uniform and concentrated loads.The steel profile is type I and can be monosymmetric or doubly symmetrical.The ABNT NBR 8800:2008, which deals with the design of steel structures, was used to model the problem and establish the beam design criteria.
The metaheuristic used is Genetic Algorithm based on the theory of evolution proposed by Charles Darwin.This algorithm performs well on highly nonlinear problems and a good sensitivity to excluding local minimums from optimal solutions.MATLAB (2016), which natively has the genetic algorithm in its library, was the platform used to develop the optimization problem.

Design variables
In the genetic algorithm the variables behave as a population formed by individuals who modify through mutations and recombinations to find an optimized solution.Figures 1 and  2 show the design variables representation for monosymmetric and doubly symmetrical profiles.
Equations ( 1) and ( 2) represent, respectively the maximum and minimum limits of the variables of steel beams with monosymmetric profiles.x(6) and x( 7) equivalent to the thickness of the upper and lower flanges in cm.The variable x(3) represents the height of the steel profile in cm.In addition to these variables related to the steel profile, the characteristics of the tendons are represented by x(4) and x( 5) that show the inclination length and the num- ber of tendons, respectively.
For doubly symmetrical profiles, Eqs. ( 3) and ( 4) define, respectively the lower bound and the upper bound of the design variables.The optimization for double symmetric profiles has a reduction in the number of variables, in relation to the monosymmetric solution, because there is no longer the possibility of differentiating the width of the profile flanges.Like this x(1) represents the width of the flanges in cm x(2) the height of the profile in cm, x(3) the number of tendons and x(4) the inclina- tion length of the tendons.
In both types of profiles all variables are continuous, except for the slope length that was defined as a vector of 5 positions ranging from 10 to 30% of the beam span, with a step of 5%.

Objective function
The objective function is the main function of the optimization problem, in this study two different objective functions were proposed, one formed by the sum of CO 2 emissions generated in the manufacture of each element and another consisting of the sum of the cost of each optimized item, that is, the steel profile and the amount of tendons.The objective functions for the optimization of cost and environmental impact, measured in Reais and kg of CO 2 emission, respectively, are presented in Eqs. ( 5) and ( 6).
Being V a the steel volume of the beam profile (m 3 ), m a the specific mass of steel (kg/m 3 ) equal to 7850, C a the cost of steel (R$/kg), E a CO 2 emission from the steel profile (kgCO 2 /kg), N s the total amount of tendons, A s the area of the tendons section (m 2 ), L s the length of the tendons (m), C s the cost of tendons (R$/kg) and E s CO 2 emission from tendons (kgCO 2 /kg).

Constraints
The constraints of the problem were defined according to ABNT NBR 8800 (2008), considering the ultimate and serviceability states limit for the design of a simply supported steel beam.The constraints of the problem are presented in Eq. ( 7). ( 7)  Being M Sd(t=∞) the bending moment requesting calcula- tion (kNm); M Sd(t=0) the bending moment requesting cal- culation in the act of prestressing (kNm); M Rd the resistant bending moment of calculation (kNm); V Sd the shear force requesting calculation (kN); V Rd the resistant shear force of calculation (kN); N Sd the normal load for calculation due to pretension (kN); N Rd the normal resistance for calculation (kN); d the height of the profile (mm); b f the width of the profile flange (mm); h the height of web; t w thickness of web; t(t=∞) the total deflection (cm); adm the permissible deflection (cm); t(t=0) the deflection in the act of prestressing (cm); cd the maximum compressive stress (kN/m 2 ); td the maximum tensile stress (kN/m 2 ); f yd the design strength of steel (MPa).
Being the constraints C(1) and C(2) verification during the use phase of the structure and at the act of stress, respectively.The constraints C(3) checks the beam for shear effort and C(4) checks the beam at normal effort.C( 5) , C(6) , C(7) and C( 8) geometric constraints of the profiles used.The equations C(9) and C(10) restrict maximum deformation on the beam during use and at the time of pretension.Constraints C(11) and C( 12) are related to the verification of the beam to combined bending during the phase of use and the act of pretension, respectively.Finally, the constraints C(13) e C( 14) are related to the limitation of maximum compres- sion and tensile stresses, respectively.
For the constraints analysis, a simple penalty method is used.If the individual is feasible, the penalty function is the fitness function.If the individual is infeasible, the penalty function is the maximum fitness function among feasible members of the population, plus a sum of the constraint violations of the infeasible individual.The values of the financial cost and CO 2 emission used in the examples are shown in Table 1.
To genetic algorithm was used an initial population with 100 individuals.To elitism and crossing taxes were used 0.05 and 0.85, respectively, whereas the mutation rate is random to accelerate to convergence process.The general flowchart of the algorithm is represented in Fig. 3.

