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Atomic Energy of Canada Limited
BAGPIPE: A PROGRAM TO DETERMINE
THERMO-ELASTIC DEFORMATIONS IN
A HOLLOW CYLINDER OF FINITE LENGTH
by
R.J.M. CROZIER
Chalk River, Ontario
April 1971
AECL-39O1
BAGPIPE: A PROGRAM TO DETERMINE THERMO-ELASTIC
DEFORMATIONS IN A HOLLOW CYLINDER OF FINITE LENGTH
By
R.J. M. CrozierApplied Mathematics Branch
A B S T R A C T
The Fortran program BAGPIPE is designed to determinethe thermo-elastic deformations which arise in a hollowcircular cylinder of finite length. The inner and outersurfaces of the cylinder are subjected to temperature changes,normal stresses and shear stresses, all of which may varyalong the length of the cylinder. The length of the cylindermay change.
The preparation of input data has been made particu-larly simple and the central processor time required toexecute a typical problem is of the order of 15 seconds.
The problem of the non-uniform heating of a fuelsheath containing fuel pellets separated by thin, highlyconductive discs is used as an illustration. It is foundthat the sheath does not experience unrealistic non-uniformdeformations when subjected to a temperature distributionproposed at Chalk River Nuclear Laboratories.
(submitted March, 1971)
Chalk River Nuclear LaboratoriesChalk River, Ontario
April, 1971
AECL-3901
BAGPIPE: Programme permettant de déterminer les déformations
thermo-élastiques d'un cylindre creux dé longueur finie
par
R.J.M. Crozier
Résumé
Le programme Fortran BAGPIPE est conçu pour déterminerles déformations thermo-élastiques se produisant dans uncylindre circulaire creux de longueur finie. La surfaceintérieure et la surface extérieure du cylindre sont assujettiesà des changements de température, à des contraintes normales età des efforts de cisaillement qui peuvent varier sur toute lalongueur du cylindre. La longueur du cylindre peut changer.
r " •
La préparation des données d'entrée a été rendueparticulièrement simple et le temps requis par l'appareil detraitement des données pour un problème typique est de l'ordrede 15 secondes.
*•-- Le problème du chauffage non-uniforme d'une gaine decombustible contenant des pastilles de combustible séparéespar des 'disques peu épais et hautement conducteurs est donnécomme exemple. On constate que la gaine ne subit pas dedéformation non-uniforme anormale lorsqu'elle est assujettieà la distribution des températures proposée par Chalk River.
(Manuscript soumis en mars 1971)
L'Energie Atomique du Canada, Limitée
Laboratoires Nucléaires de Chalk River
Chalk River, Ontario
AECL-3901
CONTESTS
Page
Section
1. Use of BAGPIPE 1
1.1 Introduction 1
1.2 Equations Solved 1
1.3 Boundary Conditions 2
1.4 Type of Solution 4
1.5 System of Units 5
1.6 Preparation of Input 5
1.7 Control of computation 9
1.8 Running the Program 9
1.9 Output 10
2. Sample Problem - Sheath Deformation 11
3. Method of solution 27
3.1 Introduction 27
3.2 Derivation of the Temperature Distribution 27
3.3 Derivation of the Elastic Displacements 30
BAGPIPE: A PROGRAM TO DETERMINE THERMO-ELASTIC
DEFORMATIONS IN A HOLLOW CYLINDER OF FINITE LENGTH
By R . J . M . C r o z i e r
1. Use of BAGPIPE
1.1 Introduction
Thermo-elastic stresses are important factors to be
considered in the design of nuclear fuel elements.
In certain fuel element, assemblies, fuel pellets are
separated from each other by thin discs of a highly conductive
material in order to reduce the bulk temperature of the fuel.
As a result the sheath experiences a non-uniform temperature
distribution along its length, those parts of the sheath opposite
the highly conductive discs becoming considerably hotter than
other parts of the sheath. Non-uniform stresses develop in the
sheath which is deformed into a bamboo-like shape, the points
of greatest deformation being adjacent to the highly conductive
discs. The program calculates these deformations together with
the associated strains and stresses.
