Earthquake Design of Reinforced Concrete Buildings Using NSGA-II

In the present study, the optimal seismic design of reinforced concrete (RC) buildings is obtained. For this purpose, genetic algorithms (GAs) are used through the technique NSGA-II (Nondominated Sorting Genetic Algorithm), thus a multiobjective procedure with two objective functions is established..e first objective function is the control of maximum interstory drift which is the most common parameter used in seismic design codes, while the second is tominimize the cost of the structure. For this aim, several RC buildings are designed in accordance with theMexico City Building Code (MCBC). It is assumed that the structures are constituted by rectangular and square concrete sections for the beams, columns, and slabs which are represented by a binary codification. In conclusion, this study provides complete designed RC buildings which also can be used directly in the structural and civil engineering practice by means of genetic algorithms. Moreover, genetic algorithms are able to find the most adequate structures in terms of seismic performance and economy.


Introduction
e scientific advances in technology and computation resources have allowed the development of new optimization procedures in recent years, such as the genetic algorithms optimization method.
is approach was initially discussed and proposed by Holland, and it is based on the natural selection theory established by Charles Darwin [1,2].
e main characteristic of GAs is the survival, adaptation, crossing, and mutation of species through time.
e individuals with best adaptive capacity have more likelihood of surviving and obtaining descendants.For this reason, the genetic code of the best individuals is maintained, to obtain descendants with equal or better adaptation capacity, thus the species evolve.
Mathematically, the GAs technique consists in the generation of an initial population (usually random) of possible solutions represented by binary codification.e weakest or most misfit individuals are eliminated, and the strongest survive and are reproduced.e adaptation level of each individual is measured with a value assigned in one objective function [3].A typical genetic algorithm uses three basic operators: selection, crossing, and mutation: (i) Selection of individuals: it is based on qualifying each individual according to their adaptation and determining which ones survive and pass to the next generation.(ii) Crossing: the aim of the crossing is to create new individuals with the exchange of genetic information (usually binary codification) among those most adapted, similar to that used by a natural organism in sexual reproduction.(iii) Mutation: it is used to introduce random changes in the population of a generation.e mutation may be beneficial as it allows introducing diversity in a population.
Once the previous steps are completed, a new generation is obtained, and the process is repeated until reaching the number of generations desired.Currently, this method is applied in many areas, and its results depend on the complexity of the problem, number of possible solutions, size of the population, among others.e combination of GAs with structural and seismic engineering can create a tool to improve conventional structural design; the required computational time to obtain satisfactory results is small even without predesigned elements by a skilled designer.With this tool, it is possible to obtain optimal solutions that satisfied the strength and displacement structural criteria under gravitational and accidental loads.GAs have been used extensively for civil and structural engineering purposes.For example, genetic algorithms have been applied in record selection for the seismic analysis of structures [4].Designing of steel structures was done, from trusses [5][6][7][8], 25-bar space tower [9], to the optimal design of steel frames by their comparisons with the conventional design [10][11][12][13][14].For reinforced concrete structures, there is a larger number of possible solutions compared with steel buildings, due to the different amount of reinforcing steel in the sections.In this case, most of the research studies developed are focused on the design of beams [15][16][17] and a few in the frames design [18]. is method is also used for composite beams [19] and multistory steelconcrete composite building designs [20].Most of the studies aimed at designing RC structures are based on the optimization of a single objective function; however, most real-life problems have several objectives that should be satisfied.For example, a building under earthquake should be able to satisfy the resistance and displacement requirements.us, it is necessary to use multiobjective algorithms, as in the case of SPEA2, MOMGA-II, PAES, NSGA-II, among others.Kelesoglu [21] proposed a method for space truss design minimizing the weight and the displacements of the structures.Barraza [22] designed steel frames using NSGA-II and PSO techniques who searched to minimize the weight and a drift function achieving optimal results.In reinforced concrete frames, most of the studies have been carried out for 2D models or only elements such as beams design, in some cases seeking to minimize the cost and the maximum displacement [23] or to minimize the cost of repair or replacement of structural members [24].e aim of this study is to illustrate the application of GAs for multiobjective design of 3D RCframed buildings under earthquake loads obtained by the static method in accordance with the Mexico City Building Code.It is important to say that based on the literature review of the authors, there are no studies aimed at providing complete designed RC buildings which also can be used directly in the structural and civil engineering practice by means of GAs.
In the present work, the NSGA-II approach is used considering two objective functions: structural cost and maximum interstory drift (MID), besides constraint variables that help to obtain constructively viable results in a fast way.For the present study, databases of 2048 columns, 2048 beams, and 1024 slabs were made with their cost and resistant strength by means of the MCBC.
e transversal sections are represented by binary codification of 11 bits for beams and columns and 10 bits for slabs to simulate the genetic information of individuals (Chapters 3 and 5).In the next chapter, a brief description of the NSGA-II method is provided.

