doi: 10.1111/j.1460-2695.2008.01283.x A frequency-domain formulation of MCE method for multi-axial random loadings D. BENASCIUTTI 1 and A. CRISTOFORI 2 1 DIEGM, Dipartimento di Ingegneria Elettrica Gestionale Meccanica, Universit ` a di Udine, via delle Scienze 208, 33100 Udine, Italy, 2 ENDIF, Engineering Department In Ferrara, Universit ` a di Ferrara, via Saragat 1, 44100 Ferrara, Italy Received in final form 12 September 2008 ABSTRACT Many multi-axial fatigue limit criteria are formalized as a linear combination of a shear stress amplitude and a normal stress. To identify the shear stress amplitude, appropriate conventional definitions, as the minimum circumscribed circle (MCC) or ellipse (MCE) proposals, are in use. Despite computational improvements, deterministic algorithms im- plementing the MCC/MCE methods are exceptionally time-demanding when applied to “coiled” random loading paths resulting from in-service multi-axial loadings and they may also provide insufficiently robust and reliable results. It would be then preferable to characterize multi-axial random loadings by statistical re-formulations of the determinis- tic MCC/MCE methods. Following an early work of Pitoiset et al., this paper presents a statistical re-formulation for the MCE method. Numerical simulations are used to com- pare both statistical re-formulations with their deterministic counterparts. The observed general good trend, with some better performance of the statistical approach, confirms the validity, reliability and robustness of the proposed formulation. Keywords frequency-domain approach; MCC/MCE proposals; multi-axial fatigue; out-of-phase loading; random loading path. NOMENCLATURE (c u , c v ) = centre coordinates of the minimum circumscribed circle (¯ c u , ¯ c v ) = mean of centre coordinates C a = shear stress amplitude E(R 1 ) = expected major semi-axis of expected minimum circumscribed ellipse (i.e. expected radius of expected minimum circumscribed circle) E(R 2 ) = expected minor semi-axis of expected minimum circumscribed ellipse EMCC/EMCE = Expected Minimum Circumscribed Circle/Ellipse method MCC/MCE = Minimum Circumscribed Circle/Ellipse method m 0,i , m 2,i = spectral moment of zero and second order of  0,i (ω) n + 0,i = expected frequency of mean upcrossings of s i (t) N + 0,i = number of mean upcrossings counted in time interval T for load s i (t) N = normal stress Q = rotation matrix R 1 = major semi-axis of minimum circumscribed ellipse (i.e. radius of minimum circumscribed circle) R 2 = minor semi-axis of minimum circumscribed ellipse ¯ R 1 , ¯ R 2 = average major and minor ellipse semi-axes s i (t) = projected load Correspondence: D. Benasciutti. E-mail: denis.benasciutti@uniud.it c  2008 The Authors. Journal compilation c  2008 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 31, 937–948 937 Fatigue & Fracture of Engineering Materials & Structures