Shear capacity model of RC beams based on MCMC approach and reliability analysis

Abstract

The research question revolves around accurately predicting the shear capacity of RC beams and enhancing the reliability of such prediction models. This study aims to refine the accuracy of the RC beam shear capacity model. To augment the safety margin, the enhanced model underwent further calibration through a reliability analysis. The shear strength models, rooted in Bayesian theory’s Markov Chain Monte Carlo (MCMC) method and the Modified Compressive Field Theory (MCFT), were introduced. A research database was curated from test data of 782 RC beams sets under shear stress. From this database, six mechanism models were juxtaposed with the MCMC models for comparison. To meet the desired reliability indexes ( $β_{T}$ = 3.2, 3.7, 4.0) in these models, resistance factors ( $ϕ$ ) were incorporated. Both the Monte Carlo and First Order Second Moment (FOSM) methods were employed for the reliability analysis, considering inherent model errors and the probability distribution of primary variables. The effect of different elements on the target reliability index was assessed using a tornado diagram. Findings showed that the refined shear model displayed minimal variance when benchmarked against models from existing literature and national codes, yielding a V_test/ V_cal ratio close to 1.0. The shear components aligned with established mechanical mechanism. Depending on various operational conditions and load combinations, resistance factors fluctuated from 0.377 to 0.516 in the RC beams without stirrup, while resistance factors fluctuated from 0.479 to 0.621 in the RC beams with stirrup. Key determinants, such as the inherent uncertainty in computing model, the live loads, and the cylinder compression strength, significantly influenced the reliability index.

Keywords

reinforced concrete beam shear capacity shear contribution analysis MCMC reliability analysis

Introduction

Research on the shear capacity of reinforced concrete beams has spanned over a century, with numerous scholars contributing to its understanding (Committee, 1962a, 1962b; Froehlich et al., 1909; Gamble and Ramirez, 1998; Hsu, 1988, 1996; Joint ASCE-ACI Task Committee 426, 1973; Kotsovos et al., 2015; Muttoni and Fernández Ruiz, 2008). Despite the myriad of semi-empirical shear capacity models proposed, grounded in both theoretical and experimental data, certain conservative design codes still raise potential safety concerns (Anderson, 1957; Rokugo et al., 2008).

In recent years, researchers have intensively focused on enhancing the predictive capabilities of models for determining the shear capacity of RC beams. Among various methodologies, Artificial Neural Networks (ANN), Genetic Algorithms (GA), Fuzzy Inference, Monte Carlo Markov Chain (MCMC) have emerged as prevalent tools for this purpose (Bashir and Ashour, 2012; Chabib et al., 2006; Fang et al., 2022; Kara, 2011; Nasrollahzadeh and Basiri, 2014; Nehdi et al., 2007). The MCMC method, distinct from typical machine learning approaches, grounds itself in deterministic, mechanistic, and theoretical foundations, allowing for a nuanced and interpretative model analysis. Several scholars have harnessed the MCMC approach to predict shear performance in reinforced concrete. For instance, Wang et al. (2023) integrated the MCMC method to analyze the finite element model of RC beams established by ABAQUS, comparing it against static test findings. Similarly, Liu (Liu et al. (2020) constructed a probabilistic shear capacity model, utilizing MCMC, and based their findings on a vast collection of 691 test data sets for deep beams. The probabilistic model using MCMC sampling procedure was proposed for estimating the value if strut efficiency factor in the strut-and-tie model (STM). Yu et al. (2018) further expanded upon this by devising a shear capacity model using the Modified Pressure Field Model (MCFT), which incorporated both objective and subjective uncertainties via the MCMC methodology. Although these developments provide innovative solutions to calculating the shear capacity of RC beams, challenges persist. Precisely identifying the contribution of each component, from the initial cracking to the onset of shear damage, remains elusive. This necessitates a comprehensive exploration into the shear contribution, imperative for model refinement and broader implementation. Wu’s work (2018), which implemented strain gauges in RC beams’ stirrups, and the evaluations of Yuan and Yi (2020) have offered valuable insights. Nevertheless, combining the regression probability model with shear contribution metrics is a pivotal progression for detailed mechanical scrutiny. A notable omission in these studies is the lack of consideration for model reliability.

Numerous academics have delved into the reliability of design models. Nathaly (Narváez et al., 2020) applied the Monte Carlo method alongside the first order second moment reliability method to evaluate the reliability of FRP beams. These were analyzed under codes ACI 440.2R (Soudki and Alkhrdaji, 2005) and CNR-DT200 (CNR-DT 200 R1/2013, 2013) across varying load scenarios. In addition, a sensitivity analysis was conducted focusing on various factors to achieve the target reliability, further aiding design optimization. Similarly, Shahnewaz (Nasrollahzadeh and Aghamohammadi, 2018) fine-tuned the four FRP shear capacity models using Genetic Algorithms, ensuring the resistance factors of the shear design equations met a standard of structural safety. Yuan (Yuan and Yi, 2017) fine-tuned the GB50010 by employing the first order second moment reliability method. This calibration utilized an RC beam shear test database sourced from both domestic and international literature.

Building on these studies and employing the MCFT theory and the MCMC method, shear capacity models of RC beams have been developed, with an emphasis on the contributions of shear components. These models underwent assessment using both the Monte Carlo method and the first order second moment reliability method across a spectrum of load combinations. From the reliability analysis, resistance factors tailored for these models were recommended to ensure structural safety in design. Additionally, the effects of individual variables on reliability were taken into account. Subsequent to these considerations, the reliability indexes for these models were calculated, with adjustments made to align with the target reliability indexes.

Research thought and methods

Research approach

To develop shear capacity models aligned with reliability design indexes and to ensure utmost precision, the primary research methodology is conceptualized as depicted in Figure 1. Initially, basic parameter indices were sourced from a comprehensive database of RC slender beams, facilitating the determination of prior parameter distributions. Using the probability distribution of these parameters and their specific values, an initial deterministic model was constructed rooted in the modified compression field theory (MCFT). This model was then adapted into the subsequent shear capacity model by MCMC method, taking into account uncertainties related to the principal tension strain and critical crack angle. For validation, this newly proposed model was juxtaposed against established academic models and national standards. An exploration was made on how the shear contribution of the model’s components varied depending on the characteristics of different stirrups. As a final step, resistance reduction factors of the RC beam shear capacity models were set in alignment with targeted reliability indices. These factors were derived through the Monte Carlo method and the first order second moment reliability method across diverse load combinations. Additionally, a sensitivity analysis was conducted to comprehend how various variables influence reliability.

Figure 1.

Overview of RC beams shear capacity modeling and its reliability analysis.

