Thermal Stress Analyses

218 Pages·1965·6.34 MB·English

by  D. J. Johns and B. G. Neal (Auth.)

THERMAL STRESS
ANALYSES
D. J. JOHNS
Reader in Aeronautical Engineering, Loughborough College of Technology,
Loughborough, Leicestershire
(Formerly, Lecturer in Aircraft Design, College of Aeronautics, Cranfield,
Bletchley, Bucks.)
PERGAMON PRESS
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Copyright © 1965
Pergamon Press Ltd.
First edition 1965
Library of Congress Catalog Card No. 65-18374
Set in 10 on 12pt. Baskerville
and printed Great Britain
by Bell and Bain Ltd., Glasgow
This book is sold subject to the con­
dition that it shall not, by way of
trade, be lent, re-sold, hired out, or
otherwise disposed of without the
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which it is published.
PREFACE
THERMAL stress problems occur in many branches of engineering
and have already received considerable attention both in analysis
and design. Such stresses result when differential thermal expan­
sions are caused in a solid body, and if these stresses are high and
associated with high temperatures in the body the yield stress of the
material at these temperatures may be approached or even exceeded.
It may be seen therefore that thermal stress problems can arise in
any branch of engineering where temperature gradients are possible,
and the only essential difference between the various problems is in
the nature of the process causing the temperature gradient.
Thus, propulsive systems are inherently prone to high tempera­
ture effects resulting from the combustion process, whilst in the
nuclear and aeronautical fields extremely high temperatures, and
large gradients, can occur due to the fission process and the pheno­
mena of aerodynamic heating respectively. In a recent study of ship
failures thermal stresses of moderate to severe intensity were found
to be present in more than one-third of those studied; the causes could
be solar radiation, the heating of fuel oil or the cooling of refriger­
ated spaces.
The problems which arise fall naturally into three categories.
The first comprises the problems accompanying a comparatively
small, uniform rise in temperature ; the thermal stresses caused are
negligible and in a stress analysis due allowance must be made for
variations in material properties with temperature. The second
category is the main concern of the present book and deals with
large variations in temperature and thermal expansion which
produce thermal stresses (elastic or plastic) in materials whose
properties are time-independent. The third category involves
materials and conditions which may introduce significant time-
dependent effects such as creep. This last category involves many
complex problems and consideration must be given in any structural
analysis to such effects as the redistribution of stress due to creep,
creep buckling, etc.
The decision to omit consideration of problems in the third
category from this book, and to concentrate on the second category
xi
Xll PREFACE
is, of course, regretted, but the author feels that in a book of this
size it is preferable to give greater attention to a smaller class
of problems.
The restriction on length also prevents the presentation of more
solutions to actual problems, but the author hopes that the many
references given will enable the reader to pursue quickly, his subject
of interest. In this connexion the two, most extensive, bibliographies
listed at the end of the References contain between them over 800
references for the period up to 1963.
ACKNOWLEDGEMENTS
ACKNOWLEDGEMENT is made to the following publishers who gave
permission for figures to be reproduced in this book ;
Advisory Group for Aeronautical Research and Development.
Aeronautical Research Institute, Univ. of Tokyo.
Aeronautical Society of India.
American Institute of Aeronautics and Astronautics.
Her Majesty's Stationery Office.
Royal Aeronautical Society.
Society of Automotive Engineers.
Weizmann Science Press (Israel).
My thanks are also due to Professor B. G. Neal, Imperial College
of Science and Technology, who invited me to write the book ; to
Professor G. M. Lilley, College of Aeronautics, Cranfield, who made
comprehensive comments and suggestions on the contents of the
Appendix, and to Professor W. S. Hemp, College of Aeronautics,
Cranfield, for his encouragement during the preparation of the
manuscript and his general comments on its final form. For her
enthusiasm and skill my most sincere thanks go to Mrs. J. Carberry
who typed and prepared the manuscript.
D. J. JOHNS
College of Aeronautics, Cranfield, and
Loughborough College of Technology
December 1963 /'January 1964
X1U
PRINCIPAL NOTATION
a, b, c, linear dimensions
A area
B Biot Number ( = hdjk) ; differential operator in Chapter 6
C heat capacity per unit volume
d thickness
D flexural rigidity (= Ed*/12(1 - v2))
e; ei, strain
e dilatation (= e + e + e )
xx yy zz
E;E Young's Modulus ; secant modulus
S
F free-energy function ; resultant vertical load on shell
G shear modulus
Gi Gibbs function
h heat transfer coefficient
