Checking Shafts for Fatigue PDF

Gépelemek 1. CHECKING THE SHAFTS FOR FATIGUE Authors: Dr. Kerényi György Molnár László, Dr. Marosfalvi János, Dr. Horák Péter, & Dr. Baka Ernő Tengelyek | GÉPELEMEK 1. előadás 1 Pre-planning of axles Gépelemek 1. Main use of them is bending: Mh  = Mh bending moment causes the normal stress:...

Gépelemek 1. CHECKING THE SHAFTS FOR FATIGUE Authors: Dr. Kerényi György Molnár László, Dr. Marosfalvi János, Dr. Horák Péter, & Dr. Baka Ernő Tengelyek | GÉPELEMEK 1. előadás 1 Pre-planning of axles Gépelemek 1. Main use of them is bending: Mh  = Mh bending moment causes the normal stress: h K Where K in case of solid axle & tubular section D 3 K= 32 ( D 4 − d 4 ) 2 K=  64 D For bending the most advantageous design is (varying cross-sectional beam) Tengelyek | GÉPELEMEK 1. előadás 2 State of stress in axles Gépelemek 1. Carrying axles. Their cross-section is usually round, in special cases I-section or closedsection beam. Their typical use is bending. The nature of the fatiguing stress is pulsating. (Shearing stress also appears, but since it is much smaller than bending, it is usually neglected.) They are in steady state. M [Nm] M [Nm] Tengelyek | GÉPELEMEK 1. előadás 3 Engineering & design of shafts Gépelemek 1. Main use of them is rotary bending & twisting: Pre-planning: determining the characteristic size (diameter or wall thickness) of the shaft from the allowable stress on the material of the shaft and the stress resulting from bending, assuming a static load; Defining the design: of the shaft in accordance with the installation and operating conditions. Control: to reduced static use; to deformation; to critical speed, to fatigue & (fracture). Tengelyek | GÉPELEMEK 1. előadás 4 State of stress in shafts Gépelemek 1. Rotating shafts: Their cross-section is almost without exception a circle or ring. (tubular) The nature of the fatiguing stress is rotary folding bending (which means tensile and compressive stress is generated in the extreme fiber of the crosssection). M [Nm] M [Nm] Tengelyek | GÉPELEMEK 1. előadás 5 Fatigue phenomenom Gépelemek 1. Centuries of experience show that cracks appear in steel components above a certain stress level, even though they are suitable for static stress. As a result of repeated stress, cracks nucleate than cracks propagate along the cross-section, and in the final stage, when the remaining cross-section is reduced to the limit of the static strength, the components break. The propagation surface of the crack is smooth and shell-like (the static fracture shows a porous appearance). Cracks usually originate from a stress collection point, from a local material defect. The propagation speed of the created crack depends on the sensitivity of the material, the stress gradient, the life cycle of the part and many other factors Tengelyek | GÉPELEMEK 1. előadás 6 Fatigue & fracture of shafts Gépelemek 1. Rotating „folding” bending stress concentration point: circle groove Rotating „folding” bending stress concentration point : keyway Tengelyek | GÉPELEMEK 1. előadás 7 Checking for fatigue Gépelemek 1. In the following, we deal with sinusoidal stress with constant amplitudes, the fact that more complex stress sequences can be traced back to this case by means of Fourier analysis Pure oscillating stress Pure pulsating stress Tensile-compressive mean fluctuating stress Tensile-mean fluctuating stress Tengelyek | GÉPELEMEK 1. előadás 8 Preloaded bolt joints under dynamic loads (example) Gépelemek 1. a) Pure pulsating stress b) Fluctuating stress between a min & a max value. Tengelyek | GÉPELEMEK 1. előadás 9 Statistical background of fatigue Gépelemek 1. Due to the uncertainty of the crack appearance