## 2 Geometrical and Mechanical Characteristics of LRPD  
The device studied in the present paper is an evolution of that reported in other foregoing papers (see, e.g., [25; 26; 27; 28; 29; 30; 31]) and consists of a steel element with suitably assigned features aimed to substitute a portion of a given standard I-shaped steel profile. In Figure 1, the typical standard steel profile is sketched and its geometry is reported in Table 1.  
In Figure 2, a scheme of the device is reported. As it is possible to observe, the overall device is assumed to be inscribed in a parallelepiped of dimensions \(\ell\times b_{p}\times h_{p}\). The device is made up by three different parts: two outers (equal each other) and one inner. From a geometrical point of view, the device possesses the following geometrical features: (i) it is characterized by symmetry with respect to three orthogonal barycentric planes; (ii) the thickness of flanges of the two outer portions is greater than that of the inner portion; (iii) the flanges of all portions possess a unique common medium plane; (iv) the thickness of web is the same in all the portions. It is important to emphasize that the actual version is an evolution of previous models (see, e.g., [31]) and, in particular, the main novelties are: (a) the flange width \(b_{i}\) of the inner part is assumed as different with respect to that of the outer portions; (b) the thickness of web is the same in all the portions. For a complete geometrical representation of the device, in Figure 3 the sketch of both the outer and inner portions, as well as a lateral view of the device, are reported while in Table 2 the adopted symbols are described. \begin{table}
\begin{tabular}{c c} \hline
**Adopted Symbol** & **Description** \\ \hline \(b_{p}\) & flange width \\ \(h_{p}\) & total height \\ \(t_{w,p}\) & web thickness \\ \(t_{f,p}\) & flange thickness \\ \hline \end{tabular}
\end{table} 
Table 1: Geometrical characteristics of the standard steel profile. Figure 1: Typical standard I-shaped steel profile. It is worth noting that the imposed overall geometry for the device implies that: \(b_{o}=b_{p}\), \(h_{o}=h_{p}\) (see Figure 2). In addition, as a further novelty, in the present new device the web thickness is constant for all the portions and equal to one of the original standard steel I-shaped profiles: i.e., \(t_{w,o}=t_{w,i}=t_{w,p}\). From a mechanical point of view, the connection with the structural elements is generally thought of as a bolted plate and back-plate system, hereafter assumed as a perfect rigid joint, as reported in Figure 4, where bolts are not shown for the sake of simplicity. The analysis of the mechanical behavior of this connection, together with its influence of the overall behavior of the device, is not faced in the present paper and it will be treated in future developments. Figure 3: Device geometry: (**a**) cross sections of the inner and outer portions; (**b**) lateral view. Figure 2: Device scheme. Among the geometrical characteristics of the device reported above, the welding size \(s\) has been introduced (Figure 3). In fact, from a technological point of view, the device is thought of as being obtained by the welding of steel plates with suitable thicknesses (for further details see [31]). For the developments of the paper, it is necessary to define the cross-section area \(A\), the moment of inertia \(I\), the elastic resistance modulus \(W_{el}\) and the plastic resistance modulus \(W_{pl}\), respectively, for each portion of the device. These quantities can be easily derived from basic geometry definitions and are reported in detail in [30; 31]. Plane frames analysis will be adopted in this paper; therefore, the constitutive elements are generally subjected to an axial force, shear force and bending moment. By neglecting the influence of the shear force, as for flexural systems, the limit elastic and plastic behavior of the device will be described, making reference just to the axial force and bending moment.