## Case Studies on Tensile Specimens without Bolts
### Tensile Specim with Nocthes  
As mentioned in the introduction, the focus of the presented research is mainly on bolted steel plates in connections. Nevertheless, also a tensile specimen with notches, as shown in Figure 9a, was also investigated to check the behaviour of the DLO approach. The failure mechanism depends on the size and shape of the notch. Therefore, similar to a bolted plate, only circular notches were considered (mild notch). Figure 7: Collapse mechanisms (blue: Slip displacements, red: Normal displacements) from DLO for the tensile specimen with a hole (only the left side of the symmetry model is shown): (**a**) Nodes; (**b**) slip line mechanism; (**c**) refined node distribution; (**d**) refined slip line mechanism, (**e**) yield zone mechanism. Figure 9a shows the same specimen as used in Section 3.2. Instead of the hole, two circular notches at the outside of the specimen with diameter \(d=40\) mm were considered. A possible _lower bound_ solution is identical to Equation (11) and results in \(F_{\text{Tresca}}=F_{\text{Mises}}=170.4\) kN. The slip line mechanism in Figure 9b was chosen similar to Figure 3b. For simplicity, the work of the small yield zone was neglected again. Furthermore, it was assumed that first yielding always starts at the point with the highest stress concentration, which is the notch in the particular case. After some calculus, the load function for the _upper bound_ solution of this slip line mechanism can be found with:  
\[ \begin{array}{l}F\ u=2\ \tau_{y}\ t\ (L_{1}v_{1}+L_{2}v_{2})\\ L_{1}=\frac{b-d}{2\cos\alpha_{1}},L_{2}=\frac{b}{2\cos\alpha_{2}}\\ v_{1}\sin\alpha_{1}+v_{2}\sin\alpha_{2}=u\\ v_{1}\cos\alpha_{1}-v_{2}\cos\alpha_{2}=0\end{array} \mathrm{F}(\alpha_{1}, \alpha_{2})=\tau_{y}\ t\ \frac{b(2+\cos 2\alpha_{1}+\cos 2\alpha_{2})-d(1+\cos 2 \alpha_{2})}{2\sin(\alpha_{1}+\alpha_{2})\ \cos\alpha_{1}\ \cos\alpha_{2}} \tag{15}\]  
\(L_{1}\) and \(L_{2}\) are the lengths of the lower and upper slip line, respectively. For the particular case, the lowest loads were calculated numerically by varying each angle from \(0\) to \(\pi/2\), and were found to be  
Figure 8: Failure mechanisms (equivalent plastic strain) from a FEA for the tensile specimen with a hole: (**a**) Mesh; (**b**) yield surface according to the Tresca criterion was used; (**c**) yield surface according to the Mises criterion was used. Figure 9: Tensile specimen with notches: (**a**) Loading and boundary conditions; (**b**) slip line mechanism; (**c**) yield zone mechanism. \(\mathrm{F_{Tresca}}=220.0\) kN and \(\mathrm{F_{Mises}}=254.0\) kN. The corresponding angles are in both cases \(\alpha_{1}=52\) and \(\alpha_{2}=38\). The calculation of the yield zone mechanism in Figure 9c is identical to Equation (14) and the _upper bound_ solution results in \(\mathrm{F_{Tresca}}=\mathrm{F_{Mises}}=170.4\) kN, which are 29% lower than the load from the slip line mechanism with the Tresca criterion. Furthermore, the load is identical to the load from the _lower bound_ solution, which means that the ultimate load was found. With the node distribution from Figure 10a, the DLO procedure for slip line mechanisms only, results in the mechanism shown in Figure 10b, with \(\mathrm{F_{Tresca}}=220.6\) kN and \(\mathrm{F_{Mises}}=254.7\) kN. Using another node distribution, the mechanism in Figure 10c with \(\mathrm{F_{Tresca}}=221.9\) kN and \(\mathrm{F_{Mises}}=256.2\) kN was obtained. The small deviation to the calculated load from Equation (15) comes from the slightly different inclinations of the slip lines (\(\alpha_{1}=57\) and \(\alpha_{2}=39\)), due to the node distribution. The obtained slip line mechanisms are similar to the assumed mechanism from Figure 9b. Furthermore, the loads are quite the same. It can be seen that the node distribution, and especially the inclinations of the discontinuities, have a significant influence on the mechanism, but the resulting loads are very similar. Nevertheless, the DLO approach found a (slightly) better failure mechanism than it was assumed first by hand calculation.