PSI - Issue 44

Christian Salvatori et al. / Procedia Structural Integrity 44 (2023) 520–527 Christian Salvatori et al./ Structural Integrity Procedia 00 (2022) 000–000

524

5

where A j and i j are the cross-section area and the spacing of the floor joists or roof tie-beams; t p , A p , I p , and L p are the thickness, cross-section area, moment of inertia, and length of a plank; χ = 1.2 is the shear factor; E t = 10 GPa and G t = 0.75 GPa are the assumed Young’s and shear modulus of timber; and s n = 9 cm is the nail spacing. k ser = 678.7 kN/m has been calculated according to Eurocode 5 (CEN, 2004) for 3-mm-diameter nails without pre drilling, assuming a timber density of 415 kg/m 3 . Table 2 summarizes the membrane properties. The stiffness of each roof pitch was simulated by a pair of diagonal linear truss elements of cross-section area: = , 4 cos 2 ( ) (2) where L t (4.43 m and 5.00 m) and α t (49.4° and 51.2°) are the length of a truss element and its angle with respect to the shaking direction; E t = 10 MPa its Young’s modulus; and k db,H (3331 kN/m and 2441 kN/m) the lateral stiffness of the actual roof structures (Salvatori, 2020), calibrated against the experimental response through nonlinear dynamic analyses. This resulted in areas of 8.35 cm 2 and 7.71 cm 2 for the North and South units, respectively.

Table 2. Parameters for the floor diaphragm four-node orthotropic membranes. Diaphragm A j [cm 2 ] i j [cm] t p [cm] L p [cm] A p [cm 2 ] I p [cm 4 ] E 1 [MPa]

E 2 [MPa]

G 12 [MPa]

1 st and 2 nd floor 3 rd floor - North 3 rd floor - South

160 200 200

53 48 43

2 2 2

53 48 43

240 240 240

288 288 288

25063 10000 10000

10000 30921 33166

25.48 28.32 31.35

3.3. Advanced 3D model During the incremental shake-table test, the specimen exhibited the activation of an out-of-plane overturning mechanism of the façades orthogonal to the shaking direction. However, a common assumption with equivalent-frame modeling is not to account for this mode of response, assuming implicitly that it is inhibited by original or retrofit details. To better reproduce the complete experimental behavior, an unconventional strategy was employed. The transverse walls were subdivided in: (i) edge macroelements working in their planes, connected with the longitudinal façades at the intersections, to capture the so-called flange effect; and (ii) inner macroelements oriented perpendicular to the transverse walls, belonging to a fictitious wall P6 along the shaking direction, to model the vertical out-of-plane behavior (Fig. 3a and b). Horizontal truss elements completed the equivalent frame of fictitious wall P6. The longitudinal stiffness of the floor diaphragms was equally distributed between the 4-node membranes and these truss elements. Additional vertical and horizontal beam elements were assigned to the transverse walls, to make-up for the in-plane stiffness of the masonry portions attributed to the out-of-plane piers (Salvatori, 2020).

(a)

(b)

(c)

Fig. 3. Out-of-plane modeling: (a) plan view with piers of fictitious wall P6 in red; (b) fictitious frame for wall P6; (c) triangular membranes.