Failure analysis into root fillet region of spline · the AGMA standard is more equation based,...

MEC3098 Individual Project

Failure analysis into root fillet

region of spline

Date:

15/05/2017

Author:

Murray Wankling

Academic Supervisors:

Mr Tom Reavie, Mr Giorge Koulin

II | P a g e M u r r a y W a n k l i n g Newcastle University

Declaration

This Report is submitted as part of the requirements for the Degree of Mechanical

and Low Carbon Transport Engineering at the University of Newcastle upon Tyne

and has not been submitted for any other degree at this or any other University. It is

solely the work of Murray Wankling except where acknowledged in the text or the

Acknowledgements below. It describes work carried out at the University of

Newcastle upon Tyne which is entirely recorded in a Project Logbook which has

been made available for examination. I am aware of the penalties for plagiarism,

fabrication and unacknowledged syndication and declare that this Report is free of

these.

Dated: 15/05/2017

Signed:

Acknowledgements

Thanks to my supervisors Tom Reavie and Giorge Koulin, Design Unit Newcastle

University, who provided direction and assistance.

III | P a g e M u r r a y W a n k l i n g Newcastle University

Abstract

This project analyses a spline shaft which has failed under certain loads, determining

whether the root fillet radius was the reason for failure using analytical and FEA models.

Various standards and papers are compared for the observed failure, using relations

from a study by Yoshitake and equation from the standard by SAE. Followed by four

FEA models of the spline using different numbers of teeth. Observable relations

between the number of teeth and shear stress for both analytical and FEA methods

conclude that further testing is required to identify the effect of the number of teeth on

the failure stress of a spline. However, all models show that the root fillet was the

cause of failure when the root fillet radius was under 0.45mm. Confirming the failure

of the spline was due to the increase in stress at the root fillet.

Nomenclature

Rf Root fillet radius (mm)

Dre Minor diameter (mm)

Kt Stress concentration factor (SCF)

T Torque (Nm)

Ss Shear stress (MPa)

Se Endurance limit (MPa)

σnom Nominal stress (MPa)

σmax Maximum stress (MPa)

IV | P a g e M u r r a y W a n k l i n g Newcastle University

Table of Contents

1.0 Introduction ...................................................................................................................................... 1

1.1 Standards review .......................................................................................................................... 1

1.2 Dimensions and application .......................................................................................................... 2

1.3 Material ......................................................................................................................................... 4

1.4 Aims and objectives ...................................................................................................................... 4

2.0 Theoretical analysis of root fillet ...................................................................................................... 5

2.1 Stress Concentration Factor .......................................................................................................... 5

2.1.1 SCF literature review .............................................................................................................. 5

2.2 Shaft with shoulder SCF predictions ............................................................................................. 6

2.3 Current root fillet failure methodology’s ...................................................................................... 7

2.3.1 Gear teeth .............................................................................................................................. 8

2.3.2 Spline teeth ............................................................................................................................ 9

2.4 Root fillet failure predictions ...................................................................................................... 11

2.5 Conclusion of Analytical methodology’s ..................................................................................... 12

3.0 Preliminary finite element analysis ................................................................................................. 13

3.1 Shaft with shoulder ..................................................................................................................... 13

3.2 Results, comparison, discussion and conclusion ........................................................................ 15

4.0 Spline model ................................................................................................................................... 16

4.1 Meshing ....................................................................................................................................... 17

4.2 Loading and measurement ......................................................................................................... 19

5.0 Results and observations ................................................................................................................ 20

5.1 SCF for FEA models ..................................................................................................................... 20

5.2 SCF for FEA and Analytical .......................................................................................................... 21

5.3 Maximum stress with failed spline ............................................................................................. 22

6.0 Discussion ........................................................................................................................................ 25

6.1 FEA models .................................................................................................................................. 25

6.2 FEA and Analytical ....................................................................................................................... 25

6.3 Models with failed spline ............................................................................................................ 27

7.0 Conclusion ....................................................................................................................................... 27

8.0 References ...................................................................................................................................... 28

9.0 Appendix ......................................................................................................................................... 29

9.1 Appendix A .................................................................................................................................. 29

9.2 Appendix B .................................................................................................................................. 30

1 | P a g e M u r r a y W a n k l i n g Newcastle University

1.0 Introduction

An involute external spline designed to that of the BS 3550 [1] standard failed in shear.

The shear failure was consistent with a fatigue failure stemming from a crack in the

root fillet region where the hob run out joins the minor diameter of the shaft.

The root fillet is the cause of failure and its geometric impact on the performance of

the spline is the investigation of this report. Similar studies have shown that fillet radii

of a larger radius fail at a higher stress [2]. Different modelling techniques using finite

element analysis (FEA) enable comparisons between the models and analytical

predictions. This results in an increased understanding of spline modelling and

highlights areas of improvements to analytical methods. Similar studies show a good

agreement between FEA and photoelastic experimentation [3], and more recently it

has been shown to be more accurate than analytical methods [4].

A preliminary case of a shaft with a shoulder and fillet radius will be used to test and

verify the methods that will be used to analyse the spline, and provide a better

understanding of modelling techniques.

1.1 Standards review

The relevant spline standards take very different approaches to arriving at the

dimensions for a working spline. Dudley’s paper creates a spline design largely based

on application [5], which is then scrutinised in a later paper on stress control against

theoretical stress equations for: shaft stresses, shear stresses in teeth, compressive

stresses and bursting stresses. This paper forms the basis of many spline design

methods, Equation 1 is from the paper and is used to estimate the shear stress in the

shaft [6], and is relevant to this investigation.

