Comprehensive Empirical Modeling of Shear Strength Prediction in Reinforced Concrete Deep Beams

1. Introduction

A deep beam, conventionally defined by a span-to-depth ratio (h/L) of ≤ 4 or with a shear span smaller than twice the depth, is primarily governed by shear strength rather than flexure, given sufficient longitudinal reinforcement utilization, as depicted in Figure 1(b) [1]. Additionally, deep beams with a span ratio (h/L ≤ 2.5) are classified as such and find extensive applications in constructions like squat walls, foundation pile caps, and deep foundations, as illustrated in Figure 1(c) [2].

Numerous studies have explored the structural behavior of reinforced concrete (RC) deep beams, employing experimental, analytical, and numerical approaches. Eyad et al. (2018) [3], investigated a simply supported deep beam subjected to a uniform distributed load, providing a comprehensive analysis of cracking effects and ultimate shear strength.

Albidah (2023) [4], conducted tests on six metakaolin-fly ash-based geopolymer concrete beams, considering parameters such as steel fiber content and shear reinforcement percentage. The study demonstrated a significant enhancement in shear strength by 16.7% and 31.6% with the addition of steel fibers at rates of 0.35% and 0.70%, respectively. Eyad et al.[5], explored the impact of confining the strut region through the use of struts reinforcement, while Eyad et al. [6], proposed an empirical formula for the strut efficiency factor (B_s) in RC deep beams, derived from a comprehensive analytical study based on the strut and tie model. Other researchers have numerically investigated main parameters influencing the behavior and shear capacity of RC deep beams [7,8].

Despite numerous experimental and numerical investigations into reinforced concrete deep beams, which have considered factors such as concrete compressive strength, (a_v/d) ratio, and reinforcement directions in relation to shear capacity, the current ACI 318R-5 [1] code and BS 8110 [9] codes still do not incorporate these factors comprehensively. The formulas that provided in these codes are primarily limited to the concrete compressive strength, web width (b_w) and depth (d) factors only as presented in equations 1, 2 and 3, respectively. Due to this fact, the current study found that the predictive accuracy of ACI 318R-5 code [1] and BS 8110 code [9] is restricted, with coefficients of variation (CV) for shear capacity prediction. This study offers a thorough examination of deep beams, taking into account pivotal factors that influence the concrete compressive strength (f_c'), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)). The assessment relies on an extensive dataset comprising 198 deep beams sourced from 15 investigations [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. These studies were chosen for their detailed information on test conditions and material properties, thus forming a robust database conducive to scrutinizing code provisions and affirming the proposed predictive model.

Introducing a novel model for predicting the shear strength of reinforced concrete deep beams generated from the analysis of 198 experimental simply supported RC deep beams subjected to concentrated and uniform loads. The proposed model demonstrates a remarkable improvement in accuracy, outperforming the predictions of both ACI 318R-5 and BS 8110 by a remarkable percentage. This significant enhancement can be credited to the model's comprehensive consideration of (10) various factors influencing the shear strength of reinforced concrete deep beams, unlike the restrictive focus solely on concrete compressive strength found in the ACI and BS standards.

2. Methodology

To achieve the research objective, twelve (12) empirical equations were developed to theoretically predict the shear strength of reinforced concrete deep beams. A total of 198 experimental simply supported RC deep beams subjected to concentrated and uniform loads have been used to establish the empirical factors. The effects of several parameters (Table 1) were considered, such as the concrete compressive strength (fc'), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)). The conducted results from the experimental tests were used to verify the developed empirical formula.

To develop the empirical equations, various input data from the selected parameters, as well as the results from experimental simply supported RC deep beams were exported manually into Microsoft Excel software. The study suggests writing program using the ‘Microsoft Visual Basic’ (MVB) in Microsoft Excel software to calculate relevant results/values required to establish twelve (12) empirical equations. In addition, the MVB was used to compute the relevant results based on codes methods formulas in order to be compared with the developed empirical formula.

To improve the readability and clarity, this study suggests organizing the methodology in three separate phases, existing experimental investigations, codes methods for determining RC deep beam shear strength, and an examination of the statistical properties of the dataset.

