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

523

4

N351 3005

n352

E1332 3006

N353

50 3

3007

n354

E1334 3008

N355

n312

n314

N311

3001

E1132 3002

N313

50 1

3003

E1134 3004

N315

E1331

E1333

E1131

E1133

322

323

324

321

325

122

123

124

121

125

E1321

E1322

E1323

E1324

E1121

E1122

E1123

E1124

5023

N251

N253

N255

n252

n254

N211

n212

N213

5021

n214

N215

312

313

314

112

113

114

311

315

111

115

E1311

E1312

E1313

E1314

E1111

E1112

E1113

E1114

5013

N151

N153

N155

n152

n154

N111

n112

N113

5011

n114

N115

1

2

3

4

301

302

303

304

305

101

103

102

E1301

E1302

N51

N53

N55

n52

n54

N11

N13

N15

(a)

(b)

(c)

Fig. 2. Numerical model: (a) overall 3D model; (b) East façade equivalent frame; (c) West façade equivalent frame.

Macroelement mechanical properties were calibrated against the nonlinear response of piers subjected to in-plane quasi-static cyclic shear-compression tests (Senaldi et al., 2018), and are summarized in Table 1. This required increasing the compressive ( f c ) and tensile ( f t ) strengths by factors of 2.75 and 1.1 respectively, and dividing the Young’s modulus ( E ) by a factor of 1.5. The shear modulus was taken as G = 0.3 E . Because the macroelement by Penna et al. (2014) concentrates flexural deformations at the member ends, multiplying the Young’s modulus by 3.0 was necessary in order to capture the correct stiffness. Moreover, the macroelement by Penna et al. (2014) requires dividing the shear modulus by the shear factor χ = 1.2, as it considers the full cross-section also in shear. The Turnšek and Sheppard’s (1980) criterion was adopted for the shear strength. Since the shear formulation of the Penna et al. (2014) macroelement is based on a Coulomb-like criterion, equivalent cohesion ( c eq ) and friction coefficient ( μ eq ) had to be provided by linearizing the desired criterion at the static axial compression. On the other hand, the improved version proposed by Bracchi et al. (2021a,b) is able to calibrate such parameters automatically; therefore, the tensile strength of masonry ( f t ) was directly assigned. Finally, parameters Gc t and β , which govern the nonlinear shear response at and beyond the peak strength, complete the description of the macroelements.

Table 1. Parameters for the masonry macroelements. Element f c [MPa] f t [MPa] E [MPa]

G [MPa]

ρ [kg/m

3 ]

eq [MPa]

μ eq [-]

Gc t [-]

β [-]

c

Piers

3.58 3.58

0.187

2310 6930

690 575

1950 1950

-

-

10 10

0.5 0.0

Spandrels

-

0.17

0.15

3.2. Floor and roof diaphragms Timber floor diaphragms were simulated through four-node linear-elastic orthotropic membranes. These finite elements are thoroughly characterized by defining the Young’s modulus in the principal ( E 1 ) and orthogonal ( E 2 ) direction, the Poisson’s coefficient ( ν ), and the shear modulus ( G 12 ), which mainly influences the capability of redistributing lateral forces among masonry walls. The diaphragms of the specimen consisted of a single layer of planks nailed to the floor joists or to the tie-beams of the roof trusses. The membranes were assigned the thickness of the planks and the equivalent stiffness properties from equation (1). The shear modulus was calculated according to Brignola et al. (2009), considering the three in series contributions of flexural and shear deformation of planks, and the rigid rotation of planks due to nail slip:

⎩⎪⎨ ⎪⎧ 1 = + 2 = 12 = � 2 + + 2 12 � −1

(1)