Nondominated Sorting Genetic Algorithm (NSGA-II)
e technique NSGA-II proposed by Deb et al. [3] and Deb and Gulati [5] is used in this study for the multiobjective optimization design of 3D structural concrete buildings under seismic forces.
e main idea of the NSGA-II approach is to find a set of solutions that are good among the different objectives, obtaining several satisfactory solutions [25].For example, let us suppose that it is necessary to minimize all objective functions in a multiobjective problem. Figure 1 shows all the feasible solutions of the optimization problem; note that the nondominated solutions correspond to those which are not worse than other solutions by considering all the objectives, or if the solutions are better than others in at least one objective function, and these solutions represent the Pareto frontier or Pareto optimal solutions (POS).In general, NSGA-II is implemented with an effective sorting method based on individual ranking by nondominated sorting and a crowded distance sorting which evaluates the population density of solutions in the same rank.
e typical steps of the NSGA-II approach are as follows: (1) An initial parent population P0 is randomly generated, and the nondominated sorting is implemented on P0 where each individual is ranked based on the dominance relation in the objective space.
(2) Individual within each rank is sorted again based on the crowded distance where the population density is evaluated.For further information about the crowded distance, see [5]

Design Parameters and Database for the RC Elements
Although several studies aimed at obtaining an optimal design through genetic algorithms, as it was previously discussed, most of them use steel structures (especially trusses) and only reduce the maximum displacement and the total weight without taking into account if the final system or design can be built in the real world (constructive feasibility).Moreover, usually the studies select defined loads.us, there is a huge gap to obtain earthquake-resistant structures designed via GAs.In order to obtain safety buildings under earthquakes by means of genetic algorithms, in this study, three RC structures are designed accounting for all the design parameters suggested by the MCBC.For example, the security of a structure must be verified for the effect of permanent, variable, and accidental actions combinations considering the following equations: where DL, LL, and SL are dead, live, and seismic loads, respectively.
To calculate the seismic loads, the MCBC suggests that the effect of an earthquake in a structure must be obtained by simulating horizontal forces acting on each of the points where the mass is supposed to be concentrated at each floor.
e seismic forces are combined taking into account two horizontal directions of the earthquake, thus the structure is analyzed considering 100 percent and 30 percent of two orthogonal directions.Equation ( 2) is used in the present study to estimate the seismic design forces along the selected RC buildings.
where S Fi is the horizontal force acting on story i, c is the seismic coefficient, W i h i is the weight of the level i multiplied by its height, Q is the seismic behavior factor or ductility, W i is the sum of the weights of all the stories, and  W i h i is the sum of the weights multiplied by their heights of all stories.Most seismic codes use the concept of design spectrum defined by dividing the elastic response spectrum through a reduction or behavior factor Q. e value of Q depends on the parameters that directly affect the energy dissipation capacity of the structure, namely, ductility, added viscous damping, strength reserves coming from its redundancy, and overstrength of individual members [26].In the MCBC regulation, the behavior factor can take values of 1, 2, 3, or 4 for elastic analysis according to the constructive system used.
e frames that will be designed in this work correspond to a behavior factor equal to 3.
e values of Equations ( 3)-( 5) are obtained from Table 1, where c is the maximum value of the pseudoacceleration (Sa) design spectrum in units of Sa/g, a 0 is the initial value of the spectrum, T a and T b are the characteristics periods, and r is an exponent.
e equations used to calculate the Sa design spectra are ( e buildings to be designed are located in the lake area of Mexico City corresponding to zone III b and represented by the following spectrum (Figure 2).
It is important to say that the RC buildings are constituted by structural elements as slabs, beams, and columns.For this reason, a large database was developed that takes into account the materials and labor cost of each one.To represent the genetic information of each section, binary code is used.A 1-bit code can represent 2 sections since it can only have 2 different configurations: 0 or 1. e number of individuals that a binary code can represent depends on the number of bits, and it is obtained by the expression: 2 bits .If we work with 10 bits, the first section is represented by and a total of 1024 (2 10 ) sections can be represented.
In the case of concrete, it is considered with the compressive strength of 250 kg/cm 2 and the yield strength of reinforcement steel of 4200 kg/cm 2 .e main characteristics of the beams, columns, and slabs selected to create the database are the following: Beams.
e heights of the transversal sections vary from 1.5 to 2.2 times the base using a multiple of 5 cm starting Slabs.Heights from 10 cm to 25 cm were proposed with different reinforcements in the upper and lower parts.Corrugated bars number 3 and 4 are used.A total 1024 sections are considered.e total cost of the frame is obtained by multiplying the unit price of each structural element by its volume.

