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Two Problems of Thermal Stress in the Infinite Solid

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TWO PROBLEMS OF THERMAL STRESS
IN THE
INFINITE SOLID
A Thesis
Presented to the Faculty of the Graduate School
of Cornell Unlrersity for the Degree of
Dootor of Philosophy
By
Nils Otto Myklestad
June, 1940
ProQuest N um ber: 10834652
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(1)
Blographloal Sketoh of the Author#
The author was born In Wllliston, North Dakota
on March 24th, 1909, but reoelved moot of hia education
in Nomay and Danmark and graduated from junior college
(gymnasium) in Oslo, Norway In 1926#
In the fall of
the same year he entered the Royal Technical College
in Copenhagen9 Denmarkt graduating from this school
in Marcht 1932*
During this period he also worked one year,
1928*29, at the South Philadelphia plant of the
Westlaghouse Electric and Manufaturing Company, and
again went to work for this company in August, 1932*
He was then continuously engaged in engineering work,
with various companies, until August, 1937*
In 1937*38 he was teaching assistant and graduate
student in Mechanical Engineering at the University of
California and, since 1938 instructor and graduate
student in Mechanics of Engineering at Cornell university*
(2 )
History of the Thermal StressProblem,
The general equations for determining thermal
stress were probably first derived by J.r.C.^uhamel
and published in Paris in 1838 (r,l).
He develops
the general thermo-elastic equations by considering
a solid body as a system of particles and, as the
sphere and the cylinder, with radial stress variation
only, are the simplest cases of thermal stress, he
derives the expressions for the stresses in these two
oases.
For the sphere he arrives at the very interesting
result that whatever the law of radial temperature
variation may be, the extreme fibres have the same
length as If the whole sphere were at the average
temperature•
Independently, and at about the same time as
Duhamel, Franz Neumann also arrived at the thermoelastic equations.
However, he was primarily interested
In the effect of uneven temperature distribution upon
the double refraction of light, and he worked out the
strains for the cases of a sphere and a long circular
eyllnder.
His work on this subject was published in
Berlin in 1841 (r,2).
(3)
Since Duhamel and Neumann very little work haa
been done on the development of the theory; moat papera
on the subject have dealt with the aame problems in
different ways or developed the theory more fully for
epeelal oases*
Almost the only problems dealt with
to date are the sphere, the oyllnder and plates*
The
following brief abstraots of some of the papers pub­
lished will perhaps Illustrate this*
C*W*Borchardt (r*3) develops a method of solving
the thermo-elastlo equations by means of potential
theory*
He arrives at rather cumbersome expressions
for the displacements u f v, w In terms of potential
fnnotions, and he treats particularly the sphere and
the olroular disc with arbitrary temperature distribution*
J* Hopklnson (r*4) brings the temperature term
into the equation of equilibrium and applies the
equations thus found to the problem of a sphere with
nonuniform radial temperature distribution*
Rayleigh (r*5) is mostly interested in the double
refraction of glass due to temperature stresses; and he
treats the sphere, cylinder and flat plate*
He states
that the investigation of temperature stresses was
first attaoked by J* Hopklnson (r*4), while in re­
ality all conclusions reached by Hopklnson were to
(4 )
he found in the muoh earlier paper by Duhamel (r.l).
It seems that neither Rayleigh nor Hopklnson knew
that was being done on the oontlnent.
Alfons Leon (r.6) develops the general equations
for a body of revolution, and proeeeds from these to
treat the sphere and the cylinder for the oase of
linear temperature variation with the radius.
Lorenz (r#7) develops the equations for displace­
ments and stresses for an infinite hollow eyllnder
with only radial variation in temperature.
For a
logarithmic temperature distribution he determines
the maximum stresses at the inside and outside of the
cylinder.
Lees (r.8) treats a hollow circular eyllnder in
a more general way and gives a simple graphical method
for determining the stresses for any temperature dis­
tribution and any law of expansion.
Barker (r.9) develops charts for the rapid cal­
culation of temperature stresses in tubes.
Kent (r.10) treats a finite, thin walled, hollow
eyllnder, with free and damped end oondltlons, by
the method of assuming an element, out from the cyl­
inder wall along Its entire length, to be a beam on
am elastic foundation.
He finds the stresses and def­
lections due to variations of temperature both radially
and axially.
(5)
Kent (r*ll) also determines the distribution of
stresses in spheres and long oiroular oylinders,
both solid and hollow, when the temperature varies
with time aooording to the law of heat conduction*
He starts with the well known formulas for the stresses
in spheres and cylinders when the temperature varies
with the radius only*
Then he determines the temperature
distribution under certain conditions of cooling and
heating of the outer surfaoe, and substitutes these
in the formulas for stress*
He also plots numerous
ourves for the variation af temperature and stress
with the radius*
Biot (r*12 and r*15) treats, as part of a some­
what more general theory, the temperature stresses
set
up in a cylinder of arbitrary oross seotion and with
a steady state temperature variation*
lhe surfaoe
temperature varies in any manner along the boundary
of a oross seotion, but is constant along a generator*
He ehows that, for this case, a solid oyllnder has
only axial stress| but a hollow oyllnder will, in
general, also have radial and tangential stresses,
unless the oyllnder can be slit lengthwise without
producing any relative motion between the two sides
of the slit*
(6 )
Ueulbetsoh (r*14) develops the formulae for def»
I m U q m tad momenta in triangular end square plates,
which are simply supported at the edges and subjeoted
to eonstant| but different, temperatures on the upper
end lover sides*
Be shows, that for these oases, the
stresses are Independent
of
the thickness of the
plate*
Goodler (r*15) investigates the thermal stress
set up by unequal heating of a flat strip, the tem­
perature varying along the length and through the
thiokness, but not aoross the width * He treats par­
ticularly a strip that is heated uniformly through
the thiokness, but only along a narrow transverse bandj
and a strip that is heated uniformly along a narrow
transverse band, but only on one side, the temperature
variation through the thiokness being linear*
Dsn HOrtog (r*16) treats the following oases,
all of vhldh to a certain degree resemble welding
processes t 1* A reotangular plate heated along a
center line;
£•A large flat plate with a central
1 circular hot spot;
Z. A tube with a hot olrole around
its middle end oool at the ends;
4* A tube with a
hot generating line*
Starting with a sinusoidal tsmperature distribution
he derives sens interesting results.
