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C. Ronchei et alii, Frattura ed Integrità Strutturale, 34 (2015) 74-79; DOI: 10.3221/IGF-ESIS.34.07 78 A good agreement between experimental and theoretical results is in general observed, since the value of RMS T is lower than 3 (note that if all the calculated results fell within the scatter band 2, the value of RMS T would be equal to 2). Moreover, the analysis of the results in terms of the mean square error for the examined materials indicates that: a) for 30CrNiMo8 Steel, higher accuracy is gained for the orientation of the critical plane computed by means of 4  . Note that, by implementing Eq.(5) in the modified C-S criterion, the value of RMS T decreases to 6% in comparison to that determined by applying the 1  expression; b) for 6082 - T6 Aluminum Alloy, the same accuracy is deduced by using the five different  expressions, since the value of  is essentially the same; c) for S335J0 Alloy Steel, the most accurate result is provided by using the 5  expression to determine the critical plane, with a decrease of the RMS T value up to 20.4% with respect to that deduced by computing the critical plane orientation through Eq.(1). Therefore, the implementation of the  relationships (proposed by Łagoda) in the modified C-S criterion yields, only for materials characterized by fatigue limit ratio typical of hard and very hard metals, fatigue lifetime results different from those determined through the original  expression. In particular, the 4  and 5  expressions, respectively for 30CrNiMo8 Steel and for S335J0 Alloy Steel, provide better results than those deduced by employing the other relationships. C ONCLUSIONS n the present paper, the orientation of the critical plane, linked to the averaged principal stress directions, is computed by taking into account different expressions of the rotation angle  . In particular, such relationships have been implemented in the modified C-S criterion in order to estimate the fatigue lifetime by varying the critical plane orientation. The comparison with some experimental data related to stress-controlled fatigue tests of specimens under biaxial loading appears to be satisfactory. In particular, better estimations in terms of fatigue life are obtained for experimental data related to hard metals by using some of the modified  expressions (instead of the original one). R EFERENCES [1] Socie, D.F., Marquis, G.B., Multiaxial Fatigue, Society of Automative Engineers, Warrendale, USA (1999). [2] Marquis, G.B., Karjalainen-Roikonen, P., Long-life multiaxial fatigue of a nodular graphite cast iron, in: A. Carpinteri, M. de Freitas, A. Spagnoli (Eds.), Biaxial/Multiaxial Fatigue and Fracture, Elsevier, Amsterdam, (2003) 383-400. [3] Susmel, L., Lazzarin, P., A stress-based method to predict lifetime under multiaxial fatigue loadings, Fatigue Fract. Engng Mater. Struct., 26 (2003), 1171-1187. DOI: 10.1046/j.1460-2695.2003.00723.x. [4] Łagoda, T, Ogonowski, P., Criteria of multiaxial random fatigue based on stress, strain and energy parameters of damage in the critical plane, Mat. Wiss. U. Werkstofftech, 36 (2005) 429–37. [5] Anes, V., Reis, L., Li, B, Freitas, M., Crack path evaluation on HC and BCC microstructures under multiaxial cyclic loading, Int. J. Fatigue, 58 (2014) 102-113, DOI: 10.1016/j.ijfatigue.2013.03.014. [6] Wang, C., Shang, D.G., Wang, X.W., A new multiaxial high-cycle fatigue criterion based on the critical plane for ductile and brittle materials, J. Mater. Eng. Perform., 24 (2015) 816-824. DOI: 10.1007/s11665-014-1335-7. [7] Carpinteri, A., Karolczuk, A., Macha, E., Vantadori, S., Expected position of the fatigue fracture plane by using the weighted mean principal Euler angles, Int. J. Fracture, 115 (2002), 87–99. DOI: 10.1023/A:1015737800962. [8] Karolczuk, A., Macha, E., A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials, Int. J. Fracture, 134 (2005) 267-304. DOI: 10.1007/s10704-005-1088-2. [9] Carpinteri, A., Spagnoli, A., Vantadori, S., Multiaxial fatigue assessment using a simplified critical plane-based criterion, Int. J. Fatigue, 33 (2011) 969-76. DOI: 10.1016/j.ijfatigue.2011.01.004 [10] Carpinteri, A., Spagnoli, A., Multiaxial high-cycle fatigue criterion for hard metals, Int. J. Fatigue, 23 (2001) 135-45. DOI: 10.1016/S0142-1123(00)00075-X. I