# Probabilistic Models for the Shear Strength of RC Deep Beams
## Glossary  
\(a\) = shear span;  
\(a\)/\(d\) = the shear span-to-depth ratio;  
\(A_{\text{sv}}\) = the total area of vertical web reinforcement in the same section;  
\(A_{\text{sh}}\) = the total area of horizontal web reinforcement in the same section;  
\(b\) = the width of specimens;  
\(b_{\text{w}}\) = the width of web;  
\(d\) = the effective depth of beam;  
\(E[f(x)]\) = the expectation of arbitrary function;  
\(f(x)\) = the arbitrary function;  
\(f_{\text{c}}^{\prime}\) = the specified compressive strength of concrete;  
\(f_{\text{cd}}\) = the design value of cylinder compressive strength of concrete;  
\(f_{\text{ck}}\) = the compressive cylinder strength of concrete;  
\(f_{\text{cu}}\) = the cube compressive strength of concrete;  
\(f_{\text{t}}\) = the tensile strength of concrete;  
\(f_{\text{y}}\) = the yield strength of longitudinal reinforcement;  
\(f_{\text{s}h}\) = the yield stress for horizontal web reinforcement in beam;  
\(f_{\text{yo}}\) = the yield stress for vertical web reinforcement in beam;  
\(f(\mathbf{\Theta})\) = the posterior distribution;  
\(h\) = the depth of specimens;  
\(h_{0}\) = the effective depth of beam;\(h_{i}(x)=\) the explanatory functions;  
\(L(\theta,\sigma\mid Y,\mathbf{X})=\) the likelihood function;  
\(L(\mathbf{\Theta})=\) the likelihood function;  
\(l_{0}/h=\) the shear-depth ratio;  
\((l_{p})_{\rm E}=\) the width of the bottom loading plate;  
\((l_{p})_{\rm P}=\) the width of the top loading plate;  
\(S_{\rm h}=\) the spacing of horizontal web reinforcement;  
\(S_{\rm n}\)= the variance of arbitrary function;  
\(S_{\rm v}=\) the spacing of vertical web reinforcement;  
\(\rm M=\) iteration times  
\(m=\) the number of samples when the Markov chain reaches stationary distribution;  
\(n=\) the total number of samples generated by simulation;  
\(p=\) the number of explanatory functions;  
\(p(\mathbf{\Theta})=\) the joint probability density function of a prior distribution;  
\(\textit{VAR}[f(x)]=\) the variance of arbitrary function;  
\(V(\mathbf{X},\mathbf{\Theta})=\) a probabilistic model;  
\(V_{\rm d}=\) an existing deterministic model;  
\(V_{i}=\) calculation of probability model of \(i\)th test;  
\(V_{n}=\) the shear strength of deep beams with different calculation models;  
\(w_{s}=\) the width of a strut perpendicular to the axis of the strut;  
\(w_{t}=\) the effective height of the bottom nodal zone  
\(w_{t}{}^{\prime}=\) the effective height of the top nodal zone;  
\(\mathbf{X}\)= the vector of input parameters;  
\(\alpha_{\rm R}=\) FORM (First Order Reliability Method) sensitivity factor;  
\(\beta=\) the elected target reliability index;  
\(\beta_{\rm S}=\) the factor used to account for the effect of cracking and confining reinforcement on the effective compressive strength of the concrete in a strut;  
\(\gamma_{\rm Rd}=\) the partial factor;  
\(\gamma(\mathbf{X},\mathbf{\theta})=\) the correction term for the bias;  
\(\varepsilon=\) the normal random variable with the zero mean and unit variance;  
\(\varepsilon_{1}=\) the main tensile strain;  
\(\varepsilon_{8}=\) the longitudinal tensile strain in mid span of beams;  
\(\theta\)= the vector of uncertain model parameters;  
\(\theta_{(-)}=\) the vector of uncertain model parameters excluding eighth parameter;  
\(\theta_{(k)}^{(i)}\)= the \(i\)th iterative value of the \(k\)th parameter;  
\(\theta_{s}=\) the smallest angle between the strut and the adjoining ties;  
\(\tilde{\theta}=\) the mean value of each uncertain model parameter;  
\(\kappa=\) the normalizing factor;  
\(\lambda=\) the shear span-to-depth ratio;  
\(\mu_{i}=\) the mean value of the \(i\)th parameter;  
\(\pi(\theta,\sigma)=\) the prior distribution of parameters;  
\(\pi(\theta,\sigma\mid Y,\mathbf{X})=\) a posteriori distribution function;  
\(\rho=\) longitudinal reinforcement ratio;  
\(\rho_{\rm h}\)= the horizontal web reinforcement ratio;  
\(\rho_{\rm v}\)= the vertical web reinforcement ratio;  
\(\sigma=\) the unknown model parameter representing the magnitude of the model error;  
\(\sigma_{i}=\) the standard deviation of the \(i\)th parameter;  
\(\sigma^{2}=\) the variance;  
\(v=\) coefficient of the experiment-prediction ratios;  
\(\varphi(\cdot)=\) the joint probability density function (PDF) of the standard normal;  
\(\Phi(\cdot)=\) the cumulative distribution function (CDF) of the standard normal distribution;  
\(\mathbf{\Theta}=\) a set of model parameters introduced to fit the model to the test results.