Issue 48

J. F. Barbosa et alii, Frattura ed Integrità Strutturale, 48 (2019) 400-410; DOI: 10.3221/IGF-ESIS.48.38 402 However, the analysis of Mu's work was limited to testing the model only for the T300/QY8911 carbon/epoxy composite. Knowing that this model has a wide adjustment capacity, in this research it will be applied to represent the fatigue behaviour of metallic materials from the ancient bridges, and in this way verify the performance of the S-N Logistic formulation. A comparative study of the performance of the S-N adjustment equations, using models such as Kohout-Věchet, Logistic, ASTM and generalized Power law, will be applied to fatigue data from the old Portuguese riveted metallic bridges (Eiffel, Luiz I, Fão and Trezói). By means of a graphical adjustment analysis and the mean quadratic error, it will be possible to find the model that best fits with the experimental data. The results will be presented and discussed for a better recommendation on using the model in predicting fatigue of old bridges. M ETHODS USED IN THE MODELLING OF THE S–N CURVES Power Law he generalized power law model is a derivation of the power law model of two terms, commonly used for the interpretation of fatigue data of composite and metallic materials. These models are of direct application and not based on any assumptions, even in limited databases. The estimation of the model parameters is based on the linear regression analysis that can be performed by simple calculations [17]. The generalized power law is given by Eqn. 1:   max log( ) log C A B N    (1) where ௠௔௫ is the maximum stress amplitude parameter, N is the number of cycles until the material failure, whereas , , and are the parameters of the fatigue model derived by linear regression analysis, resulting from the adjustment of the equation to the experimental data. The constant is an adjustment exponent that can smooth the S-N curve in the low- cycle fatigue region. The S-N curve proposed by ASTM E739 standard 1 [1] is widely used by researchers for their reliable and simple modelling process. This model does not recommend an extrapolation outside the experimental data region. The representation of the model can be done by linearized form (log-log) given by Eq. 2:     max log log N A B    (2) The equations of the power law and ASTM E739 standard have similar structures for estimating the parameters of the curve in the linearized model (log-log); however, these models have two relevant differences. The first difference presented by ASTM E739 standard when compared with the Power Law is to consider fatigue stress as a dependent variable, while the power law considers the fatigue stress an independent variable (the number of cycles to failure, ௙ , is assumed as dependent variable). The second difference is the presence of a constant C , included in power law, able to smooth the fit in the low- cycle fatigue region. Logistic Function The logistic S-N curve model, developed by Mu [16], uses a logistic function to describe fatigue life behavior of composite materials, since this function is very similar to the S shape, commonly observed in S-N curves (lin-log). The logistic function is adapted to model the S-N curve and is given by Eqn. 3:     log 1 1 N b N c c a ae        (3) where a , b and c are the material constants, obtained by nonlinear least squares, ே is the normalized stress amplitude  N =  max /  ult and N is the number of cycles until failure.  ul is the ultimate tensile strength. Kohout-Věchet Model The full-amplitude S-N curve, based on the stress-damping parameter proposed by Kohout and Věchet, has been increasingly used in assessing the fatigue life of existing bridge structures [18]. The Kohout and Věchet S-N curve is a model based on geometric technical adjustment of fatigue behavior, based on stress or other damage linked parameter, that can T