2.1. Existing Experimental Investigations

The dataset of the current study, generated from 15 literature references [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], consists of 198 simply supported RC deep beam subjected to shear testing. The main variables that imported from these tests including concrete compressive strength (f_c'), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)), as presented in Table 1. The study found that, the resulting range of ultimate shear forces (V_u) varied from 77.8 kN to 6294 kN, providing a comprehensive dataset for assessing the shear behavior of RC deep beams. This study reviewed the literature to selects all these key variables which considered influential in shear failure. By assembling this diverse dataset, this research facilitates a comprehensive exploration of how these variables impact the shear capacity of deep beams. This approach affords deep understanding of the relationship between these variables and the ultimate shear strength, thereby enhancing the ability to develop new empirical equation to predict and interpret accurate shear failure. Figure 2 shows the typical design, geometry, and failure pattern characterized by a shear diagonal fracture in a representative deep beam, all the variable that considered in this study were pointed in the figure as well.

2.2. Codes Procedure, for Calculating the Resistance of RC Beams

When determining the shear strength of RC deep beams, different design codes provide unique methods for assessment. The upcoming sections detail the approaches outlined by two standards, ACI 318 R 15 [1] and BS 8110 [9], explaining the formulas for computing the nominal shear strength. These methods are available, in assisting engineers to design RC beams safely against shear force. The equations of both codes were offers limited variables form the factors that affected on the nominal shear strength of the deep beams such as: shear span-to-depth ratio (a_v/d), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)).

• ACI 318 R-15 [1]

The ACI 318R-15 code delves into the details of shear strength (V_{n ACI}) determination, limited to a few factors such as concrete compressive strength (f’_c), web width (b_w), and depth (d) as presented in Equation (1).

$${\mathrm{V}}_{\mathrm{n}\mathrm{ACI}}=\frac{5}{6}\sqrt{{f\text{'}}_c}{\mathrm{b}}_{\mathrm{w}}\mathrm{d}$$

(1)

• BS 8110 [9]

BS 8110 code is a very popular analytical method for calculating the nominal shear strength in RC deep beams. the formulas of this code were presented through Equations (2) and (3), involves considerations of shear stress limits based on the concrete compressive strength (f’_c), web width (b_w), and depth (d), highlighting safety and efficiency in FC deep beams structural design.

$${\mathrm{v}}_{\mathrm{n}\mathrm{BS}}\text{≤}\left\{\begin{array}{c}0.8\sqrt{{\mathrm{f}}_{\mathrm{cu}}}\\ \mathrm{o}\mathrm{r}\\ 5\mathrm{N}/{\mathrm{mm}}^2\end{array}\right.$$

(2)

$${\mathrm{V}}_{\mathrm{n}\mathrm{BS}}={\mathrm{v}}_{\mathrm{n}\mathrm{BS}}{\mathrm{b}}_{\mathrm{w}}\mathrm{d}$$

(3)

2.3. Statistical Properties of Dataset

The study suggests presenting very simple equation for predicting the nominal shear strength (V_n). The suggested equation of calculating the nominal shear strength of the deep beams can be divided into two terms, concrete shear strength (V_c) and shear reinforcement strength (V_s) as presented in equation (4):

$${\mathrm{V}}_{\mathrm{n}}={\mathrm{V}}_{\mathrm{c}}+{\mathrm{V}}_{\mathrm{s}}$$

(4)

It was suggested to analyze the effect of each term (V_c and V_s) in separate sections, to provide a clear understanding of how this study suggests developing the new empirical equation encompassing various factors affecting the shear strength of RC deep beams, including the concrete compressive strength (f’_c), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)).

• Concrete Shear Strength Term (Vc)

The concrete shear strength (V_c) term is formulated in eight equations (Equations 5-8 and Equations 11-14), each tailored to specific conditions. These equations will be divided into three different stages, each stage carrying the effect of different variables.

