The Idealized Smeared Reinforcement (ISR) method for the optimization of concrete sections: a survey of the state-of-the-art and analysis of potential computational approaches

Latest revision as of 21:31, 14 February 2021

Abstract

The present work aims to define formally the method termed as “Idealized Smeared Reinforcement” (ISR) for optimization in the design of reinforcing steel in concrete structures, its boundaries, background and potentials. Such method has been extensively used implicitly in many different studies of reinforced concrete structures related with optimization and mechanical structural behaviour, but it has not yet been formally established as a method itself even though it represents a great solution approach when designing optimally concrete sections, which is a tendency so vital nowadays for sustainability in construction projects. A general survey of such method of the ISR will be presented on in this document, presenting relevant background research related to the use of the method and optimization of reinforcing steel in general for concrete structures, thereafter proposals of different optimization methods, both classical optimization and meta-heuristic methods will be regarded as possible approaches to apply the ISR, specifically Gradient Descent Optimization methods when one variable for the ISR is considered and meta-heuristics for more than one variable involved given their versatility and flexibility to adapt to different problems, particularly the Particle Swarm Optimization PSO method and the Genetic Algorithm GA. At the end, these solution approaches for the application of the ISR method will be compared for the testing of rectangular solid geometries with different analysis parameters in order to show how adaptable and feasible such ISR method might be using a proper optimization algorithm and analysis for its approach when designing reinforcing steel.

Keywords: Classical Optimization, Meta-heuristic Optimization ,Reinforcing Steel, Computational Methods, Concrete Structures, Idealized Smeared Reinforcement

(¹) Autonomous University of Queretaro, Faculty of Engineering

(²) Centro Universitario, Cerro de las Campanas s/n, Cp. 76010, Santiago de Querétaro, Querétaro, México, 2021

1 Introduction

Ever since Whitney and Cohen [2] many years ago in 1956 proposed a guide to design reinforcing steel in concrete structures subject to axial force and bending moment by ultimate strength based on the ACI code many different approaches have taken place from that point on to design optimally reinforced concrete sections. What Whitney and Cohen presented then was a series of charts based on N-M interaction diagrams normalized considering symmetric and uniform reinforcement over a cross section. In recent years however, new more realistic considerations for the design and analysis have been carried on, for instance, influence of confinement and flexural behaviour [3], or non-symmetric reinforcement [4] in which a particular solution of a problem is provided for elements subjected to flexure-compression stresses. Also, the concept of strength design has been challenged by new design approaches [5] which generate a family of solutions for arbitrary combinations of imposed axial load and moment, requiring even a non-symmetric distribution of reinforcement over the cross section. Moreover, theorems for optimal section reinforcement have been also developed [6] based on the understanding of all these studied optimal solutions, encompassing a general formulation to determine the minimum total reinforcement area required for adequate resistance to axial load and moment.

Through all of these conducted research works the idealization of a continous distributed plate along the faces of a rectangular cross section has already been created. But it is specifically in [9] by Anschheim et. al. when the concept of a “Smeared reinforcement” idealization is given formally, in which an optimization for the values of each of the four continuous plates areas (smeared reinforcement) distributed over a rectangular cross section (upper, lower, right and left faces) is carried on, establishing its due design restrictions according to the ACI-318 code, thus finding minimum reinforcement required areas for a given load combination related to the orientation of the axis direction of the cross section geometry. In recent years, this very approach made by officially by Anschheim et. al. in 2008 has also been carried on by other authors [12] although differing by means of solutions generated or by the analysis method itself.

L. Verduzco [12] made a research related directly with this very topic in which a called “Idealized Steel Profile” concept was introduced, idealizing the reinforcement steel as a continuous steel PTR structural profile embedded into the concrete element with a uniform width $t$ from which an optimization analysis was carried on to seek an optimal arrangement of reinforcing steel bars over the cross section element (either the most economical or structurally efficient), with certain restrictions such as equal diameter for the reinforcing bars according to the normative of the region. What it is also relevant to stress from this research is the purely mathematical analysis approach that was made, by defining blocks of stress and strain over the section as the depth value of the neutral axis varies, thus defining equations of resistance for different cases in which the neutral axis depth might have been located. While on one hand Anschheim et. al. generated a family of solutions given by means of percentage of reinforcing steel with respect of the cross section element net area, on the other L. Verduzco designed a mathematical approach through a computational optimization method to seek an optimal reinforcement area from which to transform directly to reinforcing steel bars arrangements. Even though both approaches differ one from another, the idea is the same, and it is of special importance to promote this concept and method by giving it a whole research work focused on its boundaries, limitations, potentials, different possible analysis and optimization approaches.

This very optimization domain presented hereby for RC structures design corresponds to a vital element of the systematic flow of research for RC structures suggested in various literature [14] Fig. 1 in which this problem is directly related to the stage of “Integration of promising techniques” for the improvement of performance of optimization strategies in detailed design of RC structures. Therefore, it can be justified from this perspective the importance and relevance of this such research work, to strengthen this very scenario for optimization in RC structures.

Figure 1: System flow for the area of optimization in RC structures,[14].

2 Background

Resuming the content of the most recent research related with optimization of reinforcing steel, it is of vital relevance to stress the development and introduction of a so called Reinforcement Sizing Diagram (RSD), formulated in [5], which displays a minimum required steel reinforcement area in one axis for a respective neutral axis depth in the other axis, given a combination of axial load and moment Fig. 2. Another relevant development is the called ``Load Combination Reinforcement Diagram (LCRD) [7] which on the other hand plots reinforcement solutions in a two-dimensional space Fig. 3a (left) defined by the coordinates $(A_{s},A_{s}^{'})$ corresponding to the reinforcement area on each of the faces of a rectangular cross section element Fig. 3 (right) allowing an engineer to easily determine an optimal reinforcement solution from given load combinations on structural concrete elements subject to biaxial flexure-compression stresses. Such (LCRD) are obtained by collecting different values of reinforcement area from a (RSD) corresponding to different values of the axis depth $c$ . It is what in multi-objective optimization would be called a Pareto Front.

]

Figure 2: A typical Reinforcement Sizing Diagram for a given load combination. [5]

It is important to stress that in the development of the LCRD was also introduced the concept of asymmetrical reinforcement uniformity over an element cross-section, based on studies which concluded that for optimal solutions an asymmetric distribution of reinforcing steel was more likely to take place [7], therefore different faces of an element cross-section were picked to assign them different values of reinforcing steel area, as seen in Fig. 3 (right), and to have a better approximation for a minimum required reinforcement, taking reference from the such smeared reinforcement concept made firstly in [9].

	]
Figure 3: A typical Load Combination Reinforcement Diagram at the left and its corresponding reinforced cross section element from which $A_{s}$ and $A_{s}^{'}$ are taken as smeared reinforcement steel. [7]

Such diagrams have been used in many further research works [8], in which even though the typical approaches and hypotheses first formulated many years ago to design reinforced concrete sections [10], [11] (for rectangular geometries) are still taken on account, other different assumptions are carried on in order to optimize the reinforcing steel, such as asymmetric reinforcement following the research made by [9] as well Fig. 4, but focusing merely on design outcomes, addressing more flexible design considerations.

Figure 4: Idealization of reinforcing steel proposed by [8] for each axis direction of a rectangular column.

Aschheim et. al. [9] used non-linear conjugate gradient search methods to obtain optimal solutions for these such mentioned smeared reinforcing steel areas, applied to a structural model for different values of rotation of the structural rectangular cross section $\phi$ and ratio between axial force and bending moment $\xi$ Fig. 5. The results generated were plotted in a contour graph Fig. 6 for the minimum reinforcement areas thereby found, by means of steel area percentage in relation with the concrete area, which may be of great use to standardize design precesses.

]

Figure 5: General formulation reference of cross section for the analysis and calculation of resistance of a reinforced concrete element. [9]

]

Figure 6: Contour graph for minimum reinforcement area corresponding to different values of

and

. [9]

Based on this contour graphs, or for the generation of such, constraints may be imposed regarding the type of reinforcement sought (symmetrical or non-symmetrical) either for all faces or opposite ones. By setting equal values of $A_{st}=A_{sb}$ and $A_{sl}=A_{sr}$ corresponding to reinforcement areas on opposite faces, two variables to minimize are obtain, or on the other hand, by setting all areas equal to one another obtaining only one variable to minimize. This method of optimization is capable of saving as much as up to 10% in relation with the conventional methods for design and conventional restrictions for reinforced columns of symmetric and uniform steel reinforcement based on the [1] code, depending also on the decision made by the engineer or contractor.

Regarding other optimization approaches L.Verduzco [12] idealization of the such mentioned “Smeared reinforcement” as a continuous steel structural profile of uniform width $t$ which he referred to as ``The Idealized Steel Profile, embedded into a concrete element from which a mathematical approach was carried on through a constant length step computational optimization method for both rectangular and circular solid cross sections Fig. 7 presented advantages of execution time when big geometries are analysed, due to the mathematical analysis in which only a few conditions and operations are required in order to calculate the resistance of a given value of $t$ (steel profile width) for a respective depth of the neutral axis. Even though such mathematical approach may not be quite applicable to circular cross sections due to the complexity it enhances in the analysis. For designs fully based on normative where so many restrictions are imposed this approach might be of great potential for any geometry, the problem comes up when seeking optimal unsymmetrical reinforcement when a bending moment condition is significantly mayor for one of the axis in a rectangular section, for which such uniform idealization of steel does not really guarantee a good approximation for the required reinforcement area due tu the constant variation of the width $t$ on both axis directions.


Figure 7: The “Idealized Steel Profile” method [12] for rectangular and circular solid cross sectioned concrete elements

3 Analysis approaches for optimization with the ISR

Different formulations for the analysis of a problem with the ISR method might be carried on, either regarding mechanical analysis (flexure-compression for short or slender elements) possibly using the Bresler's formula [10] or by computing an interaction surface diagram by rotating the cross section element Fig. 8 on its own longitudinal axis (depending on the required approximation accuracy); or with relation of the optimization analysis approach adopted, merely with classical optimization or mathematical programming methods, such as Gradient Descent Based methods as formulated by Anschheim et. al. [9] for two steel area values (or variables), or when using the Idealized Steel Profile proposed by L. Verduzco [12] for only one variable a simple numeric search method might be feasible for small cross section columns. Meta-heuristic algorithms of optimization might also be viable to adopt, for instance, taking Evolutionary Algorithms, Swarm Algorithms, Multi-objective optimization and such. Meta-heuristic optimization algorithms might be more feasible than classical optimization methods when formulating more than one different steel area values or when massive cross section elements take place, given the complexity a mathematical optimization approach may enhance.

]

Figure 8: Reference system for the rotation of a rectangular cross-section when computing a surface diagram. [9]

There has also been research and development regarding general theorems for an optimal reinforcement design [6], based on all of the research previously mentioned involving the idealization of Smeared Reinforcement, which states that the minimum total required reinforcement area for adequate resistance to axial load and moment can be identified as the minimum admissible solution among five discrete analysis cases from the infinite set of potential solutions. Such theorem is of great potential when designing for common parameters most often used in structural engineering practice and it is also of importance to mention its existence.

3.1 Optimization approaches for the ISR method

As following, different optimization methods of possible potential for the application of the ISR, such as the Particle Swarm Optimization Algorithm along with the GA for multiple values of width $t$ to optimize, the Multiple Linear Regression method for a single variable of width $t$ to estimate an initial value of with $t$ from which to start iterating in a Simple search¹ to get to the required structural efficiency without a complicated formulation, but only with experimentation and collection of data. optimization method, as well as a Gradient Descent Method are carried on and compared. It was considered that meta-heuristic algorithms could be of greater potential for this engineering problem rather than classical optimization methods, such as Newton-Raphson or a Gradient Descent Based Method given the complexity of analysis they imply when formulating a problem for more than 2 variables on place which are more needed when the steel reinforcing area is to be reduced as much as possible.

3.1.1 Single variable width value t of the ISR

For a single variable of width $t$ for the ISR a general function to evaluate the efficiency must be established, by referring to [12] such efficiency is determined for any depth value $c$ of the neutral axis by computing an Interaction Diagram for both axis directions of a cross section geometry (if a rectangular cross section geometry is taken) Fig. 9, and then to estimate $M_{R_{x}},M_{R_{y}}$ reduced with the Resistance Reduction Factor from the ACI code [1] for every flexure-compression load condition to determine their respective efficiency using Analytical Geometry (using the line intersection formula). This way a most critical condition can be determined for each iteration to take as reference of evaluation as the width $t$ of the ISR changes (decreases or increases). In structural engineering it is a common requirement for this such structural efficiency to take values between 85% to 100%

Thus, an efficiency function of $t$ subjected to a constraint $g(t)$ might be expressed as 1:

(1)

Such function for a single variable of a width $t$ value shows the following behaviour Fig. 10, presenting a non-linearity for the variation of structural efficiency as $t$ changes, forming a concave graph, with the horizontal axis corresponding to $t$ could go on until the ISR would become a solid rectangular steel profile obtaining very little non-negative values of structural efficiency, which is not practical at all, given that the resulting reinforcing bars transformed out of this steel area would not fit into the column.

Figure 9: Reference system for the determination of structural efficiency of a any flexure compression condition

Figure 10: Behaviour of the Efficiency function of a single width value (t) of the ISR

When computing this such Efficiency function, a mathematical approach might be formulated as in [12] Fig. 11 where for rectangular solid cross section geometries quite good amount of computation might be saved than when a discrete analysis of the Idealized Steel Reinforcement is done, as formulated also by [12] for circular cross sections Fig. 12.

This formulation of a single variable of width $t$ for the ISR is useful as stated previously when a design based merely on codes and normative takes place, maintaining uniformity of reinforcement over the cross-section regarding a homogeneous type of reinforcing bars for each element. When a more accurate design is required, then as stated by Aschheim et. al. [9] a non-uniformity consideration might be best to carry on in order to obtain different reinforcement area for each face of the rectangular cross-section and converge better towards a minimum required total reinforcement area.

Figure 11: Mathematical formulation fo the ISR method. Ideal for computational saving applied to rectangular cross-sections

Figure 12: Discrete formulation for the ISR method for circular cross-sections. Ideal for complex geometries.

For this work and specifically for this particular case of a single variable width $t$ of the ISR the mathematical formulation for rectangular cross-sections for the efficiency function was used and performed through Multiple Linear Regression and a Gradient Descent method. The objective of the Multiple Linear Regression formulations is to get a formula so that for each structural model at hand an initial value $t$ close enough to the required one may be determined and minimize the Simple search optimization iterations. To do so, different experimental computational runs were performed varying the values of each of the variables involved in the analysis, such as $b$ and $h$ (corresponding to the section dimensions), and $P_{u},M_{ux},M_{uy},f'{c},cover,Efficiency$ . As for the Gradient Descent Method, given the particular concave form of the graph function for $t-Efficiency$ , it could be well adapted to obtain for each structural model a value of $t$ for which the derivative of the convcave function would be less than a preestablished one Fig. 13. This way, really good approximations to the $t$ optimum cuold be gotten. The Bresler formula (2) [10] was used with the respective conditions and typical values of $a=1$ :

(2)

In order to be able to apply (2) the following condition has to be satisfied to evaluate the axial loads against the design axial resistance:

${\frac {P_{u}}{P_{R}}}<=0.1$

Multiple Linear Regression:

Regarding the experimental runs to apply Multiple Linear Regression to estimate an initial value of $t$ , three different experimentations were performed, one with a total of 100 runs randomly registered with different values for each of the main variables involved already mentioned. The following coefficients were obtained for each interpolation method:

(3)

Which enhances not really good approximations for initial values of $t$ to start iterating when applying a 1t-ISR, only for certain parameter values, given the linearity of the formulation as well as the high number of variables involved. The summary of the experimentation is presented in detail in the results section 4.3 for ten structural models.

Although this Multiple Linear Regression formula of $t$ may be used altogether with a Gradient Descent Optimization Method to minimize even more the number of iteration needed for a good approximation of structural efficiency wanted, as is following presented.

Steepest Gradient Descent method

Figure 13: Slope criteria for the Steepest Descent Optimization Method formulation.

In order to obtain a range of critical slopes for the program to converge to, a total of 10 experimental models were analysed for optimization with Simple search to get to a width $t$ optimal for each model and extract its respective optimal slope. A slope range was found to be $[-0.2539,-0.8556]$ , for efficiencies between $[0.8,1.0]$ . Therefore, this slope range might be taken as a good algorithmic criteria for the Steepest Descent Method to converge to, although it would always be better to check directly for an structural efficiency range to assure a convergence as it was done in this research. The general algorithm of the program is next presented in pseudo-code.

INICIO

for Nmodels=1:nm

Compute

While $f(t_{k})>1.0$ or $f(t_{k})<0.8$

Compute

Compute search direction

:

if $f(t_{k})<0.8$

else if $f(t_{k})>1.0$

End if

Update the current

Compute

k=k+1;

End While

End for

FIN

Algorithm. 1 General algorithmic process for the Steepest Descent Optimization Method formulation

One must be careful with the initial step length $\alpha _{0}$ (it is recommended to use $[0.4<\alpha _{0}<0.6]$ to avoid high number of iterations) as the step length for this case gets smaller the as it converges.

3.1.2 Multiple variables of width t values of the ISR

Regarding meta-heuristic and evolutionary algorithms there has been relevant research for the design of reinforcing steel in cocnrete structures such as [13] in which a formulation with the GA was performed, focusing primarily on costs, restricting the structural efficiencies to a pre-established range using the ISR method with a single variable $t$ to obtain an initial steel reinforcement area as minimum to generate the reinforcing bar sets taken as individuals for the algorithm. The results of this such work are good, but not that different and better than in previous work [12] in which an optimization with Simple search was carried on considering homogeneity for the type of reinforcing bar over the column; not that different neither in cost nor in efficiency, stressing the importance first of all of improve the accuracy on the required minimum reinforcing area and its distribution over the section to minimize iterations until an optimum is found through a Genetic Algorithm particularly, and secondly the need to focus specifically in structural efficiency through a complex non-uniform arrangement of reinforcing bars to obtain better solutions.

As following a comparison in performance and formulation between the GA and PSO adapted to the ISR method for two and four width variables $t$ will be carried on in order to simplify the optimization formulation for the problem through meta-heuristics given that such analysis with classical optimization would not be that practical for the reasons stressed previously, and neither through Polynomial Interpolations or Multiple Linear Regression due to the vast quantity of required data to collect in order to obtain good approximations of each different value of widths $t$ for each face of the cross-section element.

When formulating this types of problems it is quite significant and influential the way in which the structural efficiency is analysed for each possible solution of the optimal ISR. For this case the Bresler formula (4) was also employed [10], as follows:

(4)

In order to be able to apply (4) the following condition (3.1.2) has to be satisfied to evaluate the axial loads against the design axial resistance:

${\frac {P_{u}}{P_{R}}}<=1.0$

For this particular case of optimization ${\frac {P_{u}}{P_{R}}}<=1.0$ was set given to that when evaluating the efficiency with respect of the weak axis of the cross section, the bending moment is the preponderant factor when optimizing the ISR as contrary to when a single $t$ variable problem is formulated, due to the non-uniformity of the ISR over the cross section. This way, the optimization algorithm is able to minimize as much as possible the reinforcing area or widths over the cross section faces corresponding to each axis.

Similar to the analytical geometrical procedure as for a single t formulation problem, an Interaction Diagram is computed for both axis directions of a cross section geometry in order to estimate $M_{R_{x}},M_{R_{y}}$ reduced with the Resistance Reduction Factor from the ACI code [1].

3.1.3 PSO-ISR for multi width variables

It is necessary first of all to establish the range of the search space for each particle, that is the maximum and minimum values that each width variable $t_{j}$ could take for the analysis. A very simple consideration will be stated, such that the maximum reinforcement area on any face of the rectangular cross-section element is less than the sum of the maximum number of reinforcing bars with diameter ${\frac {12}{8}}$ such that the minimum separation $(sep_{min}={\frac {3}{4}}in)$ from the code [1] is not violated. For the upper and lower face the maximum width $t_{max}$ of each idealized smeared reinforcement is given by (5).

(5)

Where $Nv_{max-hor}$ is maximum number of reinforcing bars $\#{\frac {12}{8}}$ allowed for that section face, $b$ is the width section dimension.

For the left and right cross-section faces (6):

(6)

Whereas the minimum width for any $t$ variable might be just a very little number, for instance $0.00001cm$ .

When evaluating the structural efficiency performance, a finite discrete element method of the ISR for each face of the cross section will be made. For this problem it was considered to be time-saving than to develop the mathematical formulation as when the width ${\textstyle t}$ of the ISR remains constant

In comparison with the single ISR with one variable $t$ in which only one solution for $t$ could take place complying the structural efficiency required range, when formulating a multiple $t$ variable problem an infinite set of solutions for a established required structural efficiency are possible, therefore a Multi-objective optimization formulation is required to minimize also the reinforcement steel area. When a two $t$ variable problem is formulated, something similar to a “Load Combination Reinforcement Diagram” as presented by Aschheim et. al. Fig. 3a is obtained, which is what in Computational Optimization is formally called a Pareto Optimal Set from which a “Pareto Front” is then generated [15]. In this problem to adapt the Multi-Objective Optimization problem, the reinforcing area $A_{t}$ is set as the main evaluating function to be minimized with the additional condition $Eff(t)<100\%$ , given that not necessarily one variable (taking $t$ for instance, or reinforcing area in other words) has to be in conflict with the structural efficiency as it is supposed in the majority of Multi-Objective Problems. Therefore the PSO algorithm rapidly adapts to the given objective function and its restriction with no other operation really needed. It is of importance though, to stress that this such condition has to be carefully placed within the algorithm for a good convergence to a global optimum for both reinforcement area and structural efficiency.

