Modelling non-linear behaviour of steel fibre reinforced concrete

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Yield surface

The σ-ε response for both tension and compression, discussed in the previous sections, were developed relying on a uniaxial stress state. The strength of concrete elements can be properly determined only by considering the interaction of various components of the state of stress. For example, Kotsovos and Pavlovic (1995) explained the importance of the interaction of stresses by using compressive test results of plain concrete cylinders under various levels of confining pressure. It was found that a small confining pressure, of approximately 10 percent of the uniaxial cylinder compressive strength, is sufficient to increase the load-carrying capacity of the tested cylinder by as much as 50 percent of the original value. On the other hand, a small lateral tensile stress of about 5 percent of the compressive strength of the cylinder is sufficient to reduce the loadcarrying capacity of the cylinder by approximately the same amount.
In general, the failure of structural element can be divided into crushing and cracking types. Crushing indicates the complete rupture and disintegration of the material under compression. Cracking indicates a partial or complete collapse of the material across the plane of cracking under tensile stress states (Chen, 1982). The stress state in structures is often a combination of tension and compression.
Kupfer et al. (1969) conducted and experimental investigation to study the biaxial strength of concrete. In their experiment, prismatic concrete specimens measuring 200 x 200 x 50 mm were subjected to biaxial stress combinations in the regions of biaxial compression, compression-tension and biaxial tension. Three types of concrete with unconfined compressive strength of 19, 31.5 and 59 MPa were tested at 28 days. Within each region of stress combinations four different stress ratios were chosen and six specimens were tested for each variable. Figure 2-12 shows the relationship between the principal stresses (σ1 and σ2) at failure given for the three types of concrete investigated.
Apart from the fact that the strength of concrete under biaxial compression is larger than under uniaxial compression, the relative strength increase is almost identical for the three types of concrete used. The large variation in the uniaxial strength of these three different concrete types has no significant effect on the biaxial strength. In the range of compression-tension, the compressive stress at failure decreases as the simultaneously acting tensile stress is increased. Under biaxial tension the controlling biaxial tensile stress is almost independent of the stress ratio and therefore the strength is almost the same as the uniaxial tensile strength. In other words, tension in one plane of concrete element does not affect the tensile properties of the perpendicular plane.

Chern et al. (1992) investigated the influence of the presence of steel fibres on the behaviour of SFRC under multi-axial stress. In their investigation straight steel fibres with a length of 19 mm were used. The experimental results for plain concrete and SFRC cylinders are compared. The SFRC contains approximately 80 kg/m3 of steel fibres (1 percent by volume). Figure 2-13 shows that the addition of steel fibres to concrete has an insignificant effect on the behaviour of the composite subjected to hydrostatic compression up to 70 MPa. It should be noted that this confining pressure was approximately three times the uniaxial compressive strength. It can be deduced that failure criteria, describing the compression behaviour under multiaxial stress state, that were successfully used for plain concrete are also appropriate for SFRC.

