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MODELING OF FIBRE REINFORCED CONCRETE

Appropriate material input parameters for numerical models of steel fibre reinforced concrete were identified from measured response of four-point bending beams using inverse analysis at several levels of complexity including advanced stochastic analysis and neural network technology. Based on the obtained results the optimal material input data sets are suggested for practical utilization of various numerical material models of fibre reinforced concrete in the nonlinear computer simulation of response and damage of FRC structures and structural parts.

MULTI-LEVEL OPTIMIZATION OF INPUT PARAMETERS FOR MODELING OF FIBRE REINFORCED CONCRETE Radomír Pukl, Tereza Sajdlová Červenka Consulting, Praha 5, Na Hřebenkách 55, 150 00, Czech Republic David Lehký, Drahomír Novák Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Brno, Veveří 331/95, 602 00, Czech Republic Abstract Appropriate material input parameters for numerical models of steel fibre reinforced concrete were identified from measured response of four-point bending beams using inverse analysis at several levels of complexity including advanced stochastic analysis and neural network technology. Based on the obtained results the optimal material input data sets are suggested for practical utilization of various numerical material models of fibre reinforced concrete in the nonlinear computer simulation of response and damage of FRC structures and structural parts. Keywords: Computer simulation, Fibre reinforced concrete, Nonlinear material models, Identification of material parameters, Fracture analysis 1 Introduction The nonlinear finite element simulation is recently a well-established approach for analysis of reinforced concrete structures and it has a big potential also in the field of fibre reinforced concrete (FRC) structures. Special material models at macroscopic level are available for modelling of FRCmaterial in the numerical simulation of FRC-based structures, taking into account higher ductility of FRC. This can be represented by larger fracture energy in the material models. Appropriate input material parameters for these numerical models are basic precondition for successful analysis of the FRC structures. Requested values, in particular the tensile material properties, can be identified using inverse analysis method from results of available tests on simple structures such as bending beams. 2 Nonlinear material models for fibre reinforced concrete Special constitutive material models have been developed for description of FRC-material in the nonlinear finite element analysis (Červenka (2011), Pukl et al. (2005)). They account for the high toughness and ductility of FRC, as well as other properties differing FRC material from conventional plain concrete (e.g. shape of the descending branch of the crack opening law). Several levels of FRC modelling at the material levels are available for performing the nonlinear numerical analysis. The first choice could be utilization of the material models developed for the plain concrete with appropriately adjusted material parameters (tensile strength, fracture energy). The shape of the tensile descending branch is in this case an exponential function, which is not optimal for the description of FRC response, but its use is rather pragmatic – it is of advantage that the models for plain concrete are very well verified and exhibit rather stable behaviour. In order to improve modelling of the FRC tensile behaviour material laws with special forms of the tensile descending branch more suitable for FRC were formulated and implemented. Two models are designed especially for steel fibre reinforced concrete (SFRC). They are derived from plane stress material law for plain concrete. If the fracture energy is known, an objective material law based on the crack band approach can be used. After cracking, the tensile stress drops to certain fraction of the tensile strength. In practical cases the fracture energy value is often difficult to evaluate since in the tests it is a hard task to follow the long-persisting descending branch until the zero tensile stress. In such a case a local formulation of the tensile material law is available. It is similar to the previous model but it is formulated directly in terms of strains and does not employ the fracture energy and crack band approach. The most sophisticated and most general model of FRC material represents an extension to the fracture-plastic constitutive law called CC3-User model. It describes the tensile behaviour according to the material response measured in tests point-wise in terms of the stress-strain relationship. The first part of the diagram is the usual stress-strain constitutive law. After exceeding the localization strain the material law assumed for the prescribed characteristic crack band width is adjusted to the actual crack band width. The FRC material can be in the numerical models also combined with the conventional reinforcement. 