Impact Strength Properties and Failure Mode Classification of Concrete U-Shaped Specimen Retrofitted with Polyurethane Grout Using Machine Learning Algorithms

Abstract

The inherent brittle behavior of cementitious composite is considered one of its weaknesses in structural applications. This study evaluated the impact strength and failure modes of composite U-shaped normal concrete (NC) specimens strengthened with polyurethane grout material (NC-PUG) subjected to repeated drop-weight impact loads (USDWIT). The experimental dataset was used to train and test three machine learning (ML) algorithms, namely decision tree (DT), Naïve Ba yes (NB), and K-nearest neighbors (KNN), to predict the three failure modes exhibited by U-shaped specimens during testing. The uncertainty of the failure modes under different uncertainty degrees was analyzed using Monte Carlo simulation (MCS). The results indicate that the retrofitting effect of polyurethane grout significantly improved the impact strength of concrete. During testing, U-shaped specimens demonstrated three major failure patterns, which included mid-section crack (MC), crushing foot (CF), and bend section crack (BC). The prediction models predicted the three types of failure modes with an accuracy greater than 95%. Moreover, the KNN model predicted the failure modes with 3.1% higher accuracy than the DT and NB models, and the accuracy, precision, and recall of the KNN model have converged within 300 runs of Monte Carlo simulation under different uncertainties.

1. Introduction

Concrete is a widely utilized construction material [1], featuring the desired characteristics and good durability properties. However, its brittle characteristics and low tensile strength make it easily crack under flexural loading conditions [2]. Furthermore, conventional concrete has poor deformability and weak compression stiffness, which limit its ability to sustain impact stresses. This shortcoming restricts the use of concrete in some particular buildings that are subjected to impact loads, such as mechanical platforms and airport pavement, frequently under recurrent stresses over their lifespan. To address these inadequacies, efforts have been made to improve desirable concrete qualities [3,4,5,6]. Polyurethane appears to be a promising alternative material to improve the performance of concrete. Concrete structures deteriorate due to various circumstances, such as corrosion, de-icing chemicals, and inappropriate planning or construction. Cracking in concrete is frequently accompanied by steel corrosion, reducing the structure’s load-carrying capability [7,8]. Polymer composites commonly used to repair impaired concrete components, such as floor, pipes, and pavement, where good durability and fast setting are desired [9]. Polyurethane (PU)-based polymer materials have fascinated researchers in polymeric materials due to their exceptional characteristics [10,11,12]. Compared to traditional polymers, polyurethane demonstrates excellent chemical resistance, outstanding adhesion, rapid hardening, and good hardened behavior, resulting in improved cement-based material toughness with a low percentage [8]. It has extensive uses in the maintenance, restoration, and construction of engineering structures [13], such as in pavement construction [14], as grouting material [15,16,17], as polyurethane-cement-based material [18], added in concrete as recycled aggregate materials [19], and as PU foam [20]. Lei et al. [21] reported improved impact strength and dynamic stress compared to reference concrete. Parniani and Toutanj [22] studied the fatigue behavior of concrete specimen-strengthened polymer overlaid material. Lugsum et al. [18] investigated the static and impact strength of U-shaped concrete specimens containing polyurethane binder and ML algorithms to predict the failure strength of the U-shaped specimens. The result showed that PU binders significantly enhanced the impact strength of concrete. Al-shawafi et al. [23] coated the surface of ultra-high-performance concrete (UHPC) with polyurethane grouting The result indicated that the retrofitting effect effectively increased the dynamic properties of the UHPC.

ML algorithms have been effectively used to solved many engineering problems such as prediction, regressions, and classification [14,24,25,26,27]. Mangalathu and Jeon [28] explored the capability of ML algorithms to classify the failure modes in the reinforced concrete column using the DT, ANN, random forest, KNN, and NB models. Similarly, failure mode in an RC column was classified by [29], using the ANN and DT models. The residual-based damage of reinforced concrete columns was featured and optimized using ML algorithms [30]. Nguyen et al. [31] developed an XGB model to estimate the punching shear strength of RC inner slabs, and Nadepour and Mirrashid [32] analyzed the failure modes classification of the reinforced concrete joints using DT model. The dynamic fracture growth in brittle composite was analyzed using the ML model [33]. Fatma et al. [34] developed the deep learning Deep Neural Decision Forest (DNDF) to improve intrusion detection systems in network traffic evaluation. Najeeb et al. [35] evaluated the failure mode of plant fiber-reinforced composite using non-destructive testing technique.