Experimental database of RC columns

For this research, 782 shear strength test results from RC slender beams were compiled form 66 published articles, as detailed in Appendix Table 2. These findings originated from diverse sources ranging from Ahmad et al., 1995; Alqarni et al., 2020; Arowojolu et al., 2021; Arslan and Polat, 2013; Bazant and Kazemmi, 2005; Bentz et al., 2010; Biolzi et al., 2014; Bukhari and Ahmad, 2008; Cao, 2001; Cladera and Mari, 2005; Deng et al., 2017; Dino et al., 2001; Ghannoum, 1998; Gunawan et al., 2020; Hassan et al., 2008; Hong and Han, 2005; Hu and Wu, 2017; Huber et al., 2019; Islam et al., 1998; Jain and Singh, 2013; Jeong et al., 2017; Kawamura et al., 2021; Rahal and Al-Shaleh, 2004; Kim et al., 2013, 2018; Knaack and Kurama, 2015; Lee et al., 2011, 2015, 2020; Lee and Hwang, 2010; Lee and Kim, 2008; Lim and Oh, 1999; Mahdi et al., 2015; Mawlood et al., 2021; Monserrat López et al., 2020; Morrow and Viest, 1957; Nawaz et al., 2019; Nouri et al., 2021; Papia, 2004; Perera and Mutsuyoshi, 2013; Placas and Regan, 1971; Pradhan et al., 2018; Rahal, 2006; Rahal and Elsayed, 2021; Regan and Reid, 2004; Roller and Russel, 1990; Rombach and Faron, 2019; Serna-Ros et al., 2002; Shabani Attar et al., 2020; Shah and Ahmad, 2009; Sherwood et al., 2007; Shin et al., 1999; Siew, 2011; Słowik, 2014; Sowik, 2012; Tan et al., 1997; Tavio, 2009; Thamrin et al., 2016; Vecchio and Shim, 2004; Wang et al., 2011; Yi and Liu, 2022; Yoshida, 2000; YU et al., 2013; Yun and Choi, 2016 to Zu, 2020.

The collected data includes 429 sets related to slender beams without stirrups and 353 sets concerning beams equipped with stirrups. Comprehensive parameters pertaining to these beams, such as section width, shear span-to-depth ratio, effective depth, concrete strength, and specifications regarding stirrups, were duly recorded in the database. To illustrate, stirrup ratios were observed to vary between 0.07% and 3.02%, concrete compressive strengths fluctuated between 19.7 and 197 MPa, effective depth ranges from 156 to 1925 mm, and the shear span-to-depth ratio is in the range of 2.5 to 10.4. A strict selection criteria ensured the accuracy and integrity of the dataset. The criteria prioritized RC slender beams that exhibited shear failures. In addition, only specimens that underwent one or two concentrated loads were considered, while those subjected to uniform or continuous axial loads were excluded. Moreover, the database deliberately omits beams that incorporated wastes, fibers, and FRP, apart from specific renewable coarse aggregates such as fly ash and bricks.

Modeling and calibration

Probabilistic prediction model of shear strength

According to the modified compression field theory, the cross-section shear strength of beams is a combination of diagonal tensile stresses f₁ and diagonal compressive stresses f₂. When examining beams failing in shear, a micrometric body is chosen for stress equilibrium analysis. This results in the following equilibrium equation as derived from the stress Mohr circle (Bentz and Collins, 2006; Bentz et al., 2006; Wei, 2011):

{\begin{cases} f_{x} = v \cot θ - ρ_{s} f_{s x} + f_{1} \\ f_{y} = v \tan θ - ρ_{v} f_{s y} + f_{1} \\ f_{2} = v (\cot θ + \tan θ) - f_{1} \end{cases}

(1)

where p_s and p_v represent the ratio of longitudinal reinforcement and stirrup. f_x, f_y sum up the stresses in the x-axis and y-axis, respectively. V denotes the shear capacity of RC beams, while f₁ and f₂ are the principal stresses parallel to the crack direction. The transformation of equation (1) leads to the following equation (2):

f_{1} + f_{2} = v (\tan θ + \cot θ) = \frac{V}{b d_{v}} (\tan θ + \cot θ)

(2)

d_{v} = \min (h_{0}, h - 2 s - ρ_{s} - 2 ρ_{v})

(3)

In these equations, d_v represents the minimum between the effective depth and the distance of crack-control bars. s is the stirrup spacing, and h and h₀ represent the depth and effective depth of the beam section, respectively. The vertical unbalance force, carried by the stirrups, is defined by (Wei, 2011):

A_{v} f_{yv} = (f_{2} \sin^{2} θ - f_{1} \cos^{2} θ) b s

(4)

Incorporating equations (2) and (3) results in:

\begin{array}{l} V = [\frac{f_{1}}{\tan θ + \cot θ} + \frac{1}{\sin^{2} θ (\tan θ + \cot θ)} (\frac{A_{v} f_{y v}}{b s} + f_{1} \cos^{2} θ)] b d_{v} \\ = f_{1} b d_{v} \cot θ + \frac{A_{v} f_{yv}}{s} d_{v} \cot θ \end{array}

(5)

$ε_{c r}$ is the cracking load of concrete. The concrete strength f_t and average principal tensile strain $ε_{1}$ can be derived as (Wei, 2011):

f_{1} = {\begin{cases} E_{c} ε_{1} ε_{1} \leq ε_{c r} \\ \frac{0.33 \sqrt{f_{c}^{'}}}{1 + \sqrt{500 ε_{1}}} ε_{1} \geq ε_{c r} \end{cases}

(6)

ε_{1} = \frac{ε_{x} (1 + \cot^{2} θ) + 4 \times 10^{- 5} \cot θ}{1 + 0.0034 \cot^{4} θ}

(7)

Further, the relationship linking average longitudinal strain with the principal tensile strain is (Wei, 2011):

ε_{x} = \frac{\frac{M_{u}}{d_{v}} + V_{u} - V_{p} + 0.5 N_{u} - A_{ps} f_{p 0}}{2 (E_{s} A_{s} + E_{p} A_{ps})}

(8)

where

M_{u}

is the cracking bending moment;

V_{u}

is the ultimate shear capacity of beams; V_p is the shear strength generated by the prestressing force loaded on the beams. N_u is the axial load perpendicular to the section; E_s is the modulus of elasticity of the reinforcement; A_s is the Cross-sectional area of longitudinal tension reinforcement.

Figure 2 illustrates the stress balance conditions and the Mohr circle for microelements in reinforced concrete. It’s evident that determining average principal tensile strain and the critical crack angle requires intricate iterative calculations.

Figure 2.

Stress balance condition and stress Mohr circle of microelement. (a) Concrete principal stresses (b) web tensile reinforcement (c) concrete stress Mohr circle of microelement.

Drawing from theoretical derivations and taking into account material properties, geometric dimensions, and uncertainties, a model was proposed to predict the principal tension strain and critical crack angle. This model is represented by:

V_{p} = \frac{0.33 \sqrt{f_{c}^{'}}}{1 + \sqrt{500 ε_{1}}} b d_{v} \cot θ + \frac{A_{v} f_{yv} d_{v}}{s} \cot θ = \frac{0.33 \sqrt{f_{c}^{'}}}{1 + α_{1}} b d_{v} α_{2} + \frac{A_{v} f_{yv} d_{v}}{s} α_{2}

(9)

where

α_{1} = \sqrt{500 ε_{1}}

;

α_{2} = \cot θ

According to Bayesian theory, and using available experimental data, it’s inferred that the prior $α_{2}$ follows a Normal distribution with a mean of 1.648 and a standard deviation of 0.05. $α_{1}$ is uniformly distributed within [0.593, 1.515]. Utilizing this information, we can determine the posterior distribution of $α$ as given by Marin and Robert, 2014:

π (α | V) = \frac{π (α) p (V | α)}{\int π (α) p (V | α) d α}

(10)

where

π (α | V)

is the probability density function of the posterior distribution of

α

;

π (α)

is the priori probability density function of

α

;

\int π (α) p (V | α) d α

is the full probability density function, whose computation entails high-dimensional integrals and is complicated to solve;

p (V | α)

is the likelihood function of

α

; To compute this, the MCMC method is employed. Here are the fundamental steps of the MCMC method:

1) Choose the appropriate prior distribution $q (x | θ^{(t)})$ , and decide an initial value $θ^{(0)}$ .