This additional condition to evaluate the performance of a particle in the PSO algorithm is most recommended based on our results to be placed both when determining the best position (reinforcement area) in the current swarm and a global best position; given that a global minimum reinforcing area is preponderant to seek rather than a maximum structural efficiency (in the scale of $0-100\%$ ) as long as the structural critical efficiency requirement is covered. In fact, this structural efficiency requirement was set in the first place to minimize the reinforcement area, given that it was assumed that the less the reinforcement area is used the less efficient the structural element is, which in multiple widths $t$ is not really the case as mentioned before.

The general algorithm for this process is resumed as following in pseudo-code. A nested PSO algorithm was generated to best estimate a global optimum reinforcement area, substituting for each iteration of the PSO the best position so far found in the previous iterations into the current run (adding such optimal position into the initial generated positions of each particle at the beginning of the PSO), thus enhancing an optimization process over that given best position to accelerate in a way the optimal results.

INICIO

Generate a

for i=1:numberPSOiterations

PSO-algorithm

Initial positions and velocities (the previous best position-

) is introduced

for j=1:numberOptimIterations

Update of positions and velocities

endfor

End PSO-algorithm

Extract a best new t-values (position)

in terms of reinforcement area

End for

globalBest-tvalues=

FIN

Algorithm. 2 General algorithmic process for the nested PSO-ISR

3.1.4 GA-ISR for multi width variables

For the GA a similar consideration as for the PSO is considered regarding the range of the variables involved; both for $t_{max}$ :

(7)

(8)

And $t_{min}=0.0001cm$ for instance. As for the objective function also the reinforcing area $A(t)$ is seek to be minimized with its restriction $Ef(t)<100\%$ . The program for the application of the GA is next presented in pseudo-code:

INICIO

for i=1:numberGenerations

for j=1:numberPopulation

Decode chromosomes

Evaluate individuals (Objective function)

endfor

Create next generation

Selection (tournament)

Crossover

Mutation

Replace individuals

End for

globalBest-tvalues=

FIN

Algorithm. 3 General algorithmic process for the nested GA-ISR

(¹) The term “Simple search” here refers to an iteration process where the step length remains constant

4 Results and discussion

4.1 Results from the application of the PSO-ISR formulation

Good results were obtained for the 4t-PSO-ISR algorithm as following presented for the given parameter values:

Structural parameters:

$b(width-Section)=50cm$

${\textstyle h(height-Sectin)=80cm}$

${\textstyle rec(steel-cover)=5cm}$

${\textstyle E=2.1e6{\frac {Kg}{cm^{2}}}}$

${\textstyle f_{y}=4200{\frac {Kg}{cm^{2}}}}$

${\textstyle f'c=280{\frac {Kg}{cm^{2}}}}$

${\textstyle maxEfficiency=99.99\%}$

Load Conditions

Table. 1 Load conditions
${\textstyle Ton}$	$Ton\cdot m$	$Ton\cdot m$
${\textstyle P_{u}}$	$M_{ux}$	$M_{uy}$
${\textstyle -35.00}$	$22.00$	$45.00$
$36.00$	$25.00$	$32.00$
$-304.00$	$19.00$	$65.00$
$-46.00$	$12.00$	$76.00$
$-187.00$	$10.00$	$45.00$

PSO parameters

$\alpha =1.0$

${\textstyle c_{1}=2}$

${\textstyle c_{2}=2}$

${\textstyle dt=1.0}$

${\textstyle inertiaWeigth=1.3}$

${\textstyle \beta =0.99}$

${\textstyle number_{o}f_{p}articles=20}$

${\textstyle PSO-iteration_{number}=20}$

${\textstyle number_{total-iterations}=10}$

The evolution of the reinforcing along the iterations is as follows:

Figure 14: Progression of reinforcing area for each iteration of the PSO-ISR algorithm

With a global optimum reinforcing area of $109.63cm$ , $t-values=[0.080,0.051,0.936,0.555]cm$ , $Efficiency=95.97\%$ .

Generating the following interaction diagrams:

Figure 15: Interaction diagram in the X-axis direction for the global optima 4t-ISR

Figure 16: Interaction diagram in the Y-axis direction for the global optima 4t-ISR

It is of relevance to stress that it takes considerable time (depending on the graphics wanted and the PSO algorithm parameters) to get to a good approximation of the global optima reinforcing area. MatLab 2017b software was used for this research, therefore the execution time may be reduced with parallel computing in another faster programming language, more likely when a higher number of structural elements are to be analysed. In any case, the results obtained are much better than with the application of a 1t-ISR Steepest Gradient Descent formulation regarding the resulting optimum area as it will be compared next.

4.2 Results from the GA-ISR formulation

The Genetic Algorithm is also a viable option, although an little higher amount of computational execution time is necessary. Very similar results were obtained compared to the PSO algorithm. A computational experiment was also carried on with the GA and the results were as following presented with the same structural parameters and the next GA parameters with the same load conditions applied.

$number_{generations}=150$

${\textstyle number_{individuals}=20}$

${\textstyle population-size=30}$

${\textstyle number_{genes}=60;}$

${\textstyle prob_{mutation}=0.015}$

${\textstyle tournamente-selection-parameter=0.6}$

${\textstyle tournament-size=2}$

Figure 17: Evolution of the optimal reinforcing area with the GA-ISR

With ${\textstyle t=[0.856,0.015,0.818,0.412]}$ , ${\textstyle Efficiency=97.73\%}$ , ${\textstyle minimum-area=121cm^{2}}$ and the following interaction diagrams:

Figure 18: Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA

Figure 19: Interaction diagram in the Y-axis direction for the global optima 4t-ISR-GA

As it may be observed from the previous graphics obtained with the GA-ISR, it takes almost 50 generations or iterations to reach a very similar result from the PSO-ISR formulation, although still a little higher area. This results may vary from a structural model to another, and even from run to run. The mutation probability is of great influence in the final results. A further comparison between these two formulations is made in detail in further sections, as well as a general comparison between all of the formulations here presented for the ISR.

4.3 Results from the 1t ISR formulation with Multiple Linear Regression

Ten experimental random structural models were tested to register the number of iterations it would take for each to reach a wanted structural efficiency range given an initial width $t$ value previously estimated with the results from the application of Multiple Linear Regression formulation. The results are following presented.

Table. 2 Error estimations for the Multiple Linear Regression formula
	$cm$	$cm$	$Ton$	$Ton\cdot m$	$Ton\cdot m$	${\frac {Kg}{cm^{2}}}$		$cm$	$cm$	$cm$	$\%$
Point	b	h	$P_{u}$	$M_{ux}$	$M_{uy}$	${f'}_{c}$	$Eff$	$cover$	$t_{est}$	$t_{real}$	$error$
1	$93$	$61$	67.857	185.7708	146.0661	393	0.89	4.5	$3.85$	$1.72$	$55.3$
2	93	67	-647.700	27.3106	144.2455	164	0.89	3.5	$3.72$	$1.56$	$58.0$
3	228	34	-427.914	180.9444	121.9733	470	0.89	4.0	$1.13$	$0.95$	$15.8$
4	75	198	-273.442	190.3260	184.0664	131	0.83	3.9	2.60	$0.51$	$80.3$
5	177	35	-636.092	38.0866	73.7833	376	0.94	4.7	$3.01$	$0.37$	$87.5$
6	93	412	75.236	85.0518	62.5437	196	0.80	4.1	$0.23$	$0.12$	47.8
7	109	53	-619.890	81.5238	163.9962	531	0.97	4.0	3.35	$0.84$	74.7
8	98	93	-479.857	91.4848	175.0743	410	0.88	3.8	2.74	$0.22$	91.7
9	53	172	-610.107	95.8926	127.8633	426	0.89	4.8	3.6	$0.36$	90.0
10	58	73	-444.694	37.9420	99.0011	188	0.80	4.3	4.40	$2.06$	$53.1$

As shown in the table, the error in most cases is even higher than $50\%$ . Only two initial estimations were close to the final value of $t$ which makes the method no so quite efficient for this case. It is relevant to stress that, even though the estimation with the Multiple Linear Regression may diverge, still would take much less time to get to the optimum value of $t$ from that estimation than with the application of the PSO-4t-ISR or GA-4t-ISR, although, clearly not with less reinforcement area by any means, as it would be compared further in this text.

4.4 Results from the 1t ISR formulation with the Steepest Gradient Descent method

The same ten structural models studied for the formulation of the Multiple Linear Regression method were again taken to register the maximum number of iterations it would take for the Steepest Gradient Descent Method to converge.

Table. 3 Max number of iterations for the Steepest Gradient Descent Method with Multiple Linear Regression with an initial step length equal to 0.5
	$cm$	$cm$	$Ton$	$Ton\cdot m$	$Ton\cdot m$	${\frac {Kg}{cm^{2}}}$		$cm$	$cm$
Point	b	h	$P_{u}$	$M_{ux}$	$M_{uy}$	${f'}_{c}$	$Eff$	$cover$	$t_{est}$	$iterations$
1	$93$	$61$	67.857	185.7708	146.0661	393	0.92	4.5	$1.47$	$11$
2	93	67	-647.700	27.3106	144.2455	164	0.91	3.5	$1.28$	$10$
3	228	34	-427.914	180.9444	121.9733	470	0.94	4.0	$0.77$	$3$
4	75	198	-273.442	190.3260	184.0664	131	0.93	3.9	$0.35$	$16$
5	177	35	-636.092	38.0866	73.7833	376	0.9	4.7	$0.28$	$15$
6	93	412	75.236	85.0518	62.5437	196	0.91	4.1	$0.11$	16
7	109	53	-619.890	81.5238	163.9962	531	0.94	4.0	$0.71$	11
8	98	93	-479.857	91.4848	175.0743	410	0.96	3.8	$0.07$	16
9	53	172	-610.107	95.8926	127.8633	426	0.91	4.8	$0.20$	22
10	58	73	-444.694	37.9420	99.0011	188	0.90	4.3	$1.47$	$13$

As for the same structural model taken in the PSO-ISR and GA-ISR results, the following convergence for this Steepest Gradient Descent method was found:

Structural parameters:

$b(width-Section)=50cm$

${\textstyle h(height-Sectin)=80cm}$

${\textstyle rec(steel-cover)=5cm}$

${\textstyle E=2.1e6{\frac {Kg}{cm^{2}}}}$

${\textstyle f_{y}=4200{\frac {Kg}{cm^{2}}}}$

${\textstyle f'c=280{\frac {Kg}{cm^{2}}}}$

${\textstyle maxEfficiency=99.99\%}$

${\textstyle P_{u}=-46Ton}$

${\textstyle M_{ux}=12Ton\cdot m}$

${\textstyle M_{uy}=76Ton\cdot m}$

Figure 20: Convergence of the Steepest Gradient Descent method for the 1t-ISR

Figure 21: Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA

Figure 22: Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA

Yielding a total reinforcing area equal to $121.58cm^{2}$ , very similar to the one obtained with the GA-ISR, not so much for the PSO-ISR which yields $109.53cm^{2}$

4.5 Comparisons between the different ISR optimization formulations

As mentioned before, the performance of the GA depends a great deal on the mutation probability parameter. In order to see the effects of such parameter on the final results a total of nine experimentation runs were done, based on the same structural model used previously with ${f'}_{c}=280{\frac {Kg}{cm^{2}}},b=50cm,h=80cm,rec=5cm,P_{u}=-46Ton,M_{ux}=12Ton\cdot m,M_{uy}=76Ton\cdot m$ , changing the mutation probability from $p_{mu}=0.01$ to $p_{mu}=0.05$ for 100 generations Fig. 23.

Figure 23: Experimentation results with the GA-ISR for different values of the mutation probability from

to

The summary of the relevant results are presented in the next table:

Table. 4 Summary of the results found for different mutation probability parameters for 100 generations with the GA-ISR
		$\%$	$cm^{2}$	$cm$	$cm$	$cm$	$cm$
Run	$p_{mu}$	$Eff$	$Area_{est}$	$t_{1}$	$t_{2}$	$t_{3}$	$t_{4}$
1	0.01	98.64	$131.7$	0.002	0.240	1.234	0.510
2	0.015	99.88	$166.4$	1.795	1.112	0.561	0.154
3	0.02	97.20	$130.5$	1.764	0.009	0.052	0.799
4	0.025	98.7	$153.8$	0.072	1.081	0.620	0.917
5	0.03	99.9	$150.7$	0.828	0.910	0.325	0.835
6	0.035	99.5	$126.51$	1.030	0.231	0.713	0.374
7	0.04	90.69	$144.34$	2.151	0.437	0.070	0.512
8	0.045	99.8	$128.9$	1.818	0.050	0.012	0.512
9	0.05	99.5	$163.21$	1.313	1.237	0.861	0.013

From the results presented can been observed that there is not a clear pattern for the area found for each run, although the minimum area thus found corresponds for a mutation probability of 0.035, followed by one of 0.45. It could be recommended to use mutation probabilities of $[0.03<p_{mu}<0.45]$ for faster and better convergence.

As for the PSO-ISR formulation, it is of interest to see its efficiency of convergence for different values for the parameter $number_{total-iterations}$ mainly, higher than 10, as it was experimented previously. The following table presents the results obtained for 6 experimentation with the same structural model ${b=50cm,h=80cm,rec=5cm,f'}_{c}=280{\frac {Kg}{cm^{2}}},P_{u}=-46Ton,M_{ux}=12Ton\cdot m,M_{uy}=76Ton\cdot m$ .

Table. 5 Minimum area found for different number of iterations for the PSO-ISR
		$\%$	$cm^{2}$	$cm$	$cm$	$cm$	$cm$
Run	$N_{total_{r}uns}$	$Eff$	$Area_{est}$	$t_{1}$	$t_{2}$	$t_{3}$	$t_{4}$
1	10	98.64	$118.91$	0619	0.300	0.736	0.437
2	15	99.88	$111.55$	0.539	0.187	0.570	0.608
3	20	97.20	$112.73$	0.673	0.067	0.803	0.384
4	25	98.7	$103.47$	0.186	0.010	0.739	0.575
5	30	99.9	$99.71$	0.574	0.011	0.573	0.517
6	35	99.5	$112.70$	0.048	0.053	0.995	0.557

With its respective convergence graphs for each experiment Fig. 24

Figure 24: Experimentation results with the PSO-ISR for different number of total iterations

It is clear to see that the more the total of iterations for the PSO-ISR nested formulation, the lower the reinforcing areas, but with a much higher computational time. It could be observed from the table above that the difference between the resulting reinforcing area with 10 iterations is not really significant compared with the resultant one with 30 iterations (3 times more computational power and only a few square centimetres). It would be recommended to take 20 iterations with quite good results (much better than with the GA-ISR).

4.5.1 Additional commentaries and further research

The following step once the an optimal reinforcing area with the ISR has been obtained is to transform such area to steel bars. There might be different ways to do that, either using another optimization algorithm to optimize topologically the arrangement, position and diameter of each bar over the cross section of an element along each face or simply considering a homogeneous distribution of bars with only one diameter along each cross section face, as it would be more practical. In [13] for instance, the GA was used to optimize topologically the reinforcing steel bars once the 1t-ISR method had been used with acceptable results, but regarding a 4t-ISR method, given that a whole optimization procedure formulation has already been carried on, it would suffice to pass from this reinforcing area for each face of the cross section to a homogeneous distribution of reinforcing bars Fig. 25 minimizing thereafter the construction costs even in a greater deal.

Although the number of possible combinations of reinforcing bars could still be high, it would surely take less time to find a good optimum arrangement of reinforcing bars than employing a whole topological optimization as done by [13]. This way of reinforced concrete optimization could be a good matter of further research with its application on 3D concrete frames to better appreciate its advantages.

Figure 25: Transformation from the 4t-ISR formulation to reinforcing bars.

5 Conclusions

The different formulations for the ISR developed happen to be very simple to code for anyone having basic understanding of optimization methods. The time required is evidently higher for the meta-heuristic optimization formulations, being the PSO-ISR the most efficient and convenient to apply, due to its rapid convergence compared to the GA-ISR; as for a 1t-ISR formulation the Steepest Gradient Descent method would be the one more likely to adopt for common structural design of concrete structures, given that despite of its simplicity of formulation the results it evokes are quite acceptable, furthermore, it could be highly improved by adding different other conditions of convergence through a more accurate convergence analysis. Moreover, for the application of meta-heuristics for this ISR method, could be of greater impact in huge structural systems, being able to reduce its required execution time by adapting them to Multi-Objective optimization procedures taking on account simultaneously different structural features of a structural system, such as displacements, ductility of certain elements, costs or section dimensions.

In any case, the global application fo this method, ant its official recognition in structural engineering could enhance great advantages in the way concrete structures are designed.

BIBLIOGRAPHY

[1] Building Requirements for structural concrete and commentary, American Concrete Institute ACI 318, 2019

[2] Whitney CS. Cohen E., ``Guide for ultimate strength design of reinforced concrete. ACI J, 1956; 28(5):445-90. [Proceedings V.53].

[3] Au FTK, Bai ZZ, Effect of axial load on flexural behaviour of cyclically loaded RC columns. Comput Concr 2006: 2(4):261-84

[4] Walther R. Miehlbardt M., Dimensionnement des structures en Béton. Lausanne: Press Polytechniques et Universitaires Romandes; 1990

[5] Hernánde Montes, Gil-Martin LM, Aschheim M., “The design of concrete members subjected to uniaxial bending and compression using reinforcement sizing diagrams”, ACI Struct J 2005;102(1):150-8

[6] Hernández Montes E, Gil-Martín LM, Pasadas-Fernández M, Aschheim M., “Theorem of optimal reinforcement for reinforced concrete cross sections”, Struct Multidiscip Optim 2008;36(5):509–21.

[7] Ho Jung Lee, Mark Aschheim, Hernández Montes E, Gil-Martín LM, Pasadas-Fernández M, “Optimum RC column reinforcement considering multiple load combinations”, Struct Multidiscip Optim 2009;39;153-170.

[8] L. M. Gil-Martín, E. Hdez-Montes, M. Aschheim, “Optimal reinforcement of RC columns for biaxial bending”, Materials and Structures Journal (2010), 43:1245-1256

[9] Aschheim M, Hernández-Montes E, Gil-Martín LM (2008), “Design of optimally reinforced RC beam, column and wall sections”, J Struct Eng 134(2):169–188

[10] Bresler B., Design criteria for reinforced columns under axial load and biaxial bending, ACI Proc 57:481–490

[11] Gouwens AJ (1975), Biaxial bending simplified, reinforced concrete columns, ACI Special Publication SP-50. American Concrete Institute, Detroit

[12] Luis F. Verduzco, A. Hernández, “Optimization of reinforcing steel for the design of concrete columns”, University of Guanajuato, Bachelor Thesis to obtain the grade of Civil Engineer, Mexico, 2019

[13] Luis F. Verduzco, “Optimization of reinforcing steel with genetic algorithms for the design of concrete columns”, 14th Research Colloquium, Autonomous University of Queretaro, Mexico, 2020

[14] M. Afzal, Y. Liu, Jack C.P.Cheng, V. J.L. Gan, “Reinforced concrete structural design optimization: A critical review”, Journal of Cleaner Production-Elsevier, 2020

[15] A. Zhou, Bo-Yang Qu, Hui Li, Shi-Zheng Zhao, P. Nagaratnam, Q. Zhang, “Multiobjective evolutionary algorithms: A survey of the state of the art”, Journal of Swarm and Evolutionary Computation, doi:10.1016/j.swevo.2011.03.001

Latest revision as of 21:31, 14 February 2021

Abstract

1 Introduction

2 Background

3 Analysis approaches for optimization with the ISR

3.1 Optimization approaches for the ISR method

3.1.1 Single variable width value t of the ISR

Multiple Linear Regression:

Steepest Gradient Descent method

3.1.2 Multiple variables of width t values of the ISR

3.1.3 PSO-ISR for multi width variables

3.1.4 GA-ISR for multi width variables

4 Results and discussion

4.1 Results from the application of the PSO-ISR formulation

4.2 Results from the GA-ISR formulation

4.3 Results from the 1t ISR formulation with Multiple Linear Regression

4.4 Results from the 1t ISR formulation with the Steepest Gradient Descent method

4.5 Comparisons between the different ISR optimization formulations

4.5.1 Additional commentaries and further research

5 Conclusions

BIBLIOGRAPHY

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@@ Line 1: / Line 1: @@
+<!-- metadata commented in wiki content
-==“El Perfil idealizado de acero”, un método computacional para el diseño de acero de refuerzo en estructuras de concreto==
+==“The Idealized Smeared Reinforcement (ISR) method for the optimization of concrete sections: a survey of the state-of-the-art and analysis of potential computational approaches”==
-==“The Idealized steel profile”, a computational method for the design of reinforcing steel in concrete structures==
+<span style="text-align: center; font-size: 75%;">Luis Fernando Verduzco Martínez[1]<span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]]
-<span style="text-align: center; font-size: 75%;">Luis Fernando Verduzco Martínez[1], Alejandro Hernández Martínez, Humberto Esqueda Oliva [3]
 </span>
@@ Line 11: / Line 10: @@
 <span style="text-align: center; font-size: 75%;">[1]lf.verduzcomartinez@ugto.mx
-[2] alejandro.hernandez@ugtomx.onmicrosoft.com
+</span>
-[3] esquedah@yahoo.com  <span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]]   Resúmen
+Autonomous University of Queretaro<span id="fnc-2"></span>[[#fn-2|<sup>2</sup>]].
+-->
-Se presenta el desarrollo y modelación matemática y numérica de un nuevo método computacional llamado “El perfil idealizado de acero” para el diseño de acero de refuerzo en estructuras de concreto reforzado. En el presente trabajo se mostrará su aplicación para columnas de concreto prismáticas sólidas de sección rectangular y circular sometidas a flexo-compresión biaxial, de acuerdo a los componentes y criterios de diseño y análisis para este tipo de estructuras, adaptando la modelación numérica para ambas geometrías, así como respetando los criterios que establecen las <span id='citeF-3'></span>[[#cite-3|[3]]].
+==Abstract==
-Se probará al final dicho método para el diseño de diversos modelos experimentales de concreto reforzado bajo ciertas combinaciones de cargas a partir de un análisis estructural previo, haciendo comparaciones de los resultados de acuerdo a eficiencia estructural y geometría en la variación de varios paramétros de diseño y objetivos específicos.
+The present work aims to define formally the method termed as ''“Idealized Smeared Reinforcement” (ISR)'' for optimization in the design of reinforcing steel in concrete structures, its boundaries, background and potentials. Such method has been extensively used implicitly in many different studies of reinforced concrete structures related with optimization and mechanical structural behaviour, but it has not yet been formally established as a method itself even though it represents a great solution approach when designing optimally concrete sections, which is a tendency so vital nowadays for sustainability in construction projects. A general survey of such method of the ISR will be presented on in this document, presenting relevant background research related to the use of the method and optimization of reinforcing steel in general for concrete structures, thereafter proposals of different optimization methods, both classical optimization and meta-heuristic methods will be regarded as possible approaches to apply the ISR, specifically Gradient Descent Optimization methods when one variable for the ISR is considered and meta-heuristics for more than one variable involved given their versatility and flexibility to adapt to different problems, particularly the Particle Swarm Optimization PSO method and the Genetic Algorithm GA. At the end, these solution approaches for the application of the ISR method will be compared for the testing of rectangular solid geometries with different analysis parameters in order to show how adaptable and feasible such ISR method might be using a proper optimization algorithm and analysis for its approach when designing reinforcing steel.
-Palabras clave: criterios de diseño, acero de refuerzo, concreto reforzado, formulación matemática, métodos computacionales    Abstract
+Keywords: '''Classical Optimization, Meta-heuristic Optimization ,Reinforcing Steel, Computational Methods, Concrete Structures, Idealized Smeared Reinforcement'''
-It is presented the mathematical and numerical devolpement and modelling of a new computational method named “The idealized steel profile” for the design of reinforcing steel in concrete structures. Hereby it will be shown its application for solid prismatic concrete columns of both rectangular and circular cross sections subjected to flexo-compression stresses in two directions, according to the design components and analysis criteria stablished for this type of structures following the <span id='citeF-3'></span>[[#cite-3|[3]]].
-At the end such method will be tested for various experimental structural models under certain load combinations from a structural analysis made previously, making comparisons between the results according to certain design criteria and requirements such as structural efficiency, costs and geometry following various specific design goals.
-keywords: design criteria, reinforcing steel, reinforced concrete, mathematical formulation, computational methods
 <span id="fn-1"></span>
-<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) Universidad de Guanajuato, Aula CIMNE-UG, Guanajuato, Guanajuato, México 2019. </span>
+<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) Autonomous University of Queretaro, Faculty of Engineering</span>
-==1 Introducción==
-En los proyectos de diseño y análisis estructural de elementos de concreto reforzado se requiere de análisis numéricos que evocan muchas iteraciones para llegar a un resultado que sea tanto estructural como económicamente eficiente; con el uso de métodos numéricos podemos llegar a estas soluciones muy rápidamente.
-El presente trabajo está orientado precisamente a usar programación y métodos numéricos en la practica diaria en el diseño de análisis de infraestructura civil, en éste caso se determinará el acero de refuerzo necesario en columnas de concreto para que resistan ciertas condiciones de carga.
-En la práctica común, la determinación del acero de refuerzo necesario en elementos de concreto se realiza mediante aproximaciones empíricas basadas en la experiencia; el proceso se vuelve bastante tardado y cansado.
-Expondremos pues, el desarrollo de un nuevo método computacional, cuya principal función es la de determinar una configuración de varillas de acero de refuerzo para elementos de concreto sometidos a flexo-compresión, que resulte la más eficiente estructuralmente o la más económica. Para ello el diseñador únicamente tendrá que definir las dimensiones en función de los requerimientos estructurales y/o de espacio. Pueden ser estos elementos columnas, pilas, pilotes o incluso dados de cimentación, con secciones circulares y rectangulares sólidas. Usualmente, por cuestiones constructivas se requieren geometrías de secciones que no sean complejas y que sean uniformes.
-Como resultdos de la programación de este método se obtienen tanto propuestas como recomendaciones de disposiciónes y arreglos del refuerzo, es decir, no solo se obtienen las opciones de número de varillas con su respectivo diámetro, sino también el acomodo que pudieran tener éstas e indicando además la eficiencia crítica estructural para cada arreglo. El programa selecciona ya sea el arreglo más eficiente etructuralmente o el más económico (de acuerdo a las preferencias del constructor) para cada columna. Diversos softwares comerciales de análisis y diseño estructural no hacen esto, de ellos se obitenen únicamente resultados de un análisis con las geometrías y el refuerzo que el diseñador propone inicialmente.
-Las aplicaciones y alcances que puede tener éste enfoque de análisis son de gran potencial en el diseño de estructuras de concreto, ya que podría adaptarse a cualquier geometría, cambiando la forma en que se diseñan y analizan elementos de concreto reforzado de hoy en adelante, permitiéndo optimizar los volúmenes de materiales usados, tal que los costos de construcción sean mínimos.
-==2 Objetivo==
-Diseñar un programa que sea capaz de determinar los posibles arreglos de acero de refuerzo para cualquier número de columnas que se trate, así como seleccionar la opción más eficiente o barata para cada columna, calculando los volumenes de materiales requeridos para finalmente integrar la cotización del proyecto.
-==3 Metodología general==
-Se presenta la siguiente metodología de manera general para el desarrollo de las funciones del programa de optimización (consideraciones para análisis estructural, optimización, exportación de reportes y resultados).
-* Desarrollar la formulación matemática y numérica para la programación en un lenguaje de programación, tanto para secicones de geometría ciruclar y rectangular, de acuerdo a los criterios de diseño y análisis que establecen las <span id='citeF-3'></span>[[#cite-3|[3]]]
-* Hacer un estudio de precios unitarios y costos de proveedores de acerero, así como de equipo y mano de obra en general para armar el acero de refuerzo
-* Insertar estos costos en el programa al momento del diseño
-* Realizar las simulaciones y ejecución de experimentos requeridos para analizar el comportamiento de diferentes modelos estructurales de columnas ante diferentes condiciones de carga, haciendo comparaciones entre los diversos resultados referente a costos finales, eficiencias estructurales y geometrías
-==4 Materiales y métodos==
-===4.1 Métodos numéricos computacionales con aplicación a diseño de estructuras===
-Los métodos numéricos constituyen técnicas mediante las cuales es posible formular problemas matemáticos con resolución mediante operaciones aritméticas. Con el desarrollo de las computadoras digitales eficientes y rápidas, el papel de los métodos numéricos en ingeniería aumento considerablemente.
-Antes de las computadoras se gastaba bastante energía en la técnica misma de solución, en lugar de usarla en la definición del problema. Esto debido al trabajo monótono que se requería para obtener resultados numéricos aceptables. Ahora, al usar la potencia de una comutadora se obtienen soluciones directamente, sin tener que recurrir a simplifaciones o soluciones análiticas que pueden resultar complejas. Gracias a esto, es posible dar más importancia ahora a la interpretación de la solución y su incorporación al sistema total del problema.
-El desarrollo de los métodos numéricos más famosos por aquellos grandes científicos, matemáticos e ingenieros han servido de inspiración para muchos investigadores en la ingeniería y ciencia más que para desarrollar la aplicación de los existentes a problemas, desarrollar los suyos propios con la potencia de las computadoras.
-Se han desarrolaldo desde el uso de las computadoras innumerables métodos computacionales en el análisis y diseño de las estructuras. Tal vez el más famoso y usado de todos es el Método del Elementos Finito <span id='citeF-6'></span>[[#cite-6|[6]]] cuyas aplicaciones van desde el análisis elástico de elementos estructurales, hasta la simulación numérica de fluidos. Desde su primera aparición en 1956 el mismo método ha evulocionado y desarrollado para crear diferentes versiones del mismo dependiendo de su aplicación.
-Sin embargo, de manera general, cada vez que se quiere hacer una mejora en un análisis y/o diseño estructural en cuánto a rápidez y aproximaciones de soluciones deseadas se crea un nuevo método numérico computacional. Ya que con solo hecho de implementar la potencia de una computadora para encontrar una solución de un problema se requiere de un ajuste de datos, desarrollo de modelos matemáticos, encontrar coeficientes o iterar operaciones aritméticas, y eso, en sí, lo vuelve un método numérico computacional.
-==5 Antecedentes de construcción con concreto reforzado==
-Tan solo desde el punto de vista ambiental, en la construcción, la tecnología del concreto armado se antepone a filosofías de sustentabilidad; la manipulación de aceros de construcción ASTM A615 (barras lisas y corrugadas) en forma desmedida desde la concepción de los diseños de refuerzos previos hasta su empleo en la construcción origina altos grados de desperdicios de acero, por efectos de cortes, doblados ineficinetes, o simplemente por una falta de diseño óptimo de este en las estructuras, tal que cumpla con todas las restricciones de eficiencia estructural al mismo tiempo que se use el mínimo posible.
-===5.1 El acero de refuerzo en la industria de la construcción===
-En obras de concreto armado, se generan entre el 7% al 27% de desperdicios de aceros ASTM A615 en forma de barras corrugadas. Estas barras se fabrican tanto de acero laminado en caliente como acero trabajado en frío. Los diámetros usuales de barras producidas en México varían de <math>\frac{1}{4}pulg.</math>, a <math>1\frac{1}{2} pulg.</math>, aunque algunos productores han fabricado barras corrugadas de <math>\frac{5}{16}pulg</math>, <math>\frac{5}{33}pulg</math> y <math>\frac{3}{16}pulg</math>.
-El acero de refuerzo se produce además en otras más diversas formas para todo tipo de elementos de concreto, también las hay barras lisas, barras de acero helicoidales para la forticiación y reforzamiento de rocas, taludes y suelos a manera de perno de fijación. Incluso se fabrican como mallas cuadriculares (transpuestas en doble sentido) principalemtne para estructuras de losas y cimentaciones.
-El tener un mejor control en la cantidad de acero que se usa para este tipo de estructuras en la construcción es primordial hoy en día para mitigar los impactos ambientales y económicos a que se enfrenta un país, pues la industria de la construcción siempre será de gran peso e influencia en cualquier región del mundo. Es imprescindible el desarrollo de tecnología que pueda favorecer el control en diseño y construcción de estas estructuras.
-==6 Columnas de concreto reforzado==
-Es el elemento estructural vertical empleado para sostener la carga de la edificación o estructura principal que se trate. Se utiliza principalemente por la libertad que proporciona para distribuir espacios al tiempo que cumple con la función de soportar el peso de la construcción.
-Las formas, armados y las específicaciones de las columnas estarán en razón directa del tipo de esfeurzos y condiciones en general a que estas sean expuestas.
-===6.1 Clasificación de columnas de concreto===
-Las columnas de concreto reforzado pueden clasificarse como columnas cortas, intermedias y columnas largas.<span id="fnc-2"></span>[[#fn-2|<sup>1</sup>]] Es necesario tener esto en cuenta para entender cómo funciona el programa, ya que el análisis de las columnas preceden del mismo mecanismo de falla de estos tipos de columnas.
-Columnas cortas: La carga que pueden soportar está regida por las dimensiones de su sección transversal y por la resistencia de los materiales de que está construida.
-Columnas intermedias: La falla es por una combinación de aplastamiento y pandeo.
-Columnas largas: La capacidad de carga axial en estás columnas se ve reducida a causa de los momentos secundarios resultates debidos a la deformacion por flexión de la columna.
-El presente trabajo trata únicamente con columnas cortas.
 <span id="fn-2"></span>
-<span style="text-align: center; font-size: 75%;">([[#fnc-2|<sup>1</sup>]]) Jack C. McCormac, Rusell H. Brown, “Diseño de concreto reforzado”, 14va edición, Alfaomega(2015), p.257.</span>
+<span style="text-align: center; font-size: 75%;">([[#fnc-2|<sup>2</sup>]]) Centro Universitario, Cerro de las Campanas s/n, Cp. 76010, Santiago de Querétaro, Querétaro, México, 2021</span>
-===6.2 Tipos de columnas de concreto===
+==1 Introduction==
-El tipo de columnas de concreto depende de la forma de la sección  transversal, así como de su tipo de refuerzo <span id="fnc-3"></span>[[#fn-3|<sup>1</sup>]]. Hay columnas de concreto con refuerzo en forma de varillas longitudinales de acero, con restricción lateral (helicoidal o con estribos cerrados).
+Ever since Whitney and Cohen <span id='citeF-2'></span>[[#cite-2|[2]]] many years ago in 1956 proposed a guide to design reinforcing steel in concrete structures subject to axial force and bending moment by ultimate strength based on the ACI code many different approaches have taken place from that point on to design optimally reinforced concrete sections. What Whitney and Cohen presented then was a series of charts based on N-M interaction diagrams normalized considering symmetric and uniform reinforcement over a cross section. In recent years however, new more realistic considerations for the design and analysis have been carried on, for instance, influence of confinement and flexural behaviour <span id='citeF-3'></span>[[#cite-3|[3]]], or non-symmetric reinforcement <span id='citeF-4'></span>[[#cite-4|[4]]] in which a particular solution of a problem is provided for elements subjected to flexure-compression stresses. Also, the concept of strength design has been challenged by new design approaches <span id='citeF-5'></span>[[#cite-5|[5]]] which generate a family of solutions for arbitrary combinations of imposed axial load and moment, requiring even a non-symmetric distribution of reinforcement over the cross section. Moreover, theorems for optimal section reinforcement have been also developed <span id='citeF-6'></span>[[#cite-6|[6]]] based on the understanding of all these studied optimal solutions, encompassing a general formulation to determine the minimum total reinforcement area required for adequate resistance to axial load and moment.
-Generalmente las columnas con estribos tienen una sección transversal rectangular, mientras que las columnas zunchadas suelen tener sección trasnversal circular, pero también pueden fabricarse con secciones rectangulares, octogonales, entre otras formas. Las espirales en comparación de los estribos incrementan en mayor medida la resistencia debido al aumento de la eficiencia del efecto de confinamiento del refuerzo transversal; aunque, en algunos casos, aumentando también los costos de construcción.
+Through all of these conducted research works the idealization of a continous distributed plate along the faces of a rectangular cross section has already been created. But it is specifically in <span id='citeF-9'></span>[[#cite-9|[9]]] by Anschheim et. al. when the concept of a ''“Smeared reinforcement”'' idealization is given formally, in which an optimization for the values of each of the four continuous plates areas (smeared reinforcement) distributed over a rectangular cross section (upper, lower, right and left faces) is carried on, establishing its due design restrictions according to the ACI-318 code, thus finding minimum reinforcement required areas for a given load combination related to the orientation of the axis direction of the cross section geometry. In recent years, this very approach made by officially by Anschheim et. al. in 2008 has also been carried on by other authors <span id='citeF-12'></span>[[#cite-12|[12]]] although differing by means of solutions generated or by the analysis method itself.
-<span id="fn-3"></span>
+L. Verduzco <span id='citeF-12'></span>[[#cite-12|[12]]] made a research related directly with this very topic in which a called ''“Idealized Steel Profile”'' concept was introduced, idealizing the reinforcement steel as a continuous steel PTR structural profile embedded into the concrete element with a uniform width <math>t</math> from which an optimization analysis was carried on to seek an optimal arrangement of reinforcing steel bars over the cross section element (either the most economical or structurally efficient), with certain restrictions such as equal diameter for the reinforcing bars according to the normative of the region. What it is also relevant to stress from this research is the purely mathematical analysis approach that was made, by defining blocks of stress and strain over the section as the depth value of the neutral axis varies, thus defining equations of resistance for different cases in which the neutral axis depth might have been located. While on one hand Anschheim et. al. generated a family of solutions given by means of percentage of reinforcing steel with respect of the cross section element net area, on the other L. Verduzco designed a mathematical approach through a computational optimization method to seek an optimal reinforcement area from which to transform directly to reinforcing steel bars arrangements. Even though both approaches differ one from another, the idea is the same, and it is of special importance to promote this concept and method by giving it a whole research work focused on its boundaries, limitations, potentials, different possible analysis and optimization approaches.
-<span style="text-align: center; font-size: 75%;">([[#fnc-3|<sup>1</sup>]]) Jack C. McCormac, Rusell H. Brown, “Diseño de concreto reforzado”, 14th edition, Alfaomega (2015), p. 258.</span>
-===6.3 Conceptos generales de diseño===
+This very optimization domain presented hereby for RC structures design corresponds to a vital element of the systematic flow of research for RC structures suggested in various literature <span id='citeF-14'></span>[[#cite-14|[14]]] '''Fig. [[#img-1|1]]''' in which this problem is directly related to the stage of ''“Integration of promising techniques”'' for the improvement of performance of optimization strategies in detailed design of RC structures. Therefore, it can be justified from this perspective the importance and relevance of this such research work, to strengthen this very scenario for optimization in RC structures.
-Es necesario entender las hipótesis empleadas en el diseño de concreto reforzado, para comprender de donde provienen algunos factores, y números incluídos en el análisis.
+<div id='img-1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|-
+|[[Image:Review_942980062263-sys_flow_rc.png|422px|System flow for the area of optimization in RC structures,<span id='citeF-14'></span>[[#cite-14|[14]]].]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" | '''Figure 1:''' System flow for the area of optimization in RC structures,<span id='citeF-14'></span>[[#cite-14|[14]]].
+|}
-====6.3.1 Carga axial====
+==2 Background==
-Si se somete un espécimen de concreto simple con relación alto ancho de dos a una carga axial de compresión, la carga máxima se llegará a una deformación unitaria del orden de 0.002. Arbitrariamente se toma como 100 por ciento la resistencia de un espécimen con relación de esbeltez igual a dos. <span id='citeF-4'></span>[[#cite-4|[4]]]