The first stage comprises four equations (Equation 5 to Equation 8), which are related to concrete compressive strength (f’c) and longitudinal reinforcing percentage (p_s) variables, a phenomenon known as dowel action.

$$\mathrm{A}*\mathrm{f}\text{ʻ}{\mathrm{c}}^{\mathrm{B}}{*\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}}$$

(5)

$$\mathrm{A}*(\mathrm{f}\text{ʻ}{\mathrm{c}}^{\mathrm{B}}+{\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}})$$

(6)

$$\mathrm{A}*(\mathrm{f}\text{ʻ}{\mathrm{c}}^{\mathrm{B}}+\mathrm{C}{\mathrm{*}\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}})$$

(7)

$$\mathrm{A}*\mathrm{f}\text{ʻ}{\mathrm{c}}^{\mathrm{B}}{*\mathrm{K}}^{\mathrm{C}}$$

(8)

The value of the uncracked compression zone depth (K) is introduced as an essential parameter in this term, it can be calculated based on multiplying the longitudinal reinforcing percentage (p_s) by the ratio of the modulus of elasticity of longitudinal reinforcement (E_s) to the modulus of elasticity of concrete (E_c) as presented in equation (9) and equation (10).

$$\mathrm{K}=\sqrt{{\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}}{\mathrm{n}}^2+2{\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}}\mathrm{n}}-{\mathrm{p}}_{\mathrm{s}}^{\mathrm{C}}\mathrm{n}$$

(9)

$$\mathrm{n}=\raisebox{1ex}{${\mathrm{E}}_{\mathrm{s}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_{\mathrm{c}}$}\right.$$

(10)

Derived from deep beam geometry, the second stage comprises three equations (Equation 11, 12 and 13) which are related to the shear span to effective depth ratio (a_v/d) variable.

$$({\mathrm{a}}_{\mathrm{v}}/\mathrm{d}{)}^{\mathrm{F}}$$

(11)

$$\frac{\mathrm{D}}{\mathrm{E}+({\mathrm{a}}_{\mathrm{v}}/\mathrm{d}{)}^{\mathrm{F}}}$$

(12)

$$\mathrm{D}+\frac{\mathrm{E}}{({\mathrm{a}}_{\mathrm{v}}/\mathrm{d}{)}^{\mathrm{F}}}$$

(13)

Accounting for the size effect of diagonal shear strength in deep beams, the third stage comprises one Equation (Equation 14) which related to the effective depth (d) variable.

$${\left(\frac{\mathrm{G}}{\mathrm{d}}\right)}^{\mathrm{H}}$$

(14)

• Shear Reinforcement Strength Term (V_s)

The shear reinforcement strength (V_s) term is expressed in four equations (Equation 15 to Equation 17), each customized for specific conditions. These equations will be segmented into two different stages, with each stage reflecting the influence of distinct variables.

The first stage comprises two equations (Equation 15 and Equation 16) related to the transverse vertical shear reinforcement (p_v) which can be presented as following.

$${\mathrm{k}}_{\mathrm{v}}*{\mathrm{p}}_{\mathrm{v}}*{\mathrm{f}}_{\mathrm{yv}}$$

(15)

$${\mathrm{k}}_{\mathrm{v}}=\left(\frac{1+\left(\frac{{\mathrm{a}}_{\mathrm{v}}}{\mathrm{d}}\right)}{6}\right)$$

(16)

The second stage comprises two equations as well (Equation 17 and Equation 18) but these equations were related to the transverse horizontal shear reinforcement (p_h), which can be represented as following:

$${\mathrm{k}}_{\mathrm{h}}*{\mathrm{p}}_{\mathrm{v}}*{\mathrm{f}}_{\mathrm{yv}}$$

(16)

$${\mathrm{k}}_{\mathrm{h}}=(\frac{5-\left(\frac{{\mathrm{a}}_{\mathrm{v}}}{\mathrm{d}}\right)}{6})$$

(17)

The coefficients (k_v) and (k_h) are determined based on the ratio (a_v/d) instead of clear span to effective depth ratio (l_n/d), reflecting the influence of shear reinforcement on the deep beam. The sum of k_v and k_h is constrained to unity (k_v+k_h=1) emphasizing the proportionality of vertical and horizontal shear reinforcement. Figure 3 shows the variation of the coefficient (k_v and k_h) with respect to (a_v/d).

