## 4 Case Studies on Tensile Specimens with a Single Bolt
### Bearing Capacity from Codes  
The given bearing resistances in codes are usually higher than the bearing capacities from FE-simulations carried out with a bilinear material behaviour. Especially for large edge distances, very high bearing resistances can be obtained from experiments. Furthermore, the given formulas in codes are often based on the tensile strength \(\mathrm{f_{u}}\) (including the partial safety factor \(\gamma_{\mathrm{M2}}=1.25\)) and not on the yield strength \(\mathrm{f_{y}}\). The modification factor \(\mathrm{k_{fy,c}}\) can be obtained by comparing the bearing resistance \(\mathrm{F_{b,Rd}}\) from the Eurocode EN 1993-1-8 with the DLO approach to consider the higher resistances. Within this approach, also the bolt-hole clearance can be considered now. Assuming a typical steel grade S355, the modification factor for the maximum bearing resistance results in:  
\[\mathrm{F_{b,Rd}}=\mathrm{k_{1}}\alpha_{\mathrm{b}}\frac{\mathrm{f_{u}}}{ \gamma_{\mathrm{M2}}}\mathrm{d}\ t\stackrel{{!}}{{=}}\mathrm{k_{ fy,c}}\mathrm{f_{y}}\mathrm{d_{0}}t\Rightarrow\mathrm{k_{fy,c}}=\mathrm{k_{1}} \alpha_{\mathrm{b}}\frac{\mathrm{f_{u}}}{\gamma_{\mathrm{M2}}\mathrm{f_{y}}} \frac{\mathrm{d}}{\mathrm{d_{0}}}=2.5\cdot 1.0\cdot\underbrace{\frac{490}{355 \cdot 1.25}\frac{\mathrm{d}}{\mathrm{d_{0}}}}_{\mathrm{0}}=2.75\frac{\mathrm{d }}{\mathrm{d_{0}}} \tag{16}\]  
\[\mathrm{with\ k_{1}}=\min\!\left(2.8\frac{\mathrm{e_{2}}}{\mathrm{d_{0}}}-1.7; 2.5\right)\!\!=2.5\ \mathrm{and}\ \alpha_{\mathrm{b}}=\min\!\left(\frac{\mathrm{e_{1}}}{\mathrm{3d_{0}}};\frac{ \mathrm{f_{u}}}{\mathrm{f_{u}}};1.0\right)=1.0\]  
According to the code, also the proof of the net cross-section under tension is necessary. This can be implemented in the DLO procedure with the same approach, which results in:  
\[\mathrm{F_{net,Rd}}=0.9\frac{\mathrm{A_{net}f_{u}}}{\gamma_{\mathrm{M2}}} \stackrel{{!}}{{=}}\mathrm{k_{fy,f}}\mathrm{f_{y}}\mathrm{A_{net}} \Rightarrow\mathrm{k_{fy,t}}=0.9\frac{\mathrm{f_{u}}}{\mathrm{f_{y}}\gamma_{ \mathrm{M2}}}=0.9\frac{490}{355\cdot 1.25}\simeq 1.0 \tag{17}\]  
This means that no additional modification factor \(\mathrm{k_{fy,t}}\) for tension forces is necessary for the DLO procedure. For comparison, also an approach from [PERSON](2018) was used. This approach is less conservative than the Eurocode and independent from the edge distance \(\mathrm{e_{2}}\), with:  
\[\mathrm{F_{b,Rd}}=\mathrm{k_{b}}\alpha_{\mathrm{d}}\frac{\mathrm{f_{u}}}{ \gamma_{\mathrm{M2}}}\mathrm{d}\ t\ \mathrm{with\ k_{b}}= \begin{array}{c}1.0\ \mathrm{for\ S235};\\ 0.9\ \mathrm{for\ S690};\end{array} \ \mathrm{and}\ \alpha_{\mathrm{d}}=\frac{ \mathrm{e_{1}}}{\mathrm{d_{0}}} \tag{18}\]  
\[\mathrm{F_{net,Rd}}=\frac{\mathrm{A_{net}f_{u}}}{\gamma_{\mathrm{M2}}} \tag{19}\]  
With the given dimensions, the modification factor results in \(\mathrm{k_{fy,c}}=2.75\cdot 20/22=2.50\) (including the bolt-hole tolerance, \(\mathrm{d_{0}}/\mathrm{d}=1.1\)). For comparison, also the ultimate loads for _fit bolts_ (\(\mathrm{d_{0}}/\mathrm{d}=1.0\)) with \(\mathrm{k_{fy,c}}=2.75\cdot 22/22=2.75\) were calculated. With these assumptions, the ultimate loads of the four cases from Section 4.2 were calculated and compared. The results are summarized in Table 4 and the failure mechanisms are shown in Figure 18.  
With the DLO approach, now the bearing capacity for case 1 is significantly higher but still lower than the value from the code or literature. The reason can be seen from Figure 18a. The failure mechanism shows no longer a pure bearing mechanism. The resistance of the yield zone in compression is now much higher than the resistance of the obtained slip line mechanism. This means that it is not possible to reach the bearing capacity from the code with the DLO approach. The behaviour of the DLO procedure seems to be correct. In experiments usually, huge deformations occur, while the load is still increasing slowly. Therefore, the given resistances in codes are often based on a deformation criterion.