Master's Degree Thesis

ISRN: HK/R-IMA-EX--1997/D-01--SE

Static Characteristics ofFlexible Bellows

Madeleine Hermann

Anders Jönsson

Department of Mechanical Engineering

University of Karlskrona/Ronneby

Karlskrona, Sweden

1997

Supervisor: Göran Broman, Ph.D. Mech.Eng.

Static Characteristics ofFlexible Bellows

Madeleine Hermann

Anders Jönsson

Department of Mechanical Engineering

University of Karlskrona/Ronneby

Karlskrona, Sweden

1997

Thesis submitted for completion of Master of Science in MechanicalEngineering with emphasis on Structural Mechanics at the Department ofMechanical Engineering, University of Karlskrona/Ronneby, Karlskrona,Sweden.

Abstract:

Theoretical expressions for stiffness and maximum stress were determinedfor axial, bending and torsion load by using a combination of analytical,statistical and finite element calculations. Experimental verification showedvery good agreement.

Keywords:

Static characteristics, Flexible, Bellows, Stiffness, Stress, Experimentalverification.

2

3

Acknowledgement

This work was carried out at the Department of Mechanical Engineering,University of Karlskrona/Ronneby, Sweden, under the supervision of Dr.Göran Broman.

The work was initiated in 1996 as a co-operation project between AP PartsTorsmaskiner Technical Center AB and the Department of MechanicalEngineering at the University of Karlskrona/Ronneby.

The purpose of the project was to strengthen the knowledge about flexiblebellows at AP Parts Torsmaskiner Technical Center AB. Financial supportwas given by Blekinge Research Foundation, which is gratefullyacknowledged.

We wish to express our sincere appreciation to Dr. Göran Broman for hishelpful advice and guidance throughout the work. We also thank Dr. StefanÖstholm, head of department, for valuable discussion and support.

We also gratefully acknowledge the support from AP Parts TorsmaskinerTechnical Center AB, and especially the Research & DevelopmentManager, M.Sc. Kristian Althini who always took time to help us.

We finally want to express our gratitude to Mikael and Emma for theirpatience and support, which made this work possible to accomplish.

Madeleine Hermann

Anders Jönsson

4

Contents

1. Notation 5

2. Introduction 8

3. Basic Relations and Limitations 103.1 Bellow Geometry 103.2 Loads 11

4. Theoretical models 124.1 Axial Load 12

4.1.1 Model I: Restrained Radial Displacement 144.1.2 Model II: Free Radial Displacement 224.1.3 Finite Element Model 244.1.4 Corrected Model 27

4.2 Bending Load 314.3 Torsion Load 36

4.3.1 Flank Calculations 364.3.2 Top and Bottom Element Calculations 404.3.3 Summary of Torsion Load 41

5. Experimental Verification 445.1 Axial Load Verification 45

5.1.1 Experimental Set-up 455.1.2 Results 46

5.2 Bending Load Verification 485.2.1 Experimental Set-up 485.2.2 Results 49

5.3 Torsion Load Verification 515.3.1 Experimental Set-up 515.3.2 Results 52

6. Conclusions 53

7. References 54

1. Notation

a Radius

B Constant

C Constant

c Constant

D Constant

d Diameter

E Young’s modulus

e Displacement

f Flank distance

G Shear modulus

I Area moment of inertia

J Polar area moment of inertia

k Stiffness

L Length

M Bending moment

N Force

n Number

P Force

T Torque

R Correction function

r Radial co-ordinate

s Thickness

z Axial co-ordinate

x Co-ordinate

AB Edge

6

BC Edge

CD Edge

AD Edge

α Coefficient of thermal expansion

β Angle

ε Normal strain

γ Shear strain

ν Poisson’s ratio

σ Normal stress

τ Shear stress

θ Bending angle

ϕ Circumferential co-ordinate

φ Stress function

Indicesb Bottom

f Flank

i Inner

k Number, stiffness

m Mean

r Radial direction

t Top

o Outer

z Axial direction

Y Yield

AD Edge

corr Corrected

7

FEM Finite Element Method

max Maximum

min Minimum

ϕ Circumferential direction

θ Bending angle

8

2. Introduction

When the emission demands on personal cars became higher in the late80´s, the automotive industry had to find a flexible and gas tight connectionbetween the engine and the exhaust system. This kind of element has beenused for a long time in marine engine installations and in the processingindustry. However, when implemented into cars, the operating conditionswere different from what the elements where designed for. This lead tofailure during normal operation of the cars.

The flexible elements are operating in a wide temperature range, between-40 and 900 °C, and the forces acting on the elements are complex due tothe engine and the exhaust system movements.

The element is connected to the engine via the branch pipe, which has littledamping effect. The large engine movements are mainly due to inertialforces in the engine and shifting of gear.

There are also vibrations with smaller amplitudes transmitted from theengine. Reduction of the transmitted engine movements and vibrations tothe exhaust system are important characteristics of the flexible element.

The other end of the element is directly connected to the exhaust system.The exhaust system can give rise to large movements due to the flexibleconnection to the car chassis. Road vibrations can be transmitted via theexhaust system into the flexible element.

The manufacturing is also complex, which can lead to initial stresses in theelement. See figure 2.1 for some examples of different element designs.

9

Figure 2.1. Examples of different flexible elements.

The gas tight bellow in the flexible element has a critical function. Cracksin the bellow are a common failure reason. General descriptions of bellowcharacteristics have not been found in the literature.

The aim of this work is to develop theoretical models of a typical flexiblebellow under static axial-, bending- and torsion load.

The models will contain essential design parameters so that their influenceon bellow characteristics can be studied. An experimental verification of themodels will be carried out.

The theoretical models will probably also be useful for analysis of dynamiccharacteristics of flexible bellows.

