An efficient improved Gradient Boosting for strain prediction in Near-Surface Mounted fiber-reinforced polymer strengthened reinforced concrete beam

Abstract

The Near-Surface Mounted (NSM) strengthening technique has emerged as a promising alternative to traditional strengthening methods in recent years. Over the past two decades, researchers have extensively studied its potential, advantages, and applications, as well as related parameters, aiming at optimization of construction systems. However, there is still a need to explore further, both from a static perspective, which involves accounting for the non-conservation of the contact section resulting from the bond-slip effect between fiber-reinforced polymer (FRP) rods and resin and is typically neglected by existing analytical models, as well as from a dynamic standpoint, which involves studying the trends of vibration frequencies to understand the effects of various forms of damage and the efficiency of reinforcement. To address this gap in knowledge, this research involves static and dynamic tests on simply supported reinforced concrete (RC) beams using rods of NSM carbon fiber reinforced polymer (CFRP) and glass fiber reinforced polymer (GFRP). The main objective is to examine the effects of various strengthening methods. This research conducts bending tests with loading cycles until failure, and it helps to define the behavior of beam specimens under various damage degrees, including concrete cracking. Dynamic analysis by free vibration testing enables tracking of the effectiveness of the reinforcement at various damage levels at each stage of the loading process. In addition, application of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) is proposed to optimize Gradient Boosting (GB) training performance for concrete strain prediction in NSM-FRP RC. The GB using Particle Swarm Optimization (GBPSO) and GB using Genetic Algorithm (GBGA) systems were trained using an experimental data set, where the input data was a static applied load and the output data was the consequent strain. Hybrid models of GBPSO and GBGA have been shown to provide highly accurate results for predicting strain. These models combine the strengths of both optimization techniques to create a powerful and efficient predictive tool.

Graphical abstract

Keywords

NSM technique / fiber-reinforced polymer rods / static and dynamic analysis / GB / PSO / GA / finite element analysis

1 Introduction

The use of composites made of fiber-reinforced polymer (FRP) in civil engineering has become common over the past few decades due to their high strength-to-weight ratio and corrosion resistance, and FRP has been proven to be an effective alternative to steel for reinforced concrete (RC) structures [1–4]. There are many kinds of fibers used for reinforcement, including glass fiber in glass fiber reinforced polymer (GFRP), which is known for its ductility, and carbon fiber in carbon fiber reinforced polymer (CFRP), which has high strength. In spite of its lower tensile strength and elastic modulus than CFRP, good reinforcement material is provided by GFRP, both alone and in combination with CFRP, because of deformability, impact resistance, and break resistance properties [5].

In recent years, innovative methodologies have emerged aimed at enhancing the structural integrity and rigidity of pre-existing RC structures. Among these approaches, Near-Surface Mounting (NSM) technique employing FRP materials has garnered substantial attention. Previous scholarly contributions have comprehensively discussed the merits of NSM. Such work has included fortification of FRP materials against extraneous environmental factors and the augmentation of reinforcement effectiveness within regions subjected to negative moments [6,7]. A myriad of experimental investigations has been conducted to evaluate the efficacy of this technique in the context of RC structures. Nevertheless, there is still a question about the connection between concrete and FRP rods [8–10]. Compared to External Bonded Reinforcement (EBR), FRP materials are more effective at strengthening existing RC structures when used in NSM, due to the reduced frequency of debonding phenomena and greater use of the tensile strength of the FRP materials [11–13]. Capozucca et al. [14–16], have studied the static and vibration responses of reinforced RC beams with CFRP and GFRP in real scale and in small scale. Several other studies have explored composite materials based on FRP, as documented in Refs. [17–20].

