A Predictive Model for the Shear Capacity of Ultra-High-Performance Concrete Deep Beams Reinforced with Fibers Using a Hybrid ANN-ANFIS Algorithm

Abstract

Ultra-high-performance concrete (UHPC) has attracted considerable attention from both the construction industry and researchers due to its outstanding durability and exceptional mechanical properties, particularly its high compressive strength. Several factors influence the shear capacity of UHPC deep beams, including compressive strength, the shear span-to-depth ratio (λ), fiber content (FC), vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), and longitudinal web reinforcement (ρ_s). Considering these factors, this research proposes a novel hybrid algorithm that combines an adaptive neuro-fuzzy inference system (ANFIS) with an artificial neural network (ANN) to predict the shear capacity of UHPC deep beams. To achieve this, ANN and ANFIS algorithms were initially employed individually to predict the shear capacity of UHPC deep beams using available experimental data for training. Subsequently, a novel hybrid algorithm, integrating an ANN and ANFIS, was developed to enhance prediction accuracy by utilizing numerical data as input for training. To evaluate the accuracy of the algorithms, the performance metrics R² and RMSE were selected. The research findings indicate that the accuracy of the ANN, ANFIS, and the hybrid ANN-ANFIS algorithm was observed as R² = 0.95, R² = 0.99, and R² = 0.90, respectively. This suggests that despite not using experimental data as input for training, the ANN-ANFIS algorithm accurately predicted the shear capacity of UHPC deep beams, achieving an accuracy of up to 90.90% and 94.74% relative to the ANFIS and ANN algorithms trained on experimental results. Finally, the shear capacity of UHPC deep beams predicted using the ANN, ANFIS, and the hybrid ANN-ANFIS algorithm was compared with the values calculated based on ACI 318-19. Subsequently, a novel reliability factor was proposed, enabling the prediction of the shear capacity of UHPC deep beams reinforced with fibers with a 0.66 safety margin compared to the experimental results. This indicates that the proposed model can be effectively employed in real-world design applications.

1. Introduction

Due to its exceptional mechanical properties, particularly high compressive strength, durability, and toughness, ultra-high-performance concrete (UHPC) has been recognized as a reliable composite material in the construction industry [1]. Concrete with a compressive strength exceeding 120 MPa is classified as UHPC [2]. To mitigate crack propagation, incorporating fiber into the UHPC mix design is an excellent solution [3].

As a reliable structural element, deep beams are commonly used in bridges, offshore platforms, and high-rise buildings to carry extensive loads [4]. Based on the ACI [5] definition, if the distance between concentrated load is less than twice the beam depth and four times the beam depth is greater than the beam’s clear span, this beam is considered a deep beam.

Several factors can influence the shear capacity of UHPC deep beams, including compressive strength, fiber content (FC), vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), longitudinal web reinforcement (ρ_s), and the shear span-to-depth ratio (λ) [6]. Incorporating 2% steel fiber can increase the shear capacity of UHPC deep beams by up to 84.4% [7]. In another study, changes in UHPC mix designs led to an enhancement in the ultimate shear capacity of UHPC deep beams by up to 18.8% [8]. An increase in UHPC compressive strength of up to 37.9% can enhance the shear capacity of UHPC deep beams by up to 33.38% [9]. Comparing two specimens, one with no vertical web reinforcement (ρ_sv) and the other with 0.38% reinforcement, indicated that the shear capacity of UHPC deep beams was enhanced by up to 24.13% [10]. In another study, a comparison of two UHPC deep beams, one reinforced with 0.78% vertical web reinforcement (ρ_sv) and the other without any vertical web reinforcement (ρ_sv), showed that there was an 11.5% increase in shear capacity [11]. Additionally, an increase of 48.15% in longitudinal reinforcement ratios (ρ_s) can enhance the shear capacity of UHPC deep beams by up to 19.2% [12]. In another study, a 17.17% enhancement in the shear capacity of UHPC deep beams was observed when comparing a specimen reinforced with 0.78% vertical web reinforcement (ρ_sv) to its counterpart without any vertical web reinforcement (ρ_sv) [13]. Moreover, a 6.25% improvement in the shear capacity of UHPC deep beams can be achieved when the ratio of longitudinal web reinforcement (ρ_s) is increased by 47.24% [14]. Additionally, an improvement of up to 19.92% in the shear capacity of UHPC deep beams can be achieved when the horizontal web reinforcement (ρ_sh) is increased by up to 50% [14]. Furthermore, the shear capacity of UHPC deep beams can be enhanced by up to 16.03% when the ratio of vertical web reinforcement (ρ_sv) is increased by up to 44.05% [15]. A 61.29% reduction in the shear span-to-depth ratio (λ) can lead to an increase of up to 49.29% in the shear capacity of UHPC deep beams [16]. Increasing the fiber content (FC) from 1% to 2% can enhance the shear capacity of UHPC deep beams by up to 16.19% [17]. It can be concluded that an increase in compressive strength, fiber content (FC), vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), and longitudinal web reinforcement (ρ_s) enhances the shear capacity of UHPC deep beams. However, a decrease in the shear span-to-depth-ratio (λ) also leads to an increase in shear capacity [7,8,9,10,11,12,13,14,15,16,17].

