Keywords: normal strength concrete, high strength concrete, steel fibres, pre-stressed beams, high performance concrete.

High strength concrete (HSC) was one of the interesting and challenging innovations in concrete structures in the last decades. Later it was became common to speak about high performance concrete (HPC) instead of HSC. It was no agreement regarding the definition for HSC. A strong definition of HSC has been given by the authors [1]. Following this definition, HSC should reach a compressive strength of f_c >= 70 MPa. But the material is brittle, and to utilize the HSC strength capacity and to ensure the sections necessary ductility, steel fibres (SF) are added to the concrete matrix. It is logically to use two-layer beams consisting of steel fibred HSC in the section's compressed zone and normal stress concrete (NSC) in the tensile zone [1].

Two-layer beams are effective as bending pre-stressed elements. Using SF significantly improves the bond between the concrete matrix and the pre-stressed reinforcement and reduces the pre-tensioned members transfer length. It means that two-layer pre-stressed beams become HPC ones. Pre-stressed two-layer steel fibred beams design forms a new class of problems: obtaining the HSC class for a pre-stressed beam according to a given NSC one; calculation of the SF volume ratio; calculation of required pre-stressed section's ductility as a function of the SF ratio. Usually selection of that ratio was done based on engineering experience, technological requirements, etc. Most experimental investigations have been carried out for SF ratios that were selected in this way and varied from 0.5 to 2.5% [2]. This paper is focused on theoretical analysis of the above mentioned problems.

The paper deals with definition of a lower HSC class for two-layer pre-stressed beams. It is also focused on low ductility of a pre-stressed RC section without fibres and defines the minimum SF volume ratio. A method for calculation of the SF volume fraction vs. required ductility is proposed and its efficiency is demonstrated in a calculation example. It is shown that a ductility factor for both compressive HSC and tensile pre-stressed NSC zones is very small and near to 1.0, i.e. the bending element section is a brittle one. Therefore it is necessary to use SF to obtain high performance concrete (HPC) in the section's compressive zone. The SF do not affect pre-stressing reinforcement. In addition, a tensile concrete action is neglected, so it is not necessary to use an HPC in a section tensile zone.

Minimum SF volume fractions for HSC classes are given in the paper. It shows that fraction testing values (0.5 ... 2.5%) are close only to the minimum SF volume, corresponding to concrete transverse tensile stresses. But these values do not ensure the section's request ductility. A real SF volume fraction value is a function of the required ductility, which, in turn, is a calculation parameter. Besides, the total value of the added plastic deformations (influence of fibres confining effect) may not be more compared with ones without fibres. Based on these requirements, the linear graph of a function "SF volume fraction vs. required ductility" is given. It enables the calculation of an SF volume fraction for a given ductility.

1

I. Iskhakov and Y. Ribakov, "A design method for two-layer beams consisting of normal and fibered high strength concrete", Materials and Design, 28, 1672-1677, 2007. doi:10.1016/j.matdes.2006.03.017

2

J. Magnuson and M. Hallgern, "Reinforced high strength concrete beams subjected to air blast loading", In: N. Jones, C.A. Brebbia, "Structures under shock and impact", VIII, 8th ed. WIT Press, 2004.

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