Previous studies [36,37] have suggested that polyurethane is a sustainable coating material that can improve the dynamic resistance of concrete structure. The concrete’s detrimental effect and disintegration under impact loads can be minimized [38]. The use of stiffness and elastic materials effectively increased concrete’ energy absorption capacity [38,39]. As a result, polymer grout overlay on concrete component mitigates the detrimental effects of impact loads [23,40]. Similarly, utilization of elastomeric material to retrofitted concrete structures can improve the impact resistance of the structural component [23,40]. Therefore, attention has been paid in this study to classify the failure modes of concrete U-shaped specimens retrofitted with polyurethane grouting materials subjected to multiple drop-weight impact loads. Machine learning algorithms were employed, including the decision tree, Naïve Bayes, and K-nearest neighbor models. Moreover, the uncertainties of the failure modes under different uncertainties degree were analyzed using Monte Carlo simulation technique. The prediction skills can prevent the need for carrying out high-cost experimental work and save time.

2. Materials and Methods

2.1. Materials

Grade 42.5R ordinary Portland cement was used to produce the concrete mixture following Chinese national standard JTG55-2011 [41]. Natural river sand was utilized as fine aggregate; it had a fineness modulus of 2.63. Medium particle sizes (10 mm) obtained from crushed stone were employed as coarse aggregate. The polycarboxylate-based superplasticizer was incorporated into the concrete mixes at 0.15% by cement weight to obtain good workability.

The polyurethane binder was prepared by mixing castor oil and polyaryl polymethylene isocyanate (PAPI) using a 6: 1 mixing ratio, placed in a container, and vigorously mixed with a small mechanical mixer for about 2 min at room temperature until a homogenous solution was achieved [14,42].

2.2. Specimen Preparation

Table 1. Mix proportion for normal concrete (NC)/kg/m³ and PUG material.

Specimen ID	Cement	Fine Aggregate	Median Aggregate	Water
NC	425	718	966	170
Polyurethane grouting materials
	PU: Sand	PU matrix/200 g
		Castor oil (g)	PAPI (g)	diluent (g)
PU grout	1:0.5	167.00	33.00	8.40

displays the mixture design for preparing the normal concrete and PU grouting materials. The U-shaped specimens and beam were cast and compacted using a vibrating table. After casting, all specimens were kept at room temperature for 24 h before demolding. Then, the specimens were cured for 28 days.

The 28-day-cured specimens were allowed to air dry for 7 days to totally remove the moisture content on the specimens. The dried specimens were then put back into the clean molds to form composite specimens. The freshly PUG materials obtained by mixing quartz sand and PU binder at a 1:0.5 mixing ratio by weight were then cast on surfaces of the concrete U-shaped at 5 mm and 10 mm thicknesses according to the configuration, as shown in

Table 2. Configuration of NC-PUG specimens.

		PUG Overlaid (mm)
Specimen ID	Configuration	Top Surface	Bottom Surface
NC-PU0	-	-	-
NC-PUT5	T	5	-
NC-PUTB5	T&B	5	5
NC-PUT10	T	10	-

. The composite U-shaped specimen configurations are tested for repeated drop-weight impact at room temperature. The typical normal concrete and composite U-shaped specimens retrofitted with the PUG are shown in Figure 1.

