NAVAL POSTGRADUATE SCHOOL• • Monterey, Caiifornia
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ID DDC
THEHSH
FINITE ELEMENT SOLUTION FOR
AXISYMMETRIC TRANSIENT THERMAL STRESSES
by
Manouchiehr Bakhshandehpour
iThesis Advisor: R.E. Newton
Re po., -I. SE June 1972S~NATIONAL TECHNICALFIINFORMATION SERVICE
1) S D'o~~ort -•t of Coo'-erceSi~r-•€l,14,I A 77'151
AXISYMvIETR TRANIEN THERMAL Stez;d66Tin bRi~ESSE
UNCLASSIFIEDSpSec itv Ca.ifi ,,cation
DOCUMENT CONTROL DATA- R & DiSecuruty classuifeation of title. body o! abstract and indexing annotation niu.,t be onteed wýhen the overall rep.,rt is tenssIed)
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3 REPORT TITLE
FINITE ELEMENT SOLUTION FOR AXISYMMETRIC TRANSIENT THERMAL STRE..SES
A. DESCRIPTIVE NOTES (7`'pe I report and.inclusive dotes)
j. Mechanical Engineer's Thesis; June 19725. AUTHORISI (First name, middle initial, last name)
Manouchehr Bakhshandehpour; Lieutenant, Imperial Iranian Navy
0. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REPF.
June 1972 174 98a. CONTRACT OR GRANT NO. go. ORIGINATOR'S REPORT NUMUEFR(S$
b. PROJECT NO.
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Approved for public release; distribution unlimited.
II. SUPPLCMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate School-Monterey, California 93940
13. ABSTRACT
A finite element formulation for solving axisymmetrictransient heat conduction and thermal stress problems isdeveloped in this thesis. The governing equations ofuncoupled, linear, isotropic thermoelasticity are discretizedusing quadratic isoparametric elements. A FORTRAN IV program,using double precision arithmetic, is presented. Compactstorage techniques for banded symmetric matrices are used.
Comparisons between exact and computer solutions demon-strate close agreement for a number of test problems. De-tailed instructions for using the program are included.
DFORM (P _ __
- ,.NoV 5-3 UNCLASSIFIEDS/N 0101-807-6811 Security Clarsification•..a.,,-A-31406
UNCLASSIFIEDSecurity Classification
14 LINK A LINK B LINK C
* ROLE WT ROLE WT POLE WT
AXISYMMETRIC TRANSIENT HEATCONDUCTION
FINITE ELEMENT HEAT CONDUCTION
FINITE ELEMENT TRANSIENT THERMALSTRESSES
AXISYMMETRIC TRANSIENT THERMALSTRESSES
FODD R•° 1473 (BACK) UNCLASSIFIED/ ,. ICI •607"6•I Security Classification
Finite Element qution for
Axisymmetric Transien', Thermal Stresses
af by
Manouchehr Ba.• ishandehpourLieutenant, Imp,•rial Iranian NavyB.S., Italian VOi'al Academy, 1960
Submitted in partial fulfillment of therequirements for the degrees of
MECHANY CAL ENGINEER
and
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLJune 1972
Author 2Z
Approved by: _.-Thesis Advisor
:Chaitman, Department of Mechanical Engineering
.Academic Dean
"A1 STRACT
A finite element formulation for solving axisymmetric
transient heat conduction and thermal stress problems is
developed in this thesis. The governing equations of
uncoupled, linear, isotropic thermoelasticity are discretized
using quadratic isoparametric elements. A FORTRAN IV program,
using double precision arithmetic, is presented. Compact
storage techniques fir banded symmetric matrices are used.
Comparisons between exact and computer solutions demon-
strate close agreement for a number of test problems. De-
tailed instructions for using the program are included.
1/
' -,
TABLE OF CONTENTS
I. INTRODUCTION - 12
II. FINITE ELEMENT FORMULATION OF HEATCONDUCTION IN AXISYMMETRIC BODIES ---------------- 14
A. METHOD OF FORMULATION --------------------- 14
B. THERMAL BOUNDARY CONDITIONS ------------------ 16
C. EXACT TIME SOLUTION WITH SPATIAL
DISCRETIZATION ------------------------------- 18
D. TIME INTEGRATION ----------------------------- 20
E. ESTIMATION OF EXTREME EIGENVALUES ------------ 26
III. ONE-DIMIENSIONAL HEAT CONDUCTION ------------------ 29
IV. STRESS PROBLEM ----------------------------------- 32
A. STIFFNESS MATRIX ----------------------------- 33
B. THERMAL LOAD VECTOR -------------------------- 34
C. PRESSURE LOAD VECTOR ------------------------- 36
D. CENTRIFUGAL LOAD VECTOR ---------------------- 37
E. STRUCTURAL BOUNDARY CONDITIONS --------------- 38
F. SYSTEM EQUATION SOLVER ---------------------- 39
G. PRINCIPLE OF SUPERPOSITION ------------------- 39
H. STRESS EVALUATION ---------------------------- 39
V. ONE-DIMENSIONAL TRANSIENT STRESSES --------------- 41
VI. TEST PROBLEMS ----------------------------------- 42
Vli. CON&ZUSIONS AND RECOMMENDATIONS ------------------ 54
APPENDIX A: APPLICABLE FORMULAS AND EQUATIONS .........- 56
SAPPENDIX B: LIST OF THE PROGRAM ------------------- 59APPENDIX C: USER'S MANUAL ---------------------------- 135
3
APPENDIX D: PROGRAMMING ------------------------------- 160
LIST OF REFERENCES ------------------------------------- 171
INITIAL DISTRIBUTION LIST ------------------------------ 172
DD FORM 1473 ------------------------------------------ 173
IiI
A•
Ct
g- ,
r
k'J
LIST OF FIGURES
1. Slab with One Face Insulated and Anotherin Contact with Fluid --------------------------- 29
2. Quadrilateral Element Representation --------------- 33
3. Boundary Element Under Pressure -------------------- 36
4. Two Radial Elements Representatien inThick Cylinder ----------------------------------- 43
5. Five Radial Elements Representation inThick Cylinder ---------------------------------- 43
6. Stresses in Thick Cylinder Under ThermalLoading ------------------------------------------ 44
7. Stresses in Thick Cylinder Under UniformInternal Pressure -------------------------------- 45
8. Stresses in Rotating Thick Cylinder --------------- 47
9. Hollow Semi-sphere 32 Elements Representation 48
10. Thermal Stresses in Hollow Sphere ------------------ 49
11. Nozzle Geometry ------------------------------------ 50
12. Fluid Temperature-Time Histories ------------------- 52
13. Element Representation in Nozzle ------------------- 53
