Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Fast evaluation of stress state spectral moments Claudio Braccesi, Filippo Cianetti ⁎ , Lorenzo Tomassini University of Perugia, Department of Engineering, via G. Duranti, 93, Perugia 06125 Italy ARTICLE INFO Keywords: Random loads Frequency domain analysis Modal approach Power spectral density Spectral moments Vibration fatigue ABSTRACT In frequency domain evaluation of fatigue damage, (Vibration Fatigue) most of the evaluation criteria are based on spectral moments of the power spectral density function (PSD) of stress signal. In numerical simulation this means to perform a dynamic analysis in frequency domain of the whole model (i.e. Finite Element Model) and, consequently, to evaluate the stress PSD function of each element or node. In this paper the authors introduce a method to fast evaluate the spectral moments of the stress power spectral density functions of a finite element model analyzed by modal approach. The authors theoretically demonstrate and validate that the statistical properties (spectral moments) of the power spectral density functions matrix of the stress tensor of each element of a generic numerical modal model are obtainable only by the evaluation of spectral moments of the power spectral density functions matrix of the model modal coordinates and by a simple linear combination with their modal stress shapes. The proposed method obtains the same results than the standard approach, resulting much faster. Only the evaluation, one time, of the spectral moments of the modal coordinates PSD functions matrix is needed. By a simple linear combination of these with the stress mode shapes is then possible to directly obtain, for each element, stress spectral moments, avoiding a lot of integral calculations, proportional to the elements numerousness. 1. Introduction In this paper the authors propose a tool considered by themselves a step forward inside the, so called, Vibration Fatigue (VF) or Fatigue Spectral Methods (FSM). Starting from a finite element (FE) model and frequency domain dynamic analysis performed by modal approach [1], the standard frequency domain procedure to evaluate damage recon- structs, by looping among all elements, the power spectral density matrix of the stress tensor, a matrix 6×6 for each frequency [1–3], and, then, summarizes its content in a single power spectral density function, for example by using Preumont's approach [4,5], or identifies a single component of the stress PSD functions matrix as the funda- mental representation of the stress state. When this stress state (i.e. Preumont's equivalent stress, single component of the stress PSDs matrix) satisfies the hypothesis of Gaussian stationary ergodic signal, the literature shows a lot of approaches that, starting from its power spectral density function (PSD) and the relative spectral moments [1,6,7], allow to directly obtain an estimation of the damage [8–10]. Dirlik's approach [8] is considered by authors as reference criterion, which, if compared to the other ones, shows a greater applicability to a wide range of PSDs [9].A lot of approaches, then, try to adapt results and methods, developed for uniaxial stress conditions, to multiaxial stress states with encouraging results [11–13]. The aim of the present paper is to show how it is possible to obtain the spectral moments of the power spectral density (PSD) functions matrix of the stress tensor of mechanical components modeled by modal approach and subjected to random dynamic loads in a faster way than the standard approach does. This work theoretically demonstrates and validates how the statis- tical properties (spectral moments [6,7]) of the PSD matrix of the model modal coordinates, together with the stress mode shapes, are necessary and sufficient to achieve those of the stress tensor of each element, and thus allow, for all direct approaches developed in the frequency domain and based on the spectral moments of the stress PSD function, to assess the damage with no margin for error. It is demonstrated that it is not necessary, for each element, to evaluate the PSD function of the stress state. Only the evaluation, one time, of the spectral moments of the modal coordinates PSD functions matrix is needed. By a simple linear combination of these with the stress mode shapes is then possible to directly obtain, for each element, its stress spectral moments, avoiding a lot of integral calculations, proportional to the elements numerousness. http://dx.doi.org/10.1016/j.ijmecsci.2016.11.007 Received 30 March 2016; Received in revised form 4 October 2016; Accepted 12 November 2016 ⁎ Corresponding author. E-mail address: filippo.cianetti@unipg.it (F. Cianetti). International Journal of Mechanical Sciences 127 (2017) 4–9 Available online 16 November 2016 0020-7403/ © 2016 Elsevier Ltd. All rights reserved. MARK