## 2 Geometrical and Mechanical Characteristics of LRPD
Respecting the contents of the most recent international structural standards (see, e.g., [21; 22; 23; 24]), referring as previously indicated just to I-shaped Class 1 cross-sections, \begin{table}
\begin{tabular}{c c} \hline \hline
**Adopted Symbol** & **Description** \\ \hline \(b_{o}\) & Outer portions \\ \(h_{o}\) & \begin{tabular}{c} flange width \\ total height \\ web thickness \\ flange thickness \\ length \\ \end{tabular} \\ \(\ell_{o}\) & Inner portion \\ \(b_{i}\) & \begin{tabular}{c} flange width \\ total height \\ web thickness \\ length \\ \end{tabular} \\ \(t_{i}\) & \
\begin{tabular}{c} flange thickness \\ length \\ \end{tabular} \\ \(\ell_{i}\) & Overall device \\ \(\ell\) &
\begin{tabular}{c} total length \\ welding size \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical characteristics of the device. Figure 4: 3D view of the device and of the connection with the beam. the boundary of the elastic domain (see Figure 5) is simply described by the following dimensionless conditions:  
\[\left|\frac{N}{N_{el}}+\frac{M}{M_{el}}\right|=1,\quad\left|\frac{N}{N_{el}}- \frac{M}{M_{el}}\right|=1 \tag{1}\]  
where \(N_{el}=A\ f_{y}\) is the elastic limit value of the axial force (coincident with the plastic limit one) and \(M_{el}=W_{el}\ f_{y}\) the elastic limit value of the bending moment, with \(f_{y}\) the material yield stress. The boundary of the plastic domain, always related to I-shaped Class 1 cross-sections is reported in Figure 6 and, always according with the referenced international structural standards, is described by the following dimensionless equations:  
\[\left|\frac{N}{N_{pl}}+(1-0.5a)\frac{M}{M_{pl}}\right|=1,\quad\left|\frac{N}{N _{pl}}-(1-0.5a)\frac{M}{M_{pl}}\right|=1 \tag{2}\]  
\[\left|\frac{M}{M_{pl}}\right|=1 \tag{3}\]  
where \(N_{pl}=A\ f_{y}\) is the plastic limit value of the axial force, \(M_{pl}=W_{pl}\ f_{y}\) is the plastic limit value of the bending moment and where the parameter \(a\) is defined as follows:  
\[a=\left(A-2b_{p}t_{f,p}\right)/A\leq 0.5 \tag{4}\]  
Figure 5: Dimensionless elastic domain. Figure 6: Dimensionless yield domain: the value of \(n\) depends on the cross-section profile (\(n=0.5a\) ). It is worth noting that the size of the plateau of the domain in Figure 6 is \(2n\), and, utilizing Equations (2) and (3), it can be deduced. \[n=\frac{N}{N_{pl}}=0.5a \tag{5}\]  
Due to the imposed constraint on parameter \(a\), it always results \(n=N/N_{pl}\leq 0.25\), according to the referenced standards. The dimensionless yield domain sketched in Figure 6, and also reported in the international standards, represents a good reference from a practical point of view even if the outline of such a domain is linearized and approximated. The real yield domain boundary is nonlinear and it can be drawn, as described in the following. As it can be observed by referring to Figure 7, where the typical yield domain of an I-shaped cross section is sketched in the \(N,M\) plane, its boundary is symmetric with respect to the coordinate axes and, as a consequence, for its analytical determination reference can be made just to the first quarter of the plane. For each position of the neutral axis \(-\frac{h_{p}}{Z}\leq z_{n}\leq\frac{h_{p}}{Z}\) (see Figure 8), the entire cross section is assumed to be plasticized and the related values for the yield axial force and yield bending moment read:  
Figure 8: Standard profile geometry for yield domain: (**a**) neutral axis crossing the web; (**b**) neutral axis crossing the connection between web and flange; (**c**) neutral axis crossing the flange. Figure 4:0 Dimensionless yield domain for beam with H cross-section profile. for \(0\leq z_{N}\leq\frac{h_{p}}{2}-t_{f,p}-r\) (Figure 8a)  
\[ \begin{array}{c}N=2\ t_{w,p}\ z_{n}\ f_{y};\\ M=M_{pl}-t_{w,p}\ z_{n}^{2}\ f_{y}\end{array}  \tag{6}\]  
for \(\frac{h_{p}}{2}-t_{f,p}-r\ \leq z_{n}\leq\ \frac{h_{p}}{2}-t_{f,p}\) (Figure 8b)  
\[