10

3. Basic Relations and Limitations

3.1 Bellow Geometry

Although there are many different variants of flexible elements on themarket, they all have the same basic design, see figure 3.1.

Figure 3.1. General flexible element design.

There is often some sort of heat protection, a so-called innerline, inside theelement. The next layer is the gas tight bellow and on the outside, there is abraid to protect the bellow from outer mechanical violence. The three partsare connected at the ends with end caps.

Figure 3.2. U-shaped bellow.

The flexible bellow consists of a number of convolutions, n, see figure 3.2.The bellow that will be studied is of U-type, which has a flank, f, that

braid gas tight bellow

innerline

end cap

L

r

z

one convolution

11

connects the inner and outer radii, a, see figure 3.3, where a polar co-ordinate system is also defined for future use. The length, L, is given by

naL 4= (3.1)

The radius, a, and the diameters, di and do, are distances to the middle of thematerial. The thickness, s, of the material is small compared to the otherdimensions and does not affect the geometry description.

Figure 3.3. Dimensions of one half-convolution.

3.2 Loads

The bellow is operating under complex conditions. Two main groups aremovement of the bellow boundaries and temperature changes. The effect oftemperature changes will not be analysed in this work.

The relative movement between the engine and the exhaust system givesrise to stresses in the bellow. During operation the bellow is subjected tocombined loads. In this work analyses will be made for three separate loads:axial, bending and torsion, see sections 4.1 through 4.3.

s

a

a

f

dido

Centerline of bellow

z

r

12

4. Theoretical models

4.1 Axial Load

In this section expressions for the stiffness and the stresses during axialloading will be described, see figure 4.1. Those expressions will contain thegeometry and material properties of the bellow.

The axial stiffness of the bellow is

kP

ezz

ztot

= (4.1)

It is assumed to be the same for push and pull loads.

Advantage is taken of the axi-symmetry and that all convolutions areidentical. To simplify the calculation, the bellow is sliced in the longitudinaldirection and treated as a plate with convolutions. The plate has the samelength as the bellow and the width is the average circumference, see figure4.2.

This approximation is probably acceptable when the difference between theinner and outer diameter is much smaller than the average diameter, dm =½(do+di).

P P

eztot

Figure 4.1. Axially loaded bellow.

13

Figure 4.2. Approximated bellow for axial loading.

Due to symmetry of the convolution itself it is only necessary to consider aquarter of one convolution. This approximation of the bellow is treated withtwo different boundary conditions.

In model I, section 4.1.1, radial displacement is completely restrained and inmodel II, section 4.1.2, radial displacement is completely free. The aim ofusing those two boundary conditions is to find the interval in which thestiffness of the real bellow must be, and to get a general idea of how thegeometry and material properties influence the stiffness of the bellow.

Model I will give a too high stiffness and model II will give a too lowstiffness. A bellow without any simplification of the geometry is solved byusing the Finite Element Method in section 4.1.3.

Lz

r

14

4.1.1 Model I: Restrained Radial Displacement

The freebody-diagram is shown in figure 4.3. The moment at the symmetry-line is zero.

It is assumed that the symmetry-line, i.e. the average diameter, and the

top/bottom of the convolution, i.e. the outer and inner diameter, has noradial displacement. This explains the presence of the force N. This must bethought of as an outer force in the model of figure 4.3, since symmetryotherwise demands it to be zero. In reality, the restraint radial displacementis because such displacement also implies a change of the circumference ofthe bellow.

By combining the elementary load cases in figure 4.4 and the correspondinggeometrical displacement in figure 4.5, an expression for the axial stiffnesscan be derived.

Pz

N

Mo,i

N

ez

Pz

symmetry-line

Figure 4.3. Freebody-diagram of ¼-convolution.

15

er1

ez1

a

β1

Pz

Pz

er4

ez4

0,5

f

er3

ez3

a

N

β3

M1

er2

ez2

a

β2

1 2

3 4

Figure 4.4. Elementary load cases.

er5

ez5

0,5

fβ1

er7

ez7

0,5

fβ3

er6

ez6

0,5

fβ2

5 6 7

Figure 4.5. Geometrical displacements.

16

The radial and longitudinal displacements in figure 4.4 and 4.5 are given inequations 4.2 through 4.15. Small displacement theory is assumed valid.The displacements for the elementary load cases can be found in books ofbeam theory formulas, for example [1]. The curvature, a, is large comparedtoo the thickness of the beam, s. The area moment of inertia for all curvedbeams is therefore the same as for a straight beam, i.e. Im.

e =P a

2EIr1z

3

m

−(4.2)

eM a

EIrm

21

2

21=

−− +

π π s

24a

2

2 (4.3)

eNa

EIrm

3

3 3

42=

−− +

π π s

24a

2

2 (4.4)

eP

EIrz

m4

17

180= (4.5)

ef f f

r5 112 2

21

4 4= − = = ⋅

( cos )β

β P a

EIz

2

m

(4.6)

ef f f M a

EI

s

arm

6 222

12

2

2

21

4 4 21

12= − = = +

( cos )β

β π(4.7)

ef f f Na

EI

s

arm

7 332 2 2

2

2

21

4 4 21

24= − = = − +

( cos )β

β π π(4.8)

e P a

EIzz

m1

3

4=

π(4.9)

eM a

EIzm

21

2

= (4.10)

eNa

EIzm

3

3

2= (4.11)

17

eP f

EIzz

m4

3

24= (4.12)

efP a

EIzz

m5 2

=2

(4.13)

ef M a

EI

s

azm

61

2

22 21

12= +

π(4.14)

ef Na

EI

s

azm

7

2 2

22 21

24= − +

π π(4.15)

where

M Pf

z1 2= (4.16)

and

Is d d

mo i=

+π 3

24

( )(4.17)

The restraint at the symmetry-line gives an equation for the normal force, N,as a function of Pz.

e N P C C P C P Crkk

z z z=

∑ = ⇒ = − − −1

7

1 12

22

30 ( ) (4.18)

C1, C2 and C3 are constants containing the geometry and the materialproperties of the bellow according to equations 4.19 through 4.21.