The maximum
(*l
i l i m to « m
l| I toi 4 m m h
t o n t o hot toad
to m y
W 9 N » omd to t o n t o t |«r somewhat larger)#
la m m
• t o m>iU—
t o n i t o M m i u m at the edge
of toa hat elreular ration, and la £ s « t regardless
of atoa of plate or alaa of hot spot*
Caaaa 1 and 4
f i n riaa to aaoh aoro aarloua atroaaaa than oaaa Z
for eempareble dtoanalona of the hot band*
Ooodlar (r*17) gives a sunmary of tha formulas
for thasaal atraaa in Zlatas, rods and oyllnders
arranoad in tha form of design data*
Ooodlar (r*18) also investigates the thermal
atraaa In thin vailed oyllnders of any oross seotion,
for internal and external temperatures varying In any
■aimer round tha olroumferanoa, but not along a gen­
erator*
ferenoe*
Tha thiokness also may vary round the ciroumThe olroular oyllnder is treated in detail*
Ooodlar (r*19) makes the most Important contri­
bution to the theory of thermo-elastio stress slnoe
Duhamel and Neumann*
He shows how all stresses and
displacements In a heated body can be derived in a
vary simple manner from a single function ^ 9 vhloh
ha oalls tha thermo-elastic "Displacement Potential"*
Tha aquation for f
is of the same form as Polsson9s
aquation and its solution is facilitated by the vail
known theory of tha Newtonian Potential of a distri-
(8)
button of natter*
He also treats the particular
eases of plain stress and plain strain, and finally
applies the method to the two problems of a plate
with cun elllptlo hot spot and a plate with a reotangular
hot spot*
The shore theory by Dr* Goodler forms the basis
for the two thermal stress problems of this thesis,
and It will therefore be dealt with In more detail
under "General Theory"*
Porltsky (r*20) treats a cylindrical tube by
means of complex variables, very nruoh like Blot (r*13)*
Of books that oontain material on thermal stress
Timoshenko's "Theory of Elasticity" (r*28) Is the most
oomplete*
It contains the fundamental thermo-elastlo
equations and applications to plates, dlsos, spheres
and cylinders.
Almost the same material Is treated
In F^ppl's *Vorle8ungen iiber Teohnlsche Meohanlk” (r*29)*
Neumann (r*22) derives the thermo-elastic equations,
but gives no useful applications*
Love (r*27) barely
mentions the thermal stress problem, and the other
books referred to only treat the circular oyllnder
and elementary problems*
(9)
General Theory*
When an elaatlo solid, which Is unstressed at
a uniform temperature, Is given a nonuniform temperature
T(x,y,z) that Is not linear In (x,y9z), the natural
expansion of a volume element will he restricted by
the surrounding material and a state of stress will
ensue*
The difference between the actual strain and
the natural expansion 01T of a volume element Is related
to the stress through Hooke’s lawj oi being the linear
eoeffiolent of thermal expansion*
For simplicity It
Is assumed that the original uniform temperature Is
zero*
This makes no difference In the theory because
a uniform change of temperature creates no stress In
an Isotroplo solid*
It Is also assumed that the
elastlo constants of the material do not vary consider­
ably over the temperature range In question, although
a variation of oc with temperature oan be taken oare
of by carrying « T through as a single variable quan­
tity*
let u,v,w be the total displacements in the x,y,z
directions respectively; then the components of normal
and shear strain are (r*28, p*7),
Also tha volume expansion e »
.
ii a uniform change of temperature of a small Tolume
element does not oreate any angular distortion of the
element the shear stresses vlll be unaffected by the
term otT or9
etc., where G«
18 the
modulus of elasticity in shear , B the modulus of elas­
ticity in tension and compression (assumed equal) and 1/
Poison's ratio# The normal stresses, however, are deter­
mined by the following equations (r«28, PP*11 and SOS)
C* * 777“|/P57 e * * * ~ F I T * T]
and two more by interchanging x with y and z«
The three equations of equilibrium (r*£8, p»£08)
take the form
( * + * ) & + e v ' u - 1% r £ L = o
and two more by interchanging x with y and z9 and u
with v and w.
Also
(U)
Substituting for A and G these equations become
+(l-iv)V*u
f
eto%*
Equations (1) and (2) are the fundamental equations
of the theory, but the boundary conditions and the con­
ditions of compatibility between the strain components
must also be satisfied*
The compatibility equations
In terms of strain components (r*28, p*196) are seen
to be satisfied automatically when the strains are ex­
pressed in terms of the displacements u,v,w acoordlng
to (a).
This is also quite obvious from a physloal
standpoint as there Is no restriction upon the displace­
ment of a point of the solid as long as the material
Is not being stretched beyond the elastic limit*
The
boundary conditions are mostly given as freedom from
surface forces*
Should any external forces aot on the
surface of the solid the stresses resulting from these
can be determined separately and, by the principle of
superposition, added to the stresses already found*
In order to obtain particular Integrals of (2),
a funotion ip is introduced and defined in such a
manner that
41
4%
i
(12)
By m n i of (a) the atrains oan be expressed in terms
of the funotion f
as follows:
(b)
e» v*r.
(c)
Equations (2) oan now be written
£
eto..
[
(l + v ) e t T ] = 0
It Is seen that equations (2) are all satisfied
when
(3)
Evidently any funotion f
, which satisfies (3), pro*
Tides a particular integral of (2).
The state of stress
and strain represented by this funotion
will ordinarily
require oertain surface foroes at the boundary of the
solid.
These surfaoe foroes oan be determined from the
conditions of equilibrium at the boundary and the stresses
there, expressed in terms of f
• To oomplete the problem
It Is necessary to apply equal and opposite foroes to the
surfaoe of the solid and determine the stresses from
these* as in an ordinary (isothermal) boundary value problem.
(13)
By superposition the complete stress resulting from
unequal heating of the solid oan then he determined*
The funotion V is oalled the "Displacement Po­
tential" (r*£7, p.38).
By means of (3) equations (1) oan now be written
in the form
(ia)
eto.*
It Is seen that only the seoond derivatives of
¥ are necessary to determine the stress components*
The equation (3) is of the same form as Polsson's
equation 7*V*-4Tff (r*299 p*148)9 and a particular
Integral is given by the Newtonian Potential of a
distribution of matter of density f per unit of volume9
In this equation V is the potential at the point
(x9y9z)9 the coordinates
£ refer to a point
in the solid where the density is f 9and the inte­
gration is throughout the solid*
The distance r9
between the points (x9y 9z) and (tf
by the equation r * ^ (x-jf f +
Putting
ccT » -4Tff
£!▼»
(y-f )z + (z-JT) •
the following equation for
(14)
ip Is obtained:
W M , *) =- £ 7
/ f f T (* -y<-f ) d S d J d t
.
When the Inequalities of temperature are of a
purely looal nature In an infinite solld9 T = 0 outside
the hot part of the solid 9
beoause £ 9y ,
will vanish at infinity
are bounded 9 and consequently r goes
to infinity vhen the point (z9y 9z) goes to infinity.
As T (<$%y 9{f ) is independent of (z9y 9z)9 differentiation
under the integral sign shows that the displacements
will vanish as r~* and the stresses as r’’5 9 when the
point (z9y 9z) goes to infinity (r.30v p.28).
As this
satisfies the boundary conditions for an infinite
solid9 the solution in this case is represented by (4).
Discontinuous temperature distribution.
The
general equations (2) imply the ezlstenoe of the first
derivatives of the temperature9 and the validity of
any solution at a surface S of temperature discontinuity
therefore requires examination.
