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BiaxiaVMultiaxial Fatigue and Fracture
           Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.)
           0 Elsevier Science Ltd. and ESIS.  All rights reserved.              203




           CRITICAL PLANE-ENERGY BASED APPROACH FOR ASSESSMENT OF BIAXIAL
             FATIGUE DAMAGE WHERE THE STRESS-TIME AXES ARE AT DIFFERENT
                                        FREQUENCIES


                                       A. Varvani-Farahani
             Department of Mechanical, Aerospace and Industrial Engineering, Ryerson University, 350
                           Victoria Street, Toronto, Ontario, MSB 2K3, Canada




           ABSTRACT

           A new fatigue parameter has been developed to assess the fatigue lives under in-phase and out-
           of-phase  biaxial  constant  amplitude  fatigue  stressing  where  the  stresses  are  at  different
           frequencies.  In  this  parameter,  the  normal  and  shear  stress  and  strain  ranges  have  been
           calculated from the largest stress and strain Mohr’s circles. The total damage accumulation in a
           block loading history has been calculated from the summation of the normal and shear energies
           on the basis of a cycle-by-cycle analysis. The normal and shear energies used in this parameter
           are  divided  by  the  tensile  and  shear  fatigue  properties,  respectively,  and  the  proposed
           parameter,  unlike  many  other  parameters,  does  not  use  an  empirical  fitting  factor.  The
           proposed  fatigue  parameter  has  successfully correlated biaxial  fatigue  lives  of  thin-walled
           EN24 steel tubular specimens tested under in-phase and out-of-phase biaxial fatigue stressing
           where stresses were at different frequencies and included mean values.

           KEYWORDS

           Fatigue damage parameter, loading frequency ratio, block loading history, critical plane, shear
           and normal energies


           INTRODUCTION

           Many engineering components that undergo fatigue loading experience multiaxial stresses, in
           which  two  or  three  principal  stresses fluctuate  with  time,  i.e.  the  corresponding principal
           stresses are out-of-phase or  the principal directions change during a cycle of  loading. These
           components  include  car  axles,  helicopter propeller  shafts,  and  airplane  wings  subjected  to
           bending and torsion, which can be out-of-phase, and at different frequencies. To assess fatigue
           life of the components, many attempts have been made to derive theories based on equivalent
           stressktrain,  elastic-plastic  work/energy and  critical  plane  approaches. General  reviews  of
           multiaxial fatigue life prediction methods are presented by Garud [l], Brown and Miller [2],
           and You and Lee [3]. As yet there is no universally accepted approach. It was shown that the
           von Mises criterion was limited to correlating multiaxial life data under proportional loading in
           the  high  cycle fatigue regime  [4-51. Energy approach has been  applied in  conjunction with
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