1.2 Equations Solved
The program BAGPIPE solves the thermo-elastic equations
in cylindrical coordinates with axial symmetry. It is assumed
that temperature changes are small, so that material parameters
remain constant. It is also assumed that strains are small, so
that the classical elastic equations hold.
- 2 -
The temperature distribution, T ( r , z ) , in a tube with
inner radius r = a, outer radius r = b and height h, is governed
by the second order partial differential equation
—p- + + —2f+ H = 0 : a ^ r ^ b , o <; z < h
ar2 r ar az \
where k is the thermal conductivity of the material and H is
the rate of production of heat per unit volume.
The radial displacement, u(r,z), and the axial displace-
ment, w(r,z), are governed by the simultaneous pair of second
order partial differential equations
ifu A 9u u , c ifji , (1_c) -llwar3 r ar r2 az2
a2w i _awiar2 r ar
wherel-2v
a <. r
c =
P = (3-4c)a
in which v is Poisson's ratio and a is the coefficient of linear
expansion.
1.3 Boundary Conditions
The function T(r,z) requires four conditions to be
specified on the boundaries of the tube. The temperatures on
the inner and outer surfaces of the tube are given in the form
T = Ta(z) on r = a, o<;z<;h
T = Tb(z) on r=b, o s z s h .
- 3 -
Also, it is assumed that the ends of the tube are
thermally insulated so that
3T— = 0 on asrsb. 2=03z *
dT— = 0 on asr«b, z = h .uZ
These end conditions can be interpreted as symmetry conditions
if one imagines that at each end of the tube there is attached
another tube which is the mirror image of the first tube.
The functions u(r,z) and w(r,z) require a total of
eight conditions to be specified on the boundaries of the tube,
First, on the inner and outer surfaces of the tube the normal
and shear stresses are given in the form
o ^ = - p = { z ) on r = a , o ^ z s h
a r r = ~ P b ( z ) o n r = b > o s z < s h
a z r = - s a ( z ) on r = a , o s z s h
a = - s - j 3 ( z ) on r = b , o ^ z < h .
Second, the end conditions are of the form
-—• = 0 on air^b, z=o,hoz
w = 0 on asrsb, z = o
•w = Cja. on a^rsb, z=h
where 5 is a constant. From this last condition we can see
that the tube height increases by £h.
- 4 -
1.4 Type of Solution
We can see from the boundary conditions imposed on the
temperature distribution that it can be expressed in the form
CD
T(r,z) = ) T(r) c o s — : a s r s b , osz<h .n=0
Also, we can see from the boundary conditions imposed on the
displacement distributions that they can be expressed in the
following forms:
00
, . V / « mrz , ,u(r,z) = > ^(r) cos-r— : a <. r < b, o < z =s h
n=0
: a^r^b, o < z < h .£J. " nn=l
With this type of basic solution for the temperature
and displacements, the normal and shear stresses can be expressed
in the form
= - I Pn<r> C O S 1 T :o r r = - 2 , P n ^ r ' c o s ~ h ~ : a ^ r < b , o s z < hn = 0
V / \ n7TZ ,a
z r = - 2i s n * r ) s i n r h ~ : a s r s b , o < z < h
in which the terms Pn(r) and sn(r) are functions of (r) and
wn(r) ana their derivatives.
- 5 -
1. 5 System of Units
The program BAGPIPE requires input parameters to be
expressed in the International System of units. Distances
must be in metres, stresses in Newtons per square metre, tem-
peratures in degrees Celsius, thermal conductivity in Watts
per metre per degree Celsius, and heat production in Watts per
cubic metre.
For the purposes of computation the stress-free state
has been taken at 0°C.
1.6 Preparation of Input
The input parameters for a problem are supplied on a
deck of punched cards.
The first card of the deck is a problem name card and
is followed by a card containing constants and one containing
data used to control the computation. Finally there is a series
of cards giving the boundary conditions at different specified
axial positions. This series of boundary condition cards must
be in a specific order. The card containing the conditions on
z = h (h>o) must be first. The subsequent cards must be in
order of decreasing values of z. The last card must contain
the conditions on z = o. These boundary condition cards need
not be at equal intervals of z but must be at axial positions
which are close enough so that the boundary conditions are
smooth functions of z.
The data deck will now be described in detail.
- 6 -
First Card. Format (8A10). Problem Name.
