ATOMIC ENERGY WffR L`ÉNERGIE ATOMIQUE OF CANADA

Transcription

AECL-6476
ATOMIC ENERGY
OF CANADA UMITED
WffR
U B f
L'ÉNERGIE ATOMIQUE
DU CANADA LIMITÉE
ELASTIC STRESSES IN U-SHAPED BELLOWS
Contraintes élastiques dans les soufflets en forme de U
P. JANZEN
Chalk River Nuclear Laboratories
Laboratoires nucléaires de Chalk River
Chalk River, Ontario
May 1980 mai
ATOMIC ENERGY OF CANADA LIMITED
by
P. Janzen
Chalk River, Ontario KOJ 1JO
1980 May
AECL-6476
L'Energie Atomique du Canada, Limitée
Contraintes élastiques dans l e s s o u f f l e t s en forme de II
par
P. Janzen
Résumé
Ce rapport présente des r e l a t i o n s décrivant les niveaux de contraintes
élastiques méridionales et c i r c o n f é r e n t i e l l e s à l'embase et à l a t ê t i è r e à
cause de l a pression externe et de l a f l e x i o n a x i a l e des s o u f f l e t s en forme
de U. La d é r i v a t i o n est basée sur l ' a n a l y s e s t a t i s t i q u e de données
théoriques obtenues à p a r t i r d'une analyse des éléments f i n i s de
c o n f i g u r a t i o n s choisies de s o u f f l e t . Les formules mathématiques et
diverses représentations graphiques sont proposées comme aides pour l a
conception et l ' a n a l y s e des s o u f f l e t s .
Laboratoires nucléaires de Chalk River
Chalk River, Ontario KOJ 1J0
Mai 1980
AECL-6476
ATOMIC ENERGY OF CANADA LIMITED
by
P. Janzen
ABSTRACT
This report presents relations describing the meridional
and circumferential elastic stress levels at the root
and crown due to external pressure and axial deflection
of U-shaped bellows. The derivation is based on a
statistical analysis of theoretical data obtained from a
finite element analysis of selected bellows configurations.
The mathematical formulations and various graphical
representations are proposed as aids to bellows design
and analysis.
Chalk River, Ontario KOJ 1J0
1980 May
AECL-6476
iii
TABLE OF CONTENTS
page
1.
INTRODUCTION
1
2.
PERSPECTIVE
1
3.
REFERENCE CONVOLUTION SHAPE
4
4.
STRESS ANALYSIS
4
5.
APPROACH TO STATISTICAL STUDY
6
6.
REGRESSION ANALYSIS
7
7.
RESULTS AND DISCUSSION
11
8.
CONCLUSION
12
9.
ACKNOWLEDGEMENT
13
10.
REFERENCES
13
TABLES
FIGURES
iv
LIST OF SYMBOLS
e, exp
exponential function (base of Napierian
d
Z
nc
n
p
logarithmic system)
convolution depth
bellows' active (convoluted) length
bellows' number of convolutions
bellows' number of material plies
external pressure
r
convolution crown torus radius
c
r.
r =
r
t
E
F
K
S
S.
S
°S
S, =
_
S =
m
Y
C
r. + d/2
os " S is
j
S
+ S.
os
is
T
z
bellows convolution inner radius
bellows convolution mean radius
convolution root torus radius
material ply thickness
material Young's modulus
stress function correction factor
bellows axial spring rate
stress function
inner surface tangential stress
outer surface tangential stress
S
i
—
r
i
YF = t
~±
a., i • 0,1,••',5
3.» i = 0,1,»••,4
bending stress component
.
membrane stress component
non-dimensional configuration parameters
exponents of configuration parameters
exponents of non-dimensional configuration
parameters
ô
axial deflection per convolution
r
À. = —
m
radius ratio
X- = —p-
thickness ratio
y = /l2(l-v 2 )A 1 A 2
shape parameter
v
material Poisson's ratio
<)>
angle measured in the meridional plane
A = n c<5
total bellows axial deflection
1.
INTRODUCTION
Although bellows have numerous applications,
frcm large,low-pressure expansion joints in air ducts
and pipelines to small, high-pressure bellows stem
seals in valves, little information on design or
analysis of bellows can be found in published literature. The information which is available is based
on approximate solutions which frequently result in
unacceptable discrepancies.
This report derives equations describing the spring
constant, and the meridional and circumferential
elastic stress levels at the root and crown of Ushaped bellows due to pressurization and axial deflection. The derivation is based on a statistical analysis
of theoretical data obtained from a finite element analysis of
selected bellows configurations. These mathematical
formulations and various graphical representations are
proposed as aids to bellows design and analysis.
2.
PERSPECTIVE
The analysis of bellows to determine its response to
various applied loads has received attention over
many years. Selected contributions will be mentioned
to place this subject in perspective.
Theoretical treatments of this problem have exploited
two basic mathematical techniques designated "approximate solutions" and "asymptotic solutions" (1) .
Each approach has a limited range of validity related
to some geometric parameter. Parameters recurring in
published literature include a "radius ratio"
\± = rc/rm, a "thickness ratio" \2 = rc/t and a
combined shape parameter
/12(1-V2) A ^
= À2(l-v 2 )r c 2 /r m t.
Figure 1 defines the geometric variables and r m
+ d/2.
- 2 -
The 'approximate solution" employs a truncated infinite
series expansion of the governing variables to solve
the exact theory equations. Its applicability is
limited to small values of X^(Al<0.1) and an approximate range of A 2 from 5 to 3 5
For large values of A2(^2>:1-5), the differential
equations of the exact theory can be reduced to permit
a closed form solution in terms of higher transcendental
functions. This approach yields the "asymptotic
solution".
R.A. Clark has published asymptotic solutions for an
"Omega" piping expansion joint subjected to axial load,
a corrugated pipe subjected to axial displacement and
internal pressure (2), and a U-shaped bellows subjected
to axial load (3) .
The same case of a complete torus slit at the inner
edge and welded to a stiff cylindrical pipe subjected
to an axial load was also considered by N.C. Dahl (4).
However, he developed an approximate solution utilizing the principle of minimum complementary energy and
a four-term series to approximate the change in the
meridional tangent angles. Over a range of values of
p his solution corresponds to that of R.A. Clark.
Beyond u = 30 the solutions diverged, with the
asymptotic solution possessing greater accuracy.
U-shaped expansion joints have been the subject of
more recent papers. A. Laupa and N.A. Weil (1) used
an energy method for the toroidal sections, and the
theory of symmetrical bending of circular plates
augmented by the thick-walled cylinder analysis for
the annular plate connecting the two toroidal sections.
Their general solution permitted different radii to
be assigned to the inner and outer toroidal sections
and independent changes in the average thickness of
the toroidal sections and the interconnecting annular
plate. Axial and pressure loading were both considered. T. Ota, M. Hamada, T. Takezono, Y. Inove,
T. Nakatani and M. Moriishi (5,6,7,8,9) extended this
analysis slightly and included design charts to
illustrate the behaviour of the maximum meridional
bending and circumferential membrane stresses as
functions of geometric parameters.