2.3. Testing Methods

Multiple Drop-Weight Impact Tests of Composite U-Shaped Specimens

The experimental setup of multiple drop-weight impact test methods is depicted in Figure 2. The test was performed based on the modified testing principles of the ACI 544 2R testing procedure [43]. Multiple drop-weight impact tests have been widely adopted in many studies to investigate the impact resistance of concrete. The USDWIT was carried out following previous research by Haruna et al. [14,36], and a hammer weight of 1.25 kg was used to simulate the impact of composite U-shaped specimens, which was constantly falling from a 457 mm height. Therefore, the number of drops to induce the first crack (N₁) and failure strength (N₂) was recorded. As shown in Figure 2, a steel base with a mass of 45 kg was used to support the test specimen, which was constructed to meet the requirements of the U-shaped geometry, loading system, and elastic–plastic deformation in order to provide a more accurate impact resistance result.

2.4. AI Modeling Procedure

2.4.1. Decision Tree

A supervised ML method called a decision tree is frequently employed for classification and regression problems. It creates a model with a structure similar to a tree using trained data [44]. The test on a feature is represented by a root node at the start of the tree-structured algorithm, which then expands to b branches that represent the test’s result. Compared to other models, the DT technique is more prone to overfitting the data since a single decision-tree model has a huge variance [45]. The creation of a decision tree involves two steps:

(1)

Split up the training dataset area t₁, t₂, t₃, …t_p into H separate and overlapping areas K₁, K₂, K₃, …K_p. The Gini index (GI) is utilized for the region’s split binary operation [46], which specifies how the overall variance across all j classes was measured.

$$GI={\displaystyle \sum _{J=1}^3\widehat{P}nj(1-\widehat{P}nj)}$$

(1)

where $\widehat{P}nj$ denotes the group of training sets in the nth area that are members of the jth class.

(2)

Cost-complexity pruning can be used to implement the preventative actions for overfitting or potentially complicated trees, taking into account a collection of trees groups of trees organized by nonnegative tuning index α. Given that the GI of the holdout data is decreased during the procedure, the max α-value might be obtained by the k-fold cross-validation method.

2.4.2. Naïve Bayes (NB)

Naïve Bayes (NB) algorithm is a stochastic classifier developed according to the Bayes theorem [47]. The classifier acknowledges and indicates a set to which the target variable belongs. The parameter f_g (X) denotes the density function of X for a subset in a gth class, and ${\eta }_g$ is a randomly selected probability of the variable related to the gth class. Thus, the probability that X in a gth class is described as

$$P(Y=g\left|X=x)\right.={\displaystyle \frac{{\eta }_g{f}_g(x)}{{\displaystyle {\sum }_{i=1}^3{\eta }_i{f}_i(x)}}}$$

(2)

2.4.3. K-Nearest Neighbors

K-nearest neighbors (KNN) is a statistical approach for classifying data [48,49]. For a numeral K and observed data x, the KNN algorithm will determine the K points belonging to the train data nearest to the x that N_k first. Therefore, the KNN is regarded as allocating the K-nearest neighbors a weight of 1/k and all other zero weights. The contingent probability of x belonging to the class k is then calculated as

$$P_k(x)=\mathrm{Pr}(Y=k\left|X\right.=x)={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i\in N_k}I(y_i}=k)$$

(3)

KNN allocates x to the class k with the highest probability. The selection of K controls the selection border between the k-classes and the model’s accuracy, as shown in Equation (5), and a value of 5 was employed in this work. The low-bias–high-variance classifier corresponds to the lower K, and the low-variance–high-bias classifier to the higher K. Figure 3 presents the architecture of the developed models.

2.4.4. Performance Evaluation

Three evaluation indices, accuracy (A), recall (R), and precision (P), were utilized in this study to evaluate the effectiveness of the created models objectively. These indices are given in Equations (4)–(6).

$$\mathrm{Precision} \left(\mathrm{P}\right)={\displaystyle \frac{\mathrm{T}\mathrm{Q}}{\mathrm{T}\mathrm{Q}+\mathrm{F}\mathrm{Q}}}$$

(4)

$$\mathrm{Recall} \left(\mathrm{R}\right)={\displaystyle \frac{\mathrm{T}\mathrm{Q}}{\mathrm{T}\mathrm{Q}+\mathrm{F}\mathrm{N}}}$$