Results and discussions
Two examples of beams with external pretension were analyzed, using tendons with straight or polygonal tracing, in which we sought to optimize the CO 2 emission and cost, verifying one which one would have the best result.Figure 4 shows the straight and polygonal tracing of the tendons, with "L" being the length of the beam and "x" the inclined length.The inclined length varies according to the beam span, which can be 10%, 15%, 20%, 25% and 30% of the span.The positioning of the cable, is the  distance from the cable to the top face of the bottom table, being positive above and negative below.

Example 01 Abbas et al. (2018)
The first example analyzed was extracted from Abbas et al. (2018) and these are prestressed beams with monosymmetric steel profile.Abbas et al. (2018) performed a first analysis using Ansys, in which the exact values of the properties of the materials were entered as input data.Subsequently, an Ansys optimization algorithm was used to find the optimized solution of steel beams with external pretension.The authors analyzed two objective functions, which are the minimization of tension and the minimization of the total volume of steel of the beam.The constraints applied were referring to the maximum stresses in the steel profile and the tendons, maximum shear stress in steel and the maximum deflection in the middle of the beam span.
The beam and loads used in example 01 can be seen in Fig. 5.For this example, as well as Abbas et al. (2018), we considered steel profile ASTM A36, tendons CP 190 diameter of 9.5 mm.Three possible values were adopted for the losses of the pretension, 0%, 10% or 20%.Although the loss value of 0% does not exist in practice, the analysis was made only to have comparative parameters.Table 2 shows the input data from the optimization algorithm.
The models analyzed were named according to the symbology of Fig. 6, and the first letter referring to the profile type, monosymmetric (M) or double symmetrical (D), the second letter represents the tracing, straight (R) or polygonal (P), and the symbols after the hyphen indicate the percentage of losses of 0% (P0), 10% (P10) or 20% (P20).
In Table 3, it can be noted the optimal values presented by Abbas et al. (2018) and the values when the CO 2 emission is optimized for the monosymmetric and doubly symmetrical profiles.It is noteworthy that in the study used as reference was not specified number of tendons but the area necessary for external pretension, being this value of 130.3 mm 2 of tendon steel area.
Table 3 shows that cable consumption increased by approximately 344% since the steel profile decreased by approximately 30%.
For all models analyzed, Table 4 shows the CO 2 emission values of the steel and tendons profile, as well as the total CO 2 emission of the prestressed beam and the ratio between the total CO 2 emission of each model in relation to that of Abbas et al. (2018).It is observed that the most interesting solution from the point of view of CO 2 emission is the beam prestressed with monosymmetric profile, regardless of the type of stroke.It should also be noted that the loss of stress has little influence on the total value of CO 2 emission.As can still be observed in Table 4, the best result found had a reduction of 29% in relation to the literature, the worst case had a reduction of 15%.
Table 5 shows the main values obtained for the design and cost analysis variables and Table 6 shows the cost values of the steel and tendons profile, as well as the total cost of the prestressed beam and the ratio between the total cost of each model compared to that of Abbas et al. (2018).The most interesting solution from an economic point of view is also the prestressed beam with monosymmetric profile.
Comparing the best result found with the solution of Abbas et al. (2018) in Tables 5, it can be observed that the cable area was approximately 344% larger and the area of the steel found had a cost reduction of approximately 27%, the worst case of approximately 15% According to Tables 4 and 6, the steel profile is the main responsible for the final composition of both cost and CO 2 emissions.This underscores the importance of using the prestressing cables to minimize the sections of the steel profile.
It is observed that the straight tracing cables led to a lower cost, but the type of tracing did not impact the CO 2 emissions.It is emphasized that in the analyses, the cost and CO 2 emission of the deviators necessary for the tracing of polygonal cables were not accounted for.
Figure 7 shows the normalized values of CO 2 emission and cost in relation to those of Abbas et al. (2018).The best result was obtained when the CO 2 emission was optimized, because although the amount of cables is the same as that obtained in cost optimization, the area of the steel profile was smaller.In addition, it had a reduction of around 27% and 29% in CO 2 emissions and in the cost of the ideal model found.The worst result found was the beam prestressed with polygonal cable, with 20% losses and double symmetrical profile.
According to Fig. 8, the optimization of CO 2 emissions showed the best relationships except for the DR-P20 and DP-P20 models.That is, there is no standard for obtaining the best solution, and it will always be necessary to analyze the needs to be met and compare the results obtained.
Figure 8 presents an analysis of the predominant collapse modes according to the constraints that govern the problem.For each model and each type of design constraint, the utilization index (ratio between request and resistance) was plotted.As it can be observed, the three modes of collapse that most influenced the analysis were the bending moment, the deflection and the combined bending, since the ones that least influenced the modes of collapse were the bending moment in the act of pretension, the deflection in the act of pretension and the shear effort.