Since the aforementioned factors have not been studied comprehensively in a single research study and it is not economically feasible to evaluate the impact of all factors experimentally, the importance of utilizing machine learning (ML) becomes evident. Machine learning enables researchers to predict the shear capacity of UHPC deep beams by inputting relevant variables. One of the most widely used approaches for predicting the mechanical properties of UHPC is the use of ANNs [18]. Abbas et al. [19] evaluated the shear deficiency of ultra-high-performance fiber-reinforced concrete (UHPFRC) beams using an ANFIS. The accuracy of the ANN algorithm in predicting the shear capacity of reinforced concrete (RC) deep beams was reported as R = 0.99 [20]. The shear capacity of RC beams without shear reinforcement was predicted using the ANN algorithm. The findings showed that the ANN model could predict the shear capacity with high accuracy [21]. Incorporating 15 input features significantly improved the accuracy of predicting the shear capacity of RC deep beams using the ANN algorithm. [22]. A combination of the ANFIS and genetic algorithm (GA) was proposed as a hybrid model to predict the shear strength of concrete beams with high accuracy [23].

In ACI 318-19 [24], dimensional limits are specified for deep beams, considering parameters such as depth (d) and width (b_w), as given in Equation (1):

$$V_u\le {\phi }^{\prime }10\sqrt{f_c^{\prime }}{b}_{w}d$$

(1)

Furthermore, the minimum vertical and horizontal reinforcements in deep beams, represented by A_v,min and A_h,min, can be calculated using Equations (2) and (3), respectively, based on ACI 318-19 [24]:

$$A_{v,min}=0.0025b{s}_v$$

(2)

$$A_{h,min}=0.0025b{s}_h$$

(3)

An in-depth literature review revealed that no studies have evaluated the shear behavior of UHPC deep beams using an ANN or a hybrid algorithm combining an ANN and ANFIS. Therefore, this study aims to employ the ANN and ANFIS algorithms individually and as a hybrid approach to predict the shear capacity of UHPC-DBs.

2. Experimental Dataset

To predict the shear capacity (SC) of UHPC deep beams using machine learning (ML), data from 63 previously tested UHPC deep beams reported in the literature [9,10,11,12,13,14,15,16,17,25,26,27,28,29] were analyzed. Six key factors influencing shear capacity were experimentally investigated in prior studies, as summarized in

Table 1. Experimental dataset of factors affecting shear behavior of UHPC deep beams.

References	Beam Dimensions (l × w × d) (mm)	Compressive Strength (MPa)	ρs (%)	ρsv (%)	ρsh (%)	λ	a (mm)	Fiber Content (kg/m3)		Ultimate Shear Capacity (kN)
								Steel	Synthetic
[9]	1200 × 150 × 600	86.5	3.62	0.19	0.16	0.923	500	0	0	609.42
		113.4						0.5		873.31
		125.6						1		1089.40
		151.4						2		1356.96
		173.6						3		1407.18
		138.5				0.554	300	1		1409.21
						0.739	400			1411.80
						0.923	500			1208.20
			3.19							1150.70
			3.35							1176.40
			3.62	0						1056.30
				0.38						1392.24
				0.15						1180.60
				0.19	0.24					1270.6
					0.31					1358.70
[10]	1000 × 100 × 500	114.2	3.49	0.6	0.33	0.8	400	0	0	915
		88.1		0	0			1	0.25	566
		84.7						2	0.5	571
[11]	1000 × 150 × 300	139	2.01	0	0	0.961	250	1.5	0	1115
				0.78						1260
[12]	1500 × 150 × 500	60	0.42	0.6	0.349	0.87	400	0	0	583
			0.51							659
			0.81							720
			0.51							511
[13]	1000 × 150 × 300	139	2.01	0	0	0.961	250	1.5	0	1104.9
				0.78						1334.0
[14]	1600 × 200 × 600	73.16	0.67	0.33	0.33	0.13	75	0	0	1321
			1.05							1401
			1.27							1409
			1.05		0.25					1226
					0.5					1531
[15]	1000 × 80 × 400	172.9	3.67	0.47	0	0.79	276.5	1.5	0	890
				0.84						1060
				0.47		0.94	329			830
				0.84						910
				1.68						1010
[16]	800 × 150 × 300	158.8	7.05	0	0	1.2	293.4	2	0	737.5
	1100 × 150 × 300					1.8	440.1			538
	1700 × 150 × 300					3.1	758.0			374
	800 × 150 × 300	126.1				1.2	293.4	0		434
	1100 × 150 × 300	126.1				1.8	440.1			112.5
	1700 × 150 × 300	126.1				3.1	758			173
	1800 × 200 × 350	160.3	7.62			2	582	2		1250
		155						1.5		931
		113.8						0		459.5
	1800 × 200 × 400	160.3	6.67			2	682	2		1083.5
		155						1.5		801.5
		113.8						0		397
[17]	2000 × 150 × 225	140	2.62	2.33	0	1.8	328.5	1	0	172.5
		150						2		186
		140		1.40				1		147.5
		150						2		176
		140	3.67					1		155.5
		150						2		182.8
		140	2.62			2.6	474.5	1		104.0
			3.67							114.5
		150	2.62					2		116
			3.67							125.05
[25]	750 × 150 × 400	67	1.38%	0.419	0	0.75	300	0	1	85.32
[26]	1900 × 80 × 400	132.5	9.5	0.708	0.236	0.875	350	1.5	0	1140
[27]	2700 × 200 × 500	150	3.44	0.5	0	2.7	1350	2.5	0	717
[28]	1200 × 150 × 400	84.63	1.05	0	0	1	400	1.5	0	361
[29]	1000 × 80 × 400	132.10	5.46	0.35	0	0.79	276.5	1.5	0	760