Soft computing now offers machine learning (ML) approaches and optimization methods for the identification and resolution of civil engineering issues [21–28]. Nikbakht et al. [29] published a review paper that discussed the analysis of functionally graded structures in the context of buckling, static, dynamic, and free vibration, using methods that rely on Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Their work focused on the integration of stimulating techniques like Artificial Neural Networks (ANN) or Adaptive Neuro-Fuzzy Inference System with a metaheuristic algorithm. Additionally, they proposed potential future research directions in this field. Khatir et al. [30] applied a novel combination of PSO and the YUKI algorithm to predict double cracking in cantilever CFRP beams using a Radial Basis Function (RBF) neural network. The results indicate that a PSO-YUKI approach is more resilient in damage prediction than PSO alone. They also used the RBF neural network for crack depth prediction in RC beams [31]. Nikoo et al. [32] introduced a novel approach that amalgamated ANN and the Bat Algorithm to predict the shear strength of FRP RC beams. Their investigation demonstrated the superior practicality and enhanced precision of this method, in contrast to conventional techniques relying on GA and PSO. This approach has been subsequently adopted in various research endeavors, including the utilization of ANN in conjunction with the Butterfly Optimization Algorithm for structural health monitoring of beam models [33], the incorporation of Arithmetic Optimization Algorithm for damage assessment in Functionally Graded Material composite plates [34], the application of the Grey Wolf Optimizer for predicting crack width in FRP RC slabs [35], and the employment of the Firefly Algorithm for shear strength prediction in concrete beams reinforced with FRP [36].

Prado et al. [37] suggested utilizing modal analysis techniques to detect the harm caused by shear in NSM CFRP RC beams. Based on these findings, it is thought that the NSM CFRP approach would enhance the strength of the structural component without substantially increasing its stiffness. Yang et al. [38] examined the effectiveness of pultruded GFRP composites in constructing a large-scale space frame for pedestrian bridge applications. The study involved analyzing the dynamic and fatigue performances of the structure, with modal parameters obtained through free vibration testing using peak-picking and stochastic subspace identification techniques . The results indicate that the structure could potentially withstand the fatigue requirements for a 50-year design life of a pedestrian bridge over a rural interstate freeway. Shear strength prediction, by deep learning, of FRP RC slabs has been proposed by Ref. [39], using a new technique called tabular generative adversarial network. Deep learning was also used by Refs. [40–44], to solve composite structures problems. Refs. [45–51] discussed novel models, prediction methods, and innovative systems for assessing the structural integrity, load-carrying capacity, and behavior of RC panels, bent-up bars systems, and castellated steel beams. The focus was often on the response of these systems to extreme events, and presented new approaches, models, and methods for enhancing their performance and safety. Some investigations have proposed focus on comparing numerical methods for optimizing structures or use optimization methods like Genetic Programming involving structural analysis and performance evaluation, with one study [52] considering rigid connections and another [53] considering the performance of RC tunnels under internal water pressure. Dadgar-Rad and Firouzi [54] developed nonlinear finite element formulations for analyzing hyperelastic beams, focusing on visco-hyperelasticity and introducing a new beam element with various parameters and comparing hyperelastic models for beam analysis [55].

In this study, our primary focus is on assessing the effectiveness of composite materials used in the NSM technique for reinforcing RC beams. This assessment makes a comprehensive approach, combining both experimental and numerical simulation techniques, which will be described in the subsequent sections. Furthermore, we introduce a novel aspect, centering on the utilization of machine learning hybrid models. These models are a distinctive feature of our investigation and are based on a fusion of PSO, GA, and Gradient Boosting (GB) algorithms. Through these innovative models, we seek to accurately predict the strain induced by the applied static loads in each considered beam model. The novelty of this research offers unique contributions.