Objective Functions
e aim of the present study is the seismic design of reinforced concrete buildings using a multiobjective genetic algorithm.A parameter used to determine the performance of buildings is the maximum interstory drift.For this reason, in this study, the first objective function is aimed at designing frames with MID close to the limit allowable provided by the Mexican Code, to guarantee an adequate seismic performance.As the second objective, it is proposed to minimize the total cost of the volume of materials in the frame.e mathematical expressions that calculate the peak drift and cost objective functions proposed are where F 1 and F 2 are the objective functions of drift and cost, respectively, and all the C terms are designed constraints or penalty functions that are described below.Due to the large number of restrictions that exist in the proposed study, the objective functions had to be calibrated and evaluated up to obtain one that provide an evolution of the individuals with the best results.e selected exponents in the equations let the algorithm quickly discard those that do not comply with slab and connection constraints.For this reason, the algorithm works with those individuals who depend on the other 2 restrictions.en the program focuses on those individuals who should satisfy the criteria of resistance and control of the maximum interstory drift.
Function F 1 has the objective to find the lightest sections comparing the maximum interstory drift with a target drift (TD), as shown in Equation (9).In this study, the TD is 0.012 as suggested by the Mexican City Building Code: If the MID is large than TD, then the following expression is used: where I MID is the objective function of MID.If TD and MID are equal, I MID will be equal to 1. erefore, values close to 1 represent frames with displacements close to the limit allowable established by the MCBC regulation.Moreover, with the function F 2 , it is intended to obtain the most economical sections taking into account the materials and labor cost of the building: where C r , C c , C l , and C are reinforcement, concrete, labor, and total costs, respectively.e other parameters are used as design constraints if they do not satisfy the requirements of displacement, strength, stiffness, or constructive feasibility.
ese parameters start with a value equal to 1, and are obtained with the following equations.
If the slab is slender than a minimum, or 2 cm thicker than the required, where C slab is the constraint function for the slab, Mst is the minimum adequate thickness for the slab, and St is the slab thickness.If the beams show excessive deformation, where C d is the constraint function of displacement, Pd is the permissible deformation, and Md is the maximum deformation of the beams.

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If the beam-column or column-column joint is not possible to be made in the engineering practice construction, where C con is the constraint function for the connection, hd is the higher dimension of section, and ld is the lower dimension of section (beam or column).
When the demand on the structural members is larger than the strength, where C s is the constraint function for the strength, mf is the maximum force acting in the section, and strength is the maximum allowable strength.
As observed in Equations ( 12)-( 16), all the constraints are dimensionless; further, function F 1 is dimensionless since it is the ratio between MID and TD.On the contrary, the objective function F 2 has cubic root units of US dollars to avoid larger values of F 2 after they are penalized according to the constraints.
Note that through the generations, it is expected to obtain structural buildings that are not penalized, thus, they will have a value of one aimed at obtaining feasible designs.