14. Longitudinal Cross Section ------------------------ 137
15. Subdivision into Elements ------------------------- 138
16. Element and Node Numbering ------------------------ 139
17. Entry Temperature-Time Variation ------------------ 151
18. Deck Set-up for Using CHECK Program ---------------- 155
19. Deck Set-up for Using AXITTS Program --------------- 157
20. Deck Set-up for Obtaining a Listing ofthe Program AXITTS ------------------------------- 157
21. Deck Set-up for Obtaining a Punched Deckof the Program AXITTS --------------------------- 158
5
22. Functional Flow Chart of CANDY --------------------- 162
23. Fluid Node Representation for Inside Flow ---------- 163
24. Functional Flow Chart of STIFF --------------------- 167
25. Functional Flow Chart of F0RMF -------------------- 168
I
LIST OF TABLES
I. Attenuation Factor Comparison, Fourth OrderRunge-Kutta Algorithm --------------------------- 23
II. Attenuation Factor Comparison, TrapezoidalIntegration ------------------------------------ 25
III. Effect of Using Irons' Correction at Every10 Steps of Integration ------------------------- 27
IV. Comparison of One-dimensional TransientTemperatures ----------------------------------- 31
V. Units for Input Data ----------------------------- 136
VI. Maximum Values for Program Parameters ------------- 170
I- 7
LIST OF SYMBOLS
Note: A single underline is used to denote a column vectorand a double underline denotes a rectangular matrix.The symbols used in computer program are described inthe beginning of Appendix B.
a Entrance fluid cross-sectional area
A Linear combination of C and Y matrices
b A constant vector
B Standard rectangular strain-displacement matrix
C Thermal capacitance matrix
c Specific heat
D Standard elasticity matrix
E Young's modulus of elasticity
e Superscript designating element contribution
F Load vector
F Element thermal load vector
Fe Element pressure load vector
Fe Element centrifugal load vector
SLinear combination of C and Y matrices
h Surface heat transfer coefficient
I Identity matrixL'J Jacobian coordinate transformation matrixK System stiffness matrix
Ke Element'stiffness matrix
k Therme. conductivity
L Thickness of slab
Arc length along the side of quadrilateral
N. Shape function
m Total number of nodes
n Outwa-7d normal or number of nodes per element
P Pressure
R Radial coordinate
S Surface area
T Nodal tfnmperature vectorT Temperature or, when used as a superscript,
transpose of a matrix
Tavg Average temperature
Tf Fluid temperature (used in one-dimensional example)
U A vector defined as u Radial displacement
v Right-hand side vector in conduction equation
V Volume
W Work done by loads
W Modal matrix
w Axial displacement
w Eigenvector
+,Y Thermal admittance matrix
y Dependent variable
Yij Element ij of the m~trix Y*
[ ] Matrix representation
< > Row vector
V Gradient operator
a Coefficient of thermal expansion
0 Constant coefficient vector
6 Nodal displacement vector
6_e Element displacement vector
{ - Strain vectorEigenvalue of one-.dimensional transient temperaturesolutionThermal str.iin vector
L•e Element strain vector
Local element coordinate
f v Poisson's ratioX "Eigenvalue
p Material density
a Stress vector
3 • Local element coordinate
T Time
AT Step size of numerical time integration
TRZ Shearing stress component
8 Fluid temperature, or angle
A Spectral matrix
Speed of rotation
SI
'• 10
ACKNOWLEDGEMENTS
F The author would like to express his gratitude to theImperial Iranian Navy for having made it possible for him
to undertake the course of graduate study leading to the
present text.
To Dr. Robert E. Newton, Professor of Mechanical
Engineering, the author wishes to express deep appreciation
for his inspiring advice and guidance. Without his diligence
and patience this study could not have been accomplished in
its present form. The author is obliged to Dr. Gilles
Cantin, Professor of Mechanical Engineering, for his generous
advice during the course of study and computer programming.
Thanks are also due to the personnel at the computer center
of Naval Postgraduate School.
Finally the author should thank his wife for her
patience and understanding throughout his work at this
institution.
.V
I. INTRODUCTION
Thermal stresses have become increasingly important in
engineering practice during recent years. In power genera-
tion higher cycle temperatures and use of nuclear fission
are largely responsible for this trend. This thesis describes
a computer program for finding temperatures and stresses in
bodies having axisymmetric geometry and loading. The govern-
i equations are those of isotropic, uncoupled, quasi-static,
linear thermoelasticity. They are discretized by using the
finite element method. A FORTRAN IV computer program using
double-precision arithmetic has been written to solve
problems of the following kinds.
A. TEMPERATURE PROBLEMS
The transient temperature vector, evaluated at the nodal
points, may be obtained for an axisymmetric body with the
combination of insulated, convection, or constant temperature
boundary conditions. For the convection thermal boundary
condition, however, we may have fluid flowing with entry
temperature prescribed as a linear function of time (RAMP).
The program can handle up to 15 different ramps, each having
a different flow velocity, and with discontinuities between
successive ramps.
B. STRESS PROBLEMS
The program will generate load vectors for pressure load-
ing, centrifugal loading, and axial force. Provision is
4 made for direct input of one additional load vector. Stressesmay be found for any combination of these loadings.
C. THERMAL STRESS PROBLEMS
Thermal stresses may be found for as many as 20 different
temperature vectors which may be output of the temperature
problem or direct input. In short, in this part any combina-
tion of the temperature and stress problems may be used.
