6 Bayesian Updating of a Cracking Model for Reinforced Concrete Structures Subjected to Static and Cyclic Loadings 47 Table 6.1 Input parameters of mechanical model Parameters Notation Value Elasticity modulus (MPa) E Lognormal (μ=35,000; COV=0.12) Tensile strength (MPa) ft Lognormal (μ=3.5; COV=0.13) Concentrate load (N) F Uniform on the interval [5.4; 18] ×103 Damage initial threshold (Jm−3) Y0 1000 Poisson ratio ν 0.2 Brittleness in tension ADir 1×10−2 Brittleness in compression AInd 5×10−4 Kinematics hardening (Pa) γ0 7×10 9 Nonlinear hardening (Pa) a0 7×10−7 Fig. 6.2 Bayesian network setup Table 6.2 Discretization of nodes and a priori information for the Bayesian network Nodes Number of states Prior distribution Boundaries E(MPa) 20 LN(μ=35,000; COV=0.12) [20; 50] ×10 3 ft (MPa) 20 LN(μ=3.5; COV=0.13) [1; 5] F(N) 20 Uniform [5.4; 18] ×103 w(mm) 40 – [3; 0] σVM (MPa) 20 – [10; 80] ×10 3 6.3.2 Mechanical Model The beam was modeled in Castem finite element software in 2D case with the assumption of plane stresses. The input parameters of the model are presented in Table 6.1. The Young’s modulus, the tensile strength, and the applied load are modeled as random variables whose laws and parameters are also defined in Table 6.1. The other parameters (brittleness in tension and in compression, kinematics, and nonlinear hardening) of the model are taken from [1, 3, 6] for ordinary concretes because we don’t have information on this. These parameters are not easy to measure during experimental tests unlike others, such as the deflection whose observations will be used as described in the following sections. One thousand Monte Carlo simulations were performed for each random variable. The outputs of the finite element code are the values of the mid-span deflection (w) and the stress of Von Mises (σVM) in the direction of loading. In order to integrate the test data into the Bayesian network (Sect. 6.3.3), the properties (material characteristic, geometry, loading, etc.) of the numerical model are the same as those of the real beam. 6.3.3 Bayesian Configuration and Updating The configuration of the Bayesian network is presented in Fig. 6.2. It is constructed from the input random variables and output data estimated from the finite element code. The input random variables (E, F, and ft) are modeled as parent nodes, and the output parameters (w, σVM) as child nodes. All nodes are considered discrete and divided into a number of states within predefined bounds. The details of the discretization and the a priori information of the different nodes are summarized in Table 6.2.
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