## 5 Proposal and Evaluation of New Shear Capacity Formula  
### New Shear Capacity Formula  
According to the above research, the ultimate shear capacity \(\mathbf{V_{u}}\) of the keyed dry joints is not affected by friction force on the joint interface, or by the dimension parameters of the key. The shear resistance of the concrete key root is the only contributor to \(\mathbf{V_{u}}\). Namely, \(\mathbf{V_{u}}=\tau\mathbf{A_{k}}\), \(\tau\) is the average shear stress, \(\mathbf{A_{k}}\) is the sectional area of the key root. Distribution of the maximum principal tensile stress on the key root section of the simulation example in Figure 1 is shown in Figure 16. Even under the ultimate state, the principal stress on the whole key root section did not reached concrete tensile strength everywhere. Therefore, the principal tensile stress on the key root section could be obtain by \(\zeta\mathbf{f_{f}}\). Here, \(\mathbf{f_{i}}\) is the concrete tensile strength, \(\zeta\) is the reduction coefficient. Suppose the shear stress distribute evenly along key root width, then the stress status of any point in the key root could be illustrated as the Mohr stress circle in Figure 17. In which, \(\tau\) is the vertical shear stress on the key root section, \(\sigma_{n}\) is the lateral pre-compression stress applied on the joint, \(\mathbf{\alpha}\) is the angle between the principal tensile stress and the horizontal plane. According to the geometric relationship illustrated in Figure 17  
\[\frac{\tau}{sin2\alpha}=\frac{\tau}{tan2\alpha}+\sigma_{n}+\zeta f_{t} \tag{11}\]  
then  
\[\tau=cot\alpha(\sigma_{n}+\zeta f_{t}) \tag{12}\]  
Figure 16: Distribution of the maximum principal tensile stress on the key root section. Figure 17: Mohr stress circle. \[cot\alpha=tan\varphi=\frac{\tau}{\zeta f_{t}-\sigma_{n}} \tag{13}\]  
According to the maximum principal stress strength criterion, there is:  
\[\sigma_{1}=\frac{\sigma_{n}+\sigma_{y}}{2}+\sqrt{\frac{\left(\sigma_{n}-\sigma_{ y}\right)^{2}}{4}+\tau^{2}}=\zeta f_{t} \tag{14}\]  
where, \(\sigma_{y}=\frac{V}{BH}\cdot\frac{H}{W}=\eta\tau\). \(\eta=\frac{H}{W}\) is the aspect ratio of the shear compression zone, and \(W\) is the width of the vertical compression zone. Thus, Equation (13) can be converted into the following  
\[\frac{\sigma_{n}+\eta\tau}{2}+\sqrt{\frac{\left(\sigma_{n}-\eta\tau\right)^{2 }}{4}+\tau^{2}}=\zeta f_{t} \tag{15}\]  
then  
\[\tau=\frac{1}{2}\bigg{[}-\eta(\zeta f_{t}-\sigma_{n})+\sqrt{\eta^{2}{(\zeta f _{t}-\sigma_{n})}^{2}+4\zeta f_{t}{(\zeta f_{t}-\sigma_{n})}}\bigg{]} \tag{16}\]  
substitute Equation (13) into Equation (16), there is  
Figure 17.0  
let \(t=\sigma_{n}/\zeta f_{t}\), \(cot\alpha\) can be regarded as the function of \(t\), there is  
\[f(t)=\frac{1}  
In Table 6, data in the brackets are the ratios of shear capacity calculated according to each formula to corresponding test results, \(R_{Av}\) is the average value of the ratios in each column. Obviously, \(R_{Av}\) of the proposed formula Equation (20) is 0.998, which is the closest to 1.0 compared with other formulas. It indicates the calculated results obtained by Equation (20) are highly consistent with the existing experimental results. In the second place, calculated results obtained according to AASHTO also showed decent consistency with test results. Poor applicability of the other formulas is demonstrated with over 20% difference between calculated results and test results. Through regression analysis, the relation between experimental and predicted joint shear strength of different formulas is presented in Figure 18, values of \(R^{2}\) are also listed in Table 6, which reflects the prediction accuracy of the formulas. Similarly, \(R^{2}\) of Equation (20) proposed in this paper is the 0.982, which showed higher accuracy in predicting the shear capacity of single keyed dry joint than other existing formulas.