𝜎𝑛𝑜𝑚 =16T

πDre3 (1)

BS 3550 creates a very basic design, stating where the tolerance stack ups will occur,

introducing a clearance classification system and then provides all the dimensions for

a working spline [1]. It also produces an extensive section for how to analyse the spline

to make sure it complies with the tolerances needed. Although it doesn’t provide

backing for the tables provided, with no equations for stresses as in Dudley.


This leads to the conclusion that to understand spline design, using the Dudley

standard is better, but to measure a spline and ensure it’s at a satisfactory tolerance,

the BS 3550 standard is better by far. Most designers would probably use a

combination of the two. The Dudley standard may be used in the preliminary stages

of design to ensure the spline will not fail in accordance with its application, then the

BS 3550 standard to be confident that with modern manufacturing methods the spline

will perform as designed.

Another notable standard is by SAE, which provides a mix of tables and equations.

But this standard has limited detail and is evidently suited to splines of higher numbers

of teeth, with diameter recommendation of up to 99 teeth. As an automotive standard,

this is no surprise.

The unique quality of this standard is the inclusion of an equation for the stress

increase at the root fillet region at various radii [7].

Further standards such as the DIN 5480 [8] and AGMA [9] also provide methods for

spline design, DIN 5480 follows a similar vein to that of the BS 3550 standard, whereas

the AGMA standard is more equation based, quoting equations by Dudley for shear

stress capacities.

As there is variation between the standards it can be confusing when designing a

spline and identifying the point of failure, leading to a misunderstanding of certain

parameters and early failure.

This leads to the equations from Dudley being of primary use, along with the equation

for the stress increase at the root fillet by SAE.

1.2 Dimensions and application

The spline differs from the BS 3550 standard in several dimensions. Mainly that of the

internal spline which is not subject to investigation, but also the major diameter of the

external spline [1]. It is not known why the spline differs from the standard, but the

discrepancies will largely be ignored as the dimensions that differ are not obviously

related to the cause of failure. One dimension not provided was the root fillet radius of

the spline. As this is the dimension under investigation this value will vary, though

direct comparisons will be made against the value given in the standard.


A full set of dimensions can be found in 9.1 Appendix A. All dimensions given were

in Imperial units, that of the BS 3550 standard. However, this project will use metric

units and therefore dimensions have been converted, Table 1. Also provided were

the torque, speed and carrier cycles, Table 2.

Table 1: Dimensions to understand analysis

Table 2: Torque and speed values given

This provides some information about the operating conditions of the spline, and while

the number of carrier cycles is known. It is not known how many of these were torque

starts, nor the torque at which it failed. Therefore all torques will be considered.

The application of the spline is not known, however observations about the design of

the spline can lead to potential applications that may be useful when analysing the

failure.

The spline is made to a flat root side fit, with contact on the side of the tooth and a

clearance at both the major and minor diameter, consistent with applications of high

misalignment and slow rotational speed [5]. The pressure angle of the spline is 30°

which combined with the involute profile of the tooth supports an automotive

Key dimensions Metric (mm) Imperial (in)

Minor shaft diameter Dre 52.7812 2.078

Root fillet radius Rf 0.508 0.02

Material EN26W

Tensile strength 940MPa

Torque (Nm) Speed (rpm) Carrier cycles

1500 350 1500000

2000 200 160000

3500 120 40000

6400 60 20000


application. This could be useful for ascertaining factors such as contact wear and

misalignment.

1.3 Material

The shear fatigue failure is consistent with an oscillating stress [10], a repeated

opening and closing of a crack leading to the shear of the shaft [11]. This is most likely

caused by bending at the end of the shaft, matching a working condition of high

misalignment [11]. This would also agree with the failure occurring where the hob run

out starts, as there would be a change in stiffness at that point resulting in a localised

area of increased stress. However, this occurred after a crack initiated, so bending at

the end of shaft will not be considered in this investigation, though it was a significant

factor leading to the failure of the shaft.

Material EN26W, as stated in Table 1 has tensile strength 940MPa [12]. As the model

of the spline will assume pure torsion, the shear strength accurately represents the

failure stress due to torsion. Distortion-energy theory places the shear stress at 0.577

of the tensile strength [13]. Thus 542.38MPa will be used as the shear stress, with the

endurance limit being between 262.2MPa and 317.4MPa [14].

1.4 Aims and objectives

The aim of this paper was to use analytical and FEA methods to determine the

localised increase in stress at the root fillet region. In addition to this, comparing the

FEA models with one another will lead to an increased understanding of spline

modelling. Also, comparison of FEA models with analytical methods allows for

analysis of the validity of these methods and helps to ascertain the best practise.

Finally, this paper compares all methods to the dimensions of the failed spline and

the primary reason for failure.


2.0 Theoretical analysis of root fillet

2.1 Stress Concentration Factor

The stress in a structure is usually approximated by simple calculations, however at a

sudden geometry change there will be a localised area of stress increase, unique to

that geometry [14] such as a root fillet. The stress increase is defined by a

dimensionless quantity, the stress concentration factor (SCF), and in the elastic range

it is defined as [14]:

𝐾𝑡 =𝜎𝑚𝑎𝑥

𝜎𝑛𝑜𝑚 (2)

That is, the maximum stress at a sudden geometry change, over the nominal stress

of the body. This was confirmed by Neuber [15] using Hookian stress for sharply

curved notches.

SCF are used in some design methods of gears and splines. ISO and AGMA both use

forms of SCF in their design of gear teeth [16], as do SAE for splines [7]. All three

concerning the stress increase at the root fillet, and therefore applicable to the failure

of the spline discussed in this paper.

