# The Size Effect on Flexural Fracture of Polyolefin Fibre-Reinforced Concrete
## 1 Introduction  
Concrete is the most extensively used material in building and civil construction [1]. Although as a structural element it exhibits excellent behaviour for compressive actions, it lacks sufficient resistance for tension loadings and behaves as a quasi-fragile material does [2]. To address this weakness, steel reinforcing embedded bars or, more recently, fibres randomly distributed in the concrete mass have been used. In the steel-bar option, commonly called reinforced concrete, bars assume the tension stresses that may appear in the member sections. As it improves crack control and the effect of concrete shrinkage [3], the option of fibre reinforcement (fibre-reinforced concrete, FRC) may be a partial or total substitution of the steel bars as long as they have been considered in the design calculations [4; 5], complying with the requirements of the pertinent structural design codes [6; 7]. Nonetheless, both reinforcement options make the concrete behave as a ductile material does. The most popular types of fibres used nowadays for structural concrete are made of steel or polyolefin. Given that use started in the mid-20 th century [8], steel fibres have been widely studied with a significant amount of research work being reported to assess the relevant properties [9]. However, use of polyolefin fibres started only in the late 20 th century as a result of the development of the chemical industry. Given that the lack of experience in use of this fibre type has led to a relatively low awareness of the fibre mechanical properties, deeper understanding of use of such a fibre type when it is combined with concrete is required. A short comparison of steel and polyolefin fibres shows that the latter improve certain aspects that should be considered when the choice of one or the other is under discussion: the absence of oxide deterioration [10], lack of sensitivity of magnetic fields, absence of safety problems due to protruding fibres at the element surface [11] and lower wear of machinery for the fabrication, transport and placing of concrete when compared with use of steel fibres [12; 13]. One of the problems for transferring the research findings to the industrial applications of new techniques or materials is that the direct use of the laboratory conclusions and the required formulation parameters for member design is not always possible. In a quasi-brittle material such as concrete, it was observed that when the size of the specimen increased and the shape proportions maintained, the level of collapse stresses (that is to say, the specimen strength) decreased. This is known as the size effect [14]. Regarding such an effect, it is important to note that there are three theories involved in its study: statistical, deterministic and that based on fractals [15], with the deterministic one being the most used [16]. The size effect is defined as the deviation of the real strength capacity compared with the load predicted by plastic analysis or another classical structural strength theory based on critical stress states [17]. This effect can be graphically represented, as in Figure 1, being at least three sizes necessary in order to fit the size effect law of the material. Although the existence of the size effect is widely accepted by the scientific community, there remain discrepancies regarding assessment in FRC. Some research campaigns have been carried out with regard to the influence of the maximum aggregate size and the added fibre volume fraction, with studies revealing the existence of a size effect in such a concrete [19]. Fracture and brittleness data was obtained in the research performed by [PERSON] [14]. In addition, there is also published research which concludes that fibre-reinforced concrete with a high content of ductile steel fibres shows an almost negligible size effect [20]. This conclusion is consistent with other papers which argue that with high-volume fibre content the size effect is small due to the increment of ductility provided by the fibres [21].