2) Generate random samples from the chosen prior distribution $θ^{'}$ .

3) Evaluate samples against specific criteria. If they meet the criteria, accept $θ^{'}$ . If the sample is accepted, then $θ^{(t + 1)}$ = $θ^{'}$ ; if rejected, $θ^{(t + 1)}$ = $θ^{(t)}$ .

4) Repeat the above steps until the desired sample size is achieved.

After 50,000 iterations, a selected subset of top values, as shown in Table 1.

Table 1.

Statistical characteristics for posterior distribution of RC beams with stirrup.

Parameter	Mean value	Standard deviation	Coefficient of variation	2.5% quartiles	97.5% quartiles
$α_{1}$	0.593	0.0005	0.0008	0.5929	0.5932
$α_{2}$	1.358	0.0006	0.0004	1.358	1.36

The results were presented in Table 1, were retained for further analysis. Using this data, the posterior model of RC beams equipped with stirrups can be deduced as:

V_{ps} = 1.358 (\frac{0.33 \sqrt{f_{c}^{'}}}{1 + 0.593} b_{w} d_{v} + \frac{A_{v} f_{yv} d_{v}}{s})

(11)

For beams without stirrups, a prior model to predict shear capacity is formulated:

V_{pn} = \frac{0.33 \sqrt{f_{c}^{'}}}{1 + \sqrt{500 ε_{1}}} b d_{v} \cot θ = \frac{0.33 \sqrt{f_{c}^{'}}}{1 + β_{1}} b d_{v} β_{2}

(12)

Reapplying the Bayesian framework and using available experimental data, the prior

α_{2}

obeys a normal distribution with a mean of 1.648 and a standard deviation of 0.05, while

α_{1}

has a uniform distribution within [0.615, 1.404]. Substituting this into equation (11) and updating with the MCMC method, we obtain the posterior results presented in Table 2. Consequently, the posterior model for shear capacity of beams without stirrups is:

V_{pn} = 1.645 \frac{0.33 \sqrt{f_{c}^{'}}}{1 + 1.073} b d_{v}

(13)

Table 2.

Statistical characteristics for posterior distribution of RC beams without stirrup.

Parameter	Mean value	Standard deviation	Coefficient of variation	2.5% quartiles	97.5% quartiles
$β_{1}$	1.073	0.062	0.058	0.9492	1.074
$β_{2}$	1.645	0.049	0.030	1.547	1.646

Comparison

This section presents a comparative analysis using Figures 3–4, which illustrates the shear capacity results for beams with stirrup and without stirrup. Figures 3–4 depicts the x-axis as the sequence of model numbers and the y-axis as the normalized result—specifically, the ratio of the tested to the calculated shear strength ( $V_{test} / V_{cal}$ ). These models were assessed through statistical indicators like mean, standard deviation, coefficient of variation, which offer insights into data dispersion and central tendency, respectively. In Figures 3–4, the red scatter marks the ratio of $V_{test} / V_{cal}$ ;the box’s lower and upper boundaries correspond to the x_25% and x_75% percentiles, while the whiskers extend to the 2.5% quartiles (x_2.5%) and 97.5% quartiles (x_97.5%) percentiles. The central line within the box pinpoints the mean ratio. A ratio near 1.0 suggests minimal prediction error, signaling an accurate model. Ratios above 1.0 indicate conservative predictions, underestimating actual shear strength, while those below 1.0 suggest an overestimation.

Figure 3.

Overall prediction results for beams with stirrup.

Figure 4.

Overall prediction results for beams without stirrup.

The analysis encompasses six mechanisms, including models from various design codes—ACI318 (ACI Committee 318-19, 2019), JSCE (JSCE Guidelines for Concrete No. 3. Structural Performance Verification, 2002), GB50010 (GB 50010-2010, 2010), and formulas by Zsutty (Zsutty, 1968), Gunawan (Gunawan et al., 2020), Bazant-Kim (Bazant and Kim, 1984), compared against the MCMC model with stirrups. According to Figure 3, the MCMC model’s mean value of 0.994 is remarkably close to experimental results. The models GB50010 (GB 50010-2010; 2010) and Bazant-Kim (Bazant and Kim, 1984) also demonstrate high prediction accuracy with their respective mean, standard deviation and coefficient of variation values variation values for the models are highly accurate, reflected by the ratios of 1.022, 0.292, and 0.286, respectively. Additionally, another set of models exhibits ratios of 1.032, 0.198, and 0.192, respectively. However, the ACI318 (ACI Committee 318-19, 2019) and JSCE (JSCE Guidelines for Concrete No. 3. Structural Performance Verification. 2002) models yield mean values of 1.228 and 1.269, respectively, leaning towards conservative estimates. When examining the data dispersion, the scholars’ models and national codes exhibit comparable levels, but Bazant-Kim’s (Bazant and Kim, 1984) model stands out for its minimal dispersion. It’s noteworthy that the MCMC model without stirrups outperforms the other models, as evidenced in Figure 4, with Gunawan’s model (Gunawan et al., 2020) showing commendable stability and robustness with a standard deviation around 0.23. This robustness might stem from the model accounting for the initiation and growth of diagonal cracks. The GB50010 model (GB 50010-2010, 2010) without stirrups tends to significantly overpredict shear capacity, as evidenced by its mean value of 0.9275. In contrast, this model gives a more balanced prediction when stirrups are included.

Figures 3–4 demonstrates that stirrups influence the predictive accuracy of various models to different extents. Nonetheless, the MCMC, Zsutty (Zsutty, 1968) and Bazant-Kim (Bazant and Kim, 1984) models maintain consistent ratios close to 1.0, irrespective of stirrup presence. In stark contrast, the ACI (ACI Committee 318-19, 2019) and JSCE (JSCE Guidelines for Concrete No. 3. Structural Performance Verification, 2002) models invariably predict conservatively for both beam types. For the GB50010 (GB 50010-2010; 2010) model, the prediction is overly optimistic for beams without stirrups but is well-aligned for those with stirrups. These variations are crucial for a comprehensive discussion on how well the MCMC model aligns with the underlying shear mechanism.

Shear contribution analysis

The above discussion underscores the prediction accuracy and dispersion of various models’ results, both with and without stirrups. To elucidate the rationality of these models, their shear contribution is assessed in accordance with the shear mechanism theory. The shear strength of RC beams is attributed to two main components: the concrete (V_c) and the stirrups (V_s). These substructures in the expressions from the cited models (Ma et al., 2023) represent distinct contributions to the shear section. The analysis proceeds in two stages. The first stage compares the results of the MCMC model with six other models to verify consistency with the shear mechanism. The second stage examines the decision-making basis of the MCMC model for strict adherence to shear mechanics principles.