When the ratio (a_v/d) is low, the angle (θ) between vertical reinforcement and the failure line (diagonal shear crack) is minimal. In such cases, horizontal reinforcement proves more effective in resisting tension stresses (those that cannot be borne by the concrete) compared to vertical reinforcement, (k_v > k_h), as the ratio (a_v/d) increases, the significance of vertical reinforcement in resisting tension stresses becomes more pronounced. When ((a_v/d) = 2), both types of reinforcement exhibit equal effectiveness, with (k_v = k_h = 0.5).

3. The Proposed Empirical Equations

This study suggests presenting 12 different combinations to develop the empirical equation for estimating the nominal shear strength (V_n) of RC deep beams, as presented in Table 2. Then, collinear regression analysis served as the cornerstone for determining the coefficients (A-H) in the proposed empirical equations. This intricate process involved the utilization of Microsoft Excel Software which replaced the test results for the nominal shear strength of the selected 198 deep beams (V_n) in these calculations. The main objective of this study was to increase the accurate and find reliable estimation of the nominal shear strength through the proposed equations. So, the study suggests adopting the error values and calculate it to assess the precision and efficacy of each suggested term in each developed equation. Three key statistical metrics were employed to verify the developed equations, mean absolute error (MAE), root mean square error (RMSE), and the coefficient of multiple determinations (R²).

MAE represents the average absolute discrepancy between expected and observed values in the verification model. It is a collinear score that weighs each variable similarly. RMSE represent the root mean square error, which it is a squared and averaged measure of the difference between expected and observed values. MAE and RMSE work in tandem to diagnose erroneous variations in predicted values, with individual mistakes in a sample always being equal to or greater than RMSE. The coefficient of multiple determinations (R²) gauges how much variability the regression model can account for. A value of (R² = 1) suggests that the regression models accurately describe the data, while (R² = 0) indicates otherwise. As the optimum empirical equations, the recommended models are chosen based on having the lowest MAE and RMSE values and the highest (R²) value, aligning with the desired precision and reliability. These coefficients were calculated using Equations 19, 20 and 21 respectively.

$$\mathrm{MAE}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}\left|{\mathrm{x}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right|$$

(19)

$$\mathrm{RMSE}=\sqrt{\frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{\mathrm{N}}}$$

(20)

$${\mathrm{R}}^2=1-\frac{\mathrm{SSE}}{\mathrm{SST}}=1-\frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{x}}_{\mathrm{i}}-\stackrel{\mathrm{-}}{\mathrm{x}}\right)}^2}$$

(21)

Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, showed the relationship between the effect of the main parameters on the shear resistance prediction and the value of V_exp/V_n .

Despite variations in f’_c between 16.08 and 47.6 MPa as presented in Figure 4, the proposed method (proposal 7) exhibits minimal change, contrasting with other methods (ACI and BS) that yield significantly uneconomic strength predictions with increasing f’_c).

Figure 5 shows little change in prediction for the proposed method with (a_v/d) ranging from 0.19 to 2.5. Conversely, other methods experience a decline in the ratio of (V_exp/V_n) with the highest values of (a_v/d).

Figure 6, Figure 7 and Figure 8 highlight the substantial influence of (p_vfyv), (p_sfy), and (p_hfyv) on the proposed method, varying between 0.053 and 10.29 MPa, 0.405 to 14.11 MPa, and 0 to 5.56 MPa, respectively. This influence surpasses that of other methods.

Following a meticulous regression analysis, the proposed Equation (22) is selected as the forecast model for the nominal shear strength of deep beams (V_{n Prop}) (proposal 7 in Table 2). This decision is grounded in its exceptional performance, boasting the lowest MAE and highest RMSE values, coupled with an (R²) value that closely approaches unity. The study suggests presenting the redemption factors that adopted in the proposed equation in a tabular form (Table 3) for clear understanding.