2 Improved Gradient Boosting methodology

GB is a popular machine learning technique for regression and classification tasks. It involves combining multiple weak or base learners (typically decision trees) to form a strong prediction model [56]. The idea behind GB is to iteratively add new models to the ensemble, each one trained to correct the errors made by the previous models. This is achieved by adjusting the target values for each subsequent model, based on the difference between the predicted and actual values of the previous model. The key to successful GB is the use of a loss function that measures the difference between predicted and actual values, and the optimization of this function to minimize the overall error of the ensemble. In GB, the input features are processed by a feature engineering and selection step to generate new features that are fed into the model. The model then predicts the target value encoding, and this prediction is compared to the actual target value using a loss function. Gradient descent is used to minimize the difference between the predicted values and the actual measurements, and the resulting gradient is used to update the model. This process is repeated iteratively with the goal of minimizing the loss function. Each iteration adds a new tree to the ensemble, which combines the predictions of all the trees in the ensemble to produce a final prediction. Finally, the output is generated based on the predictions made by the multiple trees. Fig.1 shows the Gradient boosting architecture.

Mean Absolute Error (MAE), is a commonly used loss function in Gradient Boosting. It measures the average absolute differences of the target variable’s actual and predicted values. In GB, the algorithm iteratively fits models to the data, and MAE is used to evaluate the quality of each model’s predictions. The MAE is calculated by taking the sum of the absolute differences between the actual and predicted values and dividing by the number of observations. It is a measure of the average magnitude of the errors in the predictions made by the model. The use of MAE as a loss function in Gradient Boosting has some advantages over other loss functions. For example, MAE is less sensitive to outliers than other loss functions such as Mean Squared Error (MSE), which makes it more robust in situations where the data contains outliers. Additionally, MAE is easier to interpret than MSE, since it measures the absolute error rather than the squared error.

In this study, it is suggested that the GB hyperparameters should be tuned using PSO and GA for concrete strain prediction. The experimental data collected from static tests (Section 3) is used to train both a basic and improved GB model, where the input is the applied load and the output is the strain. The process begins with input data that are split into training and testing sets. Gradient Boosting Hyperparameters Optimization is performed to find the best hyperparameters for the GB model using PSO or GA. The GB model is then trained using the best hyperparameters found. The Gradient Boosting using Particle Swarm Optimization (GBPSO) and Gradient Boosting using Genetic Algorithm (GBGA) models are then trained using the GB model with the best hyperparameters found and with the PSO or GA algorithm. Finally, the GBPSO and GBGA models are evaluated using the testing set, and model output. The diagram of Fig.2 describes this process.

3 Experimental models study

This chapter presents an analysis of four beam models, un-damaged and strengthened with NSM FRP rods. Subsection 3.1 of the chapter provides a description of the tested specimens and the mechanical properties of the materials used in the experiment. Subsection 3.2 considers the theoretical static and dynamic behaviors of the beam models. In Subsection 3.3, the numerical analysis of RC beam specimen under free vibration is discussed. The four specimens tested had identical dimensions, with cross-section of 120 mm × 160 mm and 2200 mm length. They also had identical steel reinforcements including four longitudinal steel bars of 10 mm diameter and stirrups of 6 mm diameter, with a wheelbase of 60 mm at the ends and 130 mm at midspane. The stirrups were opened at the bottom to create a groove with appropriate dimensions for the insertion of FRP reinforcement, as depicted in Fig.3.

In our empirical investigations, diligent efforts were made to mitigate the influence of extraneous factors. The experimental configuration and data gathering procedures were structured to limit noise. Stringent environmental regulations were followed throughout the testing, including the minimization of vibrations and the management of temperature fluctuations. Furthermore, we used high-grade sensors and data acquisition apparatus to ensure the precision of our measurements.

Four beam models, first in their initial and un-damaged condition and then strengthened with NSM FRP rods, are analyzed.

3.1 Experimentation for NSM models

This section explains the mechanical characterization of the materials employed in the experiments and describes the tested specimens. Following the initial experimental tests that were useful in determining the beams’ behavior in their initial state, a 20 mm × 20 mm groove was created at the bottom side of four specimen beams, for the full length, to accommodate FRP rod reinforcement bars. The reinforcement bar could be either a 9.53 mm GFRP bar or a 9.7 mm CFRP bar. The reinforcement process involved the following steps.