Relative contribution

In RC beams with stirrups, the stirrup component (V_s) contributes increasingly to the predicted shear strength (V_cal) as the stirrup characteristic value $ρ_{v} f_{yv}$ rises. This correlation, along with the distribution of $V_{s} / V_{cal}$ , is depicted in Figure 5. The horizontal axis denotes the stirrup characteristic value, whereas the vertical axis corresponds to the $V_{s} / V_{cal}$ ratio. Values at the 2.5%, 25%, 75%, and 97.5% percentiles demonstrate the data’s range and central clustering. The term $[x_{25 %}, x_{75 %}]$ and $[x_{2.5 %}, x_{97.5 %}]$ are of particular interest, indicating the middle 50% and the outer 95% of the dataset, respectively.

Figure 5.

Analysis of the relative variation of stirrup contribution with the stirrup characteristic. (a) MCMC (b) Bazant-Kim, (c) Zsutty, (d) Gunawan, (e) AC1318, (f) JSCE, (g) GB50010.

Analyzing the mean, maximum and minimum contribution of V_s, here is a consistent pattern where an increase in $ρ_{v} f_{yv}$ leads to an upsurge in $V_{s} / V_{cal}$ . This nonlinear increase in $V_{s, MCMC} / V_{p}$ from 10.6% to 87.7% is noteworthy. Comparatively, these findings are parallel to those observed by Yuan (Yuan and Yi, 2020), and the average contribution of the stirrup component is found to be 40.9%, which correlates with the findings by Yi (Yi and Tan, 2022). The tensile stress behavior of RC beams influences these results. Stressed stirrups distribute tensile forces across the concrete, curbing cracking by engaging the stirrups’ stress capacity. When cracks are imminent, the stirrups act as a reinforcing bridge, bolstering the structural integrity during this critical juncture. This illustrates the significant role of stirrups in the shear contribution of the concrete component.

The analysis of concrete component’s shear capacity, as formulated in equation (12), reveals a downward trend as the shear-to-span ratio escalates, paralleling the experimental outcomes reported by Wu (Hu and Wu, 2017, 2018). This trend confirms the varying performance of concrete under different loading conditions.

Absolute contribution

In the RC beams, the beam database, beams without stirrups serve as the baseline for comparison. Beams with stirrups, under identical conditions, are considered the control group. To gauge the shear capacity contributed by stirrups, the capacities of beams without stirrups are subtracted from those with stirrups under the same conditions. Same conditions implies consistent loading systems, material properties, and geometric dimensions across the groups, with the only variable being the stirrup ratio. A collective analysis involves 150 beam sets. Shear-span ratio $a / d$ range from 2.5 to 5.5. Beam widths vary from 122 mm to 750 mm, and effective section heights span from 130 mm to 700 mm. Concrete strengths $f_{c}$ are concentrated within a range of 19.7 MPa to 102 MPa. Stirrup yield strengths $f_{yv}$ and ratios $ρ_{v}$ range between 215 MPa and 669.6 MPa and from 0.089% to 1.42%, respectively.

Figure 6 illustrates the findings, with the x-axis representing the stirrup characteristic value and the y-axis showing the ratio of experimental to theoretical shear strength. National codes and scholars’ equations are typically conservative in predicting stirrup shear capacity. This conservatism is pronounced at $ρ_{v} f_{y v}$ values below 100 MPa. In beams with low shear-span ratios, stirrups may not reach full yield, resulting in the concrete bearing most of the shear load. Conversely, beams with higher shear-span ratios rely more on stirrups for shear strength, especially after significant cracks appear. This is reflected in the ratios of tested to calculated shear capacity, suggesting that the test shear capacity of stirrups does not keep pace with their calculated capacity as the parameter increases $V_{s, test} / V_{s, cal}$ , which decrease with increasing. This discrepancy might be due to non-yielding stirrups and cracking in the concrete. According to equation (10), the $V_{s, test} / V_{s, cal}$ results for the GB50010 model and Zsutty’s equation (Zsutty, 1968) hover around 1.124 to 1.129, showcasing close agreement. However, Gunawan et al. (2020) present exceedingly cautious and conservative results.

Figure 6.

Analysis of the absolute variation of stirrup contribution with the stirrup characteristic.

Reliability analysis

Definition of the problem

Reliability analysis is utilized to ascertain the probability of structural failure by incorporating the influence of uncertainty. This section presents equations (10) and (12), which are formulated for reliability evaluation using the Monte Carlo and FOSM methods, in accordance with the engineering reliability design codes such as (ACI Committee 318-19, 2019; GB 50153-2008, 2009). The resistance factor ( $ϕ$ ) is established to meet the specified reliability indices.

The core challenge in structural reliability is ensuring that the structural strength exceeds the demands imposed by applied loads. The reliability function $G (Z)$ considers various influencing factors encapsulated by z. $G (Z)$ defines three states: the failure region

{\begin{cases} G (Z) < 0 (Failure region) \\ G (Z) = 0 (limit state) \\ G (Z) > 0 (Safe region) \end{cases}

(14)

G (Z)

< 0, the limit state

G (Z)

= 0, and the safe region

G (Z)

>0. If

G (Z)

is negative, the structure is deemed to have failed. Conversely, a positive

G (Z)

suggests that the structure is in a secure state, while

G (Z)

= 0 indicates the threshold of failure. The failure probability,

P_{f}

, is calculated by integrating the multivariate probability density function g(z) over the failure domain.

P_{f} = P [G (Z) \leq 0] = \int_{G (Z) \leq 0} g (Z) d x

(15)

Z is a vector of all the random variables involved, defined by:

Z = g (R, S) = R - γ_{0} (S_{G} + S_{Q}) = 0

(16)

where R represents the structural resistance. and

γ_{0}

is the importance factor. Based on the safety level,

γ_{0}

takes values of 1.1 for first-class structures, and 1.0 and 0.9 for second and third-class structures, respectively. The dead and live load combinations are calculated for a designed service life of 50 year as per (GB 50153-2008, 2009). The combined load effect S is computed using the following equation:

S = \max {\begin{cases} 1.2 S_{Gk} + 1.4 S_{Qk}, & 1.35 S_{Gk} + 1.4 \times 0.7 S_{Qk} \end{cases}}

(17)

where

S_{Gk}

and

S_{Qk}

are the standard values for dead and live loads, respectively.

