PSI - Issue 13

Tuan Duc Le et al. / Procedia Structural Integrity 13 (2018) 1702–1707 T.D. Le, P. Lehner, P. Konečný / Structural Integrity Procedia 00 (2018) 000 – 000

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2. The 2D probabilistic model for chloride ingress to heterogeneous RC structures

According to the 2 nd Ficks Law of Diffusion, the penetration rate of chloride into concrete is generally modeled as a function of depth and time: ( , ) = 2 ( , ) 2 (1) where: • C(x,t) = the chloride ion concentration (%) at a distance x from the surface of concrete in time t ; • D c = effective diffusion coefficient (m 2 /s), which characterizes the concrete ability to withstand the penetration of chlorides. The simplified version of transient Finite Element model (Lehner et al., 2014) with regular triangular mesh was applied in order to evaluate the relationship (1) in case of the 2D chloride ingress. It is well known that the concrete properties are improving along with the progress of concrete hydration process that improves/reduces the chloride diffusion coefficient (Pack et al., 2010). Therefore, time dependent diffusion coefficient, D c,t , was proposed to be determined as the following formula (Thomas et al., 1999): , = ,28 ( 28 ) (2) where: • D c,28 = chloride diffusion coefficient (m 2 /s) measured at selected concrete age; • t 28 = age of concrete measured at period of 28 days; • t = concrete age (years); • m = aging factor. As above mentioned, a model capable of covering spatial variability of material parameters is aimed. In the expected model, these variable parameters will be defined under the scheme of random field via correlation length. Thus, the incorporation of the random field into the expected diffusion model is now discussed herein. In 2015, Eliáš and his colleagues integrated the random fields into a discrete model (Eliáš J. et al., 2015). In that work, the dependence between two arbitrary points of random field was defined by a squared exponential autocorrelation function. This was adopted here in the form of the correlation coefficient, ρ ij : = [− ( − ) 2 ] (3) where: • x i , x j = location vectors of point i and point j ; • l c = correlation length. It is important to underline that equation (3) is applicable for Gaussian random field only. In this study, Karhunen Loeve expansion was exploited to generate the random field with spectrally decomposed covariance matrix C and Latin Hypercube Sampling (LHS) approach was used for the sampling of the independent standard Gaussian variables . To deal with extremely computationally expensive evaluation of random field because of large C , EOLE- method was adopted. EOLE allows random field evaluation on a regular orthogonal grid of nodes with the spacing of the grid is about one-third of correlation length. Value at several interparticle facet of the model can be derived from values on the grid by using following relation: ̂ ( ) = ∑ √ =1 (4) where: • ̂ ( ) = the c th realization of the correlated random field ̂ ; • K = number of eigenmodes summed up; • = eigenvalues of covariance matrix of the grid nodes; • = eigenvectors of covariance matrix of the grid nodes;