$${\mathrm{V}}_{\mathrm{n},\mathrm{Prop}.}=0.004({{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}^{0.17}+{0.65\mathrm{p}}_{\mathrm{s}}){\left(\frac{{\mathrm{a}}_{\mathrm{v}}}{\mathrm{d}}\right)}^{-0.3}{\left(\frac{1}{\mathrm{d}}\right)}^{0.17}){\mathrm{b}}_{\mathrm{w}}\mathrm{d}+\left(\left(\frac{1+\left(\frac{{\mathrm{a}}_{\mathrm{v}}}{\mathrm{d}}\right)}{6}\right){\mathrm{p}}_{\mathrm{v}}+{(\frac{5-\left(\frac{{\mathrm{a}}_{\mathrm{v}}}{\mathrm{d}}\right)}{6})\mathrm{p}}_{\mathrm{h}}\right)*{10}^{-6}{\mathrm{f}}_{\mathrm{y}\mathrm{v}}{\mathrm{b}}_{\mathrm{w}}\mathrm{d}$$

(22)

4. Evaluation of the Developed Empirical Equation

As a result, this study assesses the performance of nominal shear strength predictions of RC deep beams by comparing the proposed method with existing approaches, through a comprehensive evaluation of existing experimental results.

In the comparison shown in Figure 9, the proposed empirical equation demonstrates a strong correlation between experimental and theoretical results. The data points of the proposed equation are more convergent compared to other methods, indicating its superior predictive accuracy.

Mean value (Mean) Equation (23): Represents the average of ratios of experimental (V_exp) to predicted shear strength values (V_n) for all deep beams. Where, (N) is the total number of deep beams, equal to 198 in this study.

$$\mathrm{Mean}={\sum }_{\mathrm{i}=1}^{\mathrm{N}}({\mathrm{V}}_{\exp }/{\mathrm{V}}_{\mathrm{n}}{)}_{\mathrm{i}}/\mathrm{N}$$

(23)

Standard deviation (S.D.) Equation (24): Measures the dispersion of values of (V_exp/V_n), Avg representing the average of the V_exp/V_n.

$$\mathrm{S}.\mathrm{D}.=\sqrt{\frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left(\right(V_{exp}/V_n{)}_i-Avg.)}^2}{\mathrm{N}}}$$

(24)

Coefficient of variation (CV %) Equation (25): Indicates the relative variability of values of (V_exp/V_n), Avg representing the average of the V_exp/V_n.

$$\mathrm{CV}\left(\%\right)=\frac{\mathrm{S}.\mathrm{D}.}{\mathrm{Avg}.}\mathrm{x}100$$

(25)

Maximum value (Max.): Represents the maximum shear strength ratio.

Minimum value (Min.): Represents the minimum shear strength ratio.

Range value (Range) Equation (26): Indicates the spread between the maximum and minimum values.

$$\mathrm{Range}=\mathrm{Max}./\mathrm{Min}.$$

(26)

The detailed comparison involves examining the ratio of the shear resistance of tested beam (V_exp) to the calculated nominal shear resistance based on different methods of prediction (V_n), denoted as (V_exp/V_n). This evaluation is detailed in Appendix A, and Table 3 presents the outcomes for all 198 tested beams using different prediction methods. The last column in Table 3 illustrates the results of the proposed method (Equation 7). Notably, the coefficient of variation (CV %) values range between 29.03% and 29.53% for ACI and BS methods. However, by incorporating the effects of vertical and horizontal reinforcement ratios, the proposed method significantly improves CV% to a value of 27.08%.

Table 4. Comparative Analysis of Shear Strength Ratios.

Details	ACI Method [1]	BS 8110 Method [9]	Proposed method
Equation	(1)	(3)	(7)
Mean	1.15	1.10	1.15
Standard deviation	0.34	0.32	0.31
CV %	29.53	29.03	27.08
Max. ratio	2.49	2.25	2.23
Min. ratio	0.43	0.45	0.44
Range (max/min)	5.55	4.97	5.11
Number of tested beams that Vexp ˂ Vn	70	80	65

Table 4. Comparative Analysis of Shear Strength Ratios.

