ࡱ> Q28P N/7^1`!8P N/7  ȽXJtxcdd``a!0 ĜL  312Ec21BUs30)0)0Qcgb  P#7T obIFHeA*CT f0 PeDdThOB  S *A2B277CH# <`!277CH# @T8AJ#X(iB Ӏ)\Xvwvog >m0. ueV(FESTIMATION OF FATIGUE STRENGTH AT OPERATIONAL LOAD D. JELASKA, S. PODRUG Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, R. Boakovia b.b., 21000 Split, Croatia ABSTRACT The simple way for estimation of fatigue strength at operational load on the basis of constant amplitude testing results is presented in this paper. It is necessary only to multiply the magnitude of constant amplitude endurance limit of symmetrical cycle by three correcting factors taking into account the shape of the spectrum, its asymmetry factor, static pre-stress, stress concentration and fatigue life. Each of these factors is derived and expressed in the form of analytical function. The explicit formula for fast estimation of fatigue life of the parts subjected to the acting of stress spectrum is obtained too. KEYWORDS Stress spectrum, stress amplitude, asymmetry factor, fatigue life, static pre-stress. INTRODUCTION The safest way to obtain the correct value of the fatigue limit of machine parts and structural components subjected to operational load is their testing at operational conditions. Such tests are time-consuming and expensive, and not always feasible in design stage, especially for the large size components of steel constructions, particularly offshore constructions, pressure vessels, stirrers, etc. In such, a very often cases, the only way to predict the fatigue limit and consequently the fatigue life, is their analytical estimation by help of the results of constant amplitude testing. This methodology, based primarily on Palmgren- Miner cumulative damage linear hypothesis, is described in detail in a number of papers. It ignored many important physical and chemical changes which take place during fatigue life, e.g. change of fracture location with change of load intensity caused by stress redistribution, build-up and decay of residual stresses at critical area and change of load transfer characteristic in e.g. bolted or riveted joints. It was a reason for lot of improvements of original Palmgren-Miner rule. A significant improvement was made by introducing the relative Miner rule whose basic idea was to apply a damage sum previously experienced and different from unity. It is understood testing with similar components under similar stress-time histories. Therefore, the need arose for fast estimation of fatigue strength without additional testing, i.e. on the basis of the constant amplitude testing only. It includes the problem of fatigue of steady pre-stressed parts which has not been studied at the level enabling the reliable estimation of the fatigue limit or fatigue life. PROPOSITION FOR FATIGUE STRENGTH ESTIMATION AT OPERATIONAL LOAD The greatest amplitude of the boundary stress spectrum having r asymmetry number, N fatigue life, and ke equivalence factor, might be estimated, similarly to (1(, by help of constant fatigue strength amplitude of the same part at r=-1 stress cycle asymmetry number and infinite life:  EMBED Equation.2  (N,ke,r)=KN(KS(Kr( EMBED Equation.2  (1) where fatigue life factor KN , stress spectrum shape factor KS and stress spectrum asymmetry factor Kr are defined as follows:  KN= EMBED Equation.3  (2) KS = EMBED Equation.3  (3) Kr=  EMBED Equation.3  (4) Derivation of Fatigue Life Factor It is evident from Eq.2 that fatigue life factor depends upon fatigue life only, i.e.  EMBED Equation.3 =fN(N). (5) After definition and in accordance with S-N curve equation, it is easy to obtain fN(N)=(Ngr/N)-b' (6) Whereas for finite fatigue life the elastic-plastic strain region is reached, this factor has to be corrected with KC factor taking into account the change of fatigue stress concentration factor in that region. This factor is defined as ratio of fatigue strength concentration factors in elastic and elastic-plastic region: KC= EMBED Equation.3  (7) Involving in this equation the expression for K( presented in (2(, it was obtained KC= EMBED Equation.3 , (8) where b/b ratio and calculation factor kC were completely determined by help of only the K(,el stress concentration factor, which is usually available information. Finally, fatigue life factor might be expressed as KN=KC EMBED Equation.3 = EMBED Equation.3  (9) Derivation of stress spectrum shape factor Whereas S-N curves for both C.A. and V.A. spectrum have the same fatigue strength exponent, it is clear (Fig.1) that logRa-logRDa=log(max-log(e (10) where from, in accordance with definition after Eq.3, follows KS= EMBED Equation.3  . (11)  Fig.1 To determining of stress spectrum shape factor Equivalent C.A. stress of V.A. stress spectrum depends on spectrum shape and