Master's Degree Thesis
ISRN: HK/R-IMA-EX--1997/D-01--SE
Static Characteristics of
Flexible 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 of
Flexible 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 Mechanical
Engineering with emphasis on Structural Mechanics at the Department of
Mechanical Engineering, University of Karlskrona/Ronneby, Karlskrona,
Sweden.
Abstract:
Theoretical expressions for stiffness and maximum stress were determined
for axial, bending and torsion load by using a combination of analytical,
statistical and finite element calculations. Experimental verification showed
very good agreement.
Keywords:
Static characteristics, Flexible, Bellows, Stiffness, Stress, Experimental
verification.
2
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 Parts
Torsmaskiner Technical Center AB and the Department of Mechanical
Engineering at the University of Karlskrona/Ronneby.
The purpose of the project was to strengthen the knowledge about flexible
bellows at AP Parts Torsmaskiner Technical Center AB. Financial support
was given by Blekinge Research Foundation, which is gratefully
acknowledged.
We wish to express our sincere appreciation to Dr. Göran Broman for his
helpful advice and guidance throughout the work. We also thank Dr. Stefan
Östholm, head of department, for valuable discussion and support.
We also gratefully acknowledge the support from AP Parts Torsmaskiner
Technical Center AB, and especially the Research & Development
Manager, M.Sc. Kristian Althini who always took time to help us.
We finally want to express our gratitude to Mikael and Emma for their
patience and support, which made this work possible to accomplish.
Madeleine Hermann
Anders Jönsson
3
Contents
1. Notation
5
2. Introduction
8
3. Basic Relations and Limitations
3.1 Bellow Geometry
3.2 Loads
10
10
11
4. Theoretical models
4.1 Axial Load
4.1.1 Model I: Restrained Radial Displacement
4.1.2 Model II: Free Radial Displacement
4.1.3 Finite Element Model
4.1.4 Corrected Model
4.2 Bending Load
4.3 Torsion Load
4.3.1 Flank Calculations
4.3.2 Top and Bottom Element Calculations
4.3.3 Summary of Torsion Load
12
12
14
22
24
27
31
36
36
40
41
5. Experimental Verification
5.1 Axial Load Verification
5.1.1 Experimental Set-up
5.1.2 Results
5.2 Bending Load Verification
5.2.1 Experimental Set-up
5.2.2 Results
5.3 Torsion Load Verification
5.3.1 Experimental Set-up
5.3.2 Results
44
45
45
46
48
48
49
51
51
52
6. Conclusions
53
7. References
54
4
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
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
Indices
b
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
6
FEM
Finite Element Method
max
Maximum
min
Minimum
ϕ
Circumferential direction
θ
Bending angle
7
2. Introduction
When the emission demands on personal cars became higher in the late
80´s, the automotive industry had to find a flexible and gas tight connection
between the engine and the exhaust system. This kind of element has been
used for a long time in marine engine installations and in the processing
industry. However, when implemented into cars, the operating conditions
were different from what the elements where designed for. This lead to
failure 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 to
the engine and the exhaust system movements.
The element is connected to the engine via the branch pipe, which has little
damping effect. The large engine movements are mainly due to inertial
forces in the engine and shifting of gear.
There are also vibrations with smaller amplitudes transmitted from the
engine. Reduction of the transmitted engine movements and vibrations to
the 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 flexible
connection to the car chassis. Road vibrations can be transmitted via the
exhaust system into the flexible element.
The manufacturing is also complex, which can lead to initial stresses in the
element. See figure 2.1 for some examples of different element designs.
8
Figure 2.1. Examples of different flexible elements.
The gas tight bellow in the flexible element has a critical function. Cracks
in the bellow are a common failure reason. General descriptions of bellow
characteristics have not been found in the literature.
The aim of this work is to develop theoretical models of a typical flexible
bellow under static axial-, bending- and torsion load.
The models will contain essential design parameters so that their influence
on bellow characteristics can be studied. An experimental verification of the
models will be carried out.
The theoretical models will probably also be useful for analysis of dynamic
characteristics of flexible bellows.
9
3. Basic Relations and Limitations
3.1 Bellow Geometry
Although there are many different variants of flexible elements on the
market, they all have the same basic design, see figure 3.1.
braid
gas tight bellow
innerline
end cap
Figure 3.1. General flexible element design.
There is often some sort of heat protection, a so-called innerline, inside the
element. The next layer is the gas tight bellow and on the outside, there is a
braid to protect the bellow from outer mechanical violence. The three parts
are connected at the ends with end caps.
L
r
z
one convolution
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
10
connects the inner and outer radii, a, see figure 3.3, where a polar coordinate system is also defined for future use. The length, L, is given by
L = 4na
(3.1)
The radius, a, and the diameters, di and do, are distances to the middle of the
material. The thickness, s, of the material is small compared to the other
dimensions and does not affect the geometry description.
a
s
f
a
r
do
di
z
Centerline of bellow
Figure 3.3. Dimensions of one half-convolution.
3.2 Loads
The bellow is operating under complex conditions. Two main groups are
movement of the bellow boundaries and temperature changes. The effect of
temperature changes will not be analysed in this work.
The relative movement between the engine and the exhaust system gives
rise to stresses in the bellow. During operation the bellow is subjected to
combined loads. In this work analyses will be made for three separate loads:
axial, bending and torsion, see sections 4.1 through 4.3.
11
4. Theoretical models
4.1 Axial Load
In this section expressions for the stiffness and the stresses during axial
loading will be described, see figure 4.1. Those expressions will contain the
geometry and material properties of the bellow.
P
P
eztot
Figure 4.1. Axially loaded bellow.
The axial stiffness of the bellow is
kz =
Pz
eztot
(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 are
identical. To simplify the calculation, the bellow is sliced in the longitudinal
direction and treated as a plate with convolutions. The plate has the same
length as the bellow and the width is the average circumference, see figure
4.2.
This approximation is probably acceptable when the difference between the
inner and outer diameter is much smaller than the average diameter, dm =
½(do+di).
12
r
z
L
Figure 4.2. Approximated bellow for axial loading.
Due to symmetry of the convolution itself it is only necessary to consider a
quarter of one convolution. This approximation of the bellow is treated with
two different boundary conditions.
In model I, section 4.1.1, radial displacement is completely restrained and in
model II, section 4.1.2, radial displacement is completely free. The aim of
using those two boundary conditions is to find the interval in which the
stiffness of the real bellow must be, and to get a general idea of how the
geometry and material properties influence the stiffness of the bellow.
Model I will give a too high stiffness and model II will give a too low
stiffness. A bellow without any simplification of the geometry is solved by
using the Finite Element Method in section 4.1.3.
13
4.1.1 Model I: Restrained Radial Displacement
The freebody-diagram is shown in figure 4.3. The moment at the symmetryline is zero.
It is assumed that the symmetry-line, i.e. the average diameter, and the
N
Pz
Mo,i
Pz
symmetry-line
ez
N
Figure 4.3. Freebody-diagram of ¼-convolution.
top/bottom of the convolution, i.e. the outer and inner diameter, has no
radial displacement. This explains the presence of the force N. This must be
thought of as an outer force in the model of figure 4.3, since symmetry
otherwise demands it to be zero. In reality, the restraint radial displacement
is because such displacement also implies a change of the circumference of
the bellow.
By combining the elementary load cases in figure 4.4 and the corresponding
geometrical displacement in figure 4.5, an expression for the axial stiffness
can be derived.
14
2
1
a
a
M1
Pz
er1
er2
β2 ez2
β1 ez1
3
4
0,5 f
a
er3
β3 ez3
Pz
N
ez4
er4
Figure 4.4. Elementary load cases.
ez5
ez6
er6
er7
Figure 4.5. Geometrical displacements.
15
ez7
0,5 f
β3
0,5 f
β2
0,5 f
β1
er5
7
6
5
The radial and longitudinal displacements in figure 4.4 and 4.5 are given in
equations 4.2 through 4.15. Small displacement theory is assumed valid.
The displacements for the elementary load cases can be found in books of
beam theory formulas, for example [1]. The curvature, a, is large compared
too the thickness of the beam, s. The area moment of inertia for all curved
beams is therefore the same as for a straight beam, i.e. Im.
− Pz a 3
er1 =
2EI m
(4.2)
π
π s2 
 −1+