13
II. FINITE ELEMENT FORMULATION OF HEAT CONDUCTIONIN AXISYNIETRIC BODIES
A. METHOD OF FORMULATION
For bodies of revolution under axisymmetric loading the
mathematical problems presented are two-dimensional. The
governing equation for non-steady heat conduction is
V-kVT = pci, (1)
where k is the thermal conductivity, p the density, c the
specific heat, T the temperature and V the gradient operator.
The superior dot denotes a time derivative.
Applying Galerkin's principle [I] gives
f NiV'kVT dV = f pc N. T dV, (2)
V V
where the integral is over the volume V of the conducting
body and Ni is a "shape function" used in representing the
temperature distribution. If S is the surface of the body
and n the outward normal to surface, then using Gauss'
theorem we can write
f V- '(N. kVT) dV = f N. k 'T dS. (3)
V S n 3n
Since
f V"(N. kVT) dV = f (Ni)'(kVT)dv'4 V 1
+ f NiV.(kVT)dV, (4)
Vwe can combine Eqs. 2, 3' and 4 to get
DTf cNi dV + f (VNi)'(kVT)dV= f Nik•-in dS. (5)V V S
Each node of the solid region has a separate discretized
S~linear equation calculated from Eq. 5 using the appropriate
shape function Ni Thus each of the volume integrals on the
left hand side of Eq. 5 yields a square coefficient matrix
in the assembled set of equations. Calculation of these
matrices is a standard process. Details are given by
Zienkiewicz [1].
The discretized set of equations takes the form
C T+ Y T = v, (6)/
where there is a term by term correspondence with Eq. 5.
The real symmetric matrices C and Y represent, respectively,
the thermal capacitance and thermal admittance. The elements
of the vector T are nodal temperatures. The vector v, dis-
cussed in the following section, depends upon the thermal
boundary conditions.
In the present development piccewise constant material
properties have been assumed, i.e., k, p and c are constant
within each element, but may vary from element to element.
Also two-dimensional isoparametric elements are used. Appli-
I cable equations are summarized in Appendix A.
In this text a double underline denotes a rectangularmatrix and single underline denotes a column vector.
i~~~~ -S . ....
B. THERMAL BOUNDARY CONDITIONS
Thermal boundary conditions affect only those scalar
equations of Eq. 6 which correspond to boundary nodes.
Accordingly, the vector v is sparse. Also, in a single
problem it is common to have different thermal boundary con-
ditions on individual portions of the boundary. In what
follows the subvectors of v (distinguished by individual
superscripts) which correspond to separate boundary condi-
tions are treated individually.
1. Insulated
It is clear that for insulated boundary conditions
the subvector vM1 ) of the right hand side of Eq. 6 corres-
ponding to this boundary condition is zero, since T _n
2. Convection
The heat transfer mechanism occurs in the interface
of the solid and fluid. If the fluid temperature is e and
the heat transfer coefficient is h, then equating heat con-
ducted away from the surface to the efflux from the solid
[2] gives
-k (E) = h(T-0), (7)
where the subscript S means that the derivative is evaluated
at the surface.
For constant h:
f Nik 'T dS = h f Ni(O-T) dS. (8)S Sn 1
16
In what follows the fluid temperature e is taken to be a
specified function of position and time. For purposes of
discretization, the fluid temperature is specified at a
discrete number of fluid 'tnodes." If 0. represents the
fluid temperature at fluid node j, then the fluid temperature
along the boundary may be represented by
6 = Z N.OP (9)
where the N. are one-dimensional forms of the shape functions
used for the solid. Substitution in Eq. 8 gives
8T *f N. k E-I dS = E y. (0j-Tj)' (10)
1 3n 13 J 3
where the summation extends over the surface nodes and the,
coefficients Yij are given by
Yij = h f NiN. dS. (11)S
Assembling the contributions from Eq. 10, the subvector
v (2) for the convection boundary condition may be written
v (2) = Y e. (12)
The contributions -Z Yij j from Eq. 10 are included by
augmenting the matrix Y (see Eq. 13 below).
3. Constant Temperature
If 0 represents the constant tempe1 ature desired at
the wetted surface, then we can use the convection boundary
condition and replace h by a big number (say 1020). Since h
is very large, then for thermal equilibrium the temperature
17
T at the surface will be forced to equal e. So the subvector
V3 for the portion corresponding to the constant temperature
boundary condition can be obtained from Eq. 12.
Upon the application of these boundary conditions in
a single problem, the right-hand side vector v will be com-
bined from the corresponding subvectors and the finite ele-
ment discretized equation becomes
CT + _Y T =, (13)
Swhere Y+ +
C. EXACT TIME SOLUTION WITH SPATIAL DISCRETIZATION
We consider only the solution of Eq. 13 for v = constant
with T = a at time T = 0. Let T be a particular solution
(steady state) with is = 0 so that
is ) 1. (14)
For the homogeneous equation
C. + Y' T = 0 (15)
the assumption T = w exp(-XT), where w is a vector and X is
a scalar, yields the form
j+ w = X * w (16)
It is apparent that Eq. 16 defines an eigenvalue problem.
Let A be the spectral matrix and I the modal matrix with
normalization according to
SWTW IV = IW , (17)
18
where I is the identity matrix of the same order as C. Now
let
T_ = _w exp(-=AT) b (18)
where b is a constant vector. Substituting this in Eq. 15
gives
Y+ W exp(-AT) b = C W A exp(-rAT) b. (19)
Now Eq. 19 is satisfied for all b if
W T Y+ W=A, (19')
and this is guaranteed to be satisfied since A and W are
spectral and modal matrices for the eigenvalue problem of
Eq. 16 with W normalized according to Eq. 17.