The resistance factor for individual shear equations is determined under various load combinations, based on the shear capacity models equations (10) and (12). The final expressions for shear strength V_p of RC beams, with and without stirrups, are:

{\begin{cases} V_{p} = ϕ [(\frac{0.33 \sqrt{f_{c}^{'}}}{1 + 0.593} b d_{v} + \frac{A_{v} f_{yv} d_{v}}{s}) 1.358] for RC beams with stirrup \\ V_{p} = ϕ [1.645 \frac{0.33 \sqrt{f_{c}^{'}}}{1 + 1.073} b d_{v}] for RC beams without stirrup \end{cases}

(18)

Calculation of relevant parameters

Material uncertainty

Nowak (Szerszen and Nowak, 2003) carried out a substantial analysis on data regarding cylinder compressive strength as well as reinforcement strength, concluding that both the cylinder compressive strength (

f_{c}^{'}

) and the reinforcement yield strength of reinforcement (

f_{yv}

) follow Normal distributions. The deviation factor is defined as the ratio of the measured strength to the ACI318-provided nominal value (ACI318-19, 2019). The determination and computation of material uncertainty coefficients are in line with recommendations from established experimental data (Gao et al., 1987; Research Reports on Reinforced Concrete Structures, 1977; Yuan and Yi, 2017). The stirrup deviation coefficient

k_{f_{yv}}

is 1.156, reflecting a coefficient of variation of 0.082. These parameters influencing the cylindrical compressive strength of concrete are summarized in Table 3.

Table 3.

Statistical parameters of uncertainty of the cylinder compressive strength.

$f_{c}^{'} / MPa$	22.4	31	38.8	46.5	58.6
$K_{f_{c}^{'}}$	1.28	1.175	1.213	1.165	1.125
$δ_{f_{c}^{'}}$	0.091	0.082	0.088	0.098	0.095

Uncertainty in geometric features

Acknowledging the dimensional variability in RC beams, the recommended deviation and variation coefficients are drawn from literature and standards (Gong and Zhang, 2022; GB 50153-2008, 2009), as indicated in Table 4.

Table 4.

Statistical parameters of uncertainty in geometric features.

Name	$b_{w}$	$h_{0}$	$A_{sv}$	$s$
$K$	1.00	1.00	1.00	0.99
$δ$	0.02	0.03	0.03	0.07

Uncertainty in computing models

The uncertainty in the computational models captures the discrepancy between experimental and modeled results. The random variable

K_{p}

is derived from the equation (Yuan, 2017):

K_{P} = R_{test} / R_{cal}

(19)

where

R_{test}

is the experimental resistance and

R_{cal}

represents the resistance calculated by codes or scholarly models. Table 5 displays the comparison between equations (10) and (12) and their corresponding experimental values.

Table 5.

Statistical parameters of uncertainty in computing mode.

Name	$μ_{K_{p}}$	$σ_{K_{p}}$	$δ_{K_{p}}$
Equation (10)	0.994	0.279	0.281
Equation (12)	1.003	0.330	0.329

where $μ_{K_{p}}$ = mean value of $R_{test} / R_{cal}$ , $σ_{K_{p}}$ = standard deviation of $R_{test} / R_{cal}$ , $δ_{K_{p}}$ = variation coefficient of $R_{test} / R_{cal}$ .

Uncertainty of resistance

The resistance deviation coefficient

K_{R}

and its variation coefficient

δ_{R}

can be calculated using (Yuan and Yi, 2017):

{\begin{cases} K_{R} = \frac{μ_{R}}{R_{d}} = \frac{μ_{K_{p}} \cdot μ_{R_{p}}}{R_{d}} \\ δ_{R} = \sqrt{δ_{K_{p}}^{2} + δ_{R_{p}}^{2}} \end{cases}

(20)

where

R_{p}

is calculated based on design codes or models developed by experts, which incorporate material properties and geometrical parameters. This design expression of shear capacity, denoted as

R_{d}

, is derived from established codes or academic models. The mean value and variation coefficient of

μ_{R_{p}}

δ_{R_{p}}

, represented as

R_{p}

respectively (Yuan, 2017), are calculated using equation (21):

{\begin{cases} μ_{R_{p}} = R_{p} (μ_{x_{1}}, μ_{x_{2}}, . . . ., μ_{x_{n}}) \\ \begin{array}{l} σ_{R_{p}} = \sqrt{\sum_{i = 1}^{n} (\frac{\partial R_{p}}{\partial X_{i}} | {}_{μ}σ_{x_{i}})} \\ δ_{R_{p}} = \frac{σ_{R_{p}}}{μ_{R_{p}}} \end{array} \end{cases}

(21)

where

σ_{R_{p}}

is the standard deviation of each input variable, and

R_{p}

\frac{\partial R_{p}}{\partial X_{i}} |_{μ}

signifies the partial derivative of

R_{p}

. Equation (20) presumes that these values average out over the distribution of inputs. The structural resistance R is often assumed to follow a Lognormal distribution, as per (GB 50153-2008, 2009). The following Table 6 demonstrates that the results of

K_{R}

and

δ_{R}

align with the Lognormal distribution assumption.

Table 6.

Statistical parameters of uncertainty of resistance.

Name	With stirrup					Without stirrup
$f_{c}^{'} / MPa$	22.4	31	38.8	46.5	58.6	22.4	31	38.8	46.5	58.6
$K_{R}$	1.232	1.165	1.194	1.160	1.154	1.284	1.179	1.217	1.168	1.128
$δ_{R}$	0.291	0.291	0.291	0.291	0.290	0.342	0.340	0.342	0.344	0.343

Uncertainty of load

The statistical parameters and probability distributions for dead load as well as live loads over a 50-year design reference period conform to the recommendations of GB50153 (GB 50153-2008, 2009). These parameters are detailed in Table 7, which provides a comprehensive overview of the assumed variabilities for each type of load.

Table 7.

Statistical parameters of uncertainty of loads.

Name	$k$	$δ$	Distribution
Dead live	1.060	0.070	Normal
Live load (office)	0.524	0.288	Extreme value type I
Live load (residential)	0.644	0.233	Extreme value type I

Results from reliability analysis

The load effect value is defined by selecting the larger value of the two calculated values from equation (16). It is assumed that the load effect on structural member S is equivalent to the shear capacity resistance R. The load effect ratio is defined as $θ = S_{Qk} / S_{Gk}$ , which replaces the absolute value of the load effect. According to Yuan (2017), reliability indexes for each shear capacity model are calculated across a comprehensive range of $θ$ values, from 0.1 to 2.0.

The results for the reliability indexes from equations (10) and (12) are depicted in Figures 7–8, showing that the reliability index is inversely proportional to the load effect ratio under different conditions. As the concrete strength improves, the reliability index initially increases and then decreases. Nevertheless, equations (10) and (12) do not align with the reliability index recommended by GB50153 (GB 50153-2008, 2009). Introducing a resistance index is necessary to achieve an acceptable level of structural safety, as discussed by Szerszen and Nowak (2003). Table 8 displays the average values of the resistance index for various design requirements, and Figures 7–8 demonstrates the change in these indexes under different working conditions when targeting a reliability index $β$ of 3.7.

Figure 7.

Reliability index in RC beams versus varied load. (a) Beams without stirrup under office live loads (b) beams without stirrup under residential live loads (c) stirrup beams under office live loads (d) stirrup beams under residential live loads.

Figure 8.

Optimized reliability index in RC beams versus varied load. (a) Beams without stirrup under office live loads (b) beams without stirrup under residential live loads (c) stirrup beams under office live loads (d) stirrup beams under residential live loads.

Table 8.

Average resistance factor for target reliability indexes.