time and the differences in the propagation speed, the lifetime of the parts, and even of the steel specimen prepared for the purpose of laboratory fatigue testing, show large deviations. Assume that a large number of test specimen are loaded with the same repetitive load that causes fracture (σa=const.). frequency [piece] σa=const. N [number of repetitions] Tengelyek | GÉPELEMEK 1. előadás 10 Statistical background of fatigue Gépelemek 1. In such cases the most commonly used is the Gaussian distribution, called the normal density function: 𝑦= where s standard deviation...(of distribution) μ mean value ...(of distribution) 1 𝑠 2𝜋 𝑥−μ 2 − ⋅ 𝑒 2𝑠2 Integral of the above function is proportional with the probability of fracture, called the normal distribution function: probability of fracture 1,0 0,5 0,0 N10 N50 N90 N [pieces] Number of repetitions related with the N10, 50, 90 a 10, 50 és 90 % probability of failure. Tengelyek | GÉPELEMEK 1. előadás 11 Statistical background of fatigue Gépelemek 1. August Wöhler (1819 –1914) was a German railway engineer, best remembered for his systematic investigations of metal fatigue of train shafts & axles. 1858 Tengelyek | GÉPELEMEK 1. előadás 12 Statistical background of fatigue (Wöhler-curve) Gépelemek 1. The figure below shows the stress amplitudes acting in the fracture crosssection in the case of pure oscillating stress as a function of the number of load cycles associated with a (10% probability of failure as EU practice), according to the St50 DIN standard. [MPa] a = low cyrcle fatigue, b = finite life fatigue, c = high cyrcle fatigue. a b c 700 600 500 400 300 200 N 100 1 10 2 10 4 10 6 10 8 Tengelyek | GÉPELEMEK 1. 10 10 [cycles] előadás 13 Statistical background of fatigue (Wöhler.curve) Remarks: Gépelemek 1. 1. The presented curve is the so-called Wöhler curve. Wöhler performed fatigue tests on railway car axles in 1866. 2. The low-cycle section, which starts from the yield point, is not used in mechanical engineering practice. In this section, we are calculating for static load case. 3. The fatigue limit (σv) is mainly a steel specific number. Other metals do not really have a so-called fatigue limit. The characteristic of the fatigue limit is that at the stress level below it, the steel components have a practically unlimited service life. 4. As a result of these we introduce 2 ways of calculation: − Calculation for (fatigue limit) endurance limit − Calculation for lifetime Tengelyek | GÉPELEMEK 1. előadás 14 Checking for endurance limit (guidelines) Gépelemek 1. • In the case of steels, at stress levels below the horizontal (c) section of Wöhler curve, fracture does not occur even after practically unlimited cycles. • When calculating to endurance limit, we check whether the applied stress is lower than the endurance limit. • The load case on a machine can be approximated by a sinusoidal function taken between the highest and lowest repetitive stress values. • The dominant cause of fatigue is that stress oscillates, and the fatigue limit is usually specified as an oscillation amplitude. • The biggest problem in calculation is determining the endurance limit values & characteristics. Tengelyek | GÉPELEMEK 1. előadás 15 Checking for endurance limit (Haigh diagram) Gépelemek 1. The fatigue limit of Wöhler curves taken at different mean stresses (at 10% probability of fracture) is represented on the below figure with the mean stress (σm ) on the horizontal axis and the amplitude stress (σa) on the vertical axis, we get the Haigh diagram. The area enclosed by the two straight tick lines is called the fatigue safety zone of the material!!! Safety zone Tengelyek | GÉPELEMEK 1. előadás 16 Checking for endurance