While manufacturers have had some design formulas
at their disposal for some time, either of a proprietary
or published nature (10), the proliferation of thin-ply
- 3 -
formed bellows, especially in finite life applications,
has revealed a need for more exact design criteria.
A number of reports have been issued in an attempt
to correct this deficiency. R. A. Winborne (11) and
W.F. Anderson (12) have presented design procedures
based on asymptotic shell theory and charts for
"curvature corrections". A more comprehensive report
by T.M. Trainer (13) summarized a three-year
project whose objective was "to establish design
procedures, stress analysis methods, and other factors
essential to the successful design and fabrication of
metallic bellows". Theoretical results of the bellows
response to axial and pressure loads were obtained
utilizing a multi-segment numerical-integration
technique. A computer program, capable of linear
elastic axisymmetric and non-symmetric deformation
analysis and of non-linear axisymmetric deformation
analysis was developed. Parameters pertinent to
bellows design and fabrication were identified and some
parametric curves to permit convolution shape
optimization were presented. Although the report
was extensive, its usefulness as a practical design
aid is limited. No mathematical design formulas
are presented and the parametric curves are not given
in sufficient detail to aid in design.
Many of the above publications, in addition to comparing theoretical results with previously published
results, also reported some experimental results in
support of their analysis. One of the more enlightening papers on experimental results was prepared by
C E . Turner (14). He utilized an approximate solution
for the range of y<5 as well as a numerical solution
for an extended range of JJ from 0.4 to 35. Of most
significance, he noted a number of sources for the
discrepancies between theoretical and experimental
results. Among these were the following:
1.
The strain gage thickness is not negligible
in comparison with the ply thickness;
2.
The gage grid spans an appreciable arc length
of the convolution torus. Since the gage
measures an average strain over the grid length,
it will not indicate the maximum strain in a
region of a strain gradient;
3.
Bellows generally possess a variable ply thickness along a convolution profile which is
difficult to anticipate analytically;
4.
Nominally similar convolutions exhibited different
strain levels at corresponding locations.
- 4-
Even after attempting to account for all the known
experimental errors, C.E. Turner (14) estimated potential
errors between theory and experiment in the range
of + 10%, in extreme cases + 20%.
Similar limitations of experimental studies have been
observed by the author. In addition, an effect due
to ply interaction in multi-ply bellows has been
noted.
3.
REFERENCE CONVOLUTION SHAPE
Although convolution shapes can be categorized, they
are far from standardized. This is mainly due to
the proprietary manner in which bellows have been
developed: methods of fabrication and production
machinery itself have often been developed in-house.
Because other than shape parameters can affect
bellows performance, optimum convolution configurations may vary from one manufacturer to another.
Localized changes in ply thickness and material
properties, for example, will influence bellows
performance. These differences in convolution
characteristics are especially apparent in small,highpressure multi-ply bellows.
For the purpose of this study, an idealized,axisymmetric,
U-shaped convolution with uniform material properties
and thickness will be adopted as the reference
configuration. The physical parameters describing
this shape are the internal radius, r^, depth of
convolution, d, root and crown torus radius, r r and
r c , and thickness, t (Figure 1 ) .
4.
STRESS ANALYSIS
Stress states in bellows resulting from axial
compression and external pressure were determined
by the finite element method. Typical surface stress
distributions are depicted in Figures 2 and 3.
Thus, subjected to axial compression, a U-shaped
bellows exhibits peak meridional surface stresses at
root and crown. Resultant circumferential surface
stresses are lower and more variable, not necessarily
peaking at root and crown. All surface stresses
approach zero at midspan.
- 5 -
External pressure causes peaks in all surface stresses
at three locations: root, crown and midspan. Although
of slightly different magnitudes, each principal
stress has the same sign at root and crown which is
opposite to that at midspan.
Theoretical analyses of stress states in thin shells
generally assume a linear strain variation with
distance normal to the neutral surface. In bellows
convolutions, the ratio of torus radius to thickness
can be small enough to yield a significantly nonlinear strain distribution at root and crown. A
good approximation of the resulting stress distribution
is given by the Winkler-Bach formula for curved beams.
The stress distribution has a hyperbolic pattern,
attaining a maximum value at the concave surface in
the case of pure bending. A study of the stress
distribution obtained from a finite element analysis
of a bellows bears this out.
Due to the large circumferential radius of curvature
the thickness ratio, r/t, is very large. The stress
distribution resulting from loading should be
practically linear and this is corroborated by the
finite element analysis of bellows.
While the maximum meridional stress in bellows due to
axial deflection frequently occurs at mid-crown and
mid-root (<{>=0) , this is not always the case.
N.C. Dahl (4) found that for toroidal bellows the
location of maximum stress moved away from <f>=0 for
U>5, approaching <J>=ir/2 for large y. According to
the results of the study of U-shaped bellows by T.M.
Trainer (13), the location of maximum stress is at
(J>=0 for y<0.75. For a transition range of y from
0.75 to 2 an approximately constant maximum stress
level exists over a portion of the torus. For values
of v>>2 the location of the maximum stress tends to
<J>=7r/2 with a significant decrease in stress occurring
as
A similar situation exists for the case of bellows
subjected to external pressure with the difference
that the maximum meridional stress occurs at (j>=0 of
the crown and root for p<5, the transition range is
5<y<10, and for \i>10 the location of maximum stress
tends to (j)=ir/2, again with a significant decrease in
stress occurring as <(>-*-0.
The distribution of circumferential stress is more
complex. In the case of axial compression, extreme
- 6 -
levels generally occur near <J>=0 in root and crown.
Depending on the bellows geometry, however, the
location of peak stress may shift to <j>>0. External
pressure can cause the circumferential stress state
to exhibit several locations of extreme levels.
5.
APPROACH TO STATISTICAL STUDY
The study of the elastic states of stress in bellows
due to axial compression and external pressure reveals
a complex picture: the stress distribution changes
with change in any geometric parameter; the location
of the extreme meridional stress levels changes from
<|)=0 for small y to 0»ir/2 for large y; external
pressure causes an extreme meridional stress level at
midspan as well as the crown and root; except for
small y the locations of extreme meridional stress
levels due to axial deflection and external pressure
do not coincide; the circumferential stress distribution is difficult to categorize. Some assumptions
must be stated to reduce the problem to manageable
proportions.
It will be assumed that the meridional and circumferential stress levels at <j>=0 characterize the
bellows performance capability. Bellows which exhibit
extreme meridional stress levels due to axial
deflection and external pressure at cj>=0 fall in the
range of y<2. Such bellows have widespread application, generally at low pressure and where axial
flexibility is important. Higher pressure may be
readily accommodated by increasing the number of
plies. The associated circumferential stress does
not generally peak at <J>-0, but the resultant stress
due to the combination of axial deflection and
external pressure will be a maximum near cf>=0.
The objective is to derive a response surface for
the stresses at root and crown in terms of bellows
geometrical parameters. Five geometrical parameters
were considered essential to the statistical study:
rc» *r> *i> d, t (Figure 1 ) . Selected bellows
configurations were analysed by the finite element
method to generate the relevant data.