CEI

fa

s

a

s

a

m1

2

2

2

2

2

2

3

42

24

21

24

=− +

− +

π π

π π(4.19)

18

CEI

fa

a fas

a

s

a

m2 4

3 22

2

2

2

245

17 90 902

124

21

24

=− − − +

− +

π π

π π(4.20)

C

af s

a

as

a

3

22

2

2

22

2

2

41

12

21

24

=+ +

− +

π

π π(4.21)

Adding all displacement contributions in the z-direction and multiply by 4ngive the total z-displacement for a bellow with n convolutions

( )4 41

7

1 2n e e e n P D N Dzkk

ztot ztot z=

∑ = ⇒ = + (4.22)

D1 and D2 are constants containing the geometry and the material propertiesof the bellow according to equations 4.23 and 4.24.

D

a fa f f as

a

EIm1

3 2 3 22

26 24 3 112

24=

+ + + +

π π

(4.23)

Da

EIa f

s

am2

2 2

22 21

24= + − +

π π(4.24)

Equation 4.1, 4.18 and 4.22 give an expression for the axial stiffness

( )( )kP

n P D D C C P C P Cz

z

z z z

Ι =+ − − −4 1 2 1 1

22

23

(4.25)

It is clear that kzI is dependent on the size of the load, Pz, since N is notlinearly related to Pz.

This non-linear relation can be linearised by considering the materialproperties of the bellow and thereby limiting the interval of Pz. Themaximum stress in a convolution must be less than the yield strength, σY, ofthe material in the bellow. The maximum stress occurs at Mmax. The force,

19

N, and the shear stress are neglected at this point and Mmax occurs roughly atthe bottom and top of the convolution as

M Pf

azmax = +

2

(4.26)

With the minimum area moment of inertia

Is di

min =π 3

12(4.27)

the maximum bending stress becomes approximately

σmaxmax

min

=M

I

s

2(4.28)

Equation 4.26 through 4.28 and the material property σmax=σY give anapproximate maximum load of Pz. With σY = 400 MPa the maximum loadbecomes Pz=300 N.

Figure 4.6 shows the non-linear function

N P C C P C P Cz z z( ) = − − −1 12

22

3 (4.18)

and the approximate linear function

N*(Pz) = c�Pz (4.29)

The geometry and material properties for the convolution is:

a = 2.0 mm s = 0.25 mm

do = 87 mm f = 6.25 mm

di = 66 mm E = 210 GPa

The load Pz is limited to the interval 0-300 N and the constant, c, becomes-3.7.

20

The diagram shows that the non-linear function N(Pz) can be replaced witha linear function N*(Pz) = c� Pz with very good accuracy. The constant ofproportionality, c, is negative and therefore N is a compressive force forpushing loads and not a tensile force as in figure 4.3.

From equation 4.1, 4.22 and 4.29 the axial stiffness now becomes

kn D cDzΙ =

+1

4 1 2( )(4.30)

The moment distribution is obtained from equation 4.31 and 4.32 and isshown in figure 4.7.

M x P x xf

z( ) ,= ≤ ≤02

(4.31)

M x P x P c a a xf f

xf

az z( ) ,= + − − −

≤ ≤ +

2

2

2 2 2(4.32)

The maximum bending stress becomes

0 50 100 150 200 250 3001500

1000

500

0

N(Pz)N*(Pz)=cPz, c=-3.7

Pz [N]

N [

N]

Pz [N]N(Pz)N*(Pz) = cPz, c=-3.7

N [

N]

Figure 4.6. Linear approximation of N(Pz) to N*(Pz).

21

σmaxmax

Ι =M

I

s

m 2(4.33)

The magnitude and location of Mmax can be calculated by solving

dM

dx= 0 (4.34)

However, equation 4.34 is no simple expression. The expressions ofequation 4.31 and 4.32 are shown in figure 4.7.

Pz

cPz

M

x

f/2+a

f/2

Mm

ax

a

Mo,i

Pz Mo

,icPz

Figure 4.7. Moment distribution for ¼-convolution withgiven geometry and c=-3.7.

22

4.1.2 Model II: Free Radial Displacement

The freebody-diagram is shown in figure 4.8.

As in the previous model the moment at the symmetry-line is zero and thesymmetry-line, i.e. the average diameter, is not changing its position.However, in this model it is assumed that the top/bottom of theconvolution, i.e. the outer and inner diameter, is changing their positions inradial direction freely. This explains why there is no N-force in this model.

By combining the elementary load cases 1, 2 and 4 in figure 4.4 and thegeometrical displacement 5 and 6 in figure 4.5 an expression for the axialstiffness can be derived.

Adding all displacement contributions in the z-direction, from equations 4.9through 4.14, and multiplying by 4n give the total z-displacement for abellow with n convolutions.

4 41 2 4 5 6

1n e e e nP Dzkk

ztot ztot z=∑ = ⇒ =, , , ,

(4.35)

D1 is a constant containing the geometry and the material properties of thebellow according to equation 4.23 in section 4.1.1.

Pz

Mo,i

ez

Pz

symmetry-line

Figure 4.8. Freebody-diagram for ¼-convolution.

23

From equation 4.1 and 4.35 the axial stiffness now becomes

knDzΙΙ =1

4 1

(4.36)

Compared to the previous model, this model has a much simpler expressionfor the stiffness, due to the linear relation between the Pz-force and thedisplacement, eztot. The moment distribution is obtained from equation 4.37and is shown in figure 4.9.