Analysis of ground slabs

The thickness design of concrete ground slabs is the same as for other engineered structures where the aim is to find the optimum thickness that will result in minimal cost and adequate performance.
In the assessment of SFRC ground slabs, ductility plays a decisive role in load-carrying capacity and deformation behaviour of these slabs. Existing analytical models for structural design of ground slabs can generally be divided into three categories:
(1) Methods based on elasticity theory, assuming an un-cracked structure.
(2) Methods based on yield line theory, assuming a cracked structure.
(3) Methods based on non-linear analysis. In these methods, the P-Δ response is computed from which the ultimate load can be determined.
Westergaard (1926) was the pioneer in developing analytical models for analysing plain concrete slabs supported on a Winkler foundation. Ioannides et al. (1985) conducted a study to re-examined Westergaard’s equations. In their study, several empirical adjustments were considered and the slab size requirements for the development of infinite slab responses were established. The Westergaard model only enables us to determine when localised failure starts in perfect slabs, but it does not tell about the consequences of this localised failure. In other words no indication is given whether this leads to total failure or only to formation of harmless, small cracks (Henrik and Vinding, 1995). Methods based on elastic analysis (assumes concrete deforms linearly up to failure), can however hardly be applied to SFRC ground slabs, as they do not account for the post-cracking strength of the SFRC. In fact, steel fibres become active after cracking of the concrete matrix so that the uncracked option is not appropriate for SFRC slabs.
Modern structural design codes of practice have abandoned “permissible stress” concepts in favour of utilising the actual capacity of materials and members. A design approach based on the yield line theory may provide an improved approximation of the load-carrying capacity of SFRC slabs compared to the elastic theory approach. Models developed by Meyerhof (1962), Losberg (1978) and Rao and Singh (1986) are based on yield line theory. These models were originally developed to estimate the load-carrying capacity for plain and conventionally reinforced concrete ground slabs. For these models to be used in designing SFRC, the strength term is changed and represented as the sum of the post-cracking strength and the cracking strength (refer to Appendix A). This modification will account for the stress redistribution as the result of incorporating steel fibres (Kearsley and Elsaigh, 2003). However, there are two basic prerequisites for the yield line theory to provide a good approximation for the ultimate load carried by concrete ground slabs. The first is that the material behaviour is ideally plastic to allow for bending moment redistribution (Holmgren, 1993). This is not the case with the SFRC often used for ordinary ground slabs (Meda and Plizzari, 2004). The second is that the yield lines are correctly hypothesized. This prerequisite is crucial because the magnitude of ultimate load is dependent on the pattern on the yield lines.
Falkner et al. (1995a) suggested a combination of the elastic theory and the yield line theory. They proposed some procedures for adjusting the Westergaard formulae to model SFRC ground slabs. In their proposed formula the load-carrying capacity is calculated by multiplying the load results obtained from the Westergaard formula with a factor to account for the effect of the post-cracking strength of the SFRC. These factors were determined from full-scale ground slab test results and finite element analyses.
Conflicting opinions exist regarding the applicability of the numerical models discussed here for SFRC ground slabs. Evaluation of these models is normally conducted by comparing measured results from full-scale slab test to calculated load-carrying capacity using these numerical models. It is often stated that certain models underestimate or accurately estimate the load-carrying capacity of the SFRC slab (Kaushik et al., 1989; Beckett, 1990, Falkner et al., 1995a and Chen, 2004).
However researchers should be cautious when results from slab model tests are used to validate numerical models that can be used to design pavement slabs, which differ from the model slabs tested in the laboratory. The lack of edge restraint that is normally the case for slab models, allows the slabs to lift up from the supporting layers at the edges and corners. This is often not the case for pavement slabs as they are usually restrained by the next slab at joints (Bischoff et al., 2003).
If we are to seek a greater exploitation of SFRC, analysis should proceed beyond the initial cracking point. Recently, non-linear finite element methods were implemented to analyse SFRC ground slabs with different levels of success (Falkner et al. 1995b, Barros and Figueiras, 2001 and Meda and Plizzari, 2004). The advantage of using non-linear finite element methods is that the behavioural aspects can be obtained throughout the loading process. For example, the complete P-Δ response and associated stresses within the SFRC slabs can be obtained which is not achieved when using elastic theory or yield-line theory. Hence, an improved understanding of the behaviour of the SFRC structure, greater safety and improved economy can be achieved when utilising nonlinear finite element methods. The accuracy of these methods is much dependent on the appropriateness of the material constitutive model and the representation of the cracks.

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1.1 General
1.2 Problem statement
1.3 Research objectives and limitations
1.4 Brief description of work
1.5 Research structure
2.1 Introduction
2.2 Why use steel fibre reinforced concrete
2.3 Failure of SFRC ground slabs. ..
2.4 Cracking models for concrete
2.5 Load-deflection behaviour of SFRC ground slabs
2.6 Flexural properties of SFRC
2.7 Constitutive relationships for SFRC
2.8 Analysis of ground slabs
2.9 Summary and remarks
3.1 Introduction
3.2 Materials for concrete mixture
3.3 Slab test setup
3.4 The beam test
3.5 Cube and cylinder tests
4.1 Introduction
4.2 Analysis method
4.3 Parameter study
5.1 Introduction
5.2 A brief description of the finite element programme
5.3 Stress-strain relationship
5.4 Finite element analysis of a single element
5.5 Finite element analysis of SFRC beam
6.1 Introduction
6.2 Modelling the plate-bearing test
6.3 Model for SFRC ground slab
6.4 Implementation of the modelling approach on ground slabs tested by other agencies
7.1 Introduction
7.2 Models for the SFRC ground slab
7.3 Effect of changing strength of concrete
7.4 Effect of changing steel fibre content
7.5 Effect of changing support stiffness
7.6 Effect of slab thickness
7.7 Summary and remarks on the parameter study on the SFRC ground slabs
8.1 Conclusions
8.2 Recommendations


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