24a 2 
2
(4.3)
π s2 
− Na 3  3π


er 3 =
−2+
EI m  4
24a 2 
(4.4)
− M 1a 2
er 2 =
EI m
er 4 =
17 Pz
180 EI m
(4.5)
fβ 2
f
f  P a2 
er5 = (1 − cos β1 ) = 1 = ⋅  z 
2
4
4  EI m 
2
fβ22
f
f  M 1π a 
s2  

er 6 = (1 − cos β2 ) =
=
1 +

2
4
4  2 EI m  12a 2  
fβ32
f
f
er 7 = (1 − cos β 3 ) =
=
2
4
4
 Na 2

 EI m
(4.6)
2
π
π s2  
 −1+

24a 2  
2
(4.7)
2
(4.8)
π Pz a 3
ez 1 =
4 EI m
(4.9)
M 1a 2
ez 2 =
EI m
(4.10)
Na 3
ez 3 =
2 EI m
(4.11)
16
Pz f 3
ez 4 =
24 EI m
(4.12)
fPz a 2
ez 5 =
2 EI m
(4.13)
ez 6 =
f M 1π a 
s2 
1 +

2 2 EI m  12a 2 
(4.14)
ez 7 =
π s2 
f Na 2  π
 −1+

2 EI m  2
24a 2 
(4.15)
where
M 1 = Pz
f
2
(4.16)
and
Im =
π s 3 (d o + d i )
24
(4.17)
The restraint at the symmetry-line gives an equation for the normal force, N,
as a function of Pz.
7
∑e
k =1
rk
=0
⇒
N ( Pz ) = C1 − C12 − Pz C2 − Pz 2 C3
(4.18)
C1, C2 and C3 are constants containing the geometry and the material
properties of the bellow according to equations 4.19 through 4.21.
 3π
πs 2 
−2+


