## 3 Optimal Design Formulation  
The present section is devoted to the formulation of a new improved version of the optimal design problem for the proposed device. The essential novelty of the formulation regards a reduced and more compact form of the design variable vector and the constraint on the desired Class of the cross section of the inner portion of the device. As previously reported, the LRPD is constituted by an inner portion of length \(\ell_{i}\), with geometrical I-shaped cross section features reported in Section 2, and by two symmetrically placed outer portions, both of length \(\ell_{o}\) and equal I- shaped cross section with geometrical features as described in Section 2. The device volume is chosen as the objective function to be minimized. It is worth noting that, to ensure the occurrence of appropriate plastic deformation fields in the inner portion, the length of the inner part \(\ell_{i}\) must satisfy a suitable lower bound [28]. In the referenced paper, the length \(\ell_{i}\) is evaluated as a linear function of the height of the original beam cross section introducing an appropriate scalar factor \(\beta\): \(\ell_{i}=\beta l_{p}\) where \(0.5\leq\beta\leq 1\) is suggested. In Figure 9 all the geometrical characteristics of the device are indicated and, in particular, the assumed design variables are highlighted in red. In the referenced Figure, as previously stated, \(h_{o}=h_{p}\), \(b_{o}=b_{p}\) and \(t_{w,o}=t_{w,j}=t_{w,p}\) have been assumed. Finally, the distance between the medium planes of the flanges \(h^{*}\) is considered as a relevant variable:  
\[h^{*}=h_{o}-t_{f,o}=h_{i}-t_{f,i} \tag{16}\]  
As a consequence, the design variable vector is the following one:  
\[d^{T}=\left|h^{*}\;\;b_{i}\;\;t_{f,i}\;\;t_{f,o}\;\;\ell_{o}\right| \tag{17}\]  
The objective function to be minimized is the following:  
\[f(d)=A_{i}\ell_{i}+2A_{o}\ell_{o}=\left[2b_{i}t_{f,i}+t_{w,p}\left(h^{*}-t_{f,i }\right)\right]\ell_{i}+2\Big{[}2b_{p}t_{f,o}+t_{w,p}\left(h^{*}-t_{f,o}\right) \Big{]}\ell_{o} \tag{18}\]  
\[\qquad\qquad\qquad=\Big{[}2\;b_{i}\;t_{f,i}+t_{w,p}\left(h^{*}-t_{f,i}\right) \Big{]}\beta h_{p}+\Big{[}4b_{p}t_{f,o}+2t_{w,p}\left(h^{*}-t_{f,o}\right) \Big{]}\ell_{o}\]  
Figure 9: Device geometry and design variables: (**a**) cross sections of the inner and outer portions; (**b**) lateral view. The design variables must satisfy appropriately selected bounds, reported in the following relations:  
\[0\leq h^{*}\leq h_{p}-t_{f,p} \tag{19}\]  
\[3t_{w,p}\leq b_{i}\leq b_{p} \tag{20}\]  
\[0\leq t_{f,i}\leq t_{f,p} \tag{21}\]  
\[t_{f,p}\leq t_{f,\rho}\leq\sideset{}{{}^{h_{p}}}{\left/{\vphantom{{h_{p}}}2} \right._{2}} \tag{22}\]  
\[0\leq\ell_{o}\leq\infty \tag{23}\]  
where \(t_{f,p}\), as already defined in Section 2, is the flange thickness of the original beam element cross section. Furthermore, the device geometry implicates:  
\[d^{T}=\left|h^{*}\;\;b_{i}\;\;t_{f,i}\;\;t_{f,o}\;\;\ell_{o}\right| \tag{17}\]0  
If the I-shaped cross section of the inner portion is imposed to belong to the relevant Class 1, the following relation must be satisfied (see [21; 24]). \[d^{T}=\left|h^{*}\;\;b_{i}\;\;t_{f,i}\;\;t_{f,o}\;\;\ell_{o}\right| \tag{17}\]1  
In addition, the device is designed to be in a limit resistance condition for assigned couples of values of the axial force \(N_{a}\) and of the bending moment \(M_{a}\). Therefore, the following mechanical condition must be respected:  
\[d^{T}=\left|h^{*}\;\;b_{i}\;\;t_{f,i}\;\;t_{f,o}\;\;\ell_{o}\right| \tag{17}\]2  
if the neutral axis crosses through the web cross section, or  
\[d^{T}=\left|h^{*}\;\;b_{i}\;\;t_{f,i}\;\;t_{f,o}\;\;\ell_{o}\right| \tag{17}\]3  
if the neutral axis crosses through one of the flanges of the cross section. Finally, the equivalent bending stiffness of the device with total length \(\ell=\ell_{i}+2\ell_{o}\) must be not less than the one characterizing a portion of the original beam element with the same length.