Optimization of existing equations using a new Genetic Programming algorithm: Application to the shear strength of reinforced concrete beams

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Abstract

A method based on Genetic Programming (GP) to improve previously known empirical equations is presented. From a set of experimental data, the GP may improve the adjustment of such formulas through the symbolic regression technique. Through a set of restrictions, and the indication of the terms of the expression to be improved, GP creates new individuals. The methodology allows us to study the need of including new variables in the expression. The proposed method is applied to the shear strength of concrete beams. The results show a marked improvement using this methodology in relation to the classic GP and international code procedures.

Introduction

On certain occasions there are contrasted theoretical formulations that allow finding a solution to a particular engineering problem, but there is not often a proven theoretical solution and it is necessary to resort to empirical formulations that are inferred from experimental results. The evolutionary computation is a tool that is capable to solve on its own and from experimental results, numerous problems in different fields as, for example, in Civil Engineering [1]. In this study field it appears different interests where artificial intelligence techniques can help to the science enrichment. In most of the problems a physical phenomenon is abstracted in a mathematical problem to simulate and predict such phenomenon. Since in most of the case study there has already been some available knowledge about a particular phenomenon, that is, there have already been different models that try to adjust the physical/chemical behavior through equations, the use of artificial intelligence techniques is of great interest for the optimization or improvement, if anything, of such models.

In scientific literature there are numerous approximations for the optimization of several processes. If we concentrate on the example field (structural engineering), most of the optimization processes are focused on the resource optimization, that is, on the execution of a specific element with the minimum of resources that are used but always guarantying the element security. An example is the job made by Perera and Vique [2]. In this paper the authors use the genetic algorithms for automatically producing optimal strut-and-tie models for the design of reinforced concrete beams. For this, they look for minimizing the axial force product, the length and axial strain of the truss elements.

Another example to quote is the one developed by Sonebi and Cevik [3]. In this case the authors use the Genetic Programming technique to find an equation for modelling the fresh properties and the compressive strength of self-compacting concrete (SCC) containing pulverized fuel ash (PFA), highlighting the obtaining of good results in spite of the fact that there are available few data.

As well as the evolutionary computation techniques, the artificial neural networks (ANNs) can be used to improve the physical model involved in a process. In this aspect, it is important to point out the job of Cladera and Marí [4], who uses the ANN for the analysis of the shear strength in concrete beams without shear reinforcement. In this case, and afterwards the training and verification process, the ANNs were used as a virtual laboratory, predicting test values that were not made physically. With the one that was developed, they get to study the dependence type facing each of the variables, finally formulating two design expressions that improve noticeably any of the ones developed by other authors or by other national or international codes. The main inconvenient in the use of ANN is the impossibility to give expression explicitly to the result, that is, the result that was obtained through the learning is a data recorder which only gives results according to the input stimulus, without relating explicitly the input values to the output values at no time. On the other hand, it is impossible to apply restrictions as the ones presented in the article through the use of ANN.

Nearer to our case study, it is found the job made by Ashour et al. [5]. In this case they obtain an expression through GP that, from the previously standardized variables, is capable of predicting the shear strength in concrete beams. This example differ mainly from the one presented here in two questions. In the first place, the variables have been standardized. In the second place, the search process is not directed anyway. Although it is obtained better results with a priori standardized data, it would entail not being able to apply the resultant formula immediately since it would be necessary to apply the standardization to the data. In any case, they get good adjustments from a database of only 141 beams tests indexed to scientific literature, although they do not compare them to the current codes of practice in spite of mentioning them.

Regarding the tendencies in the field of Genetic Programming, related to the orientation of the search process, they are synthesized in syntactic restrictions. For example, Koza uses this type of restrictions when generating new individuals [6]. There is a mechanism developed likewise by Koza [7], called “Automatically Defined Functions (ADFs)”, that it could be explained as a particular case of syntactic restrictions, since the ADF are functions or subroutines that are “reusable” by the Genetic Programming algorithm of a fixed structure that can evolve. Another type of restrictions would be the ones that involve the type of data, or the dimensional coherence of a result. In this case, Montana [8] proposes a “Strongly Type Genetic Programming method (STGP)” with it is achieved, for example, that the operator “sine” is only applied to variables that contain angles. Finally, there are the techniques based on “Grammar Guided Genetic Programming (GGGP)”, in which the genetic operations are conditioned by grammar that is defined by the user. In this grammar, called “Context Free Grammar (CFG)”, it lies the expert knowledge in the study area. For example, García-Arnau et al. [9] develops a method called “Grammar Based Initialization Method (GBIM)” that he uses with GGGP for classification tasks in Breast Cancer. More related to the case study of this job, Ralte and Sebag [10] use GGGP to create a behavior model of a material from experimental data.

