PSI - Issue 41

8

Umberto De Maio et al. / Procedia Structural Integrity 41 (2022) 598–609 Author name / Structural Integrity Procedia 00 (2019) 000–000

605

The Young’s modulus and Poisson’s ratio of concrete are strength parameters, in terms of compressive and tensile strength, are , respectively. The steel reinforcement consists of Ø14 rebars and Ø6 stirrups whose mechanical properties, in terms of Young’s modulus and yielding stress are equal to 210.5 GPa s E  and 632.3 MPa y f  , respectively. In Table 3 are reported the strength parameters required by the adopted mixed-mode constitutive law, in terms of normal/tangential critical interface stresses ( c  and c  ) and mode-I/II fracture energies ( I c G and II c G ), together with the dimensionless parameter  and the initial normal/shear stiffness parameters ( 0 n K and 0 s K ). For this example, the mode-II parameters, which cannot be directly identified by the experimental test, have been set equal to 2 s t f f  and II I 10 c c G G  , in agreement with the interesting results obtained by (De Maio et al., 2019b). The following numerical simulations are conducted under plane stress state and quasi-static loading conditions, adopting a displacement-control solution scheme with constant increments of the mid-span deflection equal to 2e-3 mm. 24 GPa c E  and 0.18 c   , respectively, while the 49.4 MPa c f  and 2 MPa t f 

Table 3. Cohesive parameters required by the adopted traction-separation law. c  (MPa) c  (MPa) Ic G (N/m) IIc G (N/m) 0 n K (N/m 3 ) 0 s K (N/m 3 )  2 2.83 100 1000 3.13E14 1.57E14 5

The computational domain has been discretized by performing a Delaunay tessellation consisting of three-node triangular elements (see Fig. 6). A mesh refinement has been performed within the critical region dominated by the shear/flexural stress state by prescribing a maximum element size of 12.5 mm. In this region, zero-thickness four node cohesive elements, highlighted by blue lines in Fig. 6, are inserted between the concrete bulk elements. Two node truss elements whose size is equal to the corresponding edge of the adjacent concrete bulk element are employed to discretize the steel reinforcement. To avoid stress concentrations under both supports and load application points, rigid steel plates 60 mm length and 15 mm height are modeled.

Fig. 6. Computational discretization of the tested beam.

Fig. 7 shows the global structural response of the tested beam in terms of loading versus deflection curve and bending moment versus curvature curve. Both curves are in good agreement with both experiments ((Gribniak et al., 2016)) and numerical results obtained by a numerical model based on a smeared crack approach ((Rimkus et al., 2020)). The loading curve reported in Fig. 7a shows the typical trilinear behavior in which three different branches are detected coinciding with the elastic stage, the flexural/shear crack propagation stage, and the stage associated with the rebar yielding. On the other hand, the ultimate bending moment, coinciding with the initial point of the yielding stage in the loading curve, is very close to that obtained by the experiment (see Fig. 7b). The crack patterns predicted by the proposed model at different load levels, coinciding with the point A, B, and C of Fig. 7a are reported in Fig. 8. We can see that the adopted diffuse interface model allows the crack propagation and all cracking phenomena, including crack branching and coalescence, to be easily simulated. Moreover, the use of the embedded truss model, used in combination with the above-described bond-slip interface elements, allows the pass through the reinforced