2.1.1 SCF literature review

The Lewis equation [17], which is a widely used analysis method, viewed the tooth

as a cantilever beam and, introduced the Lewis form factor to account for the

bending stress at the root fillet region. It is still used today for gear design in the

AGMA standard for gears it was developed further by Dolan and Broghamer [18],

who found that the principle factors influencing stress concentration at the fillet were:

radius of the fillet, thickness of the tooth at base of tooth, height of load position and

pressure angle [18]. The paper concluded with two equations for a SCF for two

pressure angles, neither of which were 30°. However, it states the geometric impact

will be similar [18], and that it is a better approximation than the Lewis equation for

maximum stress which was previously used. The paper showed an increase of

stress in the tension region of the root fillet as the fillet radius was decreased,

congruent to the study in 1.0 [2, 18].


Similar to Dolan and Broghamer, Yoshitake studied the effect of root fillet size using

photoelastic experiments, resulting in a correlation of SCF against root fillet over shaft

major diameter [19]. The test was completed using Equation 1, with 8 teeth and

specified tooth height and width. As it hasn’t been analysed outside its stated

parameters, similar to Dolan and Broghamer earlier it will most likely not be directly

comparable, though the geometric impact will be comparable.

SAE provide an equation for the SCF at the root fillet. It is not known how this has

been ascertained and seemingly does not have any boundaries for which the relation

is invalid outside of, though this is unrealistic. The SCF is determined by the root fillet

radius and the tooth height [7], providing a direct comparison with the FEA.

SCF can be calculated from these various papers and standards and used as a

comparison to the failed spline, determining the accuracy of the papers and standards.

2.2 Shaft with shoulder SCF predictions

The preliminary model of a shaft with a shoulder used to verify the testing methods

and accuracy of the model will also be compared with SCF from analytical methods.

This is due to the shoulder similar to the root fillet, being a sudden geometry change.

Roark’s [14] provides a SCF for this scenario using ratio h/r which has experimental

boundaries from 0.25 to 4 [14]. The SCF for the preliminary model in torsion is:

𝐾𝑡 = 𝐶1 + 𝐶2 (2ℎ

𝐷) + 𝐶3 (

2ℎ

𝐷)2

+ 𝐶4 (2ℎ

𝐷)3

(3)

𝐶1 = 0.953 + 0.680√(ℎ/𝑟) − 0.053ℎ/𝑟

𝐶2 = −0.493 − 1.820√(ℎ/𝑟) + 0.517ℎ/𝑟

𝐶3 = 1.621 + 0.908√(ℎ/𝑟) − 0.529ℎ/𝑟

𝐶4 = −1.081 + 0.232√(ℎ/𝑟) + 0.0665ℎ/𝑟

For 0.25 ≤ h/r ≤ 4.0

D =Outer shaft diameter (mm)

r =Fillet radius at shoulder (mm)

h =Shoulder height (mm)

D and h are constant and so Equation 3 rearranged for SCF against fillet radius is

plotted in Figure 1 with fillet radius 0.75mm – 6.00mm, D =50mm and h =12.5mm.


It shows a steady increase in gradient the smaller the radius. This is comparable with

the expected results and similar studies [3, 19], and will be compared to the FEA

model.

2.3 Current root fillet failure methodology’s

Theoretical analysis of splines generally considers either pure torsion, pure bending

or pure tension, however this is seldom the case in practise. Gear teeth are considered

to be point loaded which includes bending [17], and studies have concluded that the

bending stress from a point load, contributes to half the force required for the tooth to

fail at the root fillet [18], and therefore it should be factored into the calculation of the

SCF. Hence forms of this are used in various standards [4], however the effects of

bending are commonly left out of spline standards, instead focusing solely on the

torsion of the shaft [11].

This could be due to spline teeth having a larger contact area than gear teeth, and

therefore the bending effects seen in gear teeth are reduced. However, this is curious

as spline teeth are expected to deflect over time to increase the contact surface area

[20]. This is because the entire load of the shaft will first be on one tooth due to

manufacturing variations, with more teeth engaging over time as they deflect in turn.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0.5 1.5 2.5 3.5 4.5 5.5 6.5

SCF

fillet radius(mm)

Fillet radius effect on SCF

Figure 1: Stress concentration factor variation as fillet radius decreases


This enables more teeth to come into contact, thereby sharing the load [21]. This

deflection process would seem to be accounted for by standards generally only

considering half the teeth of a spline to be in contact. Thus the overall load capacity of

the spline is reduced through this deflection process [21], and hence why some

standards have factors of a half in equations for the shear stress in teeth, for example

Dudley and AGMA assume only half the teeth are in contact [6, 9].

This therefore points to gear methodologies considering bending to be of greater

influence to the failure stress of a tooth than spline methods do, potentially being a

gross misunderstanding.

2.3.1 Gear teeth

ISO and AGMA geometric stress increase factors are as follows.

ISO considers complex stress state at tooth root and stress concentration caused by

the fillet [16]:

𝑌𝑆 = (1.2 + 0.13𝐿)𝑞𝑠𝑎 (4)

L =SFn/hFe

qs =SFn/(2ρF)

a =[1.21+2.3/L]-1

AGMA considers complex stress state at tooth root and stress concentration caused

by the fillet [16]:

𝐾𝑓 = 𝐻 + (𝑆𝐹𝑛

𝜌𝐹)𝐿

(𝑆𝐹𝑛

ℎ𝐹𝑒)𝑀

(5)

H =0.331−0.436αn

L =0.324−0.492αn

M =0.261+0.54 αn

SFn =Tooth thickness at critical section (mm)

hFe =Bending moment arm (mm)

ρF =Radius of curvature of fillet curve (at critical point) (mm)

an = Pressure angle (rad)

Ys, Kf = Geometric stress increase factor

As expected both equations consider the fillet radius of the tooth. Along with the

thickness of the critical section and the distance to the bending arm of the point load.