(5)

$$\mathrm{Accuracy} \left(\mathrm{A}\right)={\displaystyle \frac{\mathrm{T}\mathrm{Q}+\mathrm{T}\mathrm{N}}{\mathrm{T}\mathrm{Q}+\mathrm{T}\mathrm{N}+\mathrm{F}\mathrm{Q}+\mathrm{F}\mathrm{N}}}$$

(6)

TQ is the number of “true positive” predictions made correctly at a particular level. FQ stands for the number of false positives or incorrect forecasts at a certain level. The correct number of predictions made at another stage relative to the genuine negative level is known as the TN. False negatives, or FNs, are the incorrect number of predictions made on another stage relative to TPs. Equations (4)–(6) were developed for two categorization issues. The macro-average method was used for multiclass classification issues like the one used in the study. Precision or recall has the three failure modes’ average value.

3. Discussion and Results

3.1. Failure Mode of Composite U-Shaped Specimens

This study adopted a U-shaped configuration to evaluate the impact resistance of composite concrete material under multiple drop-weight impact tests. This configuration aimed to control crack formation. It was anticipated to occur in a predetermine location on the specimen, conversely to the cylinder specimens used in the ACI 544-2R testing method [43], in which cracks were formed anyplace and any direction on the test specimen. In this study, three major failure patterns, including mid-section crack (MC), crushing foot (CF), and bend section crack (BC), were observed according to the specimen condition. All reference specimens exhibited the typical single crack originated at the bottom surface of the mid-section of the specimen that grew to the top surface, which eventually dissected the specimen into two parts, as shown in Figure 4a. Upon reviewing the previous literature [14,50,51], similar behavior was observed. However, U-shaped fiber-reinforced concrete exhibited minor cracks at the bend sections of the U-shaped specimen, as found in the literature [51], which is consistent with finding of this study, depicted in Figure 4c. Some U-shaped specimens retrofitted with polyurethane grouting material at the top surface also demonstrated major cracks at the bend section. Crushing of footing (CF) of the test specimen characterized by fragmentation was observed in many specimens retrofitted at the top surface or combined top–bottom surface. Due to the retrofitting effect of PU grout at the top or bottom surface, the specimen resisted the high-impact load before the forming major crack instead the occurrence of crushing of footing characterized by fragmentation of the ordinary concrete at the footing section prior to the occurrence of major single crack at the either mid-section or bend section of the U-shaped specimen.

3.2. Impact Resistance of Composite U-Shaped Specimens

The impact resistance of concrete U-shape specimens retrofitted with PU grout material at the two cracking stages, namely the first crack (N₁) and failure strength (N₂), is depicted in Figure 5, Figure 6, Figure 7 and Figure 8 under four specimen conditions.

Figure 5 shows the impact strength of the reference specimens. In most specimens, initial cracks occurred within the range from one to three drops, and specimens underwent complete failure with an additional one to two drops. However, a significant improvement of impact strength at the post-crack stage was observed in the composite concrete U-shaped specimen due to the influence of PU grout overlaid cast on either the top or bottom surface, as depicted in Figure 5, Figure 6, Figure 7 and Figure 8. U-shaped NC-PUGT5 (specimen with 5 mm overlaid thickness) showed little or no effect on the first crack strength, but a significant increase in the failure strength was observed. The N2 value was in the range from 25 to 107 drops among the test specimens, as shown in Figure 8. Similar behaviors were also observed with specimens retrofitted with 10 mm overlaid thickness at the top surface (NC-PUGT10), with the number of drops to cause first crack not exceeding three drops among all specimens. The post-crack strength was drastically improved with a range from 59 to 322 drops exhibited by U7 and U4, respectively (see Figure 7). These results indicated that the retrofitting effect of PU grouting, regardless of the thickness, had little or no effect on the first crack stage, as the overlaid was cast on the compression surface of the specimen.