Details	ACI Method [1]	BS 8110 Method [9]	Proposed method
Equation	(1)	(3)	(7)
Mean	1.15	1.10	1.15
Standard deviation	0.34	0.32	0.31
CV %	29.53	29.03	27.08
Max. ratio	2.49	2.25	2.23
Min. ratio	0.43	0.45	0.44
Range (max/min)	5.55	4.97	5.11
Number of tested beams that Vexp ˂ Vn	70	80	65

5. Discussion

The findings of this study underscore the significance of considering multiple factors such as: concrete compressive strength (f’_c), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), yield strength of vertical stirrups (f_yv), and the concrete area (web width × depth (b_w×d)). By analyzing a comprehensive dataset consists of 198 experimental simply supported RC deep beams, this study proposes a novel empirical equation for predicting the nominal shear strength of RC deep beams, outperforming the existing codes such as ACI and BS.

Twelve different sets of empirical equations were developed to create a new empirical equation to estimate the nominal shear strength (V_n) of RC deep beams. Hence, collinear regression analysis served as the cornerstone for determining the coefficients (A-H) in the proposed empirical equations.

The coefficient of multiple determinations (R²) was used to make a comprehensive comparison between the results of the proposed variables. The results of R² showed a difference in the range from 0.001 to 0.4, which is considered less than 0.5, but the study suggested specifying the coefficients (A-H) and applying them to the proposed components of the equation. Then, the coefficient (R²) was calculated for the results of the proposed empirical equation, and it was equal to 0.94, which is considered a truly acceptable value, especially when compared to the value of R² for ACI and BS, which was equal to 0.8 and 0.7, respectively.

The study presented a detailed comparison between the current empirical equation and the ACI and BS codes equations to evaluate the ratio of the nominal shear strength of the experimental beam to the calculated nominal shear strength (V_exp/V_n). Based on the comparison results, the study found that the proposed equation outperformed the equations of the ACI and BS codes in all details, such as: Mean, Standard deviation CV%, Max. ratio, Min. ratio and Range(max/min).

The findings of this study bring positive improvements for code development and structure design. By using the proposed empirical equation, the structural designers can improve the effectiveness of structural designs by obtaining higher accuracy in predictions the nominal shear strength for the RC deep beams. As presented earlier, code committees may think about including the examined factors that effect on the shear behavior into the existing standards, like ACI 318R-15 and BS 8110.

6. Conclusions

This study suggests investigating the effect of various parameters on the shear strength capacity of RC deep beams, including the concrete's compressive strength (f’_c), shear span-to-depth ratio (a_v/d), web width (b_w), ratios of longitudinal (P_s), vertical (P_v), and horizontal (P_h) reinforcement, depth (d), vertical stirrups' yield strength (f_yv), and the concrete area (web width × depth (b_w×d)). This inquiry goes beyond what can be done with current codes, such BS 8110 and ACI 318R-5, which have a large coefficient of variation (CV) when it comes to forecasting shear capacity.

Through a comprehensive evaluation of 198 deep beams, imported from an extensive dataset around 15 investigations, this research proposes a novel predictive empirical equation for shear strength. The proposed equation (22), take into account all the mentioned key parameters and it is rigorously assessed through collinear regression analysis and statistical metrics, MAE, RMSE and R².

The results proved that the proposed empirical equation was significantly enhanced the prediction accuracy compared to the ACI and BS codes achieving a CV, SD, Max. ratio, Min ratio equal to 27.08%, 31.10%, 2.23% and 0.44% respectively, and the range was equal to 5.11. Both vertical and horizontal reinforcement ratios was considered in the proposed empirical equation and it leads for this improvement. The suggested empirical equation outperforms Aci and BS codes in a wide range of cases and exhibits robustness to changes in f'_c and a_v/d.

Furthermore, the comparative study highlights how were the accuracy of the suggested model, with better data point convergence and little susceptibility to changes in important parameters. This present that the suggested model has a higher degree of predictability for shear strength, which makes it a useful tool for structural engineers and practitioners.

This study advances the knowledge of shear strength in the simply supported RC deep beams by providing a more precise and adaptable predictive empirical equation that takes into account the complexities of real-world applications. The results of this study have significance for improving design techniques and raising the security and effectiveness of deep beam structural systems.