Returning to the original problem (Eq. 13), the com-
plete solution may be written as
Y = W (B + exp(-AT) b) (20)
where
T= W , (21.)
and B is a constant vector. Now 8 may be found (using
Eq. 14) to be
= A 1 _wT_ . (22)
Substituting this result into Eq. 20 and using the initial
condition gives the result
b = W-1 a - A" 1 WT v. (23)
The general solution of Eq. 13 may thus be written
as
T =. W- exp(-AT)) ,,-I WTV + exp(-AT)W- la. (24)
For purposes of the present program non-zero compo-
nents of vector v are to be specified as piecewise linear
functions of time. During each segment of time history of v
an analytical solution of Eq. 13 is possible in the form of
a particular solution plus a complementary solution such as
Eq. 20. At each node the corresponding time variation of
temperature will consist of a linear part contributed by the
particular solution and a sum of n terms representing the
complementary part. Each of these n ternis decaysexponen-
tially with a separate time constant. In principle it is a
straightforward process to find each particular solution and
accompanying complementary solution.
Contemplated problems may typically have from 10 to
40 segments required to represent the piecewise linear
variation of v. The number of body nodes n will be of the
order of 100 or more. In view of the number of particular
solutions required, each accompanied by an individual comple-
mentary solution of the form given by Eq. 20, it was concluded
that a numerical solution of Eq. .13 would be considerably
more economical than an analytic one such a's that given by
Eq. 24.
D. TIME INTEGRATION
in this section the relative m,:rits of the Runge-Kutta
and trapezoidal. methods of time integration are discussed.
Since either of these methods will g;ve an exact result if
the solution is a linear' function of time, investigation
is confined to performance on a single scalar equation
ý + Xy = 0 (25)
whose solution y = y0 exp(-XT) is of the same form as the
components of the complementary solution (Eq. 20 ).
1. Runge-Kutta Method
A method introduced by Runge and subsequently elabo-
rated by Heun and Kutta [3] is widely used for the numerical
solution of first order ordinary differential equatiois.
This algorithm prescribes a sequence of calculations for
determining the ordinate yi+l at time T i+l = T i+AT in terms
of yi and values of 5 at intermediate and end points of the
interval AT. The fourth-order form, which requires four
evaluations of y, gives for Eq. 25 the result
Yi+l _ 1 X + 2 (XAT)3 (XAT)41 A (XA-iT (26) +(~Yi 2! 3! 4! " (26)
The right-hand side of Eq. 26 represents the first five terms
of the Taylor expansion of the exact solution
(Yi+I/Yi = exp(-XAT)) so we may conclude that the relative
error in each time step is less than modulus of the next
term: (XAT)5/5!.
In addition to providing the apparent prospect for
high precision indicated by this error bound, the Runge-Kutta
method also permits changes of time increment during the
integration process without requiring additional computa-
tionally expensive matrix decompositions. The attractiveness
of these two features dictated a thorough exploration of the
potential of this method for the present application. The
21
disqualifying defect which emeiged after studying a number
of examples is readily appreciated from examination of
Table I. For values of XATr less than 0.5 it is apparent
that Runge-Kutta scheme affords acceptable engineering
accuracy. However, when the method is applied to solution
of Eq. 13 we must deal with a number of Ps equal to the
number of nodes (see Eq. 20 ). This number may be greater
than 100 and the ratio of the largest X to the smallest may
easily exceed 1000. Although the solution is dominated by
the contributions of the eigenvectors corresponding to the
smaller Xs, it is clear that the solution will be unstable
if the largest XATr exceeds about 2.7. Because an unaccept-
ably small AT is required in typical problems, the Runge-
Kutta method was rejected.
2. Trapezoidal Method
The trapczoidal method estimates yi+l from the formula
Yi+i= Yi + 1(i + Yi+l) (27)
Substituing for i andi+l from Eq. 25 and rearranging
gives
Yi+l 2 - XAT (28)Yi 2 + XAT
Series exijansion of the right-hand side provides an error
bound (per step):
(XAT) 3/12.
From the point of view of the size of XAT this method has no
stability limit, but has slow attenuation with alteration
22
TABLE 'I
ATTENUATION FACTOR COMPARISON
Fourth Order Runge-Kutta Algorithm
SYi+i/Yi exp(-XA'r)(Runge-Kutta) (Exact)
.0001 .9ý99 .9999
.001 .9990 .9990
.01 .9901 .9901
.1 .9048 .9048
.2 .8187 .8187
.5 .6068 .6065
1.0 .3750 .3679
2.0 .3333 .1353
2.5 .6484 .0821
3.0 1.3750 .V198
4.0 5.0000 .0183
8.0 110.3333 .0003
10.0 291.0000 .0000
20.0 5514.3333 .0000
50.0 240784.3333 .0000
100.0 4004901.0000 .0000
2
23
in sign for large XAT. Table II shows this behavior.
Since a wide usable range of XAT is essential and
the stability of trapezoidal integration is guaranteed,
this method is chosen for the present program.
Applying the trapezoidal algorithm to Eq. 13 yields
A Ti+l G Ti + L- (vi+l + v1 i (29)
2
where
A =C +AT Y+S• ~2 '
G= C AT y# +
S 2
and the superscripts denote evaluation at discrete time
intervals AT. If m is the order of the capacitance and
admittance matrices, C and Y, then once a certain step size
3.AT is chosen, it requires m /3 operations to perform the
needed t:r z•ngular decomposition of A. Thus, for large m, a
change of step size AT becomes costly from the point of view
of computer time. Accordingly, in the piesent program only
one time step size is used thloughout each problem.
Also, for assuring sufficiently rapid attenuation of the
components corresponding to the large XAT, the following
correction is utilized.
?. Irons' Correction
Irons proposed a scheme [4) for augmenting the atten-
uation of the contributions of those eigenvectors for which
XAT is large. Define
24
TABLE II
ATTENUATION FACTOR COMPARISON
Trapezoidal Integration
XAT Yi+I/Yi e-fAT(Trapezoidal) (Exact)
.0001 .9999 .9999
.001 .9990 .9990
.01 .9901 .9901
.1 .9048 .9048
.2 .8182 .8187
.3 .7391 .7408
.5 .6000 .6065
1.0 .3333 .3679
2.0 .0000 .1353
2.5 -.1111 .'0821
3.0 .2000 .0498
4.0 -. 3333 .0183
8.0 -. 6000 .0003
10.0 -. 6667 .0000
20.0 -. 8182 .0000
50.0 -. 9231 .0000
100.0 -. 9608 .0000
25
S .25 yi- + -. yi + .25 yi+l, (30)
where yi and yi+l are obtained from yi-l by trapezoidal
integration.