+Resuming the content of the most recent research related with optimization of reinforcing steel, it is of vital relevance to stress the development and introduction of a so called ''Reinforcement Sizing Diagram (RSD)'', formulated in <span id='citeF-5'></span>[[#cite-5|[5]]], which displays a minimum required steel reinforcement area in one axis for a respective neutral axis depth in the other axis, given a combination of axial load and moment '''Fig. [[#img-2|2]]'''. Another relevant development is the called ''``Load Combination Reinforcement Diagram (LCRD)'' <span id='citeF-7'></span>[[#cite-7|[7]]] which on the other hand plots reinforcement solutions in a two-dimensional space '''Fig. [[#img-3a|3a]] (left)''' defined by the coordinates <math>(A_{s}, A^{'}_{s})</math> corresponding to the reinforcement area on each of the faces of a rectangular cross section element '''Fig. [[#img-3|3]] (right)''' allowing an engineer to easily determine an optimal reinforcement solution from given load combinations on structural concrete elements subject to biaxial flexure-compression stresses. Such (LCRD) are obtained by collecting different values of reinforcement area from a (RSD) corresponding to different values of the axis depth <math>c</math>. It is what in multi-objective optimization would be called a Pareto Front.
-Para relaciones de esbeltez mayores que dos, la resistencia baja, hasta llegar al 85 por ciento, aproximadamente. Por consiguiente, la resistencia de un elemento de concreto simple sujeto a compresión axial puede estimarse como el producto del 85 por ciento del esfuerzo medido en cilindro de control <math>{f'}_{c}</math>, ensayado en las mismas condiciones, multiplicado por el área de la sección transversal del elemento. Este factor de reducción, 0.85, es solo un promedio de resultados de ensayes en miembros colados verticalmente.
+<div id='img-2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|-
+|[[Image:Review_942980062263-rsd.png|271px|A typical Reinforcement Sizing Diagram for a given load combination. <span id='citeF-5'></span>[[#cite-5|[5]]]]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" | '''Figure 2:''' A typical Reinforcement Sizing Diagram for a given load combination. <span id='citeF-5'></span>[[#cite-5|[5]]]
+|}
-Cuando se le adiciona el refuerzo longitudinal a un espécimen de concreto simple y se le añade también el refuerzo transversal necesario para mantener las barras longitudinales en su posición durante el colado, la carga máxima se obtendrá ahora a una deformación unitaria del orden de 0.0021. La falla, en cambio, se produce a una deformación unitaria de 0.003 o 0.004, si el ensaye es de corta duración.
+It is important to stress that in the development of the LCRD was also introduced the concept of asymmetrical reinforcement uniformity over an element cross-section, based on studies which concluded that for optimal solutions an asymmetric distribution of reinforcing steel was more likely to take place <span id='citeF-7'></span>[[#cite-7|[7]]], therefore different faces of an element cross-section were picked to assign them different values of reinforcing steel area, as seen in '''Fig. [[#img-3|3]] (right)''', and to have a better approximation for a minimum required reinforcement, taking reference from the such smeared reinforcement concept made firstly in <span id='citeF-9'></span>[[#cite-9|[9]]].
-La resistencia adicional sobre la de un prisma de concreto simple debido a la adición del refuerzo longitudinal en compresión se puede estimar como el producto del área del acero por el esfuerzo de fluencia <math>f_{y}</math>
+<div id='img-3a'></div>
+<div id='img-3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|-
+|[[Image:Review_942980062263-lcrd.png|242px|]]
+|[[Image:Review_942980062263-section_lcrd.png|145px|A typical Load Combination Reinforcement Diagram at the left and its corresponding reinforced cross section element from which Aₛ and A<sup>'</sup>ₛ are taken as smeared reinforcement steel. <span id='citeF-7'></span>[[#cite-7|[7]]]]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" | '''Figure 3:''' A typical Load Combination Reinforcement Diagram at the left and its corresponding reinforced cross section element from which <math>A_{s}</math> and <math>A^{'}_{s}</math> are taken as smeared reinforcement steel. <span id='citeF-7'></span>[[#cite-7|[7]]]
+|}
-Por lo tanto, la resistencia o carga máxima a compresión que un prisma de concreto con refuerzo longitudinal puede alcanzar esta dada por la siguiente expresión:.
+Such diagrams have been used in many further research works <span id='citeF-8'></span>[[#cite-8|[8]]], in which even though the typical approaches and hypotheses first formulated many years ago to design reinforced concrete sections <span id='citeF-10'></span>[[#cite-10|[10]]], <span id='citeF-11'></span>[[#cite-11|[11]]] (for rectangular geometries) are still taken on account, other different assumptions are carried on in order to optimize the reinforcing steel, such as asymmetric reinforcement following the research made by <span id='citeF-9'></span>[[#cite-9|[9]]] as well '''Fig. [[#img-4|4]]''', but focusing merely on design outcomes, addressing more flexible design considerations.
-<span id="eq-1"></span>
+<div id='img-4'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|
+|[[Image:Review_942980062263-steel_plates.png|207px|Idealization of reinforcing steel proposed by <span id='citeF-8'></span>[[#cite-8|[8]]] for each axis direction of a rectangular column.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
-|-
+| colspan="1" | '''Figure 4:''' Idealization of reinforcing steel proposed by <span id='citeF-8'></span>[[#cite-8|[8]]] for each axis direction of a rectangular column.
-| style="text-align: center;" | <math>  P_{oc} = (0.85)f{'}_{c}(A_{c}-A_{s})+f_{y}(A_{s}) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
 |}
-Cabe resaltar que se está considerando el área de acero en la contribución de la resistencia del concreto a compresión (primer término de la  Ecuación [[#eq-1|1]], que puede influir considerablemente en los cálculos.
+Aschheim et. al. <span id='citeF-9'></span>[[#cite-9|[9]]] used non-linear conjugate gradient search methods to obtain optimal solutions for these such mentioned smeared reinforcing steel areas, applied to a structural model for different values of rotation of the structural rectangular cross section <math>\phi </math> and ratio between axial force and bending moment <math>\xi </math> '''Fig. [[#img-5|5]]'''. The results generated were plotted in a contour graph '''Fig. [[#img-6|6]]''' for the minimum reinforcement areas thereby found, by means of steel area percentage in relation with the concrete area, which may be of great use to standardize design precesses.
-Por otra parte, la carga máxima a tensión que el elemento puede soportar está dada por:
+<div id='img-5'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-<span id="eq-2"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+|[[Image:Review_942980062263-inclined_crossection.png|166px|General formulation reference of cross section for the analysis and calculation of resistance of a reinforced concrete element. <span id='citeF-9'></span>[[#cite-9|[9]]]]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
-|-
+| colspan="1" | '''Figure 5:''' General formulation reference of cross section for the analysis and calculation of resistance of a reinforced concrete element. <span id='citeF-9'></span>[[#cite-9|[9]]]
-| style="text-align: center;" | <math>  P_{ot} = (f_{y}(A_{s})) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
 |}
-En la que solo interviene el acero, ya que el concreto se agrieta y no contribuye a la resistencia.
+<div id='img-6'></div>
-====6.3.2 Flexo-compresión====
-====Consideraciones generales de análisis====
-<span id="fnc-4"></span>[[#fn-4|<sup>1</sup>]]
-<br />a. Se aborda el análisis mediante propuestas de geometría de la sección de la columna que permanecerán fijas, haciendo variar únicamente el área de acero de refuerzo.
-<br />b. Un elemento puede alcanzar su resistencia bajo innumerables combinaciones de carga axial y momento flexionante. Estas combinaciones varían desde una capa axial máxima (<math>P_{oc}</math>) sin momento de flexión, hasta un momento (<math>M_{o}</math>) sin fuerza axial.
-<br />c. El lugar geométrico de las combinaciones de carga axial y momento flexionante con las que un elemento puede alcanzar su resistencia se representa gráficamente por medio de un ''diagrama de interacción" Figura [[#img-1|1]].
-<br />d. Cuando al aumentar la carga externa, el momento y la carga axial crecen en la misma proporción, la historia de carga queda representada por una recta desde el origen, con una pendiente igual al cociente <math display="inline">P/M=1/e</math>. Figura [[#img-1|1]].
-<br />e. Existen solo dos modos principales de falla de elementos sujetos a flexo-compresión: falla en compresión y falla en tensión:
-<br />1. Falla a compresión: Se produce por aplastamiento del concreto. El acero del lado más comprimido fluye, en tanto que el lado opuesto no fluye en tensión
-<br />2. Falla en tensión: Se produce cuando el acero de un lado fluye en tensión antes de que se produzca el aplastamiento del concreto en el lado opuesto más comprimido.<p>
-</p>
-<br />f. El diagrama de interacción de un elemento puede obtenerse a partir de las hipótesis descritas para el cálculo de la resistencia de elementos sujetos a flexión pura, considerando que ahora la sumatoria de fuerzas debe ser igual a la carga P aplicada.<p>
-</p>       NOTA: Lo anterior aplica para cualquier geometría.
-<div id='img-1'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig4_1.png|396px|Diagrama de interacción típico. (Dibujo propio.)]]
+|[[Image:Review_942980062263-contourplot_minimumarea.png|399px|Contour graph for minimum reinforcement area corresponding to different values of ξ and ϕ. <span id='citeF-9'></span>[[#cite-9|[9]]]]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 1:''' Diagrama de interacción típico. (Dibujo propio.)
+| colspan="1" | '''Figure 6:''' Contour graph for minimum reinforcement area corresponding to different values of <math>\xi </math> and <math>\phi </math>. <span id='citeF-9'></span>[[#cite-9|[9]]]
 |}
-====Fórmula de Bresler====
+Based on this contour graphs, or for the generation of such, constraints may be imposed regarding the type of reinforcement sought (symmetrical or non-symmetrical) either for all faces or opposite ones. By setting equal values of <math>A_{st}=A_{sb}</math> and <math>A_{sl}=A_{sr}</math> corresponding to reinforcement areas on opposite faces, two variables to minimize are obtain, or on the other hand, by setting all areas equal to one another obtaining only one variable to minimize. This method of optimization is capable of saving as much as up to 10% in relation with the conventional methods for design and conventional restrictions for reinforced columns of symmetric and uniform steel reinforcement based on the <span id='citeF-1'></span>[[#cite-1|[1]]] code, depending also on the decision made by the engineer or contractor.
-Bresler desarrolló una expresión muy simple para columnas rectangulares para calcular los valores máximos de la carga de compresión que actúa a excentricidades <math>e_{x}</math> y <math>e_{y}</math> en secciones rectangulares con refuerzo simétrico. Ecuación [[#eq-3|3]]. Dicha solución surgió ante la necesidad de evitar realizar los cálculos para determinar una superficie de interacción<span id="fnc-5"></span>[[#fn-5|<sup>2</sup>]], reduciendo el problemas a una combinación de soluciones más simples; dos de flexo-compresión en un plano de simetría y una de compresión axial.
+Regarding other optimization approaches L.Verduzco <span id='citeF-12'></span>[[#cite-12|[12]]] idealization of the such mentioned ''“Smeared reinforcement”'' as a continuous steel structural profile of uniform width <math>t</math> which he referred to as ''``The Idealized Steel Profile'', embedded into a concrete element from which a mathematical approach was carried on through a constant length step computational optimization method for both rectangular and circular solid cross sections '''Fig. [[#img-7|7]]''' presented advantages of execution time when big geometries are analysed, due to the mathematical analysis in which only a few conditions and operations are required in order to calculate the resistance of a given value of <math>t</math> (steel profile width) for a respective depth of the neutral axis. Even though such mathematical approach may not be quite applicable to circular cross sections due to the complexity it enhances in the analysis. For designs fully based on normative where so many restrictions are imposed this approach might be of great potential for any geometry, the problem comes up when seeking optimal unsymmetrical reinforcement when a bending moment condition is significantly mayor for one of the axis in a rectangular section, for which such uniform idealization of steel does not really guarantee a good approximation for the required reinforcement area due tu the constant variation of the width <math>t</math> on both axis directions.
-<span id="eq-3"></span>
+<div id='img-7'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|
+|[[Image:Review_942980062263-Fig6.png|129px|]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|[[Image:Review_942980062263-fig23.png|181px|The ''“Idealized Steel Profile”'' method <span id='citeF-12'></span>[[#cite-12|[12]]] for rectangular and circular solid cross sectioned concrete elements]]
-|-
+|- style="text-align: center; font-size: 75%;"
-| style="text-align: center;" | <math>  \frac{1}{P_{R}}=\frac{1}{P_{rx}}+\frac{1}{P_{ry}}-\frac{1}{P_{oc}} </math>
+| colspan="2" | '''Figure 7:''' The ''“Idealized Steel Profile”'' method <span id='citeF-12'></span>[[#cite-12|[12]]] for rectangular and circular solid cross sectioned concrete elements
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
 |}
-Donde: <math>P_{R}</math>=Carga normal resistente sobre la sección transversal del elemento actuando a excentricidades <math>e_{x}</math> y <math>e_{y}</math>.
+==3 Analysis approaches for optimization with the ISR==
-<math>P_{rx}</math>=Carga normal resistente a una excentricidad de <math>e_{y}</math>.
+Different formulations for the analysis of a problem with the ISR method might be carried on, either regarding mechanical analysis (flexure-compression for short or slender elements) possibly using the Bresler's formula <span id='citeF-10'></span>[[#cite-10|[10]]] or by computing an interaction surface diagram by rotating the cross section element '''Fig. [[#img-8|8]]''' on its own longitudinal axis (depending on the required approximation accuracy); or with relation of the optimization analysis approach adopted, merely with classical optimization or mathematical programming methods, such as Gradient Descent Based methods as formulated by Anschheim et. al. <span id='citeF-9'></span>[[#cite-9|[9]]] for two steel area values (or variables), or when using the Idealized Steel Profile proposed by L. Verduzco <span id='citeF-12'></span>[[#cite-12|[12]]] for only one variable a simple numeric search method might be feasible for small cross section columns. Meta-heuristic algorithms of optimization might also be viable to adopt, for instance, taking Evolutionary Algorithms, Swarm Algorithms, Multi-objective optimization and such. Meta-heuristic optimization algorithms might be more feasible than classical optimization methods when formulating more than one different steel area values or when massive cross section elements take place, given the complexity a mathematical optimization approach may enhance.
-<math>P_{ry}</math>=Carga normal resistente a una excentricidad de <math>e_{x}</math>.
+<div id='img-8'></div>
-La Ecuación [[#eq-3|3]] verifica los ensayes disponibles dentro del 20 por ciento de aproximación, y representa una familia de planos que aproximan los puntos de la superficie de interacción. Figura [[#img-2|2]].  <div id='img-2'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig4.png|420px|Superficie de interacción típica. Tomada de:]]
+|[[Image:Review_942980062263-probform.png|334px|Reference system for the rotation of a rectangular cross-section when computing a surface diagram. <span id='citeF-9'></span>[[#cite-9|[9]]]]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 2:''' Superficie de interacción típica. Tomada de:
+| colspan="1" | '''Figure 8:''' Reference system for the rotation of a rectangular cross-section when computing a surface diagram. <span id='citeF-9'></span>[[#cite-9|[9]]]
 |}
-Otro enfoque para analizar columnas sujetas a carga axial y flexión en dos planos Ecuación [[#eq-4|4]] a es partir de la cual se desprende otra expresión simplificada para calcular la eficiencia de un elemento cuando la relación <math>P_{R}/P_{oc}<0.1</math>. Ecuación [[#eq-9|9]] (que es la se presenta en las NTC-2017 [ntc17]) a las cuáles se hará referencia a continuación.
-<span id="eq-4"></span>
+There has also been research and development regarding general theorems for an optimal reinforcement design <span id='citeF-6'></span>[[#cite-6|[6]]], based on all of the research previously mentioned involving the idealization of Smeared Reinforcement, which states that the minimum total required reinforcement area for adequate resistance to axial load and moment can be identified as the minimum admissible solution among five discrete analysis cases from the infinite set of potential solutions. Such theorem is of great potential when designing for common parameters most often used in structural engineering practice and it is also of importance to mention its existence.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>   (\frac{P_{u}-P_{nb}}{P_{ot}-P_{nb}})+(\frac{M_{ux}}{M_{nbx}}^{1.5})+(\frac{M_{uy}}{M_{nby}}^{1.5})=1.0 </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
-|}
-Donde: <math>P_{u}</math>=Carga axial nominal aplicada
+===3.1 Optimization approaches for the ISR method===
-<math>P_{nb}</math>=Resistencia nominal a carga axial en la condición balanceada.
+As following, different optimization methods of possible potential for the application of the ISR, such as the Particle Swarm Optimization Algorithm along with the GA for multiple values of width <math>t</math> to optimize, the Multiple Linear Regression method for a single variable of width <math>t</math> to estimate an initial value of with <math>t</math> from which to start iterating in a ''Simple search''<span id="fnc-3"></span>[[#fn-3|<sup>1</sup>]] to get to the required structural efficiency without a complicated formulation, but only with experimentation and collection of data. optimization method, as well as a Gradient Descent Method are carried on and compared. It was considered that meta-heuristic algorithms could be of greater potential for this engineering problem rather than classical optimization methods, such as Newton-Raphson or a Gradient Descent Based Method given the complexity of analysis they imply when formulating a problem for more than 2 variables on place which are more needed when the steel reinforcing area is to be reduced as much as possible.
-<math>M_{nbx}</math> y <math>M_{nby}</math>=Momentos nominales resistentes en la condición balanceada alrededor de los ejes  X y Y recpectivamente.
+====3.1.1 Single variable width value t of the ISR====
-<math>M_{ux}</math> y <math>M_{uy}</math>=Momentos nominales aplicados alrededor de los ejes  X y Y respectivamente.
+For a single variable of width <math>t</math> for the ISR a general function to evaluate the efficiency must be established, by referring to <span id='citeF-12'></span>[[#cite-12|[12]]] such efficiency is determined for any depth value <math>c</math> of the neutral axis by computing an Interaction Diagram for both axis directions of a cross section geometry (if a rectangular cross section geometry is taken) '''Fig. [[#img-9|9]]''', and then to estimate <math>M_{R_{x}},M_{R_{y}}</math> reduced with the Resistance Reduction Factor from the ACI code <span id='citeF-1'></span>[[#cite-1|[1]]] for every flexure-compression load condition to determine their respective efficiency using Analytical Geometry (using the line intersection formula). This way a most critical condition can be determined for each iteration to take as reference of evaluation as the width <math>t</math> of the ISR changes (decreases or increases). In structural engineering it is a common requirement for this such structural efficiency to take values between 85% to 100%
-<span id="fn-4"></span>
+Thus, an efficiency function of <math>t</math> subjected to a constraint <math>g(t)</math> might be expressed as [[#eq-1|1]]:
-<span style="text-align: center; font-size: 75%;">([[#fnc-4|<sup>1</sup>]]) González Cuevas, Fco. Robles Fernández,“Aspectos básicos del Concreto Reforzado”, 4ª edición, Limusa (2005). Capítulo 6, p. 127-155.</span>
-<span id="fn-5"></span>
+<span id="eq-1"></span>
-<span style="text-align: center; font-size: 75%;">([[#fnc-5|<sup>2</sup>]]) Es el espacio geométrico de los valores de carga axial que la sección de un elemento es capaz de resistir, en todos sus planos.</span>
-===6.4 Normativa===
-Anteriormente se presentaron hipótesis generales de diseño con concreto, porque son de tales hipótesis en que los reglamentos de construcción se basan. Para desarrollar éste proyecto se hizo referencia a las Normas Técnicas complementarias para el diseño y construcción de estructuras de concreto del Reglamento de Construcciones de la Ciudad de México (NTC-2017), que aunque se basan en las hipótesis anteriores no presentan tal cual los mismos criterios de diseño, y que son cuestionados mucho por la sociedad ingenieril. Veamos que proponen estas normas.
-====6.4.1 Carga axial====
-En las NTC-2017 se considera necesario hacer una modificación en el valor de <math>f{'}_{c}</math> mediante Factores de carga <math>F_{c}</math> y Factores de Resistencia <math>F_{R}</math><span id="fnc-6"></span>[[#fn-6|<sup>1</sup>]]. En el apartado de concreto se específica que para el cálculo de resistencias se utilice una resistencia reducida a la compresión del concreto denominada <math>f^{''}c</math>. Cuyo valor es:
-<span id="eq-5"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 256: / Line 140: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  f{''}c=0.85(f{'}c) </math>
+| style="text-align: center;" | <math>E(t)=f(t),</math>
+|-
+| style="text-align: center;" | <math>         \begin{array}{ll}[g(t)=(0.85<=f(t)<=1.0)]         \end{array}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
 |}
-De modo que la ecuación Ecuación [ [[#eq-1|1]]] se transforma a:
+Such function for a single variable of a width <math>t</math> value shows the following behaviour '''Fig. [[#img-10|10]]''', presenting a non-linearity for the variation of structural efficiency as <math>t</math> changes, forming a concave graph, with the horizontal axis corresponding to <math>t</math> could go on until the ISR would become a solid rectangular steel profile obtaining very little non-negative values of structural efficiency, which is not practical at all, given that the resulting reinforcing bars transformed out of this steel area would not fit into the column.
-<span id="eq-6"></span>