l: length of deep beam; w: width of deep beam; d: depth of deep beam; a: distance between concentrated load and nearest beam support.

: compressive strength (CS), fiber content (FC), vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), longitudinal web reinforcement (ρ_s), and the shear span-to-depth ratio (λ).

Notably, the typical configurations of vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), longitudinal web reinforcement (ρ_s), and the shear span-to-depth ratio (λ) are illustrated in Figure 1. As shown in Figure 1, λ is defined as the ratio of shear span to depth.

The mean, standard deviation (SD), and maximum and minimum values of these factors are summarized in

Table 2. The descriptive statistics of the factors used for developing the ML model.

	CS	ρs	ρsv	ρsh	λ	FC	SC
Mean	128.2211	3.7175	0.4946	0.1005	1.2632	1.0794	825.2937
SD	33.478	2.1392	0.5822	0.1357	0.7332	0.7891	421.2709
Minimum	60	0.42	0	0	0.13	0	134.30
Maximum	173.60	9.5	2.33	0.5	3.10	3	1536.80
Type	Input	Input	Input	Input	Input	Input	Output

.

The linear correlation coefficients of the aforementioned factors, along with their significance based on the compiled data from the literature, were analyzed using Pearson’s linear correlation, which is commonly used to assess the relationship between input and output data. These results are illustrated in Figure 2. Figure 2 shows that the three factors most strongly correlated with shear capacity are compressive strength (CS), fiber content (FC), and vertical web reinforcement (ρ_sv), with correlation coefficients of 0.8466, 0.6369, and 0.2809, respectively. As illustrated in Figure 2, there is a negative correlation between the shear span-to-depth ratio (λ) and the shear capacity of UHPC deep beams, indicating that as λ increases, the shear capacity decreases.

3. Shear Capacity of UHPC Deep Beams Based on ACI 318-19 [25]

In ACI 318-19, the strut-and-tie model (STM) is used to calculate the shear capacity of UHPC deep beams. In this model, shear forces are transferred through the compression zone, while tensile forces are carried by the flexural reinforcement, as illustrated in Figure 3 [30].

The calculation of the shear strength of deep beams involves three components: the strength of the struts, the strength of the ties, and the strength of the nodes. These can be determined using Equations (4)–(6), respectively.

$${\phi }^{\prime }{F}_{ns}\ge F_{us}$$

(4)

$${\phi }^{\prime }{F}_{nt}\ge F_{ut}$$

(5)

$${\phi }^{\prime }{F}_{nn}\ge F_{us}$$

(6)

In the above equations, φ denotes the strength reduction factor, F_us represents the factored compressive force, F_ns indicates the nominal strength of the strut, F_ut corresponds to the factored tensile force, F_nt signifies the nominal strength of the tie, and F_nn represents the nominal strength of the nodal zone.

Notably, F_ns for deep beams reinforced with longitudinal rebars can be calculated using Equation (7) as follows:

$$F_{ns}=F_{ce}{A}_{cs}+A_s^{\prime }{f}_s^{\prime }$$

(7)

In Equation (7), $f_s^{\prime }$ represents the stress in the compression reinforcement, $A_s^{\prime }$ denotes the area of the compression reinforcement, A_cs corresponds to the cross-sectional area at the end of the strut, and F_ce indicates concrete’s effective compressive strength.

Also, F_ce can be calculated using Equation (8) as follows:

$$F_{ce}=0.85{\beta }_c{\beta }_s{f}_c^{\prime }$$

(8)

In this equation, the strut and node confinement modification factor is indicated by β_c, and β_s denotes the strut coefficient. These values depend on the geometry of the strut.