Multiobjective Seismic Design Procedure
Using NSGA-II For the design of reinforced concrete frames (structural buildings), a computer program in the Visual Basic language was developed by the authors that uses the following procedure: (1) Initial Population.e rst generation is randomly created.Each individual corresponds to a set of sections of beam, column, and slab with its corresponding reinforcement area, where each one will be represented by binary code.A number n of individuals is proposed; in this case, it is 100.(2) Design Parameters.Since the population is created, permanent, variable, and accidental actions are calculated for each individual.A structural analysis is carried out where displacements and strengths are reviewed by the regulation.In the MCBC regulation, the MID permissible is 0.012.(3) Objective Functions.e aim of the objective function is to obtain a set of frames with better seismic performance-cost relation.For this, two functions are used: one that depends on the peak interstory drift and another related to the cost of the structure.Both functions should be penalized in case of not complying with some constraints or design criteria (chapter 4).( 4) Selection.
e individuals are then separated into frontiers and crowding distances are calculated, where a total of n/2 individuals belonging to the main frontiers are selected.ese frames will cross to obtain the remaining individuals of the new generation.
e selected individuals will cross each other, to designate the couples; the binary tournament is used, which consist by taking two random individuals and having them compete to de ne the parents.e crossing consists of combining the genetic information (binary code) of both parents into two children.When the pair of parents is obtained, a random number is generated between 1 and the number of bits with which they are working.Assuming 10 bits, individuals 407 and 725 are crossed as follows: e following individuals are crossed from bit number 6: Generate two individuals with the following codes: 405 0 1 1 0 0 1 0 1 0 0 727 1 0 1 1 0 1 0 1 1 0 (6) Mutation.It is used to guarantee the diversity of the structural RC frames obtained in each generation.It is applied for all generations and consists of the change of a speci c bit of the code of an individual.A probability of mutation of 5% of the total individuals is proposed.(7) New Generation.After all the evolutionary process, a new generation is obtained and the process returns to step 2 to begin a new generation to complete until it is nished with the n number of generations.
is procedure must be repeated several times in order to obtain the POS (best individuals).Figure 3 shows a owchart of the multiobjective procedure using NSGA-II.

Multiobjective Seismic Design of the RC Buildings Using NSGA-II: Numerical Examples
In chapter 5, the multiobjective procedure for the seismic design using NSGA-II has been described.In this chapter, the methodology is used for the seismic design of 3D RC frames.ree RC 3D-framed buildings designated to o ces of 3, 6, and 9 stories are earthquake-resistant designed considering a seismic behavior factor of 3. e rst two structural models have 3 and 6 stories (RC-3 and RC-6), 3 bays in both horizontal directions with 7 m of length and a story height of 3.5 m for all the stories (Figures 4 and 5).
e third building under consideration with a 3 m of story height and 3 bays of 5 m in both directions has 9 stories (model RC-9, Figure 6).For the structural design, it was proposed the use of one section of beam and column for every 3 stories and one slab for frame.For the application of  e numerical results for the reinforced concrete building with 3 stories are illustrated in this chapter.As it was previously described, this model has 3 bays in both horizontal directions with 7 m of length and a story height of 3.5 m for all the stories.For this example, beams, columns, and slabs have the same transversal section and steel reinforcement.For this reason, the binary codification of this model is represented by 30 bits, by means of the previous data and the NSGA-II procedure (Figure 3).In Figures 7 and 8, the behavior of the objective functions of the first execution of the program is shown.It is observed that the values of the objective functions decreased with the number of generations; in general, RC buildings more economical, lightweight and safety are obtained through the generations.It can be observed that before the generation number 10, no penalized individuals are observed.is is because only three variables (one beam, one column, and one slabs for all the stories) are used to represent this structural building; for the others models, a large number of generations will be required to obtain no penalized individuals, thus to minimize the objective functions (convergence).It is important to say that at the beginning of the procedure (first generation), the random approach provides some unrealistic combinations of transversal sections or solutions.Nevertheless, the first viable solution (without restrictions) randomly created in generation one compared with the solution at the end of the algorithm was improved considerably.In general, the objective functions are reduced as shown in Table 2. Notice that each solution is subjected to different S Fi (seismic forces) that represent the earthquake, because of the differences between the dimensions of the sections (structural weight).Since the objective is to find solutions with maximum interstory drift close to the limit allowable of the MCBC, the optimal solution has a higher drift than the corresponding first solution obtained (Table 2).e algorithm first found the most economical slab; this is because its influences is about 50% of the total cost of the frame and minimizes the seismic forces S Fi .In general, S Fi depends on the dimensions of beams and columns where different combinations modify the MID and stresses on each element.With the objective functions F 1 and F 2 , these variations are evaluated reaching the combination of MID, stresses, S Fi , and dimensions of the sections that achieve optimal results.Once the program concludes the procedure, several building designs are obtained for the Model RC-3.

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In order to find the POS, the procedure of Figure 3 must be executed several times.A similar behavior of the objective functions through the generations was observed by applying the NSGA-II approach; however, the solutions obtained in each program execution provided different earthquakeresistant building designs.
e best results are shown in Table 3 and Figure 9.In this model, the POS only resulted with 2 individuals.is is because the frames are composed

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only by 3 di erent sections and there are few optimal combinations.e cost in US dollars of the structural buildings is illustrated in Table 3.
e reinforcement areas and dimensions of the sections obtained from the best individuals are very similar, up to a maximum of 10 cm in beams.As the slab contributes 50% of the total cost, the section turned out to be the same in the best 2 individuals (Figure 10).e transversal sections for beams and columns of the most economical individual are shown in Figure 11.