Conversely, U-shaped NC-PUGTB5 (specimens with 5 mm overlaid thickness at the top and bottom surface) showed great effects at both cracking stages (N₁ and N₂). The bottom overlaid had prevented an early occurrence of the initial crack. U10 and U11 revealed the lowest number of drops to cause the first crack, with recorded values of 8 and 7, respectively. U5 exhibit up to 28 drops before the initial crack occurred (see Figure 8). Moreover, these specimens also revealed improved impact strength at the failure stage, and specimens endured high repeated impact loads before complete failure, as noted in Figure 8 A minimum and maximum number of 88 drops and 213 drops were recorded for U7 and U12, respectively. This result showed that retrofitting effect at both top-to-bottom surfaces was more effective in improving the impact resistance of concrete. Under this configuration, both the tension and compression zone had been taken care of by the PU grout material, which had viscoelastic properties.

4. AI-Based Model Result

AI-based models were employed to predict the three types of failure modes of U-shaped NC-PUG specimens tested under multiple drop-weight impact programs using the DT, NB, and KNN algorithms. The input parameters included the first crack strength (N₁ blows), failure strength (N₂ blows), PU thickness (T mm), mid-span deflection (λ mm), and maximum load (P kN). The targets were the three types of failure modes: mid-section crack (MC), crushing of footing (CF), and bend-section crack (BC).

Table 3. Statistical parameters of the experimental data.

Parameters	First Crack Strength	Failure Strength	Thickness	Midspan Deflection	Max. Load
	N1 (Blows)	N2 (Blows)	(T) mm	λ (mm)	P (kN)
Max	28	322	12.5	2.6	18.26
Min	1	1	0	0.23	10.58
Mean	5.04	91.67	6.88	1.48	13.14
St.D	5.95	82.80	4.80	0.81	2.37
Kurtosis	4.56	−0.06	−1.45	−1.32	−0.49
Skewness	2.10	0.74	−0.28	−0.26	0.77

presents the statistical analysis of the experimental data. The Kurtosis and Skewness of the data were within the acceptable range. The correlation between the input and the target variables is displayed using the correlation matrix in

Table 4. Pearson correlation matrix of the experimental data.

	N1 (Blows)	N2 (Blows)	(T) mm	λ (mm)	P (kN)	MC/CF/BC
N1 (blows)	1
N2 (blows)	0.2786	1
T (mm)	0.5825	0.7884	1
λ (mm)	−0.1199	0.4606	0.4634	1
P (kN)	0.0907	−0.4625	−0.5595	−0.7378	1
MC/CF/BC	0.4497	0.8257	0.9813	0.5882	−0.6560	1

. The result indicated that the PU thickness had highest correlation value with the output parameters. All the input variables had a reasonable correlation with the target failure modes. The experimental data were divided into training (70%) and testing (30%) data to test the developed model performance. The models were implemented in the MATLAB version: 9.13.0 (R2022b) simulation interface. A five-fold value was employed to prevent the overfitting problem.

The effectiveness of the generated models was examined through a confusion matrix. A square matrix represented the relation between the experimental and predicted failure modes. The off-diagonal portions were correctly predicted by the ML algorithm, and the diagonal elements of the confusion matrix defined the real classified failure mode. Additionally, performance metrics, including recall, accuracy, and precision, assessed the effectiveness of the confusion matrix. The proportion of correctly predicted failure modes for all three, as seen in the square matrix’s principal diagonal, was defined as the model’s accuracy. The confusion matrix’s fourth column described the precision as the percentage of each projected failure mode’s accuracy prediction. A recall was the percentage of an observed failure mode correctly predicted, as seen in the confusion matrix’s fourth row.

Figure 9 presents the performance of the trained models using a confusion matrix. All the trained models could predict the three types of failures with an accuracy greater than 95%. The KNN model predicted the failure modes with 3.1% higher accuracy than the DT and NB models. The performance of the developed DT, NB, and KNN models was tested using 30% testing data and presented in Figure 10. All the developed models could predict the failure modes with 100% accuracy. The excellent performance of the models was related to the performance of the developed models and the excellent correlation of the input variables with the target failure modes. However, more data are needed to validate further the developed models’ performance.