NAME The problem name card must be present but may be
blank. It is reproduced at the head of output.
Second Card. Format (8E10.3). Constants.
A Inner radius of tube (metres). Must be specified.
B Outer radius of tube (metres). Must be specified.
E Young's modulus of material under operating con-
ditions (Newtons per square metre). Must be
specified.
P Poisson's ratio of material under operating con-
ditions. Must be specified.
AL Coefficient of linear thermal expansion of material
under operating conditions (per degree Celsius).
Must be specified.
COND Thermal conductivity of the material under operating
conditions (Watts per metre per degree Celsius).
If not specified, COND is set equal to 1030.
HEAT Rate of generation of heat per unit volume (Watts
per cubic metre) . If not specified, HEAT is set
equal to 0.
DH Allowable increase in the height of the tube (metres)
If not specified, DH is set equal to 0.
Third Card. Format (4E10.3. 4110). Control Data
EPST Controls the accuracy of the two series for the
temperature on the inner and outer surfaces
- 7 -
(degrees Celsius). Must be specified. When the
difference between a specified boundary condition
and the sum of its equivalent Fourier series is less
than EPST, no further terms in the series are cal-
culated. When all series have converged to the
required accuracy, computation ceases. A warning
message is printed when a series has not converged
satisfactorily. In this case the quantities printed
are compatible temperature and deformation distri-
butions, and comprise the solution to a slightly
different (usually smoother) problem from the one
posed.
EPSP Controls the accuracy of the two series for the
normal stress on the inner and outer surfaces
(Newtons per square metre). Must be specified.
EPSS Controls the accuracy of the two series for the
shear stress on the inner and outer surfaces (Newtons
per square metre). Must be specified.r
SECS Maximum allowed central processor time in seconds for
this problem. If the current problem has not already
been completed, computation stops after (SECS-5)
seconds to allow the results obtained so far to be
printed. If not specified, SECS is set equal to 20.
LP Controls options on output data, normal amount (0),
minimum (-1) or maximum (1). If not specified, LP
is set equal to 0.
- 8 -
LSM Smoothing integer. The interval oizsh is divided
into (LSM-1) equal intervals for the purpose of
determining accurate Fourier integrals. Must be
greater than or equal to 4*KMAX and less than or
equal to 400. If not specified, LSM is set equal
to 4*KMAX. (See section 1.7).
NMAX Specifies the maximum number of terms in any one
series. May be used to stop computation after NMAX
terms. For computational reasons, NMAX must be less
than or equal to LSM. If not specified, NMAX is
set equal to 0.
KMAX Specifies the number of boundary cards following
this card. Must be greater than or equal to 4 and
less than or equal to 100. Must be specified.
Subsequent Cards. FormattElO.S.lOX.SElO.S). Boundary Conditions
Z Axial position of this data point (metres).
TA Temperature on r= a for this value of Z (degrees
Celsius).
TB Temperature on r =b for this value of Z (degrees
Celsius).
PA Normal pressure on r = a for this value of Z
(Newtons per square metre).
PB Normal pressure on r= b for this value of Z
(Newtons per square metre).
SA Shear pressure on r= a for this value of z
(Newtons per square metre) .
- 9 -
SB Shear pressure on r=b for this value of z
(Newtons per square metre).
If required, two or more data decks can be run in the
same job, simply by stacking subsequent decks immediately
after the first deck. Computation stops when an end of file
(7-8-9) card is read or earlier if an error is detected while
reading data cards.
1.7 Control of Computation
In order to compute Fourier integrals such as
r Ta(z) cos(7z)dzo
the range o^z^h is divided into (LSM-1) equal sub-intervals.
A new array TAL is derived (at LSM equally spaced points) from
the original array TA (known at KMAX points, not necessarily
equally spaced) by cubic interpolation. This new array is
smoother than the original array since LSM is significantly-
larger than KMAX and is used to compute all the required Fourier
integrals associated with Ta(z).
The same procedure is used for the other" five boundary
conditions, Tb(z), pa(z), pb(z), sa(z) and s^z) .
1.8 Running the Program
In its present form, BAGPIPE requires 6000B words of
core storage on the CDC 6600 at Chalk River.