M x P x xf

az( ) ,= ≤ ≤ +

0

2(4.37)

The maximum bending stress then becomes

σmaxmax

min minΙΙ = = +

M

I

s P

I

fa

sz

2 2 2(4.38)

or with

fa

d do i

2 4+

=

−(4.39)

Pz

M

x

f/2+a

f/2

Mm

ax

aMo,i

Pz

Figure 4.9. Moment distribution for ¼-convolution.

24

and equation 4.27 the maximum stress can be expressed as

( )σ

πmaxΙΙ =−3

2 2

P d d

s dz o i

i

(4.40)

4.1.3 Finite Element Model

One model of the bellow was solved numerically by using the FiniteElement Method (FEM). The FEM-module in I-DEAS Master Series 4 wasused for this calculation. The advantage of this model is that the realgeometry of the bellow can be used. The disadvantage is that the results isonly valid for one specific geometry and it is not possible to directly get ananalytic expression for the stiffness and stresses.

The FE-model will be used for verification of the analytical expressions forstiffness and stresses. In section 4.1.4, it will also be used for an estimationof correction functions for these analytical expressions.

The FE-model is shown in figure 4.10.

zϕ

10

A

B C

D

r

Figure 4.10. Finite Element model of ½-convolution.

25

When treating a ½-convolution restrictions on radial displacements shouldnot be set explicitly. In Model I radial displacement were completelyrestrained at the inner-, outer- and average diameter and in Model II radialdisplacement were completely restrained at the average diameter. In the FE-model the actual restrictions on radial displacements due to circumferentialstrain resistance are automatically imposed.

Since the bellow is axi-symmetric, it is possible to use axi-symmetricelements for the FE-model. However, in this model thin shell elements wereused together with boundary conditions imposing axi-symmetry.

In this model a slice of 10° were modelled with 300-400 eight-nodeisoparametric shell elements to give sufficient convergence.

The boundary conditions are as follows:

Edge Translation displacement Rotation displacement

r ϕ z r ϕ z

AB free constant free constant Free constant

BC free free* constant constant Constant constant

CD free constant free constant Free constant

AD free free* free constant Constant constant

*Those boundary conditions do not affect the model. The results will be the same with freeor constant translation displacement.

The load is applied at edge AD in the z-direction with the magnitude

P PAD z=°°

10

360(4.41)

and the stiffness for the FE-convolution is

kP

ezFEMz

zFEM

= (4.42)

Figure 4.11 shows the force-displacement characteristics for a ½-convolution according to the two analytic models (with or without radialdisplacement), the FE-model and the measured results for a real bellow.The slope of the curves are the stiffness for ½-convolution.

26

The stiffness of the FE-model corresponds very well to the measuredstiffness of the real bellow. Due to this agreement, it is probably correct toassume that the stresses in the FE-model are close to the stresses in the realbellow.

k zI=9

50 N

/mm

k zFEM=600 N

/mm

k zreal=620 N

/mm

kzII=430 N/mm

load

Pz

[N]

0 0.1 0.2 0.3 0.40

50

100

150

200

analytic model I, restrained radial displacementFE-modelreal bellowanalytical model II, free radial displacement

displacement 1/2 convolution (mm)

Figure 4.11. Force-displacement characteristic for ½-convolution, withgiven geometry: di=66 mm, do=87 mm, s=0.25 mm, a=2 mm.

27

4.1.4 Corrected Model

The analytical models describe the stiffness of the bellow in a rough way.This is shown in figure 4.11. The first model, kzI, has about 50 % higherstiffness than the real bellow and the second model, kzII, has about 30 %lower stiffness than the real bellow. Non of these discrepancies areacceptable. This implies that the analytical expressions for maximum stressare probably also too inaccurate.

It is possible to correct the analytical expressions with correction functions.The corrected expressions for the stiffness and stresses then become

k R a s d d kzcorr k i o z= ( , , , ) (4.43)

and

σ σσmax max( , , , )corr i oR a s d d= (4.44)

where the correction functions, Rk(a,s,di,do) and Rσ(a,s,di,do), are functionsof the geometry of the bellow. Those functions will be derived by usinglinear regression. For a discussion on this topic, see for example [2]. Theprocedure used to find a correction function is as follows:

1. Choose one analytic model to start from.

The analytical model II is chosen because of its simplicity, seeequation 4.36 and 4.40, which are repeated below.

knDzΙΙ =1

4 1

(4.36)

( )σ

πmaxΙΙ =−3

2 2

P d d

s dz o i

i

(4.40)

Limit the dimensions of the bellow.

The inner diameter is constant because it has to fit to the tubes of theexhaust system. The other dimensions varies as follows:

2.

di = 66 mm

do = 80 - 90 mm

a = 1.0 - 3.0 mm

s = 0.20 - 0.30 mm

28

3. Find the real stiffness and stresses of a number of bellows withinthe limited dimensions.

The real stiffness and stresses of the bellow are calculated with theFE-model, because it is very close to the real bellow. Using the FEMinstead of measuring the stiffness and stresses of real bellows reducescosts and are more time efficient. Problems with inaccuracy in themeasurement and quality variation from manufacturing are alsoavoided.

25 calculations on ½-convolutions within the limited dimensions weredone. A force of 10 N was applied to the ½-convolution. Thedisplacement and the maximum von Mises stresses were listed foreach calculation.

