# Assessment of the Durability of Threaded Joints
## 4 Shakedown of the Threaded Joints
### Crack-Free Joints
Consequently, the longitudinal force \(F\) is equilibrated to the stresses acting in zone \(B\). Bending moment \(M\) equilibrates with the stresses acting in zones \(B\) and \(C\). In the task analyzed \(\sigma_{ij*}^{0}=\sigma_{z}\) it is assumed that the failure mechanism is a uniform elongation of the stud:  
\[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV>0\]0  
Respectively:  
\[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV>0\]1  
\[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV>0\]2 Cross-section of the stud and stress distribution under \(M\) and \(F\) forces. Whereas:  
\[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV>0\]3  
Since bending moment \(M\) varies according to a symmetric cycle and \(\Delta e_{z0}^{\prime\prime}>0\), one can be obtained:  
\[\sigma_{ij*}^{0}=\left\{ \begin{array}{c}\sigma_{y}-\frac{M}{I_{x}}y,\frac{d}{ 2}\geq y\geq 0\\ \sigma_{y}+\frac{M}{I_{x}}y,0\geq y\geq-\frac{d}{2}\end{array} \right. \tag{18}\]  
Plastic energy dissipation:  
\[D=\int\limits_{V}\sigma_{ij*}^{0}\Delta e_{ij0}^{\prime\prime}dV=4\int\limits _{0}^{l_{0}}dz\int\limits_{0}^{d/2}dx\int\limits_{0}^{\sqrt{\frac{d}{2}-x^{2} }}(\sigma_{y}-\frac{M}{I_{x}}y)\frac{\Delta u_{z0}}{l_{0}}dy \tag{19}\]  
The progressive shape change condition can be written down as follows:  
\[F\cdot\Delta u_{z0}=4\Delta u_{z0}\bigg{[}\sigma_{y}\frac{\pi d^{2}}{16}- \frac{1}{2}\frac{M}{I_{x}}\frac{d^{3}}{12}\bigg{]} \tag{20}\]  
Dimensionless loading parameters \(n=F/F_{y}\) and \(m=M/M_{y}\) are introduced with the threshold parameter values:  
\[F_{y}=\sigma_{y}\frac{\pi d^{2}}{4}=\sigma_{y}\cdot\pi\cdot r^{2},M_{y}= \sigma_{y}\frac{d^{3}}{6}=\sigma_{y}\frac{(2r)^{3}}{6} \tag{21}\]  
After that, these are entered into Equation (20), which results in the following expression of the progressive shape change:  
\[n+\frac{64}{9\pi^{2}}m=1\text{ or }an+bm=1 \tag{22}\]  
Plastic failure is obtained by comparing the values of components of variable stresses with the yield stress:  
\[2\frac{M}{I_{x}}\frac{d}{2}=2\sigma_{y} \tag{23}\]  
After the dimensionless parameters are entered, the following is obtained:  
\[\frac{32}{6\pi}m=1 \tag{24}\]  
Plastic failure is unreal in threaded joints; hence, only the progressive shape change condition is analysed in the present work. The shakedown diagram of the threaded joints M52\(\times\)4 was designed according to dependencies (22) and (24). Figure 13 shows the shakedown diagram built according to [46] (p. 42) and the experimental data according to the established pattern, that is, after about 50 cycles, when the stress is constant in the joints. The coordinates analyzed (point \(D\)) are \(n_{d}\) and \(m_{d}\), if the threaded joint is subject to axial force \(F\) and bending moment \(M\). Limit point \(L\) coordinates \(m_{\text{lim. }}\), \(n_{\text{lim. }}\) The inclination angle of the radius for similar cycles is  
\[tg\beta=\frac{m_{\text{lim}}}{n_{\text{lim}}}\text{ or }tg\beta=\frac{m_{d}}{n_{d}} \tag{25}\]  
The safety factor of the progressive shape change is the ratio:  
\[\eta=\frac{m_{\text{lim}}}{n_{\text{lim}}}\text{ or }\eta=\frac{m_{d}}{n_{d}} \tag{26}\]Using Equation (22), it could be written down that:  
\[\eta=\frac{1}{an_{d}+bm_{d}} \tag{27}\]  
The methods used for low-cycle fatigue could be used for the statistical evaluation [47; 48] of progressive shape change. The workpieces used in the production of the threaded joints are made of the same alloy and use the same heat treatment. Indicators of the static mechanical properties (\(\sigma_{y}=930\div 1010\) MPa) were determined for each of them. These were used to select the tightening and cyclic loading conditions. The differences in mechanical properties suggest that the metal contained defects and that the metallurgical and heat treatment processes resulted in a heterogeneous structure.