A "halfreplicate two level fractional factorial design"
was chosen so that the number of test cases required
would be minimized. Then, to permit an evaluation
of the quadratic terms in the response function a
"star design" (15) with center point was superposed
on the fractional factorial. A frequent practice
- 7 -
is to make such a design "rotatable" by a proper
choice of the length of the axis arm of the star
design, thus reducing the complexity of mathematical
computation. The preferred choice of the variable
levels precluded this practice. Moreover, the
least squares method was to be utilized in the
regression analysis of the data and rotatability
was not an essential characteristic. The result
is a non-rotatable "central composite design" in
five variables as shown in Table 1.
Two such designs were considered in the study: one
in a range of low values of all parameters; one in
a range of higher values of all parameters. Table 2
gives the ranges of the parameters for each design.
Taken together these arrays of tests permitted the
inclusion of higher order terms in the mathematical
model of the response surface.
In the first design the values of y ranged from 0.09
to 2.64. In the second design y ranged from 0.29
to 2.28.
6.
REGRESSION ANALYSIS
Conventional formulas for the maximum meridional
stress in a bellows are generally expressed as the
product of operational (pressure or axial deflection),
physical (number of plies or convolutions) and
geometrical parameters, the last raised to a power.
Such an expression was adopted as the mathematical
model for the response surface:
S
a
a
a
a
a
a
= e ° r l r 2 d 3 r.* t 5
c
r
i
...(1)
Here S is the stress, and a.,i=0,l,...,5 are unknown
exponents to be evaluated for a best fit of the
function to the data.
In the case of axial deflection, stress is proportional
to deflection and Young's Modulus. Thus
_
(3 a
a
a
a
a
f • ie • r ' i ! d > r.» t !
£i
c
r
...(2)
l
This expression is dimensionless.
Then the sum of
- 8 -
oti, 1 • 1,...,5 must add to -1.
A simplification
may be introduced by dealing in the dimensionless
ratios :
Y
Y
B " ^
C
Then
„
S.
Ci
*
=
•^ _
r .
3
e
B
0
y
A
3
l
3
2
Y
3
3
V
ii
y
l>
!
&
If the bellows under consideration possessed n c
convolutions and experienced a total deflection
A = n c 6, then
n
r S
—
—
A E
B
6
- e ° Y
e
X
A
B
1
2
Y
Y
B
B
Y
Y
C
3
6
Y "
Y
E
Length and deflection are common design constraints.
Further, the number of convolutions per length and
hence the deflection per convolution, will depend on
convolution pitch. It is therefore not immediately
obvious which convolution configuration will have the
more favourable stress state.
Therefore, designers may find it advantageous to relate
the bellows axial deflection to length rather than
the number of convolutions.
For U-shaped bellows the relation between length, i,
and the number of convolutions is
% = 2nc(rc + rr)
.. .(6)
The stress parameter can then be rewritten
s*2
2
0
A(Y. + Y,)E
A
a
s
1 e
3_
B
0
2
Y
A
Y
3
1
3
*B
Y
Y
Y
B3 B
C
I
Y
Y
E
- 9 -
A similar line of reasoning yields the. relation for
stress resulting from external pressure!, p.
3
3
6
Y
A
3
Y
B
3
Y
C
3
Y
i
E
i
•••(»>
where n n is the number of plies in the bellows.
P
I
While the mathematical models, Eq.(7) ar>d (8) are
not linear, they can be made so by taking their
logarithm. A multiple linear regression! analysis
utilizing the least-squares method is thjen employed.
This analysis minimizes the logarithm of; the errors
rather than the actual errors.
These formulations were adopted for both membrane
and bending stresses. Estimates of these stress
components were based on the surface stresses obtained
from the finite element method:
S
=
m
OS
=
+ Si s
2
...(9)
S
S
b
os
- S
is
2
where S o s , S±s, are the outer and inner surface
stresses, S m , Sb the membrane and bending components
of surface stresses.
In view of the non-linear
stress distribution, S m and Sb in the meridional
direction are fictitious quantities. The circumferential components however should be accurate estimates.
Design of structural components involving bellows
often requires a knowledge of forces developed due to
axial deflection. To meet this need a response
surface for bellows spring rate was derived based on
the mathematical model
n K
FT
r
i
E
oK
=
-2—
r? (YA + Y n )E
3
3
3
3
e
- ° V V V
A
B
C
- 10 K is the spring rate.
While the response functions (7), (8) and (10) give
acceptable accuracy for many applications, an
improved fit will result through a subsequent
derivation of a correction factor
F = exp(f(Yi))
. . .(11)
The function f(Yi) is a polynomial of products of
the geometric parameters. Terms of f(Yi) significant
in improving the fit of the response function were
identified by a computer program using a multiple
linear regression analysis with a forward stepwise
algorithm. Within the range of geometrical parameter
levels considered some of the circumferential stresses
changed signs. This precluded a direct application
of Equations (7) and (8). Instead, absolute values
of stress were first introduced into Equations (7)
and (8) to evaluate the unknown exponents. Subsequently a correction factor in the form of a polynomial
in the geometrical parameters, F = f(Y^), was introduced
to allow for the change in sign.
The response functions then take the form
-
-
3. 33. 3
c
AE
3
°V
(Y A +Y f i )AE
c
for axial deflection;
n_ S
3
3
3
3 3
o v lv 2 v^s Y^U
...(12)
for external pressure;
,-
3
F
4
(YA+YB)E
for the axial spring rate.
e
3
Y
A
Y
3
Y
B
3 3
C
- 11 -
7.
RESULTS AND DISCUSSION
The regression analysis according to the mathematical
models Equations (7), (8) and (10) yielded the results
in Tables 3(a) and 3(b). These oversimplified models
exhibit acceptable fit for many applications and
indicate the effect of changes in the various geometrical parameters.
According to the sum of 3., i=l,...4, all meridional
stress components, S, due to axial compression and
external pressure are practically independent of
inner radius, rj.. This is also the case for
circumferential bending stress components. The circumferential membrane stresses due to axial compression
are approximately inversely proportional to inner
radius. Under external pressure the circumferential
stress components are approximately proportional to
inner radius.
All bending stress components decrease at root and
crown for respective increases in torus radius. A
change in root torus radius has little effect on
bending stresses at the crown, and vice versa. This
is also the case for all meridional membrane stress
components. The circumferential membrane stress
components are affected by changes in either torus
radius. In the case of axial deflection, these
relationships are valid for unit deflection per
convolution. If axial deflection is set per unit
length the effect is more complicated.
Convolution depth is the most influential configuration parameter, closely followed by ply thickness.
An increase in convolution depth or ply thickness
results in a decrease and increase, respectively, of
most stress components due to axial compression and
vice versa for external pressure. Some circumferential
membrane stress components decrease in magnitude for
increase in both configuration parameters.
The axial spring rate is approximately proportional
to the inner radius and practically independent of
torus radii when expressed in units per convolution.
Ply thickness and convolution depth are the dominant
configuration parameters.
Tables 4(a) to 4(d) present formulas for the correction
factor F, which includes the sign of the stress
components. For the range of parameter levels
considered, the magnitude of F is close to 1.
- 12 -
Because the regression analysis approximates the
meridional non-linear stress distribution across
the ply thickness by a linear stress distribution,
the membrane and bending stress components, separately,
will be slightly in error. Should an accurate
estimate of the membrane stress be desired, Table 4{e)
presents appropriate formulas for the effect of
external pressure acting on the convoluted surface
only. For axial deflection, the axial spring rate
may be used.