4. Calculate the relation between the FEM and the analytical resultsof stiffness and stress.

For every FE-calculation, the corresponding analytic stiffness andstresses will be determined. The relation between the FEM and theanalytical results are

Rk

kkFEM

z

=ΙΙ

(4.45)

R FEMσ

σσ

=maxΙΙ

(4.46)

5. Make a linear regression analysis.

The linear regression analysis is done by using a statistical computerprogram, SPSS [3]. The input is the geometry (a,s,do) and the resultsfrom the FE-calculation (Rk, Rσ). It is also necessary to suggest a typeof function, for example

R a s d b b a b s b dk o o( , , ) = + + +0 1 2 3 (4.47)

The program determines the best values of the coefficients b0 throughb3. Data about how well the function describe the dependent variableRk is also presented.

29

The correction functions for the stiffness, equation 4.48, and the stress,equation 4.49, were achieved after a qualified guess of function types.

R a d a dk o o( , ) . .= + ⋅ −4 602 6 10 4317 3 (4.48)

R a s a sσ ( , ) . . .= − +0874 2137 7137 (4.49)

The stiffness for a bellow with n convolutions is attained by combiningequation 4.23, 4.36, 4.43 and 4.48.

++++

−⋅+=

== ΙΙ

2

22323

37

12132464

)1.43106602.4(24

),(

a

safffaan

daEI

kdaRk

om

zokz

ππ(4.50)

The maximum stress for a bellow during an axial load, Pz, is attained bycombining equation 4.40, 4.44 and 4.49.

( )( )saddds

P

aR

ioi

z 7.7137.213874.02

3

)(

2

maxmax

+−−=

== ΙΙ

π

σσ σ

(4.51)

When calculating the stiffness and maximum stress with equation 4.50 and4.51 SI-units must be used.

Figure 4.12 shows the force-displacement characteristic for the correctedmodel, the analytical models and the FE-model. For the given geometry, thecorrected model has stiffness that only deviate with 4% from the stiffness ofthe FE-model.

The error of the stiffness is less than 5% for most combinations ofgeometry, within the limited intervals. If the outer diameter, do, has valuesclose to upper and lower limit, the error can be up to 16%.

The error of the maximum stress is less than 10% for most combinations ofgeometry, within the limited intervals. If the outer diameter, do, has valuesclose to the lower limit, the error can be up to 23%.

30

0 0.1 0.2 0.3 0.40

50

100

150

200

analytic model I, without radial displacementFE-modelanalytic model II, free for radial displacementcorrected analytical model

displacement 1/2 convolution (mm)

load

Pz

[N]

load

Pz

[N]

k zI=9

50 N

/mm

k zFEM=600 N

/mm

k zcorr=575 N

/mm

k zII=430 N/mm

Figure 4.12. Force-displacement characteristic for ½-convolution, withgiven geometry: di=66 mm, do=87 mm, s=0.25 mm, a=2 mm.

31

4.2 Bending Load

In this section expressions for the stiffness and the stresses during bendingwill be described, see figure 4.13. Those expressions will contain thegeometry and material properties of the bellow.

The bending stiffness for the bellow is

kM

θ θ= (4.52)

When a bending moment is applied to a bellow, the maximum deflectionsoccur at opposite sides of the bellow, i.e. 180 degrees apart. This deflectioncan be divided into two cases, as shown in figure 4.14.

M

θ

Figure 4.13. Bellow during bending.

32

Case I includes only the tension/compression of the convolutions and caseII includes the bending of the convolutions. To simplify the calculation,case II is neglected. With this approximation, the bellow can be thought ofas a thin walled pipe with the same axial stiffness as the bellow.

The first advantage of this approximation is that the theory for pure bendingis valid, see figure 4.15. All cross sections perpendicular to the axis of thepipe remain plane and all planes of the cross sections passes through O. Thesecond advantage is that the axial stiffness, kz, can be used to determine thebending stiffness. (The axial stiffness is derived in section 4.1.)

An angular displacement, θ, is applied to the bellow. The geometricalrelations for the approximated bellow are shown in figure 4.16. The crosssection is shown to the right and the left part illustrates the total angulardisplacement, θ, for the bellow. An element force, dP, is actingperpendicular to the cross section at the average diameter, dm, over a

+

+

=

Case I Case II

Figure 4.14. Bending cases.

O

M

M

θ

Figure 4.15. Theory of pure bending.

33

distance of 0.5dmdϕ. The maximum deflection at the tension side of thebellow is ezmax. To determine the moment necessary to bend the bellow, thedeflection force, which varies around the circumference, must be multipliedby the corresponding levers and integrated around the circumference.

The necessary force, P, for a displacement, ez, is

P e kz z= (4.53)

The axial stiffness was derived in section 4.1 and can be expressed as

k BIz m= (4.54)

where

BE a d

n a fa f f as

a

o=+ ⋅ −

+ + + +

24 4 602 6 10 4310

4 6 24 3 112

7 3

3 2 3 22

2

( . . )

π π(4.55)

and

Id d s d s

mo i m=

+=

π π( ) 3 3

24 12 (4.56)

y'

x', x0.5dm

dϕϕ

dP

θ

ez

ezmax

d my

z

dP

y'

z'

0.5dmsinϕ

0.5d

msi

nϕ

Figure 4.16. Geometry during bending.

34

An element force, dP, at any location at the average diameter is

dP e BId

z m=ϕπ2

(4.57)

From figure 4.16 it can be seen that

e ed

dz zm

m

= max

. sin

.

