Modeling the confined compressive strength of CFRP-jacketed noncircular concrete columns using artificial intelligence techniques

Abstract In this paper, an extensive literature search has been employed to extract multiple data on the confined compressive strength of carbon fiber reinforced polymer (CFRP) concrete columns with noncircular cross-sections. The values collected are related to width (b), length (h), radius of corner (r), thickness of fiber (t), unconfined concrete strength (f’co), tensile strength of fiber (ftf), elastic modulus of fiber (Ef) and the confined strength of the CFRP-jacketed concrete columns (f’cc). The database was used to propose predictive models by artificial neural network (ANN-BP, -GA & -GRG), genetic programming (GP) and the evolutionary polynomial regression (EPR) techniques. The sum of squares errors (SSE), root mean square errors (RMSE) and coefficient determination (R2) performance indices were used to evaluate the performance accuracy and efficiency of the models. At the end of the exercise, the GP and EPR produced closed form equation with performance indices of 0.623 (28%) and 0.815 (20.9%), respectively, and these did not come close to the performance of ANN-BP, -GRG and GA which performed in that order with 0.967 (9.4%), 0.960 (10.3%) and 0.957 (10.6%), respectively. Last, the relative importance of the parameters conducted showed that f’co has the greatest influence on the f’cc of the concrete structure.