Consider first the displacement potential
the infinite solid9 equation (4).
of
Let the subscript
i refer to the limit of a funotion when the surfaoe
S is approached from its interior9 and the subscript
(15)
• refer to the limit when s ie approaohed from it*
exterior*
The temperature discontinuity on s may then
he written T|-Tt , whioh is a funtion of position on
8* At any point F on S take right handed reetangular
coordinates
$jr suoh that ct
lie in the tangent
plane and it along the outward normal*
The oondltlons that must be satisfied if (4) is
to he valid, in spite of the temperature discontinuityt
are that, on s, the displacement must he continuous
and the surfaoe foroes must be continuous*
These
require that the first derivatives of ¥ must be
continuous and the stress components 05,9 Zm* end Zp*
must be continuous*
Now 9 9 as given by (4)9 is the
potential of a distribution of matter f •-
9
and it is shown in works on the theory of the potential
that a distribution f 9 whioh is bounded and lntegrable9
has a potential suoh that its first derivatives are
continuous everywhere9 Including surfaces there f
is discontinuous, and its second derivatives are
continuous exoept at surfaces where f is discontinuous
(r*30, paragraphs 24,25,29,61,73,91,92,93,94)*
In
a coordinate system or,^ ,^ , as defined above, all
second derivatives are continuous on a surfaoe where f
Is discontinuous, exoept
* The discontinuity
**
of this last second derivative is given by the
equation (r.30, p«175)
( & )
The displacements (first derivatives of ^ ) derived
from (4) will therefore be continuous*
From (la)
the normal stress components in the new coordinate
system will be
etc.*
The stress components <7^ and <7^ are not continuous*
By means of (lb) the discontinuity may be written as
°*f" <r^ ‘ ^ 6 [ (4c^Ji ~ ( l ^ ) e ~ f t “ ft 'T«)] .
As ~
is continuous, ( 0 ) . - ( 4 - f j ^ o , and
Similarly the same expression is obtained for the
discontinuity of (fy •
The stress component
ls9 howevert oontinuous*
9 normal to the surface S,
From (lb)
(17)
Also
and consequently <r*< -
* 0*
tinuous on the surfaoe d*
The shear stresses on the
That is, <Tr is con­
surface s are immediately seen to be oontinuous, and
the solution (4) is therofore valid when T has surface
discontinuities*
It follows that any component of normal stress,
noting on an element of area normal to, and at, the
surfaoe of discontinuity is discontinuous by an amount
while the component of noxmal stress acting on an
element of the surface s Is continuous.
All these conclusions remain true even when the
solid has finite boundaries*
The boundary foroes,
coresponding to (4), are to be cancelled by superpos­
ing an isothermal stress distribution having equal
and opposite boundary values*
But any distribution of
boundary foroes, whether continuous or not, produces
a stress that is oontinuous in the interior*
The die*
(IB)
continuity of stress duo to a temperature dlaeontlsulty
remains unaffooted9 although the absolute values of the
stresses on either side of s will In general be modified*
It Is not neoessary that S lie entirely within the
boundary*
It may out It in one or more ourves*
then
It doest the complementary surfaoe foroes on the boun­
dary are discontinuous at these ourves• When it does
not| these foroes are oontinuous all over the boundary*
This is part of the general theory developed by
Professor j.N.Goodler (r*19), and the original part of
this thesis oonslsts of its application to two new
problems*
In these problems the temperature Is con­
sidered equal to zero throughout the infinite solid
exoept within a oertaln region*
The temperature in
this region is uniform* but higher than in the rest
of the solid*
There is then a temperature discontin­
uity at the boundary of the hot region*
In the first
problem the hot region is an ellipsoid of revolution*
In the seoond problem the hot region is a semi-infinite
oireular oyllnder, and here the effect of the sharp
corners are particularly interesting*
(19)
Problem 1,
Stresses aat up in an Iaotroplo Infinite solid
when a part of it, bounded by an Ellipsoid of Revolution,
la at a higher uniform temperature than the rest of
the solid*
Let the ellipsoid be formed by rotating an ellipse,
with semi-axes a and o, about the z axis*
The equation
for the ellipsoid of revolution then becomes
c*
•
In the oold exterior of the infinite solid it is
shown under "General Theory" that the stresses are
found immediately from the potential af an ellipsoid
of uniform mass density
p a -J ± !L £ il
T
h*
41T .
The temperature in the ellipsoid is considered equal
to T and in the oold exterior equal to zero*
The normal stresses are given by equations (la),
and as T * 0 for the oold exterior, the stresses in
this part of the infinite solid will be
to )
t*'
The shear stresses are
T = -L - —
11 " !♦*/
r
- J - lll
L»* “ 14* JgiZ
,
.
First oonslder tha aasa of an Oblato Spharold.
*>o, for whioh tha potantlal la (r.30, p.68)
and
k
is determined as heing the positive root in
the equation
(£1)
***** . j1
*ci«7
f | |
By introducing tha eooentrlolty e« V '£ C " » o oan
ha eliminated*
This gives o*» a*(l-a*) and a*-o*« 0*0*f
where e la a dlmenslonless quantity.
Also put
y** **• r* and (7) gives, by solving for *,
(a)
c * * x = j [ r a- * * * * *
e b *)*+ 4 e2a * z 2 ]
In differentiating (6) with respeot to x, y and z, X
oan be treated as a oonstant beoause gjj- * 0 (r.30,
p.63).
This is easily verified when it is reoalled
that from (7)
***** C*+K
z ± _ *I. U0
whioh is a faotor in the above derivative.
The partial derivatives of (6) then beoome
4 * , <1Trailhe* f 'jc**X
1Z
e»
L *•**
I r!rt~f a t
at *,n
J* ,
4V M A i r f a f c P t I r?r,-< a e
4z
e*
L««
1„
1~
»
)
ia**x P I k J
la different lating equations (9), however, X
oust be
treated as a funotion of (x,y,z), whioh funotion la
Immediately given by either of equations (8),
The second partial derivatives of (6) become
d*v
£vri>4i/i-e*
Jk*
e*
• -L.
la*+x
a*
4 *y ■ £WPaJi-ei \ y/c*+K L
> y* ’
e*
L a * + x ~ ae
_ * ir w E P fz
4z*~
In whioh
e*
eiS 'ae
♦
*****
J* 7
\fa*+x (am+*} \/c*4>x d* «*,
in- t a e
.
tfr*7x
z
.
ela * y
J* ]
(ox* * H c * t x 4<i J,
e*a*z
4*1
f i X S i F * ! * ' * J»
* - f i -
4%
L
dx _ r».
rV * * ‘
h
e*4*)*+ 4e*a*z*
J '
/■*♦«*«* _ i_ .
Jl l1 h r ± ' W . 4 ' & ? r
’
and
d*V
tTTPat \/I-ei
V .
