PSI - Issue 62

Gianluca Bottin et al. / Procedia Structural Integrity 62 (2024) 177–184 G. Bottin et al./ Structural Integrity Procedia 00 (2019) 000–000

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3.3. Predictive FE numerical simulations A finite element model of the two beams has been developed. For the sake of brevity, the results on the central beam are reported in this work. The Open System for Earthquake Engineering (OpenSees) framework (McKenna et al., 2010) was adopted with the pre- and post- processor Scientific ToolKit for OpenSees (STKO) (Petracca et al., 2017). The numerical models were realized using a distributed plasticity approach by means of force-based nonlinear elements with fiber discretization of cross sections. The Gauss-Lobatto scheme with 5 integration points was selected for numerical integration. The model was discretized with 4 beam elements, that is the minimum number required to simulate a four-point bending scheme. As well known, no further discretization is necessary for force-based beam column elements. The applied constraints simulate the designed experimental setup. Concerning the material constitutive laws, concrete was modelled with a simple Kent-Park model (i.e. Concrete01 in OpenSees), assuming a strain at peak stress equal to the widely accepted value of -0.002. Tensile strength of concrete is disregarded. Reinforcement steel was modelled with an elastic-plastic law (i.e. Steel01 in OpenSees). The material parameters adopted in the model were obtained from the available experimental results and are reported in Table 1. The hardening parameter b of the reinforcement was evaluated from the experimental measures of the ultimate strength and the ultimate strain. To account for the uncertainty in the parameter’s estimation, all the parameters have been assumed as random variables, the type of the statistical distribution, the mean value, the standard deviation and the Coefficient of Variation (COV) are reported in the same table. It is worth mentioning that elastic modulus was measured only on tests of bars with 14 mm of diameter and the same value is adopted for bars with 28 mm of diameter. Furthermore, since only two tests were performed for reinforcement with 28 mm of diameter, the COV for yield strength � and hardening parameter is assumed to be the same as the reinforcement with 14 mm of diameter. Finally, since no information was available experimentally for ultimate strain �� of concrete, a medium-high COV of 0.25 was assumed. The standard deviation of hardening parameter was estimated with usual uncertainty propagation for uncorrelated variables. More experimental tests will be performed in the future, extracting more bars after the full scale test on the beam, to collect more data, and improve the robustness of the statistical parameters and assumptions.

Table 1. Material parameters (N: Normal distribution, LN: Log-Normal distribution). Material Parameter Distribution Mean Std. dev.

COV

Elastic modulus ��� N Yield strength � , ��� N Hardening ratio ��� LN Elastic modulus ��� N Yield strength � , ��� N Hardening ratio ��� LN Ultimate strain �� N N

Reinforcement  14

216722 MPa

28752 MPa

0.133 0.116 0.721 0.133 0.116 0.721

382 MPa 0.002346

44 MPa 0.001692

Reinforcement  28

216722 MPa

28752 MPa

311 MPa 0.002256

36 MPa 0.001627

Compressive strength �

Concrete

29.89 MPa

6.31 MPa

0.21 0.25

0.006

0.0015

The first analysis was carried out assuming the mean values of all the material parameters. The results obtained in terms of load vs. midpoint displacement curve is represented by the blue curve in Fig. 8. Yielding occurs at about 20 mm of midpoint displacement and about 695 kN of force. The peak force is of about 830 kN reached for a corresponding midpoint displacement of about 625 mm. After the peak an evident softening branch occurs due to the crushing of concrete in the flange. As expected, after the linear branch, curvature shows a localization in the middle part of the beam, specifically in the beam portion between the forces (i.e. in the zero-shear force part of the structure). Another important aim of the numerical analysis was the estimation of the maximum horizontal displacement of sliding support and the displacement field in order to validate the experimental setup and avoid possible contacts