Elsevier

Engineering Structures

Volume 32, Issue 11, November 2010, Pages 3452-3466
Engineering Structures

Optimal adjustment of EC-2 shear formulation for concrete elements without web reinforcement using Genetic Programming

https://doi.org/10.1016/j.engstruct.2010.07.006Get rights and content

Abstract

This paper presents the improvement of the EC-2 shear strength formulation for concrete beams without shear reinforcement. The method used is based on the modification of the Genetic Programming (GP) technique configured to generate symbolic regression from a set of experimental data. Starting from the EC-2 formulation, several points of the equation are subject to optimization, together with the set of restrictions that must be fulfilled, achieving in this way to control and direct the searching process. It is specifically pursued to improve the term of size effect, the influence of the amount longitudinal reinforcement and the bending-moment–shear-force interaction. For the development and checking of the models it has been used in about 1200 experimental tests on concrete beams from the literature. The four expressions obtained are analyzed in depth through GP and, finally, three expressions of great simplicity are proposed that improve, significantly, the shear strength prediction with respect to the proposals of the Eurocode 2 and ACI 318-05 Code.

Introduction

The shear strength in concrete beams without shear reinforcement has been, undoubtedly, one of the most controversial aspects bound to the Ultimate Limit States. It exists a wide range of theoretical models of high complexity that, due to the difficulty of capturing the importance of all the variables involved, they result in different ruling proposals, being to a great extent about formulation cases of empirical origin, as is the case of the European code EC-2 [1] and the American code ACI 318-05 [2].

Once the diagonal tension cracks develop in the web of a slender beam without shear reinforcement, there exists several transfer mechanisms: the shear stress in the no-cracked concrete of the compression head, the shear transferred in the surface of the fissures and the dowel action of the longitudinal reinforcement [3], [4]. These different mechanisms are translated in a simplified way into the internal strength developed in a cracked beam, as is shown in Fig. 1.

Kani showed the importance of arch action in not very slender beams [5]. Its importance is inversely proportional to the relation between the shear span and the effective depth, a/d. In beams with a coefficient a/d lower than 2.5, slanting fissures are developed and, after an internal redistribution of stress, the beams are capable of resisting a significant load increase because the applied strength can be transmitted directly to the supports through the appearance of compressed struts in the concrete. In the case of beams with a/d higher or equal to 2.5, this effect loses importance, as is observed in Fig. 2.

The importance of the size effect was not suggested until 1967. Kani [7] showed experimentally that raising the depth of a beam, keeping the rest of the parameters constant, reduced the failure shear stress. Raising the depth of the beam, the fissure width tends to increase. Some authors consider that this entails a reduction of the effect of aggregate interlock leading to a decrease of the shear strength [8]. Fig. 3 shows the result of Shioya’s test [9] in which the great influence of the size effect can be observed. Besides, some authors [8], [10], [11] relate the size effect to the compression resistance of the concrete, suggesting that the higher the concrete compressive strength the more the reduction of tangential stress at cracking due to the size effect is emphasized.

The influence of the aggregate maximum size is also shown in Fig. 3. A decrease of the aggregate maximum size produces a fall of the shear crack stress. However, this parameter is not always perfectly known to engineers during the designing process, and most concrete codes do not take into account this influence in the shear strength formulations. This is the case for the Eurocode 2 [1], the ACI 318-05 Code [2] and the Spanish Concrete Code EHE-08 [12]. However, this influence is considered in more complex models [13], [14].

According to the experimental results, the different transfer mechanisms depend mainly on the concrete resistance, on the ratio of longitudinal reinforcement, on the effective depth and on the shears span to depth ratio a/d.

Due to its simplicity and the generally acceptable correlation against experimental results, the equation given in the Eurocode 2 is being extended to different national codes of practice, as it is the case of the Spanish Concrete Code EHE-08 [12]. However, one of the limitations that pose for elements without shear reinforcement, is the fact that the EC-2 procedure does not take into account the bending-moment–shear-force interaction, except for the need to check that the longitudinal tension reinforcement is able to resist the additional tensile force caused by shear. To a given section, according to the EC-2 formulation, the shear strength is independent from the concomitant bending moment if the last is kept away from which produces the plastification of the longitudinal reinforcement. On the contrary, most complex models as the Modified Compression Field Theory (MCFT) [13] predict a reduction in the shear strength as the concomitant bending moment increases for any value of the bending moment (Fig. 4).

The treatment of the influence of the longitudinal reinforcement also varies noticeably from one code of practice to another. The formulation given by the Eurocode 2 propounds that shear strength is proportional to the amount of longitudinal reinforcement. However, other models propose that the shear strength is proportional to the ρlVd/M value, as in the case of one of the methods proposed in the ACI Code. In the real sizing of beams, the amount of longitudinal reinforcement grows proportionally with the concomitant bending moment, and for that reason, the ρlVd/M parameter is practically constant. However, it is usual in laboratory test to use disproportionately raised ρl values to avoid bending failures and, even to make series of tests in which a/d decreases, therefore decreasing the concomitant bending moment, without varying the longitudinal reinforcement. For this reason, some authors [14] supports that the adjustment of equations with these unrealistic tests can cause deviations for real elements with the usual combinations of the different variables. In the case of the given formulation by the Eurocode 2, an increase of the longitudinal amount would always mean an increase in the shear strength regardless of the concomitant bending moment.