Therefore, this is not useful for this investigation as it considers bending as well as

torsion. Furthermore, the bending arm length would have to be estimated, as it is not

known and could introduce further uncertainties.

2.3.2 Spline teeth

SAE and Yoshitake SCF are as follows.

SAE states an equation for the SCF of the root fillet region [7],

𝐾𝑡 = 1 +1

2√

ℎ

𝑅𝑓 (6)

h =spline tooth whole depth

= ½ (major dia – minor dia)

= ½ (59.6189-52.7812) = 3.41885mm.

Mentioned above is also the SCF factor by Yoshitake [19].

𝐾𝑡 = 6.083 − 14.775 (10𝑅𝑓

𝐷𝑟𝑒) + 18.250 (

10𝑅𝑓

𝐷𝑟𝑒)2

(7)

Valid for 0.01 ≤ Rf/Dre ≤ 0.04

In contrast to the gear methodologies above, the critical section and the bending arm

are not used in the calculation of the SCF for splines. Confirming earlier statements

0

2

4

6

8

10

0 0.01 0.02 0.03 0.04 0.05

Str

ess c

oncentr

ation f

acto

r

Rf/Dre

Analytical SCF variation

SAE

Yoshitake

Figure 2: Theoretical stress concentration factor variation against radius size:


of spline methods not considering bending to be of significant impact. This leads to

the conclusion that equations for SCF of splines only consider pure shear as no

bending factors are included. This could be a misunderstanding of bending effects

on spline teeth as already mentioned.

For ease of comparison the equation from the SAE standard has been adapted to a

format similar to the equation used by Yoshitake, this allows for a comparable variation

of the root fillet radius, as shown in Figure 2.

Table 3 shows the result of the above equations using dimension from Table 1.

For a root fillet radius of 0.508mm taken from the BS3550 standard, SAE and

Yoshitake have quite different SCF, with Yoshitake roughly twice as high as SAE as

seen in Table 3.

Note the Rf/Dre ratio in Table 3 is out of the bounds stated by Yoshitake by

0.00375mm. As this is an extremely small variation, the Yoshitake relation will still be

compared.

Table 3: SCF using SAE and Yoshitake at Rf value 0.508mm

Rf/Dre SCF

Yoshitake SAE

0.009625 4.830016 2.322067


2.4 Root fillet failure predictions

Table 4 summarises the nominal stresses at each torque using torques from

Table 2, relation in Equation 1 and dimensions from Table 1:

Table 4: Nominal stress applied at each torque

Taking the above nominal stress values, SCF from Table 3 and relation for maximum

stress, Equation 2, stress predictions at the four torques are outlined in

Table 5.

Table 5: Stress predictions using SCF from SAE and Yoshitake

As noted earlier the number of torque starts is not known and so the percentage of life

used up in accordance with Miner’s law is not known [10]. Therefore, to give a rounded

view of what will happen, using the torques provided the maximum stress will be tested

against the shear stress for EN26W and the Endurance limit. Predicting one of three

scenarios:

T σnom

Torque (Nm) Carrier Cycles Nominal shaft stress (MPa)

1500 1500000 51.95

2000 160000 69.27

3500 40000 121.23

6400 20000 221.67

T σnom

Kt (SCF)

σmax

Ss Se

Torque

(Nm)

Nominal

stress

(MPa)

SAE Yoshitake SAE

(MPa)

Yoshitake

(MPa)

Shear stress Endurance limit

SAE Yoshitake SAE Yoshitake

1500 51.95 2.32 4.83 120.64 250.94 Pass Pass Pass Pass

2000 69.27 2.32 4.83 160.86 334.59 Pass Pass Pass Fail

3500 121.23 2.32 4.83 281.50 585.53 Pass Fail Fail Fail

6400 221.67 2.32 4.83 514.74 1070.68 Pass Fail Fail Fail


1. Immediate failure

2. Fatigue failure during lifetime

3. Theoretical infinite use

Hence the two limits in Table 5 are the endurance limit and shear strength of EN26W.

These limits ignore any environmental effects, such as stress corrosion cracking which

could lead to an earlier failure [10].

The endurance limit fail at 3500Nm by SAE (Table 5) is because the endurance limit

bound is between 262.2MPa and 317.4MPa. At that point it exceeded the lower bound

but not the upper bound, suggesting that it may have infinite use, though it will be

thought to have an extended life for simplicity in paper.

2.5 Conclusion of Analytical methodology’s

The results from Table 5 would indicate that according to SAE the spline is built so it

will not immediately fail under any of the torques provided. But it does have a set

design life as at the higher torques the endurance limit is exceeded, agreeing with

observed fatigue failure. In contrast Yoshitake predicts an immediate failure at the

higher torques of 3500Nm and 6400Nm, with most other torques exceeding the

endurance limit. Indicating that even without an immediate failure the design life of the

spline is very short.

Considering the failure observed in 1.0 was that of a fatigue shear failure and not an

immediate failure, the analytical methods should predict that. Exceeding the

endurance limit but not the shear failure limit. However, as stated above and seen in

Table 5 Yoshitake disagrees.

This leads to the conclusion that predictions by SAE are more accurate to that of

Yoshitake, as was expected due to the experimental nature of Yoshitake’s prediction.


3.0 Preliminary finite element analysis

The FEA models have the potential to be more accurate than the analytical methods

[3]. For the two case scenarios presented, shaft with shoulder and failed spline,

corresponding FEA models have been constructed.