In the program presented in Appendix B, Eq. 30 is used
after every 10 steps of time integration. Table III shows
the resulting modifications.
E. ESTIMATION OF EXTREME EIGENVALUESAnalytic results for one-dimensional heat conduction
give, for an eigenvalue,
r2k 1(31)4pc d2
where d is the distance between points of extreme tempera-
ture and zero temperature.
If we use this to estimate the smallest X in cylindrical
coordinates, two modifications are recommended.
1. Assume that the point of zero temperature is in
the fluid at a distance from the wall equal to k/h,
where h is the surface heat transfer coefficient.
2. If there are two approximately orthogonal paths for
heat flow from the (single) maximum temperature point,
then replace l/d2 in the above formula by
1 1 + d (32)
d mi. max.
For estimating the largest X, the surface heat transfer
coefficient has no significant effect. We may continue to
26
TABLE III
EFFECTS OF USING IRONS' CORRECTION
AFTER 10 STEPS OF INTEGRATION
XAr exp(-10XAT) Yl0/Y0 y* 1 0 /y 0Exact Trapezoidal Corrected
0.00010 0.99900 0.99900 0.99900
0.00020 0.99800 0.99800 0.99800
0.00100 0.99005 0.99005 0.99005
0.01000 0.90484 0.90484 0.90486
0.10000 0.36788 0.36757 0.36849
0.20000 0.13534 0.13443 0.13579
0.30000 0.04979 0.04866 0.04978
0.50000 0.00674 0.00605 0.00645
1.00000 0.0000S 0.00002 0.00002
2.00000 0.00000 0.00000 0.00000
4.00000 0.00000 0.00002 -0.00001
8.00000 0.00000 0.00605 -0.00040
10.00000 0.00000 0.01734 -0.00072
20.00000 0.00000 0.13443 -0.00136
50.00000 0.00000 0.44914 -0.00072
100.00000 0.00000 0.67028 -0.00027
I
• 27
use the same formula for d2 , but now consider only the
smallest element and take
d _ length of smallest sidedmin 4
(32')
d max length of largest side4
A comparison of estimates based on Eq. 31, 32, 32' with
the exact solution of the eigenvalue problem has been carried
out for several examples. Based on these comparisons, it is
believed that these estimates are sufficiently accurate for
choosing a time step and estimating the time of occurrence
of the extreme stresses.
28
III. ONE-DIMENSIONAL HEAT CONDUCTION
For comparison of numerical (time) integration methods,
studies of one-dimensional heat conduction were made. In
this section numerical results for trapezoidal time integra-
tion using Irons' correction are compared with the exact
transient temperature solution.
Consider a flat slab of thickness L with conductivity k,
density p, specific heat c, zero initial temperature, one
face insulated and the other in contact with fluid at tem-
perature Tf (Fig. 1). The surface heat transfer is h. The
exact transient heat conduction solution is available ES] as
SSin~iL Cospix Vk '-TT=Tf 1- 2 E L+SinpiL Cos•iL e (33)
-b •Temperature T f
Slab . .
Figure 1. Slab with One Face Insulated and Anotherin Contact with Fluid.
29
where k/pc is the diffusivity and the •i are the solutions
of the transcendental equation
ShL ITan UL = hL 1 (34)
For the finite element comparisons we subdivide the
distance L into m one-dimensional 3-noded elements and use
the corresponding isoparametric shape functions. (The work-
ing equations, shape functions and the element capacitance
and admittance matrices are given in Appendix A.)
In the course of this investigation separate computer
programs were written to evaluate nodal transient tempera-
tures using the following methods:
(a) exact transient temperature solution, Eq. 33;
(b) exact time solution with spatial discretization• -(section II.C, Eq. 24);
(c) Runge-Kutta time integration (section II.D,part a);
(d) trapezoidal time integration (section II.D,part b);
(e) trapezoidal time integration with Irons'correction (section II.D, part c).
For an initial step change of fluid temperature from
zero to 1, transient temperatures have been found. For these
comparisons the parameters (in consistent units) were taken
to be:
L = 8, p = 25, k = 8, c = 5 and h = 5
The distance L was subdivided into m.= 3 elements. For the
present purpose, comparison is confined to the exact solu-
tion (item (a)), arid the finally adopted system (item (e)).
30
In Table IV, temperatures at the two faces (x = 0, x = 8)
are compared for various times. The trapezoidal integration
has been performed using the constant time increment unity
and the Irons' correction is applied after every 10 incre-
ments.
It is believed that Table IV demonstrates that the
numerical integration method gives adequate accuracy for
engineering applications.
TABLE IV
COMPARISON OF ONE-DIMENSIONAL TRANSIENT TEMPERATURES
Method x Time=0 Time=10 Time=20 Time=30 Time=50 Time=70
Exact 0. .00000 .00000 .00000 .00002 .00095 .00560Trapezoidal 0with Ironsl .00000 .00028 .00007 .00066 .00207 .00619
Exact 8. .00000 .38431 .47684 .53284 .60264 .64685
Trapezoidalwith Irons'V 8. .00000 .38720 .47716 .53262 .60256 .64686
'Irons' correction is used after every 10 steps of trapezoidalintegration.
31
IV. STRESS PROBLEM
For bodies of revolution deformed symmetrically under
axisymmetric loading, the stress distribution is two-
dimensional. Since deformation is symmetric about the axis
of revolution, cylindrical coordinates (R,Z,O) are used.
It follows that the stress components are independent of the
angle e and all derivatives with respect to O are zero.
Also the components of shearing stress TRe and TZO vanish
on account of the symmetry. But since any radial displace-
ment induces a strain c in the circumferential direction,
this non-zero component of strain and the three in-plane
components (cZ' CR' YRZ)' complete the state of strain at a
:, $. point in any axisymmetric situation. Hence the state of
stress for an axisymmetric body under axisymmetric loading
is given by
T (35)
- 0 RZ
In this chapter the stiffness matrix of an axisymmetric
body and the thermal, pressure, and centrifugal load vectors
are formulated and, finally, evaluation of stresses at a
point is discussed. The treatment closely follows that of
Zienkiewicz [(] and this reference should be consulted for
further details.