24a 2 
2 EI m  4
C1 =
2
fa  π
πs 2 
 −1+

24a 2 
2
(4.19)
17
C2 =
EI m
45 fa 4

π
πs 2  
 17 − 90a 3 − 90 fa 2  − 1 +

24a 2  
2

π
πs 2 
 −1+

24a 2 
2
 fπ 
s2  

a +  1 +
2
 4  12a  
2
(4.20)
2
2
C3 =
π
πs 2 
a  − 1+

24a 2 
2
(4.21)
2
2
Adding all displacement contributions in the z-direction and multiply by 4n
give the total z-displacement for a bellow with n convolutions
7
4 n∑ ezk = eztot
⇒
k =1
eztot = 4n( P z D1 + N D2 )
(4.22)
D1 and D2 are constants containing the geometry and the material properties
of the bellow according to equations 4.23 and 4.24.

s2 
6πa 3 + 24 fa 2 + f 3 + 3 f 2 aπ  1 +

 12a 2 
D1 =
24 EI m
(4.23)
π
a2 
πs 2  
a + f  − 1+
D2 =

2 EI m 
24a 2  
2
(4.24)
Equation 4.1, 4.18 and 4.22 give an expression for the axial stiffness
k zΙ =
(
(
Pz
4n Pz D1 + D2 C1 − C − Pz C2 − P C3
2
1
2
z
))
(4.25)
It is clear that kzI is dependent on the size of the load, Pz, since N is not
linearly related to Pz.
This non-linear relation can be linearised by considering the material
properties of the bellow and thereby limiting the interval of Pz. The
maximum stress in a convolution must be less than the yield strength, σY, of
the material in the bellow. The maximum stress occurs at Mmax. The force,
18
N, and the shear stress are neglected at this point and Mmax occurs roughly at
the bottom and top of the convolution as
f

M max = Pz  + a
2

(4.26)
With the minimum area moment of inertia
I min
π s 3d i
=
12
(4.27)
the maximum bending stress becomes approximately
σ max =
M max s
I min 2
(4.28)
Equation 4.26 through 4.28 and the material property σmax=σY give an
approximate maximum load of Pz. With σY = 400 MPa the maximum load
becomes Pz=300 N.
Figure 4.6 shows the non-linear function
N ( Pz ) = C1 − C12 − Pz C2 − Pz 2 C3
(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.
19
0
NN [N]
[N]
500
1000
1500
0
50
100
150
Pz
[N]
P [N]
200
250
300
z
N(P
N(Pz)
z)
N*(P
N*(Pz)=cPz,
c=-3.7
z) = cPz, c=-3.7
Figure 4.6. Linear approximation of N(Pz) to N*(Pz).
The diagram shows that the non-linear function N(Pz) can be replaced with
a linear function N*(Pz) = c˜ Pz with very good accuracy. The constant of
proportionality, c, is negative and therefore N is a compressive force for
pushing 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
k zΙ =
1
4n( D1 + cD2 )
(4.30)
The moment distribution is obtained from equation 4.31 and 4.32 and is
shown in figure 4.7.
M ( x ) = Pz x , 0 ≤ x ≤
f
2
(4.31)
2

f  

2

M ( x) = Pz x + Pz c a − a −  x −  

2 

The maximum bending stress becomes
20
,
f
f

≤ x ≤  + a (4.32)


2
2
σ max Ι =
M max s
Im 2
(4.33)
The magnitude and location of Mmax can be calculated by solving
dM
=0
dx
(4.34)
However, equation 4.34 is no simple expression. The expressions of
equation 4.31 and 4.32 are shown in figure 4.7.
Mo,i
x
cPz
Pz
f/2+a
Mo,i
a
Pz
cPz
Figure 4.7. Moment distribution for ¼-convolution with
given geometry and c=-3.7.
21
M
Mmax
f/2
4.1.2 Model II: Free Radial Displacement
The freebody-diagram is shown in figure 4.8.
Pz
Mo,i
Pz
symmetry-line
ez
Figure 4.8. Freebody-diagram for ¼-convolution.
As in the previous model the moment at the symmetry-line is zero and the
symmetry-line, i.e. the average diameter, is not changing its position.
However, in this model it is assumed that the top/bottom of the
convolution, i.e. the outer and inner diameter, is changing their positions in
radial 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 the
geometrical displacement 5 and 6 in figure 4.5 an expression for the axial
stiffness can be derived.
Adding all displacement contributions in the z-direction, from equations 4.9
through 4.14, and multiplying by 4n give the total z-displacement for a
bellow with n convolutions.
4n
∑e
zk
k =1, 2 , 4 ,5, 6
= eztot
⇒ eztot = 4nPz D1
(4.35)
D1 is a constant containing the geometry and the material properties of the
bellow according to equation 4.23 in section 4.1.1.
22
From equation 4.1 and 4.35 the axial stiffness now becomes
k zΙΙ =
1
4nD1
(4.36)
Compared to the previous model, this model has a much simpler expression
for the stiffness, due to the linear relation between the Pz-force and the
displacement, eztot. The moment distribution is obtained from equation 4.37
and is shown in figure 4.9.
f