3 Inverse analysis of FRC material parameters A set of four-point bending tests made by company Betotech CR was used for inverse analysis. The tested beams were casted from concrete C55/67 with fibres of length 60 mm and diameter 0.8 mm in amount of 40 kg/m3. The scheme and dimensions of the tested and simulated specimen is shown in Fig. 1. Finite element model of the four point bending beam created in ATENA 2D shows Fig. 2. Load-deflection diagrams from the performed tests have been documented in Smiřinský (2012). Three groups of beams were tested. The material of all beams was the same, only dates of casting were different. -1.000E-03 -1.000E-03 Fig. 1 Scheme of the four-point bending tests Fig. 2 Numerical model of the FRC beam in ATENA 2D with finite element mesh The basic material parameters were generated in ATENA from the mean compressive strength of 74 MPa which is based on the characteristic cubic compressive strength of concrete C55/67. The generated material parameters are summarized in Table 1. Table 1 Generated material parameters for plain concrete C55/67 Material model SBETA 3D NLC2 E [MPa] 38000 38000 µ [-] 0.2 0.2 ft [MPa] 4.2 4.2 fc [MPa] -63 -63 Gf [N/m] 106 106 wd [m] -0.0005 -0.0005 RC [-] 0.8 0.2 In the next step, the material input parameters were randomized using advanced probabilistic system SARA. Regular stochastic distributions with the mean values corresponding to the adjusted ones have been used for selected material parameters. The results from the stochastic analysis were compared with the measured response for the specimens made with the fibre contents of 40 kg/m3 (see Fig. 3), and the optimal sets of material parameters have been derived (Table 2). Fig. 3 Load-deflection diagrams of FRC beams from stochastic analysis, comparison with the measured response Table 2 Identified material parameters of fibre reinforced concrete with 40 kg/m3 of fibres Material model SBETA SBETA SFRC 3D NLC2 3D NLC2 User E [MPa] 38000 38000 38000 38000 µ [-] 0.2 0.2 0.2 0.2 ft [MPa] 2.15 4.23 2.15 4.23 fc [MPa] -63 -63 -63 -63 Gf [N/m] 2700 3300 2700 - wd [m] -0.0005 -0.0005 -0.0125 - RC [-] 0.8 0.8 1 1 The relationship between the material input parameters and fibre contents has been determined. The multiplicative factor for the fracture energy of the plain concrete is from 25 to 30 for the FRC with the fibre contents of 40 kg/m3. For the tensile strength this factor is 0.5 in model SBETA and 3D NLC2. Note that the resulting material parameters for fibre reinforced concrete do not necessarily represent the real material properties but rather just appropriate input parameters for the numerical material model of FRC material. Similar procedure with comparable results has been performed using neural network adaptive system for automatic adjustment of the optimal set of input material parameters. Another level of approach is combination of the basic models for plain concrete with smeared reinforcement representing the steel fibres, including their bond properties to the concrete. Stochastic procedures based on random variables and random fields technologies are employed in order to account for random distribution of the steel fibres. This kind of FRC modelling is recently under progress and development. Load-displacement diagrams of FRC beams from numerical simulations using various material models are compared with the measured ones in Fig. 4. It shows very good agreement between the laboratory tests and their numerical simulation, in particular if the material 3D NLC based models are employed. Fig. 4 Comparison of load-deflection diagrams for FRC with fibre contents of 40 kg/m3 4 Conclusions Nonlinear finite element analysis is an efficient tool for investigation and design of the fibre reinforced structures. Advanced material models for numerical simulation of fibre reinforced concrete are available, but determining appropriate input material parameters suitable for realistic analysis lies above the usual testing methods. The required values can be efficiently determined using inverse analysis based on stochastic analysis at several levels of complexity. In particular, input tensile parameters of the material models (tensile strength, fracture energy) should be adjusted to represent behaviour and damage of fibre reinforced concrete. Inverse analysis is a feasible way to identify optimal set of input parameters for modelling the fibre reinforced concrete material and structures. Acknowledgement The presented results are based on research performed within the fundings P105/10/1156 “COMOCOS” and TA01011019 “SimSoft”. Authors gratefully acknowledge this support. References Červenka V. (2011), Design of fibre reinforced concrete structures based on nonlinear analysis. Fibre Concrete 2011, Prague, Czech republic: Czech technical university in Prague, pp. 11-12. Pukl R.; Červenka, V.; Červenka, J.; Novák, D.; Vořechovský, M.; Lehký, D. (2005), Deterministic and statistical models for nonlinear FE analysis of FRC-based structures. CCC 2005, Graz, Austria, pp. 130-133. Pukl, R.; Havlásek, P.; Sajdlová, T.; Červenka, V. (2011), Advanced modelling of fibre reinforced concrete structures. CCC 2011, Balatonfured, Hungary: fib Hungary, pp. 381-384. Sajdlová, T. (2012), Identification of material parameters for nonlinear modelling of fibre reinforced concrete structures (in Czech). Bachelor thesis, Czech Technical University in Prague, Czech republic. Smiřinský, S. (2012), Report about results of fibre reinforced concrete experiments. (in Czech) Beroun, Czech republic: BETOTECH