32
A. STIFFNESS MATRIX
The elements used are bodies of revolution (about the
Z axis). For analysis it is sufficient to describe thecross-section in the R,Z plane. In Fig. 2 such an element
and the local •,r coordinates are shown.
I4
+/
IW
-. Z
Figure 2. Quadrilateral Element Representation.
If u and w are the displacement components at a point in the
directions of R and Z respectively, then these displacement
components may be defined in terms of the nodal displacements
by the appropriate isoparametric shape functions as
8 8u Z Niui , w = Z Niwi , (36)
i~l 11i=l11
where Ni, a function of 9 and n , is the shape function for
element node i and ui'wi, are the nodal displacement compo-
nents. The strain-displacement relations [6] can now be used
to obtain the components of the strain vector. Thus
33
c = B 6 , (37)
where B is the standard rectangular strain-displacement
matrix of any finite element formulation, a function ofthe local coordinates ý and n, and 6 is the vector of nodal
displacement. (See Appendix A, part 3, where the applicable
formulas and useful equations are summarized).
If the elasticity matrix for an isotropic material is
D, then the stress vector o at a point is given by
oaDe . (38)
Now, by evaluation of the total strain energy in the element,
the element stiffness matrix can readily be obtained as
Ke - f BT D B dV , (39)
where the integration extends over the volume of the
element.
In the present program the upper triangle of each ele-
ment stiffness matrix is evaluated by numerical integration
using four Gauss points within the range of ý and of n [1].
The element contribution is immediately placed in the system
stiffness matrix, which is stored in banded form.
B. THERMAL LOAD VECTOR
If we denote T as the difference between local tempera-
ture and reference temperature, then the thermal strain fo
is given as
c UaT, (40)
34
whereU =< 1 1 O>T
and a is the coefficient of thermal expansion.
The thermal load vector Fe is given by Zienkiewicz [1]
as
E f T Dco dV. (41)
From the point of view of the numerical evaluation it is
interesting to note, however, th-t the product D e in
Eq. 41 will reduce to
D e = Ea Tu (42)L-o 1-2.v ,
where E is the modulus of elasticity and v is Poisson's
ratio. Thus
Fe_ Ea f T BT U dV (43)T 1-2v
nwhere T = E NiTi and n is the number of nodes in each
i=lelement.
In the attached computer program in Appendix B the advan-
tage of the simplicity of Eq. 42 has been utilized. Also,
since B has some zero components, in the process of multi-
TTplication of BT U, simply the addition of the appropriate
non-zero components of each column of the B has been per-
formed. Finally, Eq. 43 has been integrated with four Gauss
points.
35
C. PRESSURE LOAD VECTOR
Consider a quadrilateral element as in Fig. 3 on the
boundary of the axisymmetric cross-section where constant
pressure P is applied. The infinitesimal force dF due to
the normal pressure acting on the inner infinitesimal cir-
cumferential surface dS is
dF = PdS = 2rRP dt , (44)
where dt is the infinitesimal length along the side of
quadrilateral.
p- Z,W
Figure 3. Boundary Element Under Pressure.
Let FR and Fz be the components of the pressure force in the
R and Z directions respectively, then
dFR = dF Cos 0, dFz = -dF Sin 0 , (45)
where Sin = dR and Cos 0 = d Therefore
dFR = (2-RP)dZ dFz = -(27r RP)dR. (46)
- 36
Since the total work W done by the normal force F is equal
to the sum of the work done in R and Z directions,
W = fu dFR + 1w dFZ
S2'rP (fR u dZ -. IR w dR). (47)
Now, by using the appropriate shape functions, each compo-
nent of Eq. 47 may be defined in terms of the nodal values.
Since
.W =( 6 _e) e; (48)
(48'
where F is the element pressure load vector
vector contributed by node j is explicitly given as
g F R -e1N zF. 2,fP f (ENiRi) Nj dý (49)"JP _F zj -1i DN Nip L - • R•
where N. in Eq. 49 is the appropriate shape function for
node j.
D. CENTRIFUGAL LOAD VECTOR
Refer again to Fig. 2 and assume that the body is rotat-
ing about the Z axis with angular speed Q. Then the centri-
fugal force per unit volume at a point distant R from the Z
axis will be pQ2 R, where p is the density of the material.
The work done in this case is
W pP2 R udV (50)
V
37
For constant pP2 the corresponding element centrifugal load
vector is readily obtained by evaluation of the components
of the integral in Eq. 50 in terms of the nodal variables,
i.e.,
F = 21Tpg 2 f f (ZNiRi) det J N. d~dn, (51)3c -i -i =
where det J is the determinant of the Jacobian coordinate
transformation matrix (see Appendix A).
k E. STRUCTURAL BOUNDARY CONDITIONS
The structural boundary conditions implemented in the
program are:i (a) one or more nodes prevented from moving axially;
(b) one or more nodes prevented from moving radially;
(c) the right-hand end cross-sectional plane remainsplane and the transmitted axial force is specified.Hereinafter this will be referred to as the plane-end boundary condition.
The computer program presented in Appendix B has the
capability of handling any combination of the above men-
tioned structural boundary conditions. For boundary condi-
tions of the types (a) and (b), simpfsly multiplying the
corresponding diagonal component of the stiffness matrix byco20 gives zero displacement [8] (for practical purposes).
For the boundary condition of type (c) both ends are initially
fixed axially for all solutions. An additional solution is
obtained for uni\. axial displacement of one end. The axial
., force is evaluated for each solution. Superposition is per-
formed by adding the displacement vectors for the given
, •38
loadings (thermal plus mechanical), plus an appropriate
fraction of the vector found for unit axial displacement.
This fraction is chosen so that the resultant axial load has
the specified value.
