Elsevier

Applied Soft Computing

Volume 19, June 2014, Pages 112-120
Applied Soft Computing

Linear genetic programming for shear strength prediction of reinforced concrete beams without stirrups

https://doi.org/10.1016/j.asoc.2014.02.007Get rights and content

Highlights

  • We have introduced LGP algorithm for shear capacity modeling of RC beams without stirrups.

  • An extensive experimental database including 1938 test results gathered from literature.

  • A simplified LGP based formula is obtained for different kinds of concrete.

  • Our results are better than the nine different code models.

Abstract

A new design equation is proposed for the prediction of shear strength of reinforced concrete (RC) beams without stirrups using an innovative linear genetic programming methodology. The shear strength was formulated in terms of several effective parameters such as shear span to depth ratio, concrete cylinder strength at date of testing, amount of longitudinal reinforcement, lever arm, and maximum specified size of coarse aggregate. A comprehensive database containing 1938 experimental test results for the RC beams was gathered from the literature to develop the model. The performance and validity of the model were further tested using several criteria. An efficient strategy was considered to guarantee the generalization of the proposed design equation. For more verification, sensitivity and parametric analysis were conducted. The results indicate that the derived model is an effective tool for the estimation of the shear capacity of members without stirrups (R = 0.921). The prediction performance of the proposed model was found to be better than that of several existing buildings codes.

Introduction

Although several research programs have been conducted to predict the shear capacity of concrete (e.g. [1]), there is still no clear expression to predict the shear failure mechanisms of concrete elements. Most of the available shear design expressions have different forms and do not provide a consistent factor of safety against shear failure. Thus, the behavior of concrete beam has been extensively investigated during the last three decades. Numerous theoretical models have been established in recent years to investigate the interaction between several forces including axial, shear, bending, and torsion [2], [3].

Recently, application of machine learning has attracted much attention for solving structural engineering problems. The machine learning systems are powerful tools for design of computer programs. They automatically learn from experience and extract various discriminators [4]. Artificial neural networks (ANNs) are the most widely used branch of machine learning. There have been some researches focusing on the application of ANNs to the evaluation of the shear strength of reinforced concrete beams without reinforcement. Recently, Choi et al. [5] used another machine learning method, namely fuzzy logic (FL) for the modeling of the shear strength of slender reinforced concrete beams. Although ANNs and FL are successful in prediction, they are not usually able to produce practical prediction equations. Furthermore, for the ANN-based modeling, the structure of the network should be identified a priori. Besides, determination of the fuzzy rules in FL is a difficult task. These methods are mostly appropriate to be used as a part of a computer program [6]. Genetic programming (GP) [7] is a developing subarea of the machine learning techniques. GP is known as an extension genetic algorithm (GA) where the solutions are computer programs rather than fixed length binary strings [6]. Classical (standard) GP and its variants have been recently employed to derive greatly simplified formulations for structural engineering problems and especially concrete structures modeling (e.g. [8], [9], [10]). Linear genetic programming (LGP) [11] is a new subset of GP. LGP operates on computer programs that are represented as linear sequences of instructions of an imperative programming language [6]. In contrast with ANNs, GA and classical GP, application of LGP in the field of civil engineering is quite new and restricted to a few areas [6], [12], [13], [14], [15], [16]. It is worth mentioning that classical GP and new variants of GP have been, respectively, used by Ashour et al. [17] and Gandomi et al. [18], [19] to predict the load capacity of RC deep beams. Gandomi et al. [20] have applied LGP to the modeling of fibrous RC beams. Recently, Kara [21] employed GP for the prediction of the shear strength of FRP-reinforced concrete beams without stirrups upon a limited number of experimental test results. Moreover, Pérez et al. [22] applied GP to the optimal adjustment of EC-2 shear formulation for RC beams without web reinforcement.

However, applications of the GP-based approaches to directly obtain a simple formula to predict the shear strength of RC beams are conspicuous by its absence. There are approaches which present simple formulation, but based on other advanced approaches [23]. The main purpose of this study is to utilize the LGP technique to build a simple predictive model for the shear strength of RC beams without stirrups. The shear strength was formulated in terms of shear span to depth ratio, concrete cylinder strength at date of testing, amount of longitudinal reinforcement, lever arm and maximum specified size of coarse aggregate. The predictions made by models were further compared with those provided by several well-known building codes.

Section snippets

Machine learning

Machine learning is a branch of artificial intelligence essentially inspired from biological learning. The machine learning approach deals with the design of computer models that are able to automatically learn with experience [4], [24]. The machine learning methods extract knowledge and complex patterns from machine readable data [4]. The major focus of the machine learning research is on data mining problems, difficult-to-program applications, and software applications customizing to the

Formulation of the shear strength of RC beams without stirrups

To develop the LGP model, the shear strength of RC beams without stirrups was considered to be a function of the following influencing parameters:vu=ffc,ad,ρ,Z,agwhere vu (MPa) is the shear strength (vu = Vu/(bwd)); Vu (kN) is the ultimate shear force; bw (mm) is the web width; d (mm) is the effective depth; Z is the mechanic arm (Z = 0.9d); fc (MPa) is the 28-day concrete compressive strength; a/d is the shear span to depth ratio; (%) is the amount of longitudinal reinforcement (ρ = A/bwd); A is

Parametric analyses

For verification of the LGP-based prediction model, a parametric analysis was performed in this study. The parametric analysis investigates the response of the predicted shear strength from the LGP model to a set of hypothetical input data generated over the ranges of the minimum and maximum data used for the model training. The methodology is based on the change of only one input variable at a time while the other variables are kept constant at the average values of their entire data sets. A

Model validity

Smith [53] suggested the following criteria for judging performance of a model:

  • if a model gives |R| > 0.8, a strong correlation exists between the predicted and measured values.

  • if a model gives 0.2 < |R| < 0.8 a correlation exists between the predicted and measured values.

  • if a model gives |R| < 0.2, a weak correlation exists between the predicted and measured values.

In all cases, the error values (e.g., MAE) should be at the minimum [54]. The model can therefore be judged as very good. Based on the

Comparative study

A comparative study was conducted to benchmark the proposed LGP model against several codes of practice including ACI 446 [58], ACI 318 [59], ASCE-ACI 445 [60], CEB-FIB [61], CSA [62], EC-2 [63], NZS [64] and ICC [65]. In order to evaluate the capabilities of the models, the RMSE, MAE and R performance measures were used. The prediction performance of different models on for entire database is summarized in Table 6. Besides, Fig. 8 visualizes the histogram plots of the ratio of the experimental

Conclusions

The LGP technique was utilized to formulate the strength capacity of RC beams. A reliable database from the literature was used for developing the model. The proposed design equation gives reliable estimations of the strength capacity of RC beams without stirrups. The validity of the model was tested for a part of test results beyond the training data domain. Furthermore, the derived prediction model efficiently satisfies the conditions of different criteria considered for its external

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