4.1. Monte Carlo Simulation (MCS)

MCS techniques are commonly used by researchers and construction industries to evaluate the divergence of the input space. The technique defines the divergence of the output result through statistical prediction model analysis. The MCS technique was successfully used in the probabilistic analysis of concrete members under shear stress [52], randomly estimating the pore effect and the size of the aggregate [53], material characteristics, and geometric size of reinforcement embedded in UHPC [54]. The Monte Carlo method is particularly dependable and efficient when using numerical AI models to determine how input variability affects output outcomes. [55,56]. The main idea behind the Monte Carlo approach is to simulate a condition in the input space repeatedly while determining the related output using a simulation model. [57,58]. Using a two-dimensional input space and a common probability distribution, Figure 11 demonstrates the fundamental concept of the Monte Carlo method. This section covered the statistical convergence and probability density function.

The performance of the developed AI models was evaluated using the MCS method. The random sampling process typically impacts how well the models perform in training and testing. In order to evaluate the three AI models strength, a Monte Carlo simulation was run. For the simulation to produce more representative results, acceptable instances were used. MSC was used in this study to evaluate the KNN algorithm under stochastic conditions. Five hundred (500) runs of MSC were conducted under five uncertainty conditions, namely So = 0.05, 0.0, 0.15, 0.20, and 0.25. The performance of the KNN in predicting the failure mode was evaluated in terms of accuracy, precision, and recall. The testing outcomes were used for model evolution because they directly reflected the algorithm’s performance.

Table 5. MSC results.

Uncertainties		Accuracy	Precision			Recall
			MC	CF	BC	MC	CF	BC
0.05	Max	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	Min	0.429	0.111	0.000	0.000	0.800	0.000	0.000
	Mean	0.958	0.998	0.927	0.949	0.999	0.912	0.951
	StD	0.050	0.041	0.133	0.109	0.012	0.149	0.110
0.1	Max	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	Min	0.857	0.667	0.000	0.000	0.667	0.000	0.000
	Mean	0.956	0.999	0.906	0.944	0.999	0.920	0.938
	StD	0.044	0.015	0.154	0.132	0.016	0.143	0.133
0.15	Max	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	Min	0.857	0.750	0.000	0.000	0.750	0.000	0.000
	Mean	0.960	0.999	0.919	0.947	0.999	0.921	0.951
	StD	0.045	0.014	0.153	0.124	0.015	0.138	0.122
0.2	Max	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	Min	0.429	0.200	0.000	0.000	0.667	0.000	0.000
	Mean	0.958	0.998	0.917	0.948	0.999	0.918	0.945
	StD	0.052	0.037	0.143	0.131	0.019	0.152	0.130
0.25	Max	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	Min	0.857	0.833	0.000	0.000	0.750	0.000	0.000
	Mean	0.956	0.999	0.910	0.941	1.000	0.913	0.939
	StD	0.046	0.012	0.155	0.143	0.011	0.151	0.143

presents the performance of the KNN under different uncertainty conditions. The min, max, mean, and standard deviation of the accuracy, precision, and recall of each failure mode are presented, and it can be observed that the accuracy of the KNN algorithms was not affected by increasing the uncertainty. The maximum accuracy of the model under each uncertainty was 100%, and the average mean accuracy was 95.8%, which shows the robustness of the developed model. The average mean precision of the MC, CF, and BC failure modes was 99.8, 91.7, and 94.6%, respectively. The average mean of the failure mode recall was 99.9, 91.6, and 94.5%, respectively.

4.2. Convergence Plot

The convergence analysis of the model evaluation metrics, namely the accuracy, precision, and recall of the KNN on each failure mode, was conducted under five different uncertainties. The convergence criteria of the KNN were analyzed using Equation (7).

$$f(n_{MC})={\displaystyle \frac{1}{\underset{¯}{Y}}}{\displaystyle \frac{1}{n_{MC}}}{\displaystyle \sum _{i=1}^{n_{MC}}{Y}_i}$$

(7)

where, Y is the random variable, and $\underset{¯}{Y}$ is the average value of Y. The n_MC represents the number of MCS runs.

The convergence of KNN under So = 0.05, 0.10, 0.15, 0.20, and 0.25 is presented in Figure 12. The precision and recall of mid-crack (MC) failure mode converged within the first 100 runs of MCS, while the precision and recall of the crushing foot (CF) and bend-crack (BC) failure modes converged within first 300 runs. The accuracy of the failure modes converged between 100 and 200 runs. Therefore, 300 runs were adequate to meet the convergence requirements.