F. SYSTEM EQUATION SOLVER
Vi Once the desired structural boundary conditions are
applied, then the problem is to find the nodal displacement
vector 6, corresponding to a given number of load vectors.
,;f We have
K 6 F , (52)
where K is the system stiffness matrix in banded form and F
is a load vector. In the present computer program a single
Cholesky decomposition is performed on K. Then, by a process
of forward and back substitution, each load vector is
replaced by the corresponding displacement vector.
G. PRINCIPLE OF SUPERPOSITION
Upon the evaluation of the displacement vectors due to
the various types of loading, the principle of superposition
can be applied on the displacement vectors. On each thermal
displacement vector the displacement due to any other type
of loading is superimposed and, as the result, the number of
displacement vectors is reduced to the number of thermal load
1' vectors.H. STRESS EVALUATION
From the system displacement vector 6, the displacement
vector of each element _e may be obtained easily. Then the
39
corresponding element strain vector se at any point can be
found from
5e B 6e (53)
Finally, the corresponding element stress vector Oe is
obtained by
a e D (C e L- ) *(54)
Since normally the stresses on the inner and outer surfaces
of axisymmetric bodies are desired, in the computer program
presented in the Appendix B provision has been made to cal-
culate the stresses at the two Gauss points corresponding
to • = ± 1_L_ on the inner and outer boundaries of eachVT
element (where n = ±1).
Upon the evaluation of the stresses at each point the
mean stress and the octahedral shearing stress [7] are
calculated. The program gives as output the extreme values
of these stresses, the R and Z coordinates of the corres-
ponding points, and the times of occurrence.
40
V. ONE-DIMENSIONAL TRANSIENT STRESSES
In this section a one-dimensional comparison of stresses
is made between exact and finite element results. The tran-
sient temperature problem is the one previously described in
Section III.
If the slab edges are free to translate in the plane of
the slab, but are prevented from rotating, the exact solu-
tion for thermal stress [9] is
a Ea -(T T) (55)Oy -z 1-V Tavg
where T is the local temperature (Eq. 33), andTavg is the average temperature in the slab
If we choose E = 2, a = .50, and v = 0 (all in consistent
units), then the maximum stress obtained from the exact solu-
tion is
a ma -. 477786
and it occurs at x = 8, 1 = 73.
Using the finite element technique with trapezoidal time
integration and Irons' correction every 10 steps, the maximum
stress is found to be
amax - -. 477778
and it also occurs at x = 8, T = 73, as before.
f It is observed that the method chosen gives excellentresults.
41
VI. TEST PROBLEMS
Program integrity and accuracy have been verified by
solving a number of test problems. Since stresses, whose
evaluation depends upon derivatives of displacements, are
known to be less accurate than temperatures, comparisons
with exact results are confined to stresses. Individual
problems are described below.
A. THICK CYLINDER
Consider a thick cylinder with inside radius 30 inches
and outside radius 50 inches and the following material
properties.
Modulus of elasticity E = 28.9 x 106 Psi
Poisson's ratio v = .28
Coefficient of thermal expansion a = 7.22 x 10"6 I/F 0
Thermal conductivity k = 28. hr.ftuF
Density p = 489. Lbm/ft3
Specific heat c = .111 LBm.tuF
An arbitrary length of 25 inches has been selected for the
cylinder and it has been subdivided into two different
element representations as in Figs. 4 and 5. The plane-end
boundary condition with zero axial force is used. The
stresses for various types of loading are compared with the
corresponding exact analytic solutions as described below.
42
Figure 4. Two Radial Elements Representationof Thick Cylinder.
Figure 5. Five Radial Element Representation
of Thick Cylinder.
1. Thermal Loading
For a linear variation of the temperature T = 20R
the stresses obtained by the finite element method for the
above representations are compared with the exact analytic
solution in Fig. 6. The TRZ for this problem clearly is
zero and the one obtained by the program was 10-9. The
accuracy of the other results is clearly satisfactory.
2. Pressure Loading
A uniform pressure of 1000 psi. acts on the inner
surface of the cylinder. Again, T RZ is zero and the program
gives 10 -0. In Fig. 7 the other stresses induced by this
uniform pressure are compared with the exact solutions.
Here also the accuracy of the results, even with two radial
elements, is adequate.
43
AKSl.
60
50
-EXACT.
40 42 A ELEMENTS.
30 o 5 ELEMENTS.
20
10
030 32 34 36 38 42 44 46 46 50 RIN.
10
20
30 FIG.
THICK CYL INDER UNDER THERMAL
LOADING, T= 20R
40
450
• 44
PI Z
2000
1600
1200
- EXACT.800 A 2 ELEMENTS
0 5 ELEMENTS
400
Az
-400
-600.
30 34 39 42 46' 50 JIcJnch
FIG. 7 THICK CYLINDER? UNDER INTERAL PRESSUE,
45
3. Centrifugal Loading
The speed of rotation has been assumed to be 500
revolutions per minute. The stresses obtained by the pro-
gram are compared with the analytic solution in Fig. 8. In
this case also the results obtained by the program, even
with only two radial elements, are very close to the exact
solution.
B. HOLLOW SPHERE
We consider a hollow sphere with the same material
properties as in the thick cylinder with inside spherical
radius 30 inches and outside 50 inches. The loading is
thermal with T = 20 -(spherical radius). Symmetry permits
using only half of the sphere for the computer analysis.
The elements representation is given in Fig. 9. Since the
program gives stresses in cylindrical coordinates, these
have been transformed to the spherical coordinates for
comparison with the exact solution in Fig. 10. The accuracy
of the results is noteworthy.
C. THERMAL STRESSES IN NOZZLE
This problem concerns thermal stresses near the inter-
section of a cylindrical pipe and the spherical vessel.
Fig. 11 gives the cross-section of the structure which is
to be analysed. The material properties are:
Modulus of elasticity E = 29.3 x 106 Psi
Poisson's ratio v = .30
Thermal expansion coefficient a = 7.6 x 10-6 1/F0
46
4..iKSI.6
4.-
A
3.
--. EXA C T.
A 2 ELEMENTS.
2. o 5 ELEMENTS .
1.5
FIG. 8
ROTATING CYLINDERI.