Modified equations	Performance function	$β_{T}$	$ϕ$
RC beams with stirrup	Dead load + office live load	3.2	0.621
		3.7	0.530
		4.0	0.494
	Dead load + residential live load	3.2	0.600
		3.7	0.516
		4.0	0.479
RC beams without stirrup	Dead load + office live load	3.2	0.516
		3.7	0.435
		4.0	0.391
	Dead load + residential live load	3.2	0.476
		3.7	0.417
		4.0	0.377

Table 8 compares the $β$ index results for equation (18) from Monte Carlo and FOSM methods under various conditions. The comparison shows that the relative differences between the two methods are typically below 1%, validating the precision of the FOSM method. An exception is found at $θ$ = 1 in RC beams with stirrups, where FOSM yields a 2.1% higher reliability index than Monte Carlo simulations. For beams without stirrups under dead load with residential live loads, the FOSM method’s reliability indexes are consistently higher than those from the Monte Carlo method. For all other conditions with $θ$ = 1, the FOSM method results are slightly lower, indicating the method’s accuracy for equation (18).

Sensitivity analysis

Sensitivity analysis plays a crucial role in evaluating the influence of various variable on reliability, serving as an essential reference for structural reliability design. This analytical approach, extensively investigated by researchers focusing on uncertainty in seismic collapse fragility (Baker and Cornell, 2003; Eduardo and Hesameddin, 2003; Porter et al., 2002), has found its place in construction engineering due to the effective methodology it provides. The tornado diagram, for instance, is a widely applied tool that facilitates the analysis of sensitivity (Yin et al., 2012; Yu and Lu, 2012). It allows for one variable to randomly vary within a specific range while keeping other variables constant, enabling the quantification of sensitivity between the variable and the parametric model through comparative calculations.

Determining the distribution type for the variable

K_{p}

involves Kolmogorov Smirnov (KS) test, where commonly utilized distributions include Normal, Lognormal, Weibull and Gamma. For the KS test, a significance level of 0.05 leads to critical values for

K_{p}

in RC beams with and without stirrups being 0.078 and 0.087 respectively. The D value below these critical values suggests that the hypothesis of the distribution fitting the data can be accepted. Empirical testing has shown that the Normal distribution best fits the data for RC beams both with and without stirrups, having D values of 0.1 and 0.128, respectively. Consequently, the Normal distribution is chosen for

K_{p}

in both scenarios. (Table 9).

Table 9.

Optimized reliability indexes for the different methods and performance function.

Type	Performance function	$θ$	$β$
Type	Performance function	$θ$	FOSM	Monte Carlo
RC beams with stirrup	Dead load + office live load	0.1	4.319	4.326
		0.25	4.316	4.311
		0.5	4.280	4.255
		0.75	4.217	4.188
		1	4.136	4.049
		1.25	4.049	4.015
		1.5	3.969	3.940
		1.75	3.895	3.867
		2	3.835	3.811
	Dead load + residential live load	0.1	4.184	4.189
		0.25	4.187	4.177
		0.5	4.162	4.138
		0.75	4.109	4.070
		1	4.042	4.007
		1.25	3.958	3.923
		1.5	3.896	3.862
		1.75	3.838	3.804
		2	3.781	3.752
RC beams without stirrup	Dead load + office live load	0.1	4.422	4.425
		0.25	4.405	4.403
		0.5	4.347	4.307
		0.75	4.268	4.241
		1	4.178	4.152
		1.25	4.091	4.059
		1.5	4.014	3.980
		1.75	3.944	3.917
		2	3.885	3.858
	Dead load + residential live load	0.1	4.317	4.313
		0.25	4.305	4.291
		0.5	4.259	4.223
		0.75	4.193	4.163
		1	4.121	4.084
		1.25	4.051	4.026
		1.5	3.989	3.964
		1.75	3.934	3.911
		2	3.888	3.864

In the context of RC beams subjected to dead load and office live loads, examples will be used to illustrate the analysis with concrete strength set at C30 and the load effect ratio at 1.0. The quantile values for variables, provided at 97.5% and 2.5% intervals, are presented in Table 10. These values are incorporated into equation (20) to derive the upper and lower limits of the reliability indexes, depicted in Figure 9. The figure illustrates the varying degrees to which each variable influences the

β

index. Notably, the variables that predominantly affect the reliability index are the mode of computing

K_{p}

, the live load

Q

, and the cylinder compressive strength

f_{c}

. The latter shows high sensitivity due to its significant impact on the concrete component’s contribution to shear capacity. Although the geometric features and the stirrup component are key in determining

β

values, they demonstrate lesser sensitivity. Given that the

K_{p}

calculation depends heavily on the experimental data and chosen model, it reflects the necessity of basing analyses on extensive data sets and accurate predictive models.

Table 10.

Probabilistic characteristics of different variables.

Name	Mean	COV	Distribution	$X_{2.5 %}$	$X_{97.5 %}$
$f_{c}$	1.175	0.082	Normal	0.986	1.364
$f_{y v}$	1.156	0.082	Normal	0.964	1.342
$b_{w}$	1	0.02	Normal	0.961	1.039
$d_{v}$	1	0.03	Normal	0.941	1.059
$s$	0.99	0.07	Normal	0.854	1.126
$K_{p}$	1.003	0.328	Normal	0.358	1.648
$G$	1.06	0.07	Normal	0.915	1.205
$Q$	0.524	0.288	Extreme value type I	0.438	1.024

Figure 9.

Sensitivity factors of the input variables. (a) RC beams with stirrup (b) RC beams without stirrup.

Conclusion

The shear capacity models for RC beams have been proposed based on the MCFT and MCMC method, supported by 782 sets of experimental data. The rationality of the proposed models was discussed through the contribution of each shear component. Additionally, in order to meet the target reliability indexes, resistance factors were introduced, and the models were analyzed for their reliability. Key conclusions include:

(1) The mean values of $V_{test} / V_{cal}$ for shear capacity models of RC beams, both with and without stirrups, are 0.994 and 1.003, respectively, and the standard deviations are 0.279 and 0.33. The other six models predict shear capacity accurately for RC beams with stirrups, with mean values of $V_{test} / V_{cal}$ ranging from 1.022 to 1.269 and standard deviations from 0.198 to 0.301. While the mean values of $V_{test} / V_{cal}$ range from 0.928 to 1.282 and standard deviations from 0.231 to 0.333 for RC beams without stirrups in the other six models. The proposed models have demonstrated accuracy in predicting the shear capacity of RC beams, validating their robustness across diverse datasets.

(2) Evaluating the contribution of the concrete component and the stirrup component, highlights the soundness of the model’s rationale from relative and absolute contribution. In RC beams with stirrups under shear load, the concrete component is the primary shear component, contributing on average 59.1%. The stirrup component is essential, accounting for 40.9% of the shear capacity. The range of stirrup component contribution varies between 10.6% and 87.7%, which is conservative compared to experimental values. The trends between the proposed model and the other six models are consistent, while the interval ranges are also highly comparable. These results indicate a solid adherence to the established shear mechanism.