M ( x ) = Pz x , 0 ≤ x ≤  + a 
2

(4.37)
x
Pz
f/2+a
Mo,i
a
M
Mmax
f/2
Pz
Figure 4.9. Moment distribution for ¼-convolution.
The maximum bending stress then becomes
σ max ΙΙ =
M max s
P f
s
= z  + a
2
I min 2 I min  2
(4.38)
or with
f
 do − di
 + a =
2

4
(4.39)
23
and equation 4.27 the maximum stress can be expressed as
σ max ΙΙ =
3 Pz (d o − d i )
(4.40)
2πs 2 d i
4.1.3 Finite Element Model
One model of the bellow was solved numerically by using the Finite
Element Method (FEM). The FEM-module in I-DEAS Master Series 4 was
used for this calculation. The advantage of this model is that the real
geometry of the bellow can be used. The disadvantage is that the results is
only valid for one specific geometry and it is not possible to directly get an
analytic expression for the stiffness and stresses.
The FE-model will be used for verification of the analytical expressions for
stiffness and stresses. In section 4.1.4, it will also be used for an estimation
of correction functions for these analytical expressions.
The FE-model is shown in figure 4.10.
B
C
A
D
10
ϕ
r z
Figure 4.10. Finite Element model of ½-convolution.
24
When treating a ½-convolution restrictions on radial displacements should
not be set explicitly. In Model I radial displacement were completely
restrained at the inner-, outer- and average diameter and in Model II radial
displacement were completely restrained at the average diameter. In the FEmodel the actual restrictions on radial displacements due to circumferential
strain resistance are automatically imposed.
Since the bellow is axi-symmetric, it is possible to use axi-symmetric
elements for the FE-model. However, in this model thin shell elements were
used together with boundary conditions imposing axi-symmetry.
In this model a slice of 10° were modelled with 300-400 eight-node
isoparametric 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 free
or constant translation displacement.
The load is applied at edge AD in the z-direction with the magnitude
PAD = Pz
10°
360°
(4.41)
and the stiffness for the FE-convolution is
k zFEM =
Pz
(4.42)
ezFEM
Figure 4.11 shows the force-displacement characteristics for a ½convolution according to the two analytic models (with or without radial
displacement), the FE-model and the measured results for a real bellow.
The slope of the curves are the stiffness for ½-convolution.
25
The stiffness of the FE-model corresponds very well to the measured
stiffness of the real bellow. Due to this agreement, it is probably correct to
assume that the stresses in the FE-model are close to the stresses in the real
bellow.
200
N/
mm
=
0
62
mm
N/
=9
=
k zII
I
=6
k
E
zF
0
M
al
k zre
50
100
kz
load Pz [N]
150
m
/m
N
0
N
43 0
m
/m
50
0
0
0.1
0.2
0.3
displacement 1/2 convolution (mm)
0.4
analytic model I, restrained radial displacement
FE-model
real bellow
analytical model II, free radial displacement
Figure 4.11. Force-displacement characteristic for ½-convolution, with
given geometry: di=66 mm, do=87 mm, s=0.25 mm, a=2 mm.
26
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 % higher
stiffness than the real bellow and the second model, kzII, has about 30 %
lower stiffness than the real bellow. Non of these discrepancies are
acceptable. This implies that the analytical expressions for maximum stress
are 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 zcorr = Rk (a , s, d i , d o ) k z
(4.43)
σ max corr = Rσ (a , s, d i , d o )σ max
(4.44)
and
where the correction functions, Rk(a,s,di,do) and Rσ(a,s,di,do), are functions
of the geometry of the bellow. Those functions will be derived by using
linear regression. For a discussion on this topic, see for example [2]. The
procedure 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, see
equation 4.36 and 4.40, which are repeated below.
k zΙΙ =
1
4nD1
σ max ΙΙ =
2.
(4.36)
3 Pz (d o − d i )
(4.40)
2πs 2 d i
Limit the dimensions of the bellow.
The inner diameter is constant because it has to fit to the tubes of the
exhaust system. The other dimensions varies as follows:
di = 66 mm
do = 80 - 90 mm
a = 1.0 - 3.0 mm
s = 0.20 - 0.30 mm
27
3.
Find the real stiffness and stresses of a number of bellows within
the limited dimensions.
The real stiffness and stresses of the bellow are calculated with the
FE-model, because it is very close to the real bellow. Using the FEM
instead of measuring the stiffness and stresses of real bellows reduces
costs and are more time efficient. Problems with inaccuracy in the
measurement and quality variation from manufacturing are also
avoided.
25 calculations on ½-convolutions within the limited dimensions were
done. A force of 10 N was applied to the ½-convolution. The
displacement and the maximum von Mises stresses were listed for
each calculation.
4.
Calculate the relation between the FEM and the analytical results
of stiffness and stress.
For every FE-calculation, the corresponding analytic stiffness and
stresses will be determined. The relation between the FEM and the
analytical results are
5.
Rk =
k FEM
k zΙΙ
(4.45)
Rσ =
σ FEM
σ max ΙΙ
(4.46)
Make a linear regression analysis.
The linear regression analysis is done by using a statistical computer
program, SPSS [3]. The input is the geometry (a,s,do) and the results
from the FE-calculation (Rk, Rσ). It is also necessary to suggest a type
of function, for example
Rk (a , s, d o ) = b0 + b1 a + b2 s + b3 d o
(4.47)
The program determines the best values of the coefficients b0 through
b3. Data about how well the function describe the dependent variable
Rk is also presented.
28
The correction functions for the stiffness, equation 4.48, and the stress,
equation 4.49, were achieved after a qualified guess of function types.
Rk (a , d o ) = 4.602 + 6 ⋅ 10 7 a 3 − 431
. do
(4.48)
Rσ (a , s) = 0.874 − 213.7a + 713.7 s
(4.49)
The stiffness for a bellow with n convolutions is attained by combining
equation 4.23, 4.36, 4.43 and 4.48.
k z = Rk (a , d o )k zΙΙ =
=
24 EI m ( 4.602 + 6 ⋅ 107 a 3 − 43.1d o )