consequently the spectrum shape factor depends on spectrum shape only. If the operational conditions of load results with stress history, which can be described as stationary quasi-gaussian and ergodic random process in which stress amplitude always exceeds a certain level p(called spectrum density factor and equals share of minimum spectrum amplitude), KS factor might be obtained after (3( : KS=3a(1-p) , (12) where a is material constant found to equal a=1,14 for welding joints. Derivation of Stress Spectrum AsymmetryFactor The fatigue limit line in schematized Smiths diagram for infinite life of part subjected to acting of C.A. stress spectrum (Fig.2) can be expressed as  EMBED Equation.3 . (13) The fatigue limit is defined (Fig.2a) as intersection of this line and r=const line  EMBED Equation.3  (14) after which the maximum stresses of stress cycle change. It was obtained:  EMBED Equation.3  (15) and  EMBED Equation.3 = EMBED Equation.3  . (16) Stress spectrum asymmetry factor is now: Kr= EMBED Equation.3 = EMBED Equation.3 . EMBED Equation.3  (17)  Fig.2 Determining of fatigue limit for infinite life of parts subjected to C.A. stress spectrum having r asymmetry number: (a) for not pre-stressed parts; (b) for previously pre-stressed parts For the pre-stressed parts, it was showed in e.g. (4( that the origin of Smiths diagram (Fig.2b) is moved at point ( EMBED Equation.3 ), and consequently the equation of r=const line was obtained to be  EMBED Equation.3 . (18) The fatigue limit is presented now as intersection of this straight line and fatigue limit line. In the similar way as before, it was obtained:  EMBED Equation.3  (19)  EMBED Equation.3  (20) It is easy now to obtain a relation for determining the stress spectrum asymmetry factor for pre-stressed parts:  EMBED Equation.3 . (21) By involving in Eq.1 the greatest amplitude of acting stress spectrum instead Ra, it is easy to derive the simple formula for fast estimation the fatigue life of a part: N=Ngr EMBED Equation.3  . (22) CONCLUDING REMARKS The simple way to estimate the fatigue strength and fatigue life of the parts and structural elements subjected to operational conditions of load by help of results of C.A. testing, has been presented, and unique formula for the fast estimation of fatigue strength amplitude is proposed. This estimation is good in so far as parameters existing in derived relations are reliable. Therefore it is necessary to make an effort in order to obtain the correct values of this parameters. This particularly refers to determining of stress spectrum, which presents the response of the part to load spectrum and might be significantly different from load spectrum, especially in case of vibrations. The necessary attention has to be paid to influence of possible elevated temperatures, which could be taken into account e.g. by introducing the additional temperature factor in the fatigue strength amplitude formula. This might be a subject of some referring paper in the future. REFERENCES Paetzold, H., Petershagen, H.(1984): Untersuchungen zur Betriebsfestigkeit geschweisster Schiffbaukonstruktionen, Shiff und Hafen 11, p.71-74. 2. Jelaska, D.(1998) : Estimation of Fatigue Stress Concentration in the Zone of Elastic-plastic Strain, Proc.Int.Symp. Fatigue Design 1998, Espoo. 3. Jelaska, D.(1997) : Estimation of Fatigue Life under Random Loading, Proc.14 th Symp. "Danubia-Adria" on Exper. methods in solid mech., Pore  1997., p.212-214. 4. Jelaska, D.(1994): Estimation of Fatigue Limit at Operational Load (in Croatian), Strojarstvo 36(3,4),p.127- 130. NOMENCLATURE b-fatigue strength exponent (slope of smooth specimen S-N curve) b'-slope of notched specimen S-N curve C.A.- constant amplitude KN-fatigue life factor Kr-stress spectrum asymmetry factor KS-stress spectrum shape factor  EMBED Equation.2 -fatigue stress concentration factor in the region of elastic-plastic strain  EMBED Equation.2 -fatigue stress concentration factor in the region of elastic strain  EMBED Equation.2 -stress concentration factor at the boundary of quasistatic fracture region kC- calculation factor ke- stress spectrum equivalence factor, = EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.2 -slope of N( EMBED Equation.2  isoline in schematized Smith's diagram, =2(1-RDa/RD0)(2(1-R-1/R0)=0,71,0 N-fatigue life (number of endurable cycles) p-spectrum density factor; = EMBED Equation.2  R-strength (generally), MPa r-stress cycle asymmetry number V.A.- variable amplitude (I- frequency of occurance of i-th stress block  EMBED Equation.3   EMBED Equation.2 -stress (generally), MPa  EMBED Equation.3 - equivalent stress, = EMBED Equation.3   EMBED Equation.2 -static pre-stress, MPa. 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