(3) For the models to achieve the target reliability indices $β_{T}$ , the required resistance factors $φ$ for RC beams with stirrups must be 0.621, 0.53, and 0.494 for $β_{T}$ values of 3.2, 3.7, and 4.0, respectively. For beams without stirrups, the corresponding $φ$ values are 0.516, 0.435, and 0.391. For beams subject to dead and residential live loads, $φ$ values of 0.6 and 0.476 are necessary for a $β_{T}$ of 3.2, while $φ$ values of 0.516 and 0.417 meet a $β_{T}$ of 3.7, and $φ$ values of 0.479 and 0.377 for $β_{T}$ of 4.0. The optimized models exhibit higher reliability when using these resistance factors.

(4) The factors that significantly impact the reliability values are the uncertainty in the mode $K_{p}$ , the live load $Q$ and the cylinder compressive strength $f_{c}$ . Monte Carlo simulation results are more conservative compared to those obtained by the FOSM method, with the relative error between the two methods not exceeding 1%.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the Fundamental Research Funds for the Central Universities, CHD (Grant No. 300102282503), Tianshan Talent Training Program of Xinjiang Province (Grant No. 2023TSYCCX0110), National Natural Science Foundation of China (Grant No.52478215), and the NSF of Xinjiang Province (Grant No. 2022E02095).

ORCID iDs

Cailong Ma

Guohua Xing

Appendix

Appendix Table 1 Appendix Table 1.

Summary of shear capacity models for RC beams.

Model	Formulas
ACI318	$V_{u} = (0.158 \sqrt{f_{c}^{'}} + 17.2 ρ_{v} \frac{V_{u} d}{M_{u}}) b_{w} d$
GB50010	$V_{u} = a_{cv} f_{t} b h_{0} + f_{yv} \frac{A_{sv}}{s} h_{0}$
JSCE	$\begin{array}{l} V_{u} = \frac{β_{d} β_{p} f_{vcd} b_{w} d}{1.3} + \frac{A_{v} f_{yv}}{1.1 s} z (\sin α + \cos α) \\ β_{d} = \sqrt[4]{1000 / d}; β_{p} = \sqrt[3]{100 ρ}; f_{vcd} = 0.2 \sqrt[3]{f_{ck}} \end{array}$
Zsutty	$V_{u} = 2.2 {(f_{c}^{'} ρ \frac{d}{a})}^{1 / 3} b_{w} d + \frac{A_{v} f_{yv}}{s} d$
Gunawan	$V_{u} = 0.2 {(f_{c}^{'} 100 ρ)}^{1 / 3} {(\frac{d}{1000})}^{- 1 / 4} (0.75 + \frac{1.4}{a / d}) b_{w} d + \frac{A_{w} f_{yv} z}{s}$
Bazant-Kim	$V_{u} = \frac{10 ρ^{1 / 3}}{\sqrt{1 + \frac{d}{25 d_{a}}}} (0.083 \sqrt{f_{c}^{'}} + 20.69 \sqrt{\frac{ρ_{1}}{{(\frac{a}{d})}^{5}}} b_{w} d + \frac{A_{v} f_{y} d}{s})$

Note. $f_{c}^{'}$ = cylinder compressive strength; $ρ_{v}$ = stirrup ratio; $V_{u}$ = the ultimate shear strength; $M_{u}$ = the ultimate moment; $b_{w}$ = the beam width; $a$ = shear span; $d$ = the effective depth; $s$ = the stirrups’ spacing; $A_{v}$ = the cross section area of the stirrup; $β_{d}$ = coefficient to consider effect of effective depth on shear capacity; $β_{p}$ = coefficient to consider effect of longitudinal reinforcement on shear capacity; $z$ = distance between resultant compressive strength and centroid of tension reinforcement; $f_{ck}$ = the compressive strength of concrete with prism specimens of $150 \times 150 \times 300$ mm; $α$ = the inclination of the stirrups; $d_{a}$ = maximum aggregate size; $ρ_{1}$ = longitude reinforcement ratio; $f_{y}$ = the yield strength of the stirrups.

Appendix Table 2.

Experimental database of RC slender beams.