First, to track the deformation during the bending test, a piezoelectric strain gauge was positioned in the middle of the FRP bar. Then, the groove in the whole lower faces of four beams was created and cleaned using a metal bristle brush and compressed air. A two-component epoxy resin compound was prepared in accordance with the supplier’s advice (ratio 1:1), and a first continuous layer of the epoxy was applied. The FRP rod was inserted into the groove with light pressure to let some resin to flow laterally and avoid creating internal cavities. The groove was coated in further epoxy resin in layers until the intrados of the reinforced element were reached, and any excess adhesive was removed (Fig.4).

This study focuses on the investigation of four specimens, denoted as follows: B represents an un-strengthened control beam serving as the reference element, while B1 and B2 represent beams reinforced through the utilization of NSM GFRP bars. Additionally, B3 signifies a reinforced beam achieved with NSM CFRP bars.

3.2 Mechanical characterization of the materials

Crushing tests were conducted on ten standardized concrete specimens measuring 150 mm × 150 mm × 150 mm after they had been cast for 119 d, using a hydraulic press (Fig.5). The concrete was found to have a characteristic cubic compressive strength of 44.31 MPa. The elastic modulus of the concrete {(E}_{\mathrm{c}\mathrm{m}})was found according to NTC 2018 at §11.2.10.3 [57], using Eq. (1) below, based on the experimental characteristic strength.

(1)E_{\mathrm{c}\mathrm{m}}=22000{\left(\frac{f_{\mathrm{c}\mathrm{m}}}{10}\right)}^{0.3}=34493.7\mathrm{M}\mathrm{P}\mathrm{a},

where f_cm is the compressive strength of concrete.

Flexural examinations, with further crushing tests, were conducted on three concrete specimens with dimensions of 40 mm × 40 mm × 160 mm. The tensile strength was ascertained through the flexural tests, followed by compression assessments on six samples procured from the fractured specimens. The average tensile strength was determined to be 0.528 MPa.

The reinforcement of the beam specimens was achieved with B450C steel bars. Deformation control tensile tests were performed on three Ø8 rebars, each measuring approximately 600 mm in length, using a universal testing machine. The computed average yield stress was established at 503.45 MPa.

For the reinforcement application, the semifluid FLK epoxy structural adhesive provided by AhRCOS was used. Its attributes were characterized by subjecting three specimens, each with dimensions of 40 mm × 40 mm × 160 mm, to compression tests. In this assessment, two strain gauges were employed to measure both vertical and horizontal strains.

The FRP bars from MAPEI, as used for reinforcement, were pultruded bars with either carbon or glass fibers. The MAPEROD carbon fiber reinforcement in the CFRP bar had a high tensile strength, whereas the MAPEROD glass fiber reinforcement in the GFRP bar had enhanced adhesion. The bars were normally 9.525 and 9.7 mm in diameter, with a length of 2000 mm. Tab.1 shows the bars’ mechanical characteristics as provided in the technical data sheet from the supplier.

3.3 Experimental static tests

The section describes the experimental setup and procedure for conducting static bending tests on four beam models (Fig.6(a)). The aim of the tests was to analyze the behavior of strengthened and un-strengthened specimens. The test setup involved a four-point load test type, with two support points and two load points. The Linear Variable Displacement Transducer (LVDT), with sensitivity of 0.01 mm, was used to record the vertical deflection of specimens (Fig.6(b)). Piezoelectric strain gauges were used to measure strains on the lower rebars and FRP bars at midspan, and strain gauges were positioned at extrados to measure concrete surface deformations (Fig.6(c)).

Beam specimens were subjected to loading and unloading cycles, until failure. Four load cycles were applied to the un-strengthened beam specimens, with peak forces of 4, 8, and 16 kN. The strengthened beam specimens with NSM FRP underwent the same loading cycles but also received additional peak loading of 24 kN for B2 and 24 and 30 kN for B3, before eventually failing.

