A probabilistic fatigue model based on the initial distribution to consider frequency effect in plain and fiber reinforced concrete Luis Saucedo a , Rena C. Yu b,⇑ , Arthur Medeiros b , Xiaoxin Zhang b , Gonzalo Ruiz b a Department of Materials, University of Oxford, Parks Road, OX1 3PH Oxford, UK b ETSI de Caminos, C. y P., Universidad de Castilla-La Mancha, Avenida Camilo José Cela s/n, 13071 Ciudad Real, Spain article info Article history: Received 8 August 2012 Received in revised form 19 November 2012 Accepted 20 November 2012 Available online xxxx Keywords: Weibull cumulative distribution function Loading frequency Secondary strain rate abstract The objective of this work is twofold. First, we aim to develop a new fatigue model valid for quasi-brittle materials like concrete, which properties have considerably larger standard deviation than metals. Hav- ing this in mind, we fit the measured strength data with a three-parameter Weibull cumulative distribu- tion function and in turn take it as the initial distribution for an asymptotic fatigue model in concrete. Second, we endeavor to take into account the observed influence of frequency and stress ratio on the fati- gue life in concrete, both plain and reinforced with fibers. The developed model is validated against fati- gue tests in compression on cubic specimens for different stress ratios and loading frequencies. All the parameters have found physical meaning in the extensive experimental tests performed for two plain high strength concretes and two concretes reinforced with fibers. The secondary strain rate is found to be correlational with the number of cycles to failure. Finally, a reduced test procedure is proposed for fati- gue strength characterization. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Interest in the fatigue of concrete began more than a hundred years ago with the development of reinforced concrete bridges. Since then, numerous experiments have been conducted to study the influence of different fatigue parameters, for instance, see [1– 11,13,14,17,19–21,23] and the references within. These parame- ters are either set by the fatigue test conditions, such as the mini- mum stress r min , the maximum stress r max and the loading frequency f, or determined by the material properties, for example the static material strength r c , which can be the compressive strength f c or the tensile strength f t , or any other critical stress de- fined accordingly. Other parameters include the stress ratio R, de- fined as r min /r max , the stress amplitude or stress range Mr, calculated as r max  r min , or the stress level, defined as r max /r c . For metals, the stress range plays an important role, and the fa- tigue life (the number of cycles N resisted before failure) is often described by the Wöhler curve (or S–N curve). For concrete, how- ever, the influence of the stress ratio, loading frequency and stress level has been observed to be important [6,13,17,23]. The fatigue equation has evolved accordingly to illustrate the role of those parameters. For instance, Aas-Jakobsen [6] proposed to include the effect of the stress ratio R as follows r max r c ¼ 1 ð1  RÞb ln N ð1Þ where b is a material parameter. The same relation was confirmed by Tepfers and Kutti [10] and Tepfers [11] for fatigue strength of concrete in compression and in tension for splitting tests of cubes. Even though the influence of loading frequency (or time) has been observed as early as 1960s by Rusch [4] and confirmed by Awad and Hilsdorf [7], Sparks and Menzies [8] and Holmen [12] in 1970s, it was not included in the fatigue equation until Hsu [13], Furtak [14] improved Eq. (1) by including the loading time and fre- quency respectively. Zhang et al. [19] further improved the equa- tion of Furtak by redefining the stress ratio R when there is stress reversal. However, none of them considered the marked dispersion of the static strength r c in concrete. The first consideration of the statis- tical distribution of concrete strength properties for fatigue tests was by Zhao et al. [21], who considered a normal distribution as suggested in the design codes. Recently, Przybilla et al. [22] consid- ered the statistical feature of the characterized strength for brittle materials and derived the primary three-parameter Weibull cumu- lative distribution function (CDF) of fracture stress from three- and four-point bending tests. Weibull distribution was also used to fit the fatigue life of concrete at various stress levels by Oh [17] and to fit the flexural fatigue life of concrete containing nano-particles by Li et al. [20]. 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2012.11.013 ⇑ Corresponding author. E-mail address: rena@uclm.es (R.C. Yu). International Journal of Fatigue xxx (2012) xxx–xxx Contents lists available at SciVerse ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue Please cite this article in press as: Saucedo L et al. A probabilistic fatigue model based on the initial distribution to consider frequency effect in plain and fiber reinforced concrete. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.11.013