Introduction
For reinforced concrete to perform and be as safe as possible, the behavior of columns is critical (Rocca et al., 2009). Concrete buildings need to be strengthened and/or retrofitted on a regular basis, with beams and columns being the most common components. Many factors can cause concrete structures to deteriorate, including the age of the structures, design flaws, changes in code requirements, or changes in facility usage, among others. In the case of damaged or faulty structures, inadequate reinforced concrete must be used to repair damaged or faulty columns. There has been significant research performed on the compression behavior of columns with traditional steel and reinforced concrete jackets (Farghal, 2018;Jahangir & Esfahani, 2022;Soleymani & Esfahani, 2019).
Many researchers advocate using advanced composites as a viable alternative to traditional methods of strengthening columns, especially those working in the field of fiber-reinforced polymers (FRP; He & Zeng, 2022;Jin et al., 2020). In the construction industry, reinforced concrete buildings are commonly reinforced with fiber reinforced polymers. Due to FRP's resistance to premature debonding and the confinement efficacy of FRP grows considerably with increasing compression stress, wrapping FRP fabric around columns for strengthening and rehabilitation is the most effective and efficient structural element (Bagheri et al., 2019). Although the externally bonded FRP laminate technology has been explored and used for the retrofit and repair of concrete elements since the early 1980s, the American Concrete Institute (ACI) 440.2 R-02 design guide for FRP-reinforced concrete was not released until 2002 (ACI (American Concrete Institute) & ACI, 2002,Zhong-Feng et al., 2022 .The materials of fiber reinforced polymer (FRP) such as glass fiber reinforced polymer (GFRP), carbon fiber reinforced polymer (CFRP) and Aramid fiber reinforced polymer (AFRP), are widely used in engineering, such as repairs and rehabilitation, because they are corrosion resistant, light, strong and easy to manufacture (Jahangir & Esfahani, 2020;Lu et al., 2020;Yu et al., 2021).
Using bi-directional CFRP composite panels to reinforce rectangular reinforced concrete columns, Chaallal and Shahawy (Chaallal & Shahawy, 2000) studied the effects of different eccentricities on CFRP composite panels. The two haunched-head specimens had dimensions of 3.6 m length, 200 mm width, and 350 mm height (on the test section). By combining transverse and longitudinal fibers, the bidirectional composite fabrics significantly strengthened the columns. The study by Parvin and Wang (Azadeh and Wei, 2001) investigated the use of several CFRP layers over concrete columns with 108 mm x 108 mm cross-sections and 305 mm height to reinforce them and apply load to them at varying eccentricities. According to the findings, raising the eccentricity lowered the column's strength capacity while adopting CFRP increased the column's load capacity. Hatami and colleagues (Chaallal & Shahawy, 2000) and Sadeghian et al. (Pedram et al., 2010). They tested the efficacy of 1.5 m long CFRP wrapped rectangular RC columns (200 300 mm) under eccentric compressive loading in their experimental tests. CFRP-enhanced rectangular carbon fiber reinforced concrete columns with square cross-sections were evaluated under eccentric loading by Lei et al. (Lei et al., 2012) and Widiarsa and Hadi (Widiarsa & Hadi, 2013). The CFRP wrapping significantly enhanced the toughness and malleability. Yang et al. (Yang et al., 2018) constructed 16 rectangular high-strength columns, 14 of which were strengthened externally using CFRP strips. These specimens were all tested to eccentric compressive force between 50 and 100 mm. The most critical variables in this experiment were the CFRP template, the quantity of CFRP strips, and the pre-yield state. In addition, rupture causes, axial force-mid span deformation shapes, malleability determinants, and CFRP strain dispersal were investigated. According to the research results, wrapping CFRP strips on every side of the specimens raised their malleability and final capacity. Comparing to other modified specimens, the perpendicular fully covered CFRP revealed much excellent efficiencies for eccentric applied loads of 50 mm. The effects of horizontal and transversal carbon-fiber-reinforced polymers strips on the radial and bending performances of rectangular tubular structures were investigated by Al-Nimry and Neqresh (Al-Nimry & Neqresh, 2019). The squared column samples, with 1200 mm high, 200 mm width, and 12.5 mm in corner radius, were tested in eccentric compressive loading of 35, 50 and 65 mm, as well as for pure bending. Their test results revealed that using a FRP confining hoop system to increase the axial strength of eccentrically loaded pillars was feasible. Despite the apparent decline in axial resistance of the untied and confined pillars, the untied FRP panels did show a stable increase of resistance of around 12 percent over the confined pillars, regardless of the amount of bending force applied. Using CFRP wrapped reinforced concrete square columns subjected to axial, eccentric, and bending loadings, Shaikh and Alishahi (Yang et al., 2018) examined the effect of CFRP volumetric proportions on the mechanical properties and corrosion resistance. There were a total of 20 pillars analyzed. All specimens were 175x175x800 mm in dimension, with curved edges of 20 mm. In this study, three factors were considered, including layer number (0, 1, 2 and 3 layers), which corresponded to CFRP volumetric ratios of 0, 0.3, 0.6 and 0.9 percent, load granularity (0, 25, 35, and 50 mm), and bending forces alone. The findings demonstrate that regardless of eccentricity, the ultimate load capacity, axial deformation, hoop stresses, stiffness, and malleability of CFRP wrapped columns rise as the ratio increases of CFRP volumetric. Increased eccentricity, on the other side, lowered load capacity and deformation for any given CFRP volumetric ratio. Researchers Jin et al. (Jin et al., 2020) used experimental and computational techniques to study the impact of size on CFRP wrapped concrete pillars. The effects of cracking on loading style, mechanical size, and reinforcements were explored and quantified. As well, structural diversity and CFRP concrete reactions were considered using three-dimensional mesoscale modeling to study the size effects of CFRP wrapped columns. Almagsoosi et al. (Almagsoosi et al., 2021) investigated the structural behavior of eight square concrete samples of the same size (300 mm x 300 mm x 1000 mm) which were enclosed in various CFRP designs with a weak compressive strength. As a control sample, two pillars were left without CFRP consolidation, samples were enhanced with two layers of CFRP strips (full restriction), samples were enhanced with two layers of CFRP strips and CFRP strips mixed with CFRP anchors, one sample was enhanced with one layer of CFRP and CFRP strips coupled with CFRP anchors, and the ultimate sample was enhanced with CFRP strips coupled with CFRP anchorages (partial confinement). The outcomes demonstrated that CFRP composite covering in multiple confinement processes can enhance the load bearing capacity and strength properties of square concrete columns with low strength in compression (Al-Salloum, 2007;Ilki et al., 2004;Kumutha et al., 2007;Rochette & Labossiere, 2000;Shehata et al., 2002;Tao et al., 2004;Wu & Wang, 2009;Wu & Wei, 2010;Xiaobin et al., 2013) compelled to axial force, and the anchoring framework enhanced based on overall brittle fracture of confined pillars where the failure did not take place throughout the anchor's areas (Abbasnia et al., 2013;Abbasnia & Ziaadiny, 2010Chaallal et al., 2003;Demers & Neale, 1994;Gandomi et al., 2010; Rousakis et al., 2007;Syamsir et al., 2022;Tao et al., 2008;Wu & Wei, 2010), which was also the aim of the investigation.
A comprehensive literature review shows that most previous studies concentrated on reinforced concrete columns with circular cross-sections, and their compressive behavior is not so complicated as their rounded corners. Though many columns, in reality, are square or rectangle in shape, very few academic papers investigate the behavior of these kinds of columns that are square or rectangular under compression. In this paper, artificial intelligent models are proposed for the confined compressive strength of the CFRP-jacketed concrete columns with noncircular crosssections considering various concrete mixes under unconfined compression and CFRP compositions. The intelligent model techniques used in this exercise were the artificial neural network (ANN) operated with the smart algorithms of genetic algorithm, gradually reducing gradient and back-propagation, the genetic programming (GP) and the evolutionary polynomial regression (EPR) techniques.