.5
47
R
o30 35 40 50 Z
jfl Figure 9. Hollow Semi-sphere 32 Elements Representation.
48
80
70
60
50
40
30
20
I0R
10
20
20
4030
50
3 32 .34 3ý 38 4 42 44 4G 48 50 SPHERERADIUS.IN.
Figure 10. Thermal Stresses in Hollow SphereT = 20 (spherical radius).
49
LUQ
F- Lu
C) jc
LuJ
F Le
50
Thermal conductivity k = 10.25 Btu/ft.hr.°F
Density p = 530.5 Lbm/ft3"
Specific heat c = .128 Btu/LbmoF
The loading results from the thermal transient in the fluid
contained in the nozzle. This fluid is in contact with the
structure on surface "i" (Fig. 11) and has the entry temper.-
ature time variation as in Fig. 12. The fluid in the sphere,
which is in contact with the structure on surface "2" (Fig.
11), has the constant temperature 478 0 F. At T = 0 structure
has a uniform temperature of 478OF and is stress free.
Exterior surface of the structure is insulated. The flow
velocity past surface 1 is 8.5 ft/sec (inward) and
there is no flow past surface 2. The surface heat
transfer coefficients are 1393 and 2910 Btu/hr.ft 2OF for
surfaces "1" and "2" respectively.
For the structural boundary condition it is assumed
that the nodes on the left end cross-section of Fig. 13 are
prevented from any axial displacement.
The maximum thermal stresses obtained by the program
occur in element 14 of Fig. 13 as follows:
Maximum mean stress = 18.81 ksi. at time = 4 seconds;
Maximum octahedral shearing stress = 11.5 ksi. at time
= 18 seconds.
These results appear to be reasonable, but no suitable com-
parison solution is available.
S~51
0F
637
478
0 2 SEC.
VARIATION OF ENTRY FLUID TEMPERATURE
IN NOZZLE. (SURFACE-1
0
F
478
4t
0
SEC.FLUID PAST SURFACE "2.
FIG. 12. Fluid Temperature - Time Histories.
52
kNN
(D Lu
53
VII. CONCLUSIONS AND RECOMMENDATIONS
A. CONCLUSIONS
FA computer program has been developed for the solution
of axisymmetric transient heat conduction and thermal stress
problems. The system will accommodate a wide variety of
geometric arrangements, thermal and structural boundary con-
ditions, and mechanizal loadings.
Although double-precision arithmetic is employed through-
out, efficient algorithms for the manipulation and storage
of large symmetric banded matrices allow in-core solutions
with modest time requirements.
The quadratic isoparametric elements used allow accurate
representation of curvilinear boundaries and the stress
field. Examples presented show that a small number of ele-
ments is generally sufficient to determine stresses with
good precision.
B. RECOMMENDATIONS
Incorporation of several additional features would
significantly increase the program capabilities. The follow-
ing extensions are recommended.
1. Material thermal and elastic properties have been
assumed constant within each element. Since such properties
are generally temperature dependent,.provisions should be
made for periodically "updating" them during both temperature
and stress solutions.
54
2. The surface heat transfer coefficient, taken as
constant in the program, is a function of temperature and
flow velocity. Provision should be made to include these
effects.
3. The program presently starts every temperature
solution with constant initial solid temperature. Provision
for an externally specified initial temperature vector should
be included.
4. The large quantity of temperature and stress results
generated by the program is currently presented as digital
printout. Graphical output in the form of two-dimensional
contour plots of temperatures and stresses should be
provided.
L
APPENDIX A
APPLICABLE FORMULAS AND EQUATIONS
PART 1
(a): Two dimensional 8-noded (parabolic)isoparametric shape functions.
Corner nodes:
Ni=I (I + •o(1 + no(o+ no-i
4 o o o
Mid nodes:
S0, N(1 - E2) (1 + )•i= 0, N
S 1 E 2n=0, N 2=O
where
0o E i no 1n ni
(b): Temperature at a point in terms of thenodal temperatures.
8T= E N.T.i=l z 1
(c): Coordinates at a point in terms of thenodal coordinates.
8•iR E N NRi
i=l1 1
.8Z E NiZi
i=l1
(d): The Jacobian coordinate transformationmatrix.
56
j-
3Z aRJ
(e): Element of the finite element capacitancematrix.
eC. pc f Ni N. dV!1J
(f): Element of the finite element admittance matrix.
Yij = k f VNi*VN• dV
(g): Elemental volume.
dV = 2fR det J dg dn
PART 2
(a): One-dimensional 3-noded (parabolic) iso-parametric shape functions.
_1End nodes Ni - 0 o (1 + ýo)
i E2)Mid node N. = (I )
where, again, o =
(b): One-dimensional capacitance and admittancematrices.
!•4 2 -1 7 -8 1 h 0 0
Ce pcZ 2 16 21 +e kE 8 16 -8 +i = ~ ~ ~~~30 '=3, 81
3 -0 2 4I1 -8 7 L 0 0
where Z is the length of the element.
(c): The element v vector for the one-dimensional case.
Se= T
57
II
PART 3
(a): Strain-displacement relations.
_Ni
C R TR 3R -• i
Z. NU.u i
N N. UU + 211
_2-u - _ U + E IVYRZ 3Z R 3z i 3R
where Ni are the same isoparametric shape
function as in Part 1, (a).
(b): The B matrix.
IN1 aV2 DN80 0To B .......
3N1B 0 0 ....... 0
N 1 0 N2 00R R
BN1 •N1 •N2 •N2 aN8_z _- Uz • .......
(c): The element displacement vector.
6e = T
(d): Elasticity matrix D for an isotropic material.
D1 0i ~= E(I-v)) )(l+v)(1-2v) 1- -V
V V 101-v 1-v 1 0"J TV% TV%
0 0 .0 1-2v82 (1-v)•, 58
0000000000000 0 00 0 000000000r)00000000000oo0000000000
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* z W>-Z. IWI U0 ZI-')' I*U W )>O(/) '-'10 S-'W<
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000000000000000000000000000000000000000000000000
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