05

05

ϕ(4.58)

Substituting equation 4.56 and 4.58 into equation 4.57 gives

dP e Bs d

dzm= max sin

3

24ϕ ϕ (4.59)

Multiplying the element force, dP, by the lever, 0.5dmsinϕ, give the elementmoment

dM dPd e Bs d

dm z m= =2 48

3 22sin sinmaxϕ ϕ ϕ (4.60)

Integration around the circumference gives the total moment

Me Bs d

de B s dz m z m= =∫max maxsin

3 22

0

2 3 2

48 48ϕ ϕ

ππ

(4.61)

The maximum deflection at the tension side of the bellow can be written as

ed

zm

max =2

θ (4.62)

Combining equation 4.52, 4.55, 4.61 and 4.62 gives the bending stiffness

kE s d a d

n a fa f f as

a

m oθ

π

π π=

+ ⋅ −

+ + + +

3 3 7 3

3 2 3 22

2

4 602 6 10 4310

16 6 24 3 112

( . . ) (4.63)

The maximum stress for a bellow during axial load, Pz, was derived insection 4.1 and can be expressed as

( )( )σπmax . . .= − − +3

20874 2137 71372

P

s dd d a sz

io i (4.64)

35

Substitution of equation 4.53, 4.54 and 4.56 into equation 4.64 gives themaximum stresses during an axial displacement, ez.

( )( )σmax . . .= − − +e Bd s

dd d a sz m

io i8

0874 2137 7137 (4.65)

When a bending moment is applied to a bellow the maximum deflections,±ezmax, occur at its opposite sides. From equation 4.61 the maximumdeflection is

eM

B s dzm

max =48

3 2π(4.66)

The maximum stress during a bending load occurs at the maximumdeflection and by substituting equation 4.66 into equation 4.65 themaximum stress becomes

( ) ( )σπmax . . .=

−− +

60874 2137 71372

M d d

d s da so i

i m

(4.67)

36

4.3 Torsion Load

The bellow is not primarily designed to deal with torsion loads. Because ofits high stiffness, it can only stand very small rotational displacements. Inthis section the relationship between the torsion stiffness and the length ofthe bellow, and the stresses in the bellow when exposed to a rotationaldisplacement, will be obtained.

The bellow geometry is simplified to make the calculation model simpler.The radius of the convolution is replaced by straight lines, see figure 4.17.The advantage of the approximation is that the beam theory can be used forthe top- and bottom elements, and the plate theory can be used for the flankelements of the bellow.

4.3.1 Flank Calculations

The flank is treated as a plate, see figure 4.18. Due to the geometry and loadon the plate, plane stress theory can be used, (σz=0). The aim is to find therelationships between torque, rotational displacements and stresses in theplate.

s

a

a

dido

Centerline of bellow

Bottom element

Top element

Flank element

Figure 4.17. Simplified geometry description of one convolution.

37

According to for example Timoshenko [4], the solution of two-dimensionalproblems without mass forces is determined by

∆∆φ = 0 (4.68)

where ∆ is the Laplace operator for polar co-ordinate systems, see equation4.69, and φ(r,ϕ) is a stress function.

∆ = + +∂∂

∂∂

∂∂ϕ

2

2

2

2 2r r r r(4.69)

For a circular plate subjected to torsion, one possible stress function thatboth fulfils equation 4.68 and the boundary conditions, with one edgegrounded and the other edge free, is

φ ϕ ϕ( , )r C= 1 (4.70)

where C1 is a constant.

The stress components are described by a general solution of equation 4.68,see equation 4.71-4.73.

σ∂φ∂

∂ φ∂ϕ

σ∂ φ∂

τ∂φ∂ϕ

∂ φ∂ ∂ϕ

ϕ

ϕ

r

r

r r r

r

r r r

= +

=

= −

1 1

1 1

2

2

2

2

2

2

2

(4.71-4.73)

r

ri

ro

ϕ

Tro

Tri

Figure 4.18. General plate description.

38

Combining equation 4.70 through 4.73 gives

=

==

21

0

0

r

Cr

r

ϕ

ϕ

τ

σσ

(4.74-4.76)

The shear stress do not depend on ϕ and the stresses at the inner and outerradius of the plate can be written

τ

τ

rii

roo

C

r

C

r

=

=

12

12

(4.77-4.78)

These stresses give the resulting moments

T r sr

T r srri ri i i

ro ro o o

==

τ πτ π

2

2(4.79-4.80)

Combining equation 4.77 and 4.78 with 4.79 and 4.80 gives

T T sCri ro= = 2 1π (4.81)

The constant C1 is thus

CT

s1 2=

π(4.82)

The stress function, equation 4.70, and the shear stress distribution function,equation 4.76, can now be rewritten as

φ ϕπ

ϕ( , )rT

s=

2

(4.83)

τπϕr

T

sr=

2 2 (4.84)

39

To determine the rotational angle the strain components

ε∂∂

ε∂∂ϕ

γ∂∂

∂∂ϕ

ϕϕ

ϕϕ ϕ

rr

r

rr

e

r

r

e e

r

e

r

e

r r

e

=

= +

= − +

1

1

(4.85-4.87)

and Hooke´s law

( )( )

ε σ νσ

ε σ νσ

γ τ

ϕ

ϕ ϕ

ϕ ϕ

r r

r

r r

E

E

G

= −

= −

=

1

1

1

(4.88-4.90)

where G is

GE

=+2 1( )ν

(4.91)

have to be used. By using the stress function, equation 4.83, and inserting itinto equations 4.71 through 4.73, equation 4.85 through 4.90 give

∂∂

∂∂ϕ

∂∂ϕ

∂∂ π

ϕ

ϕ ϕ

e

r

e

r

e

r

e

r

e

r

e

r

T

sGr

r

r

r

=

+ =

+ − =

0

0

2 2

(4.92-4.94)

Knowing that there are no changes in the ϕ-direction gives

e

e

r

e

r

T

sGr

r =

− =

0

2 2

∂∂ π

ϕ ϕ (4.95-4.96)

40

The solution of equation 4.96 is given by

e C rT

sGrϕ π= −2 4

(4.97)

where C2 is an arbitrary constant.