To help visualize the response functions of Equation
(12), Figures 4 to 9 show the relationship of
meridional stress components at the convolution root
versus a configuration parameter for bellows subjected
to axial compression and external pressure.
Because these formulas are based on a statistical
derivation, some error is likely to be present in most
estimates. For the selected configurations of this
study the predicted meridional stresses agreed with
calculated stresses to within a maximum of five per cent.
Errors for the circumferential stresses were higher,
reaching a maximum of about seven per cent for bending and more for the membrane stresses. The latter
errors were due to the low magnitude of the calculated
stresses for the selected configuration.
Results of the analysis were also compared with those
of several sources for the example given in Â. Laupa
and N.A. Weil (1). Although the convolution configuration
possessed a characteristic 11-1.47, the dimensions
were outside the range of parameter levels considered
in this study. Nevertheless, as Tables 5(a) to 5(h)
show, the statistically derived response functions
yield results which approach those of the finite
element method more closely than many design formulas
proposed in literature or in current use.
8.
CONCLUSION
Response functions in terms of geometrical variables
*i> rr» r c » ^ an<* *• a r e derived for U-shaped bellows
with a characteristic shape parameter y<2 for bending
and membrane stress components at root and crown in
the meridional and circumferential directions due to
axial compression and external pressure, for axial
- 13 -
spring rate and for the actual meridional membrane
stress due to external pressure acting on the
convoluted surface only. Suitable for design or
analysis, these formulas represent an improvement
in the accuracy of response estimates and in the
simplicity of calculations over earlier design
formulas.
9.
ACKNOWLEDGEMENT
The finite element computer program utilized in this
study was developed by R. Shill of Atomic Energy of
Canada Limited, Chalk River.
10.
REFERENCES
A. Laupa, N.A. Weil, Analysis of U-Shaped
Expansion Joints, J. of Applied Mechanics,
March 1962, pp 115-123.
R.A. Clark, On the Theory of Thin Elastic
Toroidal Shells, J. of Mathematics and Physics,
vol. 29, 1950, pp 146-178.
R.A. Clark, An Expansion Bellows Problem, J. of
Applied Mechanics, March 1970, pp 61-69.
N.C. Dahl, Toroidal-Shell Expansion Joints,
J. of Applied Mechanics, December 1953, pp 497503.
T. Ota, M. Hamada, On the Strength of Toroidal
Shells, Bulletin of JSME, vol. 6, No. 24, (1963),
p 638-654.
M. Hamada, S. Takezono, Strength of U-Shaped
Bellows, (1st. Report, Case of Axial Loading),
Bulletin of JSME, vol. 8, No. 32, (1965),
pp "=25-531.
M. Hamada, S. Takezono, Strength of U-Shaped
Bellows, (2nd Report, Case of Axial Loading Continued), Bulletin of JSME, vol. 9, No. 35,
(1966), pp 502-513.
- 14 -
8
-
M. Hamada, S. Takeaona, Strength of U-Shaped
Bellows, (3rd Report, Case of Loading of
Internal Pressure), Bulletin of JSME, Vol. 9,
No. 35, (1.966), pp 513-523.
9.
M. Hamada, Y. Inove, T. Nakatani, M. Moriishi,
Design. Diagrams and Formulae for U-Shaped
Bellows, Int. J. of Près. Ves. and Piping, (4),
1976, pp 315-328.
10.
The M.W. Kellogg Company, Design of Piping
Systems, John Wiley & Sons Inc., Revised 2nd
Edition, 1955, pp 214-230.
11.
R.A. Winborne, Simplified Formulas and Curves
for Bellows Analysis, NAA-SR-9848, Atomics
International, California, USA, August 1964.
12.
W.F. Anderson, Analysis of Stresses in Bellows,
Part 1: Design Criteria and Test Results,
Atomics International, NAA-SR-4527, October
1964.
13.
T.M. Trainer, Final Report on the Development
of Analytical Techniques for Bellows and
Diaphragm Design, T.R. Ho. AFRPL-TR-68-22,
Battelle Memorial Institute, Columbus, Ohio,
March 1968.
14.
C E . Turner, Stress and Deflection Studies of
Flat-Plate and Toroidal Expansion Bellows
Subjected to Axial, Eccentric or Internal
Pressure Loading, J. of Mechanical Engineering
Science, vol. 1, No. 2, 1959, pp 130-143.
15.
J.S. Hunter, Determination of Optimum Operating
Conditions by Experimental Methods, Part II-3,
Models and Methods, Industrial Quality Control,
vol. 15, No. 8, February 1959.
16.
Standards of the Expansion Joint Manufacturers
Association, Inc., Fourth Edition, 1975.
-15 -
TABtE 1
*
CASE
Design Matrix Used to Derive Computer Generated Data.
CROWN TORUS ROOT TORUS
RADIUS
RADIUS
r
r
c
r
A
B
CONVOLUTION
DEPTH,
d
i
|
INTERNAL
RADIUS,
r
i
PLY
THICKNESS,
t
C
D
E=ABCD
-1
-1
-1
+1
-l
I
t
1
-1
2
+1
-1
-1
3
-1
+1
4
+1
+1
-l
5
-1
-1
+i
6
+1
-1
+i
7
-1
+1
+i
8
+1
+1
+i
!
-1
-1
+1
-1
+1
+1
9
-1
-1
10
+1
-1
-l
11
-1
+1
12
+1
+1
-1
-i
13
-1
-1
+i
14
+1
-1
+i
15
-1
+1
+i
16
+1
+1
+i
0
0
0
0
18
-1
0
0
0
0
19
+1
0
0
0
0
+1
+1
+1
+1
+1
+1
Fractional
Factorial
-1
-l
+1
25-l
-1
+1
+1
-1
+1
-1
-1
+1
+1
17
.
0
20
0
-1
0
0
0
21
o
+1
o
0
0
22
0
0
-1
0
0
0
o
23
24
25
0
o
o
!
