## Case Studies on Tensile Specimens without Bolts
### Tensile Specim with Nocthes
If yield zone mechanisms were allowed too, _upper bound_ values of \(\mathrm{F_{Tresca}}=\mathrm{F_{Mises}}=170.4\) kN were obtained, see Figure 10d. This value equals the _lower bound_ solution, and so the ultimate limit load was found. It can be seen that neglecting yield zone mechanisms would lead, in some cases, to mechanisms with much higher resistances. Figure 11 shows the results from a finite element analysis (FEA) of the same example and a vertical displacement of \(\mathrm{u}=0.5\) mm. The Tresca criterion results in an ultimate load of \(\mathrm{F_{Tresca}}=171.8\) kN and the Mises criterion in \(\mathrm{F_{Mises}}=191.7\) kN. A large yield zone between the notches was formed in both cases. If the results from the Mises criterion only were compared, it can be seen that the ultimate load from the FEA is larger than the ultimate load from the _upper bound_ solution, which is contradictory. The reason is that the yield strength was considered with \(\mathrm{f_{y}}\) in the DLO procedure. Due to the real stress state, significant stresses normal to the load direction arise between the notches. Therefore, higher stresses (above \(\mathrm{f_{y}}=355\) MPa) in load direction are necessary that the material starts yielding. The maximum possible value, according to the Mises criterion, is \(2/\sqrt{3}\,\mathrm{f_{y}}\simeq 1.155\,\mathrm{f_{y}}=410\) MPa. In the particular case, the critical stress that the material starts yielding can be recalculated by comparing  
Figure 10: Collapse mechanisms (blue: Slip displacements, red: Normal displacements) from DLO for the tensile specimen with notches: (**a**) Nodes; (**b**) asymmetrical slip line mechanism; (**c**) symmetrical slip line mechanism; (**d**) yield zone mechanism. the results from DLO and FEM. The critical stress corresponds to \(191.7/170.4=1.125\)\(\mathrm{f_{y}}=399\) MPa. This value was also reproduced by comparing the individual stress components from the FEA. In this particular case, the notches have a positive influence on the Mises criterion (and so on the ultimate load). From a practical point of view, if only the yield strength is assumed, the obtained resistance is conservative. For comparison, the same tensile specimen was now investigated with a relative small distance between the notches (deep notches). Therefore, the distance between the notches was simply reduced to a neck width of 20 mm as shown in Figure 12.  
A _lower bound_ solution can easily be found by assuming a full plastic net section, which results in \(\mathrm{F_{Tresca}}=\mathrm{F_{Mises}}=56.8\) kN. The _upper bound_ solutions were calculated with the DLO approach. Assuming slip line mechanisms only, the mechanism in Figure 12a was obtained, which is quite different from the mechanisms in Figure 10b,c. Due to the coarse node distribution, a sub-model of the neck was created with a much  
Figure 11: Failure mechanisms (equivalent plastic strain) from a FEA for the tensile specimen with notches: (**a**) Mesh; (**b**) yield surface according to the Tresca criterion was used; (**c**) yield surface according to the Mises criterion was used. Figure 12: Collapse mechanisms (blue: Slip displacements, red: Normal displacements) from DLO for the tensile specimen with deep notches: (**a**) Slip line mechanism; (**b**) sub-model with refined slip line mechanism; (**c**) yield zone mechanism. finer node distribution. The refined slip line mechanism is shown in Figure 12b, with corresponding loads of \(\mathrm{F_{Tresca}}=69.4\) kN and \(\mathrm{F_{Mises}}=81.1\) kN. Allowing also yield zone mechanisms, the DLO approach results in the mechanism shown in Figure 12c. A refined analysis with a sub-model is not necessary for this case. The load results in \(\mathrm{F_{Tresca}}=\mathrm{F_{Mises}}=56.8\) kN, which equals the _lower bound_ solution. By concerning the examples from the previous sections, it can be drawn that slip line and yield zone mechanisms should be considered both in the DLO formulation. The so obtained loads correspond in all cases to the _lower bound_ solution.