Optimum design of prestressed steel beams via genetic algorithm

Abstract The objective of this article is to present an optimization problem formulation to reduce the total structural cost of prestressed doubly-symmetric and monosymmetric I-shaped steel simply supported beams with straight tendons. The optimization problem was implemented via MATLAB’s native Genetic Algorithm. The validation and evaluation processes adopted two examples from literature. The design method follows the Brazilian standard NBR 8800:2008 for the Ultimate and Serviceability Limit States. The best result was found for a monosymmetric case by up to 20.00% and 25.70%. Saving in material weight and installation of tendons, without exceeding the security limits, was also effective. Furthermore, the results presented an efficient alternative for structural engineering, providing a significant model for similar analyzes.


Engenharia Civil
Although steel structures have more advantages than other materials, their cost is a negative point.Composite materials as well as new design methods can be developed to minimize such a factor.
It is well known that prestressed steel could provide economic advantages over traditional techniques.In this respect, such technique has been involving research for improvement since Belenya (1977).Furthermore, Belletti and Gasperi (2010) noticed the number of deviators, such as prestressing tendons and theirs position, as critical design variables that require attention, aiming for better beam performance.
Design methods must be as accurate as possible to guarantee structural safety.Therefore, a higher number of variables, combinations, and conditions are involved.Moreover, the trial and error to obtain the lowest cost of prestressed steel structures requires computational approaches.

Introduction
Optimum design of prestressed steel beams via genetic algorithm GA also presents a high capacity for working with multiple constraints without elevated computational cost.Dealing with this type of problem was approached by Tang, Tong and Gu (2005).Mixed coding, i.e., integer and continuous variables, successfully showed results on a truss design optimization.Kripakaran, Hall and Gupta (2011) proposed a GA formulation employing discrete decision variables.Furthermore, a search space was delimitated to limit the GA combinations.Such methodology is deeply explained by Rajeev & Krishnamoorthy (1997) and Gupta et al. (2005) aiming for a higher computational performance.
The Spanish, European and American codes were adopted for the optimization of steel structures by Prendes-Gero et al. (2018) .The GA was implemented to follow a 144 cross-section database and determine the optimum value for a threestory steel building.Results demonstrated the capacity of the algorithm on the internationals building design codes.Moreover, the researchers pointed out the use of discrete variables to obtain better results.
Taiyari, Kharghani and Hajihassani (2020) compared four metaheuristic optimization techniques in the design of pile wall retaining systems: Genetic, Particle swarm optimization, Bee, and Biogeography-based algorithms.The reduction on the total structural cost was selected as the objective function.OpenSees software and the MATLAB platform were used for coding.The GA proved to be able to reduce the objective function.
Alves & Ramos (2021) proved the GA effectiveness of the weight reduction on a steel-concrete composite beam.MATLAB's native GA was also adopted, using its GUI platform.The Ultimate and Serviceability Limit States were followed by the Brazilian standard NBR 8800:2008.Small spans were analyzed, i.e., from 5 to 16 meters.The objective function consists of multiple constraints.
Recently This article presents an optimization problem for prestressed I-shaped simply supported steel beams.The design model is in accordance with the standard NBR 8800:2008 and was implemented via MATLAB's platform using the GUI tool to generate an interactive graphical interface.To solve the optimization problem, this program makes use of MATLAB's native GA.Both are monosymmetric, such that their being doubly-symmetric, presented significant economic results.

Objective function
The optimization problem considered shapes as illustrated by Figure 1.Therefore, the number of prestressed tendons (n t ), the depth of cross-section (d), and the flange widths (b fs and b fi ), were considered as integer variables.Flange thicknesses (t fs and t fi ) and web thickness (t w ) were considered as con-tinuous variables, i.e., a hybrid formulation.Notice that doubly-symmetric cases are particular cases when b fs = b fi and t fs = t fi .1.The presence of deviators naturally alters the objective function, so that the length of the tendons will not be equal to the span.The C(1), C(2), C(3) and C(4) delimits the maximum efforts of steel considering its resistance.C(5) and C(6) evaluate the displacements on the Serviceability and Ultimate Limit States.C(7) (prestressing time) and C(8) (In service) limits the combining bending.C(9), C(10), C(11), The constraint functions, Equations 2 to 18, followed the Brazilian standard NBR 8800:2008.The design criteria satisfy both serviceability and strength requirements.Notice that C(7), C(8), and C(10) are coded as an if-else statements due to the crosssectional shape, i.e., doubly-symmetric or monosymmetric.The aforementioned researchers did not delimit the prestressing losses.Thus, it was convenient to vary it by 0, 5, 10, 15, and 20% to obtain different parameters of comparison.Therefore, the input data considered: 2-point loads of 120 kN applied at 10.25 and 11.75 m from the left support; L of 22 m; t w of 10.40 mm; tendons of 9.5 mm allocated The authors point out that the selfweight was considered by the program.Therefore, the lower and upper limits assigned to the GA are stated in Table 3.The initial population contains 120 individuals and the following, 60.The rate of elite individuals and crossing of the intermediate type are 0.05 and 0.8, respectively, whereas the mutation rate is random.The GA is performed primarily with an entirely random initial population, thereby obtaining an optimal local response.The MS geometries varies with prestressing losses.On the other hand, the DS shapes has presented exactly the same geometry.Thus, as expected, the MS are more able than DS to change their shapes without exceeding the constraints.Naturally, the inertia on the DS is constant.