s2  

4n 6πa 3 + 24 fa 2 + f 3 + 3 f 2 aπ 1 +
2 
 12a  

(4.50)
The maximum stress for a bellow during an axial load, Pz, is attained by
combining equation 4.40, 4.44 and 4.49.
σ max = Rσ ( a )σ max ΙΙ =
3Pz
(d o − d i )(0.874 − 213.7a + 713.7 s )
=
2πs 2 d i
(4.51)
When calculating the stiffness and maximum stress with equation 4.50 and
4.51 SI-units must be used.
Figure 4.12 shows the force-displacement characteristic for the corrected
model, the analytical models and the FE-model. For the given geometry, the
corrected model has stiffness that only deviate with 4% from the stiffness of
the FE-model.
The error of the stiffness is less than 5% for most combinations of
geometry, within the limited intervals. If the outer diameter, do, has values
close 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 of
geometry, within the limited intervals. If the outer diameter, do, has values
close to the lower limit, the error can be up to 23%.
29
=9
50
N/
mm
200
I
kz
load
[N]
load P
Pzz [N]
150
100
mm
N/
0
0
=6
M
k zFE
m
/m
N
75
=5
rr
k zco
m
N/m
0
3
=4
II
kz
50
0
0
0.1
0.2
0.3
displacement 1/2 convolution (mm)
0.4
analytic model I, without radial displacement
FE-model
analytic model II, free for radial displacement
corrected analytical model
Figure 4.12. Force-displacement characteristic for ½-convolution, with
given geometry: di=66 mm, do=87 mm, s=0.25 mm, a=2 mm.
30
4.2 Bending Load
In this section expressions for the stiffness and the stresses during bending
will be described, see figure 4.13. Those expressions will contain the
geometry and material properties of the bellow.
M
θ
Figure 4.13. Bellow during bending.
The bending stiffness for the bellow is
kθ =
M
θ
(4.52)
When a bending moment is applied to a bellow, the maximum deflections
occur at opposite sides of the bellow, i.e. 180 degrees apart. This deflection
can be divided into two cases, as shown in figure 4.14.
31
Case I
Case II
+
=
+
Figure 4.14. Bending cases.
Case I includes only the tension/compression of the convolutions and case
II includes the bending of the convolutions. To simplify the calculation,
case II is neglected. With this approximation, the bellow can be thought of
as a thin walled pipe with the same axial stiffness as the bellow.
The first advantage of this approximation is that the theory for pure bending
is valid, see figure 4.15. All cross sections perpendicular to the axis of the
pipe remain plane and all planes of the cross sections passes through O. The
second advantage is that the axial stiffness, kz, can be used to determine the
bending stiffness. (The axial stiffness is derived in section 4.1.)
M
M
θ
O
Figure 4.15. Theory of pure bending.
An angular displacement, θ, is applied to the bellow. The geometrical
relations for the approximated bellow are shown in figure 4.16. The cross
section is shown to the right and the left part illustrates the total angular
displacement, θ, for the bellow. An element force, dP, is acting
perpendicular to the cross section at the average diameter, dm, over a
32
distance of 0.5dmdϕ. The maximum deflection at the tension side of the
bellow is ezmax. To determine the moment necessary to bend the bellow, the
deflection force, which varies around the circumference, must be multiplied
by the corresponding levers and integrated around the circumference.
y
ezmax
y'
y'
dP
dP
dϕ
0.5d
m
θ
sinϕ
ez
z
0.5dmsinϕ
ϕ
x', x
dm
0.5dm
z'
Figure 4.16. Geometry during bending.
The necessary force, P, for a displacement, ez, is
P = ez k z
(4.53)
The axial stiffness was derived in section 4.1 and can be expressed as
k z = BI m
(4.54)
where
B=
24 E (4.602 + 6 ⋅ 10 7 a 3 − 4310
. do )