H internal heat generation rate
i enthalpy
I second moment of area
J mechanical equivalent of heat
k thermal conductivity
K; K bulk modulus ( = E/3 (1 — 2v) ), spring stiffness ; flexi­
bility
l, m, n direction cosines
L length ; potential energy
L complementary potential energy
M,M,Mxy bending moment resultants
x y
M Mach Number of free stream flow
a
M EajTzdz
T
N,N N force resultants
x yi xy
N EajTâz
T
P pressure
P load
q heat flux per unit area per unit time
Q heat supplied per unit volume
Q*,Qy shear force
r, Θ,ζ cylindrical coordinate system
ν,θ,φ spherical coordinate system
n bwjbs ( = ratio of web depth to skin width)
XV
PRINCIPAL NOTATION
r tw/ts ( = ratio of web thickness to skin thickness)
t
R radius of curvature
R, Θ, Z body forces per unit volume in cylindrical coordinates
R, Θ, Φ body forces per unit volume in spherical coordinates
R concentrated reaction
c
S Reynolds analogy factor
t time ; thickness
t t web thickness ; skin thickness
W9 s
T temperature rise above initial stress-free state
T temperature at initial stress-free state (absolute)
0
T' absolute temperature ( = T + T)
0
u, v, w components of displacement vector in coordinate
directions.
U intrinsic energy per unit volume
Uo strain energy
0 complementary strain energy
0
U free-stream velocity
a
U, V variables introduced in shell theory
W work done per unit volume; parameter in heat
transfer (= Ktjd2)
W loss in potential energy of surface forces
u
W loss in potential energy of body forces
t
x, y, z cartesian coordinate system
X, Y, Z body forces per unit volume in rectangular cartesian
coordinates
X, Y, Z surface forces per unit area in rectangular cartesian
s s s
coordinates
a coefficient of linear thermal expansion
ß Ea/(l — 2v) ; shell parameter [3(1 - ν2)/άΉψ*
8 displacement
€ emissivity
η entropy per unit volume
Θ rotation ; shell angular coordinate
K thermal diffusivity ; curvature change
λ, μ Lame's elastic constants (Chapter 1)
v Poisson's ratio
p mass density ; least radius of gyration
c, Gij stress
PRINCIPAL NOTATION xvii
Σ σ + Gyy + σ
χχ ζζ
T twist
φ Airy stress function ; beam rotation
ΦΝ> ΦΤ force functions
φ potential function ; shell angular coordinate
CHAPTER 1
Fundamentals of Thermal
Stress Analysis
1.1. Preliminary Remarks on Thermal Stress
Most substances expand when their temperature is raised and
contract when cooled, and for a wide range of temperatures this
expansion or contraction is proportional to the temperature change.
This proportionality is expressed by the coefficient of linear
thermal expansion (a) which is defined as the change in length
which a bar of unit length undergoes when its temperature is
changed by 1°.
If free expansion or contraction of all the fibres of a body is
permitted, no stress is caused by the change in temperature. How­
ever, when the temperature rise in a homogeneous body is not
uniform, different elements of the body tend to expand by different
amounts and the requirement that the body remain continuous
conflicts with the requirement that each element expand by an
amount proportional to the local temperature rise. Thus the various
elements exert upon each other a restraining action resulting in
continuous unique displacements at every point. The system of
strains produced by this restraining action cancels out all, or part
of, the free thermal expansions at every point so as to ensure con­
tinuity of displacement. This system of strains must be accompanied
by a corresponding system of self-equilibrating stresses. These
stresses are known as thermal stresses.
A similar system of stresses may be induced in a structure made
of dissimilar materials even when the temperature change through­
out the structure is uniform. Also, if the temperature change in a
homogeneous body is uniform and external restraints limit the
1
2 THERMAL STRESS ANALYSES
amount of expansion or contraction, the stresses produced in the
body are termed thermal stresses.
These various ideas can be illustrated by the following simple
example. Two parallel rods of different materials and lengths are
fixed at one end and are restrained to move together at their other
end, see Fig. 1.1. Movement of the combined structure is only
permitted in the direction parallel to the rod axes, and it is reacted
by an elastic spring of stiffness K. If T and T denote the rise in
1 2
Bari
Spring
K
U·-
Bar2 σ2 α2
FIG. 1.1. Simple two-bar structure.
temperature from the initial stress-free state, experienced by each
rod, the conditions of equilibrium and compatibility of strain are
given, respectively, by
σΑ + σΑ = P (1.1)
1 1 2 2
and α,Τ^Ζ,! + ——= a TL + —— = - 8 (1.2)
2 2 2
E,
Here, σ and σ are the corresponding tensile thermal stresses in the
λ 2
two rods, of cross-sectional areas A and A respectively, and P is
x 2
the compressive load in the spring which has a compressive
displacement δ (i.e. P = K8). In eqn. (1.2) the terms aTL refer to
the free thermal expansions whilst the terms ajE refer to the strains
necessary to ensure continuity of displacement (E is the modulus of
elasticity).
The solution of the above equations yields the following result
for σ
Λ