+<div id='img-9'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|
+|[[Image:Review_942980062263-diagintreferencia.png|341px|Reference system for the determination of structural efficiency of a any flexure compression condition]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" | '''Figure 9:''' Reference system for the determination of structural efficiency of a any flexure compression condition
+|}
+<div id='img-10'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-| style="text-align: center;" | <math>  P_{oc}=F_{R}((f{''}c)(A_{c}-A_{s})+f_{y}(A_{s})) </math>
+|[[Image:Review_942980062263-efficiency_function_single_t.png|252px|Behaviour of the Efficiency function of a single width value (t) of the ISR]]
-|}
+|- style="text-align: center; font-size: 75%;"
-| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
+| colspan="1" | '''Figure 10:''' Behaviour of the Efficiency function of a single width value (t) of the ISR
 |}
-De modo que la ecuación Ecuación [ [[#eq-2|2]]] se transforma a:
+When computing this such Efficiency function, a mathematical approach might be formulated as in <span id='citeF-12'></span>[[#cite-12|[12]]] '''Fig. [[#img-11|11]]''' where for rectangular solid cross section geometries quite good amount of computation might be saved than when a discrete analysis of the Idealized Steel Reinforcement is done, as formulated also by <span id='citeF-12'></span>[[#cite-12|[12]]] for circular cross sections '''Fig. [[#img-12|12]]'''.
-<span id="eq-7"></span>
+This formulation of a single variable of width <math>t</math> for the ISR is useful as stated previously when a design based merely on codes and normative takes place, maintaining uniformity of reinforcement over the cross-section regarding a homogeneous type of reinforcing bars for each element. When a more accurate design is required, then as stated by Aschheim et. al.  <span id='citeF-9'></span>[[#cite-9|[9]]] a non-uniformity consideration might be best to carry on in order to obtain different reinforcement area for each face of the rectangular cross-section and converge better towards a minimum required total reinforcement area.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<div id='img-11'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|
+|[[Image:Review_942980062263-fig12.png|295px|Mathematical formulation fo the ISR method. Ideal for computational saving applied to rectangular cross-sections]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
-|-
+| colspan="1" | '''Figure 11:''' Mathematical formulation fo the ISR method. Ideal for computational saving applied to rectangular cross-sections
-| style="text-align: center;" | <math>  P_{ot}=F_{R}(f_{y})(A_{s}) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
 |}
-Para una mejor interpretación de los factores mecionados se puede hacer referencia a la Figura [[#img-3|3]], que es el bloque de transformación equivalente de esfuerzos de las hipótesis de las NTC-2017.
+<div id='img-12'></div>
-<div id='img-3'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-Fig5.png|576px|Hipótesis de las NTC-2017 sobre la distribución de deformaciones y esfuerzos en la zona de compresión. Adaptada de: <span id='citeF-3'></span>[[#cite-3|[3]]]]]
+|[[Image:Review_942980062263-fig27.png|249px|Discrete formulation for the ISR method for circular cross-sections. Ideal for complex geometries.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 3:''' Hipótesis de las NTC-2017 sobre la distribución de deformaciones y esfuerzos en la zona de compresión. Adaptada de: <span id='citeF-3'></span>[[#cite-3|[3]]]
+| colspan="1" | '''Figure 12:''' Discrete formulation for the ISR method for circular cross-sections. Ideal for complex geometries.
 |}
-Donde: <math> C=ab(f{''}_{c}) </math> <math> 0.65<=(\beta _{1}=1.05-\frac{f{'}_{c}}{1400})<=0.85 </math>
-====6.4.2 Compresión y flexión en dos direcciones====
+For this work and specifically for this particular case of a single variable width <math>t</math> of the ISR the mathematical formulation for rectangular cross-sections for the efficiency function was used and performed through Multiple Linear Regression and a Gradient Descent method. The objective of the Multiple Linear Regression formulations is to get a formula so that for each structural model at hand an initial value <math>t</math> close enough to the required one may be determined and minimize the ''Simple search'' optimization iterations. To do so, different experimental computational runs were performed varying the values of each of the variables involved in the analysis, such as <math>b</math> and <math>h</math> (corresponding to the section dimensions), and <math>P_{u},M_{ux}, M_{uy},f'{c},cover,Efficiency</math>. As for the Gradient Descent Method, given the particular concave form of the graph function for <math>t-Efficiency</math>, it could be well adapted to obtain for each structural model a value of <math>t</math> for which the derivative of the convcave function would be less than a preestablished one '''Fig. [[#img-13|13]]'''. This way, really good approximations to the <math>t</math> optimum cuold be gotten. The Bresler formula ([[#eq-2|2]]) <span id='citeF-10'></span>[[#cite-10|[10]]] was used with the respective conditions and typical values of <math>a=1</math>:
-<span id="fnc-7"></span>[[#fn-7|<sup>2</sup>]]   Se usará la expresión siguiente (que es derivada de la fórmula de Bresler) para el cálculo de la carga máxima actuante en la sección a cierta excentricidad.
+<span id="eq-2"></span>
-<span id="eq-8"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 308: / Line 193: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>   P_{R}=\frac{1}{\frac{1}{P_{rx}}+\frac{1}{P_{ry}}-\frac{1}{P_{oc}}} </math>
+| style="text-align: center;" | <math>\frac{M_{ux}}{M_{uy}}^{1.0}+\frac{M_{R_{x}}}{M_{R_{y}}}^{1.0}<=1  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
 |}
-Para valores de <math>\frac{P_{R}}{P_{oc}}<0.1</math>, se usará la expresión siguiente:
+In order to be able to apply ([[#eq-2|2]]) the following condition has to be satisfied to evaluate the axial loads against the design axial resistance:
+<math>\frac{P_{u}}{P_{R}}<=0.1</math>
+==Multiple Linear Regression:==
+Regarding the experimental runs to apply Multiple Linear Regression to estimate an initial value of <math>t</math>, three different experimentations were performed, one with a total of 100 runs randomly registered with different values for each of the main variables involved already mentioned. The following coefficients were obtained for each interpolation method:
-<span id="eq-9"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 321: / Line 211: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  \frac{M_{ux}}{M_{rx}}+\frac{M_{uy}}{M_{ry}}<=1.0 </math>
+| style="text-align: center;" | <math>t=-0.067-0.0204(b)-0.013(h)-0.0007(P_{u})+0.003(M_{ux})-...</math>
+|-
+| style="text-align: center;" | <math>0.0013(M_{uy}{)-0.002(f'}_{c})+4.178(efficiency)+0.785(cover) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
 |}
-Donde: <math>M_{ux}</math> y <math>M_{uy}</math> son los momentos actuantes alrededor de los ejes X, Y, respectivamente.
+Which enhances not really good approximations for initial values of <math>t</math> to start iterating when applying a 1t-ISR, only for certain parameter values, given the linearity of the formulation as well as the high number of variables involved. The summary of the experimentation is presented in detail in the results section [[#4.3 Results from the 1t ISR formulation with Multiple Linear Regression|4.3]] for ten structural models.
-<math>M_{rx}</math> y <math>M_{ry}</math> son los momentos resistentes de diseño alrededor de los mismos ejes.
+Although this Multiple Linear Regression formula of <math>t</math> may be used altogether with a Gradient Descent Optimization Method to minimize even more the number of iteration needed for a good approximation of structural efficiency wanted, as is following presented.
-<span id="fn-6"></span>
+==Steepest Gradient Descent method==
-<span style="text-align: center; font-size: 75%;">([[#fnc-6|<sup>1</sup>]]) El factor de reducción <math>F_{R}</math>, que para el caso de columnas con carga axial es de 0.8 <span id='citeF-3'></span>[[#cite-3|[3]]]</span>
-<span id="fn-7"></span>
+<div id='img-13'></div>
-<span style="text-align: center; font-size: 75%;">([[#fnc-7|<sup>2</sup>]]) González Cuevas, Fco. Robles Fernández, “Aspectos básicos del Concreto Reforzado”, 4ª edición, Limusa (2005). Capítulo 6, p. 148.</span>
-==7 El método del perfil idealizado de acero==
-Empleando las hipótesis de diseño anteriormente descritas se ha idealizado el acero de refuerzo de un elemento como se muestra. Figura [[#img-4|4]]   <div id='img-4'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-Fig6.png|193px|Sección rectangular idealizada de concreto reforzado.]]
+|[[Image:Review_942980062263-steepest_descent_form.png|309px|Slope criteria for the Steepest Descent Optimization Method formulation.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 4:''' Sección rectangular idealizada de concreto reforzado.
+| colspan="1" | '''Figure 13:''' Slope criteria for the Steepest Descent Optimization Method formulation.
 |}
-Para estas secciones se desarrolló un método puramente matemático con ecuaciones que definieran la resistencia de la columna de acuerdo a la ubicación del eje neutro de esfuerzos en la sección siguiendo paramétros de geometrías que resulta un tanto complejo para programar. Por ello también se desarrolló un método numérico que resulta más sencillo de programar y se meustra a continuación.
-Tomando como referencia la siguiente figura Figura [[#img-5|5]]:  <div id='img-5'></div>
+In order to obtain a range of critical slopes for the program to converge to, a total of 10 experimental models were analysed for optimization with ''Simple search'' to get to a width <math>t</math> optimal for each model and extract its respective optimal slope. A slope range was found to be <math>[-0.2539,-0.8556]</math>, for efficiencies between <math>[0.8, 1.0]</math>. Therefore, this slope range might be taken as a good algorithmic criteria for the Steepest Descent Method to converge to, although it would always be better to check directly for an structural efficiency range to assure a convergence as it was done in this research. The general algorithm of the program is next presented in pseudo-code.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-perfilacero1.png|308px|Sección rectangular idealizada de concreto reforzado.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 5:''' Sección rectangular idealizada de concreto reforzado.
-|}
-Donde: <math>t</math> = espesor de la sección transversal del acero de refuerzo idealizado como un refuerzo constante alrededor de la sección.
-===7.1 Secciones rectangulares-modelo numérico===
-Se discretizará el perfil idealizado de acero como se muestra a continuación Figura [[#img-6|6]]:  <div id='img-6'></div>
+{| style="margin: 1em auto;border: 1px solid darkgray;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-numerico_rec_modelo.png|257px|Modelo geométrico para la metodología numérica de análisis:  Dibujo propio.]]
+|
-|- style="text-align: center; font-size: 75%;"
+:'''INICIO'''
-| colspan="1" | '''Figura 6:''' Modelo geométrico para la metodología numérica de análisis:  Dibujo propio.
-|}
-Donde: <math>dA_{ac}=dL(t)</math>
-<math>L=h-2rec</math>
-<math>dL=\frac{L}{nElementos}</math>
-<math>nElementos=</math> Número de elementos
-====7.1.1 Cálculo de la resistencia del acero====
-Metodología      Para la determinación de la resistencia de carga axial y flexión se sigue la siguiente metodología:
-* Determinar la distancia de cada segmento de perfil con respecto a la fibra superior más alejada de la sección transversal del elemento de concreto <math display="inline">d</math>.<p>   Para las partes superior e inferior del perfil se tiene:
-<span id="eq-10"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| '''for Nmodels=1:nm'''
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  F_{a}=\varepsilon _{a}E_{ac}t(b-2rec) </math>
+| <math display="inline">t_{k}=inital-t=-0.067-0.0204(b)-0.013(h)-0.0007(P_{u})+...</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
-|}
-<span id="eq-11"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| <math display="inline">0.003(M_{ux})-0.0013(M_{uy}{)-0.002(f'}_{c})+...</math>
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  F_{b}=\varepsilon _{b}E_{ac}t(b-2rec) </math>
+| <math display="inline">4.178(efficiency)+0.785(cover)</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
-|}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| Compute <math display="inline">f(initial-t)=f(t_{k})</math>
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> M_{a,b}=F_{a}(d_{1}-\frac{1}{2}h)+F_{b}(d_{2}-\frac{1}{2}h) </math>
+| '''While <math>f(t_{k})>1.0</math> or <math>f(t_{k})<0.8</math>'''
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
-|}
-Donde: <math>\varepsilon _{a}=(1-\frac{d_{1}}{c})(0.003)</math></p><p> <math>\varepsilon _{b}=(\frac{d_{2}}{c}-1)(0.003)</math></p><p> <math>d_{1}=rec+\frac{1}{2}(t)</math></p><p> <math>d_{2}=h-rec-\frac{1}{2}(t)</math></p><p> <math>E_{ac}=\frac{fy}{\varepsilon _{y}}=\frac{4200 \frac{kg}{cm^2}}{0.0021}</math></p><p> <math>E_{ac}=2e10 \frac{kg}{cm^{2}}</math>       Mientras que para los costados del perfil se tiene:
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| Compute <math display="inline">g(t_{k})=\nabla f(t_{k})</math>
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> F_{R}=2\sum _{i=1}^{nElementos}E_{ac}\varepsilon (dA_{ac}) </math>
+| Compute search direction <math display="inline">p_{k}</math>:
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
-|}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| '''if <math>f(t_{k})<0.8</math>'''
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> M_{R}=\sum _{i=1}^{nElementos}-2E_{ac}\varepsilon (dA_{ac})(\frac{1}{2}h-d) </math>
+| <math display="inline">p_{k}=1</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
-|}
-Donde: <math>t</math> = Espesor del perfil</p><p> <math>rec</math> = recubrimiento</p><p> <math>h</math> = peralte de la sección</p><p> <math>\varepsilon _{a}=(1-\frac{d}{c})(0.003)</math></p><p>       </p>
-* A partir de ahí se cálcula <math display="inline">\varepsilon </math> en función de c, cuidando únicamente la siguiente restricción <math display="inline">[-0.0021<\varepsilon <0.0021]</math>.<p>  </p>
-* Y finalmente calcular la fuerza y momento resultantes resistentes:<p>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| '''else if <math>f(t_{k})>1.0</math>'''
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> F_{R}=F_{a}+F_{b}+2\sum _{i=1}^{nElementos}E_{ac}\varepsilon (dA_{ac}) </math>
+| <math display="inline">p_{k}=-1</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
-|}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| '''End if'''
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> M_{R}=F_{a}(d_{1}-\frac{1}{2}h)+F_{b}(d_{2}-\frac{1}{2}h)+\sum _{i=1}^{nElementos}-2E_{ac}\varepsilon (dA_{ac})(\frac{1}{2}h-d) </math>
+| Update the current <math display="inline">t_{k+1}=t_{k}+\alpha _{k}(p_{k})</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
-|}
-</p>
-====7.1.2 Cálculo de la resistencia para el concreto====
-Lo que se pretende aquí es encontrar el área neta del concreto bajo esfuerzos para cualquier valor en la profundidad del eje neutro Figura [[#img-8|8]]. Al ser una sección rectangular, el análisis se simplifica a un modelo matemático puro, ya que el área variará uniformemente a lo largo de todo el peralte de la sección, ya sea respecto al eje X o Y. A partir de esa área neta se determinará el volumen del bloque de esfuerzos-deformaciones resultante de compresión en unidades de Fuerza, y posteriormente el momento de flexión resultante respecto al eje del centroide de la sección como sigue:
-<div id='img-7'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig21.png|465px|Diagrama de bloque equivalente de esuferzos de compresión en el concreto. Dibujo propio.]]
+| Compute <math display="inline">f(t_{k+1})</math>
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 7:''' Diagrama de bloque equivalente de esuferzos de compresión en el concreto. Dibujo propio.
-|}
-<span id="eq-17"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| k=k+1;
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  C=-abf{''}c </math>
+| '''End While'''
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
-|}
-<span id="eq-18"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+| <math display="inline">t_{final}=t_{k}</math>
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  M_{c}=-C(\frac{h}{2}-\frac{a}{2}) </math>
+| <math display="inline">Ef_{final}=f(t_{k})</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
-|}
-<div id='img-8'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig22.png|398px|Diagrama tridimensional del bloque equivalente de esfuerzos de compresión en el concreto. Dibujo propio.]]
+| '''End for'''
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 8:''' Diagrama tridimensional del bloque equivalente de esfuerzos de compresión en el concreto. Dibujo propio.
-|}
-==8 Columnas circulares==
-Análogo a la idealización del acero de refuerzo de columnas rectangulares, se ha idealizado el acero de refuerzo en éste tipo de columnas como se muestra en la siguiente figura Figura [[#img-9|9]].  <div id='img-9'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig23.png|302px|Idealización de una sección circular de concreto reforzado. Dibujo propio.]]
+| '''FIN'''
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 9:''' Idealización de una sección circular de concreto reforzado. Dibujo propio.
-|}
-Donde: <math>t</math> = Espesor del perfil
-<math>rec</math> = recubrimiento
-<math>d_{ma}</math> = diámetro de la sección del elemento       Nota: El recubrimiento se mide desde el borde de la sección de la columna al centro del espesor del refuerzo ficticio.
-A diferencia de la geometría rectangular, para la ciruclar resultó bastante complejo analizar la distribución de esfuerzos mediante el enfoque de análisis planteado originalmente (determinando la geometría de las configuraciones de estados de deformaciones, por lo que simplificó con el siguiente planteamiento:
-===8.1 Cálculo de la resistencia del acero===
-Lo que se hará es tratar el perfil de acero como una línea circular y subdividirla en pequeños segmentos con espesor <math>t</math>, Figura [[#img-10|10]], aunque en realidad cada segmento se tratará como un punto al que le corresponderá una cierta condición de esfuerzos dependiendo de su ubicación en el plano cartesiano de referencia y del caso del cual se trate.   <div id='img-10'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig27.png|499px|Esquema de análisis para las condiciones de esfuerzos en el acero de refuerzo. Dibujo propio.]]
+| style="text-align: center; font-size: 75%;"|
-|- style="text-align: center; font-size: 75%;"
+'''Algorithm. 1''' General algorithmic process for the Steepest Descent Optimization Method formulation
-| colspan="1" | '''Figura 10:''' Esquema de análisis para las condiciones de esfuerzos en el acero de refuerzo. Dibujo propio.
 |}
-Donde: <math>dA_{ac}=dL(t)</math>
-<math>L=\Pi (d_{ma}-2rec)</math>
+One must be careful with the initial step length <math>\alpha _{0}</math> (it is recommended to use <math>[0.4<\alpha _{0}<0.6]</math> to avoid high number of iterations) as the step length for this case gets smaller the as it converges.
-<math>dL=\frac{L}{n}</math>
+====3.1.2 Multiple variables of width t values of the ISR====
-<math>n=</math> Número de elementos
+Regarding meta-heuristic and evolutionary algorithms there has been relevant research for the design of reinforcing steel in cocnrete structures such as <span id='citeF-13'></span>[[#cite-13|[13]]] in which a formulation with the GA was performed, focusing primarily on costs, restricting the structural efficiencies to a pre-established range using the ISR method with a single variable <math>t</math> to obtain an initial steel reinforcement area as minimum to generate the reinforcing bar sets taken as individuals for the algorithm. The results of this such work are good, but not that different and better than in previous work <span id='citeF-12'></span>[[#cite-12|[12]]] in which an optimization with ''Simple search'' was carried on considering homogeneity for the type of reinforcing bar over the column; not that different neither in cost nor in efficiency, stressing the importance first of all of improve the accuracy on the required minimum reinforcing area and its distribution over the section to minimize iterations until an optimum is found through a Genetic Algorithm particularly, and secondly the need to focus specifically in structural efficiency through a complex non-uniform arrangement of reinforcing bars to obtain better solutions.
-====8.1.1 Metodología====
+As following a comparison in performance and formulation between the GA and PSO adapted to the ISR method for two and four width variables <math>t</math> will be carried on in order to simplify the optimization formulation for the problem through meta-heuristics given that such analysis with classical optimization would not be that practical for the reasons stressed previously, and neither through Polynomial Interpolations or Multiple Linear Regression due to the vast quantity of required data to collect in order to obtain good approximations of each different value of widths <math>t</math> for each face of the cross-section element.
-Para la determinación de la resistencia de carga axial y flexión se sigue la siguiente metodología:
+When formulating this types of problems it is quite significant and influential the way in which the structural efficiency is analysed for each possible solution of the optimal ISR. For this case the Bresler formula ([[#eq-4|4]]) was also employed <span id='citeF-10'></span>[[#cite-10|[10]]], as follows:
-* Determinar la distancia de cada segmento de perfil con respecto a la fibra más alejada de la sección transversal del elemento de concreto <math display="inline">d</math>.<p>  </p>
-* A partir de ahí se cálcula <math display="inline">\varepsilon </math> en función de c, cuidando únicamente la siguiente restricción <math display="inline">[-0.0021<\varepsilon <0.0021]</math>.<p>  </p>
-* Y finalmente calcular la fuerza y momento resultantes resistentes:<p>
+<span id="eq-4"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 548: / Line 304: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> F_{R}=\sum _{i=1}^{nElementos}E_{ac}\varepsilon (dA_{ac}) </math>
+| style="text-align: center;" | <math>\frac{M_{ux}}{M_{R_{x}}}^{1}+\frac{M_{uy}}{M_{R_{y}}}^{1}<=1  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
 |}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+In order to be able to apply ([[#eq-4|4]]) the following condition ([[#3.1.2 Multiple variables of width t values of the ISR|3.1.2]]) has to be satisfied to evaluate the axial loads against the design axial resistance:
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math> M_{R}=\sum _{i=1}^{nElementos}-E_{ac}\varepsilon (dA_{ac})(\frac{1}{2}d_{ma}-d) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
-|}
-</p>
+<math>\frac{P_{u}}{P_{R}}<=1.0</math>
-===8.2 Casos de análisis para el concreto===
+For this particular case of optimization <math>\frac{P_{u}}{P_{R}}<=1.0</math> was set given to that when evaluating the efficiency with respect of the weak axis of the cross section, the bending moment is the preponderant factor when optimizing the ISR as contrary to when a single <math>t</math> variable problem is formulated, due to the non-uniformity of the ISR over the cross section. This way, the optimization algorithm is able to minimize as much as possible the reinforcing area or widths over the cross section faces corresponding to each axis.
-A diferencia de las columnas de sección rectangular, en las circulares el diferencial de área al cambiar el eje neutro no será uniforme.
+Similar to the analytical geometrical procedure as for a single t formulation problem, an Interaction Diagram is computed for both axis directions of a cross section geometry in order to estimate <math>M_{R_{x}},M_{R_{y}}</math> reduced with the Resistance Reduction Factor from the ACI code <span id='citeF-1'></span>[[#cite-1|[1]]].
-También podremos transformar ésta área de compresión en un bloque equivalente de esfuerzos, y para poder hacer ésto será necesario cálcular el centroide del área real de compresión en cada posición del eje neutro y a partir de ahí calcular el momento de flexión resistente de la sección. Figura [[#img-11|11]].  <div id='img-11'></div>
+====3.1.3 PSO-ISR for multi width variables====
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-fig47.png|539px|Diagrama de distribución de esfuerzos para el concreto. Dibujo propio.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 11:''' Diagrama de distribución de esfuerzos para el concreto. Dibujo propio.
-|}
-Donde: <math>a=\beta (c)</math>
-<math>f{''}_{c}=F_{R}(0.85)f^{'}c</math>      Para calcular ésta fuerza de compresión tomaremos el siguiente sistema de referencia: Figura [[#img-12|12]], para definir integrales para calcular el área efectiva de compresión del concreto en cualquier posición del eje neutro.    <div id='img-12'></div>
+It is necessary first of all to establish the range of the search space for each particle, that is the maximum and minimum values that each width variable <math>t_{j}</math> could take for the analysis. A very simple consideration will be stated, such that the maximum reinforcement area on any face of the rectangular cross-section element is less than the sum of the maximum number of reinforcing bars with diameter <math>\frac{12}{8}</math> such that the minimum separation <math>(sep_{min}=\frac{3}{4}in)</math> from the code <span id='citeF-1'></span>[[#cite-1|[1]]] is not violated. For the upper and lower face the maximum width <math>t_{max}</math> of each idealized smeared reinforcement is given by ([[#eq-5|5]]).
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-fig48.png|567px|Sistema de referencia cartesiano para el análisis de la distribución de esfuerzos de compresión en el concreto. Dibujo propio.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 12:''' Sistema de referencia cartesiano para el análisis de la distribución de esfuerzos de compresión en el concreto. Dibujo propio.
-|}
-Subdividiremos el análisis en dos casos para simplificación:
-Cuando <math>a<\frac{1}{2}d_{ma}</math>
+<span id="eq-5"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 594: / Line 327: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> F_{co}=f{''}c \int _{-z(y_{co}=a)}^{z(y_{co}=a)}(a-y_{co})\mathrm{d}z </math>
+| style="text-align: center;" | <math>t_{max_{hor}}=Nv_{max-hor}(\frac{12}{8})^{2}(\frac{\pi }{4})\frac{1}{(b-2rec)}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
 |}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+Where <math>Nv_{max-hor}</math> is maximum number of reinforcing bars <math>\# \frac{12}{8}</math> allowed for that section face, <math>b</math> is the width section dimension.
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>\begin{array}{l} F_{co}\cdot Y_{co}=f{''}c \int _{-z(y_{co}=a)}^{z(y_{co}=a)}(a-y_{co})...\\ (\frac{1}{2}d_{ma}-\frac{1}{2}(a+y_{co})))\mathrm{d}z \end{array}</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
-|}
-Cuando <math>a>=\frac{1}{2}d_{ma}</math>
+For the left and right cross-section faces ([[#eq-6|6]]):
+<span id="eq-6"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 616: / Line 342: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{l} F_{co}=f{''}c (\Pi (\frac{1}{2}d_{ma})^{2}-...\\ ...-\int _{-z(y_{co}=d_{ma}-a)}^{z(y_{co}=d_{ma}-a)}((d_{ma}-a)-y_{co})\mathrm{d}z) \end{array}</math>
+| style="text-align: center;" | <math>t_{max_{ver}}=Nv_{max-ver}(\frac{12}{8})^{2}(\frac{\pi }{4})\frac{1}{(h-2rec)}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
 |}
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+Whereas the minimum width for any <math>t</math> variable might be just a very little number, for instance <math>0.00001cm</math>.
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>\begin{array}{l} F_{co}\cdot Y_{co}=f{''}c (-\int _{-z(y_{co}=d_{ma}-a)}^{z(y_{co}=d_{ma}-a)}((d_{ma}-...\\ ...-a)-y_{co})(-(\frac{1}{2}d_{ma}-...\\ ...-\frac{1}{2}(d_{ma}-a-y_{co}))\mathrm{d}z) \end{array}</math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
-|}
-==9 Geometría y propiedades mecánicas de los materiales==
+When evaluating the structural efficiency performance, a finite discrete element method of the ISR for each face of the cross section will be made. For this problem it was considered to be time-saving than to develop the mathematical formulation as when the width <math display="inline">t</math> of the ISR remains constant
-Recordando que el objetivo es determinar un arreglo de refuerzo tal que cumpla con un rango de eficiencia aceptable ante ciertos elementos mecánicos actuantes al que la estructura concreto estaría sometidos, manteniendo constante la geometría de las sección propuesta inicialmente. Por lo que nuestros datos de entrada serán el número de elementos a diseñar, para cada elemento se indicará la resistencia a compresión simple del concreto <math>f{'}_{c}</math>, la resistencia a la fluencia del acero <math>f_{y}</math>, y el recubrimiento <math>rec</math>.
+In comparison with the single ISR with one variable <math>t</math> in which only one solution for <math>t</math> could take place complying the structural efficiency required range, when formulating a multiple <math>t</math> variable problem an infinite set of solutions for a established required structural efficiency are possible, therefore a Multi-objective optimization formulation is required to minimize also the reinforcement steel area. When a two <math>t</math> variable problem is formulated, something similar to a ''“Load Combination Reinforcement Diagram”'' as presented by Aschheim et. al. '''Fig. [[#img-3a|3a]]''' is obtained, which is what in Computational Optimization is formally called a ''Pareto Optimal Set'' from which a ''“Pareto Front”'' is then generated <span id='citeF-15'></span>[[#cite-15|[15]]]. In this problem to adapt the Multi-Objective Optimization problem, the reinforcing area <math>A_{t}</math> is set as the main evaluating function to be minimized with the additional condition <math>Eff(t)<100%</math>, given that not necessarily one variable (taking <math>t</math> for instance, or reinforcing area in other words) has to be in conflict with the structural efficiency as it is supposed in the majority of Multi-Objective Problems. Therefore the PSO algorithm rapidly adapts to the given objective function and its restriction with no other operation really needed. It is of importance though, to stress that this such condition has to be carefully placed within the algorithm for a good convergence to a global optimum for both reinforcement area and structural efficiency.
-El programa también necesitará las fuerzas actuantes en la columna para su diseño. Dichas fuerzas se determinan normalmente mediante un análisis estructural, el cual es común que se realice en algún software para tal fin. A partir de éste punto se determinará un valor inicial del espesor del perfil idealizado de acero <math>t</math>, acomodado en la secicón de concreto de tal manera que se respete el recubrimiento deseado, para así comenzar con las iteraciones de actualización de los diagramas de interacción de las columnas hasta que la condición mecánica actuante más crítica sea cubierta.
+This additional condition to evaluate the performance of a particle in the PSO algorithm is most recommended based on our results to be placed both when determining the best position (reinforcement area) in the current swarm and a global best position; given that a global minimum reinforcing area is preponderant to seek rather than a maximum structural efficiency (in the scale of <math>0-100%</math>) as long as the structural critical efficiency requirement is covered. In fact, this structural efficiency requirement was set in the first place to minimize the reinforcement area, given that it was assumed that the less the reinforcement area is used the less efficient the structural element is, which in multiple widths <math>t</math> is not really the case as mentioned before.
-==10 Cálculo de los diagramas de interacción==
+The general algorithm for this process is resumed as following in pseudo-code. A nested PSO algorithm was generated to best estimate a global optimum reinforcement area, substituting for each iteration of the PSO the best position so far found in the previous iterations into the current run (adding such optimal position into the initial generated positions of each particle at the beginning of the PSO), thus enhancing an optimization process over that given best position to accelerate in a way the optimal results.
-Se establece un cierto número de puntos para determinar los diagramas de interacción, determinado los valores <math>P_{ot}</math>, <math>P_{oc}</math> como los límites del rango de las fuerzas del diagrama de interacción, pudiendo extraer una fuerza respectiva a cada punto del mismo que tendrá el papel de raíz en el método numérico a emplear para el cálculo de su respectivo momento resistente.
-El método numérico empleado para la aproximación de ráices es el llamado ''Método de la Falsa posición. Anexo 3 p. actual
+{| style="margin: 1em auto;border: 1px solid darkgray;"
+|-
-===10.1 Cálculo de la eficiencia mecánica-estructural===
+|
+:'''INICIO'''
-Para cada condición de fuerza actuante en la columna se determinará la respectiva eficiencia estructural que hay entre ésta condición y la resistencia que el diagrama de interacción índica.
+|-
+| Generate a <math display="inline">t_{0}=[]</math>
+|-
+| '''for i=1:numberPSOiterations'''
+|-
+| '''PSO-algorithm'''
+|-
+| Initial positions and velocities (the previous best position-<math display="inline">t_{i-1}</math>) is introduced
+|-
+| <math display="inline">x_{ij}=t_{ij}=t_{min}+r(t_{max}-t_{min})</math>
+|-
+| <math display="inline">v_{ij}=\frac{\alpha }{\Delta t}(-\frac{t_{max}-t_{min}}{2}+r(t_{max}-t_{min}))</math>
+|-
+| '''for j=1:numberOptimIterations'''
+|-
+| Update of positions and velocities
+|-
+| <math display="inline">v_{ij}=v_{ij}+c_{1}q(\frac{t_{ij}^{pb}-t_{ij}}{\Delta t})+c_{2}r(\frac{t_{j}^{sb}-t_{ij}}{\Delta t})</math>
+|-
+| <math display="inline">t_{ij}=t_{ij}+v_{ij}\Delta t</math>
+|-
+| '''endfor'''
+|-
+| '''End PSO-algorithm'''
+|-
+| Extract a best new t-values (position) <math display="inline">t_{i}=[]</math> in terms of reinforcement area <math display="inline">At_{i}=(b-2rec)(t_{1}+t_{2})+(h-2rec)(t_{3}+t_{4})</math>
+|-
+| '''End for'''
+|-
+| globalBest-tvalues=<math display="inline">t[t_{1},t_{2},t_{3},t_{4}]</math>
+|-
+| '''FIN'''
-La diferencia en el cálculo de éstas eficiencias en columnas circulares y rectangulares es que en las rectangulares se deberán extraer de los diagramas de interacción calculados en el sentido de <math>x</math> y <math>y</math> la carga resistente y el momento resistente (<math>M_{rx}, M_{ry}, P_{rx}, P_{ry}</math>  [p. actual]), y para las circulares al ser su sección simétrica respecto a cualquier eje, se calculará la eficiencia respecto a una sola dirección, extrayendo la carga resistente y el momento resistente (<math>M_{r}, P_{r}</math>) del diagrama de interacción calculado.
-Para lo anterior se empleará geometría analítica con el sistema cartesiano de los propios diagramas de interacción Figura [[#img-13|13]] calculando el punto de intersección entre dos rectas imaginarias A y B, tal que A partirá del origen pasando por alguna condición de fuerza actuante prolongándose hasta el borde del diagrama de interacción, y B será tal que una el punto anterior <math>(M_{i}, P_{i})</math> y posterior <math>(M_{i+1}, P_{i+1})</math> del punto en el cual la recta A intersecta al diagrama de interacción.   <div id='img-13'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagintreferencia.png|375px|Diagrama de interacción en el plano cartesiano de referencia para el programa.]]
+| style="text-align: center; font-size: 75%;"|
-|- style="text-align: center; font-size: 75%;"
+'''Algorithm. 2''' General algorithmic process for the nested PSO-ISR
-| colspan="1" | '''Figura 13:''' Diagrama de interacción en el plano cartesiano de referencia para el programa.
 |}
-De modo que el cálculo del momento resistente para una respectiva condición se cálcula como:
-<span id="eq-25"></span>
+====3.1.4 GA-ISR for multi width variables====
+For the GA a similar consideration as for the PSO is considered regarding the range of the variables involved; both for <math>t_{max}</math>:
+<span id="eq-7"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 664: / Line 411: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  M_{r}=\frac{P_{i+1}+(\frac{P_{i}-P_{i+1}}{M_{i+1}-M_{i}})}{\frac{P_{u}}{M_{u}}-\frac{P_{i+1}-P_{i}}{M_{i+1}-M_{i}}} </math>
+| style="text-align: center;" | <math>t_{max_{hor}}=Nv_{max-hor}(\frac{12}{8})^{2}(\frac{\pi }{4})\frac{1}{(b-2rec)}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
 |}
-Y su fuerza resistente respectiva como:
+<span id="eq-8"></span>
-<span id="eq-26"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 677: / Line 422: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  P_{r}=\frac{P_{u}}{M_{u}}M_{r} </math>
+| style="text-align: center;" | <math>t_{max_{ver}}=Nv_{max-ver}(\frac{12}{8})^{2}(\frac{\pi }{4})\frac{1}{(h-2rec)}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
 |}
-Ver demostración Anexo 1. p. actual
+And <math>t_{min}=0.0001cm</math> for instance. As for the objective function also the reinforcing area <math>A(t)</math> is seek to be minimized with its restriction <math>Ef(t)<100%</math>. The program for the application of the GA is next presented in pseudo-code:
-Ahora ya se pueden aplicar las fórmulas que índican las NTC-2017 p. actual, para calcular las eficiencias de cada condición mecánica actuante.
-===10.2 Cálculo de área requerida de acero===
+{| style="margin: 1em auto;border: 1px solid darkgray;"
-Ya determinada la eficiencia mecánica estructural para cada condición de carga actuante se determinará a continuación la más crítica, y a partir de ésta se cálculará el espesor del perfil de acero simplemente iterando el valor de <math>t</math> con un cierto incremento <math>dt</math>, hasta llegar a un rango de eficiencia crítica requerida, establecida a priori, pues está será la eficiencia que nos servirá de comparación para seguir iterando o no. Figura [ [[#img-14|14]]]
-<div id='img-14'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-condcritica.png|359px|Condición de carga crítica de diseño.]]
+|
-|- style="text-align: center; font-size: 75%;"
+:'''INICIO'''
-| colspan="1" | '''Figura 14:''' Condición de carga crítica de diseño.
-|}
-==11 Determinación del acero de refuerzo==
-Una vez que se ha determinado el área de acero a emplear para que se alcance la eficiencia deseada, se debe transformar esa área ficticia de acero en varillas corrugadas, distribuyéndolas uniformemente en la sección (obedeciendo las hipótesis de diseño de la Sección [[#6.3.2 Flexo-compresión|6.3.2]]), por lo que sólo se admitirán números pares de varillas.
-El programa genera un arreglo matricial de propuestas de número de varillas, desde 4 (para columnas rectangulares) o 6 (para columnas circulares), calculando para cada diámetro el número de varillas requeridas, tomando como referencia los tipos de varillas disponibles comercialmente.
-El usuario podrá elegir no sólo el diámetro del acero de refuerzo, sino también la distribución de éstas en la sección (solo para las columnas rectangulares, ya que para circulares la distribución es simétrica respecto a cualquier eje de simetría), de manera que siempre se respete la separación mínima o distancia libre entre varillas <math>S_{min}=\frac{3}{2}TMA</math>. Aunque por defecto lo que usualmente se busca es la opción que resulte la más eficiente o la más económica. De manera que el usuario también podrá elegir que criterio es el más conveniente.
-Cabe mencionar que la distribución del acero de refuerzo tiene influencia en la eficiencia mecánica de los elementos, y para cada opción de número de varillas/diámetro le corresponderá un buen número de configuraciones de éstas en las sección, por lo que el programa hará también un análisis de eficiencia de cada opción disponible, pero ahora tratando al acero de refuerzo precisamente como varillas de acero, para verificar cuáles cumplen con el rango requerido de eficiencia que se establecio a priori, una vez hecho esto, el programa seleccionará la opción más eficiente.    </span>
-{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-1'></span>Tabla. 1 Tabla de tipos de varillas comerciales en México
 |-
-| style="text-align: left;" |   Tipo (octavo)
+| '''for i=1:numberGenerations'''
-| style="text-align: left;" | <math>diam(pulg)</math>
-| style="text-align: left;" | <math>diam(cm)</math>
-| style="text-align: left;" |  área<math>(cm^{2}</math>
 |-
-| style="text-align: left;" |   <span style="font-size: 75%;"><math>\# 3</math> </span>
+| '''for j=1:numberPopulation'''
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.375</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.9525</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.7126</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 4</math> </span>
+| Decode chromosomes <math display="inline">t_{min}+\frac{2t_{max}}{1-2^{-k}}\sum _{j=1}^{j=k}(2^{-j}g_{j})</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.5</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.27</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.2668</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 5</math> </span>
+| Evaluate individuals (Objective function) <math display="inline">Efficiency(t)</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.625</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.5875</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.9793</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 6</math> </span>
+| '''endfor'''
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>0.75</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.905</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>2.8502</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 8</math> </span>
+| '''Create next generation'''
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.0</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>2.54</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>5.067</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 9</math> </span>
+| Selection (tournament)
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.125</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>2.8575</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>6.413</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 10</math> </span>
+| Crossover
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.25</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>3.175</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>7.917</math></span>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>\# 12</math> </span>
+| Mutation
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>1.5</math> </span>
+|-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>3.81</math> </span>
+| Replace individuals
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>11.400</math></span>
+|-
+| '''End for'''
+|-
+| globalBest-tvalues=<math display="inline">t[t_{1},t_{2},t_{3},t_{4}]</math>
+|-
+| '''FIN'''
+|-
+| style="text-align: center; font-size: 75%;"|
+'''Algorithm. 3''' General algorithmic process for the nested GA-ISR
 |}
-<span style="text-align: center; font-size: 75%;">
-==12 Diseño de experimento==
+<span id="fn-3"></span>
+<span style="text-align: center; font-size: 75%;">([[#fnc-3|<sup>1</sup>]]) The term “Simple search” here refers to an iteration process where the step length remains constant</span>
-Para cada tipo de geometría se probó un modelo estructural de experimentación con diferentes valores de <math>f'{c}</math>.
+==4 Results and discussion==
-===12.1 Columnas rectangulares===
+===4.1 Results from the application of the PSO-ISR formulation===
-En un análisis estructural hecho previamente se proponen las siguientes geometrías y <math>f'{c}</math>:
+Good results were obtained for the 4t-PSO-ISR algorithm as following presented for the given parameter values:
-====12.1.1 Datos de entrada====
+'''Structural parameters:'''
-</span>
+<math>b(width-Section)=50cm</math>
-{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-2'></span>Tabla. 2 Datos de entrada para el diseño de acero de refuerzo en columnas rectangulares
-|-
-| style="text-align: left;" | <math>b(cm)</math>
-| style="text-align: left;" | <math>h(cm)</math>
-| style="text-align: left;" | <math>rec(cm)</math>
-| style="text-align: left;" | <math>f'{c}(cm^{2}</math>
-|-
-| style="text-align: left;" |   <span style="font-size: 75%;"><math>40</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>75</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>5</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>280</math></span>
-|-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>50</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>50</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>5</math> </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>300</math></span>
-|}
+<math display="inline">h(height-Sectin)=80cm</math>
-<span style="text-align: center; font-size: 75%;">     Con las siguientes combinaciones de carga para ambas estructuras:
-</span>
+<math display="inline">rec(steel-cover)=5cm</math>
-{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-3'></span>Tabla. 3 Combinaciones de carga resultantes de un análisis estructural previo para ambos modelos estructurales.
+<math display="inline">E=2.1e6\frac{Kg}{cm^{2}}</math>
+<math display="inline">f_{y}=4200\frac{Kg}{cm^{2}}</math>
+<math display="inline">f'c=280\frac{Kg}{cm^{2}}</math>
+<math display="inline">maxEfficiency=99.99%</math>
+Load Conditions
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-1'></span>Table. 1 Load conditions