s2  
3
2
3
2
4n 6πa + 24 fa + f + 3 f aπ 1 +

 12a 2  

(4.55)
and
π (d o + d i ) s 3 πd m s 3
Im =
=
24
12
(4.56)
33
An element force, dP, at any location at the average diameter is
dP = ez BI m
dϕ
2π
(4.57)
From figure 4.16 it can be seen that
ez = ez max
0.5d m sin ϕ
0.5d m
(4.58)
Substituting equation 4.56 and 4.58 into equation 4.57 gives
dP = ez max B
s 3d m
sin ϕdϕ
24
(4.59)
Multiplying the element force, dP, by the lever, 0.5dmsinϕ, give the element
moment
dm
ez max Bs 3 d m2
dM = dP
sin ϕ =
sin 2 ϕdϕ
2
48
(4.60)
Integration around the circumference gives the total moment
ez max Bs 3 d m2
M=
48
2π
ez max Bπs 3 d m2
∫ sin ϕdϕ =
48
0
2
(4.61)
The maximum deflection at the tension side of the bellow can be written as
ez max =
dm
θ
2
(4.62)
Combining equation 4.52, 4.55, 4.61 and 4.62 gives the bending stiffness
kθ =
Eπs 3 d m3 (4.602 + 6 ⋅ 10 7 a 3 − 4310
. do )


s2  
3
2
3
2

16n 6πa + 24 fa + f + 3 f aπ  1 +

 12a 2  

(4.63)
The maximum stress for a bellow during axial load, Pz, was derived in
section 4.1 and can be expressed as
σ max =
3 Pz
(d − d i )(0.874 − 213.7a + 713.7 s)
2πs 2 d i o
34
(4.64)
Substitution of equation 4.53, 4.54 and 4.56 into equation 4.64 gives the
maximum stresses during an axial displacement, ez.
σ max =
ez Bd m s
(d o − d i )(0.874 − 213.7a + 713.7 s)
8d i
(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 maximum
deflection is
ez max =
48 M
Bπs 3 d m2
(4.66)
The maximum stress during a bending load occurs at the maximum
deflection and by substituting equation 4.66 into equation 4.65 the
maximum stress becomes
σ max =
6 M (d o − d i )
d i πs 2 d m
(0.874 − 213.7a + 713.7 s)
35
(4.67)
4.3 Torsion Load
The bellow is not primarily designed to deal with torsion loads. Because of
its high stiffness, it can only stand very small rotational displacements. In
this section the relationship between the torsion stiffness and the length of
the bellow, and the stresses in the bellow when exposed to a rotational
displacement, 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 for
the top- and bottom elements, and the plate theory can be used for the flank
elements of the bellow.
Top element
Flank element
a
s
a
Bottom element
do
di
Centerline of bellow
Figure 4.17. Simplified geometry description of one convolution.
4.3.1 Flank Calculations
The flank is treated as a plate, see figure 4.18. Due to the geometry and load
on the plate, plane stress theory can be used, (σz=0). The aim is to find the
relationships between torque, rotational displacements and stresses in the
plate.
36
Tro
ro
ϕ
Tri
r
ri
Figure 4.18. General plate description.
According to for example Timoshenko [4], the solution of two-dimensional
problems without mass forces is determined by
∆∆φ = 0
(4.68)
where ∆ is the Laplace operator for polar co-ordinate systems, see equation
4.69, and φ(r,ϕ) is a stress function.
∂2
∂
∂2
∆= 2 +
+
r∂r r 2 ∂ϕ 2
∂r
(4.69)
For a circular plate subjected to torsion, one possible stress function that
both fulfils equation 4.68 and the boundary conditions, with one edge
grounded and the other edge free, is
φ (r , ϕ ) = C1ϕ
(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.

1 ∂φ 1 ∂ 2 φ
σ r = r ∂r + r 2 ∂ϕ 2


∂ 2φ
σ ϕ = 2
∂r


1 ∂φ 1 ∂ 2 φ
τ rϕ = 2 ∂ϕ − ∂ ∂ϕ
r r
r

(4.71-4.73)
37
Combining equation 4.70 through 4.73 gives

σ r = 0

σ ϕ = 0

τ rϕ = C21

r
(4.74-4.76)
The shear stress do not depend on ϕ and the stresses at the inner and outer
radius of the plate can be written
C1

τ
=
ri

ri 2


τ = C1
 ro ro 2
(4.77-4.78)
These stresses give the resulting moments
Tri = τ ri 2πri sri

Tro = τ ro 2πro sro
(4.79-4.80)
Combining equation 4.77 and 4.78 with 4.79 and 4.80 gives
Tri = Tro = 2πsC1
(4.81)
The constant C1 is thus
C1 =
T
2πs
(4.82)
The stress function, equation 4.70, and the shear stress distribution function,
equation 4.76, can now be rewritten as
 T 
φ (r , ϕ ) = 
ϕ
 2πs 
(4.83)
T
2πsr 2
(4.84)
τ rϕ =
38
To determine the rotational angle the strain components

