## 4 Application  
In this section the optimal design for the LRPD described in the foregoing section is applied to the cases of two I-shaped cross section standard steel (S235) profiles. The selected profiles are the IPE400 and HEA300 which represent, in the authors' opinion, two meaningful examples of a typical beam element that is adopted in building constructions. The first step in applying the proposed optimal design is to select the values of the ultimate axial force and bending moment for the device. In order to do this, the elastic and plastic domains for both the selected profiles have been obtained and reported in Figure 10 by utilizing Equations (1) and (6)-(13). On the basis of the domains reported in Figure 10, for both the selected commercial profiles two different loading conditions have been selected to perform the optimal design: the first one is characterized by a high bending moment and a small axial force; in the second one the situation is reverted with a high axial force and a relatively small bending moment. Clearly, both situations are such that the mechanical behavior of the standard profile is always elastic. These situations have been referred to as DP1 and DP2 in the following and are reported in Figure 11 together with the elastic and yield domains already reported in the previous figure. The optimal designs of the devices related to the chosen points (DP1 and DP2) are determined by solving problem (30)-(35). The input values for each problem are reported in Table 3 and in Table 4 the obtained results are reported. Figure 11: Dimensionless elastic (red line) and yield (black line) domains with selected design points (DP1 orange point; DP2 green point): (**a**) IPE400; (**b**) HEA300. Figure 10: Dimensionless elastic (red line) and yield (black line) domains: (**a**) IPE400; (**b**) HEA300. As is easy to recognize by an examination of Table 3, the selected value of \(\beta=\ell_{i}/h_{p}\) is different for IPE and HEA profiles. Specifically, it has been assumed as the lower limit of the suggested interval \(\beta=0.5\) (see [28]) for the IPE profile and the upper one \(\beta=1\) for the HEA profile. This choice is due to the remark that the web flexural contribution for IPE profile is certainly greater than that for HEA profile. Consequently, for the beam with the IPE cross section the plastic deformations easily spread along the web. To check the reliability of the proposed optimal design, the elastic and yield domains of the inner part of the device have been determined by utilizing Equations (14) and (15). These domains are reported in Figure 12 for the case of the IPE400 profile and in Figure 13 for the case of the HEA300 profile, respectively. An examination of these Figures confirms that the selected design points belong to the yield domain of the inner part of the device designed for that assigned point. The next step consists of checking the mechanical behavior of the designed devices by means of a suitable FEM analysis performed in an ABAQUS environment. In Figure 14 the model for the LRPD designed for the IPE400 profile and design point 1 is sketched (models for the other LRPDs are similar). The model is constituted by four different parts, sketched in different colors in Figure 14. The central part (sketched in dark green) is the LRPD whose geometrical characteristics have been obtained by the optimal design. The first and third parts (sketched in dark white and dark red, respectively) are equal to each other and the related geometrical characteristics are equal to those of the outer parts of LRPD. The length of these parts is equal to that of the LRPD so that the overall length of the model, without the loading plate, is equal to \(3\ell\). The last part (sketched in dark blue) is a plate where the loadings are applied with a thickness of \(t=30\) mm. Each part has been tied to the next one to ensure a perfect kinematic compatibility. The overall model is that of a cantilever beam so that the white part is clamped at its end section while the dark blue part is free.