3.4 Experimental dynamic tests

Vibration tests were employed as a non-destructive technique for the purpose of beam model control. After completion of each loading cycle, dynamic assessments were carried out to ascertain the intrinsic frequencies and to monitor the impact of reinforcement and structural impairment on the dynamic behavior of the beams. These assessments were executed utilizing an accelerometer in conjunction with an impact hammer to quantify the beam’s response. During the dynamic evaluations, denoted as mark i, where i ranged from 1 to 9, the accelerometer was systematically positioned in nine distinct locations, with ten measurements obtained at each location, and subsequently averaged.

The instrumentation included a Type 8202 Brüel & Kjær (BK) produced impact hammer, accelerometer, and a random waveform exciter in order to linearize any nonlinear behavior and cover the low frequency range. The data acquisition system LAN-XI TYPE 3050 BK, controlled by ‘BK CONNECT 2018-PULSE’ software, and developed by BK, was used to process and convert the accelerometer signals in the frequency domain.

Using an instrumented hammer, the experimental beam models were tested with ten impacts at the same point after each loading step (Fig.7), and the responses were captured by placing an accelerometer at one of the nine measurement points (mark i) each time. Using the Fast Fourier Transformation analyzer, dynamic responses were recorded and stored, and then transferred to the laptop connected to the acquisition system, where they were displayed in frequency response functions (FRFs), which were then analyzed with the BK connect software.

4 Results and disscussion

4.1 Bending static test

Following the results of the static experiment and the acquired values, shown in Tab.2, the behavior of models of beams strengthened with the NSM technique can be classified in different states of damage due to cracks. D_i refer to the different levels of the damage. The instrumentation used to measure the response of the beams provides valuable data on their behavior during the main stages of loading: cracking, yielding, and failure (at stage D_f). However, it should be noted that in some cases, inaccuracies in measurements may have occurred due to malfunctioning or detachment of the measuring instruments during the later load cycles. Such inaccurate measurements are omitted from the table and the diagrams.

Overall, the results can be used to improve the real-world design and performance of RC beams strengthened with the NSM.

4.1.1 Load–strain

The load and deformations relationship of compressed concrete at extrados during each loading−unloading cycle is illustrated in Fig.8. The load–strain curves of concrete acquired at midspan reveal that all of the strengthened beam models showed an almost straight-line behavior until they reached the maximum load, without any significant indications of concrete cracking or tension steel yielding.

Fig.9, the load–strain relationship of the tensile steel, shows how the deformation of the tensile steel changes with the load for each specimen that was subjected to bending loading. This figure is useful for analyzing the performance of the specimens and comparing different strengthening techniques; its data can be used to evaluate the load capacity and deformation behavior of the specimens under bending loads.

4.1.2 Moment–curvature

For each beam model, Fig.10 presents the moment versus curvature graphs. The curvature was determined at the midspan section and calculated using the following Eq. (2):

(2)\chi =\frac{{\epsilon }_{\mathrm{c}}+{\epsilon }_{\mathrm{s}}}{d},

where ε_c and ε_s represent the strains measured at the compressed concrete extrados and the tensile steel intrados, respectively, while d is the working depth of the section, which is the distance between the extrados concrete surface and the centroid of lower rebars.

The tangent to the loading and unloading cycles is shown in red in Fig.10. The purpose of including this tangent is to help visualize the behavior of the beam models under loading and unloading conditions. The data from this figure can be used to evaluate the stiffness and ductility of the beam models and compare their performance under bending loads.

Based on the analysis of the data collected, it can be concluded that the curvature values for beam models reinforced with GFRP rods are similar or slightly lower than those of the unreinforced specimen, at the same load levels. As for specimen B3, which was reinforced with CFRP, its curvature values are about half of the other specimens during the first two load cycles. Additionally, the tangent trend analysis reveals that there is no plastic phase observed beyond the yielding point of the tensile steel in this specimen. Instead, its behavior is primarily linear until it ultimately fails. Fig.11 illustrates the moment–curvature curves of each beam model.