Example 02 Mageveske et al. (2021)
For example, 02, was used as reference the study presented by Mageveske et al. (2021), in which the cost of double symmetrical I profile with external pretension, prestressing losses of 20%, tendons diameter of 15.2 mm and straight tracing was optimized and genetic algorithm was used to find the optimization problem solution.The section profile was optimized, except for the thickness of the upper and lower table and the web, which remained constant.This example was originally taken from Rezende (2007), but a comparative analysis will be made with the best result of Mageveske et al. (2021) and the values found in this study.
The uniform load used is shown in the Fig. 9. ASTM A572-55 steel was considered for steel profiles and CP210 tendons with 15.2 mm diameter.Table 7 shows the input data for the study.Only monosymmetric sections will be analyzed, with the variation of losses by 0%, 5%, 10%  and 20%.The web thickness will be the same in all analyses, equal to 8 mm, a value found in the optimization of Mageveske et al. (2021).
The beams analyzed are designated according to Fig. 10, the first letter being the type of stroke, straight (R) or polygonal (P), the second installment the value of the loss of pretension, being 0% (P0), 5% (P5), 10% (P10) or 20% (P20) and the third letter the objective function of optimization, the cost (C) or the emission of CO 2 (E).
Table 8 shows the main values obtained for the project variables as well as the total CO 2 emission and total cost.The model that generated the best result was the monosymmetric profile with straight cables and 0% losses, optimizing the cost.Comparing these results with those obtained in Mageveske et al. (2021) it can be observed that the number of cables was lower, generating 2 fewer cables.The values found for the best solutions were approximately 31% lower for Cost and 54% lower for CO 2 emissions.
Figure 11 presents the CO 2 emission and the total cost plotted with normalized values in relation to the solution of Mageveske et al. (2021).With this analysis it becomes possible to conclude that the best result found was when the cost was optimized.However, when the two optimizations are compared, there is greater efficiency in optimizing CO 2 emissions.
For each model and each type of sizing constraint, the utilization index (ratio between request and resistance) is plotted in Fig. 12.It can be noted that the modes of collapse that governed the analyzed example were combined bending with all loads, the bending moment and the deflection.On the other hand, those who least interfered in the modes of collapse were the deflection in the act of pretension, the shear effort, the deflector moment in the act of pretension.

Conclusions
This study aimed to propose the formulation of cost optimization and CO 2 emission of steel beams, using monosymmetric or double symmetric profiles, with external pretension, using cables with straight or polygonal tracing.For this, two examples were analyzed, and in the first one the loss of pretension was varied, and both monosymmetric and double symmetric profiles were analyzed.In the second example, losses were also varied, but only monosymmetric profiles were analyzed, considering that it was the profile that presented the best solution in Example 01.
For all optimized models, better solutions were obtained than the examples in the literature.With the first example, it is concluded that monosymmetric profiles generate more economical results with lower CO 2 emissions than doubly symmetrical profiles.In addition, there was a reduction of    In the second example, the values found were approximately 31% lower for the cost and approximately 54% lower for CO 2 emissions when the CO 2 emission was optimized.
For all models analyzed, because they are steel beams, the option of inclined cable did not influence the final solution of the problem, being the best solutions obtained with the use of straight cables.
Regarding the structural behavior, it was noted that the modes of rupture that governed the examples were combined bending, the bending moment and the deflection.On the other hand, the modes of rupture that least influenced were the deflection in the act of pretension, the shear effort and the deflection in the act of pretension.
Acknowledgements The authors acknowledge the Brazilian Federal Government Agency CAPES for the financial support provided during

Fig. 8 Fig. 9
Fig. 8 Constraints analysis to CO 2 emissions optimization of Example 01 CO 2 emissions and 29% in cost in the best solutions found.

Fig. 11
Fig. 11 Graph of the normalized solutions of Example 02

Table 1
Cost and CO 2 emission values for each component of the structure a SINAPI (2022); b World Steel Association (2022)

Table 2
Input data from Example 01

Table 5
Cost optimization results-example 01

Table 6
Cost values of Example

Table 7
Input data from Example 02 Fig. 10 Designation of the models analyzed in Example 02