In the most of the studies it is necessary to analyze test databases. In order to do this, A.I. techniques can be used, which are no longer unconnected with the field of the Civil Engineering. In scientific literature there are several contributions that use different techniques applied to the model of shear strength of concrete beams without shear reinforcement. It is important to mention examples of three different techniques: in the first place, Cladera and Marí [8], [15] and Jung and Kim [16] use the Artificial Neural Networks (ANNs) for their analysis. In the case of Cladera and Marí, the ANNs, after training according to the available experimental results, are used as a virtual laboratory, predicting values of tests that are not made physically. In this way, they manage to study the influence of each of the variables in the shear strength, formulating in the end two models. Jung and Kim use ANN as a retrieval mechanism of information, altogether with a shear database, to imitate problem-solving strategy of human. As a demonstration, they develop two models, the first one estimates shear strength and the second model systematically provides conservative estimation. In the second place, Choi et al. [17] develop a model that uses the fuzzy sets to predict the shear strength. Finally Ashour et al. [18] get an expression through Genetic Programming (GP) that, from previously standardized variables, is capable of predicting the shear strength in concrete beams.

All these approaches present essential improvements against the current codes of practice but, at the same time, they have deficiencies when applying the obtained method. The main drawback in the use of ANN is the impossibility to give an explicit expression to the result, that is, that the result obtained after the learning process is a black box that only gives results as the input stimulus are given, without relating at no time the input variables to the output value in an explicit way. The direct application and without restrictions of GP makes the complexity of the formulation to increase immeasurably for the obtaining of a good adjustment. Besides it is difficult to give a “physical” sense to the obtained formulation. In the cases mentioned, the application of ANN and GP is made standardizing the variables, although in GP would not be strictly necessary. The pre-processing and the post-processing of normalizing and denormalizing the variables respectively, gives complexity to the direct application of the expressions obtained about the data. And another characteristic factor is that guidance in the search process is not given for the obtaining of new expressions.

In this paper it is presented a method that enables to improve a mathematical expression, specifically the formulation given by the EC-2 for the shear strength of elements without shear reinforcement. Through specific restrictions it has been achieved to guide the search process of the models. Finally, it is obtained four expressions that improve significantly the behavior of the current formulations before real experimental tests that have not been used in the design, through training, of the expressions.

Section snippets

Eurocode 2

In the Eq. (1) it is shown the formulation adopted by the Eurocode 2 [1] in the case of beams without axial load and once the security factor has been removed. The value of the result of applying the equation always must be higher than the minimum value given by the Eq. (2). In Table 1 the different variables used in this formulation are shown. Vc=0.18k(100ρlfc)1/3bwdVRd,cmin=0.035k3/2fc1/2bwd.

ACI 318-05

ACI Code [2] presents two possible formulations for the calculation of shear strength in

Genetic Programming

Genetic Programming is a subset of search techniques of solution framed within the term of Evolutive Computation (EC). EC includes a set of methods based on models that emulate certain features of nature, fundamentally the capability that human beings possess to adapt themselves to their environment. This feature of human beings had already been embodied by Charles Darwin in his theory of evolution according to the principle of natural selection of species [20]. Darwin supports that those

Genetic Programming to improve the model of shear strength

As it has been said, one of the abilities that GP techniques possess is the symbolic regression on data. This means that given a set of data (input–output), the GP is capable of relating algebraically one to each other through an expression more or less complex that does not guarantee the dimensional integrity. This technique, applied to a lot of cases within Civil Engineering, it is the one followed by Ashour et al. [18] to obtain an expression that, from the previously normalized variables

Global comparison

In the first place, it has been carried out an analysis of the predictions by the different formulations for the whole of the DB2 database.

In Table 11 it is observed the results obtained. It is important to mention that it has been followed the modus operandi suggested by Collins [28], who considers asymmetric the distribution of the values Vtest/Vpred, with a median value generally lower than the average. Thereby, in the table it is presented the coefficient of variation (COV) for all the

Conclusions

It has been developed a GP algorithm valid for the adjustment of existent expressions, applied to the shear formulation for elements without shear reinforcement given by the Eurocode 2. This methodology goes further the mere adjustment of numerical values within an equation, since it allows the inclusion of expressions, if it is necessary, in different parts of the initial equation fulfilling the established restriction all the times.

The innovation of the developed algorithm resides in its

Acknowledgements

This work was partially supported by the Spanish Ministry of Education and Science (Ref. BIA2007-60197) and grants from the General Directorate of Research, Development and Innovation (Dirección Xeral de Investigación, Desenvolvemento e Innovación) of the Xunta de Galicia (Ref. 07TMT011CT, Ref. 08TIC014CT and Ref. 08TMT005CT). The work of Juan L. Pérez is supported by an FPI grant (Ref. BES-2006-13535) from the Spanish Ministry of Education and Science (Ministerio de Educación y Ciencia).

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