4.3. Sensitivity Analysis

The sensitivity of the KNN on the uncertainty conditions was investigated using the equation:

$$CV={\displaystyle \frac{\sigma }{\mu }}$$

(8)

where σ and µ are the standard deviations and mean values of the accuracy, precision, and recall of the KNN model, and the effect of different degrees of uncertainty, s₀ = 0.05, 0.10, 0.15, 0.20, and 0.25, were tested. Figure 13 presents the sensitivity of KNN on the different degrees of uncertainties in terms of accuracy, precision, and recall. The CV of the KNN accuracy when s₀ = 0.05, 0.10, 0.15, 0.20, and 0.25 was 0.052, 0.046, 0.047, 0.054, and 0.048, respectively. The degree of uncertainties had less effect on the effect accuracy of the KNN model. Similar results were obtained in terms of precision and recall. Therefore, the results indicate the robustness of the KNN model.

4.4. Probability Density Function

The performance of the KNN model was further investigated using another probabilistic technique, the probability density function (PDF). It is a ball-shaped curve of the normal distribution that is commonly accepted as it agrees with the central limit theorem (CLC). The PDF can be expressed as

$$\phi (x)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}exp\left(-{\displaystyle \frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}\right)$$

(9)

Figure 14 shows the PDF of the KNN model exposed to different degrees of uncertainties. It can be observed that by increasing the degree of uncertainties, the stochastic bound (σ) increased, and the confidence interval of the KNN accuracy slightly increased. Increasing the degree of uncertainty from 0.05 to 0.1 and to 0.15 increased the stochastic bound (σ) of the precision of the MC failure (Figure 14b) model and decreased the confidence interval. A similar effect was also observed for the recall of the MC failure mode (Figure 14e). The degree of uncertainties on the P-CF, P-BC, R-CF, and R-BC was less regarding the stochastic bounds (σ) and confidence interval, as presented in Figure 14a,c,d,f,g.

5. Conclusions

The study evaluated and classified the failure modes of U-shaped concrete specimens retrofitted with polyurethane grouting materials of various thicknesses and positions. The composite U-shaped specimens were tested under multiple drop-weight impact tests. The experimental dataset was used to train and test three machine learning algorithms, namely decision tree, Naïve Bayes and K-nearest neighbors, to predict the three failure modes exhibited by U-shaped specimens during testing. The uncertainty of the failure modes under different uncertainties degree were analyzed using Monte Carlo simulation. Therefore, the following conclusions were derived.

(1)

The retrofitting effect of polyurethane grout resulted in significant improvement in the impact strength of the U-shaped specimen. The U-shaped specimen strengthened at the top surface showed little or no effect on the number of drops to originated first crack. However, the specimen strengthened at the top-to-bottom surface revealed significant improvement at both the first crack and failure crack stages.

(2)

Three major failure patterns, including mid-section crack (MC), crushing foot (CF), and bend section crack (BC), were observed according to the specimen’s condition. Control specimens exhibited the typical single crack at the mid-section of the specimens. On the other hand, composite specimen demonstrated failure at the bend section and crushing of footing due to the retrofitting effect of PU grout overlaid, resulting the high endurance of the composite specimen to multiple drop-weight impact load.

(3)

All the trained models could predict the three types of failures with an accuracy greater than 95%. The KNN model predicted the failure modes with 3.1% higher accuracy than the DT and NB models. The excellent performance of the models was related to the performance of the developed models and the better correlation of the input variables with the target failure modes.

(4)

The accuracy, precision, and recall of the KNN model converged within 300 runs of Monte Carlo simulation. Similarly, the developed KNN model predicted the failure modes with 95% accuracy under different uncertainties. The maximum accuracy of the model under each uncertainty was 100%, and the average mean accuracy was 95.8%, which shows the robustness of the developed KNN model. The average mean precision of MC, CF, and BC failure modes was 99.8, 91.7, and 94.6%, respectively. The average mean of the failure mode recall was 99.9, 91.6, and 94.5%, respectively.