∂er
ε r =
∂r


1 ∂eϕ er
+
εϕ =
r
r
∂ϕ


∂eϕ eϕ 1 ∂er
γ rϕ =
−
+

r r ∂ϕ
∂r
(4.85-4.87)
and Hooke´s law
(
(
1

εr = E σ r − νσ ϕ

1

σ − νσ r
εϕ =
E ϕ

1

γ rϕ = G τ rϕ
)
)
(4.88-4.90)
where G is
G=
E
2(1 + ν )
(4.91)
have to be used. By using the stress function, equation 4.83, and inserting it
into equations 4.71 through 4.73, equation 4.85 through 4.90 give
 ∂er
=0

 ∂r
 er ∂eϕ
=0
 +
 r r∂ϕ
 ∂e
∂eϕ eϕ
T
 r +
−
=
 r∂ϕ ∂r
r
2πsGr 2
(4.92-4.94)
Knowing that there are no changes in the ϕ-direction gives
er = 0

 ∂eϕ eϕ
T
 ∂r − r = 2πsGr 2

(4.95-4.96)
39
The solution of equation 4.96 is given by
eϕ = C2 r −
T
4πsGr
(4.97)
where C2 is an arbitrary constant.
Assuming that the inner edge of the plate is grounded and the outer edge is
subjected to a torsion moment, T, gives the boundary conditions
r = ri

eϕ = 0
(4.98-4.99)
Equation 4.97 can now be written as
eϕ (r ) =
T  r 1
 − 
4πsG  ri 2 r 
(4.100)
The torsion stiffness, (Nm/rad), of the plate is given by
ro T
4πsGri 2
1
k=
= 4πsG
=
2
1
1
eϕ (ro )
 ri 
−
ri 2 ro2 1 −  r 
o
(4.101)
The total number of flanks is 2n, see figure 4.17, which gives the total
stiffness for all flanks in the bellow
k ftot =
4πsGri 2
2πsGri 2
=
  r  2
  r  2
i
2n 1 −    n 1 −  i  
  ro  
  ro  
(4.102)
By using equation 4.84 and 4.102 the stress distribution and the torsion
stiffness 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 using
the beam theory. The polar moment of inertia is determined by equation
4.103 for the top element, and by equation 4.104 for the bottom element.
J t = 2πro3 s
(4.103)
40
J b = 2πri 3 s
(4.104)
The stresses in the elements are given by
τt =
T
r
Jt o
4.105)
τb =
T
r
Jb i
(4.106)
The total top and bottom element lengths are determined by
Lt = Lb = 2na
(4.107)
The total torsion stiffness for the elements is determined by equations 4.108
and 4.109.
k ttot =
GJ t
Lt
(4.108)
k btot =
GJ b
Lb
(4.109)
4.3.3 Summary of Torsion Load
The total torsion stiffness of the bellow can now easily be obtained by using
the relationship below.
1
1
=∑
k tot
k
(4.110)
Combining equation 4.102, 4.108 and 4.109 and using equation 4.110 gives
the total torsion stiffness
k tot =
G