:
0
+1
0
0
-1
0
0
0
+1
0
26
0
;
0
0
27
o
;
0
0
>
:
0
-2
0
+2
Center Point
Modified
Star
Design
- 16 -
TABLE 2 : Variable Levels Corresponding to
Elements of the Design Matrix
DESIGN 2
DESIGN 1
VARIABLE
SYMBOL,
UNITS
LOW
(-1)
HIGH
(+1)
LOW
(-1)
HIGH
( + 1)
Crown Torus
Radius,
rc
A» nun
0.762
1.524
2.540
4.445
Root Torus
Radius,
r
r
B, mm
0.762
1.524
2.540
4.445
Convolution
Depth,
d
C> mm
3.175
6.350
8.890
17.78
D, mm
12.70
63.50
76.20
152.4
E j mm
0.203
0.305
0.356
0.457
Inner Radius,
r
i
Ply
Thickness,
t
TABLE 3(a) :
VALUES OF EXPONENTS IN THE RESPONSE FUNCTIONS FOR MERIDIONAL STRESS
COMPONENTS AND AXIAL SPRING RATE:
r
S
"E i
is
^
Axial Compression
AE
(Y.+YÏAE
B
B B
B *
n S
F e ° Y . 1 Y 2 Yv ' Yv "
External Pressure
A 'B
C
E '
P
ncK
Axial Spring Rate
X.K
2ir r ± E
STRESS
COMPONENT
LOCATION
B
r
i(YA+YB)E
8i
B2
B
0
B
it
CROWN
-1.160
-1.082
-0.006
-1.980
2.033
ROOT
-1.063
-0.032
-1.100
-1.932
2.053
CROWN
-0.879
-0.490
-0.018
-1.699
1.082
ROOT
-0.818
-0.029
-0.548
-1.655
1.111
CROWN
-0.781
-0.532
-0.020
1.462
-0.918
ROOT
-1.036
-0.038
-0.626
1.636
-0.988
CROWN
-1.878
-0.531
0.028
2.255
-1.901
ROOT
-1.760
0.019
0.519
2.353
-1.926
-0.850
-0.028
-0.001
-2.620
2.896
MEMBRANE
AXIAL
COMPRESSION
BENDING
MEMBRANE
EXTERNAL
PRESSURE
BENDING
AXIAL SPRING RATE
*
F determines the sign of stress
TABLE 3(b) :
VALUES OF EXPONENTS IN THE RESPONSE FUNCTIONS FOR CIRCUMFERENTIAL
STRESS COMPONENTS:
"c
Axial Compression:
r
i
AE
S
IS
Fe
B
B
1
Y.
A YB*
B
B
Y C3 Y*E ** *
External Pressure:
STRESS
COMPONENT
LOCATION
B
B
B2
B3
0
3It
CROWN
-1.647
1 .636
0 .452
-1.624
-0. 430
ROOT
-0.702
0 .299
0 .746
-1.604
0.334
CROWN
-2.143
-0 .496
-0 .026
-1.697
1. 086
ROOT
-2.096
-0 .029
-0 .561
-1.655
1. 112
CROWN
1.566
0 .099
0 .509
-1.139
-0. 425
ROOT
1.538
0 .188
0 .478
-0.899
-0. 566
CROWN
-3.091
-0 .534
0 .031
2.243
-1 891
ROOT
-2.954
0 .025
-0 .522
2.333
-1 912
MEMBRANE
AXIAL
COMPRESSION
I
BENDING
MEMBRANE
EXTERNAL
PRESSURE
BENDING
F determines the sign of stress.
05
-
TABLE 4 (a)
1.
:
19 -
Formulas for Correction Factor F for Meridional
Stresses due to Axial Compression, and Axial
Spring Rate.
Signs of Bending Stress Components
are for Outer Surface.
To obtain the Inner
Surface Bending Stress Component, change the sign
of F.
MEMBRANE STRESS AT CROWN:
F - -exp[0.711 x 10" 1 - 0.130 x 10<Y £ /Y c ) - 0.451 x 10 2 (Y A Yg 2 )
+ 0.297 x 1O" 2 (Y E /Y B 2 ) + 0.344(Y A 2 /Y E ) + 0.363 x 10(Y B 2 /Y c )
+ 0.188 x H f 1 <YC2/YE) - 0.207(Y A Y C /Y E ) - 0.340 x 10" 2 (Y B /Y
2.
MEMBRANE STRESS AT ROOT:
F « -exp[0.364 x 10" 1 + 0.139 x 10" 3 (Y c /Y A 2 ) - 0.104
+ 0.566(Y A 2 /Y B ) + 0.571(Y B 2 /Y E ) + 0.318 x
lO'1^
- 0.326(Y B Y C /Y E ) - 0.256 x 10" 1 (Y B /Y B Y c ) ] .
3.
BENDING STRESS AT CROWN:
F - exp[0.257-0.435 x 10" 2 (l/Y A ) + 0.820 x 10(YA Y c )
+ 0.264 x lO^CY^Yg) - 0.454(Y E /Y A ) - 0.173 x 1 0 ' 4 ( l / Y c Y £ )
- 0.753 x 10" 2 (Y B /Y c 2 ) + 0.596(Y A 2 /Y B ) - 0.677(Y A 2 /Y E )
+ 0.227 x 10(Y B 2 /Y c ) + 0.141 x 1 0 " 5 ( l / Y A 2 Y,,) - 0 . 1 8 0 x 10 3 (Y A Y c Y £ )
- 0.107(Y A YC/YE) + 0.141 x 10" 3 (Y c /Y A Y B ) ] .
4.
BENDING STRESS AT ROOT:
F - -exp[0.281-0.639 x 10~ 2 (l/Y_.) - 0.499 x 1 0 ~ 2 ( l / Y o )
3
2
+ 0.391 x ÎO^CY at,/Y ) - 0.211 x 10(Y_/Y
n ) - 0.949 x 10 (Y. Y )
Ci
(j
A
B
- 0.920 x 10(YA Y c 2 ) - 0.703 x 104(YA Yg2) + 0.696 x U f 4 (Yj/Y^)
+ 0.185 x 10 2 <Y B 2 /Y c ) - 0.143 x lOttg 2 /^) + 0.201 x 10~ 5 (l/Y B 2 Yc)
+ 0.161 x 103(YA YB Yc) + 0.364 x 104(YA Yfl YE) - 0.623 x 10" 3 (Y B /Y c
5.
AXIAL SPRING RATE:
F - exp[-0.185 + 0.257 x 10' 4 (l/Y B 2 )
+ 0.558 x 10" 2 (Y E /Y A 2 ) + 0.417(YA
- 20 -
TABLE 4(b)
1.
:
Formulas for Correction Factor F for Meridional
Stresses due to External Fressure.
Signs of
Bending Stress Components are for Outer Surface.
Inner Surface Bending Stress Components are
obtained by a change of sign of F.
F - - e x p [ - 0 . 3 4 5 + 0.870 x 10" 1 (Y c /Y A > - 0.727 x 10~ 2 (Yj/Yç 2 )
- 0.397 x 10" 3 (Y./Y. 2 ) + 0.129 x 10<Y. 2 /Y
C A
AC
2
) - 0.662 x 10" 2 (Y,, Y./Y 2 )
B CA
+ 0 . 1 6 9 x 10" 6 (Y B /Y A Y E 2 ) - 0 . 2 5 8 x 10" 1 (Y ( , Y j / Y g 2 ) + O . 1 2 1 x l O 3 ( Y A Yg 2
- 0.1A2(Y B Y C 2 /Y E ) + 0 . 1 8 7 x 1 0 "
2.
3
2
F - exp[-0.782 x 10"2 - O.19O(Y-,/Y2) + 0.638 x lO'^Y^/Y-)
be
D &
- 0.126 x 10~5(l/Y
2
Y.) + 0.104 x 10(Y 2 /Y. 2 ) - 0.998 x 10~3(Y 2/Y.2)
a
+ 0.295 x 10~2(Y
2
I*
o .
\i
LA
\J
/ Y 2 ) + 0.927 x 10" 1 0 (l/Y. 2 Y_2) + 0.215 x 10"2
A
D
•(Y E /Y B Y c ) - 0.672 x 10~ 1 (Y B ?<?/*£+
L
0.213 x 10 3 (Y A 2 Yg Y c )
- 0. 338 x 10(Y A 2 Y B /Y E ) - 0.170 x 10~ 2 (Y A 2 /Y B Y £ ) + 0.143 x 10" 3
• (YC2/YB V
3.