The amount of central processor time required for any
particular BAGPIPE calculation depends on the size and complexity
of the problem, the smoothness of the prescribed boundary
- 10 -
conditions and the required accuracy of the solution. Since
the method of obtaining a solution involves expressing the
six boundary conditions in terms of Fourier series, the central
processor time depends directly on the number of terms in
these series. A typical problem, with all six boundary con-
ditions given at 25 axial positions requires roughly 15 seconds
of central processor time to supply a solution which is the
sum of 100 terms in each of the Fourier series.
The program BAGPIPE can be run on the CDC 6600 at Chalk
River using the control cards
ATTACH (A, BAGPIPE)
A.
The first program card is
PROGRAM BAGPIPE (INPUT,OUTPUT,TAPE1=INPUT,TAPE9)
where TAPE9 is used to store additional computational details
when LP=1.
1.9 Output
All input quantities are listed before computation
begins. The problem name is printed at the head of the output.'
The parameters on the second and third cards, or their corrected
values, are then named and listed with their physical dimensions,
To complete the reproduction of the input, the boundary condi-
tions are listed.
The parameter LP determines the amount of output pro-
duced. We will describe here the normal option, LP=0.
- 11 -
As computation proceeds, term by term, the Fourier
coefficients, Tn(a), Tn(b), Pn(a), pn(b), sn(a) and sn(b)
given by equations (5), (13) and (14) are printed, for each
value of n, together with the derived displacement componentsun(a), un(b), wn(a) and wn(b). In general, all these quantities
should decrease as n increases.
When computation ceases, the temperature distribution,
displacement distributions, strain distributions and stress
distributions are printed. At the head of each distribution
a warning message is printed if required. The thickness of
the tube is divided into eight equal sections and each distri-
bution is displayed at nine radial positions and at all the
axial positions used to define the initial boundary conditions.
For the sake of clarity in displaying the results
any temperature less than EPST is set to zero,
any displacement less than E-15 is set to zero,
any strain less than E-09 is set to zero,
any normal stress less than EPSP is set to zero, and
any shear stress less than EPSS is set to zero.
2. Sample Problem - Sheath Deformation
A particular fuel assembly consists of a thin-walled
sheath containing a number of fuel pellets separated from one
another by thin discs of a highly conductive material. A large
proportion of the heat produced in a pellet travels through
the adjacent disc to the sheath. The outer surface of the
sheath is water cooled. The resulting high heat flux between
the disc and sheath causes that part of the sheath adjacent to
the disc to become considerably hotter than other parts of the
sheath.
- 12 -
When considering the deformations which arise in the
sheath it is sufficient to look at that part of the sheath
which lies between the mid-plane of the pellet (z = o) and the
mid-plane of the disc (z = h).
The sheath has an inner radius of 8.00 mm, an outer
radius of 8.43 mm and, as far as this problem is concerned,
a height of 8.40 mm. (This corresponds to a pellet height of
16.00 mm and a disc height of 0.80 mm.) The sheath is made
of an elastic material with a Young's modulus of 90 GN/m3, a
Poisson's ratio of 0.40 and a coefficient of linear thermal
expansion of 6.5xl0~6 per degree Celsius. No heat is produced
in the sheath so that the thermal conductivity need not be
specified. The sheath is known to increase in length by 0.1%
so that DH = 8.4 x 10~6 m.
The temperature distributions on the inner and outer
surfaces of the sheath are known at 11 points along the length
of the sheath and are shown in figure 1. The inner and outer
surfaces of the sheath are subjected to constant pressures of
20 MN/m2 and 10 MN/m3 respectively. No shear stresses exist
on the surfaces of the sheath.
The computation is controlled by specifying EPST, EPSP
and EPSS. The surface temperatures are known .to within 1°C
so a reasonable value for EPST is 0.5°C. Since the surface
pressures are constant, the values assigned to EPSP and EPSS
are not important, we set them equal to unity.
The temperature distributions on the surfaces of the
tubes are not as smooth as we would have liked. Therefore we
choose a relatively high value for LSM and choose NMAX=LSM=100.
Different combinations of LSM and NMAX can be tried to determine
how sensitive the solution is to these quantities.
