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

525

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The horizontal flexural stiffness of the walls excited out of plane was represented by three-node linear-elastic orthotropic membranes at each level (Fig. 3c). Their shear modulus ( G eq ) was calculated by equating the elastic shear stiffness of the membrane to the flexural stiffness of the wall portion resisting horizontal bending: = � ℎ� 3 6 2 � (3) where E = 2310 MPa is the elastic modulus assigned to the masonry; t and ℎ� are the thickness and the height of the wall portion involved in the mechanism; χ = 1.2 is the shear factor; L is half the length of the transverse wall; b is the distance between consecutive out-of-plane walls; and s = 2 cm is the membrane thickness. Coefficient β depends on the boundary conditions, ranging from 3 (simply-supported) to 12 (double-fixed); values of 9, 12, and 6 were assigned to the membranes for the North (A), Center (B, C), and South (D) walls, respectively, from calibration through nonlinear dynamic analyses (Salvatori, 2020). The main properties of these membranes are summarized in Table 3.

Table 3. Parameters for the out-of-plane three-node orthotropic membranes.

ℎ� 75

2 nd floor G eq [MPa] ℎ� [cm] t [cm] G eq [MPa] ℎ� [cm] t [cm] G eq [MPa] 28 75 15 28 37 15 14 3 rd floor

1 st floor

Membrane

b [cm]

L [cm]

[cm]

t [cm]

A B C D

271 271 291 291

262 262 262 262

15 30 30 30

163 163 163

328 306 306

148 148 148

30 30 25

298 277 161

75 75 75

30 30 25

151 141

82

3.4. Conventional 3D model The second strategy consisted of following the common modeling practice, that is, neglecting the out-of-plane response of masonry walls. In this context, all elements of the unconventional approach explained above, such as the fictitious frame of wall P6, the additional vertical beam elements, and the three-node membranes were omitted. The transverse walls were explicitly discretized in piers, spandrels, and nodes, following the actual opening layout. 3.5. Single-wall 2D models Starting from the conventional 3D model, the two façades (East and West) parallel to the shaking direction were extracted to perform 2D single-wall nonlinear static analyses. Macroelements were added at the intersection with each transverse wall, to capture the flange effect. The length of these flanges was equal to half the length of the South and Central walls. For the North façade, only the vertical strip bounded by the first opening alignment was included. Horizontal truss elements were modeled with half the longitudinal stiffness of the floor diaphragms. Moreover, additional masses were introduced to consider the out-of-plane contributions of the North wall central strip and of the three gables, not explicitly modeled. The contribution of the North wall and gable had to be considered in terms of dynamic mass only, because their weight was independently transferred to the ground; for this reason, the vertical forces generated by their masses were canceled out by applying upwards static forces. 4. Comparison between numerical and experimental results Nonlinear pushover analyses were performed on all 3D and single-wall 2D models, considering two different horizontal force distributions. The first, named “uniform”, consisted of a force distribution proportional to the nodal masses, whereas the second one, termed “modal”, represented a first-mode-type force distribution, with forces proportional to the product of the nodal masses times their height above the base. In the following figures, results from the 3D model with the out-of-plane unconventional approach are labeled “OOP”. Drift limits of 3% and 1% were assigned to both piers and spandrels for flexural and shear failure, respectively, according to the experimental behavior of masonry piers subjected to complementary in-plane cyclic shear-compression tests (Senaldi et al., 2018).