PSI - Issue 44

Paola Sorrentino et al. / Procedia Structural Integrity 44 (2023) 1308–1315 Paola Sorrentino et al. / Structural Integrity Procedia 00 (2022) 000–000

1313

6

through M-N plastic domain of columns reduced by 30% and the M-N plastic domain of beams. In Figure 3 the comparisons for the columns and beams of generic frame are shown. In particular, Figure 3a and Figure 3b show that all points representing the stresses induced by the 32 combinations in columns are inside the M-N plastic domain in y direction and are near the boundary of the domain in x direction. In Figure 3c and Figure 3d, only few points representing the stresses in beams in the end sections are outside the domains, i.e. the positive stress values in the end sections of the beams.

100 200 300 400 500

100 200 300 400 500

M [kNm]

M [kNm]

N [kN]

N [kN]

-500 -400 -300 -200 -100 0

-500 -400 -300 -200 -100 0

-1000 -500 0 500 1000 1500 2000 2500 3000

-1000 -500 0 500 1000 1500 2000 2500 3000

(a)

(b)

100 200 300 400 500

100 200 300 400 500

M [kNm]

M [kNm]

N [kN]

N [kN]

-500 -400 -300 -200 -100 0

-500 -400 -300 -200 -100 0

-1000 -500 0 500 1000 1500 2000 2500 3000

-1000 -500 0 500 1000 1500 2000 2500 3000

(c)

(d)

Figure 3. MN plastic domain in (a) in y direction and (b) x direction for columns and (c) for beams in the end sections and (d) in the center section of generic frame at first floor. 4.2. Nonlinear static analysis For the assessment of seismic vulnerability of the case study, a nonlinear static analysis has been done in compliance with N2 method (Fajfar, 1999; Fajfar, 2000) and according to current Building Code. In these earlier evaluations, shear failure mechanism of the elements and shear-induced brittle failures of nodal points have not been considered. In this case, the geometry of the brackets, typical of the gravity-load designed buildings, is inadequate to counteract the shears due to seismic forces in both directions. The application of N2 method in x and y direction is shown graphically in Figures 4 and 5. Nonlinear force-displacement relation of the MDOF system has been obtained by subjecting the structure to a monotonically increasing pattern of lateral forces, representing the inertial forces proportional to mass and to the first vibration mode. The pushover curves exhibit maximum base shears :,<=> of 466t and 97t and ultimate displacements of 5.04cm and 14.06cm, respectively in y and x direction (Figure 4a); in terms of percentage of yield force, they show maximum values of 2.75% and 3.03% and in terms of percentage of weight 16% and 3%, in y and x direction respectively; in terms of displacement expressed as a percentage of the height of the structure they exhibit maximum values of 0.30%H and 0.80%H.

100 150 200 250 300 350 400 450 500

0 0.5 1 1.5 2 2.5 3 3.5 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 F/Fy d/H X+_Mass X-_Mass Y+_Mass Y-_Mass X+_Mode X-_Mode Y+_Mode Y-_Mode

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

F [t]

F/W

X+_Mass X-_Mass Y+_Mass Y-_Mass X+_Mode X-_Mode Y+_Mode Y-_Mode

X+_Mass X-_Mass Y+_Mass Y-_Mass X+_Mode X-_Mode Y+_Mode Y-_Mode

0 50

d [cm]

d/H

0 2 4 6 8 10 12 14 16

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

(a)

(b)

(c)

Figure 4. Nonlinear force-displacement relation of the MDOF system.