# Assessment of the Durability of Threaded Joints
## 3 Experimental Results and Their Analysis
### Comparison of the Experimental Results to Benchmark Curves  
In the normative calculations for the nuclear energy installations, the low-cycle durability calculations were used to establish the actual margin and compare it with the regulated or desired value. According to PNAE norms, the mean stress in the cross-section caused by the internal pressure, tightening, and thermal forces shall not exceed 0.65\(\sigma_{y}\), or 0.85\(\sigma_{y}\) in view of bending. Stress caused by internal pressure may exceed the analog stresses permissible in ASME norms. The mean stresses caused by all loads are virtually the same, while PNAE establishes strict limitations on axial and bending stresses compared to ASME norms. The benchmark fatigue curves were obtained according to the PNAE norms where the strain concentration factor is equal to 4 (margin \(n_{\sigma}=1.5\); \(n_{N}=3\)) for the metric thread in the tensile area, and, for the elastic-plastic area, it could be calculated according to [PERSON]'s rule [34] (margin \(n_{\sigma}=1.5\); \(n_{N}=5\)), reaching 5.5. According to the PNAE norms, the admissible number of cycles \(N_{adm}\) is determined at a certain amplitude of the stress cycle, and an admissible stress amplitude \(\sigma_{a.n.tm}\) is determined by providing a number of cycles. This is implemented using two methods: according to the calculated fatigue curves provided in the norms and according to the formulas, if the number of cycles does not exceed \(10^{6}\). In the PNAE normative methodology, low-cycle durability is determined according to the first stage of failure, i.e., crack initiation in the stress concentration areas. The crack propagation stage was not considered due to the increased safety requirements. The calculation was based on binomial-type formulas [35; 36; 37; 38]; the terms whereof expressed the plastic and elastic components of the strain amplitude:  
\[ \begin{array}{l}\sigma_{a\;radm}^{*}=\frac{E.c_{c}}{n_{\sigma}(A+N_{adm})^{ m}}+\frac{\sigma_{-1}}{n_{\sigma}(1+\frac{\sigma_{-1}}{\sigma_{0}},\frac{1+ \sigma_{0}}{1-\sigma_{0}})}\\ \sigma_{a\;radm}^{*}=\frac{E.c_{c}}{n_{\sigma}(4\cdot n_{N}\cdot N_{adm})^{m}} +\frac{\sigma_{-1}}{1+\frac{\sigma_{-1}}{\sigma_{0}},\frac{1+\sigma_{0}}{1- \sigma_{0}}}\end{array}  \tag{1}\]  
The smaller value of the number of load cycles to crack initiation in the stud \(N_{0}\) was determined on the basis of the two values calculated using these formulas and was considered to be a calculation result. According to the ASME code, in the case of the high-strength study (700\(\div\)1150 MPa), the low-cycle durability is determined according to the calculated durability curves that are verified experimentally. For the threaded joints, the curves were calculated using the formula [39; 40; 41]:  
\[\sigma_{a\;radm}^{*}=\frac{A_{e}\times E}{N_{f}\times C}+S_{e} \tag{2}\]  
Figure 6 presents a comparison of the experimental results for the joints bolted using crack initiation with the fatigue curves calculated according to the norms. It could be seen that the experimental values of cyclic durability were considerably higher (2- to 10-fold) than the values reflected by the norm curves. Individual stud and bolt failure stages are not identified in ASME standards, and the safety factors are fairly large. In the case of calculation using the nominal stresses caused by internal pressure, the fatigue safety factors according to the PNAE norms are not as conservative as in the case of ASME norms. However, a direct comparison of fatigue curves, where the number of cycles does not exceed \(10^{8}\), including an increase in concentration factors, shows that the permissible amplitude and number of cycles are subject to stricter regulation in the PNAE norms than in the ASME norms. The total tensile and bending stresses in the outlying layers of the bolt may sometimes exceed the yield strength. The resulting nonelastic strain conditions along the entire length of the bolt may lead to changes in the operating conditions of the joint.