Collected database and statistical analysis
A total of 330 records were collected from experimentally tested samples of a combination of rectangular and square short columns wrapped with FRP sheets under axial compressive stresses. To give more generality to the resulted proposed models, different specimens with different properties were selected to cover the possible ranges of the input parameters with scatter diversities. Each record contains the following data: The collected records were divided into training set (260 records) and validation set (70 records) as presented in Table 1. Table 2 and Table 3 summarize their statistical characteristics and the Pearson correlation matrix. Finally, Figure 1 shows the histograms for both inputs and outputs.

Research program
Five different Artificial Intelligent (AI) techniques were used to predict the confined compressive strength of concrete short rectangular column wrapped with FRP sheets using the gathered dataset. The implemented techniques are "Artificial Neural Network with back-propagation

Figure 1. Distribution histograms for inputs (in blue) and outputs (in green).
algorithm (ANN-BP), with generalized reduced gradient algorithm (ANN-GRG) and with genetic programming algorithm (ANN-GA)", "Genetic programming (GP)" and "Evolutionary Polynomial Regression (EPR)". All these techniques were used to predict the wrapped cylinder compressive strength of rectangular concrete column (Fcc) in (MPa) using column width (b), column Length (h), corner radius (r) and total thickness of wrapping FRP sheets (t) in (mm) besides the cylinder compressive strength of un-wrapped concrete (Fco) and tensile strength of wrapping sheets (Ftf) in (MPa) and the elastic modulus of wrapping sheets (Ef) in (GPa). Each implemented technique is based on different approach mimicking human brain for ANN, optimization of mathematical regression for EPR and simulating evolution of natural creatures for GP. However, for all techniques, their accuracies were evaluated in terms of "Sum of Squared Errors (SSE)", "Root of Mean of Squared Errors (RMSE)" and "Determination Coefficient (R 2 )". The next sections present the results of each technique and its accuracy metrics.

Behaviour of the confined compressive strength with the concrete mixes
The available data has shown that the strength of the noncircular concrete columns jacketed with fiber reinforced polymer (FRP) depends for the major part on the concrete unconfined strength (f'co), which depends on the concrete mix proportions. The collected also shows that the thickness of the applied CFRP (t) had great influence on the confined strength of the jacketed concrete columns (f'cc) because increased t consistently showed increased f'cc. Also, reduced radius of corner (r) showed increased strength but consistent enough to be used as a model key factor.