Using Equation (9), the tie’s nominal tensile strength can be calculated as follows:

$$F_{nt}=A_{ts}{f}_y+A_{tp}{\Delta f}_p$$

(9)

In the above equation, ${\Delta f}_p$ corresponds to the stress enhancement in the prestressed reinforcement, $A_{tp}$ represents the prestressing reinforcement area of the tie, $A_{ts}$ signifies the non-prestressed reinforcement area, and $f_y$ indicates the specified yield strength of the non-prestressed reinforcement.

4. Predictive Model for Shear Capacity of UHPC Deep Beams

4.1. Artificial Neural Network (ANN)

In the ANN model, there are three layers: the input, hidden, and output layers. In each layer, neurons receive messages from all neurons in the previous layer. Subsequently, the generated signals are forwarded to the next layer. Notably, signals propagate from the input layer to the output layer in a one-way approach. During the data training process, which occurs in the input layer, each neuron receives the training data equally. Notably, in the training process, the bias (b) and connection weight (w) between input neurons should be considered to ensure accurate learning. Once the activation function is applied, the processed input values are transferred to the next layer, leading to the generation of ANN output values. Here, the neural network acts as a complex function that is created from layers of simple functions. The mean square error is employed as the loss function, which needs to be optimized during the training process. The loss function is represented by J, and the gradient descent algorithm is employed to train the model [31]. Regarding the gradient derivative, the gradient of the jth node in the Nth layer, extending to the last layer of the neural network, is straightforward and can be calculated using Equation (10), as follows:

$${\delta }_j^N={\displaystyle \frac{\partial J}{\partial w_j^N}}={\displaystyle \frac{\partial J}{\partial y_j^N}}·{\displaystyle \frac{\partial y_j^N}{\partial u_j^N}}·{\displaystyle \frac{\partial u_j^N}{\partial w_j^N}}$$

(10)

where the output value of the corresponding node is represented by $y_j^N$. Notably, choosing the mean square error leads to J being calculated using Equation (11), as follows:

$$J={\displaystyle \frac{1}{2}}{‖{\overline{y}}_j^N-y_j^N‖}^2$$

(11)

The input value of the node activation function, represented by $u_j^N$, can be calculated using Equation (12), as follows:

$$u_j^N=w_j^N·x_j^N·b_j^N$$

(12)

The corresponding node bias and weight are represented by $b_j^N$ and $w_j^N$, respectively. Additionally, the node input value is denoted by $x_j^N$.

The gradient can be obtained using Equation (13) when the Sigmoid function is adopted as the activation function for the nodes.

$${\delta }_j^N=-y_j^N\left(1-y_j^N\right)\left({\overline{y}}_j^N-y_j^N\right)x_j^N$$

(13)

Subsequently, based on the defined step size, the weight of this layer is updated, and the gradient for the N − 1 layer can be calculated using Equation (14), as follows:

$${\delta }_j^{N-1}={\displaystyle \frac{\partial J}{\partial w_j^{N-1}}}={\sum }_k{\displaystyle \frac{\partial J}{\partial u_j^N}}.{\displaystyle \frac{\partial u_j^N}{\partial x_j^N}}.{\displaystyle \frac{\partial x_j^N}{\partial w_j^{N-1}}}$$

(14)

It can be observed that all nodes connected to the N layer contribute to the gradient of the N − 1 layer. Additionally, it is clear that Equation (14) consists of three multiplied components.

To complete a training step, the weights of all layers need to be updated by repeating the above process. This process is known as the backpropagation algorithm.

In this study, the Levenberg–Marquardt algorithm, along with a multi-layer perceptron (MLP) feed-forward artificial neural network, was used to carry out the backpropagation learning process. As illustrated in Figure 4, the input layer comprises six neurons, each representing one of the six factors influencing the shear capacity of UHPC deep beams. This is followed by hidden layers, with the output layer representing the target variable, the shear capacity of UHPC deep beams. The number of hidden layers is crucial; a high number can lead to overfitting, while a low number can result in underfitting. After several attempts, ten hidden layers were selected, as this configuration produced the best fit for the model.

Notably, the MATLAB code used for this simulation has been included in Appendix A at the end of the paper.

Out of the 63 available UHPC-DB experimental data points, 45 were used for training, 9 for validation, and 9 for testing, as illustrated in Figure 5.

4.2. Performance Metrics

The accuracy of the predictive model was evaluated using two metrics, the coefficient of determination (R²) and root mean squared error (RMSE), whose formulas are presented in Equations (15) and (16), respectively.

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{(a_i-{\widehat{a}}_i)}^2}{\sum _{i=1}^n{(a_i-{\overline{a}}_i)}^2}}$$

(15)

$$RMSE=\sqrt{MSE}=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{(a_i-{\widehat{a}}_i)}^2}$$

(16)

where $a_i$ represents the observed shear capacity of UHPC deep beams, ${\widehat{a}}_i$ denotes the predicted shear capacity of UHPC deep beams, ${\overline{a}}_i$ signifies the average predicted shear capacity of UHPC deep beams, and n is the total value of the observed shear capacity.