 |-
-| style="text-align: left;" | <math>P(Ton)</math>
+| <math display="inline">Ton</math>
-| style="text-align: left;" | <math>M_{x}(Ton-m)</math>
+| <math>Ton\cdot m</math>
-| style="text-align: left;" | <math>M_{y}(Ton-m)</math>
+| <math>Ton\cdot m</math>
 |-
-| style="text-align: left;" |   <span style="font-size: 75%;"><math>-35</math> </span>
+| <math display="inline">P_{u}</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>22</math> </span>
+| <math>M_{ux}</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>45</math></span>
+| <math>M_{uy}</math>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>36</math> </span>
+| <math display="inline">-35.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>25</math> </span>
+| <math>22.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>32</math></span>
+| <math>45.00</math>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-304</math> </span>
+| <math>36.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>19</math> </span>
+| <math>25.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>65</math></span>
+| <math>32.00</math>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-46</math> </span>
+| <math>-304.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>12</math> </span>
+| <math>19.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>75</math></span>
+| <math>65.00</math>
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-187</math> </span>
+| <math>-46.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>10</math> </span>
+| <math>12.00</math>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>45</math></span>
+| <math>76.00</math>
+|-
+| <math>-187.00</math>
+| <math>10.00</math>
+| <math>45.00</math>
 |}
-<span style="text-align: center; font-size: 75%;">
-====12.1.2 Resultados====
+'''PSO parameters'''
-Una vez hecho el análisis de resistencia con las cargas y datos dados se obtienen los siguientes diagramas de interacción, tabla de eficiencias estructurales para cada modelo estructural con los grosores resultantes del perfil idealizado de acero, distribución óptima en costo y eficiencia de varillas de acero en cada modelo estructural y sus respectivas tablas de eficiencia con dicha opción de varillado:
+<math>\alpha=1.0</math>
-Eficiencia estructural con espesores de perfiles idealizados de acero resultantes:
+<math display="inline">c_{1}=2</math>
-<div id='img-15'></div>
+<math display="inline">c_{2}=2</math>
+<math display="inline">dt=1.0</math>
+<math display="inline">inertiaWeigth=1.3</math>
+<math display="inline">\beta=0.99</math>
+<math display="inline">number_of_particles=20</math>
+<math display="inline">PSO-iteration_{number}=20</math>
+<math display="inline">number_{total-iterations}=10</math>
+The evolution of the reinforcing along the iterations is as follows:
+<div id='img-14'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_tabla_t_colrec01.png|375px|Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 01.]]
+|[[Image:Review_942980062263-evolution_area_iteration.png|224px|Progression of reinforcing area for each iteration of the PSO-ISR algorithm]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 15:''' Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 01.
+| colspan="1" | '''Figure 14:''' Progression of reinforcing area for each iteration of the PSO-ISR algorithm
 |}
-<div id='img-16'></div>
+With a global optimum reinforcing area of <math>109.63cm</math>, <math>t-values=[0.080,0.051,0.936,0.555]cm</math>, <math>Efficiency=95.97%</math>.
+Generating the following interaction diagrams:
+<div id='img-15'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_tabla_t_colrec02.png|384px|Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 02.]]
+|[[Image:Review_942980062263-diagPerfil4t_X.png|168px|Interaction diagram in the X-axis direction for the global optima 4t-ISR]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 16:''' Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 02.
+| colspan="1" | '''Figure 15:''' Interaction diagram in the X-axis direction for the global optima 4t-ISR
 |}
-Diagramas de interacción con espesor de perfil (t) resultante para cada modelo:
-Nota:Los puntos rojos representan las condiciones de carga en ese sentido.  <div id='img-17'></div>
+<div id='img-16'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfilX_rec_col01.png|525px|Diagrama de interacción en X con espesor de perfil (t) resultante-Modelo estructural 01.]]
+|[[Image:Review_942980062263-diagPerfil4t_Y.png|168px|Interaction diagram in the Y-axis direction for the global optima 4t-ISR]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 17:''' Diagrama de interacción en X con espesor de perfil (t) resultante-Modelo estructural 01.
+| colspan="1" | '''Figure 16:''' Interaction diagram in the Y-axis direction for the global optima 4t-ISR
 |}
-<div id='img-18'></div>
+It is of relevance to stress that it takes considerable time (depending on the graphics wanted and the PSO algorithm parameters) to get to a good approximation of the global optima reinforcing area. MatLab 2017b software was used for this research, therefore the execution time may be reduced with parallel computing in another faster programming language, more likely when a higher number of structural elements are to be analysed. In any case, the results obtained are much better than with the application of a 1t-ISR Steepest Gradient Descent formulation regarding the resulting optimum area as it will be compared next.
+===4.2 Results from the GA-ISR formulation===
+The Genetic Algorithm is also a viable option, although an little higher amount of computational execution time is necessary. Very similar results were obtained compared to the PSO algorithm. A computational experiment was also carried on with the GA and the results were as following presented with the same structural parameters and the next GA parameters with the same load conditions applied.
+<math>number_{generations}=150</math>
+<math display="inline">number_{individuals}=20</math>
+<math display="inline">population-size=30</math>
+<math display="inline">number_{genes}=60;</math>
+<math display="inline">prob_{mutation}=0.015</math>
+<math display="inline">tournamente-selection-parameter=0.6</math>
+<math display="inline">tournament-size=2</math>
+<div id='img-17'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfilY_rec_col01.png|525px|Diagrama de interacción en Y con espesor de perfil (t) resultante-Modelo estructural 01.]]
+|[[Image:Review_942980062263-GA-ISR-evolution_area.png|224px|Evolution of the optimal reinforcing area with the GA-ISR]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 18:''' Diagrama de interacción en Y con espesor de perfil (t) resultante-Modelo estructural 01.
+| colspan="1" | '''Figure 17:''' Evolution of the optimal reinforcing area with the GA-ISR
 |}
-<div id='img-19'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfilX_rec_col02.png|525px|Diagrama de interacción en X con espesor de perfil (t) resultante-Modelo estructural 02.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 19:''' Diagrama de interacción en X con espesor de perfil (t) resultante-Modelo estructural 02.
-|}
-<div id='img-20'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfilY_rec_col02.png|525px|Diagrama de interacción en Y con espesor de perfil (t) resultante-Modelo estructural 02.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 20:''' Diagrama de interacción en Y con espesor de perfil (t) resultante-Modelo estructural 02.
-|}
-Opciones de distribución de varillas disponibles resultantes:
-<div id='img-21'></div>
+With <math display="inline">t=[0.856,0.015,0.818,0.412]</math>,<math display="inline">Efficiency=97.73%</math>, <math display="inline">minimum-area=121cm^{2}</math> and the following interaction diagrams:
+<div id='img-18'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-opciones_var_rec_01.png|288px|Tabla de opciones disponibles de varillado-Modelo 01.]]
+|[[Image:Review_942980062263-diagIntXGA.png|168px|Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 21:''' Tabla de opciones disponibles de varillado-Modelo 01.
+| colspan="1" | '''Figure 18:''' Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA
 |}
-<div id='img-22'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-opciones_var_rec_02.png|292px|Tabla de opciones disponibles de varillado-Modelo 02.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 22:''' Tabla de opciones disponibles de varillado-Modelo 02.
-|}
-Distribución de varillado óptima en costo y eficiencia
-<div id='img-23'></div>
+<div id='img-19'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-seccion_col01_rec.png|525px|Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 3 Tabla [[#img-21|21]]-Modelo 01]]
+|[[Image:Review_942980062263-diagIntYGA.png|168px|Interaction diagram in the Y-axis direction for the global optima 4t-ISR-GA]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 23:''' Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 3 Tabla [[#img-21|21]]-Modelo 01
+| colspan="1" | '''Figure 19:''' Interaction diagram in the Y-axis direction for the global optima 4t-ISR-GA
 |}
-<div id='img-24'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Verduzco Martinez_325285613-seccion_rec_col02.png|525px|Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 1 Tabla [[#img-22|22]]-Modelo 02]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 24:''' Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 1 Tabla [[#img-22|22]]-Modelo 02
-|}
-Tomando como referencia las distribuciones de varillas en la sección para cada modelo con las tablas de opciones disponibles de varillado, se puede notar que el programa al tomar la opción más económica, hace la selección tomando en cuenta también la mejor eficiencia estructural, que aunque no es la más eficiente de manera absoluta, si lo es con respecto al costo más bajo, ya que se pueden presentar mismos costos óptimos con diferentes eficiencias estrucrales.
-Diagramas de interacción para las opciones de varillado óptimas en costo para cada modelo estructural:
+As it may be observed from the previous graphics obtained with the GA-ISR, it takes almost 50 generations or iterations to reach a very similar result from the PSO-ISR formulation, although still a little higher area. This results may vary from a structural model to another, and even from run to run. The mutation probability is of great influence in the final results. A further comparison between these two formulations is made in detail in further sections, as well as a general comparison between all of the formulations here presented for the ISR.
+===4.3 Results from the 1t ISR formulation with Multiple Linear Regression===
+Ten experimental random structural models were tested to register the number of iterations it would take for each to reach a wanted structural efficiency range given an initial width <math>t</math> value previously estimated with the results from the application of Multiple Linear Regression formulation. The results are following presented.
-<div id='img-25'></div>
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 Error estimations for the Multiple Linear Regression formula
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVarX_rec_col01.png|525px|Diagrama de interacción en X con varillas resultantes-Modelo estructural 01.]]
+|
-|- style="text-align: center; font-size: 75%;"
+| <math>cm</math>
-| colspan="1" | '''Figura 25:''' Diagrama de interacción en X con varillas resultantes-Modelo estructural 01.
+| <math>cm</math>
-|}
+| <math>Ton</math>
-<div id='img-26'></div>
+| <math>Ton\cdot m</math>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| <math>Ton\cdot m</math>
+| <math>\frac{Kg}{cm^{2}}</math>
+|
+| <math>cm</math>
+| <math>cm</math>
+| <math>cm</math>
+| <math>%</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVarY_rec_col01.png|525px|Diagrama de interacción en Y con varillas resultantes-Modelo estructural 01.]]
+|  Point
-|- style="text-align: center; font-size: 75%;"
+| b
-| colspan="1" | '''Figura 26:''' Diagrama de interacción en Y con varillas resultantes-Modelo estructural 01.
+| h
-|}
+| <math>P_{u}</math>
-<div id='img-27'></div>
+| <math>M_{ux}</math>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| <math>M_{uy}</math>
+| <math>{f'}_{c}</math>
+| <math>Eff</math>
+| <math>cover</math>
+| <math>t_{est}</math>
+| <math>t_{real}</math>
+| <math>error</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVarX_rec_col02.png|525px|Diagrama de interacción en X con varillas resultantes-Modelo estructural 02.]]
+|  <span style="font-size: 75%;">1 </span>
-|- style="text-align: center; font-size: 75%;"
+| <math>93</math>
-| colspan="1" | '''Figura 27:''' Diagrama de interacción en X con varillas resultantes-Modelo estructural 02.
+| <math>61</math>
-|}
+| 67.857
-<div id='img-28'></div>
+| 185.7708
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 146.0661
+| 393
+| 0.89
+| 4.5
+| <math>3.85</math>
+| <math>1.72</math>
+| <math>55.3</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVarY_rec_col02.png|525px|Diagrama de interacción en X con varillas resultantes-Modelo estructural 02.]]
+| <span style="font-size: 75%;">2 </span>
-|- style="text-align: center; font-size: 75%;"
+| 93
-| colspan="1" | '''Figura 28:''' Diagrama de interacción en X con varillas resultantes-Modelo estructural 02.
+| 67
-|}
+| -647.700
+| 27.3106
-===12.2 Columnas circulares===
+| 144.2455
+| 164
-====12.2.1 Datos de entrada====
+| 0.89
+| 3.5
-</span>
+| <math>3.72</math>
-{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"
+| <math>1.56</math>
-|+ style="font-size: 75%;" |<span id='table-4'></span>Tabla. 4 Datos de entrada para el diseño de acero de refuerzo en columnas circulares
+| <math>58.0</math>
 |-
-| style="text-align: left;" | <math>D(cm)</math>
+| <span style="font-size: 75%;">3 </span>
-| style="text-align: left;" | <math>rec(cm)</math>
+| 228
-| style="text-align: left;" | <math>f'{c}(cm^{2}</math>
+| 34
+| -427.914
+| 180.9444
+| 121.9733
+| 470
+| 0.89
+| 4.0
+| <math>1.13</math>
+| <math>0.95</math>
+| <math>15.8</math>
 |-
-| style="text-align: left;" |   <span style="font-size: 75%;"><math>40</math> </span>
+| <span style="font-size: 75%;">4 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>5</math> </span>
+| 75
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>300</math></span>
+| 198
-|-
+| -273.442
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>50</math> </span>
+| 190.3260
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>5</math> </span>
+| 184.0664
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>300</math></span>
+| 131
+| 0.83
-|}
+| 3.9
-<span style="text-align: center; font-size: 75%;">     Con las siguientes combinaciones de carga para ambas estructuras:
+| 2.60
+| <math>0.51</math>
-</span>
+| <math>80.3</math>
-{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-5'></span>Tabla. 5 Combinaciones de carga resultantes de un análisis estructural previo para ambos modelos estructurales.
 |-
-| style="text-align: left;" | <math>P(Ton)</math>
+| <span style="font-size: 75%;">5 </span>
-| style="text-align: left;" | <math>M_{x}(Ton-m)</math>
+| 177
+| 35
+| -636.092
+| 38.0866
+| 73.7833
+| 376
+| 0.94
+| 4.7
+| <math>3.01</math>
+| <math>0.37</math>
+| <math>87.5</math>
 |-
-| style="text-align: left;" |   <span style="font-size: 75%;"><math>-35</math> </span>
+| <span style="font-size: 75%;">6 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>22</math></span>
+| 93
+| 412
+| 75.236
+| 85.0518
+| 62.5437
+| 196
+| 0.80
+| 4.1
+| <math>0.23</math>
+| <math>0.12</math>
+| 47.8
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>36</math> </span>
+| <span style="font-size: 75%;">7 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>25</math></span>
+| 109
+| 53
+| -619.890
+| 81.5238
+| 163.9962
+| 531
+| 0.97
+| 4.0
+| 3.35
+| <math>0.84</math>
+| 74.7
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-304</math> </span>
+| <span style="font-size: 75%;">8 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>19</math></span>
+| 98
+| 93
+| -479.857
+| 91.4848
+| 175.0743
+| 410
+| 0.88
+| 3.8
+| 2.74
+| <math>0.22</math>
+| 91.7
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-46</math> </span>
+| <span style="font-size: 75%;">9 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>12</math></span>
+| 53
+| 172
+| -610.107
+| 95.8926
+| 127.8633
+| 426
+| 0.89
+| 4.8
+| 3.6
+| <math>0.36</math>
+| 90.0
 |-
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>-187</math> </span>
+| <span style="font-size: 75%;">10 </span>
-| style="text-align: left;" |  <span style="font-size: 75%;"><math>10</math></span>
+| 58
+| 73
+| -444.694
+| 37.9420
+| 99.0011
+| 188
+| 0.80
+| 4.3
+| 4.40
+| <math>2.06</math>
+| <math>53.1</math>
 |}
-<span style="text-align: center; font-size: 75%;">
-====12.2.2 Resultados====
+As shown in the table, the error in most cases is even higher than <math>50%</math>. Only two initial estimations were close to the final value of <math>t</math> which makes the method no so quite efficient for this case. It is relevant to stress that, even though the estimation with the Multiple Linear Regression may diverge, still would take much less time to get to the optimum value of <math>t</math> from that estimation than with the application of the PSO-4t-ISR or GA-4t-ISR, although, clearly not with less reinforcement area by any means, as it would be compared further in this text.
-Una vez hecho el análisis de resistencia con las cargas y datos dados se obtienen los siguientes diagramas de interacción, tabla de eficiencias estructurales para cada modelo estructural con los grosores resultantes del perfil idealizado de acero, distribución óptima en costo y eficiencia de varillas de acero en cada modelo estructural y sus respectivas tablas de eficiencia con dicha opción de varillado:
+===4.4 Results from the 1t ISR formulation with the Steepest Gradient Descent method===
-Eficiencia estructural con espesores de perfiles idealizados de acero resultantes:
+The same ten structural models studied for the formulation of the Multiple Linear Regression method were again taken to register the maximum number of iterations it would take for the Steepest Gradient Descent Method to converge.
-<div id='img-29'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-3'></span>Table. 3 Max number of iterations for the Steepest Gradient Descent Method with Multiple Linear Regression with an initial step length equal to 0.5
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_cir_01_perfil.png|237px|Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 01.]]
+|
-|- style="text-align: center; font-size: 75%;"
+| <math>cm</math>
-| colspan="1" | '''Figura 29:''' Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 01.
+| <math>cm</math>
-|}
+| <math>Ton</math>
-<div id='img-30'></div>
+| <math>Ton\cdot m</math>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| <math>Ton\cdot m</math>
+| <math>\frac{Kg}{cm^{2}}</math>
+|
+| <math>cm</math>
+| <math>cm</math>
+|
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_cir_02_perfil.png|237px|Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 02.]]
+|  Point
-|- style="text-align: center; font-size: 75%;"
+| b
-| colspan="1" | '''Figura 30:''' Tabla de eficiencias con espesor de perfil (t) resultante-Modelo estructural 02.
+| h
-|}
+| <math>P_{u}</math>
-Diagramas de interacción con espesor de perfil (t) resultante para cada modelo:
+| <math>M_{ux}</math>
+| <math>M_{uy}</math>
-<div id='img-31'></div>
+| <math>{f'}_{c}</math>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| <math>Eff</math>
+| <math>cover</math>
+| <math>t_{est}</math>
+| <math>iterations</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfil_cir01.png|525px|Diagrama de interacción con espesor de perfil (t) resultante-Modelo estructural 01.]]
+|  <span style="font-size: 75%;">1 </span>
-|- style="text-align: center; font-size: 75%;"
+| <math>93</math>
-| colspan="1" | '''Figura 31:''' Diagrama de interacción con espesor de perfil (t) resultante-Modelo estructural 01.
+| <math>61</math>
-|}
+| 67.857
-<div id='img-32'></div>
+| 185.7708
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 146.0661
+| 393
+| 0.92
+| 4.5
+| <math>1.47</math>
+| <math>11</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntPerfil_cir02.png|525px|Diagrama de interacción con espesor de perfil (t) resultante-Modelo estructural 02.]]
+| <span style="font-size: 75%;">2 </span>
-|- style="text-align: center; font-size: 75%;"
+| 93
-| colspan="1" | '''Figura 32:''' Diagrama de interacción con espesor de perfil (t) resultante-Modelo estructural 02.
+| 67
-|}
+| -647.700
-Opciones de distribución de varillas disponibles resultantes:
+| 27.3106
+| 144.2455
-<div id='img-33'></div>
+| 164
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 0.91
+| 3.5
+| <math>1.28</math>
+| <math>10</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_opc_cir_01.png|196px|Tabla de opciones disponibles de varillado-Modelo 01.]]
+| <span style="font-size: 75%;">3 </span>
-|- style="text-align: center; font-size: 75%;"
+| 228
-| colspan="1" | '''Figura 33:''' Tabla de opciones disponibles de varillado-Modelo 01.
+| 34
-|}
+| -427.914
-<div id='img-34'></div>
+| 180.9444
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 121.9733
+| 470
+| 0.94
+| 4.0
+| <math>0.77</math>
+| <math>3</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-ef_opc_cir_02.png|198px|Tabla de opciones disponibles de varillado-Modelo 02.]]
+| <span style="font-size: 75%;">4 </span>
-|- style="text-align: center; font-size: 75%;"
+| 75
-| colspan="1" | '''Figura 34:''' Tabla de opciones disponibles de varillado-Modelo 02.
+| 198
-|}
+| -273.442
-Distribución de varillado óptima en costo y eficiencia
+| 190.3260
+| 184.0664
-<div id='img-35'></div>
+| 131
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 0.93
+| 3.9
+| <math>0.35</math>
+| <math>16</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-seccion_cir01.png|525px|Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 2 Tabla [[#img-33|33]]-Modelo 01]]
+| <span style="font-size: 75%;">5 </span>
-|- style="text-align: center; font-size: 75%;"
+| 177
-| colspan="1" | '''Figura 35:''' Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 2 Tabla [[#img-33|33]]-Modelo 01
+| 35
-|}
+| -636.092
-<div id='img-36'></div>
+| 38.0866
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 73.7833
+| 376
+| 0.9
+| 4.7
+| <math>0.28</math>
+| <math>15</math>
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-seccion_cir02.png|525px|Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 1 Tabla [[#img-34|34]]-Modelo 02]]
+| <span style="font-size: 75%;">6 </span>
-|- style="text-align: center; font-size: 75%;"
+| 93
-| colspan="1" | '''Figura 36:''' Distribución de opción óptima de varillado en costo y eficiencia estructural correspondiente a la opción 1 Tabla [[#img-34|34]]-Modelo 02
+| 412
-|}
+| 75.236
-Diagramas de interacción para las opciones de varillado óptimas en costo para cada modelo estructural:
+| 85.0518
+| 62.5437
-<div id='img-37'></div>
+| 196
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 0.91
+| 4.1
+| <math>0.11</math>
+| 16