" °-189x
10
"2(YB
Y
C /Y A Y E ) ] -
BENDING STRESS AT CROWN
F - exp[0.596 x lO"1 - 0.245 x 10"4(l/YA2) + 0.167 x 102(YA Y ^
+ 0.205 x 10"4(Yc2/YE2) + 0.190 x 10"7<l/YA2 Y^) - 0.394 x 10(Ytt YE/YA)
- 0.226 x 10(YA2 YC/YB) - 0.226 x 10(YA2 Y C /Y E - 0.160 x 10(Yc2 Y E /Y A >],
4.
F - exp[0.117 + 0.268 x 10(Y_) + 0.151 x 10 3 (Y. Y^)
- 0.304 x 10*(Y Y 2 ) - 0.103(Y_/Y
O
fit
ti
2
) - 0.250 x U f ^ Y ^ / Y 2 )
\f
A
if
- 0.371 x 10(Y. YJV) - 0.131 x 10"1(Y_/Y. Y_) + 0.118 x 103.
•(YA Y B Y c 2 ) - 0.337 x 103(YA Y E 2 /Y C ) - 0.260 x 10"5(Yc/YA2 Y B )
+ 0.106 x 10~7(l/YA Y B 2 Y c ) - 0.278 x 10(Yg2 Yp/Yg)].
- 21 -
TABLE 4(c) :
1.
Formulas for Correction Factor F for Circumferential
Stresses due to Axial Compression. Signs of Bending
Stress Components are for Outer Surface. Inner
Surface Bending Stress Components are obtained by
a change of sign of F.
F - [0.139 + 0.280 x lO^d/Y.) - 0.765 x 10"1(Y,,/Y_)
U
IS
Ci
- 0.182 x 10 2 (Y n Y r ) - 0.570 x 10"4(l/Y,. Y_) + 0.104 x 10 2 (Y. Y
2
5
2
+ 0.375 x 10(Y B /Y A ) - 0.278 x 10" (l/Y A Y B ) - 0.916 x 10~J
•<Y E /Y A Y B ) + 0.991 x 10' 6 (l/Y A Yg Yg)].
2.
F - - 1.00
3.
F -
exp[0.318 - 0.679 x lo"2(l/YA) + 0.673 x 10"X(YA/YB)
- 0.176 x 10(Y£/Yc) - 0.465 i^^/\) + 0.240 x 10(YB2/Yc>
- 0.918 x 10~2(Yc2/YE) + 0.135 x 10"5(l/YA2 Yç) + 0.428 x 10~*
•WE»'
4.
F - -exp[-0.628 x 10"1 - 0.320 x 10"1(l/Y(;) + 0.182 x 10 2 (YA Y )
2
2
+ 0.101 (Y_/Y,J
- 0.925 x 103(Y.A Y_
) - 0.100 x 105(Y. Y_
)
a t
D
At
+ 0.951 x 10~2(YA/Yc2) - 0.806 x 10"4(YB/YE2) - 0.226 x 10"?
.(YC/YB2) + 0.407 x 10"7(l/Yc Y£2) - 0.453 x 10 3 (Y B 2 Y £ )
+ 0.142 x 10(Y2/Y.) + 0.176 x 102(Y2/Y,,) - 0.201 x 10<Y 2/X )
DA
3
O
4
Li
D
IS
+ 0.112 x 10 (YA Y B Y c ) + 0.436 x 10 (YA Y B Y £ ) + 0.199 x 10"l
(YE/YB Y c ) + 0.131 x 10~5(l/YA Y B Y c )].
2
)
- 22 TABLE 4(d) :
1.
Formulas for Correction Factor F for Circumferential
Stress due to External Pressure. Signs of Bending
Stress Components are for Outer Surface.Inner Surface
Bending Stress Components are obtained by a change
of sign of F.
F - - [ - 0 . 1 7 5 x 10 + 0.521 x 10 2 (Y A ) + 0 . 1 2 8 ( l / Y c ) - 0.160 x U f 1 (1/Yg)
+ 0.145 x 10(Y c 2 ) + 0.331(Y c /Y E ) + 0.448 x 10(Y A 2 /Y E )
- 0.393 x 10" ? (Y c 2 /Y E 2 ) + 0.354 x 10 3 (Y A YR Y c ) - 0.534 x 10(Y A Y c /
- 0.111 x 10 2 (Y A Y E /Y C 2 ) + 0.936 x 10" 2 (Yfi Yç/Yg 2 ) +0.236 x 10"?
.(Y C /Y B Y E 2 ) + 0.374 x 10" 3 (Y B /Y c 2 Y £ ) - 0.230 x l o V /
Yg/Yç)
- 0.296 (Y B 2 /Y C Y £ ) - 0.543 x 10" 2 (Y c 2 /Y A Y^,)].
2.
F - - [ - 0 . 4 7 1 X 10 + 0.113 x 10 2 (Y o ) + 0.922 x 1 0 ~ 2 ( l / Y o )
+ 0.191(Y c /Y E ) + 0.575 x 10 2 (Y E /Y c ) + 0.200 x 10" 2 (Y c /Y B 2 )
- 0.345(Y c 2 /Y A ) - O.729(Y C 2 /Y E ) - 0.134(Y A 2 /Y B 2 )
- 0.228 x 10" 2 (Y 2 /Y_ 2 ) + 0.757 x 10"1<Y_/Y. Y.) - 0.449 x 10"2.
Vi
B
CM
I»
A
•(YB/YA Y c 2 ) + 0.183 x 10" 5 (Y B /Y A Y £ 2 ) + 0.361 x l O ' V g / Y g Y,, 2 )
- 0.26 x 1 0 ~ V A / Y B 2 Y E ) - 0.519 x 10'2(YE/YB2 Yc> + 0.551 x 10~*
•(YA YC/YB Y E ) - 0.346 x 103(YE2/Yc2)l.
3.
F - exp[0.461 x 10"2 - 0.105 x 10"*(l/YA2) + 0.650 x 102(YA Y,,)
- 0.249 x 103(Y. Y_) - 0.943(Y_2/Y.)+ 0.104(Y. Y_/Y_ )
U
£•
V
A
A
t>
b
+ 0.674(YB Y E /Y C 2 ) - 0.869 x 10~6(Yc/YA Yj2) + 0.234 x 10 •
.(Y Y 2/Y ) + 0.325 x 103(Y
A
4.
D
L
Y_2/Y ) - 0.664 x 10 (Y 2 Y /Y )].
U
b
D
A
U
£
F - e x p [ - 0.124 x 10" 1 + 0.674 x 10(YB> + 0.153 x 10~ 1 (Y (; /Y B )
- 0.468 x H f V / Z Y j 2 ) + 0.151 x 10 3 (Y B 2 Y c 2 ) - 0.574 x 10*
•<YB Y c Y £ 2 ) - 0.390 x 10(Y B Yg/Yç 2 ) - 0.559 x 10" 1 (Y ( , Yg/Y A 2 )
- 0.341 x 10(Y B 2 Y C /Y E )J.