£
5w
*
tdz
rf*y
fi.
r z-e *a *
g jffa H h ^ _ f , A___ r ‘ * e*a*
' (a***)*\fc*** L'
\i(r*-e»a*),*4etaizt
tir r ^ /F e * f , r
dydz~ (a*+x)*)/c*-tx
L
r * f e *a *
y fr^ ^ *a *)*+ 4 e *a *z *
~|...
1„ ,
J
'
1 ..
-
By substituting for f aooording to (e)9 In equations
(10) and (11) the seoond partial derlvatlYes of
will be obtained.
Then(5) will yield the following
expressions for the stresses in the infinite solid
outside of the hot ellipsoid
(64)
tr -* W
* I'V
#r.
a\fbt* [<fc**K
g P ~ l**+ x
f"T tfl/Te*
* ” i-v
i ri„-i ae .
*****
**
I(5*7* ( a * * * ) * V c ‘ *M
cbl
I
S *1
I
(<**■*k )\/c M*it Jy J
/
. ^ - I ge .
L a ***
_
Va*4k
-ft1w «yg
c**~T7~~ZF~L7e *
2
1Z*T* W T *
e *a *9
.
e *a *z
a-i
rf>|
<**♦*)£'♦*)**i\
and
E *T
s
r
_
a * / h F _ f,.
/■*-*'
h* 2(a*+x)
xF7c*n<L ^(r*- e*a*)*+ 4e*a*Z *
F*T
LM - ~
X
- 1*1
a V f-e *
]x</
f > .______ r * *+e*a*
e *a *
1 .,,
\j(r*.e *a *)**+ e *a *z *J
2(S** * ! * & * * L'
^/Te5
f.
r**e *a *
f [ l 9)
_ 1 „,
Nov consider the case of a Prolate Spheroid,
o>a, for vhloh the potential Is (r«30, p.63)
#§<*vi ‘#£1
This time a will be eliminated by bringing in the
eooentrlolty e«
- - Lt-- ^
c
t o*-a*« e*o* and a2= o*(l-e*)»
(25)
The formulas for a*+x
and o* + k
then take the fora
x = j [ r l - el c2+ \J(r*+e*c*)e- 4 e * c * z * J
(U)
c z* ) k-
j[ r * + e * c * + { ( r * 7 e ° P ) ^ - 4 e * c * P J
Also
<&. s [ , .
r * t e Mc M
]r
L /(#•*+ e3^ ) * - 4 f * c V ‘ J
’
—
-/"/+
*
" L
r £+ e * c *
~|
.
*>* r ,.
r*-e V
1^
L‘ )((r*+e£c * )* -4 e * c *Z g J
The normal stresses at exterior points are
rr ~-i*l
*“
>-•»
<r -
d L £ ! 2 f - L -I'rh** e c _ >£*♦* * « * C * x £ * ]
Lee
V^*+K
c r/-« *;f I .j-L-1ec
/-.>
*e4
Lecswh& I ^
t r - - f * T cO~e,) [ *
£
*'
* e r l \ ( F Z r “ «c
«a+K
y/c^K .
«>»1
I /.,%
a *>* * < * • * & & . * «tyj
f U 5)
ec
e*c*y
e *c *z
^*1
(«*+*)(c*+*)*
and the shear stresses are
I
- -M
c •((•**)
f,.
r**e *c*
1
*»
i-»» < ^ k j 1 r c * » K I )J(rt te *c */i - 4e'ctz t J
»
-
t«r
fV /-**)
**'
r -_gw
cVz-eff
f..
r*-e*c*
i...
L hr^e'c*)*- 4e*c*zal
r,.
i-v 2 ( a * + * ) * \ f F Z i l '
«
r*-e*c*
\fr r * * e W -4 e 1 e fr l
All stresses considered above are for the oold
exterior of the infinite solid*
The stresses inside
the hot ellipsoid of revolution oan easily be found
from the potential at Interior points of an ellipsoid
of uniform mass density*
This is done on pages 50-52,
and all interior stresses are shown to be constant
for any one ellipsoid of revolution*
The most interesting stresses in the cold ex­
terior are the stresses along the axis of the ellipsoid
and along a line, through the centroid, perpendicular
to the axis*
These stresses, along the z and x axes
respectively, have been calculated for different radii
of ourvature at the ends of the semi-axes of the
ellipse
Xa
2*
I
The results are plotted in figure 1, page 49*
Appendix to Problem 1*
A*
Stresses along the x axis.
Put x * an*
Then, for an Oblate Spheroid*
€ t*+ x *a *n * #
s*
c** jc= a*(n*~ e * \
* un >
A,
Jz
°
From (12) the normal stresses become
<T = B* T f 1 s in '(e \ 1- 1/ [<
>fi) na
?/<-«*
<2 e 4
rt- - *«»T f • f . 6
'/r7*-e4 ) VT-e4
°i - —
[ e s 'rt 7T TP-
zr . ««T f
Cr* a —
L
/
t f ^
I
- T
«7 4 l-e *
s,n n j —
The radius of curvature at the end of the semi-axis
a9 of the ellipse
v2
.2
— 1*— = I
**
c*
9
is f*a(l-e*)f giving e*= ! " •
For an Indefinitely flat ellipsoidt .p*0f e * 1#
all stresses are immediately seen to be zero for all
(sa)
values of n ozoept n»l, when
<T - -
(T - O
(T s MSS.
For a sphere, p* a 9 e*0, the terms in equations (17)
will be expressed in the form of a series*
nils gives
, -i«
*
i &&
$to zr s it + 7“— i * terms of higher order
ft n 6 n*
s
terms of higher order
* n ( h j*
*
t0rms of higher order)*
Substituting these expressions In (17)f multiplying
out the equations, and then dropping terms of higher
order In e, the following formulas for normal stress
will be found
<rx = Q- _
2
EotT
3 n * 1- 1/
/
feT
*~3n* I
(T =
*
iSil
J/;* /-* .
These agree with the known results, (22), page 35,
/
( f8 )
at
p
For a Prolate spheroid put cr»
af*#» a#n* and o*f* »
s then
y ~ j )•
From (15) the normal stresses beeome
e«
fr t fa T r
*
*'17
* -« * -» >
e
®
n*
r
— _ f « T r / ' ' * * * n*
•»-—
1—
/
n//-e*j2e*
-
«
5-------- —
t e
*
«
1
I
5"*
ov = J t t f i ' - W ' - T i . Also /*
7 i
. '-«*
rtit-e *.
14
1 4
J e*
* which gives e*«
For a spheret /» a, e* 0, the terms in equations (19)
may be expressed in the form of the following series
g
£
~
h - e * + 4
/
e3
7 n > o -e * )&
=
if/1** = / - ^ e * f
Substitution in (19) yields
+
.
'
..............
'
which oheoks with (18) •
For an indefinitely thin ellipsoid« p = * o » eel.
