OPTIMIZATION OF CROSS-SECTIONAL AREAS OF REINFORCED CONCRETE COLUMNS SUBJECTED TO AXIAL FORCE AND BIAXIAL BENDING VIA GENETIC ALGORITHMS OTIMIZAÇÃO DE SEÇÕES TRANSVERSAIS DE PILARES DE CONCRETO ARMADO SUJEITAS A FORÇA AXIAL E MOMENTOS BIAXIAIS VIA

Abstract: The sizing of reinforced concrete structures is influenced by high magnitude forces and, as a result of its calculations, some designs may present specifications that are not optimum. In this case, it can occur exaggerated dimensions and an oversized structure resulting in financial losses and material wastes. Thus, it can be applied to the sizing, optimization techniques to achieve the best solution regarding, as an example, efficiency and material costs. This paper presents the optimization of cross-sectional areas of reinforced concrete columns using a Genetic Algorithm (GA) and considering the structure subjected to an axially compressive force and biaxial bending. It was developed an algorithm using the formulation of Araújo (2014) for the sizing associated with Deb’s Genetic Algorithm (2001). The developed software present solutions as cross-sectional areas of a column regarding the minimization of its costs and in which its reinforcement steel positions and diameters are optimized. Its dimensions and concrete resistance may also be optimized as a choice of the designer/engineer. The algorithm was applied to an example and had its solutions compared with other authors. Its results had achieved feasible solutions and shown similar costs.


INTRODUCTION
Reinforced concrete columns are important elements of structures that have the purpose of resisting loads of beams, slabs, and other components and forces.
As a result of the high magnitudes forces, they are frequently oversized regarding the cross-sectional areas dimensions and reinforcement steel diameters and positions. Therefore, this amount of excessive materials causes an increase in construction costs, financial losses, more significant wastes on-site production and environmental degradation.
This procedure may be classified as a conventional design which the core solution is the result of the sizing of reinforced concrete column. Its starts with an initial design predicted by the designer/engineer and it is checked for the requirements. If the result is suitable, the process can be finished. However, if it is interesting to achieve a better solution or the criterion still have not been reached, more iterations can be done. In this case, the next solution will be improved, relying on the designer past experiences and projects (Arora, 2012).
However, it is mainly distinguished from the improvement of the design through optimization theories. These provides the best solution for the specific problem regarding performance objectives and constraints. It is a non-subjective path to achieve a feasible and efficient solution. In this sense, the optimization methods can be applied to structural problems in several ways, such as the minimization of costs, weight, compliance, etc. In general, the main purpose is to be optimized and may involve the optimization of dimensions, cross-sectional areas, topology, or shape (Arora, 2012).
In this paper, the cost of the reinforced concrete columns is the objective to be optimized by using a Genetic Algorithm (GA). The optimization problem in the context of this paper is based on the mathematical formulation presented by Araújo (2014) and the Simple Genetic Algorithm code developed and provided by Deb (2001).
Several studies related to the optimization of reinforced concrete columns and the minimization of costs have been done. Chaves (2004) considered a rectangular cross-sectional area subjected to an axial force and an axial bending in which four fixed steel bars were arranged near the corners so that only its areas were optimized. The problem was analyzed by the method of Lagrange Multipliers and the Kuhn-Tucker conditions. Bordignon (2010) considered the same case of forces but used the Simulated Annealing technique in the optimization process and the Golden Section Method to finding the neutral axis position. The reinforcement steel positions and diameters were optimized regarding an approach where four bars were fixed in the corners, and others between them vary in quantity. The diameters were also up to three different values.
The case of an axial force and biaxial bending have also been analyzed. Bastos (2004) studied rectangular cross-sectional areas in which the dimensions and reinforcement steel were optimized. The author used the programming language Visual Basic and a Genetic Algorithm. Sias (2014) developed an optimization software with the function "fmincon" of MatLab that used mathematical programming. It was considered rectangular and circular crosssectional areas and the optimization of dimensions, steel bars positions, and cross-sectional areas. However, the commercial diameters were not used in his sizing. By these means, it is necessary to search for methods of design that approaches the best and feasible solution for reinforced concrete columns. This paper presents in the second section some concepts of Genetic Algorithm; in the third section the sizing of a reinforced concrete column; in the fourth section the concepts of the developed algorithm; in the fifth section the formulation of the optimization problem; in sixth section the results; and, finally, in the seventh section the conclusion.

GENETIC ALGORITHM
The optimization algorithm adopted in this paper is the Genetic Algorithm (GA). It is a metaheuristic that considers a probabilistic approach in the achievement of the best solution.
The GA search the feasible region with a population of points providing a simultaneously analysis and a more efficient approach for the global solution.

Every point of this initial population is a solution in the optimization and it is a
Revista Mundi Engenharia, Tecnologia e Gestão. Paranaguá, PR, v.5, n.2, p. 222-01, 222-16, 2020 DOI: 10.21575/25254782rmetg2020vol5n21154 4 candidate solution for the best one. A candidate has its design variables represented by genes and chromosomes (Goldberg, 1989).
The procedure of the algorithm starts with a random initial population (1) and it is followed by the: evaluation of solutions (2), the selection of candidates to reproduce (3), application of genetic operators (4), generation of a new population (5). The steps (2) to (5) set a loop in the algorithm until a stop criteria, for instance, a maximum number of generations or analysis is achieved (Goldberg, 1989). A Simple Genetic Algorithm code in C with a constraint handling technique (Deb, 2000) provided in https://www.egr.msu.edu/~kdeb/codes.shtml is adopted to solve the optimization problems discussed in this paper. The real code was adopted in this GA.