3.1 Shaft with shoulder

To give grounding to the FEA of the spline, a shaft of simpler geometry was analysed

first as a preliminary model. For this model a torsional load was applied at the visible

face of the shaft in Figure 3, torqueing the shaft at the nominal load being resisted by

the opposing end, the fixed boundary. As the nominal load was applied, the SCF

equals the maximum stress, which was recorded at the fillet region for fillet radii

between 0.75mm and 6.00mm, as in 2.2. The torque applied was taken from a

rearrangement of Equation 1, where the nominal stress is resulting in a torque of

3067.96Nm.

High density mesh. Element size = 0.3125mm

Fixed boundary condition

Low density mesh. Element size = 5mm

Moment applied on face

Figure 3: Mesh density map of shaft with shoulder annotated with mesh and boundary conditions


The main cause for error between analytical and computational modelling is the mesh

refinement that is used to determine the force path in the computational model [3].

Therefore, to ensure an accurate model, areas of high stress have been refined so

that mesh independence is reached. Areas of low stress have a coarse mesh so that

computational time is kept to a minimum and is as effective as possible. Meshing

methods used included face sizing and edge sizing as Figure 3 demonstrates.

The smallest radius of 0.75mm was used initially because if mesh independence was

attained for the smallest radius, then it would be suitable for all radii. Suitable

reductions in mesh size were modelled and as seen in Figure 4, the smaller the mesh

size the smaller the percentage variation in SCF. A mesh size of 1.25mm and

0.625mm varied by 6%, which was reduced to 1% at mesh size of 0.3125mm. The

variation is small enough to show convergence and mesh independence.

Figure 4: mesh conversion with element size

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

0.1 1 10

SC

F

mesh size (mm)

Mesh convergence for radius 0.75mm.

6%

1%

6%


3.2 Results, comparison, discussion and conclusion

A mesh density of 0.3125mm was used and suitable fillet radii from 0.75mm to 6.0mm

were tested. The results were combined with the results from the analytical method

(Figure 1), as shown in Figure 5.

In Figure 5 a consistent response is seen between the two methods. With the FEA

model showing a reduced SCF at smaller radii compared to the analytical method, but

it also showed the results were within 5% for radii over 2mm.

The difference of over 5% for radii less than 2mm between the FEA and analytical

methods could be due to the mesh not being fine enough for independence, and that

the 1% variation is too large for radii below 2mm. This could be improved in the next

model by aiming for mesh convergence of 0.5%.

However, the difference is more likely due to the limits of the analytical method. The

upper limit of 4 for the h/r ratio in 2.2 was exceeded by 0.17 for fillet radii of 3mm. This

is where the models begin to diverge in Figure 5 and this is more likely the cause.

The concurrence between the FEA model and the analytical method is consistent

enough to validate an FEA model concerning the root fillet radius of the failed spline.

Though improvements of a smaller mesh to achieve a convergence of 0.5% and a

better understanding of theoretical limits from experimental data.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0.5 1.5 2.5 3.5 4.5 5.5 6.5

SC

F

Fillet radius (mm)

Fillet radius effect on SCF

Analytical

FEA

Figure 5: Analytical and FEA stress concentration against fillet radius for shaft with shoulder


4.0 Spline model

A similar method to the shaft with a shoulder is applied to the second case, the failed

spline. However, due to its complex geometry additional analysis steps have been

included. Though using dimensions from Table 1 and 9.1 Appendix A. This resulted in

4 models.

Model A: A single tooth on a shaft

Model B: Three teeth on a shaft

Model C: Three teeth and the hob run out

Model D: The entire spline

Figure 6: Models A, B, C, D. With separate high density mesh body in red

Model C

Model A Model B

Model D

Moment applied on

face

Moment applied on

face


4.1 Meshing

The increase in geometric complexity makes meshing more difficult, therefore the root

fillet region has been modelled as a separate body, bonded to the rest of the object.

This is so a finer mesh can be applied, highlighted red in Figure 6.

Similar to the shaft with shoulder in 3.1, mesh independence must be known, so using

model A and various fillet radii the element size of the cut out region was reduced till

the percentage variation was suitably small. Using conclusions from 3.2 a percentage

variation 0.5% was desired.

4 radii were used to map out the mesh density needed: 1.5mm, 0.75mm, 0.5mm and

0.25mm. 0.25mm was the most influential radius as the mesh independent element

size for this radii will also be true for all other radii. However, for efficient computational

time, the other three points were calculated.

Figure 7 displays the variation in SCF for each of the 4 radii cases, starting at element

size 5mm, reducing till a variation below 0.5% is observed.

There is a significant variation of SCF for each radii at different element sizes, thus

confirming the significance of achieving mesh independence. This has led to the

following mesh element size for each radii in Table 6. Visual display of mesh density

differences for Rf =1.5mm in Figure 7.

Figure 7: Mesh convergence for 4 fillet radii

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

0.01 0.1 1 10

SC

F

Element size (mm)

SCF Mesh independance

r =0.25mm

r =0.5mm

r =0.75mm

r =1.5mm


Table 6: Element size for each fillet radius

Rf (mm) Element size (mm)

1.50 – 0.80 0.625

0.75 – 0.55 0.3125

0.50 – 0.25 0.15625

Model A and B have a quadrilateral mesh with a bias on the cut-out region to be finer

in the centre away from edge effects [22]. Models C and D use a triangular mesh due

to the inclusion of the hob run out which effected the cut out region so there was no

longer a sweep-able shape, therefore no quadrilaterals could be used, similarly no

bias is included.

Figure 7: Models A, B, C, D with mesh for Rf =1.5mm

Model A

Model D

Model B

Model C


4.2 Loading and measurement

The nominal load, calculated using Equation 1, taking dimensions from Table 1 gives

nominal torque of 41608Nm to be applied at the visible face with a fixed support on

the face directed away as in Figure 6.