In equations (19) it is then necessary to find the
Uniting value of
as e approaches 1*
This can be written in the form
of a fraotlon as
I
/-«*
the numerator and denominator of which both approaoh
Infinity when e approaches 1«
Considering the numer­
ator and denominator of the above fraotlon as funotlons
of e, it is shown in books on the calculus that the
Uniting value of the fraotlon will be the same as
that of another fraotlon whose numerator and denomi­
nator have been derived from those of the original
fraotlon by differentiation with respeot to e.
(91)
Differentiating both numerator and denominator with
reapeot to e glrea
I
11( b p * V/-e*
i ^ r i
nll- ‘)
.
,-e*
This is easily seen to approaoh zero ae e approaohes 1,
Then Llm(l-» e*)sinh * ■ ulti s 0
e+l
nil-e*
and equations (19) immediately give
—.
x
Bo
I
E ar
'I7Ta ~
rr
,
I
E ar
^ ’ Z J ^ T T ,
~ _ G i-O .
Stresses along the z axls>
Put z » on*
Then, for an Oblate Spheroid, put
j
a » ■~ -y • and obtain
1-9*
**+* s £*(”*+jz§k),
i t
H
Sy
=. 0
*
c*+x = c*n*t
* £ .S £ cn .
( ft
From (12) the normal stresses become
The radius of curvature at the end of the sen! axle
o9 of the ellipse
X* , z * _ »
a* * c* " 1
la / =
, giTing e*» 1-y>.
For an Indefinitely flat ellipsoid* p = c o 9 e » l # all
stresses are Immediately seen to he aero*
For a Prolate Spheroid,
*C
—
**
■-
— >= 0
*9
•.
>
C*+ k s c*n* >
4Z
s
2CR
TToa (15) the normal stresses became
(as)
<Tu* <rt
Uf)
i
-
^
e*
Also f t o d - e 1) and e1* l-~*
For an infinitely thin ellipsoid, f * 0 9 e*l, all
etreaeea are seen to be zero ezoept for n > l f when
(T S(T - J. I S il
2 1-*
,
For a sphere, f * o 9 e« 0, the stresses for both the
oblate and the prolate spheroid are found to oheok
with the ones previously found along the z axis,
equations (18).
However 9 a wholly Independent oheok on the
validity of the above developed formulas oan be had
by oaloulatlng the radial and tangential stresses,
for the spherloal oase, by means of the ordinary
theory of elasticity.
The radial and tangential stresses at the surface
(84)
Of a sphereI of radius 0| duo to on external pressure
P O N (*»M» P«9*5)§
At the surface of o spherioal oavity, of radius a,
lnoldo on lnflnlto solid, tho corresponding strossos
duo to on Internal pressure p are
If the sphere fitted snugly In the spherical cavity
at temperature zero the pressure p will be due
solely to the uniform heating of the sphere and the
tangential strains of the two surfaoes considered
■not differ by an amount ce T*
For the surface of
the sphere this tangential strain is
and for tho spherical cavity it is
Dlls gives
(35)
Oil the outside of the oomon spherioal surface , then
t r * - -S- SSS
^
<T = - Ssu
3H,
* J I-*
and In the Infinite solid, at a distance from the
oenter of the sphere of n times the radius, the stresses
w l U be
These stresses oheok with (18) when It Is remembered
that In (18) <TX corresponds to the radial stress
and both dj and OJ correspond to the tangential stress.
A further oheok on the calculations for exterior
points can be obtained from (3) and (5) when It Is
remembered that TsO.
(T* ♦ ( f y ♦
This gives
- Oj
that Is Immediately seen to oheok with equations (19)
and (20) and Is a very useful check on all numerical
ealdilations.
(36)
Numerical Calculations for Problem 1.
A.
Stresses along the i axis.
In the following calculations the radius of
curvature f
refers to the end of the semi-axis a
of the ellipse
a
*
and e is the eccentricity of the same ellipse.
When f
is smaller than a, equations (17)t fbr an
oblate spheroid,are referred to, and when p is
larger than a, reference is made to equations (19),
for a prolate spheroid.
n is the distance from the oenter of the ellipsoid
in terms of the length of the semi-axis a.
p« .la
e*= .9
1
2
n
J.
n
3
4 -I—e
sin
e = .949
4
£.(»)
5
n*+ e*
6
n*-e*
1
•949
1.250
1.318
1.9
•1
1.1
•864
1.043
1.100
2.11
•31
1.25
.759
.862
.908
2.462
.662
1.6
•655
.685
.721
3.15
1.35
2
.475
.495
.521
4.9
3.1
(37)
7
{w
Q
9
n*-(7)
(5)
(8)
10
(4)-(9) —
11
12
Ox
(7)
n*
.316
•316
6.01
-4.69
-.823
.361
.557
.674
3.13
-2.03
-.356
.460
•814
1.270
1.937
-1.029
-.180
•522
1.161
2.61
1.205
-.484
-.085
•516
1.760
7.04
.697
-.176
-.031
.440
13
14
(4)-<12)
E«T ° *
15
1
(7)
16
17
18
n*
(15)—(4)
.645
1
.697
.245
1.21
1.229
.321
•113
1.562
•036
.861
•140
•049
2.25
.014
.568
.047
.016
4
1.002
.176
3.16
.640
.112
1.797
.386
.068
.205
.081
1.84
(38)
P*#85a
e*s *75
e « *866
1
1
11
2
n
3
4
-l c
sin |
£•<»)
5
6
na* e*
al
n •o
1.75
•25
•81
1
•866
1.047
1.25
•695
•765
•883
2.31
1.5
•577
•615
.710
3.00
1*5
2
.483
•448
.518
4.75
3.25
8
9
10
11
12
7
(7)
(S)
is)
•5
•5
•90
1.405
1.645
1.224
2,75
1.804
7.23
13
14
LJL ( f
(4)-(12] E«7UU
3.5
1*21
(4)-(9) h !L (y
BUT u *
-2.29
(7)
n*
-.763
•5
-.762
