Issue 48

J. F. Barbosa et alii, Frattura ed Integrità Strutturale, 48 (2019) 400-410; DOI: 10.3221/IGF-ESIS.48.38 404       2 exp 1 1 log log n est i MSE n       (8) where ௘௫௣ and ௘௦௧ are the experimental cyclic stress and the estimated cyclic stress levels of the S-N curves, respectively, for all models considered in this research. The stress levels correspond to the same number of cycles, N , of the experimental data. Therefore the comparison between fatigue models was evaluated qualitatively and quantitatively. Acquisition of each model was examined for its accuracy in modeling, ability to extrapolate and interpolate, number of model parameters, sensitivity to available experimental data, etc. The MSE normalized calculation related with the adjustment of the presented S-N models to the experimental fatigue data, can be observed in Tab. 1. To facilitate the comparison between the S-N curve models, the results of the MSE values presented in Tab. 1 are normalized using the following expression NORM MAX MSE MSE/ MSE  (9) where MSE max is the largest MSE (Equation 8) between the Logistic, Kohout-Věchet, Power Law and ASTM E739 models when compared by the bridge model. Material Normalized mean squared error (MSE) Logistic Kohout-Věchet Power Law ASTM n Eiffel (R=0)* 0.4946 0.4943 0.5138 1 16 Eiffel (R=-1)** 0.6246 0.6741 0.6214 1 27 Luiz I (R=-1)** 0.4312 0.6450 0.4343 1 16 Fão (R=0)** 0.8618 0.9776 0.8341 1 21 Fão (R=-1)** 0.5268 0.4400 0.5705 1 14 Trezói (R=-1)** 0.5823 1.0000 0.3960 0.8455 10 All bridges(R=-1)*** 0.6682 0.7312 0.6788 1 66 * Fatigue tests under stress-controlled conditions; ** Fatigue tests under strain-controlled conditions; *** Only bridges R=-1. Table 1 : Normalized calculation of the mean squared error of the derived S-N curves. The obtained S-N curve based on ASTM E739 standard did not present a satisfactory performance when compared with the other models, such as, Logistic formulation, Kohout-Věchet model and Power law. The fatigue model proposed by generalized Power Law obtained a smaller error in four of the seven fatigue data of the analyzed bridge materials, resulting in a better approximation of the S-N curve to the experimental data set. This method obtained the best estimate considering the MSE value computed for the material from the Trezoi bridge (see Table 1). The logistic model that uses only three parameters in the equation obtained a lower MSE value when all the experimental fatigue data are analyzed together, considering only the data of the fatigue tests under strain-controlled conditions at R=-1. In the individual analysis of each bridge material it can be observed that the performance of the model was very similar to that of generalized Power Law model. For a set of fatigue experimental data with few data, it is possible to observe that some equations can’t obtain such precise estimates. This situation is verified for the material from the Trezói Bridge at R=-1 (fatigue test under strain- controlled conditions) with a sample of 10 specimens, where the Kohout-Věchet model obtained a very high MSE value when compared to Logistic formulation and Power Law. In general, the S-N curve using the Logistic formulation obtained satisfactory performance in terms of MSE values for all samples with size lower than 16. In the low-cycle fatigue region (LCF), Logistic, Khout-Věchet and Power Law curves achieved better adjustments than the ASTM standard. This can be seen in Fig. 1 related with the material from the Eiffel bridge at R=0 (fatigue test under stress- controlled conditions), where these models are able to flatten and smoothen the inflection of the S-N curve in this region.