limit (Smith-diagram) Gépelemek 1. By depicting the fatigue limit of Wöhler curves taken at different mean stress values (at 10% probability of fracture) in a common diagram whose horizontal axis is the mean stress (σm) and the vertical axis is the sum of the mean stress and the amplitude stress (σa), we get the Smith-diagram. Smith diagram and the construction of the approximate Smith chart according to the recommendation of the German engineering association VDI Tengelyek | GÉPELEMEK 1. előadás 17 Endurance limit for machine elements I. Gépelemek 1. The fatigue limit is reduced by: the absolute size of element, the change of surface roughness & the cross-section. where b1 size factor 1. Size factor: b1 b2 surface roughness factor βk inhibit factor Mére tténye ző b 1 v  = , v b1b2 k v 1,20 Carbon szénacélsteel 1,00 ötvözött acél Alloy steel 0,80 0,60 0,40 0,20 0,00 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 d [mm] Tengelyek | GÉPELEMEK 1. előadás 18 Endurance limit for machine elements II. Gépelemek 1. 2. Surface roughness factor: b2 On vertical axis surface machining prosesses are depicted. Tengelyek | GÉPELEMEK 1. előadás 19 Endurance limit for machine elements III. Gépelemek 1. 3. Inhibit factor: βk The fatigue-reducing effect of the stress contrentration points is taken into account with the inhibition factor:  k  1 +  k ( k − 1) where k shape factor k sensitivity factor the ηk sensitivity factor (smaller than 1) is dependent upon the tensile strenght of the material. Sensitivity factors: Material: carbon steel annealed steel spring steel light metals ηk 0,4 … 0,8 0,6 … 0,9 0,9 … 1,0 0,3 … 0,6 Tengelyek | GÉPELEMEK 1. előadás 20 Shape factor (load concentration factor) Gépelemek 1. Explanation of shape factor:  max k = n The factor is dependent on the size, the geometrical shape, but not on the material. Tengelyek | GÉPELEMEK 1. előadás 21 Shape factor of a shaft shoulder Gépelemek 1. Shape factor of a shaft shoulder design for bending Alaktényező hajlításra  hajl d/ D 4,00 3,50 3,00 0,90 0,80 0,70 2,50 0,60 0,5 2,00 0,4 1,50 1,00 0,10 1,00 10,00 r/t Tengelyek | GÉPELEMEK 1. előadás 22 Shape factor of a shaft shoulder Gépelemek 1. Shape factor of a shaft shoulder design for twisting Ala kténye ző csa va rásra  csav 2,40 2,20 d/ D 2,00 0,90 0,80 0,70 0,60 0,5 0,4 1,80 1,60 1,40 1,20 1,00 0,10 1,00 10,00 r/ t Tengelyek | GÉPELEMEK 1. előadás 23 Sensitivity factor (theoretical derivation) Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 24 Inhibit factor (theoretical derivation) Gépelemek 1.  tényl =  átl +  tényl  tényl =  k  max  max =  átl +  max  max =  k átl Tengelyek | GÉPELEMEK 1. előadás 25 Inhibit factor (calculation) Gépelemek 1. Calculation of the inhibition factor from the shape factor and the tensile strength in the case of a shaft shoulder: For BENDING:  hajl b1 For TWISTING:  csav b1   154     Rm   1,2 hajl 1 − 1  . + 0,1r  Rm   1 +   1370   154     Rm   1,15 csav 1 − 1  + 0,1r  Rm   . 1 +   1370 where r-in [mm], Rm –in [MPa] –to substitute Tengelyek | GÉPELEMEK 1. előadás 26 Inhibit factor (from standard tables) Gépelemek 1. Inhibit factor values for keeways With készült vertical cutter ujjmaróval reteszhorony Rm [MPa] 400 500 600 800 1000 1300 hajl csav 1,6 1,7 1,9 2,2 2,4 2,6 1,4 1,5 1,6 1,7 1,8 1,9 tárcsamaróval készült reteszhorony With horizontal cutter Rm [MPa] 400 500 600 800 1000 1300 hajl csav 1,25 1,35 1,5 1,7 1,9 2,1 1,4 1,5 1,6 1,7 1,8 1,9 Tengelyek | GÉPELEMEK 1. előadás 27 Inhibit factor (from standard tables) Gépelemek 1. Inhibit factor values for hub & shaft with interference fit Szorosan agy hengeres kúpos tengelyen Hub &illesztett shaft (taper shaft) withvagy interference fit Rm [MPa] 500-800 hajl