Random Fields for Non-Linear Finite Element Analysis of Reinforced Concrete

Abstract

With more advanced methods and increasing computational power, the simulation of reinforced concrete in a Finite Element Analysis (FEA) has become more and more realistic. In a non-linear analysis of reinforced concrete, cracking behaviour and the maximum load-capacity can be determined. Such analyses sometimes suffer from unstable behaviour, especially when large parts of the structure crack at the same time. It is expected that spatially varying concrete material properties will affect crack initialization, crack patterns and the stability of the analysis. In this report, the use of spatial variability in the material properties of concrete in a Finite Element Analysis (FEA) was investigated. To incorporate spatial variation in the Finite Element method (FEM), discretized random fields are used which are assigned to elements or integration points in the Finite Element (FE) model. In this research the following methods are implemented and have been assessed on their performance: Covariance Matrix Decomposition method (CMD), Fast Fourier Transform method (FFT), Local Avarage Subdivision method (LAS) and Expansion Optimal Linear Estimation method (EOLE). To be appropriate for the implementation in a general purpose FEM program the method has to be efficient with respect to computation time, accurate in representing the statistical characteristics of concrete and easy to implement in the program. In a literature review, a large variation was found in the used values for the statistical characteristics which are involved in the modelling of the spatial variation of concrete properties. In the assessment of the random field generators this range of values was used as input. From literature and the assessment it was found that the CMD method is easy to implement and is the most accurate in representing the statistical characteristics of concrete. With respect to efficiency, the method performs poorly when the number of nodes increases. This is the case for random fields in multiple dimensions and/or for random fields with a small correlation length. The FFT method is slightly less accurate but performs very well with respect to efficiency when the number of nodes increase. The derivation of the one sided Spectral Density Function (SDF), which is needed in the FFT method, is however quite difficult. The threshold value in the correlation function and the distribution type have the largest influence on the accuracy of the random field. If the threshold value increases, and a log-normal distribution type with a high coefficient of variation (COV) is selected, the accuracy of the different methods decreases. The FFT method is slightly more accurate in representing the statistical characteristics in that case. With a large correlation length and a threshold value, the values in the random field are strongly correlated. It was found that in such a case the assumption of ergodicity does not hold any more. In the general purpose FEM program DIANA, a random field application has been developed. The guidelines in the JCSS Probabilistic Model Code are followed and implemented as a material model in the program. This material model is used in an example to assess the model code and the influence of spatially varying material properties on a non-linear FEA. A concrete floor submitted to a shrinkage load was analysed using the JCSS material model and some variations on this material model. Unfortunately, none of the analyses reached the convergence norm in all the load steps where cracking occurs. The results are therefore not reliable since the true equilibrium path may not be followed. However, they do give insight in the influences of spatially varying material properties on a non-linear FEA. The analysis resulted in non-symmetric cracking patterns, more gradual grow in the total number of cracks and crack initialization on the weakest point in the structure. In future studies the observations from this research can be used in a probabilistic analysis where the uncertainty in the material properties, which vary in space, can be taken into account which will yield a more accurate estimate of the reliability of a structure.