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVar_cir01.png|525px|Diagrama de interacción con varillas resultantes-Modelo estructural 01.]]
+| <span style="font-size: 75%;">7 </span>
-|- style="text-align: center; font-size: 75%;"
+| 109
-| colspan="1" | '''Figura 37:''' Diagrama de interacción con varillas resultantes-Modelo estructural 01.
+| 53
-|}
+| -619.890
-<div id='img-38'></div>
+| 81.5238
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+| 163.9962
+| 531
+| 0.94
+| 4.0
+| <math>0.71</math>
+| 11
+|-
+| <span style="font-size: 75%;">8 </span>
+| 98
+| 93
+| -479.857
+| 91.4848
+| 175.0743
+| 410
+| 0.96
+| 3.8
+| <math>0.07</math>
+| 16
+|-
+| <span style="font-size: 75%;">9 </span>
+| 53
+| 172
+| -610.107
+| 95.8926
+| 127.8633
+| 426
+| 0.91
+| 4.8
+| <math>0.20</math>
+| 22
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-diagIntVar_cir02.png|525px|Diagrama de interacción con varillas resultantes-Modelo estructural 02.]]
+| <span style="font-size: 75%;">10 </span>
-|- style="text-align: center; font-size: 75%;"
+| 58
-| colspan="1" | '''Figura 38:''' Diagrama de interacción con varillas resultantes-Modelo estructural 02.
+| 73
+| -444.694
+| 37.9420
+| 99.0011
+| 188
+| 0.90
+| 4.3
+| <math>1.47</math>
+| <math>13</math>
 |}
-==13 Discusión y conclusiones==
+As for the same structural model taken in the PSO-ISR and GA-ISR results, the following convergence for this Steepest Gradient Descent method was found:
-El método aparte de ser muy fácil de programar en una computadora ofrece una nueva forma para analizar y diseñar estructuras de concreto reforzado. Sin duda que aún queda mucho por desarrollar y este puede es el comienzo de nuevas investigaciones en material de optimización de estrucruras de concreto, ya que como tal este método también se implementó en le presente trabajo una optimización, aunque con un método muy sencillo de búsqueda, más sin embargo los resultados obtenidos son los óptimos de acuerdo a los requerimientos de diseño.
+'''Structural parameters:'''
-==14 Agradecimientos==
+<math>b(width-Section)=50cm</math>
-Agradezco a la Universidad de Guanajuato, mi casa de estudios, por esta oportunidad, así como especial agradecimiento al Aula CIMNE-UG, que representó para mi un espacio de ideas y centro de exploración organizacional. A mi director de tesis el Dr. Alejandro Hernández Martínez y a mi tutor proyectos en el Aula CIMNE a la cual también dicho trabajo forma parte, el MC. Humberto Esqueda Oliva, pues con el Aula CIMNE se desarrolló el programa computacional como tal en lenguaje C principalmente con MatLab para la parte gráfica, y para la parametrización visual de los elementos estructurales y su acero de refuerzo.
+<math display="inline">h(height-Sectin)=80cm</math>
-==15 Anexo 1. Demostraciones de las variables envueltas en los casos de análisis de columnas rectangulares==
+<math display="inline">rec(steel-cover)=5cm</math>
-====Caso 1====
+<math display="inline">E=2.1e6\frac{Kg}{cm^{2}}</math>
-<div id='img-39'></div>
+<math display="inline">f_{y}=4200\frac{Kg}{cm^{2}}</math>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
+<math display="inline">f'c=280\frac{Kg}{cm^{2}}</math>
-|[[Image:Draft_Verduzco Martinez_325285613-fig49.png|332px|Demostración 1.1 para el caso 1 de columnas rectangulares.Dibujo propio.]]
-|- style="text-align: center; font-size: 75%;"
+<math display="inline">maxEfficiency=99.99%</math>
-| colspan="1" | '''Figura 39:''' Demostración 1.1 para el caso 1 de columnas rectangulares.Dibujo propio.
-|}
-Por relación de triángulos:
-<math>\frac{0.003}{d}=\frac{0.0051}{d_{1}}</math>
+<math display="inline">P_{u}=-46Ton</math>
-<math>d=\frac{30}{51}d_{1}</math>
+<math display="inline">M_{ux}=12Ton\cdot m</math>
-====Caso 2====
+<math display="inline">M_{uy}=76Ton\cdot m</math>
-<div id='img-40'></div>
+<div id='img-20'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig50.png|395px|Demostración 2.1 para el Caso 2 de columnas rectangulares]]
+|[[Image:Review_942980062263-Graph_SGD_1t.png|224px|Convergence of the Steepest Gradient Descent method for the 1t-ISR]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 40:''' Demostración 2.1 para el Caso 2 de columnas rectangulares
+| colspan="1" | '''Figure 20:''' Convergence of the Steepest Gradient Descent method for the 1t-ISR
 |}
-<math>s=h_{3}+c-d_{1}</math>
-Por relación de triángulos:
+<div id='img-21'></div>
-<math>\frac{\varepsilon }{h_{3}-s}=\frac{0.003}{c}</math>
-Entonces:
-<math>\varepsilon =\frac{0.003}{c}(h_{3}-(h_{3}+c-d_{1}))</math>
-<math>\varepsilon=0.003(\frac{d_{1}}{c}-1)</math>
-====Caso 3====
-<div id='img-41'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig51.png|181px|Demostración 3.1 para el Caso 3 de columnas rectangulares]]
+|[[Image:Review_942980062263-1t_SGD_diagIntX.png|168px|Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 41:''' Demostración 3.1 para el Caso 3 de columnas rectangulares
+| colspan="1" | '''Figure 21:''' Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA
 |}
-Por relación de triángulos:
-<math>\frac{c}{0.003}=\frac{d_{1}}{0.003-0.0021}</math>
+<div id='img-22'></div>
-Entonces:
-<math>c=\frac{30}{9}d_{1}</math>
-====Caso especial====
-Sea <math>m_{max}</math> la razón de distribución líneal de esfuerzos para el limite máximo del caso especial, y <math>m_{min}</math> la razón de distribución líneal de esfuerzos para el límite mínimo: <math>m_{max}=\frac{d_{1}}{0.003-0.0021}</math>
-<math>m_{min}=\frac{d_{2}}{0.003+0.0021}</math>    <div id='img-42'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig52.png|326px|Demostración CE.1 para el Caso Especial de columnas rectangulares]]
+|[[Image:Review_942980062263-1t_SGD_diagIntY.png|168px|Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 42:''' Demostración CE.1 para el Caso Especial de columnas rectangulares
+| colspan="1" | '''Figure 22:''' Interaction diagram in the X-axis direction for the global optima 4t-ISR-GA
 |}
-De modo que se deberá cumplir la siguiente relación:
-<math>\frac{d_{2}}{0.0051}<=\frac{d_{1}}{0.0009}</math>
+Yielding a total reinforcing area equal to <math>121.58cm^{2}</math>, very similar to the one obtained with the GA-ISR, not so much for the PSO-ISR which yields <math>109.53cm^{2}</math>
-<math>\frac{9}{51}d_{2}<=d_{1}</math>    Luego: <math>\frac{9}{51}(h-rec-\frac{1}{2}t<=rec+\frac{1}{2}t)</math>   Y simplificando, se tiene:
+===4.5 Comparisons between the different ISR optimization formulations===
-<math>\frac{60}{102}t>=\frac{9}{51}h-\frac{60}{51}rec</math>
+As mentioned before, the performance of the GA depends a great deal on the mutation probability parameter. In order to see the effects of such parameter on the final results a total of nine experimentation runs were done, based on the same structural model used previously with <math>{f'}_{c}=280\frac{Kg}{cm^{2}},b=50cm,h=80cm,rec=5cm, P_{u}=-46Ton,M_{ux}=12Ton\cdot m,M_{uy}=76Ton\cdot m</math>, changing the mutation probability from <math>p_{mu}=0.01</math> to <math>p_{mu}=0.05</math> for 100 generations '''Fig. [[#img-23|23]]'''.
-Y ahora, por relaciones trigonométricas
+<div id='img-23'></div>
-<math>\frac{c-d_{1}}{\varepsilon _{1}}=\frac{c}{0.003}</math>
-<math>\varepsilon _{1}=0.003(1-\frac{d_{1}}{c})</math>
-Y además:
-<math>\frac{0.003}{c}=\frac{\varepsilon _{2}}{d_{2}-c}</math>
-<math>\varepsilon _{2}=0.003(\frac{d_{2}}{c}-1)</math>
-====Caso 4====
-<div id='img-43'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig53.png|185px|Demostración 4.1 del Caso 4.]]
+|[[Image:Review_942980062263-9runs_GA.png|556px|Experimentation results with the GA-ISR for different values of the mutation probability from 0.01 to 0.05]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 43:''' Demostración 4.1 del Caso 4.
+| colspan="1" | '''Figure 23:''' Experimentation results with the GA-ISR for different values of the mutation probability from <math>0.01</math> to <math>0.05</math>
 |}
-Por relaciones trigonométricas:
-<math>\frac{h_{4}+d_{1}}{0.003-0.0021}=\frac{c}{0.003}</math>
+The summary of the relevant results are presented in the next table:
-Por lo tanto:
-<math>h_{4}=(1-\frac{21}{30}-\frac{d_{1}}{c})c</math>
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-4'></span>Table. 4 Summary of the results found for different mutation probability parameters for 100 generations with the GA-ISR
+|-
+|
+|
+| <math>%</math>
+| <math>cm^{2}</math>
+| <math>cm</math>
+| <math>cm</math>
+| <math>cm</math>
+| <math>cm</math>
+|-
+|  Run
+| <math>p_{mu}</math>
+| <math>Eff</math>
+| <math>Area_{est}</math>
+| <math>t_{1}</math>
+| <math>t_{2}</math>
+| <math>t_{3}</math>
+| <math>t_{4}</math>
+|-
+|  <span style="font-size: 75%;">1 </span>
+| 0.01
+| 98.64
+| <math>131.7</math>
+| 0.002
+| 0.240
+| 1.234
+| 0.510
+|-
+| <span style="font-size: 75%;">2 </span>
+| 0.015
+| 99.88
+| <math>166.4</math>
+| 1.795
+| 1.112
+| 0.561
+| 0.154
+|-
+| <span style="font-size: 75%;">3 </span>
+| 0.02
+| 97.20
+| <math>130.5</math>
+| 1.764
+| 0.009
+| 0.052
+| 0.799
+|-
+| <span style="font-size: 75%;">4 </span>
+| 0.025
+| 98.7
+| <math>153.8</math>
+| 0.072
+| 1.081
+| 0.620
+| 0.917
+|-
+| <span style="font-size: 75%;">5 </span>
+| 0.03
+| 99.9
+| <math>150.7</math>
+| 0.828
+| 0.910
+| 0.325
+| 0.835
+|-
+| <span style="font-size: 75%;">6 </span>
+| 0.035
+| 99.5
+| <math>126.51</math>
+| 1.030
+| 0.231
+| 0.713
+| 0.374
+|-
+| <span style="font-size: 75%;">7 </span>
+| 0.04
+| 90.69
+| <math>144.34</math>
+| 2.151
+| 0.437
+| 0.070
+| 0.512
+|-
+| <span style="font-size: 75%;">8 </span>
+| 0.045
+| 99.8
+| <math>128.9</math>
+| 1.818
+| 0.050
+| 0.012
+| 0.512
+|-
+| <span style="font-size: 75%;">9 </span>
+| 0.05
+| 99.5
+| <math>163.21</math>
+| 1.313
+| 1.237
+| 0.861
+| 0.013
+|}
-O también:
+From the results presented can been observed that there is not a clear pattern for the area found for each run, although the minimum area thus found corresponds for a mutation probability of 0.035, followed by one of 0.45. It could be recommended to use mutation probabilities of <math>[0.03<p_{mu}<0.45]</math> for faster and better convergence.
-<math>h_{4}=(1-\frac{21}{30}-\frac{d_{1}}{c})c</math>
+As for the PSO-ISR formulation, it is of interest to see its efficiency of convergence for different values for the parameter <math>number_{total-iterations}</math> mainly, higher than 10, as it was experimented previously. The following table presents the results obtained for 6 experimentation with the same structural model <math>{b=50cm,h=80cm,rec=5cm,f'}_{c}=280\frac{Kg}{cm^{2}},P_{u}=-46Ton,M_{ux}=12Ton\cdot m, M_{uy}=76Ton\cdot m</math>.
-====Caso 5====
-<div id='img-44'></div>
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|+ style="font-size: 75%;" |<span id='table-5'></span>Table. 5 Minimum area found for different number of iterations for the PSO-ISR
+|-
+|
+|
+| <math>%</math>
+| <math>cm^{2}</math>
+| <math>cm</math>
+| <math>cm</math>
+| <math>cm</math>
+| <math>cm</math>
+|-
+|  Run
+| <math>N_{total_runs}</math>
+| <math>Eff</math>
+| <math>Area_{est}</math>
+| <math>t_{1}</math>
+| <math>t_{2}</math>
+| <math>t_{3}</math>
+| <math>t_{4}</math>
+|-
+|  <span style="font-size: 75%;">1 </span>
+| 10
+| 98.64
+| <math>118.91</math>
+| 0619
+| 0.300
+| 0.736
+| 0.437
+|-
+| <span style="font-size: 75%;">2 </span>
+| 15
+| 99.88
+| <math>111.55</math>
+| 0.539
+| 0.187
+| 0.570
+| 0.608
+|-
+| <span style="font-size: 75%;">3 </span>
+| 20
+| 97.20
+| <math>112.73</math>
+| 0.673
+| 0.067
+| 0.803
+| 0.384
+|-
+| <span style="font-size: 75%;">4 </span>
+| 25
+| 98.7
+| <math>103.47</math>
+| 0.186
+| 0.010
+| 0.739
+| 0.575
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig54.png|147px|Demostración 5.1 para el Caso 5.]]
+| <span style="font-size: 75%;">5 </span>
-|- style="text-align: center; font-size: 75%;"
+| 30
-| colspan="1" | '''Figura 44:''' Demostración 5.1 para el Caso 5.
+| 99.9
+| <math>99.71</math>
+| 0.574
+| 0.011
+| 0.573
+| 0.517
+|-
+| <span style="font-size: 75%;">6 </span>
+| 35
+| 99.5
+| <math>112.70</math>
+| 0.048
+| 0.053
+| 0.995
+| 0.557
 |}
-Por relaciones trigonométricas:
-<math>\frac{0.003}{c}=\frac{0.0009}{h_{5}}</math>
+With its respective convergence graphs for each experiment '''Fig. [[#img-24|24]]'''
-====Caso 6====
+<div id='img-24'></div>
-<div id='img-45'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Verduzco Martinez_325285613-fig55.png|146px|Demostración 6.1 para el Caso 6]]
+|[[Image:Review_942980062263-6runsPSO.png|584px|Experimentation results with the PSO-ISR for different number of total iterations]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 45:''' Demostración 6.1 para el Caso 6
+| colspan="1" | '''Figure 24:''' Experimentation results with the PSO-ISR for different number of total iterations
 |}
-Por relaciones trigonométricas:
-<math>\frac{0.0009}{d_{2}}\frac{0.003}{c}</math>
+It is clear to see that the more the total of iterations for the PSO-ISR nested formulation, the lower the reinforcing areas, but with a much higher computational time. It could be observed from the table above that the difference between the resulting reinforcing area with 10 iterations is not really significant compared with the resultant one with 30 iterations (3 times more computational power and only a few square centimetres). It would be recommended to take 20 iterations with quite good results (much better than with the GA-ISR).
-Entonces:
+====4.5.1 Additional commentaries and further research====
-<math>c=\frac{30}{9}d_{2}</math>
+The following step once the an optimal reinforcing area with the ISR has been obtained is to transform such area to steel bars. There might be different ways to do that, either using another optimization algorithm to optimize topologically the arrangement, position and diameter of each bar over the cross section of an element along each face or simply considering a homogeneous distribution of bars with only one diameter along each cross section face, as it would be more practical. In <span id='citeF-13'></span>[[#cite-13|[13]]] for instance, the GA was used to optimize topologically the reinforcing steel bars once the 1t-ISR method had been used with acceptable results, but regarding a 4t-ISR method, given that a whole optimization procedure formulation has already been carried on, it would suffice to pass from this reinforcing area for each face of the cross section to a homogeneous distribution of reinforcing bars '''Fig. [[#img-25|25]]''' minimizing thereafter the construction costs even in a greater deal.
-==16 Anexo 2. Demostraciones del desarrollo del programa==
+Although the number of possible combinations of reinforcing bars could still be high, it would surely take less time to find a good optimum arrangement of reinforcing bars than employing a whole topological optimization as done by <span id='citeF-13'></span>[[#cite-13|[13]]]. This way of reinforced concrete optimization could be a good matter of further research with its application on 3D concrete frames to better appreciate its advantages.
-===16.1 Geometría analítica para el cálculo de eficiencias mecánicas===
+<div id='img-25'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-Ecuación de la recta A:
-{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|
+|[[Image:Review_942980062263-4t-isr-topologies.png|447px|Transformation from the 4t-ISR formulation to reinforcing bars.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
-|-
+| colspan="1" | '''Figure 25:''' Transformation from the 4t-ISR formulation to reinforcing bars.
-| style="text-align: center;" | <math> y=\frac{y_{u}}{x_{u}}x </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
 |}
-Ecuación de la recta B:
+==5 Conclusions==
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+The different formulations for the ISR developed happen to be very simple to code for anyone having basic understanding of optimization methods. The time required is evidently higher for the meta-heuristic optimization formulations, being the PSO-ISR the most efficient and convenient to apply, due to its rapid convergence compared to the GA-ISR; as for a 1t-ISR formulation the Steepest Gradient Descent method would be the one more likely to adopt for common structural design of concrete structures, given that despite of its simplicity of formulation the results it evokes are quite acceptable, furthermore, it could be highly improved by adding different other conditions of convergence through a more accurate convergence analysis. Moreover, for the application of meta-heuristics for this ISR method, could be of greater impact in huge structural systems, being able to reduce its required execution time by adapting them to Multi-Objective optimization procedures taking on account simultaneously different structural features of a structural system, such as displacements, ductility of certain elements, costs or section dimensions.
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math> y=\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}}x+(y_{i+1}+(\frac{y_{i}-y_{i+1}}{x_{i+1}-x_{i}})x_{i+1}) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
-|}
-Igualando las ecuaciones anteriores por compatibilidad:
+In any case, the global application fo this method, ant its official recognition in structural engineering could enhance great advantages in the way concrete structures are designed.
-<span id="eq-29"></span>
+===BIBLIOGRAPHY===
-{| class="formulaSCP" style="width: 100%; text-align: left;"
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>  \frac{y_{u}}{x_{u}}x=\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}}x+(y_{i+1}+(\frac{y_{i}-y_{i+1}}{x_{i+1}-x_{i}})x_{i+1}) </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
-|}
-Despejando x de la ecuación anterior:
+<div id="cite-1"></div>
+'''[[#citeF-1|[1]]]'''  Building Requirements for structural concrete and commentary, American Concrete Institute ACI 318, 2019
-<span id="eq-30"></span>
+<div id="cite-2"></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+'''[[#citeF-2|[2]]]'''  Whitney CS. Cohen E., ''``Guide for ultimate strength design of reinforced concrete. ACI J'', 1956; 28(5):445-90. [Proceedings V.53].
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>  x=x_{r}=\frac{y_{i+1}+(\frac{y_{i}-y_{i+1}}{x_{i+1}-x_{i}})}{\frac{y_{u}}{x_{u}}-\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}}} </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
-|}
-Sustituyendo la Ecuación [ [[#eq-25|25]]] en Ecuación [ [[#eq-29|29]]].
+<div id="cite-3"></div>
+'''[[#citeF-3|[3]]]'''  Au FTK, Bai ZZ, ''Effect of axial load on flexural behaviour of cyclically loaded RC columns.'' Comput Concr 2006: 2(4):261-84
-<span id="eq-31"></span>
+<div id="cite-4"></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+'''[[#citeF-4|[4]]]'''  Walther R. Miehlbardt M., ''Dimensionnement des structures en Béton. Lausanne: Press Polytechniques et Universitaires Romandes; 1990''
-|-
-|
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>  y=y_{r}=\frac{y_{u}}{x_{u}}x_{r} </math>
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
-|}
-Donde: <math>y_{u}=P_{u}</math>
+<div id="cite-5"></div>
+'''[[#citeF-5|[5]]]'''  Hernánde Montes, Gil-Martin LM, Aschheim M., ''“The design of concrete members subjected to uniaxial bending and compression using reinforcement sizing diagrams”'', ACI Struct J 2005;102(1):150-8
-<math>x_{u}=M_{u}</math>
+<div id="cite-6"></div>
+'''[[#citeF-6|[6]]]'''  Hernández Montes E, Gil-Martín LM, Pasadas-Fernández M, Aschheim M., ''“Theorem of optimal reinforcement for reinforced concrete cross sections”'', Struct Multidiscip Optim 2008;36(5):509–21.
-<math>y_{r}=P_{r}</math>
+<div id="cite-7"></div>
+'''[[#citeF-7|[7]]]'''  Ho Jung Lee, Mark Aschheim, Hernández Montes E, Gil-Martín LM, Pasadas-Fernández M, ''“Optimum RC column reinforcement considering multiple load combinations”'', Struct Multidiscip Optim 2009;39;153-170.
-<math>x_{r}=M_{r}</math>
+<div id="cite-8"></div>
+'''[[#citeF-8|[8]]]'''  L. M. Gil-Martín, E. Hdez-Montes, M. Aschheim, ''“Optimal reinforcement of RC columns for biaxial bending”'', Materials and Structures Journal (2010), 43:1245-1256
-<math>y_{i}=P_{i}</math>
+<div id="cite-9"></div>
+'''[[#citeF-9|[9]]]'''  Aschheim M, Hernández-Montes E, Gil-Martín LM (2008), ''“Design of optimally reinforced RC beam, column and wall sections”'', J Struct Eng 134(2):169–188
-<math>x_{i}=M_{i}</math>
+<div id="cite-10"></div>
+'''[[#citeF-10|[10]]]'''  Bresler B., ''Design criteria for reinforced columns under axial load and biaxial bending'', ACI Proc 57:481–490
-<math>y_{i+1}=P_{i+1}</math>
+<div id="cite-11"></div>
+'''[[#citeF-11|[11]]]'''  Gouwens AJ (1975), ''Biaxial bending simplified, reinforced concrete columns'', ACI Special Publication SP-50. American Concrete Institute, Detroit
-<math>x_{i+1}=M_{i+1}</math>
+<div id="cite-12"></div>
+'''[[#citeF-12|[12]]]'''  Luis F. Verduzco, A. Hernández, ''“Optimization of reinforcing steel for the design of concrete columns”'', University of Guanajuato, Bachelor Thesis to obtain the grade of Civil Engineer, Mexico, 2019
-==17 Referencias bibliográficas==
+<div id="cite-13"></div>
+'''[[#citeF-13|[13]]]'''  Luis F. Verduzco, ''“Optimization of reinforcing steel with genetic algorithms for the design of concrete columns”'', 14th Research Colloquium, Autonomous University of Queretaro, Mexico, 2020
-===BIBLIOGRAFÍA===
+<div id="cite-14"></div>
+'''[[#citeF-14|[14]]]'''  M. Afzal, Y. Liu, Jack C.P.Cheng, V. J.L. Gan, ''“Reinforced concrete structural design optimization: A critical review”'', Journal of Cleaner Production-Elsevier, 2020
-<div id="cite-1"></div>
+<div id="cite-15"></div>
-'''[1]'''  Adolfo Iván Jiménez P. y Dra. Sonia Elda Ruiz Gómez, Factores óptimos de cargas para el diseño de columnas esbeltas, Tesis de Licenciatura, UNAM, México, 2013   <div id="cite-2"></div>
+'''[[#citeF-15|[15]]]'''  A. Zhou, Bo-Yang Qu, Hui Li, Shi-Zheng Zhao, P. Nagaratnam, Q. Zhang, ''“Multiobjective evolutionary algorithms: A survey of the state of the art”'', Journal of Swarm and Evolutionary Computation, doi:10.1016/j.swevo.2011.03.001
-'''[2]'''  Steven C. Chapra, Raymond P. Canale, Métodos numéricos para ingenieros, Quinta edición, McGraw Hill  <div id="cite-3"></div>
-'''[[#citeF-3|[3]]]'''  Gaceta Oficial de la Ciudad de México, Normas técnicas complementarias de la Ciudad de México, 2017, Normas Técnicas Complementarias para Diseño y Construcción de Estructuras de Concreto 2017  <div id="cite-4"></div>
-'''[[#citeF-4|[4]]]'''  González Cuevas, Fco. Robles Hernández, Aspectos básicos del concreto reforzado, 4ta edición, Limusa 2005  <div id="cite-5"></div>
-'''[5]''' Jack McCormac y Rusell H. Brows, Diseño de concreto reforzado, 14va edición, Alfaomega, 2015   <div id="cite-6"></div>
-'''[[#citeF-6|[6]]]'''  Turner, M. J., Clough, R. W., Martin H. C. and Topp, L. J. Stiffness and Deflection Analysis of Complex Structures . Journal of the Aeronautical Sciences, Vol. 23 No. 9, 1956 pp. 805-823.  <div id="cite-7"></div>
-'''[7]'''  Yoeber Castro Atau, “Sistematización de detalles, habilitación y armado de acero ASTM A615 para construcciones de concreto armado: impacto técnico, económico y ambiental”, Universidad Nacional de San Cristobal de Huamanga, 2010
-</span>