-.254
.577
1.090
-.380
-.127
.544
.657
-.139
-.046
.451
16
17
15
1
(7)
18
na
(15)-(4) i^T °Z
.71
.237
2.000
.790
.527
1
.306
.102
1.111
.228
•152
1.56
.166
.055
.816
.106
.071
2.25
.067
•022
.554
.036
•024
4
(39)
Q» »8a
eas .5
1
2
n
n
3
sln-'f
• s .707
4
i'W
5
n ▼ e
6
n
e
1
•707
•785
1.110
1.5
1*25
.565
•600
••649
2.062
1.062
1.5
.471
•490
•694
2.75
1.75
2
.353
•361
•511
4.5
3.5
8
9
10
11
12
7
fit)
(4)—(9)
n**(7)
•707
.707
1.030
1.610
1.322
2.12
-1.01
4T«T *5
•5
(7^
/t*
-.714
.707
1.280
-.431
-.305
.659
2.97
.926
-.232
-.164
•583
1.870
7.48
•602
-.091
-.064
.467
13
14
16
17
(4)-(12) ilH <7£*7
15
/
(V
(15) — (4)
<r
18
n*
•403
.285
1.414
.304
.430
1
•190
.134
.971
•122
.173
1.562
•106
•075
.757
.063
.089
2.25
•044
•031
.535
.024
.034
4
(40)
e * r .5
Prn 2*
1
2
n
e*
n*
3
e » .7 0 7
4
l-e*-(2) l-e**U)
5
^(4)
6
n<(5)
1
•500
.000
1.000
1.000
1.000
1*25
•320
.180
•820
•907
1.134
1.5
•222
.278
.722
•850
1.275
2
•125
.375
•625
•790
1.580
7
8
9
10
11
(3)
(6)
e
slnh f(8)
n ft-e *
12
(7)-{10) £«T ***
*
.000
1.000
•882
•623
-.623
-.623
•159
•800
.733
•518
-•359
-•359
•218
.667
•625
•442
-•224
-•224
•237
•500
•481
•340
-.103
-.103
13
14
15
16
(S)
n
(13)-(10)
iz ± *
17
18
(10)-(16)
(6)
1.000
.377
.377
•500
•123
•246
•750
•207
•207
•441
.077
•154
•567
•125
•125
•392
•050
•100
•595
•055
•055
•317
•023
•046
(41)
.« 4*
8*. .75
1
2
n
6*
H*
3
e = .866
4
5
l-e*-(2) l-e*t (2) m
6
n*(5)
1
.750
-.500
1.000
1.000
1.000
1.25
.480
-.230
.730
•855
1.070
1.5
.535
-.083
•583
.763
1.145
2
.188
•062
•438
.662
1.324
7
8
9
10
11
(3)
(0)
e
12
(7)—(10) Id L r r
slnh f(8)
£«T
*
-.500
1.732
1.317
•381
-.881
-.587
-.215
1.385
1.129
•326
-.541
-.361
-.072
1.155
.987
.287
-.359
-.239
.047
•866
.783
.227
-.180
-.120
13
14
15
16
(£>
if
(13M10) t L fr
f*r v*
/-•«
(*)
17
18
(10)—(16) h ± r r
1*000
*019
•413
•250
•131
•175
•500
*807
*286
•233
•093
•124
•500
*221
*147
•219
•066
•091
#801
*104
*009
•189
•038
•051
(40
p » 10a
e * = .9
1
2
n
e*
n*
3
«
4
b
. 949
5
1—8^—(2) l-e*+(2) V u T
6
n*(5)
1
.900
-.800
1.000
1.000
1.000
1.25
•576
-.476
.676
•822
1.026
1.5
•400
-•300
•500
•707
1.060
2
•225
-•125
•325
•570
1.140
10
11
7
W
(6)
8
e
9
slnh '(8) ^ 4 9 )
12
(7)-(10) I S
-.800
3.000
1.819
•192
-.992
-.551
-.464
2.400
1.609
.170
-.634
-.352
-.283
2.000
1.444
•152
-.435
-.242
-•110
1.500
1.195
•126
-.236
-.131
13
14
15
(£)
n
<TL
(13M10) —
SctT 7
16
(6)
17
18
(lo)-(ie) ilt i r
£«T z
1.000
•808
.449
•100
•092
.102
•657
•487
.270
•097
.073
•081
•472
•320
.178
•093
•059
•066
•285
•159
•088
.088
•038
•042
(48)
Tho maximum shear stress along the x axis Is determined
by the formula
^
T*Mtfff * J
{ p n * m x " Grnirt) #
0g and 0J are the prinolple stresses along the
As
x axis the maximum shear stress is found by subtract­
ing the smallest of these normal stresses from the
largest one*
The result, for different values of n
and f , are shown In the following table.
fm 0
n
p * ola
f s .25a
P * .5a
.734
.645
.572
1*85
0
.147
•203
•239
1.5
0
.067
•099
•127
8
0
.084
•035
•049
/■ In
r*4 .
fa 10*
•
1
■
1
•to o
•M O
•M O
•800
•M O
.8 8 4
•M B
•BOO
•811
•810
•1 4 .
< in
•IB B
•H O
•888
•M l
•O ff
•OBB
•110
•188
All n l m
to «!>• i M n
tahl. «r» to b« Multiplied by
t o o b ta in th o M t W i i l m r t t r o o s .
(44)
B. Stresses along the z axis.
In the following calculations the radius of
ourrature p
refers to the end of the semi-axis o
of the ellipse
1
«
.
f i. 3
a * * c*
I
•
Alien p Is larger than o reference Is made to equations
(£0), for an oblate spheroid, and when p is smaller
than of equations (£1), for a prolate spheroid, are
referred to.
n Is now the dlstanoe from the center of the ellipsoid
In terms of the length of the semi-axis o*
For p *00 all stresses are zero along the z axis*
p « 10c
e*« *9
1
2
n
1-e1* i^»*
5
es *949
4
n» (5)
5
e
(4)
6
sln"*(5)
1.000
1*000
1.000
•949
1*250
1*2$
.676
•822
1*026
.925
1.181
1*0
•500
.707
1*060
*895
1*108
2
•525
.570
1*140
.855
.985
1
(45)
7
8
9
n* (2 )
/-e *
w
10
(7 )-(9 )
11
£<*r
*
12
*■«/ —
fir
*
.417
1.000
•100
.317
.176
- .3 5 2
•392
• 845
.11 8
• 276
•153
- .3 0 6
.369
.750
•133
•236
•131
- .2 6 2
•328
• 650
.154
.174
•097
- .1 9 4
P * 40
e2 = *75
e « .866
1
2
3
4
n
l-e% —*
r»a
VTil
n«(3)
5
e
(4)
6
sin^iS)
1.000
1.000
1.000
.866
1.047
1*25
.730
.855
1.070
•809
•942
1.5
•583
.763
1.145
.756
.857
2
.438
.662
1.324
.654
.713
8
9
11
12
n « (2 )
i-e *
IS )
1
7
S .I.)
10
(7)-(9)
S 7 ,r-
•605
1.000
• 250
.355
.237
-.4 7 4
•545
• 912
.274
.271
.180
- .3 6 0
•496
.875
.286
.210
•140
- .2 8 0
•412
.876
.28 5
.127
.085
-.1 9 0
(46)
P»
e*= .5
80
1
n
2
1-e**
e s .707
3
4
VT2)
n*(3)
5
e
(4)
6
8liT*(5)
1,000
1.000
1.000
.707
.765
1*25
•820
•907
1.154
.622
.671
1.5
•722
•850
1.275
•552
•586
2
•625
.790
1.580
.447
•462
7
8
9
11
12
£ ^ ( 6)
n*(2)
hJL1
(*)
1
10
(7)-(9) —
<T.