Constraint functions
Due to the impossibility of reduction on the DS geometries without exceeding the security limits, GA decreases the number of tendons.Moreover, the higher (?) on the tendons and inertia led to lower displacements.Therefore, the MS showed greater values than DS about the tensile and compressive strength.
Besides this study presents a greater number of tendons compared to Abbas et al. (2018), the total structural costs were lower than the reference for each case.The large flanges and thicknesses, as well as the smaller depth of cross-section, was not the best combination, increasing the final cost.
Figure 4 illustrates the total structural cost normalized according to Abbas et al. (2018).In general, each of the models are better choices than the reference.The greatest reduction was found on the MS0 with 20.00%.On the other hand, other models are not unsatisfactory options, considering those costs near to MS0.The large deviation is 4.00% (DS15 and DS20), which is negligible in the authors point of view.The designer will choose a model based on the local parameters of construction, e.g., architectonic project or structural constraints.The optimum results are displayed on Table 4 and its constraints are graphically represented on Figure 3.The nomenclature adopted by this example indicates the type of the beam (MS or DS) as well the related losses (L), i.e., 0, 5, 10, 15, or 20. Ferreira (2007) analyzed prestressed I-shaped beams (i) with polygonal and (ii) straight-line tendons.However, this study focused only on (ii).
The tw varied at 12.5 and 16.0 mmlower values exceeded the GA constraints.Moreover, bfs ranged as follows: 22.4, 25.0, 31.5, 37.5, and 44.4 mm.Therefore, the input data considered: 3-point loads of 150 kN applied at 11, 12.5, and 14 meters from the left support; overload of 3 kN/m; permanent load of 12.86 kN/m; serviceability overload of 15 kN/m; L of 25 m; tendons of 15.2 mm allocated 100 mm below the inferior flange bottom; fy equal to 345 MPa; E of 205000 MPa; and prestressing losses of 12.3%.
The optimum results are displayed in Table 5 and their constraints are graphically represented on Figure 5. Different from example 1, the nomenclature herein indicates only the type of the beam (MS or DS) and an identification number.The MS shapes obtained higher d than DS, according to the conclusion in Example 1 about the MS capacity of its shape changing.Furthermore, the inertia clearly increased, meaning unnecessary higher number of tendons.Thus, MS shapes support more elevated tensile and compressive strength than DS.
Unlike Example 1, the number of tendons was lower than the referenced.In general, MS got a half of Ferreira's (2007) values.This fact jointly with the smaller geometry, increases the total structural cost of Ferreira (2007).
Figure 6 illustrates the total structural cost normalized according to Ferreira (2007).
As such, the MS as well as DS models, obtained better results than the reference.Considering the design method, i.e., traditional technique, employed by the aforementioned author, lower costs were also expected.The best result was MS04 with 25.70% of reduction.However, MS03, MS06, MS08, MS10, MS12, DS10, and DS12, resulted almost the same value as MS04.Between those models, the large deviation is 0.0138%, again negligible in the authors point of view.Therefore, any of these models can be designed with a lower cost, considering the applicability will need to consider the local conditions.
The influence of b fs and t w on the total structural cost are demonstrated by Figure 7. Overall, the MS models results optimum values than DS.Furthermore, the DS bfs are proportional to the cost, different from MS.Moreover, as well as the bfs decrease, DS costs have a tendency to show almost MS costsabout the same t w .
Therefore, DS models with t w equal to 12.5 mm and b fs less than or equal to 31.50 mm, results minor total structural than MS models with t w equal to 16.0 mm.In this respect, for small values of b fs , the MS are almost equal DS geometries.This article presents an optimum design for prestressed I-shaped steel beams via MATLAB's native GA technique.Savings in material weight as well as prestressing tendons anchorage is the objective function -total structural cost.Abbas et al. (2018) and Ferreira (2007) were the examples to validate and evaluate the algorithm.Furthermore, significant models were proposed considering the objective function without exceeding the security limits.The design method followed the Brazilian standard NBR 8800:2008.Thus, the main conclusions from the are: • Example 1 (Abbas et al., 2018) resulted in 20.00% of reduction considering 0% of prestressing losses (MS0).However, the largest deviation from MS0 is 4%.Thus, the MS as well as DS models are able to be applied.
In general, monosymmetric was the better option compared to the doubly-symmetric shapes.Nevertheless, the deviation between some shapes could be neglected.For that reason, the proposed formulation was efficient to obtain the optimal solution for the prestressed I-shaped steel beams.Moreover, the GA proved to be useful with a low computational cost.

References
, Mageveske et al. (2021) have shown the possibility of savings in material weight and total structural cost on doubly-symmetric I-shaped steel beams.The researchers optimized the Ultimate and Serviceability Limit States following the Brazilian standard NBR 8800:2008.
Abbas et al. (2018) studied a Finite Element model via the ANSYS optimization package.Two objective functions were employed to minithe strain energy and the material weight of two steel girders -with

Table 3 -
GA variables: lower and upper limits.

Table 2 -
Relationships for estimating the internal forces.andC(12)governthegeometricproperties to avoid buckling.C(13) and C(14) evaluate the yield strength over the limits of compression and tension.Therefore, the optimization problem proposed was solved via MATLAB's native GA.In the analysis method, the required forces were estimated from the equilib-rium equations -Table2.The tendons eccentricity herein defined as e is considered on the pretension load.The distance from the left support is indicated by a.