46 H. M. Imounga et al. of tension/compression, or even the parameters of hardening, which are determined from experimental tests. These tests may contain uncertainties related to the implementation of the tests, the variability of the material properties, the errors in the measurements, and the used calculation methods. In this context, the main objectives of this work is to propose a methodology based on Bayesian networks that could be useful to propagate uncertainties in the damage model and that could be updated with some measurements. The first part of this work describes the constitutive equations of the used damage model (Sect. 6.2). The numerical implementation and the description of the proposed Bayesian network–based methodology are detailed in Sect. 6.3. Finally, the results are presented and discussed in Sect. 6.4. 6.2 Constitutive Equations of the Damage Model The model used is an isotropic damage model developed by Richard et al. [3] and implemented in Castem codes. In this model, the damage variable representing the degradation of the elasticity modulus, due to micro-cracking, is a scalar which varies between 0 for healthy concrete and 1 for cracked concrete. The constitutive equations are formulated in the framework of the thermodynamics of irreversible processes and the model accounts for residual deformations, hysteretic behaviors, and unilateral effects (opening - closing of the crack) of concrete [3]. The different state laws based on the inequality of Clausius–Duhem–Trusdell [4] are summarized by the following equations: ⎧ ⎨ ⎩ σ =E((1−D)<εx>++<εx>−) +2(1−D)με +2Dμ(ε −ε π) σπ =2Dμ(ε−επ) Y = E 2 <εx>2 ++με.ε −μ(ε −ε π) . (ε −επ) (6.2) where μis a shear coefficient, σπ is the frictional tensor, επ is the sliding tensor, and Yis the damage energy released rate. Equation (6.3) gives the expression of the scalar damage variable: D=1− 1 1+ ADirH <εij>+ <σij>+ +AInd 1−H <εij>+ <σij>+)) (Y −Y0) (6.3) where ADir represents the brittleness in tension, AInd the brittleness in compression, Y0 the damage initial threshold, and H is Heaviside. 6.3 Numerical Implementation and Parameters Updating 6.3.1 Problem Description The approach is illustrated based on the configuration of a test conducted by Wang et al. [5]. The specimens are reinforced concrete beams of dimensions 300 mm×120mm×1500 mm, subjected to a cyclic bending at three points (10,000 cycles), with an alternating point load varying between 5.4 kN and 18 kN. The geometry of the beam as well as the configuration of the loading is presented in Fig. 6.1. Fig. 6.1 Beam geometry configuration
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