250 450
OJ
i
FIGURE 1 TEMPERATURE DISTRIBUTIONS APPLIED TO THE INNER AND OUTER SURFACES OF THE SHEATH
- 14 _
The input card deck is reproduced in figure 2 and the
output obtained is shown in figure 3. The results were calcu-
lated to the required accuracy in less than 4 seconds and
required 14 terms in the Fourier series for Tb(z) and 48 terms
in the Fourier series for T (z).cl
Some results are plotted in Figures 4, 5 and 6.
The displacements of the inner and outer surfaces of
the sheath are shown, in a greatly magnified form, in figure 4.
The effect of the higher temperature in the region adjacent
to the disc (z = 8.0 to z= 8.4 mm) can be seen clearly. The
height of the ridge formed by such a deformation is
36.99 - 36.15 = 0.84 \m
which is negligible compared with the radius of the outer
surface of the sheath. The axial displacement is almost exactly
proportional to axial position, the total increase in height
having been imposed on the problem.
The tangential and axial strains on the inner and outer
surfaces of the sheath are shown in figure 5. The average
axial strain of 0.1% was imposed on the problem.
The tangential and axial stresses on the inner and
outer surfaces of the sheath are shown in figure 6. The axial
stress is in compression except on the outer surface adjacent
to the disc where it is in slight tension. An average axial
compressive stress of approximately 43 MN/m2 is required to
limit axial elongation to 0.1%. The effect of decreasing the
axial strain would be to increase the required compressive
axial stress by a constant value.
COL 5 0 5 0 5 0
fl.00000-3 fl.4300 0-35?*0'0000-1 1.00000*0
»;*2i0O0O-3^8400000-37.00000-36.00000-35.00000-34.00000-33.00000-32.00000-31.00000-30.00000-3
COL 5 0
SAMPLE9.OOOA*1Q1.00000*04.35660*24.30720*24.11100*23.60270*23.53750*23.55340*23.57900*23.60170*23.61780*23.62720*23.63030*2
PROBLEM FOR BAGPIPE RFPORT)-} 6.50000-6
3.29120*23»?781p*23.94350*23.07690*23.04970*23.05390*23.06220*23*06980*23.07520*23.07840*23.07940*2
2.00000*72.00000*72.00000*72.00000*72.00000*72.00000*72.00000*7?.00000*72.00000*72.00000*72.00000*7
. 1001.00000*71.00000*71.00000*7i.ogooo*7
j .oppoo*7i..o6p6o*7
1.00000*71.00000*71.00000*7
5 0
loo 11
I
5 0 5 0
FIGURE 2 INPUT DATA DECK FOR BAGPIPE
- 16 -
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FIGURE 4 RADIAL AND AXIAL DISPLACEMENTS ON THE INNER AND OUTER SURFACES OF THE SHEATH
0.0
I
FIGURE 5 TANGENTIAL AND AXIAL STRAINS ON THE INNER AND OUTER SURFACES OF THE SHEATH
8.0
7.0
6.0
5.0
4.6
3.0
2.0
1.0
1.0
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1
-A 1
i
°88 ( b ' Z )
/1
i i 1200-100 0 100
COMPRESSION TENSION MN/m2
FIGURE 6 TANGENTIAL AND AXIAL STRESSES ON THE INNER AND OUTER SURFACES OF THE SHEATH
300
I
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- 27 -
In the light of these results it is felt that the
sheath does not experience unrealistic non-uniform deformations;
manufacturing tolerances on the geometrical shape of the
"cylindrical" sheath could produce non-uniform deformations
of similar magnitude.
3. Method of Solution
3.1 General
The solution of the basic uncoupled equations of
thermo-elasticity is found in two stages. First, the tempera-
ture distribution in the tube is determined using the two-
dimensional axisymmetric form of the linear heat conduction
equation. Then the thermo-elastic deformations are determined
from the two-dimensional axisymmetric form of the linear
thermo-elastic equations.
3.2 Derivation of the Temperature Distribution
The linear two-dimensional axisymmetric heat conduction
equation in cylindrical coordinates (r,z) is
where k is the thermal conductivity of the material, H is the
rate of production of heat per unit volume and T(r,z) is the
change in temperature from the stress-free state.
