# Assessment of the Durability of Threaded Joints
## 4 Shakedown of the Threaded Joints
### Joints with a One-Sided Crack  
The experimental investigation showed that crack initiation and propagation only had specific characteristics of only threaded joints in the case that the studs were deformed, as depicted in Figure 16. When the stud was subjected to the stress and bending moment variable according to the symmetrical cycle, the crack was one-sided in the case of a certain position of the nut.  
In this case, according to the assumption of ideal elasticity, the variable components of the stresses were calculated using the following equation:  
\[\sigma_{z}^{(e)}=\frac{M}{I_{x1}}(y+y_{c}),\,-r\cdot\cos\alpha\leq y\leq r\cdot \cos\alpha \tag{32}\]  
Figure 14: Probabilistic relationships of the parameters \(m_{d}\) (1) and \(n_{d}\) (2) and their \(95\%\) probabilistic intervals for crack-free threaded joints.  
Figure 15: Statistical assessment of crack-free threaded joints: \(\bullet\)—experimental data.  
For the fictive yield surface, the stresses were calculated using the following equations:  
\[\sigma_{ij}^{0}=\left\{\begin{array}{l}\sigma_{y}-\frac{M}{L_{1}}(y+y_{c}),0\leq y \leq r\cos\alpha\\ \sigma_{y}+\frac{M}{L_{1}}(y-y_{c}),-r\cos\alpha\leq y\leq 0\end{array}\right. \tag{33}\]  
The geometric indicators of the cross-section with a one-sided crack were calculated according to Figure 16. The progressive shape-change condition can be written down as follows:  
\[F\cdot\Delta u_{z0}=4\int\limits_{0}^{l_{0}}dz\int\limits_{0}^{r\cos\alpha}dy \cdot\int\limits_{0}^{\sqrt{r^{2}-y^{2}}}\left[\sigma_{y}-\frac{M}{L_{1}}(y+y_ {c})\right]\frac{\Delta u_{z0}}{l_{0}}dx \tag{34}\]  
By the integration of Equation (34) and the introduction of dimensionless load parameters \(m\), \(n\), the following was obtained:  
\[\frac{\pi}{\pi+\sin 2\alpha-2\alpha}n+\frac{16\sigma^{4}}{3l_{x1}}\left[\frac{ \sin^{3}\alpha}{6(\pi-\alpha+\sin\alpha\cos\alpha)}+\frac{1-\sin^{3}\alpha}{3 (\pi+\sin 2\alpha-2\alpha)}\right]m=1 \tag{35}\]  
The crack depth \(h_{cr}\) to the radius ratio of the cross-section \(k=h_{cr}/r\). By using those ratios according to Figure 16 and from Equation (35), the conditions of progressive plastic strain were obtained (Table 1).  
The safety factor was calculated using Equation (27), where the probability ranged from 1% to 99%, changing from 1.073 to 1.791. The developed probabilistic dependencies on the coordinates _P-\(n_{d}\)_ and _P-\(m_{d}\)_ are presented in Figure 17.  
\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 \(k=0\) & \(n+0.72m=1\) \\ \(k=0.05\) & \(1.014n+0.735m=1\) \\ \(k=0.10\) & \(1.04n+0.768m=1\) \\ \(k=0.15\) & \(1.074n+0.808m=1\) \\ \(k=0.20\) & \(1.117n+0.852m=1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Conditions of progressive plastic strain.  
Figure 16: Loading of the stud and crack propagation.  
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 18).