As illustrated in Figure 5, the coefficients of determination (R²) for the training, validation, and testing data and the entire ANN model were 0.95, 0.95, 0.97, and 0.95, respectively. This indicates that the ANN model’s accuracy is acceptable, with a strong correlation between shear capacity and experimental shear capacity, as the R-value exceeds 0.9, despite the limited data available in the literature.

A flowchart of the machine learning model (ANN) is illustrated in Figure 6. As shown in Figure 6, 70% of the data were selected for training, while the remaining 30% were used for testing. Of this, 15% were allocated for validation and the other 15% for testing. Since the dataset is limited to 63 previously tested UHPC-DBs, where all six key factors, compressive strength, FC, ρ_sv, ρ_sh, ρ_s, and λ, were not simultaneously evaluated, only the data containing all these parameters were selected for training to ensure their impact was considered during the training process.

Notably, although the dataset of 63 previously tested UHPC-DBs is limited, another study that evaluated the shear strength of deficient UHPC beams using a machine learning model with 66 tested UHPC beams produced promising results [19]. This suggests that despite the limited dataset, accurate predictions can be obtained, provided the model demonstrates high performance.

After training, the generated model was tested using the testing dataset, employing two performance metrics: R² and RMSE. Once the model’s accuracy was verified, new samples were input into the predictive model to generate shear capacity predictions for UHPC-DBs, considering the six effective factors.

It is worth noting that although the geometrical impact on the shear capacity of UHPC-DBs is accounted for through λ, shear stress (shear capacity divided by the cross-section) was also used for training the proposed machine learning model as a control measure. As expected, no significant difference was observed between the accuracy of the predicted shear capacity and predicted shear stress, based on the performance metrics R² and RMSE. This suggests that training machine learning models using shear capacity yields high accuracy, even when variations in beam dimensions are present.

4.3. Adaptive Neuro-Fuzzy Inference System (ANFIS)

An adaptive neuro-fuzzy inference system (ANFIS) establishes a logical relationship between input and output data by combining fuzzy logic (FL) and artificial neural network (ANN) rules. Fuzzy logic enables ambiguous perceptions to be modeled with reasonable actions. Employing Boolean logic enables logical decision-making, where 1 represents true, and 0 represents false. In contrast, fuzzy logic allows for partial logical values, such as partially true or partially false, by defining membership functions. Using artificial intelligence, the adaptive neuro-fuzzy inference system (ANFIS) employs IF–THEN rules for decision-making based on these membership functions. Logical operations such as NOT, AND, and OR are applied to fulfill these rules under specific conditions. In fuzzy inference systems (FISs), membership functions are utilized by a fuzzifier to assign fuzzy values to raw input data. It is important to note that fuzzy linguistic rules guide the learning process within fuzzy inference systems (FISs). Finally, the defuzzification approach converts the fuzzy outcomes into an output vector using a defuzzifier engine [19].

In the domain X, for the variable x, set A (A ⊆ X) is defined by Equation (17), representing the membership elements (x ⊆ X), where x is not fully associated with set A. Notably, the fuzzy set A in domain X, with the membership function ${\mu }_A\left(x\right)$, is defined by Equation (18). Additionally, employing Equation (19) defines a functional consequent, where the rule constants are represented by p, q, and r [19]:

$$A=\left\{x\right|x\in X\}$$

(17)

$$A=\left\{\right\{x,{\mu }_A\left(x\right)\left\}\right|x\in X\}$$

(18)

$$\mathrm{IF}x=A\mathit{and}y=B, \text{THEN} f=\mathit{px}+\mathit{qy}+r$$

(19)

The fuzzy inference process is illustrated in Figure 7. As illustrated in Figure 7, the adaptive neuro-fuzzy inference system (ANFIS) process consists of five layers: fuzzification, rule, normalization, defuzzification, and output. The optimal structure of the adaptive neuro-fuzzy inference system (ANFIS) is determined using a trial-and-error approach. In the ANFIS algorithm, similar to the ANN approach, 63 UHPC deep beam experimental data points were utilized: 45 for training, 9 for validation, and 9 for testing.

The neuro-fuzzy inference process is illustrated in Figure 8. Using membership functions, crisp inputs are fuzzified and then processed in the FIS as input data. This process transforms the raw input data into fuzzy variables. Subsequently, the FIS applies fuzzy linguistic rules to generate knowledge-based information. Finally, the fuzzy outputs are converted into crisp values using a defuzzification engine within the FIS [19].

4.4. Hybrid Model of ANN-ANFIS Algorithm

In this novel hybrid model, two algorithms, an ANN and ANFIS, were combined to predict the shear capacity of UHPC deep beams. The performance of the hybrid model was achieved by utilizing the data generated from the ANN as input to train the ANFIS model. This approach eliminated the need for experimental data to train the ANFIS model. Out of the 63 UHPC deep beam data points generated by the ANN, 45 were used for training, 9 for validation, and 9 for testing the ANFIS model. The accuracy values of the ANN-generated data used as input for the ANFIS model for training, validation, testing, and overall performance were 0.98, 0.97, 0.98, and 0.97, respectively, as illustrated in Figure 9a–d.