Model (1)-Using (GP) technique
The developed GP model has five levels of complexity. The population size, survivor size and number of generations were 125000, 35000 and 2000, respectively. Eq. (1) presented the output formula for (Fcc), while Figure 4(a) showed its fitness. The average errors % of total dataset is (28.0%), while the (R 2 ) value is (0.623).
These three developed models were used to predict (Fcc) values. The used networks layout is illustrated in Figure (3) while the weight matrices of each model are showed in Table 4, 5 & 6. The average errors % of total dataset are (9.4%, 10.3% & 10.6%) and the (R 2 ) values are (0.967, 0.960 & 0.957), respectively. The relative importance values for each input parameter are illustrated in Figure (2), which indicated that un-wrapped compressive strength (Fco) and corner radius (r) the most important factors, then the FRP properties (t, Ef & Ftf), while column dimensions have much lower influence. The relations between calculated and predicted values are shown in Figure 4(b-d). These observations agree with the results of previous investigations (Watanabe et al., 1997, October, Abdelhamid et al., 2019Masoud & Ebid, 2019a;Spoelstra & Monti, 1999;Zaher & Ebid, 2014).

Model (3)-Using (EPR) technique
Finally, the developed EPR model was limited to cubic level, for 7 inputs; there are 120 possible terms (84 + 28 + 7 + 1 = 120) as follows: GA technique was applied on these 120 terms to select the most effective 50 terms to predict the values of (F'cc). The output is illustrated in Eq. (2) and its fitness is shown in Figure 4(e). The average error% and (R 2 ) values were (20.9%-0.815). The results of all developed models are summarized in Table 7. Figure. 5 and 6 compare the accuracies of the developed models graphically. These performance observations agree with the results of previous investigations (Watanabe et al., 1997, October, Abdelhamid et al., 2019Masoud & Ebid, 2019a;Spoelstra & Monti, 1999;Zaher & Ebid, 2014) on the application of polynomial regressions powered with the algorithms of GA. Although, this model proposed a closed-form equation making it flexible enough to be applied manually and electronically, the performance of about 81% operating with an error of a little above 20% is not good enough. þ 54t 2 þ 0:95r:t À 3:2t:Ef þ 28:3Efþ 31Fco þ 11:5t À 2:2Ftf À 11:4r À 1010 (2)

Conclusions
This research presents three models using five (AI) techniques (GP, ANN-BP, ANN-GRG, ANN-GA and EPR) to predict the wrapped cylinder compressive strength of concrete (Fcc) using column width (b), column Length (h), corner radius (r) and total thickness of wrapping FRP sheets (t) besides the cylinder compressive strength of un-wrapped concrete (Fco) and tensile strength of wrapping sheets (Ftf) and the elastic modulus of wrapping sheets (Ef). The results of comparing the accuracies of the developed models could be concluded in the following points: • GP presented the simplest and the less accurate model with accuracy of about 72%. EPR came next with more complicated and more accurate model with accuracy of about 79.1%. The three (ANN) presented more complete and more accurate models with almost the same level of accuracy of over 90%.
• The results indicated that the accuracy of the ANN model is slightly affected by the training algorithm. Back propagation (BP) and gradually reduced gradient (GRG) showed almost the same level of accuracy while the genetic algorithm (GA) showed a lower level of accuracy.
• Although, the prediction accuracy of GP and EPR models are lower than ANN models, but their outputs are closed-form equations, which could be used manually or as software unlike the ANN output, which can't be used manually.
• The summation of the absolute weights of each neuron in the input layer of the developed ANN model that un-wrapped compressive strength (Fco) and corner radius (r) the most important factors, then the FRP properties (t, Ef & Ftf), while column dimensions have much lower influence.
• The elastic modulus of FRP (Ef) didn't appear in (GP) model this may be the reason for its low level of accuracy.
• GA technique successfully reduced the 120 terms of conventional polynomial regression quadrilateral formula to only 50 terms without significant impact on its accuracy.
• Like any other regression technique, the generated formulas are valid within the considered range of parameter values, beyond this range; the prediction accuracy should be verified.

Funding
The authors received no direct funding for this research.