Assuming that the inner edge of the plate is grounded and the outer edge issubjected to a torsion moment, T, gives the boundary conditions

r r

ei=

= ϕ 0

(4.98-4.99)

Equation 4.97 can now be written as

e rT

sG

r

r riϕ π( ) = −

4

12 (4.100)

The torsion stiffness, (Nm/rad), of the plate is given by

kr T

e rsG

r r

sGr

r

r

o

o

i o

i

i

o

= =−

=

−

ϕπ

π( )

41

1 14

12 2

2

2 (4.101)

The total number of flanks is 2n, see figure 4.17, which gives the totalstiffness for all flanks in the bellow

ksGr

nr

r

sGr

nr

r

ftoti

i

o

i

i

o

=

−

=

−

4

2 1

2

1

2

2

2

2

π π(4.102)

By using equation 4.84 and 4.102 the stress distribution and the torsionstiffness can be determined for the flanks in the bellow.

4.3.2 Top and Bottom Element Calculations

The torsion stiffness for the top and bottom elements is calculated by usingthe beam theory. The polar moment of inertia is determined by equation4.103 for the top element, and by equation 4.104 for the bottom element.

J r st o= 2 3π (4.103)

41

J r sb i= 2 3π (4.104)

The stresses in the elements are given by

τ tt

o

T

Jr= 4.105)

τbb

i

T

Jr= (4.106)

The total top and bottom element lengths are determined by

L L nat b= = 2 (4.107)

The total torsion stiffness for the elements is determined by equations 4.108and 4.109.

kGJ

Lttott

t

= (4.108)

kGJ

Lbtotb

b

= (4.109)

4.3.3 Summary of Torsion Load

The total torsion stiffness of the bellow can now easily be obtained by usingthe relationship below.

1 1

k ktot

= ∑ (4.110)

Combining equation 4.102, 4.108 and 4.109 and using equation 4.110 givesthe total torsion stiffness

kG

L

J

L

J

nr

r

sr

tot

b

b

t

t

i

o

i

=

+ +

−

1

2

2

2π

(4.111)

42

In figure 4.19 the torsion stiffness for a bellow as a function of the numberof convolutions can be seen.

The stress distribution in the bellow is determined by the following threeequations

τπϕr

T

sr=

2 2 (4.112)

τbb

i

T

Jr= (4.113)

τ tt

o

T

Jr= (4.114)

The stress in the bottom element is the same as at the inner edge of theflank, and the stress in the top element is the same as at the outer edge ofthe flank. This means that it is enough to study the stress distribution in the

0 5 10 15 20 25 30 35

2 105

4 105

6 105

8 105

Number of convolutions

Tor

sion

stif

fnes

s (N

mm

/rad

)

k tot( )n

n

Figure 4.19. Torsion stiffness for different number of convolutions.

43

flank to get the total stress distribution. In figure 4.20 the stress distributionin the flank is given for T=40 Nm.

25 30 35 40 45 50

10

20

30

40

50

Radius (mm)

Sh

ear

stre

ss (

MP

a)

τ f( )r

r

Figure 4.20. Shear stress distribution.

44

5. Experimental Verification

To verify the theoretical models derived in chapter four, an experimentalinvestigation of stiffness and stresses during different loads is done. Themeasurements are done for two different bellows. The convolutiongeometry of the bellows is shown again in figure 5.1.

s

a

a

dido

Centerline of bellow

Figure 5.1. Convolution geometry for the bellows used for measurements.

45

5.1 Axial Load Verification

5.1.1 Experimental Set-up

The bellow is fixed at one end and the axial load is applied at the other end,see figure 5.2. The load is applied by using dynamometers. The axialdeformation of the bellow is measured by using a dial indicator, not shownin the figure.

The geometry values for the bellow used for verification of axial load are:a = 2 mm, s = 0.3 mm, do = 87 mm and di = 66 mm. The deformation andthe applied force give the axial stiffness of the bellow.

The strain is measured with a 0°/45°/90° rosette, which is made of threeseparate gages mounted on the top of the convolutions. The strain gageshave an active length of 0.6 mm. The results are presented in table 5.1 andin figure 5.3.

Figure 5.2. Set-up for axial load verification.

46

5.1.2 Results

Load

P [N]

Displacement

e [m]

Gage 1, 0°

ε0 [µstrain]

Gage 1, 45°

ε45 [µstrain]

Gage 1, 90°

ε90 [µstrain]

20 0.00136 57.8 62.2 62.0

40 0.00272 104 117 109

60 0.00405 152 174 161

80 0.00542 198 224 211

100 0.00672 242 274 260

120 0.00802 285 326 312

140 0.00936 327 374 361

160 0.01062 366 415 408

180 0.01183 408 463 459

Table 5.1. Test results for axial loading.

The average measured axial stiffness of the bellow is 15240 N/m. Thetheoretical axial stiffness, according to equation 4.50, is 15600 N/m. Theagreement for the axial stiffness is excellent.

The principal normal stresses, σ1,2, are calculated according to

( ) ( ) ( )σν

ε εν

ε ε ε ε1 20 90

0 45

2

90 45

2

1 2 2 1, =−

+±

+− + −

E E(5.1)

The von Mises stress at the top can now be determined by

( )σ σ σ σ σe = − + +1

2 1 2

2

12

22 (5.2)

47

A comparison between the theoretical model and the test results is shown infigure 5.3. The maximum deviation between measured and theoreticalvalues is 15%. Considering uncertainties about the material in the measuredbellow and its geometry, the agreement is very good.

0

50

100

150

200

250

0 20 40 60 80 100 120 140 160 180

Axial Load [N]

von

Mis

es S

tres

s [M

Pa]

Measuredvalues

Theoreticalmodel

Figure 5.3. Verification of stresses for axial load.

48

5.2 Bending Load Verification

5.2.1 Experimental Set-up

The set-up for measuring during bending loads is principally the same as foraxial loads except that the load is applied in a perpendicular direction. Theload is applied at a distance, L = 0.272 m from the fixed end, see figure 5.4.

