## 2 Simplified Analytical-Based Method
### Case Study  
#### 2.2.1 Description  
Let us consider the roof of an existing RC building used for sporting activities. The FE model of the roof made using SAP2000 [37] software is shown in Figure 1.  
The roof plane is bounded by two straight parallel lines in one direction and by two circumference arcs in the transverse direction. Considering a reference system with the origin located at the center of the circumference to which, ideally, the curved edges belong to, the X axis is taken parallel to the straight sides while the Y axis is orthogonal to X. Along X and Y, the roof has dimensions of about 34 \(\times\) 18 m\({}^{2}\). The bearing structure of the roof is made by 24 trusses disposed radially with respect to the origin of the reference axes and equally spaced. The trusses are, in turn, composed of three structural elements--a top beam, an inclined bottom chord, and a vertical strut--which are connected to each other to form a triangular static scheme. The top beam has a slight 6% slope with respect to the horizontal plane. The top beam is made of a composite steel-concrete element (Figure 2), with a HEA200 profile connected on the top to a rectangular cross-section RC element of dimensions 60 \(\times\) 22 cm\({}^{2}\). The connection is made with steel U-shaped connectors welded to the top flange of steel profiles along its whole length. The inclined bottom chords and the vertical struts are, instead, made of circular hollow steel profiles with a diameter of 9 cm and a thickness of 6.5 mm. Along the perimeter of the building, the radial trusses are connected to the RC perimeter beams corresponding to the nodes between the top beam and the bottom chord (Figure 1), where one of these nodes is circled in red. In the center of the roof, the nodes between the bottom chord and the vertical strut of trusses are connected to each other with 24 steel profiles, constituting a polygonal truss lying in a horizontal plane. In Figure 1, one of these nodes is highlighted with a green circle. These profiles have the same cross-section of vertical struts and bottom chords. Conversely, the nodes between the top beam and the vertical strut, one of which is circled in black in Figure 1, are connected to each other by a circular-shaped RC beam (emergent beam in Figure 1), with rectangular cross-section dimensions of 40 \(\times\) 140 cm\({}^{2}\). In this beam, there are 16 square holes of side 50 cm. Moreover, the nodes between the top beams and the vertical struts are connected to the bottom nodes of the adjacent beams belonging to the polygonal truss by means of diagonal ties, composed of steel bars 20 mm in diameter, which are represented in red in Figure 3.  
Figure 1: 3D view of the FE model of the case study’s roof. Figure 2: Connection between the floor and the top beams of the radial trusses (dimensions are in cm). The effect exerted by RC perimeter beams of the building on the radial trusses is modelled through restraints that prevent translations along the axial direction of the perimeter beams, assuming these beams are inextensible. These restraints also prevent the translations along the vertical direction due to the high flexural stiffness of perimeter beams toward vertical loads. Conversely, translations orthogonal to the perimeter beams axis are allowed due to the finite flexural stiffness of the perimeter beams under bending actions in the horizontal plane. The roof is composed of 24 floor cells, each of which is bounded by the top beams of two adjacent radial trusses and by the portions of the perimeter beam and the circular-shaped RC beam between the considered radial trusses. The floors are made of pre-cast predalles type slabs with polystyrene lightening elements placed above, completed by casted-in-place RC joists with a slab on the top. Predalles slabs are 4 cm thick and 120 cm wide and are installed with their longitudinal axis orthogonal to the bisector of the angle between the longitudinal axes of the radial trusses bounding the floor cell, resting on the bottom flanges of HEA200 profiles.