Ref	a/d	d/mm	f_c^’/MPa	f_y/MPa	ρ_l/%	f_yv/MPa	ρ_v/%	V_u/kN
Regan and Reid (Regan and Reid, 2004)	3.68∼3.75	200∼350	34.60∼41.00	475∼600	2.39∼4.19	0∼350	0.25∼0.5	50∼207
Cucchiara et al. (Papia, 2004)	2.80	219∼225	41.2	610	1.86∼1.91	0∼510	0∼0.63	40∼117
Angelakos et al. (Dino et al., 2001)	2.75∼2.92	925	21∼99	483∼550	0.50∼2.02	0∼508	0∼0.08	163∼452
Lim and Oh (Lim and Oh, 1999)	3.08	130	34	420	3.09	0∼359	0∼1.41	46∼91
Rahal and Shaleh (Rahal and Al-Shaleh, 2004)	2.73∼3.00	300∼330	60.9∼66.4	440	2.16∼3.96	0∼305	0∼0.30	125∼270
Cao (Cao, 2001)	2.80∼2.93	1845∼1925	27.5∼30.8	436	0.36∼1.52	0∼483	0∼0.07	399∼1225
Yoshida (Yoshida, 2000)	2.86∼3.07	110∼1890	35.4∼37.7	437∼550	0.74∼0.91	0∼470	0∼0.07	40∼550
Attar et al. (Shabani Attar et al., 2020)	4.12	170	30	390	0.89	0	0	26
López et al. (Monserrat López et al., 2020)	2.56∼10.40	384∼389	22.3∼36.4	218∼544	1.63∼2.29	0∼549	0∼0.21	81.8∼281
Arowojolu et al. (Arowojolu et al., 2021)	2.50	279.40	69.5	420	0.94	0∼400	0∼1.42	99∼131
Rombach and Faron (Rombach and Faron, 2019)	3.00∼5.00	302	49.2∼54.8	550	1.56	0	0	75∼84
Kim et al. (Kim et al., 2018)	2.50∼4.00	300	34	476.5	1.43	0	0	62∼107
Lee et al. (Lee et al., 2020)	3.05	442∼459	23.1	396∼408	1.66∼1.82	504	0.11∼0.22	187∼273
Jeong et al. (Jeong et al., 2017)	2.50	150∼780	33.5	455.4∼542.4	0.96∼3.44	0	0	81.7∼245
Choi and Yun (Yun and Choi, 2016)	2.50∼5.00	525	27	433	1.87	0	0	206∼259
Alqarni et al. (Alqarni et al., 2020)	3.20	250	49.4∼81.6	571	4.19	0∼599	0∼0.52	70.4∼197
Jain and Singh (Jain and Singh, 2013)	3.50	251	24.5∼28.15	566	2.67	0∼558	0∼0.46	62∼201
Huber et al. (Huber et al., 2019)	3.04	460	42.5∼63.9	566	1.17	0	0	56∼85
Perera and Mutsuyoshi (Perera and Mutsuyoshi, 2013)	3.00∼4.00	250	116∼194	750	3.04	0	0	80∼226
Wang et al. (Wang et al., 2011)	3.00	155	70.51	336.5	2.59	441.5	0.19	58
Nouri et al. (Nouri et al., 2021)	3.00	135	44∼85	439∼447	1.2∼2.3	0	0	23∼30
Rahal and Elsayed (Rahal and Elsayed, 2021)	3.00	387	44.4∼55.5	565	1.04	0	0	60∼68
Deng et al. (Deng et al., 2017)	3.00	360	31.05∼34.13	470.6	1.58	0	0	77∼99
Hassan et al. (Hassan et al., 2008)	2.80∼3.75	100∼667	47	480	1.09∼3.14	0	0	78∼340
Nawaz et al. (Nawaz et al., 2019)	3.37	260	19.8∼50.3	530	0.73∼1.08	530	0.56∼0.83	228∼271
Biolzi et al. (Biolzi et al., 2014)	2.50∼4.50	260	42.11∼74.97	516.5∼589.6	0.91∼1.03	0∼589.6	0∼0.10	34∼109
Słowik (Słowik, 2014)	2.50∼4.10	220	35	453∼545	1.8	0	0	40∼50
Hu and Wu (Hu and Wu, 2017)	2.60	270	32∼40.5	551	2.36	0∼442	0∼0.62	117∼258
Lee et al. (Lee et al., 2011)	2.50∼4.00	300∼520	25∼50.3	530∼707.1	2.7∼4.6	187.1∼758.7	0.3∼0.5	187∼758
Bukhari and Ahmad (Bukhari and Ahmad, 2008)	2.50∼6.00	267	44.8∼52.7	420	0.58∼1.07	0	0	24∼70
Yi and Liu (Yi and Liu, 2022)	2.60∼3.4	335∼338	29.93∼53.63	552	1.52∼1.84	414.3	0∼0.19	80∼215
Zu (Zu, 2020)	2.64∼3.48	201∼218	25.81	439∼443	1.13∼1.85	0	0	31∼45
Yu et al. (Yu et al., 2013)	2.50∼2.70	426∼1098	27.255∼33.81	500	0.67∼1.19	0	0	150∼630
Islam et al. (Islam et al., 1998)	3.00∼3.90	203	50.8∼83.3	532∼554	2.02∼3.22	0	0	45∼117
Khaleel and Kareem (Kawamura et al., 2021)	2.50∼3.50	220	56.09	401.8	1.52∼3.05	313.6	0.24	62∼87
Bentz et al. (Bentz et al., 2010)	3.26∼4.10	194∼937	35∼46	397∼474	0.51∼2.54	0	0	54∼232
Thamrin et al. (Thamrin et al., 2016)	3.70	219	32	550	0.97∼2.50	0	0	32∼37
Lee et al. (Lee et al., 2015)	2.56	390	33.6∼68.4	648	3.72	0∼667	0∼253.68	403∼914
Lee and Hwang (Lee and Hwang, 2010)	3.00	252∼383	26.8∼84.6	525∼1068	1.44∼6.92	508∼510	0.20∼3.02	188∼836
Arslan and Polat (Arslan and Polat, 2013)	2.50	230	20.83	420	1.16∼2.21	393	0.22∼0.54	57∼87
Gunawan et al. (Gunawan et al., 2020)	2.50∼3.50	300	40.8∼49.1	1152∼1197	1.69	0∼395	0∼0.65	57∼144
Tavio (Tavio, 2009)	3.50	200∼700	85∼102	512	3.98	0∼518	0∼0.58	69∼620
Lee and Kim (Lee and Kim, 2008)	3.0∼5.0	280∼410	19.7∼40.8	520∼550	0.93∼4.76	0∼215	0∼0.18	70∼316
Kawamura et al. (Kawamura et al., 2021)	2.86	350	20.3∼21.5	537	1.84	370∼403	0.32∼0.36	114∼171
Vecchio and Shim (Vecchio and Shim, 2004)	4.00∼5.00	457∼466	22.6∼25.9	436∼555	1.80∼3.46	325∼600	0.1∼0.202	141∼244
Rahal (Rahal, 2006)	3.00	340	24.3∼42.2	440	1.60∼2.66	445	0.2	166∼264
Mawlood et al. (Mawlood et al., 2021)	3.85	130	91.5	390	0.38∼1.69	0∼485	0∼0.22	40∼186
Shah and Ahmad (Shah and Ahmad, 2009)	3.00∼5.50	254	52	414	0.33∼2	0∼276	0∼0.1	23∼127
Bentz and Buckley (Bazant and Kazemmi, 2005)	2.94	84∼333	34.3∼36.1	400	1.55∼1.63	0	0	14∼43
Kim et al. (Kim et al., 2013)	2.50	300∼600	31.8∼34.9	529∼651.2	1.9	0	0	60∼261
Arezoumandi et al. (Mahdi et al., 2015)	3.00	407	30∼37.2	414	1.24∼2.49	0	0	111∼173
Ignjatovic’ et al. (Islam et al., 1998)	4.20	235	33.3∼37	555∼580	4.09	0∼300	0∼0.19	91∼163
Pradhan et al. (Pradhan et al., 2018)	2.60	269	42.8	508∼606	0.75∼1.31	0∼352	0∼0.32	81∼162
Knaack and Kurama (Knaack and Kurama, 2015)	3.83	200	31.2∼46.4	572	1.34	0	0	31∼44
Roller and Russell (Roller and Russel, 1990)	2.50∼3.00	559∼762	72∼125.4	436.6∼489.5	1.61∼6.89	406.7∼457.8	0.07∼1.76	301∼2267
Cladera and Mari (Cladera and Mari, 2005)	3.01∼3.08	351∼359	49.9∼87	500	2.24∼2.99	0∼540	0∼0.11	99∼308
Sowik (Sowik, 2012)	2.50∼4.10	220	35	453∼545	1.93	0	0	40∼51
Ros et al. (Serna-Ros et al., 2002)	4.00	206	24.5∼32,6	549.75∼669.6	2.2	0∼669.6	0∼0.12	182∼360
Sherwood et al. (Sherwood et al., 2007)	2.89	280∼1400	28.1∼77.3	452∼494	0.83∼1.34	494∼496	0.10∼0.101	36∼298
Ghannoum (Ghannoum, 1998)	2.50	65∼889	34.2∼58.6	385∼477	1.15∼2	-	-	42∼386
Ahmad et al. (Ahmad et al., 1995)	3.00	198∼216	38.34∼84.83	421	0.96∼4.51	0∼421	0∼0.78	23∼120
Shin et al. (Shin et al., 1999)	2.50	215	52∼73	414	3.77	0∼388	0∼0.94	56∼183
Hong and Han (Hong and Han, 2005)	3.00∼6.00	142∼915	53.7	477	1.01∼4.68	0	0	39∼332
Placas and Regan (Placas and Regan, 1971)	3.36	272	24.8∼30.3	621	0.98∼1.46	0	0	45∼54
Tan et al. (Tan et al., 1997)	2.79∼3.14	400∼448	54.7∼74.7	498.9∼538	2∼5.8	353.2∼385	0.48	135∼265
Morrow and Viest (Morrow and Viest, 1957)	2.76∼7.86	356∼372	14.7∼45.7	483	0.58∼3.79	0	0	83∼182

References

ACI Committee 318: Building Code Requirements for Structural Concrete (ACI 318-19) Commentary on Building Code Requirements for Structural Concrete and Commentary (ACI 318R-19) (2019) Farmington Hills, USA.

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