5. Results and Discussion

After training predictive algorithms, including the ANN, ANFIS with experimental results, and the hybrid ANN-ANFIS algorithm, the predictive performance of all three models was evaluated by comparing their predicted shear capacity values with 15 randomly selected UHPC deep beam experimental data points. These data points were not used during the training phase of any of the algorithms. The results indicated that the accuracy of the ANN, ANFIS, and ANN-ANFIS models, as measured by R², was 0.95, 0.99, and 0.90, respectively. Furthermore, the root mean squared error (RMSE) values for these models were 44.48, 31.78, and 78.11, respectively, as illustrated in Figure 10(a1,a2), Figure 10(b1,b2), and Figure 10(c1,c2), respectively. These findings demonstrate that the ANFIS model outperformed the ANN model in terms of accuracy, which can be attributed to its ability to allocate fuzzy values to input data rather than binary values of 1 (true) and 0 (false).

As illustrated in Figure 10(c1,c2), despite not using experimental data in the training process, the hybrid ANN-ANFIS algorithm accurately predicts the shear capacity of UHPC deep beams obtained from experimental programs. This finding indicates that even in the absence of experimental results, this novel algorithm can reliably estimate the shear capacity of UHPC deep beams by inputting six key factors: compressive strength, fiber content (FC), vertical web reinforcement (ρ_sv), horizontal web reinforcement (ρ_sh), longitudinal web reinforcement (ρ_s), and the shear span-to-depth ratio (λ).

A comparison of the experimental shear capacity of UHPC deep beams with the predicted values from the ANN, ANFIS, and the hybrid ANN-ANFIS algorithm, as well as the shear capacity calculated using the STM method according to ACI 318-19, is shown in

Table 3. Shear capacities of UHPC deep beams as determined through experimental studies, predictive models, and ACI 318-19 strut-and-tie model (STM).

Beam Type [9]	Compressive Strength (MPa)	ρs (%)	ρsv (%)	ρsh (%)	λ	Fiber Content (kg/m3)	Experimental Shear Capacity (kN) [9]	ANN-Predicted Values (kN)	ANFIS-Predicted Values (kN)	ANN-ANFIS-Predicted Values (kN)	ACI 318-19 (STM) Values (kN)
1	86.5	3.62	0.19	0.16	0.923	0	609.42	674.9	413	524.4	488.23
2	113.4					0.5	873.31	864.7	809.5	799.4	552.19
3	125.6					1	1089.40	1082.2	1109.4	1062.4	611.84
4	151.4					2	1356.96	1315	1356.9	1480.1	789.77
5	173.6					3	1407.18	1374.3	1407.2	1458	912.64
6	138.5				0.554	1	1409.21	1519.7	1421	1536.8	851.31
7					0.739		1411.80	1361.4	1398.3	1386.8	881.54
8					0.923		1208.20	1167.6	1202.6	1216.1	896.42
9		3.19					1150.70	1171	1152.7	1173.8	813.12
10		3.35					1176.40	1169.9	1173.1	1189.9	844.22
11		3.62	0				1056.30	1036.9	1069.2	1071.8	749.38
12			0.38				1392.24	1337.6	1398.9	1333.9	813.52
13			0.15				1180.60	1134.4	1169.3	1187	778.53
14			0.19	0.24			1270.6	1189.9	1270.8	1179	796.02
15				0.31			1358.70	1315.3	1358.6	1156.5	807.69

.

Additionally, the ratios of experimental data to ACI, the ANN to ACI, the ANFIS to ACI, and the combined ANN-ANFIS to ACI are presented in

Table 4. Ratios comparing experimental data to ACI predictions, ANN to ACI predictions, ANFIS to ACI predictions, and combined ANN-ANFIS to ACI predictions.

Beam Type [9]	Compressive Strength (MPa)	ρs (%)	ρsv (%)	ρsh (%)	λ	Fiber Content (kg/m3)	Exp/ACI	ANN/ACI	ANFIS/ACI	(ANN-ANFIS)/ACI
1	86.5	3.62	0.19	0.16	0.923	0	1.25	1.38	0.85	1.07
2	113.4					0.5	1.58	1.57	1.46	1.45
3	125.6					1	1.78	1.77	1.81	1.74
4	151.4					2	1.72	1.66	1.72	1.87
5	173.6					3	1.54	1.51	1.54	1.60
6	138.5				0.554	1	1.65	1.78	1.67	1.80
7					0.739		1.60	1.54	1.59	1.57
8					0.923		1.35	1.30	1.34	1.36
9		3.19					1.41	1.44	1.42	1.44
10		3.35					1.39	1.38	1.39	1.41
11		3.62	0				1.41	1.38	1.43	1.43
12			0.38				1.71	1.64	1.72	1.64
13			0.15				1.52	1.45	1.51	1.52
14			0.19	0.24			1.60	1.49	1.60	1.48
15				0.31			1.68	1.63	1.68	1.43
Average							1.546	1.528	1.515	1.52

.