Since the top (z = h) and bottom (z = o) surfaces of the
tube are thermally insulated, the general solution must be of
t h e f o r m ' - ; ° - ! ••-••• ; '••• . ' : : - •••••"•.••- ~ •-'
28 -
T(r,z) = (2)n=0
wheremr
The temperature distribution on the inner ( r = a ) and
outer (r = b) surfaces of the tube are given and must be
expressible in the form
T(a,z) = Ta(z) = 2, Tn{ a) cos(7n
z)n=0
o <; z < h
T(b,z) = Tb(z) = £ Tn( b ) cos(Tn
z) '• oszshn=0
(3)
where Ta(z) and Tb(z) are the given conditions. It is therefore'
sufficient to determine the T (r) in ec
equation (2) into equation (1) we find
sufficient to determine the T (r) in equation (2). Substituting
Hk
n= 0
o ; n > 0
where a dash denotes differentiation with respect to the
argument.
These equations have solutions
Tn(r) =
f n= 0
n > 0
(4)
where iQ and Ko are modified Bessel functions of the first and
second kind of order zero, and A , B' . A and B are constantso o n n
- 29 -
which must be determined from the boundary conditions.
Using Fourier's inversion formula, the coefficients
Tn(a) and Tfl(b) in equation (3) can be written explicitly in
terms of the boundary conditions Ta(z) and Tb(z) in the following
form.
.h
Tn(a) = -jf J Ta(z) cos(Ynz)dzo
Tn(b) = "jf- J Tb(z) cos(Tnz)dzn = 0,1,2,... (5)
where e = 1 and e =2 when n > 0.
When the constants Tn(a) and Tfl(b) have been determined
from equations (5), the constants AQ, BQ, AR and Bn are obtained
from equations (4) in the form
A =o
Bo =
1 H =
To(a) - T 1 H b 3- a*4 k , b
•In —a
(6)
An "
Tn(a)
T n ( a )
- Tn(b)
- V b )
C (v b) K (Y a)o1 rn ' o w n
: n > 0
n> 0
(7)
The solution of the heat conduction problem is now
complete. Equations (4) give the functions Tn(r) in terms of
the A and Bn and equation (2) provides the final series giving
the temperature distribution in the tube.
- 30 -
3.3 Derivation of the Elastic Deformations
The radial displacement, u(r,z), and the axial displace
ment, w(r,z), of a point with cylindrical coordinates, (r,z),
are governed by the equilibrium equations in the radial and
axial directions. These simultaneous equations can be written
in the form
drdz
r dr
a <. r(8)
where
c =1- 2v
P = (3-4c)ct
in which v is Poisson's ratio and a is the coefficient of
linear thermal expansion.
The principal strains, e r r , e e Q and e z z and the only
non-zero shear strain, e z r , are given by
6 r r dr
ue98 ~ r
_S z z "
'zr
(9)
- 31 -
The principal stresses, o r r, o Q Q and o 2 Z and the only
non-zero shear stress, azr, are given by
arr = X(err + e e e + e z z ) + 2\ierr - (3X+2n.)aT
^99 = M e r r + e e e + ezz) + 2M-eee - (3X+2(x)aT
zz = M£rr - (3\+2p.)aT
'fzr =
(10)
in which X and \i are the Lame constants given by
v E E, _ (1+v)(l-2v) 2(l+v)
in which E is Young's modulus.
Since the symmetry conditions on the end faces of the
tube are
•r = 0 on a^rsb, z=0
du _-r— = 0dz on a s r
w = 0 on
w = Qh. on
z= h
z = 0
z=h
where Q is a constant, the elastic deformations must be
expressible in the form
u(r,
w(r,
z) =
z) = Cz +
In=0
0 9
y£ wn(
r) sin(Ynz)n=l
(11)
We see that this type of solution is compatible with equations
(2) and (8) .
- 32 -
The normal and shear pressures are given on the inner
and outer surfaces of the tube. In the light of equations (9),
(10) and (11) these stresses must be expressible in the form
arr(a,z) = - p_(z) = -'rr n=0
arr(b,z) = - Pj-,(z) = -
o7T.(a,z) = - s(z) = -
n=0
'zrn=l
azr(b,z) = -
cos(Ynz)
s i n(7 nz)
o £ z <; h
o <; z <; h(12)
n(b). sin(7nz) :
where the p ( z ) , p^tz), s a(z) an<3 sb(z) a r e the given conditions,
The negative sign appears because a positive stress implies
tension, a negative stress implies compression.
