# Assessment of the Durability of Threaded Joints
## 4 Shakedown of the Threaded Joints
### Joints with a Two-Sided Crack  
For the tightened threaded joint, the position of the nuts may contribute to the development of a symmetric two-sided crack in the cyclic bending plane, as shown in Figure 16. In this case, according to the assumption of ideal elasticity, the variable components of the stresses were calculated using the following equation:  
\[\sigma_{z}^{(e)}=\frac{M}{l_{x}}y,-r\cdot\cos\alpha_{0}\leq y\leq r\cdot\cos \alpha_{0} \tag{36}\]  
For the fictive yield surface, the stresses were calculated using the following equations:  
\[\sigma_{ij}^{0}=\left\{\begin{array}{c}\sigma_{y}-\frac{M}{l_{x}}y,0\leq y \leq r\cdot\cos\alpha_{0}\\ \sigma_{y}+\frac{M}{l_{x}}y,-r\cdot\cos\alpha_{0}\leq y\leq 0\end{array}\right. \tag{37}\]  
Figure 17: Probabilistic relationships of the parameters \(m_{d}\) (1) and \(n_{d}\) (2) and their \(95\%\) probabilistic intervals for crack-free threaded joints.  
Figure 18: Statistical assessment of threaded joints with a one-sided crack: \(\bullet\)—experimental data.  
The geometric indicators of the two-sided crack cross-section were calculated according to Figure 16. The progressive shape change condition:  
\[F\cdot\Delta u_{z0}=4\int\limits_{0}^{l_{0}}dz\int\limits_{0}^{r\cos\alpha}dy \int\limits_{0}^{\sqrt{r^{2}-y^{2}}}\left(\sigma_{y}-\frac{M}{I_{x}}\right) \frac{\Delta u_{z0}}{l_{0}}dx \tag{38}\]  
By integration Equation (38) and introduction of dimensionless load parameters \(m\), \(n\), the following was obtained:  
\[\frac{\pi}{\sin 2\alpha+\pi-2\alpha}n+\frac{16r^{4}}{9I_{x}}\left(\frac{1-\sin^{3} \alpha}{\pi+\sin 2\alpha-2\alpha}\right)m=1 \tag{39}\]  
The trigonometric functions of the angle \(\alpha\) were expressed by the depth of the crack \(h_{cr}\) and the radius \(r\). Crack depth \(2h_{cr}\) to cross-sectional diameter \(d=2r\) ratio \(k=h_{cr}/r\). Using various values of the ratio, progressive plastic strain conditions were obtained (Table 2).  
The safety factor was calculated using Equation (27), where the probability varied from 1% to 99%, changing from 1.084 to 1.685. The developed probabilistic dependencies on the coordinates \(P\)-\(n_{d}\) and \(P\)-\(m_{d}\) are presented in Figure 19.  
The hypothetical probabilistic curve obtained was used to determine the probabilistic values for the parameters \(n\) and \(m\), where \(P=1\%\), 10%, 30%, 50%, 70%, 90%, and 99%. The results obtained are depicted in the shakedown diagram (Figure 20).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Relative Crack Depth** & **Conditions of Progressive Plastic Fracture** \\ \(k=h_{cr}/r\) & **Conditions of Progressive Plastic Fracture** \\ \hline \(k=0\) & \(n+0.72m=1\) \\ \(k=0.05\) & \(1.013n+0.745m=1\) \\ \(k=0.10\) & \(1.039n+0.791m=1\) \\ \(k=0.15\) & \(1.074n+0.854m=1\) \\ \(k=0.20\) & \(1.116n+0.932m=1\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Conditions of progressive plastic strain.  
Figure 19: Probabilistic relationships of the parameters \(m_{d}\) (1) and \(n_{d}\) (2) and their 95% probabilistic intervals for threaded joints with a two-sided crack.  
The limit cycle parameters (limit changes of mechanical loads and temperature fields) have maximum values, in which case the shakedown conditions are satisfied in view of the task analyzed, and minimum values, in which case the non-shakedown conditions are satisfied. The reliability of the structure increases when this kind of comprehensive assessment is used.