4.1.3 Failure mode of beam specimens

Visual observations of crack propagation and failure modes were conducted during static testing. In the case of the unstrengthened specimen B, an initial scarcity of cracks was noted, which progressively increased in number as the applied load was incrementally raised, and conformed to the conventional pattern observed in RC beams (Fig.12(a)). Strengthened specimens B1, B2, and B3 exhibited a crack development pattern similar to that of the unstrengthened specimen, albeit with notable increases in crack depth and width observed exclusively during the final loading cycles. A comprehensive description of the failure mechanisms for each beam model is provided below. The failures primarily resulted from the crushing and cracking of the compressed concrete within the tensioned region, leading to the delamination of the FRP reinforcement.

In the context of the experimental investigation, it was observed that in specimen B1, a GFRP bar experienced delamination at the midspan section when subjected to a load of 34.04 kN. This delamination event involved both the adhesive interface and the concrete cover, as illustrated in Fig.12(b). In specimen B2, the application of a load of 38.40 kN resulted in the crushing of the compressed concrete and a complete debonding of the GFRP rod. Consequently, the concrete cover detached, as depicted in Fig.12(c).

In the case of beam B3, which was reinforced with NSM CFRP, failure occurred at a load of 49.06 kN. This failure was attributed to the combined effects of compressed concrete crushing and debonding of the CFRP rod. The initiation of CFRP rod debonding was localized in the region experiencing the highest moment and propagated toward the beam’s end. Additionally, the debonding between the adhesive and concrete interfaces affected a portion of the concrete cover, causing it to displace away from the midspan section. This phenomenon is illustrated in Fig.12(d).

It is noteworthy that a distinctive audible occurrence, characterized by an explosive sound, accompanied the pull-out of the FRP rods in all the strengthened specimens. These events led to the ultimate failure of the beams under investigation.

4.2 Dynamic tests

Experimental vibration testing was conducted on beam models subjected to dynamic loading at various levels of structural damage, using an instrumented hammer and accelerometers, as previously described. Modal parameters, specifically the natural frequencies, were then extracted and analyzed from the recorded vibration data to assess the dynamic characteristics of the beam models in varying damage states. The natural frequencies for the first four vibration modes of the beams were determined by analyzing the FRF values derived from the experimental data set. The frequency values were identified in correspondence with the acceleration response peaks. The data recording software possessed the capability to discern the most stable frequencies, thereby enabling the selection of the most precise frequency values. This information proves valuable in the assessment of the FRP strengthening effectiveness and the structural health monitoring of the beam models.

Experimental FRF diagrams were generated for different stages of beam loading (D_i/i = 0,1,…,4). These diagrams represent the amplitude of vibration recorded by the accelerometer at different points (mark i) for each beam model. Some of the obtained diagrams are illustrated in Fig.13. The x-axis represents the frequency values in Hz, while the y-axis shows the amplitude of FRF in logarithmic scale. However, not all the peaks observed in the diagrams correspond to the natural frequencies of the tested samples. Therefore, the natural frequencies of the specimens were identified by selecting the frequency values with the highest peaks. Furthermore, it is observed that the constraint system has an impact on the dynamic response recording, producing results that are less clear and more contaminated by noises. This indicates that the constraint system may have a significant impact on the dynamic behavior of the specimens and needs to be taken into account when analyzing the results. Overall, the experimental FRF diagrams can provide valuable insights into the dynamic behavior of the specimens, proper identification of natural frequencies, and consideration of the constraint system, which is crucial for accurate analysis.

An overview of the main experimental frequency data obtained from dynamic tests on simply supported specimens is presented in Tab.3, which shows the average frequency measured at each accelerometer position during dynamic testing on un-strengthened beam B as well as strengthened beams B1, B2, B3. Fig.14 illustrates the frequency values obtained from the experiments for each of the four vibration modes that were studied, corresponding to different states of damage.