Pérez et al. [11] have presented an algorithm that allows to improve a mathematical expression that is controlled by an expert on the basis of experimental data, leading the search process through restrictions given by the expert in the creation of new solutions. In the current article it is carefully presented the followed methodology, and it is compared to the results that would be obtained with classic techniques of GP. Besides, it is proposed a methodology to study the necessity or not to include certain variables that were not considered in the initially chosen formulation to be optimized. As an example, and as an illustration of its functioning, it has been chosen a problem that is enshrined within the structural engineering: the shear strength phenomenon in concrete beams. Besides, in the article it is presented how the consideration of a variable that was not initially included, the relation among the shear force and the concomitant bending moment allows to establish shear-moment interaction diagrams through two simple expressions, obtaining results that have a lot in common to the ones given by one of the most developed and complex theoretical models, the Modified Compression Field Theory [12].

Section snippets

Genetic Programming

Genetic Programming is a subset of solution search techniques enshrined within the term of evolutionary computation (EC). EC includes a set of methods based on models that emulate certain characteristics of nature, mainly the capacity that living beings possess to adapt themselves to their environment. This feature of living beings had been captured by Charles Darwin to make his theory of evolution according to the species natural selection principle [13]. Darwin holds that those individuals in

Genetic Programming to improve well-known equations

The use of GP to develop mathematical expressions is probably its most extended application [15]. Its way of codifying allows these to be represented in an easy way, and they have been applied in a great deal of different fields related to science or engineering [1], [16], [17]. The results that were found have been very beneficial and the expressions that were achieved have improved in a great number of occasions compared to the previous ones existent in this field.

However, GP should not be

Problem description

With the aim of proving the good functioning of the algorithm that has been developed in a real case, it has been used a problem that is enshrined within structural engineering: shear strength in concrete beams. This problem is one of the most controversial aspects linked to ultimate limit states in structural engineering, since the great complexity of the theoretical models makes it necessary to simplify in order to obtain standardized simple expressions. In fact, nowadays the current codes of

Results

In the following tables it is shown the adjustments obtained by the current standardized formulations and the equations proposed in this article against the database used. The equations that have been developed in this research work have been subdivided into three categories. The first one represents the equations that were generated through the algorithm developed in this article; the second one corresponds to the equations that were generated with classic GP; and, finally, the equations

Influence of the maximum aggregate size

The design procedures based on the Modified Theory of Compression Field [12] include as a design variable the maximum aggregate size, ag. Its influence on shear strength is based on the fact that the aggregate size is a main parameter in the shear-friction mechanism, which is, at the same time, one of the key mechanisms of shear strength. Its influence was inferred, at that moment, from the maximum aggregate size influence in the shear-friction response in type Z elements [28]. Its influence is

Bending moment-shear force interaction

The proposed 8H1 and 8I1 equations introduce the term of V · d/M or V/M, which reflects the influence of the relationship between the concomitant moment and shear force. Fig. 14 presents the diagram of ultimate bending moment-shear force interaction obtained with Eurocode 2, the proposed equations 8H1 and 8I1, equation 7G1 and through the software Response-2000 [29], based on the Modified Compression Field Theory [12].

The non-dimensional values of Fig. 14 were obtained usingμ=Mub·d2·fcω=Vub·d·fc1/

Conclusions

The algorithm that has been presented is a valid method for improving existing expressions in certain points that were chosen by the experts with different possible restrictions. The restrictions can refer both to the choice of the data set variables that can be used, and what operators (both terminal and non-terminal) are allowed in the generation of new individuals. With the proposed solution it is gone beyond mere numerical value adjustments within an expression. Although this case would be

Acknowledgements

This work was partially supported by the Spanish Ministry of Science and Innovation (Refs. BIA2007-60197 and BIA2010-21551) and Grants from the Ministry of Economy and Industry (Consellería de Economía e Industria) of the Xunta de Galicia (Refs. 08TMT005CT and 10TMT034E).

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