Using Fourier's inversion formula, the coefficients
(a) a n d sn ^ c a n b e w r i t t e n explicitly inPn(a), pn(b) , sn(i
terms of the boundary conditions p_(z),
in the following form.
>h
, s (z) and
Pn(a) = —
en Pn^b^ = ~h~ J o
cos(«y z)dz
n= 0,1,2, (13)
en Phsn(a) = T~ s (z) sin(7^z)d11 xi j a n
• o - -•
s (b) = -r— s, (z) sinlyn h JQ °bv ^n
where eQ= 1 and en = 2 when n > 0.
n= 1,2,3, (14)
- 33 -
There remains the problem of finding the functions
un(r) and w n(r). Substituting equations (11) into equations
(8) we find the following simultaneous equations governing
un(r) and w n(r).
Un' + r Un" r2 Un ' C< Un + (1"c) V n =
(15)
where a dash denotes differentiation with respect to the
argument.
When n =0, only the first of the pair of equations (15)
applies, the second equation being identically zero. Using the
first of the expressions (4) for T (r) we can write the solution,
u Q(r), of the first of equations (15) in the form
uo(r) = EQr + -2 + G(r) (16)
where E and F are constants to be determined from the boundary
conditions and
G(r) = | p
where A Q and B Q are given by equations (6).
In this case, when n = 0, there are only two unlcnown
constants E Q and FQ, to be determined. There appear to be four
conditions, (13) and (14), to be satisfied. However the shear
stresses are automatically zero when n= 0.
Using equations (9), (10) and (16), the pressure terms
of order zero on the inner and outer surfaces of the tube can
_ 34 -
be written in the form
- 2c-§ - 2c°4^ + (l-2c)C
Q = 2(l-c)Eo - 2cg| - 2c^p) + (l-2c)C
where a bar denotes a dimensionless pressure
P = \ + 2jx
Solving for the only unknown quantities E Q and Fo we find
[Fo - ' [ 2 c{ b G ( a ) " aG(b)}- ab{5o(a) -
This completes the solution of the thermo-elastic
equations when n= 0. The radial displacement component,
uQ(r), is given by equation (16). The axial displacement
component corresponds to constant axial strain and is Qz.
The corresponding strains and stresses are found using equations
(9) and (10).
When n > 0, a similar analytical technique is used to
determine the deformation components for a particular value
of n.Consider, for n > 0, the expressions
(17)wr>(r) - ~ (E^ +2P )I - (l-c)'v rF I ," « n o „. n n 1
+ (Gn ~ 2H )K + (1-cW rH K. +~rz : {A I + B K \
^ o xt n 1 (l"~c)fy • I. n o n oj
- 35 -
where E R , F n, GR and H n are constants, A and Bn are given by
equations (7) and I and K are modified Bessel functions with
argument (Ynr). It is not difficult to verify that these
expressions satisfy the inhomogeneous equations (15).
As above, the corresponding dimensionless pressures
on the inner and outer surfaces of the tube can be obtained
using equations (9) and (10) and can be written in the form
- Pn(b) = X^n ) (b)En+ X^,
n) (b)Fn +X*,n) (b)Gn+ X^
n) 0>)Hn+ X<n ) (b)
where
X<n><r> -
xin)(r) =-2c T JKO + -i-
G rnl o Y nr
(r) =
(18)
in which the argument of the Bessel function is \7 nr), as above.
In precisely the same manner, the dimensionless shear
pressures on the inner and outer surfaces of the tube can be
obtained from equations (9) and (10) and can be written in the
- 36 _
form
- in(a) =( n )
where
(b, -
(a)
Y*"'(b)
(19)
(r) = -
4n) (r) =- 2cTn{(l-c)Ynno+l1}
The only unknown quantities in equations (18) and (19) are
E , F , G and H . These four equations can be solved to
produce the four unknown quantities.
Having determined these constants the solution of the
thermo-elastic equations is complete. The displacement com-
ponents are given by equations (17). The corresponding strains
and stresses are given by equations (9) and (10).
This completes the description of the analytic model
used in the program BAGPIPE.
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