All models were tested for Rf of 0.25mm to 1.5mm at 0.05mm intervals, measuring the

maximum stress at the centre point of each model denoted by the arrow in Figure 8.

Figure 8 and 9 display the measurement of maximum stress using Model D as an

example, though applicable to all models. Figure 9 displays a closer stress map of the

root fillet region. The blackline shows the point at which the maximum stress is present

within the root fillet region. Being near the centre of the separate body, it confirms that

the cut out body encapsulates the maximum stress region of the root fillet region.

Figure 9: Closer view of centre line from Model D, showing maximum stress distribution within region

Figure 8: Model D with centre line for stress measurement between 1 and 2


5.0 Results and observations

Models A, B, C and D ran over the course of two weeks, varying in computational time

of 5 mins per point to 30 mins per point. Figure 10-15 display the results, Figure 10

and 11 show the SCF for models A-D, with Figure 12-15 factoring in the torques

applied. Full table of FEA values in 9.2 Appendix B.

5.1 SCF for FEA models

Observations:

Expected general increase in SCF as Rf decreases.

Correlations between Model A and D, and B and C.

Model A and B are very consistent. Model C and D less so, with a larger spread

of data, however at times a consistent spread with each other.

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

SC

F

Rf (mm)

FEA of SCF for Models A-D

Model A

Model B

Model C

Model D

Figure 10: FEA SCF for each Model


5.2 SCF for FEA and Analytical

Figure 11 displays FEA methods and variant of Figure 2.

Observations:

FEA same general trend as SAE and Yoshitake

SAE underestimates by 18%

Yoshitake significantly overestimates SCF by 68%

FEA better approximated by SAE

Slight convergence with FEA and SAE at smaller Rf

Slight divergence with FEA and Yoshitake at smaller Rf

Figure 12-15 compare all methods as in Figure 11 for each Torque. Applying relation

for maximum stress Equation 2, multiplying nominal stress in Table 4 by SCF in Figure

11. Including the lower of the two endurance limits, 262.2MPa and shear stress limit,

542.38MPa to draw comparisons as in 2.4.

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

SC

F

Rf (mm)

SCF comparison between Analytical methods and FEA Models

Model A

Model B

Model C

Model D

SAE

Yoshitake

Figure 11: SCF FEA and Analytical


5.3 Maximum stress with failed spline

Figure 12 shows that none of the methods pass the threshold for the Endurance limit

except from Yoshitake at Rf =0.4mm. Figure 13 also shows the same though at a

different limit for Yoshitake, Rf =1.1mm.

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

σm

ax

(MP

a)

Rf (mm)

Max stress at torque 1500Nm

Model A

Model B

Model C

Model D

SAE

Yoshitake

Ss

Se

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

σm

ax (

MP

a)

Rf (mm)


Model A

Model B

Model C

Model D

SAE

Yoshitake

Ss

Se

Figure 12: Maximum stress for each model at torque 1500Nm



Figure 14 shows that analytical and FEA models exceed the endurance limit for all

values, except SAE for Rf above 0.65mm. With Yoshitake predicting immediate shear

failure for Rf below 0.65mm.

0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

σm

ax (

MP

a)

Rf (mm)


Model A

Model B

Model C

Model D

SAE

Yoshitake

Ss

Se

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

σm

ax (

MP

a)

Rf (mm)


Model A

Model B

Model C

Model D

SAE

Yoshitake

Ss

Se




Figure 15 shows that every model exceeds the shear stress limit, SAE at 0.45mm and

the FEA models between 0.75mm and 1.25mm.

To conclude observations from Figure 12-15:

At torque 1500Nm all methods are below shear and endurance limit except

from Yoshitake which exceeds the endurance limit at Rf =0.4mm.

At torque 2000Nm all methods are below the shear and endurance limit

except from Yoshitake which exceeds the endurance limit at larger radius

Rf =1.1mm.

At torque 3500Nm all FEA models and Yoshitake exceed the endurance

limit for all radii tested, with SAE exceeding this limit below Rf =0.65mm.

Yoshitake also exceeds the shear limit at the same Rf.

At torque 6400Nm all methods exceed the endurance limit for all radii tested.

Yoshitake predicts immediate failure for all Rf values, SAE predicts

immediate failure below 0.45mm. With FEA models predicting immediate

failure around 0.75mm to 1.25mm and under.


6.0 Discussion

6.1 FEA models

Models A-D all follow the expected trend of an increase in SCF as Rf decreases.

Model A and D were similar in SCF and trend. With and average difference of 2%

and B and C being similar, with and average difference of 1.5%. Models B and C

both used geometries involving 3 teeth, therefore the effect of a tooth either side on

the stiffness of the model could explain a concurrence in results, this may also

explain the discrepancy to models A and D, as there was a difference of 8% between

the two averages.

Models A and D were also similar in SCF and trend. If the SCF does change with the

number of teeth then model A should have a larger SCF than any other model, with

model D being in its right position. However, model A does not follow this trend,

suggesting instead that it would be a good approximation of model D. Further

research to confirm that it is a good approximation and not a coincidence, as it goes

against similar research, would be required [23].

The change in spread between models A and B, and C and D could be due to several

reasons. The similar troughs and peaks displayed clearest in Figure 10 suggest a

cause consistent either with a change in geometry or mesh. The increase in spread

as triangular mesh is used, suggests a larger distortion of the force path compared to

the quadrilateral mesh in A and B. Reducing and therefore invalidating the element

size found to be mesh independence. The other reason could be due to the

introduction of the hob run out, altering the placement of maximum stress to outside

the region measured. However, this does not explain the troughs and peaks in the

data, as a reduced spread of data would be observed. Hence the difference in mesh

is the most likely cause. As model D is of the entire spline with models A-C building

up to it using different geometries, the most reliable data is that from model D.

