Predicting ultimate condition and transition point on axial stress–strain curve of FRP-confined concrete using a meta-heuristic algorithm

https://doi.org/10.1016/j.compstruct.2022.116387Get rights and content

Abstract

Accurately predicting key reference points on the axial stress–strain curve of fiber-reinforced polymer (FRP)-confined concrete is of great importance for the pre-design and modeling of structures manufactured with this composite system. This paper presents a detailed study on the development of accurate and practical expressions for predicting the ultimate condition and transition point, as key reference points, on axial stress–strain curves of FRP-confined concrete using generic programming (GP). A comprehensive data tuning and cross-validation analysis was firstly performed to develop prediction models. Afterwards, the accuracy and performance of the developed empirical expressions were examined by sensitivity analysis, parametric analysis and model validation. Finally, a comparison was made between the performance of these proposed expressions and that of the existing best-performing expressions in the literature using statistical analysis. Based on the sensitivity and parametric analysis of the database, it is shown that: compressive strength (f’cc) and axial transition strain (ɛc1) are more sensitive to FRP lateral stiffness (Kl); ultimate axial strain (ɛcu) is more sensitive to Kl-to-unconfined compressive strength (f’co) ratio and fiber ultimate tensile strain (ɛfu); hoop rupture strain (ɛh,rup) is more sensitive to fiber elastic modulus (Ef); and axial transition strength (f’c1) is more sensitive to f’co. It is also shown that the proposed expressions provided more accurate predictions of the ultimate condition and transition point on the axial stress–strain curve of FRP-confined concrete than the existing expressions. This was achieved by using a larger number of datasets and accurately capturing the effects of the most influential input parameters in the proposed expressions.

Introduction

Accurate prediction of the mechanical behavior of fiber-reinforced polymer (FRP)-confined concrete is of paramount significance to broadly use it as a high-performance structural system in the construction and infrastructure industry [1]. Axial stress–strain relationship is the most important indicator of the mechanical behavior of FRP-confined concrete under compression [2]. Two different approaches have been proposed to predict the mechanical behavior of FRP-confined concrete under compression [2]. These approaches include: design-oriented modeling in closed-form, which interprets experimental results [3], [4], [5], [6], [7], [8], [9], [10]; and analysis-oriented modeling, which explicitly evaluates the interaction between the FRP jacket and the concrete [11], [12], [13], [14], [15].

Although predicting the complete curve of axial stress–strain by analysis-oriented models offers a versatile method to analyse the mechanical behavior of FRP-confined concrete under compression loading, the accuracy of these models notably depends on the estimation of the dilation behavior of FRP-confined concrete [16], [17]. However, the existing analysis-oriented models could not accurately determine the dilation behavior of FRP-confined concrete [14], [18], [19], [20]. Although Lim and Ozbakkaloglu [21] developed a new approach to predict the dilation behavior of FRP-confined concrete with high accuracy, the iterative nature of their approach demanded a great computational effort. On the other hand, design-oriented models offer a simple and appealing method for prediction of the key reference points on the axial stress–strain curve of FRP-confined concrete by using experimental datasets and data-driven analysis. Although the design-oriented models are easy to use, their accuracy significantly depends on the database consistency as well as the mathematical expressions used during the modeling procedure [9], [16], [22], [23].

The ultimate condition (ultimate axial stress (f’cc) and strain (ɛcu)) and transition point (the point where the first ascending region transitions to quasi-linear behavior) are two key reference points on the axial stress–strain curve of FRP-confined concrete [9]. The prediction of these two key reference points is useful to create the complete axial stress–strain curve of FRP-confined concrete [16]. More than 88 empirical models are available in the literature for predicting f’cc and ɛcu of FRP-confined concrete [2]. These models are based on either material properties or the geometry of the specimens as the independent (input) parameters. These independent parameters include the compressive strength (f’co) and the corresponding axial strain (ɛco) of the unconfined concrete, hoop rupture strain (ɛh,rup), fiber ultimate tensile strain (ɛfu), elastic modulus of fiber (Ef), fiber thickness (tFRP), the total thickness of FRP jacket (tf), nominal lateral confining pressure (flu), actual lateral confining pressure (flu,a), FRP lateral stiffness (Kl), and the concrete diameter (D). However, only some of these independent parameters were considered in the modeling, while the importance of the other influential parameters on the accuracy and complexity of the models was not evaluated. In addition, some of the existing expressions for capturing the ultimate condition were developed based on a specific FRP type (e.g., glass, aramid and carbon FRP) and some of them only considered a specific type of concrete (e.g., normal- and high-strength concrete) [16]. In addition, there are a few design-oriented models for predicting the transition point on the axial stress–strain curve of FRP-confined concrete [23], [24], [25], [26]. However, these models provided complex expressions [16]. As a result, a comprehensive study is needed to evaluate the importance of different input parameters and mathematical functions to develop simple and more accurate expressions for the prediction of the ultimate condition and transition point of FRP-confined concrete.

Hoop rupture strain (ɛh,rup) of the FRP jacket is one of the most significant parameters in predicting ɛcu of FRP-confined concrete. Ozbakkalgolu and Lim [27] provided the most accurate predictions for ɛcu using experimental ɛh,rup values, which indicated the importance of ɛh,rup in predicting the mechanical behavior of FRP-confined concrete. However, the presented experimental values in various studies were inconsistent and some of the datasets did not follow a general trend [21], [28], [29]. The previous studies on the behavior of the FRP jacket reported that ɛh,rup had a lower values compared to ɛfu, which can be attributed to the deformation localization in the cracked concrete that causes non-uniform stress distribution in the FRP jacket, leading to the premature rupture of FRP [28]. In previous studies ɛh,rup was calculated using a reduction factor (kɛ) [6], [21], [23], [30], [31], [32], [33], [34]. However, very few studies proposed an expression to predict ɛh,rup using kɛ (e.g., [9], [23]), and the majority of them only considered a constant value for kɛ in their calculations (e.g., [35], [36], [37], [38], [39]). Therefore, more studies are required to develop an accurate expression to predict ɛh,rup or kɛ.

