Prediction of moment redistribution capacity in reinforced concrete beams using gene expression programming
Introduction
In structural engineering, it is aimed to obtain the most realistic and most economical solutions while designing. Three solution methods are generally emphasized in the literature; Elasticity Theory Method, Ultimate Strength Method and Plasticity Theory Method. Material failure is occurred in the Elasticity Theory Method while sectional failure is observed in the Ultimate Strength Method. In addition, system failure occurs in the Plasticity Theory Method and this method offers more realistic results compared to the other two methods. The inelastic behavior of the material is fully utilized by using Plasticity Theory Method, and a moment transfer takes place from the sections that have reached their capacity in statically indeterminate systems to the sections that have not reached their capacity yet, by means of plastic rotations. This bending moment transfer phenomenon is called as “moment redistribution” in the literature [1]. In a RC system, the bending moment transfer takes place from the support section to the mid-span section if the stiffness of the support section is less than the mid-span section and/or the support section is exposed to more bending moment as against mid-span section. This transfer phenomenon is expressed as positive moment redistribution in the literature [2]. However, if the opposite is the case, it is stated as negative moment redistribution.
In the arrangement of reinforcement of a RC system, moment redistribution enables flexibility and this phenomenon is very useful for practical design [3]. For this purpose, while the reinforcement concentration in the support sections are decreased, the reinforcement concentration in the mid-span sections are slightly increased, and decreasing reinforcement concentration in the support sections facilitates the settlement of the concrete, facilitates the tying of reinforcement, and increases the adhesion performance of the reinforcement to the concrete in the support sections, especially beam–column joints. In addition, an economical design is performed as the reserve capacity of the material is fully utilized and savings of reinforcement occurs thanks to the moment redistribution [4], [5]. Especially, when the ratio of live load to dead load is larger, this economic design consideration can has significance [6].
Examining the various international RC design codes around the world, it is seen that these codes allow the use of the elastic analysis method in the ultimate limit state (ULS). However, these codes take into account the nonlinear behavior of RC systems by permitting a limited amount of redistribution of bending moment from one part of the structure to another [3]. In these codes, “linear elastic analysis with limited redistribution” principle is considered by using Ultimate Strength Method. It is aimed to approach the failure load obtained from system failure as much as possible thanks to this widely used method [6]. According to RC design codes, it can be expressed that considering the bending moment diagram determined by using the linear elastic analysis with limited redistribution approach is more realistic in comparison with using the bending moment diagram determined directly from the linear elastic analysis [7].
Gene expression programming (GEP) is one of the soft computing modelling tool that has been frequently used in many civil engineering studies to predict mathematical relations based on experimental studies as an efficient alternative to traditional regression. There are many studies in the literature performed on RC beams using GEP such as, prediction of flexural strength for beams reinforced with FRP bars [8], proposing a formulation for shear strength of slender RC beams [9], and proposing a model for shear strength of steel fibers RC beams [10]. However, in the literature review, it is not available any study regarding the determination of the moment redistribution with the GEP model. Therefore, in this study, it is aimed to predict the moment redistribution of RC continuous beams by GEP. For this reason, experimental data of 108 RC continuous beams collected from the literature was taken into account to determine the moment redistribution by the GEP to provide a reliable predictive model. The performance of the model was demonstrated by comparing with experimental results and the results of the equations related to the moment redistribution existing in the current design codes in the literature. Besides, a parametric study and the sensitivity analysis were also performed to investigate the effects of parameters on moment redistribution of RC continuous beams.
Section snippets
Overview of existing formulations and design codes provisions
In the international RC design codes, “linear elastic analysis approach with limited redistribution” is taken into account in the ultimate limit state (ULS) for RC beams, while the control of structural ductility is not performed. This situation can be interpreted as a deficiency in these codes in general. Most RC design codes determine the redistribution of bending moments by taking into account the neutral axis depth factor . The reason for this is that this parameter characterizes the
Summary of experimental data
The redistribution of moments in RC structures can also be determined by means of models developed with various mathematical and statistical methods. In the study by [25], moment redistribution was tried to be estimated using artificial neural networks (ANN) and support vector regression (SVR) methods, and the researchers [25] investigated the effects of the parameters of concrete compressive strength , reinforcement yield strength , neutral axis depth factor , stirrup ratio ,
Gene expression programming
Gene expression programming (GEP) is a genetic algorithm that uses populations of individuals and introduces genetic variation using one or more genetic operators. The chromosomes and expression trees are main determinants in gene expression programming [38]. The flowchart of a gene expression algorithm (GEA) is shown in Fig. 3a.
As shown in Fig. 3a, the first process begins with the creation of the chromosomes of the first population. The chromosomes are then expressed, and after each
Evaluation of proposed GEP model
The degree of agreement between the proposed GEP model and experimental results can be measured by coefficient of correlation (R). Statistically, value approaching 1 indicates strong correlation. As shown in Fig. 5, the values between the predicted GEP results for both of the training and validation datasets and the experimental results were obtained as 0.807 and 0.834, respectively. As in [41], [44], [45], values for training and validation datasets are related to the random choosing of
Sensitivity and parametric analyses
In prediction of the moment redistribution (β), the effect of the input parameters to the target model was determined through sensitivity and parametric analyses. The predicted values of obtained by GEP-based model as a function of each parameter to find the effect of each input parameter on the output parameter are presented in Fig. 8. The changes in were illustrated against the change in the value of each input parameter from maximum to minimum [46], [47], [48].
A sensitivity analysis (SA)
Conclusions
In this study, considering 108 data points collected from the literature regarding the moment redistribution of RC continuous beams, a new formulation was proposed with GEP and the outcomes of the formulation statistically was compared with the results of formulations given in the current design codes and the experimental studies existing in the literature. The proposed formulation is valid in the ranges 0.003 – 0.026 for ρ, 0.16 – 1.14 for , 18.02 – 66.23 for , 0.04 – 0.47 for
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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