Therefore comparisons with analytical data will be compared to model D.

6.2 FEA and Analytical

Using Model D as a direct comparison to analytical methods, Figure 16 (a

simplification of Figure 11) displays the two analytical methods and Model D. The FEA


models followed the same trends as that seen by Analytical methods. An explanation

for Yoshitake overestimating by 68% and SAE underestimating by 18% could be due

to the number of teeth analysed as discussed in 2.1.1. Yoshitake is based on a

photoelastic test of 8 teeth, with SAE being an automotive standard where a high

number of splines is commonly used. To examine whether the number of teeth effects

the SCF as hypothesised, further study into this would be required. FEA models of 8

teeth and many teeth to study the correlation with SAE and Yoshitake would be

suggested. An additional factor would be the relation between Rf over Dre, this was

stated in Yoshitake’s experiment and could be a better way to measure the SCF in

future studies. A better approximation of the FEA models is by SAE. This could be a

hint as to the number of teeth used in automotive applications, near to 18. Also the

magnitudes of the FEA models also hints that the relation between number of teeth

and SCF is not linear otherwise the models would be more spread apart. Alternatively

this could be as mentioned above, the linear relation between number of teeth and

SCF using a variable of Rf over Dre instead. The slight convergence and divergence

of SAE and Yoshitake is possibly due to the limits of the analytical methods. As seen

in 4.2 a divergence of results was observed when the limits of the analytical methods

was exceeded. For the smallest value we know already that limits for Yoshitake have

been exceeded. Therefore the convergence with SAE could also be due to the limits

of Equation 6, which are not known, being exceeded.

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

SC

F

Rf (mm)

SCF comparison between Analytical methods and Model D

Model D

SAE

Yoshitake

Figure 16: SCF analytical and model D


6.3 Models with failed spline

Comparing the analytical and FEA results to the dimensions and torques given for

the spline indicates that the Yoshitake prediction is unrealistic, as the number of

carrier cycles indicates that an immediate shear fail was not observed. Consequently

Yoshitake should only be used to ascertain a similar geometric impact, which it did

conform to. For Rf =0.508mm, suggested in the BS3550 standard, all the FEA

models predict an immediate shear fail with SAE predicting an immediate shear fail

at an Rf 0.05mm smaller. Therefore, assuming the shear stress limit is accurate, for

the largest torque of 6400Nm the BS3550 should not be used as would and did

result in failure. For radii smaller than 0.45mm it cannot be expected for the spline to

be suitable for use as for all methods an immediate shear failure is predicted at

torque of 6400Nm.

7.0 Conclusion

FEA of splines should use ordered quadrilateral meshes where possible, and a model

of the entire spline as best practise. As errors for models of one tooth are unknown

and there is a variation to that of 3 teeth which further modelling, specifically looking

at the stiffness change against number of teeth, would be needed. For a spline with

18 teeth it would be unwise to use the relation from Yoshitake as this significantly

overestimates the SCF in the root fillet region. Further testing to observe the effect of

reducing the number of spline teeth and comparing once again to Yoshitake’s relation

would be needed. It would also suggest the need to include the number of teeth in

future calculations for SCF, that or the relation of Rf to Dre. SAE predicts similar failure

stresses to the FEA models, though as it underestimates for all models it could lead

to an early failure. Similarly to Yoshitake it may be valid for spline teeth of more than

18 teeth, therefore further testing would be required to ascertain the number of teeth

for which it is valid for. For the failed spline it would seem sensible to conclude that the

reason for failure was due to the stress concentration increase at the root fillet region.

For methods both analytical and FEA based the spline would fail immediately in shear

at torque 6400Nm for Rf smaller than 0.45mm. However it would be a sensible choice

for torques of 1500Nm and 2000Nm, as neither the shear nor the endurance limit are

exceeded. With a torque of 3500Nm best matching up with the failure observed,

exceeding the endurance limit but not the shear, resulting in a fatigue failure.


8.0 References

1. British Standard, Specification For Involute Splines. British Standards Institution, 1963. 2. Mattingly. J, P.B., Stress concentration free spline for high torque twin screw power

transmission. 3. Wilcox, L. and W. Coleman, Application of Finite Elements to the Analysis of Gear Tooth

Stresses. Journal of Engineering for Industry, 1973. 95(4): p. 1139. 4. Lisle, T.J., B.A. Shaw, and R.C. Frazer, External spur gear root bending stress: A comparison of

ISO 6336: 2006, AGMA 2101-D04, ANSYS finite element analysis and strain gauge techniques. Mechanism and Machine Theory, 2017. 111: p. 1-9.

5. Dudley, D.W., how to design Involute Splines. 1957. 6. Dudley, D.W., When Splines Need Stress Control. 1957. 7. Chaplin, R.W.C.M.R., Design Guide for Involute Splines. Society of Automotive Engineers,

1994. 8. DIN, -. Involute splines based in reference diameters - Part 1: General. 2006. 9. AGMA, -.-B., Design Manual for Enclosed Epicyclic Gear Drives. American Gear

Manufacturers Association, 2006. 10. A, C., Fatigue and fatigue life. Newcastle University, 2017. 11. Volfson, B.P., Stress Sources and Critical Stress Combinations for Splined Shaft. Journal of

Mechanical Design-Transactions of the Asme, 1982. 104(3): p. 551-556. 12. Steel, W.Y., EN26W Alloy Steel. 2017. 13. Budynas, R.G., J.K. Nisbett, and J.E. Shigley, Shigley's mechanical engineering design. 2011,

New York: McGraw-Hill. 14. Warren C. Young, R.G.B., Roark's Formulas For Stress and Strain. 1989(Seventh ). 15. Neuber, H., Theory of Stress Concentration for Shear-Strained Prismatical Bodies With

Arbitrary Nonlinear Stress-Strain Law. Journal of Applied Mechanics, 1961. 28(4): p. 544. 16. Kawalec, A., J. Wiktor, and D. Ceglarek, Comparative Analysis of Tooth-Root Strength Using

ISO and AGMA Standards in Spur and Helical Gears With FEM-based Verification. Journal of Mechanical Design, 2006. 128(5): p. 1141.