In recent decades, many soft computing algorithms, including artificial neural network (ANN), genetic algorithm (GA), support vector machine (SVM), and fuzzy logic algorithms, have been developed by various researchers (e.g., [8], [40], [41]). These algorithms can handle complicated datasets having a large number of input variables to provide complex and nonlinear relationships between the input parameters and their corresponding output parameters in an intelligent manner without any presupposition [8]. Nevertheless, the performance of these algorithms significantly depends on their structure as well as the programming skills/knowledge of the user [8]. Furthermore, most of the soft computing algorithms are known as black-box techniques which have a complex internal structure and cannot provide practical outputs (i.e., mathematical equations and/or visual charts) [42]. Genetic programming (GP), as a subset of meta-heuristic algorithms, has been widely used in civil, geotechnical, and mining engineering fields by different researchers [8], [43], [44], [45], [46], [47], [48], [49]. GP, unlike ANN and fuzzy logic algorithms, is known as a white-box technique which can provide mathematical expressions for predicting or classifying the quantitative and qualitative parameters. In another word, GP correlates the dependent and independent parameters of a complex database using the symbolic regression (SR) analysis principles. Additionally, GP can reveal the uncertainty and variability of the proposed solutions [8]. Based on the developed models using the GP algorithm, it is possible to evaluate the influence of independent parameters on the outputs separately, and also detect the most influential parameters on the output trend [50], [51].

Several researchers applied GP algorithm to investigate the mechanical behavior of various structural systems [52], [53], [54], [55], [56], [57]. GP was also used for predicting f’cc of FRP-confined concrete [8], [43], [58], [59]. For instance, Lim et al. [8] developed an expression for predicting ɛcu and ɛh,rup of FRP-confined concrete using GP without performing sensitivity analysis, cross validation, and parametric analysis. They compared the proposed GP-based expression with the existing design-oriented models and found that their proposed expression provided a more accurate prediction [8]. Considering the high performance of GP algorithm in the development of a simple and accurate expression, there is no study in the literature that predicts the transition point of FRP-confined concrete using GP algorithm. Therefore, a study is required for predicting ɛh,rup and transition point by performing sensitivity analysis, cross-validation, and parametric analysis for accurately capturing the axial stress–strain curve of FRP-confined concrete.

To address the above-mentioned research gap, this paper presents the first study on the development of simple and accurate expressions for predicting the transition point on the axial stress–strain curve of FRP-confined concrete using GP algorithm. New accurate expressions are also developed for predicting the ultimate condition and ɛh,rup of FRP-confined concrete using GP. To this end, the influence of input variables on the accuracy of the developed GP expressions was evaluated. This was followed by the sensitivity and parametric analyses on the database and developed expressions. Finally, the predictions of the existing design-oriented models were compared with those of the developed GP-based expressions in this study using different performance indices. The findings of this study point to the possibility of the use of the developed expressions in pre-design and modeling of structures manufactured with FRP-confined concrete.

Section snippets

Genetic programming mechanism

Genetic programming (GP) is one of the foremost well-known evolutionary algorithms inspired by the Darwinian principle of the “survival of the fittest”, which was initially invented by Cramer [60] and then significantly improved by Koza [51]. In the last decade, GP has been used in various fields to solve real-world complex problems [61], [62], [63]. In GP, solutions/expressions are in the form of binary-coded strings, which are known as computer programs (CPs) and follow the LISP language [64]

Experimental database

In this study, the database compiled in Ref. [16] for ultimate condition and transition point parameters was used. Table 1 displays the total number of data points used for each key parameter of: compressive strength (f’cc), ultimate axial strain (ɛcu) and hoop rupture strain (ɛh,rup) of FRP-confined concrete as ultimate condition parameters; and axial stress at the transition point (f’c1) and axial strain at the transition point (ɛc1) on the axial stress–strain curve of FRP-confined concrete

Development of GP-based models

Before modelling, the consistency of the database for each output parameter (i.e., f’cc, ɛcu, ɛh,rup, f’c1 and ɛc1) was evaluated, and the datasets which showed inconsistent behavior or did not follow the general trend were eliminated. As the database used in this study was similar to that in Ref. [16], the details of the database evaluation were not reported in this study. In this study, the GP algorithm was coded in MatLab environment to perform the function finding task and evaluate the

Sensitivity analysis

To investigate the influence of each input variable on the output parameters in predicting the ultimate condition and transition point, the relevance factor (r) was used in this study for sensitivity analysis [66], [67], [68], [69]. r shows the positive and negative relationships between the variables, and can be used to identify the most influential parameters on the dependent variables based on the developed expressions. Eq. (10) is used to calculate r as follows:rInpk,μk=i=1n(Inpk,i-Inp¯k)(μ

Conclusions

This paper has presented new and accurate expressions developed from GP method for predicting the ultimate condition and transition point on the axial stress–strain curve of FRP-confined concrete. In this study, a comprehensive database including 836 f’cc, 571 ɛcu, 443 ɛh,rup, 256 f’c1, and 228 ɛc1 with f’co (28 to 95 MPa) and Kl (1275 to 6781 MPa) was used. Below are the main findings of this study:

  • f’co, Kl, and ɛfu are the most influential parameters on f’cc and f’c1; Kl/f’co, ɛfu, ɛco, and Kl

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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