  r 2

n 1 −  i   
L
  ro   
L
 b + t +

2πsri 2
 Jb J t





41
(4.111)
In figure 4.19 the torsion stiffness for a bellow as a function of the number
of convolutions can be seen.
5
Torsion stiffness (Nmm/rad)
8 10
5
6 10
k tot ( n )
5
4 10
5
2 10
0
5
10
15
20
25
30
35
n
Number of convolutions
Figure 4.19. Torsion stiffness for different number of convolutions.
The stress distribution in the bellow is determined by the following three
equations
τ rϕ =
T
2πsr 2
(4.112)
τb =
T
r
Jb i
(4.113)
τt =
T
r
Jt o
(4.114)
The stress in the bottom element is the same as at the inner edge of the
flank, and the stress in the top element is the same as at the outer edge of
the flank. This means that it is enough to study the stress distribution in the
42
flank to get the total stress distribution. In figure 4.20 the stress distribution
in the flank is given for T=40 Nm.
50
Shear stress (MPa)
40
30
τ f( r )
20
10
25
30
35
40
r
Radius (mm)
Figure 4.20. Shear stress distribution.
43
45
50
5. Experimental Verification
To verify the theoretical models derived in chapter four, an experimental
investigation of stiffness and stresses during different loads is done. The
measurements are done for two different bellows. The convolution
geometry of the bellows is shown again in figure 5.1.
a
s
a
do
di
Centerline of bellow
Figure 5.1. Convolution geometry for the bellows used for measurements.
44
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 axial
deformation of the bellow is measured by using a dial indicator, not shown
in the figure.
Figure 5.2. Set-up for axial load verification.
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 and
the applied force give the axial stiffness of the bellow.
The strain is measured with a 0°/45°/90° rosette, which is made of three
separate gages mounted on the top of the convolutions. The strain gages
have an active length of 0.6 mm. The results are presented in table 5.1 and
in figure 5.3.
45
5.1.2 Results
Load
Displacement
Gage 1, 0°
Gage 1, 45°
Gage 1, 90°
P [N]
e [m]
ε0 [µstrain]
ε45 [µstrain]
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. The
theoretical axial stiffness, according to equation 4.50, is 15600 N/m. The
agreement for the axial stiffness is excellent.
The principal normal stresses, σ1,2, are calculated according to
σ 1 ,2 =
E ε 0 + ε 90
E
±
1− ν
2
2 (1 + ν )
(ε
− ε 45 ) + (ε 90 − ε 45 )
2
0
2
(5.1)
The von Mises stress at the top can now be determined by
σe =
1
2
(σ
− σ 2 ) + σ 12 + σ 22
2
1
46
(5.2)
A comparison between the theoretical model and the test results is shown in
figure 5.3. The maximum deviation between measured and theoretical
values is 15%. Considering uncertainties about the material in the measured
bellow and its geometry, the agreement is very good.
250
von Mises Stress [MPa]
200
Measured
values
150
100
Theoretical
model
50
0
0
20
40
60
80 100 120 140 160 180
Axial Load [N]
Figure 5.3. Verification of stresses for axial load.
47
5.2 Bending Load Verification
5.2.1 Experimental Set-up
The set-up for measuring during bending loads is principally the same as for
axial loads except that the load is applied in a perpendicular direction. The
load is applied at a distance, L = 0.272 m from the fixed end, see figure 5.4.
Figure 5.4. Set-up for bending load verification.
This way of applying the load gives a linear moment distribution along the
bellow. To be able to compare the measured bending stiffness, where the
load is a force, with the theoretical bending stiffness, where the load is a
moment, a correction factor has to be derived.
48
By comparing the two elementary load cases for beams with one end fixed
and the other end subjected to a force or a moment, the bending stiffness
can be expressed as
kθ =
P L2
e 3
(5.3)
The equipment for measuring of displacement and strain are also equivalent
to the one used for axial load measurement and the stresses are calculated in
the same way.
5.2.2 Results
Load
Displacement
Gage 1, 0°
Gage 1, 45°
Gage 1, 90°
P [N]
e [m]
ε0 [µstrain]
ε45 [µstrain]
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/rad
and according to equation 4.65, the theoretical bending stiffness is 11.42
Nm/rad. The agreement for the bending stiffness is very good.
49
The principal stresses and the von Mises stresses are calculated according to
equation 5.1 and equation 5.2. A comparison between the theoretical model
and 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 its
geometry, the agreement is very good.
90
80
von Mises Stress [MPa]
70
Measured
values
60
50
40
Theoretical
model
30
20
10
0
0
0,273
0,546
0,819
1,092
1,365
Bending Moment [Nm]
Figure 5.5. Verification of stresses for bending load.
50
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 other
end, 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 a
convolution top in the axial direction. If the load is pure torsion, then there
will 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 to
measure the small angular changes. The torsion stiffness will therefore be
verified by comparison with FEM-calculations.
The strain is measured with a gage mounted on the top of a convolution
directed in 45° angle to the axial direction. This is done because torsion
load gives principal strains in the 45° direction. The strain gages have an
active length of 0.6 mm. The results are presented in table 5.3 and in figure
5.7.
51
5.3.2 Results
Load
Gage, 45°
[Nm]
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. The
FEM-calculated stiffness is 16100 Nm/rad. A comparison between the
theoretical 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, its
geometry and difficulties in applying a perfect torsion load, the agreement is
very good.
45
40
von Mises Stress [MPa]
35
Measured
values
30
25
20
Theoretical
model
15
10
5
0
0
25
50
75
100
Torque [Nm]
Figure 5.7. Verification of shear stresses for torsion load.
52
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 stiffness
and 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 analytic
expressions, FEM-calculations and linear regression was used to derive the
theoretical expressions for axial load. To be able to do a good regression
analysis, the bellow dimensions were limited to the dimensions that are
interesting for practical manufacturing and for use in normal car exhaust
systems according to AP Parts Torsmaskiner Technical Center AB, see
chapter 4.1.4. The axial load expressions were also used for deriving the
theoretical expressions for bending load.
The theoretical expressions derived for torsion load was done by
simplifying the bellow geometry. This was done to get analytical and simple
expressions for the stiffness and the stresses in the bellow. These
expressions do not have any theoretical limitations for the bellow
dimensions but they have only been verified for the same dimensions as the
dimensions used for axial and bending loads.
An experimental verification of the theoretical models was carried out. The
agreement between theoretical and experimental results is very good. The
expressions for the stiffness of the bellow derived in this work are probably
suitable also for investigation of the dynamic behaviour of flexible bellows.
53
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, New
York, (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).
54
Department of Mechanical Engineering, Master’s Degree Programme
University of Karlskrona/Ronneby, Campus Gräsvik
371 79 Karlskrona, SWEDEN
Telephone: +46 455-78016
Fax: +46 455-78027
E-mail: Goran.Broman@ima.hk-r.se