PSI - Issue 64

Corrado Chisari et al. / Procedia Structural Integrity 64 (2024) 199–205 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 3 • , 1 , 2 Poisson’s ratio, first invariant of the strain tensor and second invariant of the deviatoric part of the strain tensor, respectively, • maximum principal strain; • = 0 2 0 −1 −1 , with 0 ratio between biaxial and uniaxial compressive strength; • = 3(1− ) 2 −1 , with ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield. Two loading surfaces are respectively associated to the equivalent strains: − ≤0 and − ≤0 , with: = [ 0 , ] and = [ 0 , ] (4) and 0 , 0 the initial thresholds for , . The damage evolution is defined separately for two damage variables and by defining the relationship between the strength | and | : | ( | )=1− | ( | ) | (5) with E Young’s modulus. The global damage variable is finally defined as: = 1 − (1 − )(1− ) (1− ) +(1− ) (6) with =max( ⁄ ) and =max( ⁄ ) . The solution algorithm is enhanced through the use of IMPL-EX strategy (Oliver, et al., 2008) at local level. At each time step a predictor-corrector scheme is followed in which the prediction part only satisfies equilibrium. This makes the single time step converge in just one iteration, thus increasing robustness and efficiency. The material model is implemented in Code_Aster v16.4.9 (Code_Aster, 2023). 2.2 Definition of the parametric analysis A parametric study was performed, by varying the main geometrical features of a reference cross vault, defined based on a typical example (Rouhi, et al., 2022) of Italian three-nave church (Fig. 1a). The prototype consists of a cross vault with a squared plane with internal spans of the arches equal to 6 m ( s =3 m). The lateral arches defining the vaults are pointed arches. Three of the lateral arches are closed by walls having a thickness t w , thus simulating the effect of external walls in the lateral zone of the transept. The distance between the center of one arch and the center of the span (eccentricity) is indicated by the letter e . The vaults also present internal ribs having a thickness t r and a width equal to half thickness of the vault t (Fig. 1b). The loading protocol comprises a preliminary phase in which self-weight - considering a masonry density of 1600 kg/m 3 , typical of tuff masonry - is applied, followed by a stage where a horizontal displacement is imposed at the springings of the open side, simulating the effect of a relative longitudinal motion of the central nave with respect to the transept wings (Fig. 1a). 201