Cumulative-Damage Reliability for Random-Independent (Normal- or Weibull-Distributed) Fatigue Stress, Random-Fixed Strength, and Deterministic Usage

To determine helicopter component retirement intervals, it is common for fatigue engineers to utilize Miner’s rule for cumulative damage. Time consuming Monte Carlo simulations are considered the state of the art method for determining reliability in the presence of load variations. However, a new set of methods employed on a recent AHS fatigue reliability round robin problem provide an indication that near term advances are on the horizon. This paper presents the development of a set of analytical formulas for the calculation of cumulative damage per Miner’s rule for the case with randomly varying oscillatory fatigue loads applied to a given component selected randomly from a population of various components. Although further development is required to cover all material characterizations, the method shows great promise in providing efficient fatigue reliability calculations during iterations of the design process. This paper provides a detailed derivation of the method, useful charts and tables, and guidance regarding implementation of the formula in various computer applications. NOTATION    , a “ a function” derived herein      , , b “ b function” derived herein E COV endurance limit coefficient of variation S COV load amplitude coefficient of variation D accumulated fatigue damage E endurance limit    erfc complementary error function    exp exponential function    f normal probability density function    g Weibull probability density function i load cycle index j component index k material constant (factor) l binomial expansion term index m material constant (exponent) N number of cycles to failure n number of applied load cycles p load condition index     p primary polynomials used to evaluate the a function     q secondary polynomials used to evaluate the a function R component reliability S oscillatory load amplitude w “standardized” Weibull load amplitude Presented at the American Helicopter Society 65 Annual Forum, Grapevine, Texas, May 27-29, 2009. This is a work of the U.S. Government and is not subject to copyright protection in the U.S. 0 w “standardized” Weibull damaging load amplitude x generalized oscillatory load amplitude y generalized endurance limit z standardized normal load amplitude 0 z standardized normal damaging load amplitude  material constant (endurance limit adjustment)  Weibull slope (for load amplitudes) S  load amplitude Weibull slope     Gamma function      , upper incomplete Gamma function    ,  “standardized” Weibull probability density function  Weibull minimum expected value (for load amplitudes) m  small increase in m with 1 0   m   Weibull characteristic value (for load amplitudes)  mean (of load amplitudes) E  endurance limit mean S  load amplitude mean  load condition usage spectrum weighting factor  standard deviation (of load amplitudes)     standardized normal probability density function DISCLIMER: Reference herein to any specific commercial, private or public products, processes, or services by trade name, trademark, manufacturer, or otherwise, does not constitute or imply its endorsement, recommendation, or favoring by the U.S. Government. The views and opinions expressed herein are strictly those of the authors and do not represent or reflect those of the U.S. Government. The viewing of the presentation by the Government shall not be used as a basis of advertising.