csav retesz nélkül 1.7~2.1 1.3~1.7 hajl csav retesszel 2.3~2.5 1.4~1.8 Szorítógyûrûkkel felerõsített agy Hub & shaft with clamping rings Rm 500-600 hajl ~1.5 csav ~1.15 Tengelyek | GÉPELEMEK 1. előadás 28 Inhibit factor (from standard tables) Gépelemek 1. Inhibit factor values for hub & shaft with spline & gear joints Bordás Splinedtengely shaft Rm [MPa] 400 500 600 700 800 900 1000 1200 Spline profile Egyenes profil hajl 1,35 1,45 1,55 1,6 1,65 1,7 1,72 1,75  csav egyenes 2,1 2,25 2,36 2,45 2,55 2,65 2,7 2,8 evolvens 1,4 1,43 1,46 1,49 1,52 1,55 1,58 1,6 Evolvensprofile profil Involute Tengelyek | GÉPELEMEK 1. előadás 29 Construction of the safety zone for machine elements Gépelemek 1. HAIGH diagram Safety factor interpretation Safety zone interpretation if m = const. n= NA NM if m/a = const. n= OB OM Tengelyek | GÉPELEMEK 1. előadás 30 Construction of the safety zone for machine elements Gépelemek 1. SMITH diagram Safety zone interpretation Safety factor interpretation Tengelyek | GÉPELEMEK 1. előadás 31 Checking for complex (bi-directional) state of stress Gépelemek 1. The acting four uses: m & a as well as m & a . The reduced middle stresses:  mr =  + a  2 m where  mr = 2 2 F m h ReH aF = cs ReH  m2 aF2 +  m2 The reduced amplitude stresses:  ar =  a2 + av2 a2 where  ar =  vh av = cs v  a2 av2 +  a2 Tengelyek | GÉPELEMEK 1. előadás 32 Checking for complex (bi-directional) state of stress Gépelemek 1. The plotted Mohr diagram: Interpretation of safety factors: if a = const. n= EA EM if τa =const. n= DC MM if a/τa = const. Tengelyek | GÉPELEMEK 1. OB n= OM előadás 33 How to reduce the stress concentration Gépelemek 1. It can be reduced by... - Local dimension increase; - Form good shape transitions - Shading - Causing residual compression stress Tengelyek | GÉPELEMEK 1. előadás 34 How to reduce the stress concentration Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 35 How to reduce the stress concentration Gépelemek 1. reduction possibilities of drilled holes Tengelyek | GÉPELEMEK 1. előadás 36 Railway axles in motion (fatigue stresses) Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 37 Calculation for lifetime Gépelemek 1. The first step is to determine how the stress changes in time acting in the dangerous cross-section. This is usually not considered a regular, constant amplitude load, but rather an irregular curve that can be squeezed between two limits. Load versus time can not be expressed with exact equations. t Tengelyek | GÉPELEMEK 1. előadás 38 Calculation for lifetime Gépelemek 1. The load case diagram can be broken down into oscillations with constant amplitudes, and for an expected lifetime, it can be determined how many stress numbers n1, n2,...ni correspond to the oscillation amplitudes 1, 2, ...i, 1, n 1 2, n 2 3, n 3 We brake down the load case versus time into sections. Tengelyek | GÉPELEMEK 1. előadás 39 Calculation for lifetime Gépelemek 1. To mark & project to the finite life fatigue section of the Wöhler-curve: the expected usage numbers of the recorded load spectrum: ni 1, n 1 2, n 2 Finite life fatigue section 3, n 3 N 1 10 2 10 4 10 6 N1 N2 10 8 10 10 N3 According to Palmgren-Miner principle of linear cumulative damage, no fracture occurs during the recorded time if the following inequality exists: n1 n 2 n 3 + + 1 N1 N 2 N 3 Tengelyek | GÉPELEMEK 1. előadás 40 Calculation for lifetime (FKM) Gépelemek 1. A Forschungskuratorium Maschinenbau (FKM) Published in 1998, revised and supplemented in 2003 and 2010 „Rechnerischer Festigkeitsnachweis für Maschinenbauteile” (calculated yield stress for machine elements) VDI Guidelines, gives, a unified approach in calculation into the hands of engineers, in terms of strength calculation and in fatigue checking The FKM calculation guidelines were compiled based on the former TGL and former DIN standards as well as other source materials, supported by a large number of experimental and scientific studies, which were further developed based on the latest theoretical and experimental known materials. Tengelyek | GÉPELEMEK 1. előadás 41 Calculation for lifetime (FKM) Gépelemek 1. FKM engineering guidelines:  provides a uniform view in calculation of structural elements for static strength and fatigue.  