Numerical
Results for the Building RC-6.In the case of buildings with 6 stories, 5 di erent transversal sections are selected for the members: two di erent transversal sections for beams and columns-one for the stories 1 to 3 and another one for the stories 4 to 6.In all the stories, the same section of slab is used.For this model, the binary codi cation is represented by 50 bits.As shown in Equations ( 7) and ( 9),  8 Advances in Civil Engineering the objective function F 1 has the purpose of nding solutions with the maximum interstory drift close to the limit allowable established by the Mexican regulation.In Figure 12, it is observed that the average maximum interstory drift of the individuals of the RC-6 model increases through the generations and tends to the maximum permissible interstory drift of the Mexican Building Code (0.012). is means that the dimensions of the transversal sections decrease as the generation increases, as well as the cost.In other words, the RC building tends to use its maximum deformation capacity.e behavior of the algorithm was similar to the previous model, decreasing the value of the objective functions with the generations.A large number of individuals at the POS are expected due to the possible combinations of solutions.Table 4 shows the best results after executing the program several times.
MID values in the best individuals range from 0.011 to 0.012, proving that the results are at the limit allowable peak displacement, and the costs are very close to each other with a di erence of US$ 4,300.Moreover, since the bays have the same dimensions as the previous model, the same slab was obtained.POS are de ned by 3 individuals very similar to each other (Figure 13).In Figures 14 and 15, the sections of the beams and columns are shown.

Numerical Results
for the Building RC-9.For this RC building, a total of 7 objectives are used (3 columns, 3 beams, and 1 slab), and the binary codi cation of each individual is represented by 70 bits, so it is expected that there will be a wide range of results and it will be very di cult for all the frames to result with same similar transversal sections.In addition, convergence of the objectives functions is expected in a larger number of generations.Table 5 illustrates the best seismic design results.It is observed that the MID and cost are very similar for all the cases obtained.
e POS are de ned by 5 individuals with very similar objective cost function and MID close to the target 0.012 (Figure 16 and Table 5).In this model, the MID varies from 0.0117 to 0.012, and the maximum cost di erence between the best individuals is just US$ 4,265.
In Figures 17-20, the transversal sections of the economical individual are shown.As it is expected, the   Advances in Civil Engineering dimensions and reinforcement areas of beams and columns decrease at the top stories.Although the results were satisfactory, to better de ne POS, it is recommended to use a larger number of individuals and generations to consider more combinations and to nd di erent but not dominated results.

Conclusions
ree reinforced concrete buildings under earthquake loads have been designed using genetic algorithms with the aim to reduce the structural cost and to increase the seismic performance.For this objective, a computer program was developed.
e application of a genetic algorithm with multiobjective optimization using the NSGA-II approach is an excellent option for the seismic design of the reinforced concrete buildings.It was observed that through the   10 Advances in Civil Engineering generations, the designed buildings tend to reduce their cost and increase the maximum interstory drift.e MID function (F 1 ) helped to obtain less robust sections and peak drifts close to the upper limit capacity equal to 0.012.On the contrary, the cost function (F 2 ) found the most economical results.Finally, the study provides complete designed RC buildings which also can be used directly for practitioner of the structural and civil engineering.Finally, the time required for the structural optimization in a computer depends mainly on the number of elements of the building, individuals and number of generations.For example, because the RC-9 model has a large number of elements, it requires more computational time.Indeed, to define its POS, the algorithm was executed 10 times, with a computer of 16 GB of RAM and processor Intel Core I7-6700, and the results were reached after 216 hours.

Figure 3 :
Figure 3: Flowchart used for the design of three-dimensional RC buildings.

Figure 13 : 8 Figure 14 :
Figure 13: Objective functions F 1 and F 2 for the best individuals of model RC-6.

Table 1 :
Parameter values to calculate the acceleration spectra.
PeriodFigure 2: Design spectrum of Mexico City zone III b .

Table 2 :
Di erences between the rst and best solution of rst execution of the procedure for the RC-3 Model.

Table 3 :
Best individuals for model RC-3.

Table 4 :
Best individuals for model RC-6.

Table 5 :
Best individuals for model RC-9.