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Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 1 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
Index: 1. Abstract...................................................................................................................................2 2. Description of the problem.......................................................................................................2 3. Endurance limit ....................................................................................................................... 3 4. FE model and loads ................................................................................................................4 5. Stress field correction..............................................................................................................5 6. Preliminary stress study ..........................................................................................................7 7. Equivalent stresses .................................................................................................................9 8. Fatigue verifications ..............................................................................................................10 9. References............................................................................................................................ 11
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 2 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
1. Abstract
The aim of this job is to conduct a fatigue analysis using FEA software to relieve the stress.
2. Description of the problem
A shaft is subjected to a steady torsion and steady forces due to the spur gears loads. The shaft is supported by two rolling bearings. The figure below shows the shaft dimensions and the shaft loads.
fig. 2-1: Forces due to spur gears and reactions
fig. 2-2: Shaft geometry
Steady torsion is equal to 360 Nm. The reactions, due to the rolling bearing, are been calculated
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 3 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
considering the shaft like a beam. The material is C35 EN 10083-2: 2006 normalized.
3. Endurance limit
For the material specified above, the endurance limit, the yield strength and the ultimate tensile strength are:
e
1e m2 2 2
N N NS 245 ; R 275 ; R 490 mm mm mm
Considering the surface obtained from turning machine, the surface factor is equal to: ak 0.93
fig. 3-1: Surface factor
The size factor, for bending and torsion, is equal to: bk 0.82
fig. 3-2: Dimensional factor
The others modification factors can be taken 1. The endurance limit is so equal to:
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 4 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
1e a b c d e e 2
NS k k k k k S 186.8 mm
4. FE model and loads
The Fe model are depicted on figure below. The mesh has been refined on notched zones. The others pictures show the load applied to the model. Note that the forces due to the spur gears and to the reactions of rolling bearings have been modeled with Gencoz distribution.
fig. 4-1: Forces applied to the model
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 5 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
fig. 4-2: Shaft FE model for stress on groves and fillets
Note that the shaft geometry has been simplified and the keyways will be study separately. Once checked that the body is on equilibrium under the load applied have been clamped only two points on the shaft axis. So the shaft has a minimal supports. So we have two cases to resolve for fatigue life verification: CASE 1: Steady torsion mT 360 Nm CASE 2: Steady bending moment separately for the shaft and for the keyways. Obviously the steady bending moment generates an alternate stress because of the shaft rotation. So the combination for the fatigue life determination is: COMBO 1: von Mises stress of combination of Midrange shear stress due to torsion (not alternate component) plus Alternate direct stress due to bending moment
5. Stress field correction
The notch sensitivity factor can be obtained without the nominal stress determination using the Siebel and Stieler theory.
f
t ss
K 1K C
where Css is the Siebel-Stieler material constant and χ is the relative stress gradient which is equal to:
Shoulder fillets: 2 4r D d
for bending; 1 4r D d
for torsion
Groove: 2 2r d
for bending; 1 2r D d
for torsion
The relative stress gradient for the notched parts of the shaft are indicated on the figure below.
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 6 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
fig. 5-1: χ factor for relevant notches
The material constant is obtained from a chart which furnish the ratio Kt/Kf.
fig. 5-2: Kt/Kf.
For the stress field correction we need the ratio Kf/Kt because the FE analysis furnish the stress amplified by Kt. Next table summarizes the data.
χ Action Kt/Kf Kf/Kt 2.11 2.05 2.07
bending 1.18 0.85
0.54 bending 1.06 0.94 0.29 torsion 1.04 0.96
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 7 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
6. Preliminary stress study
Once applied the loads defined at the chapter 4 it’s possible to determinate the stress field and to locate the critical sections.
fig. 6-1: Tangential stress due to torsion
fig. 6-2: Maximum principal stress due to bending
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 8 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
fig. 6-3: Critical points
For the keyways the unfavorable position is for that one placed at the right on fig. 2.2. For this reason it was build a simplified model without a one keyway and with mesh refined only around the right keyway.
fig. 6-4: FE model for keyway stress study
From this study is possible to know where is located the critical point for the keyway. Figures 6.5 and 6.6 show the maximum principal and the shear stress on the keyway due respectively to bending moment and to torsion. Because of the shaft rotation the tension stress due to bending becomes a compression stress. For the keyway the Siebel-Stieler theory doesn’t furnish the relative stress gradient so we can assume prudently Kf/Kt = 1.
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 9 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
fig. 6-5: Maximum principal stress due to bending (tension)
fig. 6-6: Maximum shear stress due to torsion
7. Equivalent stresses
It’s now possible to reassume, for the three critical points, the alternate and medium normal and shear stresses
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 10 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
Point Normal alternate
component σa
Normal midrange component
σm
Shear alternate component
τa
Shear midrange component
τm
A 209.50 0 0 0 B 82.90 0 0 55.86
C keyway 136.80 0 0 55.86 Now is necessary to correct the stresses because of the notch sensitivity ratio.
Point Normal alternate
component σa
Normal midrange component
σm
Shear alternate component
τa
Shear midrange component
τm
A 178.01 0 0 0 B 77.93 0 0 53.62
C keyway 136.80 0 0 55.86 Combining by von Mises formula the equivalent stresses at points are reassumed on next table.
Point Equivalent alternate component σa,eq
Equivalent midrange component σm,eq
A 178.01 0 B 77.83 92.87
C keyway 136.80 96.75
8. Fatigue verifications
Using Sodeberg and ASME criteria for infinite life verifications is obtained:
Point
Sodeberg criterion a,eq m,eq
e e
1S R
ASME criterion 2 2
a,eq m,eq
e e
1S R
A 0.953 0.908 B 0.754 0.287 C 1.084 0.660
The Sodeberg criterion is too safely but also for ASME criterion the point A has a very low factor of safety. So it’s necessary to modify the shaft geometry or choose a more resistant material.
Calculation sheet Subject Job Rev. Date Sheet
FAT2 0 2012/01 11 of 11
ING. ANDREA STARNINI
Fatigue analysis using FEA software
Compiled by ing. Andrea Starnini
fig. 8-1: Infinite life verifications
9. References
Shigley’s Mechanical Engineering Design – McGraw Hill - 8th Edition Yung-Li Lee et al: Fatigue testing and analysis – Elsevier 2005 M. Rossetto: Introduzione alla fatica dei materiali e dei componenti meccanici – Levrotto & Bella 2000