- 23 -
TABLE 4(e) :
1.
Actual Meridional Membrane Stress due to
External Pressure Acting on the Convoluted Surface Only.
ROOT
- ° - 0 6 8 Y R ° - W Y, 1 ' 112 Y -1'050
exp( - 0 .826)Y
A
2.
CROWN
P
p S mc
/np
-b
L
£•
- 24 -
TABLE 5 : Bellows Response to Loading in the Example
of Laupa and Weil(l):
r
(a)
= r = 1 3 . 7 mm,
r
r
= 304.6 mm,
d
= 58.2 mm,
t
= 1.2 7 mm
y
= 1.47
v
= 0.3
c
Bellows Axial Spring Rate and Meridional S t r e s s
Components a t Root due to Axial Compression.
Spring Rate, Membrane
K
Stress,Sm
SOURCE
10
n
c
K
Surface Bending
Stress, Sb
1 0 2 n c riSfe
10 2 n c r., S
i m
AE
ri E
AE
Resultant Surface
Stress, S R
10 2 n c r i S R
AE
Inner
Outer
Inner
Outer
8 69
-9 17
8 .58
-9 28
6 82*
-8 82*
8 .48
-9.17
Finite Element
Analysis
2 .88 *
Statistical
Model
2.99
Laupa & Weil
(1)
2.90
-0 11
8 80
-8 80
8 .6<)
-8. 92
Salzmann (1)
3.06
-0 12
10 51
-10 51
10 .40'
-10.63
Hamada
2 .64 *
-0 10
9 91
-9.91
9 .81
-10.01
2 .92
-0 10
8 91
-8 91
8 81
-9.02
Kellogg (10)
2 .21
-0 08 t
12 48
-12. 48
12 40
-12.57
Anderson (12)
2 .62
-0 10 t
11. 73
-11. 73
-11 63
-11.83
EJMA
3 .05
-0. 09
(-0. 12) t
10.24
-10. 24
10 15
-10.33
(7)
*
(16)
-0 .11
-0 .35*
(-0 11) t
Assumed linear stress distribution.
ncK
r
i
E
=
t Derived from K
"cr iS m . 2*t ;
AE
r
4 Derived from
V
F --= KA
i
N * Sjjt, Membrane force per unit length.
F * Axial Force.
_' Bending moment per unit length.
- 25 -
TABLE 5 (Cont'd)
(b)
Bellows Axial Spring Rate and Meridional Stress
Components at Crown due to Axial Compression.
Spring Rate, Membrane
K
Stress, Sm
SOURCE
105n
r
10
K
c
iE
Statistical
Model
ViSm
102 n
AE
2.88 +
Finite Element
Analysis
Surface Bending
Stress, Sb
10 2 n c r± S R
AE
-0,09
-0.34*
(-0.10)t
2.99
c ri Sb
Resultant Surface
Stress, S R
AE
Inner
Outer
Inner
Outer
-9.00
8.51
-9.09
8.42
-8.49*
8.49*
-8.83
8.15
Laupa & Weil
(1)
2.90
-0.09
-8.68
8.68
-8.77
8.59
Salzmann (1)
3.06
-0.10
-10.51
10.51
-10.61
10.41
Hamada
2.64 *
2.92
-0.09
-0.09
-9.91
-8.71
9.91
8.71
-10.00
-8.81
9.83
8.62
Kellogg (10)
2.21
-0.07t
-12.48
12.48
-12.56
12.41
Anderson (12)
2.62
-0.08t
-11.73
11.73
-11.81
11.64
EJMA
3.05
-0.09
(-0.10)t
-10.24
10.24
-10.33
10.15
Y'
(16)
*
t
*
Assumed linear stress distribution
Derived from K
Derived from Sm
n
r
c
K
i
E
„
n rJ S
i m
c
F
-
. Zirt
r
i
(d +
r
' *- ï
i
Membrane Force perUnit
N M
AE
Sbt2
F -
Length
, Bending Moment per Unit Lengf
6
- Axial Force
-
26 -
TABLE 5 (Cont'd)
(c)
Bellows Meridional Stress Components at Root due
to External Pressure
Membrane
Stress,Sm
SOURCE
Surface Bending
S t r e s s , Sj,
n S
%Sm
P
F i n i t e Element Analysis
S t a t i s t i c a l Model
23.4
(39.6)*
23.6 t
R e s u l t a n t Surface
S t r e s s , SR
"p S R
P
P
Inner
Outer
Inner
Outer
-588.4
620.8
-565.0
644.2
(600.4)*
-560.8
640.0
(-600.4)*
Laupa & Weil
(1)
23.9
-582.?
582.9
-559.0
606.8
Hamada
(9)
136.9
-625.4
625.4
-488.5
762.3
Kellogg
(10)
-
-1050
1050
-1050
1050
Anderson
(12)
120.5
-758.3
758.3
-637.8
878.8
EJMA
(16)
22.9
-608.3
608.3
-585.4
631.2
*
Assumed Linear Stress Distribution
t
Obtained from Axial Force Equilibrivan
Root Axial Force F = S r . t
r
mi
_
27 -
TABLE 5 (Cont'd)
(d)
Bellows Meridional Stress Components at
Crown due to External Pressure
Membrane
Stress,Sm
Surface Bending
Stress, S],
Resultant Surface
Stress, S R
SOURCE
p
pSm
P
Inner
Finite Element Analysis
Statistical Model
"p S R
P
P
-22.5
(-39.0)*
-22.3 t
Outer
Inner
Outer
-588.2
555.4
-610.7
532.9
(-571.0)*
(571.0)*
-610.0
532.0
Laupa & Hell
(1)
-22.1
-591.3
591.3
-613.4
569.2
Hamada
(9)
-115.0
-625.4
625.4
-740.4
510.4
Kellogg
(10)
-
-1050
1050
-1050
1050
Anderson
(12)
-120.5
-758.3
758.3
-878.8
637.8
EJMA
(16)
-22.9
-608.3
608.3
-631.2
585.4
*
Assumed Linear Stress Distribution
t
Obtained from Axial Force Equilibrium
Crown Axial Force F
S
m(ri
+ d ) t
- 28 -
TABLE 5 (Cont'd)
(e)
Bellows Circumferential Stress Components
at Root due to Axial Compression
Membrane
Stress,Sm
SOURCE
1Q2
Vi S m
Surface Bending
Stress, S^
10 2 n
c
AE
r
iSb
Resultant Surface
Stress, S
R
10 2 n
C
AE
AE
Outer
Inner
r. SR
1
Outer
Inner
-4.59
2.67
-2.67
-1.92
-7.26
Statistical Model
-4.43
2.56
-2.56
-1.87
-6.99
Laupa & Weil
(1)
-4.35
2.64
-2.64
-1.71
-6.99
Hamada
(7)
-4.56
2.73
-2.73
-1.83
-7.28
(f)
at Crown due to Axial Compression
Membrane
Stress,Sm
SOURCE
102n r.S
c l m
Surface Bending
Stress S
' b
10 2 n
AE
c
r. S,
l b
Resultant Surface
Stress S
' R
102nc
AE
Inner
r
iSR
AE
Outer
Inner
Outer
3.84
-2.62
2.62
1.22
6.46
Statistical Model
3.75
-2.47
2.47
1.28
6.21
Laupa & Weil
(1)
4.04
-2.60
2.60
1.44
6.65
Hamada
(7)
3.92
-2.67
2.67
1.25
6.58
- 29 -
TABLE 5 (Cont'd)
(g)
at Root due to External Pressure
Membrane
Stress,S
' m
Surface Bending
Stress S
• b
Resultant Surface
Stress, S.,
K
SOURCE
n
pSm
"pSR
"p S b_
P
P
P
Inner
Outer
Inner
Outer
87.2
-181.1
181.1
-93.9
268.3
Statistical Model
86.7
-175.1
175.1
-88.4
261.8
(1)
66.6
-174.9
174.9
-108.3
241.5
(16)
97.6
-
-
-
Laupa & Weil
EJMA
1
(h)
Bellows Circumferential Stress Components at
Crown due to External Pressure
f
t
t
Membrane
Stress,Sm
np Sm
SOURCE
P
Surface Bending
Stress , Sb
Resultant Surface
Stress S
• R
n
_JL
P
"pSR
P
Inner
Outer
Inner
Outer
Finite Element Analysis .