This study compared the dynamic response of different beam models by analyzing the percentage changes in frequency values relative to the initial and previous damage conditions and evaluated the impact of different types of reinforcement on the dynamic behavior of the beams at different stages of damage. To follow the pattern of frequency fluctuations as the crack develops, the study analyzed the frequency values at the initial state (D₀) and successive damage levels for each beam model. Fig.15 illustrates the percentage frequency variations, based on Eq. (3), for the first four modes of vibration of each simply supported beam model at different damage levels.

(3)\frac{\mathrm{\Delta }{f}_r}{{f_r}^{D0}}=100\times \frac{{f_r}^{D0}-{f_r}^{Di}}{{f_r}^{D0}},

where f_r is a natural frequency of r-mode with r = 1, 2, 3, and 4.

The diagrams provide evidence that, in certain situations, the frequency values decrease as the level of crack damage increases. For samples B and B3, there is a clear reduction in frequency values as the damage level rises, but a more erratic trend in frequency is shown for samples B1 and B2. Fig.16 displays the percentage of frequency changes, calculated using Eq. (4), for the considered modes of vibration examined in a hinged condition, as damage progresses.

(4)\frac{\mathrm{\Delta }{f}_r}{{f_r}^{Di}}=100\times \frac{{f_r}^{D(i+1)}-{f_r}^{Di}}{{f_r}^{Di}}.

From Fig.16, it can be inferred that bending has a significant impact on frequency fluctuation, particularly in the cases of the two initial two loads, when the concrete starts to crack. However, the reduction in frequency values becomes less prominent for the final load cases, and in some instances, the values have even increased. Additionally, it is worth noting that the most substantial variation values were observed in the B beam sample without reinforcement, indicating that reinforcement can help alleviate crack damage.

The diagrams showing frequency variations are supplemented by additional diagrams displaying the shape for different levels of damage in each specimen, as shown in Fig.17. The diagram of FRF envelopes provides a clearer illustration that, for the beam samples tested, there existed multiple peaks in the FRF that did not necessarily align with the beam’s natural vibration frequencies.

5 Numerical model

This chapter presents an explanation of the finite element modeling performed on the examined beam specimens. Considering the material characteristics as shown in Tab.1, we conducted dynamic analysis on three-dimensional (3D) models of simply supported RC beams using the Abaqus finite element analysis software, as shown in Fig.18. These models included both strengthened and unstrengthened beams, including scenarios where damage due to concrete cracking was simulated.

To develop a mathematical model for comparison and verification of our experimental results, we implemented the following approach. The Finite Element Method (FEM) utilized 3D elements to represent the concrete beams and resin, while the FRP rods were modeled as line elements with associated cross-sectional properties. The 3D 8-node element was employed to model the structural components, and the mesh was carefully designed to create square elements ensuring adequate mesh density. A mesh consisting of 36960 elements, with 10 mm element size, was utilized to represent the structural components. The selection of the element number was based on mesh sensitivity considerations, ensuring that the mesh density was sufficient to capture the intricacies of the beam’s behavior under various conditions. Modal analysis, performed under the absence of external forces, was conducted to determine the natural frequencies and vibration modes of the system. The outcomes of the modal analysis, which include the natural frequencies and associated modal shapes, under various damage conditions, were examined and were compared with the corresponding experimental findings, providing an assessment of our finite element modeling.

Tab.4 displays the FEM vibration frequencies computed for every beam model at each stage of damage, achieved by changing the concrete’s elastic modulus. By examining the frequency values, it became evident that they decrease as the damage state becomes more severe. Occasionally, there was a minor rise in frequency values during the final loading steps, but this increase was also observed in the experimental tests. Thus, the model is suitable for dynamic analysis.

As for the experimental explanation, the frequency percentage changes between the initial condition D₁ and the preceding damage condition D_i−1 were determined after the natural frequency values of the finite element beam models were constructed. Fig.19 displays a comparison of the absolute percentage variations in frequency of experimental (EXP) models and the FEM for each mode of vibration as the damage state changes. Typically, it is possible to notice that differences between the percentage changes of the finite element simulations and those obtained from experimental investigations are lower than 15%.