L i a~
£«T
.785
1.000
.500
•285
.285
-.570
.671
1.025
.488
•183
.183
-.366
.586
1.082
.462
.124
•124
-.248
•463
1.250
•400
.063
.063
-.126
p » .5 0
• « .707
e*s .5
1
2
3
4
5
6
n
n*
(2)-e*
(1)
(1)
\[(3)
C
(57
1
1.000
.500
2.000
.707
1.000
1*25
1.562
1.062
1.176
1.030
.686
1#5
2.250
1.750
.857
1.322
.534
2
4.000
3.500
1.870
.373
.571
(47)
7
slnh J(6)
8
9
±*(7)
(4)-(8)
10
11
—
12
n
£ *T **
•882
1.248
.752
.376
-.752
1
•641
.907
.269
•135
-.269
1.25
•511
.723
.134
•067
-.134
1.5
.370
.524
.047
•024
-.047
2
JE>»_j »25o
© s #75
1
2
3
n
n*
(2)-e*
6 s #866
4
5
6
e
(S)
(1)
(3)
1
1.000
•250
4.000
.500
1.731
1.1
1.210
•460
2.390
•679
1.275
1.25
1.562
•812
1.540
.901
.960
1.5
2.250
1.500
1.000
1.224
.707
2
4.000
3.250
•615
1.804
.480
7
8
9
(4)-(8)
slnh /(6)
11
10
f«r
2
12
n
1*316
1.520
2.480
.413
-.825
1
1*063
1.228
1.162
•194
-.387
1.1
*863
.985
.555
.092
-.184
1.25
*658
•761
.239
•040
-.080
1.5
*463
*535
.080
.013
-.027
2
(48)
P»
»lo
e*s .9
1
2
3
n
n*
(2)-e*
e s .949
4
5
ill
(3)
/(ST
6
€
(S I
1
1.000
•100
10.000
•316
3.000
1.1
1.210
•310
3.550
•557
1,704
1*25
1.562
•662
1.888
•814
1.234
1.5
2.250
1.350
1.111
1«161
•817
2
4.000
3.100
.645
1.760
.549
7
8
10
9
(4)-(8)
slnh *(6)
11
/T-
£«T
12
n
1.819
1.915
8.085
•448
••898
1
1.303
1.373
2.177
•121
••242
1.1
1.038
1.094
.794
.043
••088
1.25
.746
.786
•325
.018
-.036
1.5
•525
.553
.092
•005
••010
2
The following table gives the maximum shear stress#
n
100 fm 4C ps2o
pm 0
.5c
.25c
•lc
0
1
.264
•356
.428
.500
•564
•619
•673
.75
1.25
.230
.270
.275
.256
.202
.138
•066
0
1.5
•197
•210
•186
.148
•101
•060
•027
0
2
•146
•138
.095
.062
•036
•020
•008
0
(49)
Stresses along XAxis*
.6
CTX
.4
P*44
2
I
1.8
1.6
1.4
U
1.2
6. 4
* 4
P*4 ~
1.6
1.6
I
'
*
*1
j
t
■P« 44
^ max.
rt
t
r----Z A
1.4
A|k
O
ims
.-j
XA kis
Radius of Cmvotuf,
Stresses <aton4 Z Axis
SSI
I-*
O
2
14
1.6
1.4
14
14
119J-
14
1.6
LS
I
PMN
2
(50)
Stresses at Interior Points*
The potential of a homogeneous solid ellipsoid
for interior points is (r*S09 p.49)
a** s
b +s
%j <*s
/(a*« s)(b** s )(c** s )
U i )
in which a 9 b and o are the semi-axes of the ellipsoid*
Equation (23) is seen to he the same as the equation
for the potential at exterior points (r*309 p*56)
exoept that the lower limit is zero Instead of Jc •
The normal stresses are then immediately derived from
(12) and (15) by putting
and subtracting
y
according to (la)*
For an Oblate Spheroid these stresses beoome
<ry S <rM
(24)
(51)
and for a Prolate Spheroid
' e
i | i-e *l
j l-e * ~ i - e * J 2 e * l
^ 5 = (Tx
(ssi
It Is Interesting to note that the normal stresses
are all oonstant and independent of the size of the
ellipsoid*
These stresses do9 however, depend upon
the shape of the ellipsoid of revolution, in whioh
the semi-axes a and b are equal*
The shear stresses are immediately seen to be
zero, as the first derivatives of the potetlal in
this ease are functions of one variable only, see
(9), page 22*
By putting e = 0 in equations (24) and (25) the
normal stresses inside a hot sphere are determined*
In both oases it is easily verified that the stresses
are
0^ s 0^
fhloh oheoks with what was previously found in this
oast, see page 34•
(52)
For an Indefinitely flat ellipsoid, e= 1, equations
(84) Immediately yield
O .’ ■ <r, * -
° i ‘ °,
,
whioh oheoks with what was previously found on pages
28 and 32.
For an Indefinitely thin ellipsoid, e»l, and
from page 51
Lim(l- e2 )sinh *= =
*+•
\[h e*
=0*
Then equations (25) give
_
I E<*T
^.-oi’- F —
,
£<xT
_
< 5 - —
,
which oheoks with what was previously found on pages
51 and 33•
A check on the formulas
from (3) and (la).
(24) and (25) canbe had
If the three normal stresses
are added together and
(la)
eliminated by means of (3) $
the following result is obtained
^
_
o E<XT
(Tx + (Ty + <T2 = ~ c
777
-
If the three normal stresses from either (24) or (25)
are added together it is easily verified that the result
oheoks with (26)•
(2 6 )
(53)
Remarks on Problem 1,
The maximum stress that occurs anywhere is -65LE
i-\/ i
and this stress occurs only when the ellipsoid is
Indefinitely flat (oblate spheroid) or indefinitely
thin (prolate spheroid).
In these two cases the
e^yy
maximum stress inside the hot ellipsoid is
j
everywhere, but in the cold exterior it reaches this
value only at points of the surface of the ellipsoid
where p is zero, and the stress then immediately drops
to zero outside of these isolated points.
To keep the boundaries of the ellipsoid finite,
it is assumed that a beoomes zero for an indefinitely
thin ellipsoid and that o beoomes zero for an indefin­
itely flat ellipsoid.
At points where p
is zero infinite stress
might be expeoted, but as this only ooours when the
ellipsoid is either indefinitely flat (o = 0 ) or
Indefinitely thin (a m 0 ) the amount of material that
tries to expand in these oases approaches zero, and
the result Is a finite stress of
focJ- as previously
i*t/
OSjUlMd*
N r ■ sphere the maximum stress Is j -J7 7
both ioslde M M outside of the hot reclon.
(M)
II It tl« 0 U N w i l i k r to not* that ror i prolot*
•yfceroid Ibo i i l w o oboor stress at the and of tna
• w l < u 1 i a lo eonotont B rorardless of the ?alua of p
at tblo point.
for othor potato of tha x axis, however,
tblo otrooo differs with f , being largest for an
indefinitely thin allipoold ( p * * o ) and soallest for
o opbero ( p « a).
In general, ao la seen from figure 1, page 49.
the otreas In the oold exterior is a maximum at the
ourfbee of the hot ellipsoid of revolution and falls
off rather rapidly for points farther out in the
exteriorj according to Saint-Venant*s principle.