This way of applying the load gives a linear moment distribution along thebellow. To be able to compare the measured bending stiffness, where theload is a force, with the theoretical bending stiffness, where the load is amoment, a correction factor has to be derived.

Figure 5.4. Set-up for bending load verification.

49

By comparing the two elementary load cases for beams with one end fixedand the other end subjected to a force or a moment, the bending stiffnesscan be expressed as

3

2L

e

Pk =θ (5.3)

The equipment for measuring of displacement and strain are also equivalentto the one used for axial load measurement and the stresses are calculated inthe same way.

5.2.2 Results

Load

P [N]

Displacement

e [m]

Gage 1, 0°

ε0 [µstrain]

Gage 1, 45°

ε45 [µstrain]

Gage 1, 90°

ε90 [µstrain]

2 0.00523 37.9 40.6 34.6

4 0.0103 76.2 76.8 64.3

6 0.0155 113 115 94.3

8 0.0202 152 150 124

10 0.0246 158 194 119

Table 5.2. Test results for bending loading.

According to equation 5.3, the measured bending stiffness is 10.49 Nm/radand according to equation 4.65, the theoretical bending stiffness is 11.42Nm/rad. The agreement for the bending stiffness is very good.

50

The principal stresses and the von Mises stresses are calculated according toequation 5.1 and equation 5.2. A comparison between the theoretical modeland the test results is shown in figure 5.5.

The maximum deviation between measured and theoretical values is 9%.Considering uncertainties about the material in the measured bellow and itsgeometry, the agreement is very good.

0

10

20

30

40

50

60

70

80

90

0 0,273 0,546 0,819 1,092 1,365

Bending Moment [Nm]

von

Mis

es S

tres

s [M

Pa]

Measuredvalues

Theoreticalmodel

Figure 5.5. Verification of stresses for bending load.

51

5.3 Torsion Load Verification

5.3.1 Experimental Set-up

The bellow is fixed at one end and the torsion load is applied at the otherend, see figure 5.6. The load is applied by using dynamometers and an arm.This set-up does not necessarily give a pure torque.

To be able to control that the load is mostly torsion, a gage is mounted on aconvolution top in the axial direction. If the load is pure torsion, then therewill not be any strain in the axial direction.

Figure 5.6. Set-up for torsion load verification.

The geometry values for the bellow used for verification of torsion load are:a = 2 mm, s = 0.25 mm, do = 82.5 mm and di = 66 mm.

Measuring of the torsion stiffness will not be done because it is difficult tomeasure the small angular changes. The torsion stiffness will therefore beverified by comparison with FEM-calculations.

The strain is measured with a gage mounted on the top of a convolutiondirected in 45° angle to the axial direction. This is done because torsionload gives principal strains in the 45° direction. The strain gages have anactive length of 0.6 mm. The results are presented in table 5.3 and in figure5.7.

52

5.3.2 Results

Load

[Nm]

Gage, 45°

ε [µstrain]

50 158

75 232

100 336

Table 5.3 Test results for torsion load.

The theoretical torsion stiffness, equation 4.110, is 13900 Nm/rad. TheFEM-calculated stiffness is 16100 Nm/rad. A comparison between thetheoretical model stresses and the test results is shown in figure 5.7.

The maximum deviation between measured and theoretical values is 13%.Considering uncertainties about the material in the measured bellow, itsgeometry and difficulties in applying a perfect torsion load, the agreement isvery good.

0

5

10

15

20

25

30

35

40

45

0 25 50 75 100

Torque [Nm]

von

Mis

es S

tres

s [M

Pa]

Measuredvalues

Theoreticalmodel

Figure 5.7. Verification of shear stresses for torsion load.

53

6. Conclusions

An analysis of flexible bellows, which are used in exhaust systems for cars,has been carried out in this work. Theoretical expressions for the stiffnessand the maximum stresses during three different loads have been derived.The loads that have been studied are axial, bending and torsion load.

Due to the complexity of the problem, a combination of analyticexpressions, FEM-calculations and linear regression was used to derive thetheoretical expressions for axial load. To be able to do a good regressionanalysis, the bellow dimensions were limited to the dimensions that areinteresting for practical manufacturing and for use in normal car exhaustsystems according to AP Parts Torsmaskiner Technical Center AB, seechapter 4.1.4. The axial load expressions were also used for deriving thetheoretical expressions for bending load.

The theoretical expressions derived for torsion load was done bysimplifying the bellow geometry. This was done to get analytical and simpleexpressions for the stiffness and the stresses in the bellow. Theseexpressions do not have any theoretical limitations for the bellowdimensions but they have only been verified for the same dimensions as thedimensions used for axial and bending loads.

An experimental verification of the theoretical models was carried out. Theagreement between theoretical and experimental results is very good. Theexpressions for the stiffness of the bellow derived in this work are probablysuitable also for investigation of the dynamic behaviour of flexible bellows.

54

7. References

1. Pilkey, W.D., Formulas for Stress, Strain and Structural Matrices,John Wiley & Sons, New York, (1994).

2. Weisberg, S., Applied Linear Regression, John Wiley & Sons, NewYork, (1985).

3. SPSS inc., SPSS for Windows, Release 7.0, (Dec 19, 1995)

4. Timoshenko, S.P., Theory of Elasticity, McGraw-Hill Book Company,New York, Ch. 4, 65-68 (1970).

Department of Mechanical Engineering, Master’s Degree ProgrammeUniversity of Karlskrona/Ronneby, Campus Gräsvik371 79 Karlskrona, SWEDEN

Telephone: +46 455-78016Fax: +46 455-78027E-mail: Goran.Broman@ima.hk-r.se