As summarized in Table 4, the average ratio of experimental data to the shear capacity calculated based on ACI 318-19 is 1.55, indicating a safety margin of 0.64 when comparing the experimental data to the ACI 318-19 values. Additionally, the ratio of the shear capacity values predicted by the ANN model to those calculated using ACI 318-19 is approximately 1.53, indicating that the ACI 318-19 values incorporate a safety margin of 0.65 compared to the ANN predictions. Moreover, the ratio of the predicted shear capacities from the ANFIS model to those calculated using ACI 318-19 is 1.51, indicating that the ACI 318-19 values incorporate a safety margin of 0.66 compared to the ANFIS predictions. Finally, the ratio of the shear capacity of UHPC deep beams predicted by the hybrid ANN-ANFIS model to those calculated using ACI 318-19 is 1.52. This indicates that the ACI 318-19 values incorporate a safety margin of 0.66 compared to the hybrid ANN-ANFIS predictions, despite the absence of experimental data in the training process.

The shear capacities of 15 UHPC deep beam types, as determined through experimental programs, the ANN, the ANFIS, the hybrid ANN-ANFIS algorithm, and ACI 318-19 (STM), are presented in Figure 11. This figure illustrates the impact of various factors on the shear capacity of UHPC deep beams as predicted by each method. For example, in five UHPC-DBs, increasing the fiber content from 0% to 3% results in an increase in predicted shear capacity by the ANN, ANFIS, and ANN-ANFIS models of up to 50.89%, 70.65%, and 64.03%, respectively. In beams 6, 7, and 8, an increase in λ from 0.554 to 0.923 results in a decrease in the predicted shear capacity of UHPC-DBs by 23.17%, 15.37%, and 20.87% when predicted by the ANN, ANFIS, and ANN-ANFIS, respectively. In beams 8, 9, and 10, an increase in ρ_s from 3.19% to 3.62% results in an enhancement in the predicted shear capacity of UHPC-DBs by up to 4.15% and 3.48% when predicted by the ANFIS and hybrid ANN-ANFIS algorithm, respectively. In beams 11, 12, and 13, an increase in ρ_sv from 0 to 0.38% results in an increase in the predicted shear capacity of UHPC-DBs by up to 22.48%, 23.57%, and 19.65% when predicted by the ANN, ANFIS, and ANN-ANFIS algorithms, respectively. Finally, in beams 8, 14, and 15, an increase in ρ_sh from 0.16% to 0.31% results in an increase in the predicted shear capacity of UHPC-DBs by up to 11.23% and 11.48% when predicted by the ANN and ANFIS algorithms, respectively.

Data Consistency and Reliability of Proposed Model

To reduce inconsistencies in the dataset, the following steps were taken:

• Experimental datasets that adhere to international testing standards, such as ACI, and ASTM, were selected to ensure consistency in the test procedures.
• To standardize parameters such as compressive strength, which may be obtained from different sample sizes across various standards, adjustments were made using conversion equations where applicable.
• To ensure clarity and comparability, only experimental results, including detailed material properties and mix designs, were selected.
• The parameter units were converted to the SI system to improve the generalizability of the model.
• To avoid potential bias, a sensitivity analysis was conducted to assess the impact of the effective factors on the shear capacity of UHPC-DBs.
• The performance metrics (R² and RMSE) of the proposed model were evaluated, indicating that despite dataset heterogeneity, the proposed model remains reliable.

Additionally, to apply the proposed models in real-world design, a novel reduction factor was introduced. This factor can be calculated using Equation (20), as follows:

$$\gamma ={\displaystyle \frac{ML}{ACI}}$$

(20)

where the machine learning-predicted shear capacity values of UHPC-DBs are represented by ML, and their counterparts calculated by ACI 318 (STM) are signified by ACI.

Notably, according to Table 4, the average ratio of the predicted values from the proposed models to those calculated based on ACI 318 (STM) is approximately 1.52. Therefore, this value is used to generate machine learning-corrected (MLC) values, as defined in Equation (21), as follows:

$$MLC={\displaystyle \frac{ML}{\gamma }}={\displaystyle \frac{ML}{1.52}}$$

(21)

Considering the proposed reliability factor, the corrected values for the proposed machine learning models are summarized in

Table 5. The shear capacities of UHPC deep beams determined through an experimental program, machine learning-corrected predictive models, and the ACI 318-19 strut-and-tie model (STM).

