calculation sheet - altervista · calculation sheet subject job rev. date sheet fat2 0 2012/01 3 of...

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


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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


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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:


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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


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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.


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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


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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


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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.


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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


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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.


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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

calculation sheet - altervista · calculation sheet subject job rev. date sheet fat2 0 2012/01 3 of...

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