PSI - Issue 44

P. Morandi et al. / Procedia Structural Integrity 44 (2023) 1060–1067 Author name / Structural Integrity Procedia 00 (2022) 000–000

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unreinforced pier initial stiffness and maximum base shear V max , around 240 kN for both specimens. Once shear cracks formed, the retrofit held the masonry together, significantly delaying the associated damage and exploiting the compressive strength of masonry.

a)

b)

Fig. 2. Experimental force-displacement curves: a) UBPS01; b) RBPS01.

3. DEM simulation of the in-plane lateral response of unreinforced and retrofitted masonry piers A simplified micro-modelling strategy, based on DEM was employed to simulate the in-plane lateral response of the tested specimens. In simplified micro-models, masonry units are expanded up to the half-thickness of the mortar joints; bricks are modelled as continuum blocks, while mortar joints as zero-thickness interface springs. In the following sections the numerical simulation of the response of header-bond piers, UBPS01 and RBPS01, is discussed. The presented numerical models were built using the software 3DEC developed by Itasca (2019). 3.1. Modelling strategy: contact and block constitutive models Each masonry unit can be represented explicitly using the DEM. When two block are detected to be in contact, subcontacts are generated along the block contact surface. Zero-thickness springs characterised by both normal and shear contact stiffnesses, named respectively as k n and k s , and by a constitutive law are assigned to the subcontacts at the interface between units. Contact stresses are then calculated in the normal ( σ ) and shear ( τ ) directions depending on the related contact stiffnesses. The tensile strength ft mo , cohesion c , and friction angle ϕ are also defined to characterize the nonlinear behaviour of the normal and shear springs. In this work, masonry units are modelled as deformable blocks, divided into two (2R) finite-difference (FD) regions, each of them consisting in five constant strain tetrahedral elements. In Fig. 3 a schematic outline of the adopted modelling strategy is reported. In this study, a contact model validated by Pulatsu et al. (2020) is employed. This constitutive model allows to account for softening regimes in tension and shear through the definition of fracture energies. A Mohr-Coulomb joint slip model is employed to represent the contact spring shear behaviour. The cohesion is gradually set to zero once reached the maximum shear stress, τ max . The transition from the peak to the residual shear stress, τ res , is controlled by the shear fracture energy, G f,s . Regarding the spring normal behaviour, only a tensile failure is admitted: once ft mo is attained, tensile stresses drop to zero according to the tensile fracture energy, G f,t . Masonry compressive failure is accounted for in the constitutive model defined for masonry units, since no failure in compression is allowed at interface springs. A strain-softening version of the Mohr-Coulomb plasticity model (MPM) with tension cut-off was adopted. The linear elastic behaviour of FD zones is defined by assigning proper Young and shear moduli, E b and G b . As suggested by Malomo et al. (2019), equivalent cohesion, c b,eq , and friction, ϕ b,eq , are selected to obtain a FD region diagonal failure along the edges of the internal tetrahedral mesh when the compressive stresses attain the actual masonry compressive strength fc m . Similarly, to simulate the unit flexural damage, an equivalent tensile strength ( ft b,eq ) is inferred simulating bending tests on bricks. In addition, a