Beam Type [9]	Compressive Strength (MPa)	ρs (%)	ρsv (%)	ρsh (%)	λ	Fiber Content (kg/m3)	Experimental Shear Capacity (kN) [9]	ANN-Corrected Predicted Values (kN)	ANFIS-Corrected Predicted Values (kN)	ANN-ANFIS-Corrected Predicted Values (kN)	ACI 318-19 (STM) Values (kN)
1	86.5	3.62	0.19	0.16	0.923	0	609.42	445.434	272.58	346.104	488.23
2	113.4					0.5	873.31	570.702	534.27	527.604	552.19
3	125.6					1	1089.40	714.252	732.204	701.184	611.84
4	151.4					2	1356.96	869.9	895.554	976.866	789.77
5	173.6					3	1407.18	907.038	928.752	962.28	912.64
6	138.5				0.554	1	1409.21	1003.002	937.86	1014.288	851.31
7					0.739		1411.80	898.524	922.878	915.288	881.54
8					0.923		1208.20	770.616	793.716	802.626	896.42
9		3.19					1150.70	772.86	760.782	774.708	813.12
10		3.35					1176.40	772.134	774.246	785.334	844.22
11		3.62	0				1056.30	684.354	705.672	707.388	749.38
12			0.38				1392.24	882.816	923.274	880.374	813.52
13			0.15				1180.60	748.704	771.738	783.42	778.53
14			0.19	0.24			1270.6	785.334	838.728	778.14	796.02
15				0.31			1358.70	868.098	896.676	763.29	807.69

.

As illustrated in Figure 12, the proposed machine learning models exhibit an approximate safety margin of 0.66 compared to the experimental data, suggesting that the corrected machine learning models can be reliably employed in real-world design to ensure safety during the design process.

6. Conclusions

Based on the findings of this research, it can be concluded that machine learning (ML) techniques offer an effective approach to predicting the shear capacity of UHPC deep beams, even with limited experimental data. Three distinct predictive algorithms, an ANN, ANFIS, and an innovative hybrid ANN-ANFIS model, demonstrated promising results in estimating shear capacity. Specifically, the ANN model achieved an accuracy of 0.95, consistent with previous research, while the ANFIS model outperformed the others with an accuracy of 0.99, owing to its ability to assign fuzzy values to input data. Additionally, the hybrid ANN-ANFIS model demonstrated impressive predictive performance, achieving accuracies of 90.90% and 94.74% relative to the ANFIS and ANN, respectively. These findings underscore the potential of machine learning models, especially hybrid ones, to predict the shear capacity of UHPC deep beams effectively, even without extensive experimental data in the training process.

Apart from the performance and accuracy of the proposed models, the main conclusions are as follows:

• The most effective factors in predicting the shear capacity of UHPC-DBs across all prediction models were compressive strength, fiber content (FC), and the shear span-to-depth ratio (λ), with correlation coefficients of 0.84, 0.64, and −0.36, respectively.
• The least effective factor in predicting shear capacity was identified as longitudinal reinforcement (ρ_s), with a correlation coefficient of 0.06.
• The impact of vertical web reinforcement (ρ_sv) and horizontal web reinforcement (ρ_sh) in predicting shear capacity was approximately equal, with correlation coefficients of 0.28 and 0.22, respectively.
• Compressive strength and fiber content were found to interact strongly, with a correlation coefficient of 0.89, indicating that incorporating up to 3% fiber significantly increases compressive strength. Notably, both factors were identified as the most influential on the shear capacity of UHPC deep beams (UHPC-DBs) predicted by the proposed models, highlighting their critical role in accurate shear capacity estimation.
• A novel reliability factor (γ = 1.52) was proposed, allowing the corrected machine learning shear capacity of UHPC-DBs to be calculated. This resulted in a predicted shear capacity with a 0.66 safety margin compared to the experimental values.
• Increasing the fiber content from 0% to 3% enhanced the shear capacity of UHPC-DBs predicted by the ANN, ANFIS, and ANN-ANFIS algorithms by up to 50.89%, 70.65%, and 64.03%, respectively.
• Decreasing the shear span-to-depth ratio (λ) from 0.923 to 0.554 increased the shear capacity of UHPC-DBs, as predicted by the ANN, ANFIS, and ANN-ANFIS algorithms, by up to 23.17%, 15.37%, and 20.87%, respectively.
• Increasing the longitudinal reinforcement (ρ_s) from 3.19% to 3.62% enhanced the shear capacity of UHPC-DBs, as predicted by the ANFIS and ANN-ANFIS algorithms, by up to 4.15% and 3.43%, respectively.
• An increase in vertical web reinforcement (ρ_sv) from 0 to 0.38% led to an increase in the predicted shear capacity of UHPC-DBs by up to 22.48%, 23.57%, and 19.65% when predicted by the ANN, ANFIS, and hybrid ANN-ANFIS algorithms, respectively.
• Increasing the horizontal web reinforcement (ρ_sh) from 0.16% to 0.31% resulted in an increase in the predicted shear capacity of UHPC-DBs by up to 11.23% and 11.48% when estimated by the ANN and ANFIS algorithms, respectively.