17. Lewis, W., Investigation of the strength of gear teeth. Proceeding of the Engineers Club of Philadelphia, 1892.

18. Thomas J. Dolan, E.L.B., A photoelastic study of stresses in gear tooth fillets. University of Illinois Engineering Experiment Station Bulletin Series No. 355, 1942.

19. Peterson, R.E., Stress Concentration Factors. 1974. 20. Hong, J., D. Talbot, and A. Kahraman, A semi-analytical load distribution model for side-fit

involute splines. Mechanism and Machine Theory, 2014. 76: p. 39-55. 21. Chase, K.W., C.D. Sorensen, and B.J.K. DeCaires, Variation Analysis of Tooth Engagement and

Loads in Involute Splines. Ieee Transactions on Automation Science and Engineering, 2010. 7(4): p. 746-754.

22. Besnard, G., F. Hild, and S. Roux, “Finite-Element” Displacement Fields Analysis from Digital Images: Application to Portevin–Le Châtelier Bands. Experimental Mechanics, 2006. 46(6): p. 789-803.

23. Celik, M., Comparison of three teeth and whole body models in spur gear analysis. Mechanism and Machine Theory, 1999. 34(8): p. 1227-1235.


9.0 Appendix

9.1 Appendix A

Dimensions given of the failed spline, both metric and imperial along with the

relevant table from standard BS 3550 and discrepancies between the two.

Involute spline data to BS 3550 Table 10 Class 2 fit for a flat root side fit.

in inch in mm BS 35500 (mm) Table 10

Discrepancy (mm)

Number of splines 18 18 18 0

Pitch 8/16 16.93

Pressure Angle 30 30

Pitch Circle Diameter

2.25 57.15 57.15 0

Base Diameter 1.9486 49.49444 49.49444 0

Form Diameter

Int 2.1205 53.8607 59.7027

Ext 53.8607

Major Diameter

Int 2.3522 59.74588 60.325 -0.57912

Ext 2.3472 59.61888 59.5884 0.03048

Minor Diameter

Int 2.096 53.2384 53.975 -0.7366

Ext 2.078 52.7812 53.2384 -0.4572

Measurement across pins

Int 2.6169 66.46926

Ext 2.6149 66.41846

Pin Diameter 0.24 6.096

Length of Engagement

2.625 66.675

Hob Run out 3 76.2

Circular Spline Width

Min effective 0.1949 4.95046 4.98602 -0.03556

Max effective 0.1963 4.98602 4.98602 0

Min actual 0.1931 4.90474 4.90474 0

Max actual 0.1945 4.9403 5.0673 -0.127

Fillet radius Int 0.6858

Ext 0.508


9.2 Appendix B

The results from FEA models and Analytical methods for each radius.

Rf Model

mm A B C D SAE Yoshitake

0.25 3.3798 3.5858 3.5860 3.2700 2.8461 5.4241

0.30 3.1561 3.3549 3.4228 3.1539 2.7010 5.3022

0.35 3.0529 3.2420 3.3263 2.9210 2.5841 5.1835

0.40 2.9606 3.1474 3.2046 2.9529 2.4869 5.0681

0.45 2.8269 3.0529 3.0524 2.8534 2.4043 4.9560

0.50 2.7726 2.9532 3.0085 2.7759 2.3327 4.8471

0.55 2.7263 2.9135 2.9722 2.7114 2.2697 4.7416

0.60 2.6685 2.8609 2.9208 2.6874 2.2137 4.6393

0.65 2.6213 2.8148 2.8821 2.5866 2.1635 4.5402

0.70 2.5575 2.7689 2.8270 2.6360 2.1179 4.4445

0.75 2.5216 2.7278 2.7243 2.5346 2.0764 4.3520

0.80 2.4822 2.6128 2.5858 2.3782 2.0383 4.2628

0.85 2.4489 2.5801 2.4822 2.3673 2.0032 4.1769

0.90 2.4092 2.5510 2.5666 2.3289 1.9706 4.0943

0.95 2.3742 2.5224 2.5536 2.2878 1.9402 4.0149

1.00 2.3531 2.4948 2.4800 2.2942 1.9119 3.9388

1.05 2.3217 2.4714 2.4735 2.2757 1.8853 3.8660

1.10 2.3033 2.4442 2.4674 2.2335 1.8603 3.7964

1.15 2.2985 2.4753 2.5562 2.2758 1.8367 3.7302

1.20 2.2761 2.4391 2.5392 2.2849 1.8144 3.6672

1.25 2.2658 2.4293 2.4362 2.2359 1.7933 3.6075

1.30 2.2274 2.4324 2.4805 2.2989 1.7732 3.5510

1.35 2.2134 2.4152 2.4660 2.3003 1.7541 3.4979

1.40 2.1925 2.4008 2.4536 2.2674 1.7359 3.4480

1.45 2.1874 2.3915 2.4458 2.2366 1.7185 3.4014

1.50 2.1754 2.3755 2.3990 2.2393 1.7018 3.3580

Failure analysis into root fillet region of spline · the AGMA standard is more equation based,...

Documents

Transcript of Failure analysis into root fillet region of spline · the AGMA standard is more equation based,...