Deriving stress–strain relationships for steel fibre concrete in tension from tests of beams with ordinary reinforcement
Graphical abstract
Highlights
► A technique for deriving residual stresses of SFRC in tension has been proposed. ► Separation should be made between crack width and deflection analyses of SFRC. ► For deflection analysis, a total stress model should be employed. ► For crack width calculation, residual stresses should be used.
Introduction
Two main disadvantages of concrete as a structural material are its low tensile strength and brittleness. The inclusion of steel fibres may significantly improve mechanical properties of concrete such as ductility and residual load carrying capacity (toughness) [1]. The issue of quantifying the residual tensile strength (stresses) for a cracked section is one of the most critical points in the theory of steel fibre reinforced concrete (SFRC) [2], [3], [4], [5], [6], [7], [8]. To model the behaviour of SFRC in tension, the contribution of fibres can be introduced in two different ways: (1) by addition of a term (dependent on the fibre geometry and orientation) to represent the fibre stress and fibre-reinforcement ratio in a crack and (2) by considering SFRC as a homogeneous material with higher toughness characterized by the ability of a cracked section to resist a substantial fraction of tensile strength. The latter approach, considered in the present study, due to its simplicity and numerical effectiveness is much more widely used than the first one.
In present practices, the residual stress is quantified by means of bending tests on notched or un-notched specimens (600–700 mm in length). Based on these tests and standard techniques (RILEM [9], DBV [10], etc.), the first crack strength and the residual stresses are defined. These parameters are further used for the strength, deflection and crack width analysis of SFRC members. The standard techniques are always accompanied by a large scatter of the test results [8], [11], [12], [13], [14], [15] reaching over 30%. The scatter is closely related to variation in the number of fibres across the fracture plane [16].
Due to a great diversity in the shape and aspect ratio of steel fibres and, consequently, varying bond characteristics, there are no reliable constitutive models of residual stresses available until present time [17]. This, as a consequence, limits application of steel fibre reinforced concrete. In numerical simulation, the post-cracking behaviour is modelled either by a stress–strain relationship [2], [4], [18], [19], [20] or a stress–crack width law [6], [21], [22], [23]. In less sophisticated applications, a constant value of the residual stress can be assumed [3], [9].
Average stress–average strain relationships in tension can be obtained using an innovative inverse technique proposed by Kaklauskas and Ghaboussi [24]. The innovative technique is based on the layer section model [25] and the smeared cracking conception. For a given experimental moment–curvature diagram, the equivalent stress–strain relationship is progressively computed for the extreme tension fibre of the concrete section. Recently, the inverse technique was modified [26] to eliminate the shrinkage effect from the test data of flexural reinforced concrete elements [27], [28].
The present study extends application of the inverse technique to analysis of SFRC elements. It aims at deriving the equivalent stress–strain relations for steel fibre reinforced concrete in flexural tension using test data of concrete beams reinforced with steel bars and fibres. The paper reports results of the experimental and numerical investigation of deformation and cracking behaviour of SFRC flexural members. The experimental part includes tests on eight lightly reinforced beams (longitudinal reinforcement ratio 0.3%) with different contents of steel fibres (0%, 0.5%, 1.0%, and 1.5% by volume). Based on the proposed inverse technique, the equivalent stress–strain relations for concrete in tension were defined for each of the beams and further used for the curvature and the crack width analyses.
Section snippets
Constitutive analysis of the flexural members
Present investigation is aimed at developing a numerical procedure for the residual strength analysis of SFRC in tension using moment–curvature relationships of beams with ordinary reinforcement and steel fibres. The inverse procedure is based on the direct technique, the Layer section model [25] and uses the following approaches and assumptions.
Description of test beams
The beams of rectangular cross-section with nominal length 3280 mm (span 3000 mm) were tested under a four-point bending scheme. The experimental programme consisted of eight beams with different contents of fibres: 0%, 0.5%, 1.0% and 1.5% by volume. The specimens were reinforced with three 10 mm bars of tensile reinforcement resulting in reinforcement ratio 0.3%. Main parameters of the beams are listed in Table 1, where fy and Es are the yielding strength and the elastic modulus of the bar
Deriving equivalent stress–strain relationships of SFRC in tension
The equivalent stress–strain relations of SFRC in tension were derived by the inverse technique (see Section 2.3) using the experimental moment–curvature diagrams shown in Fig. 6. The constitutive laws taken in the analysis are shown in Fig. 1f and g. An ideal elastic–plastic diagram was assumed for the bar reinforcement (see Section 3.2 and Fig. 1f). The compressive behaviour of fibre concrete was modelled using the law proposed by La Mendola and Papia [29] (see Fig. 1g). The elastic modulus
Deflection and crack width analyses
As noted above, the stresses in the derived equivalent stress–strain diagrams consist of the tension-stiffening stresses and the residual stresses. In the selection of a SFRC constitutive law in tension, separation should be made between the crack width and deflection analyses. For deformation/deflection analysis based on the smeared crack approach, the total stress model should be employed, whereas for crack width calculation, dealing with a single section analysis, the residual stresses
Concluding remarks
The paper deals with experimental and theoretical investigation of deformation behaviour of concrete beams with ordinary reinforcement and steel fibres. In the experimental part, four-point-bending test results of eight lightly reinforced beams (reinforcement ratio 0.3%) with different contents of steel fibres (0%, 0.5%, 1.0%, and 1.5% by volume) have been reported. In the theoretical part, an alternative approach for deriving residual stresses of SFRC in tension has been proposed. This
Acknowledgements
The authors gratefully acknowledge the financial support provided by the Research Council of Lithuania (Project No. MIP-083/2012). Viktor Gribniak wishes to acknowledge the support by the Research Council of Lithuania for the Postdoctoral fellowship granted within the framework of the EU Structural Funds (project “Postdoctoral Fellowship Implementation in Lithuania”). Darius Ulbinas also wishes to express his sincere gratitude to Professor Lucie Vandewalle for her advice and guidance at his
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