REINFORCED CONCRETE COLUMNS
A reinforced concrete column usually has the biggest dimension in the vertical axis and a cross-sectional area in rectangular, square or circular shapes.
This structural element can be subjected by compressive axial force and axial or biaxial bendings due to the project, construction or accidental eccentricities. The compressive force may act away from the centroid and then cause the effects of bending. The cross-sectional area considered in this paper (Araújo, 2014) is presented in Fig. 1, where the axial force is represented by , the eccentricities result in = and result in = , 0 is the position of neutral axis and is the angle of neutral axis with the x-axis. The cross-sectional area is rectangular and has the reinforcement steel position in a symmetric way with the same diameter. Its identification is adapted from Bordignon (2010).
The sizing of reinforced concrete column must satisfy the requirements of NBR 6118 (ABNT, 2014) and it is crucial that the safety is guarantee such as the subjected forces must be lower than the resistant forces. Distances in vertical ( ) and horizontal ( ℎ ) between the axis ( ) and faces ( ) of two subsequent reinforcement steels, dimensions of cross-sectional area, rates of reinforcement steel ( , , ) and a criterion to ensure the case of reinforced concrete columns (5 ≥ ℎ) must to be checked. The calculations presented in the next sections are based on Araújo (2014) formulation and in NBR 6118 (ABNT, 2014). Araújo (2014) considers a rotation in the coordinate axes ( − to ′ − ′ ) to simplify the reinforced concrete column sizing which is made considering an axis that is parallel to the neutral axis. This procedure is represented in Fig. 2, where ℎ is the height of the crosssectional area after the rotation, ′ is the y coordinate of the reinforcement steel, ′ and ′ are the higher and lower value of ′ for the cross-sectional area, respectively, and is the distance between the major compressed point and the center of the reinforcement steel.
The NBR 6118 (ABNT, 2014) introduces some basic hypotheses to simplify the calculations. These hypotheses deal with the case of proportional relation between deformation and the position of neutral axis; the perfect adherence between the concrete and the reinforcement steel that results in the same deformation for both materials; the concrete and reinforcement steel stress; and deformation domains.

Position and Slope Angle of Neutral Axis
The position and the slope angle of neutral axis are approximate from two relations and tolerances . According to Araújo (2014)

THE ALGORITHM
The algorithm was developed in the programming language C with the It starts with the input of data such as the subjected forces, the concrete resistance , the reinforcement steel resistance , the modulus of elasticity , the stirrups diameter ϕ , the reinforcement coverage , and the materials costs. Then, other parameters are defined, the forces are verified and the reinforcement steel diameter is chosen. This last step is done with the optimization of one design variable that points out for the respective diameter ( Fig. 4). For each integer value that the variable has, a different diameter is considered in the sizing. The possible diameters are: 10,0 mm; 12,5 mm; 16,0 mm; 20,0 mm; 25,0 mm and 32,0 mm.
After that, the distance between steel bars are calculated, and the maximum and minimum spaces among them. At last, the stress in the concrete and reinforcement steel as well the resistant forces are determined. After the completion of a run in the evolutionary process, the feasible solutions are stored in a text file with their design variables leading to their fitness value. Then, the determination of the best design must be done through the analysis and comparison of each feasible solution by the designer/engineer.

FORMULATION OF THE OPTIMIZATION PROBLEM
The design variables are listed in Table 1 and the optimization problem is formulated from Eqs. (24) to (43).  Minimize ( ) = 5 6 + + ( 5 + 6 ) 2 Subject to:

RESULTS
The developed algorithm was applied to the example of Carvalho and   (2014) determined solutions using his optimization software that adopts deterministic methods. The first one considered a fixed geometry of 20 cm x 40 cm (Table 2 -II); the second one, a geometry optimization (Table 2 -III). Commercial's diameters were not defined for Carvalho and Pinheiro (2009) and neither considered in the optimization process by Sias (2014). The Genetic Algorithm was run for the case of a geometry optimization keeping the concrete resistance (Table 2 -IV). The second case was the optimization of the geometry and the with values under 50 MPa (Table 2 -V).
The cross-sectional areas detail from the GA are shown in Fig. 5.

CONCLUSION
This paper presented the optimization of the cross-sectional area of the reinforced concrete column via a Genetic Algorithm (Deb, 2001) and Araújo (2014) formulation. Some of the concepts of the GA were introduced and briefly explained, and then it was presented the main part of the formulation of the optimization problem discussed in this paper.
The algorithm developed with the software Code::Blocks in the programming language C presented similar results with the reference providing rigorously feasible solutions. Despite the GA costs were higher than Sias (2014), it was defined commercial diameters for the steel bars and considered only discrete dimensions which is easiest to apply to on-site production. These facts were not considered in the sizing of Sias (2014) and may influence on the cost as a tradeoff, but the convenience and facility of the construction process make it worth.
Thus, the objective of achieving a solution with reduced costs was accomplished. Therefore, the algorithm may be used for the sizing of columns as a support tool for subsequent sizing and future studies. However, for the case of geometry optimization it is important to analyze the bonds of the reinforced concrete column with other structural elements as long as the changes in dimensions may cause different subjected forces. More detailed researches may analyze this last situation and also consider the influence of stirrups in the optimization process. Deb's Genetic Algorithm (2001) may also be applied to the sizing of beams, slabs, and other structures.