# Anchor Design of a Ring Joint Based on Reliability in a Precast Shear Wall Structure
## 5 Reliability-Based Anchoring Design
### Target Reliability Indicator  
To utilise the strength of the tensile reinforcement fully, a sufficient anchoring length is required to ensure that no bond failure occurs before the rebar yields. Therefore, the conditions at the anchor limit state are that the stress at the free end of the rebar reaches the maximum value (i.e., \(\sigma_{\rm s}=\eta f_{\rm y}\)), and the bond between the rebar and the concrete reaches the maximum value (i.e., \(\tau=\tau_{\rm u}\)). This is used as a starting point for determining the reliability of the anchoring performance, and the probability is given by Equation (13).  
\[p_{\rm fa}=\ P\Big{(}\sigma_{\rm s}\ \geq\ \eta f_{\rm y},\tau\ \geq\ \tau_{\rm u}\Big{)}=\ P\Big{(}\sigma_{\rm s}\ \geq\ \eta f_{\rm y}\Big{)}.P\Big{(}\tau\ \geq\ \tau_{\rm u}|\sigma_{\rm s}\ \geq\ \eta f_{\rm y}\Big{)} \tag{13}\]  
In GB 50068-2001, when structural members experience ductile failure and brittle failure under the second-level safety condition, the design reliability indicators are taken as \(\beta=3.2\) and \(\beta=3.7\), respectively. To ensure that the reliability index of the anchorage performance is higher than the bearing capacity reliability of other component sections, the total reliability index of the anchoring performance design and the corresponding failure probability calculation value are expressed as Equation (14).  
\[\beta_{\rm a}=3.95,\ p_{\rm fa}=4.0\times 10^{-5} \tag{14}\]  
For rebar, the force is designed based on the strength, so when the \"free end stress of the rebar reaches the maximum value (i.e., \(\sigma_{\rm s}=\eta f_{\rm y}\))\", the allowable probability and reliability indicators are based on the standard values, as expressed in Equation (15).  
\[\beta=3.2,\ p_{\rm f}=P\ (\sigma_{\rm s}=\eta f_{\rm y})=6.87\times 10^{-5} \tag{15}\]  
From Equation (13):  
\[p_{f0}=\ P\Big{(}\tau\ =\ \tau_{\rm u}|\sigma_{\rm s}=\ \eta f_{\rm y}\Big{)}= \ p_{\rm fa}/p_{f} \tag{16}\]  
From Equations (14)-(16), the allowable probability and reliability indicators are expressed in Equation (17).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Geometric** \\ **Size** \\ \end{tabular} & \begin{tabular}{c} **Symbol** \\ \end{tabular} & \begin{tabular}{c} **Anchor Length** \\ \(I_{\rm a}^{0}/I_{\rm a}^{\rm d}\) \\ \end{tabular} & \begin{tabular}{c} **Cover Layer** \\ \(c^{0}/\epsilon^{\rm d}\) \\ \end{tabular} & \begin{tabular}{c} **Rebar Diameter** \\ \(d^{0}/d^{\rm d}\) \\ \end{tabular} & \
\begin{tabular}{c} **Relative Position** \\ \(S_{\rm sw}^{0}/S_{\rm sw}^{\rm d}\) \\ \end{tabular} \\ \hline
\begin{tabular}{c} Average value \\ Coefficient of \\ variation \\ \end{tabular} & \(\delta\) & 0.077 & 0.30 & 0.018 & 0.06 \\ \hline \hline \end{tabular} Note: the superscript “0” indicates the measured value, and the superscript “d” indicates the design value.  
\end{table}
Table 6: Statistical parameters for the geometric dimensions of structural members.  
\[p_{\rm f0}=5.82\times 10^{-2},\,\beta_{\rm 01}=1.57 \tag{17}\]  
According to Schedule 2 in reference [28], the reliability index corresponding to \(P_{\rm f0}\) is \(\beta_{\rm 01}=1.57\), which is the basis for determining the design value of the anchor length in the anchoring strength problem [29].  
In summary, in the analysis of the anchoring problem, the reliability of the anchor design itself is considered to be \(\beta=3.20\), and the anchoring reliability is higher than the reliability of the section design. Relevant experiments [15] have shown that rebar with hooks and bends do not substantially fail to anchor until yielding, and thus additional reliability indices are not considered in this anchoring design.