47.7
-171.2
171.2
-123.5
218.9
Statistical Model
46.6
-166.6
166.6
-120.0
213.2
-177.4
177.4
-107.7
247.1
I
i
|
Laupa & Weil
(1)
j
69.7
EJMA
(16) ,
97.6
i
_
!
-
CONVOLUTI 9N
U
HRnWN
TORUS
RADIUS
t : PLY THICKNES:
CROWN
GAP
CROWN
WIDTH
1
1
CUFF
I
/SPAN
ROOT
WIDTH
I
\
)
CONVOLUTION
DEPTH, d
_ (OD-ID)
I \
2
r =
/ V /
ROOT
GAP
V
1_
r r : ROOT
TORUS
RADIUS
OD - OUTER D) AMETER
ID - INNER D AMETER
FIGURE 1
o
Bellows Nomenclature
00
T
i
1
ID
1
2
OUTSIDE MERIDIONAL
OUTSSDE CIRCUMFERENCE
INSIDE CIRCUMFERENCE
— INSIDE MERIDIONAL
I
0.5
1.0
1.5
2.0
CENTRELINE
FIGURE 2
2.5
DISTANCE
3.0
3.5
4.0
4.5
(mm)
Typical Bellows Stress Distribution Due to Axial Compression of
1 mm per Convolution.
OUTSIDE
200h
INSIDE
MERIDIONAL
OUTSIDE
INSIDE
CIRCUMFERENCE
CIRCUMFERENCE
MERIDIONAL
i
-150
0.0
05
1.0
1.5
2.0
CENTRELINE
FIGURE 3
2.5
3.0
DISTANCE
3.5
4.0
(mm)
Typical Bellows Stress Distribution Due to External Pressure of 1 MPa.
a) Membrane Stress: Y,
FIGURE 4
B
0.02
b) Bending Stress:
Y
= Y
0.02
B
Compressive Meridional Membrane and Outer Surface Bending Stress Components
at Convolution Roots Due to Axial Compression as a Function of Thickness
t/r i"
Parameter, Y,
L
10 . -
o.ooo
c)
10
.rot .1
Membrane S t r e s s :
9.0»
Y. = Y
B
0.04
FIGURE 4 c o n t ' d
d)
.004
Bending
Stress
= YB = 0.04
I
10
0.000
e)
Y
A
f) Bending Stress:
0.06
FIGURE 4 cont'd
0.06
10",
1
o.ooo
g)
-m
Membrane
.a»
.ou
Stress:
.016
.020
.021 .02s
0.000
Y . = Y_ = 0 . 0 8
n
a
h)
Bending S t r e s s :
Y
0.08
10
0.000
1)
Membrane Stress:
.032
Y
0.10
0.000
j) Bending Stress:
FIGURE 4 cont'd
Y,
1
LO
GO
I
o.ooo
k)
0.000
Y
= 0.12
1)
.004
.008
.012
.028
.016
Y. = Y_. = 0.12
A
.03?
J5
10*
10*
10"
1
It"
0.000
a)
Y, = Y
B
A
0.02
b)
.004
.00*
Bending
.012
.OU
.020
Stress:
Y
A
= Y
= 0.02
1J
FIGURE 5 Tensile Meridional Membrane and Outer Surface Bending Stress Components
at Convolution Roots due to External Pressure as a Function of Thickness
Parmeter, Y w =
-40-
IIIIIII i i—I»
-
—I
SSXSSSRRK
ses
—hnrrn
s
s
s
-ao
o
11
60
a
•H
c
ai
IttUJ.U J
.'J J
I11111 1 1 1
him
IUUJJJ J
1 1 1
UiUU J
U
tur
o
o
(0
to
ai
M
0)
G
M
•8
10
-C-
0.000
e)
10"'.
.004
YA « YR = 0 . 0 6
f)
Bending S t r e s s ;
Y
A
Figure 5 cont'd
= Y
= 0.06
a
.E-
0.000
g)
.O0<
.01»
.012
.0»
0.000
.0»
Membrane S t r e s s : Y = Y = 0 . 0 8
A
B
Figure 5 cont'd
h)
.004
.001
.012
.01*
.020
Y
.OU
.ait
.032
A ~YB» ° '
0 8
«.0»
i)
.OU
Y. * Y_ = 0.10
A
D
Figure 5 cont'd
j) Bending Stress:
YA = Y
- 0.10
10'. .
I
10
k)
Membrane Stress:
t
= "Ï = 0.12
Figure 5 cont'd
1)
Bending Stress:
Y
A
=
Y
B
= 0.12
I
O.D
a)
Membrane Stress:
FIGURE 6
0.008
b)
.1
Bending Stress:
Yt
0.008
at Convolution Roots Due to Axial Compression as a Function of Convolution
Depth Parameter Y_ = d/r..
:
1
1—~ i —
1
1
1
^^-Jo.02
-^*^O--0.04
0-
10*.
i
•
Y
I
:
'"'I
10'
/
10"
ON
I
1
a)
Membrane Stress:
FIGURE 7
Y n = 0.008
Bending Stress:
1
1
1
Yv = 0.008
Tensile Meridional Membrane and Outer Surface Bending Stress Components at
Convolution Roots due to External Pressure as a Function of Convolution Depth
Parameter Y n = d/r., .
10'
îo-l
i)
Membrane Stress:
FIGURE 8
Y w = 0.008
E
"r
i
L
O.OO
.02
.0*
b)
Bending Stress:
••= 0 . 0 0 8
at Convolution Roots due to Axial Compression as a Function of Root Torus
Radius Y B = r r /r ± .
I
00
0.0
a)
Membrane Stress
Y_ = 0.008
ft
FIGURE 9
b)
.02
.01
Bending Stress:
= 0.008
Tensile Meridional Membrane and Outer Surface Bending Stress Components at
Convolution Roots due to External Pressure as a Function of Root Torus
Radius Y B = rr/r..
ISSN 0067 - 0367
ISSN 0067 - 0367
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