# Probabilistic Models for the Shear Strength of RC Deep Beams
## 1 Introduction  
Reinforced concrete (RC) deep beams have been widely used for constructions such as pile caps, transfer girders, shear walls, and floor slabs [1]. In particular, the shear behavior of RC deep beams has emerged as an essential research topic [2]. Many models were developed by simplifying the rules of mechanics or regressing experimental data [3; 4; 5]. The most widely used method is the strut-and-tie model, which simplifies the complex stress state in discontinuity regions (D-regions) of concrete members into uniaxial and simple stress paths by incorporating a truss system comprising struts, ties, and nodes [6; 7]. Another standard method is the empirical model considering the multiple parameters and experimental data [8]. However, these models provide conservative strength predictions or unpredictable biases due to complex mechanical factors [9; 10].  
The Bayesian parameter estimation method has been extensively applied to design beams, columns, joints, shear walls and predict probabilistic fatigue strength [11; 12]. In contrast, the posterior parameter distribution of Bayesian estimation is mostly high-dimensional and complex, which is difficult to calculate directly [13]. The Markov Chain Monte Carlo (MCMC) method was commonly introduced to samples directly from the posterior distribution, thus solving the problem [14; 15; 16]. [PERSON] explored the Bayesian-MCMC methods for evaluating the posterior distributions of flood quantiles and flood risk, which provides complete uncertainty descriptions of parameters, qualities, and performance metrics [17]. [PERSON] applied the Bayesian method, improved by the MCMC method, to update the numerical model of a tunnel in soft soil conditional on settlement measurements [18]. This MCMC method, concerned with the simulation of high dimensional probabilistic distributions, has gained enormous prominence and revolutionized Bayesian statistics [19].  
The quantification and explicit inclusion of resistance model uncertainty are essential for reliability investigations of RC structures when the description of structural response isexcessively complex and the operational procedures depend on simplified models [20]. As an indicator of the difference between model predictions and actual structural resistances, the uncertainties in shear strength models for essential variables were investigated and described by the statistical characteristics, including the partial factor, mean, and coefficient of variation (CoV) of the experiment-to-prediction ratios [21,22].  
This paper introduced the Bayesian-MCMC method to establish the probabilistic shear strength models with an experimental database and multiple important parameters. The existing deterministic models, including ACI318-14 (2014) [23], GB50010-10 (2010) [24], CSA A23.3-04 (2004) [25], EC2 (2004) [26], CEB-FIP (2010) [27], [PERSON] and Gilbert (1998) [28], [PERSON] and [PERSON] (1974) [29], were compared and improved by corresponding bias correction terms. The probabilistic model without prior models was proposed to reduce bias and uncertainty further. Then, the performance of probabilistic models was investigated according to the evaluating indicators containing the mean, CoV, and partial factor.