the load case which determines the state of stress can be determined analytically as nominal stress or numerically as local stress. The numerical method can be a finite element method or a boundary element method. The stress determined by measurements can also be used as the local stress.  the calculation procedure is equally suitable for solving 1D (rodshaped), 2D (surface-like) and 3D (volumetric) problems.  the procedure can be used for elements made of steel, stainless steel, cast steel and cast iron, as well as light metal structural materials, between -40 − +500 degrees.  can be used for machined or non-machined parts or even welded parts Tengelyek | GÉPELEMEK 1. előadás 42 Calculation for lifetime (FKM) Gépelemek 1.  the calculation for fatigue does not require knowledge of the endurance limit characteristics defined for each material and mode of stress used in traditional calculation methods (oscillating-pulsating strength, Smith or Haigh diagrams)  the calculation to FKM refers to a 97.5% survival probability According to the FKM, the utilization factor (a) proves the adequacy of the tested component, either in the case of static or fatigue calculation. The utilization factor is the quotient of the stress state () and the limit state modified by the safety factor (R) a=  R / jerf 1 (1) where jerf is a safety factor depending on the structure and the mode of the use. The highest value of the utilisation factor can be 1. Tengelyek | GÉPELEMEK 1. előadás 43 Calculation for lifetime (FKM) Gépelemek 1. If the factor of utilization is greater than 1 (or 100%), then the component fails to meet static stress or fatigue conditions. The FKM calculation guidelines are essentially calculation algorithms built uniformly for different application cases, which consist of rules, calculation procedures, relationships and factors used for calculation There is a block diagram for calculating the components for static stress and fatigue which can be found in the document. The calculation steps of the static strength check and the fatigue check are the same. These diagrams show that for calculating, the stress state and the endurance limit state, both of them must be determined. Tengelyek | GÉPELEMEK 1. előadás 44 The Wöhler-curve (SWL Synthetische Wöhlerlinie) Gépelemek 1. Load (stress) Low cycle fatigue section Finite life fatigue section High cycle fatigue section Number of cycles ♦ low cycle or plastic: the cross section is exposed to elastic & plastic deformations ♦ finite life section: the cross section is exposed to elastic deformation , but at notches area plastic deformation is also possible ♦ high cycle section: the cross section is exposed to elastic deformation (infinite life) Tengelyek | GÉPELEMEK 1. előadás 45 SWL (Synthetische Wöhlerlinie) Gépelemek 1. The task is: to detrmine the special points of the curve: (ND,SD); (NP,SP); (1,SB); k. SB: tensile strenght SP: yield strenght NP: yield strenght cycle number SD: endurance limit stress ND: endurance limit cycle number k: power of the Wöhler-curve (gradient of the finite life section) Calculation & checking is at 50 % possibility of survival. Tengelyek | GÉPELEMEK 1. előadás 46