6 Improved GB application for strain prediction

The GB technique was trained on specific data that was collected from static tests conducted on beam models B, B1, B2, and B3. The data used in the training consisted of the static load and strain measurements recorded during the last load cycle before the beam failed, whether D₁, D₂, D₃, or D₄. The number of data points collected for each model varied, with 279 for model B, 181 for model B1, 708 for model B2, and 1020 for model B3.

To train the technique, a population of 50 was considered for both the PSO and GA methods, and the training results for predicting strain can be seen in Fig.20. The MAE values for each case are summarized in Tab.5.

The use of a design of experiments played a pivotal role in managing uncertainties and collecting valuable data for stochastic optimization. We were able to collect a limited set of selected data points, reducing the number of experiments while obtaining essential information on the variables that influence the problem or process. Furthermore, the data collected through the design of experiments facilitated the creation of a response surface.

The utilization of design of experiments also allowed us to identify interactions among various factors and determine the optimal levels of these factors. This approach not only enhanced the precision of our results but also provided a foundation for addressing multi-objective optimization in future research. By exploring this avenue, it became possible to consider not only weight optimization but also other objectives that may have been relevant to the cost-effectiveness of structures, particularly in construction projects.

The results presented here underscore the effectiveness of the new models, GBPSO and GBGA, for strain prediction. Both models consistently demonstrated strong predictive capabilities, as highlighted in our ML section. Our analysis revealed that, in most cases, the PSO model outperformed the GA model in terms of predictive accuracy. However, it is worth noting that for a specific beam model, B1, GA achieved the best performance. Moreover, a practical aspect to consider is the computational time associated with each model. GA’s computational time was observed to be longer than that of the PSO model, and this consideration could be vital when selecting the most suitable model for a particular application. If computational efficiency is a priority, the PSO model may be the preferred choice. These findings illustrate the nuanced trade-offs that engineers and researchers may encounter when selecting the optimal ML model for concrete strain prediction in NSM-FRP RC structures.

This discussion highlights the intricate relationships between experimental, numerical, and ML facets of our study, clarifying their combined influence on the outcomes, and provides valuable insights into the selection of the most appropriate ML model for practical engineering applications. Overall, the results suggest that the GB-PSO model is a promising approach for strain prediction tasks. Further research could explore the performance of this model on different data sets and in different applications.

7 Conclusions

The study evaluated the behavior of RC beams that were strengthened with FRP rods using the NSM technique. The research involved both static and dynamic tests to determine the reinforced beam models’ flexural behavior, failure modes, and different damage stages. The study also investigated the effectiveness of different types of reinforcement and the bond between reinforcement bar and concrete. For the beam models, dynamic results were compared with FEM.

Cracks in concrete cause beam models to become less stiff, affecting both static and dynamic behavior. Tests show that vibration frequency values decrease significantly when cracks are present, but strengthening the beam with NSM reinforcement can reduce this effect. The un-reinforced beam showed the greatest frequency variation, while the strengthened beams had much lower variations, around 5%. These findings suggest that NSM reinforcement can mitigate crack damage and limit frequency variations under bending conditions.

GB is a machine learning algorithm that was introduced as an artificial intelligence technique to predict RC strain using data obtained from static tests. To enhance its efficiency, the algorithm’s training parameters were further improved using PSO and GA. The use of GB in predicting concrete strain has shown promise in accurately predicting the behavior of RC structures. The results showed that both models achieved better results for strain prediction, with the PSO model performing slightly better than the GA model. However, it’s worth noting that the GA model took a longer time to compute than the PSO model. This suggests that using an optimization algorithm to fine-tune the GB model can lead to improved accuracy. With the GB improvements made through PSO and GA, the algorithm became more efficient and effective in its predictions, providing a powerful tool for engineers and researchers in the field of structural engineering. The development of this technique is a significant advancement in the field of artificial intelligence and can help improve the safety and longevity of RC structures.

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