If the indefinitely thin ellipsoid is considered
to have a finite thickness ( a finite), but Infinite
length (o Infinite), the stresses along the x axis
will he the same as for an infinite oircular cylinder.
This case oan be oheoked by the* ordinary theory of
alastloity, and this is done in problem 2 as a check
an the oaloulatlons for the semi-infinite oircular
cylinder.
The results oheok with those on page 31.
(5b)
Not all the numerical calculations are plotted In
figure 1*
However, enough curves have been plotted
to Indicate the trend of the stress variation as
p
varies from zero to infinity, and any more curves
would olutter the figure up so as to reduce its
usefulness.
It might have been slightly preferable to use
cylindrical coordinates (r,d,z) instead of the
rectangular coordinates used*
To obtain the stresses
in these coordinates it is only neoessary to put
x= r and y = 0 in the expressions for stress*
<T# would then become
and (Ty would become <TJ .
r„z would become 7>z and rrt and Tj* would be zero,
where t is the direction tangent to the olrole r
equal to a constant*
(56)
Problem 2 *
Stresses set up In an Isotropic Infinite solid
when part of it, bounded by a Semi-Infinite Circular
Cylinder, is at a higher uniform temperature than the
rest of the solid*
As far as the author has been able to asoertaln
the potential of a semi-infinite circular cylinder is
not known.
However, it is the seoond derivatives of
this potential that are required in order to find the
stress9 and the determination of these second derivatives
constitutes the major part of this problem*
The first derivatives of the potential of a homo­
geneous body can be found in terms of a surface potential
by means of Gauss* theorem (r*31, p*Q), see also
(r*SO, p.110).
Then the attraction at a point F of a
solid body of any form, when resolved parallel to any
straight line, (taken as the axis of x )9 is given as
Ui)
Where f
is the constant density of the body, r the
distance from P to any element da/ of the area of the
surface and $ is the angle the normal at dia, drawn
Inwards* makes with the positive direction of x*
(57)
This Is true both for external and Internal points.
The ooordlnate system is now located in such a
manner that the origin is at the center of the base
olrole^and the positive z axis forms the axis of the
cylinder.
The base of the semi-infinite cylinder, then,
lies in the xy plane.
For the attraction in the z direction the angle #
is zero for all elements of the base of the oyllnder
and 90° for all elements of the cylindrical surface.
The attraction in this direction, then, is equal to
which is the potential of the surface of the base of
the cylinder with a mass density p per unit of area.
This result could also have been obtained directly
by considering the difference between the potentials
of two homogeneous cylinders a distance dz
apart.
The difference in these potentials is obviously equal
to the potential of a circular disc at the base of the
oyllnder and with a surface density of pdz.
Dividing
this by dz the above result for the attraction is
obtain#**
(58)
The potential of a circular disc of radius a and
mass density p
at
per unit of area Is (r*21,p*398)
i j 1' ^ ~ T
where the dlso lies In the zy plane with center at the
origin*
and
Thenr is defined by the equation r** x*4 y*
is the algebraically larger (positive) root of
the equation
-J±
a **X
X
X I
(i9)
'
'
'
the smaller (negative) root of which is /Ij*
Equation (29) gives
(30)
*
also
£[r‘tI*- a*-f(rl*z*-a*)z* 4a*z* ]
♦ f a r*4
a* and
^ s - a* z*, which gives
(*-*.)(*-**) ■ r*-(r*+z*-a*)t The partial derivative of (28) with respect to
- 2 Pa* J h - r -
-
-?-7
—
(3l)
is
I-----
Whioh Is Immediately seen to be zero, by means of (29).
(59)
The partial derivatives with respeot to x 9 y and z of
(28) can therefore be obtained simply by differentiating
under the integral sign with respect to these quantities.
Doing this, and using (51), the following results are
obtained
i t _
M l '
!* ± a (T
t-* Z'n
i
J _____
J
t/ +
jLf _ 211 * i a gu f°°
iyii ~
I-* 2 ™
—
“J
.oo
i t
-
« T „ &y f _______ d ± ______
in which equations p has been put equal to ~'
/-o 4 Tf .
Evaluation of the elliptic integrals in (52) and
substitution into the formulas for stress (see appendix)
yields
7 - ill a *X
E
** '
C a '+ A jifP Z ,
/-W TT
T
-
z**'
TT
\ (33 )
A-f
(a *.At ) \ f t t t
everywhere in the infinite solid, and
(60)
°*S
w
{ k ^ n f ^ k n L ) *(t -KE)%]-k J
for exterior points only.
(34)
For interior points it is
only necessary to subtract
from (34), according
to (la), to obtain the normal stress in the direction
of the axis of the cylinder.
The functions entering into the above formulas
for stress are defined in the following manner:
(35)
which quantity is always smaller them 1 , as
smaller than
•
le
Also k is always taken as positive.
By means of (30) equation (35) can be written in the
following two new forms
k
________________________________________________( 3 f a )
♦ ^(r*+2*-a*)*+ + a zz *
and
k
2 ra
Then
JT
K . (*
<**
is the complete elliptic integral of the first kind,
and
(61)
B » J |fhUsiit1?
df
la the complete elliptic integral of the second kind,
With
k* * \f 1 -k 2
the following quantities are defined
ir
K,a
f *
4+
, y/-A '*sinef
jr
e
B * * J ]} l-k '* S in * V d V
o
/'
!.
—
SW f
r*___________ _
E(T,k») =
(f h
df
■n(T,k')« sin 'f
on(v,k*) * oo» /’ » / l-sn*(v,k')
dn(v,k')« ^l-k'*sn*(T,k*)
(62)
The function dn(Tfk 9) is found from the following
equation
(36)
a
and from this all the other elliptic functions, entering
into equation (34), can be found*
From (35a) it is easily seen that k is the same
for z«f m as for z> -m, which then also holds for the
elliptic functions and integrals*
It follows immediately
that the two shear stresses (33) are symmetrical with
respect to the xy plane and that the normal stress <JJ »
according to (34), has equal, but opposite, stress for
z > « m and z * -m*
For the purpose of numerical calculation the
equations for stress oan be greatly simplified by
means of (30) and (35a and b), and also by putting
r x pa and z s na •
Then it is easily verified that
rn
__ p+
k
~
k
(63)
and
(a****)\/aV3,
s (ral^/ks
.
The two shear stresses (32) oan immediately be reduced
to one by introducing cylindrical coordinates r, 0 ,z«
Putting xa r * pa, y * 0 and z r na gives
(K-e)
r
n
' 0
The elliptic functions entering into equation (34)
can now be reduced to the following:
A 1
i
3
on (y,kf)• 1-sn (vfk*)* k
and finally
/-k *
P-Ar
(64)
dn(v,k*)
'^sntVjk* )on(v,k')
I
p
If (1 -pk) (p-k)
IT^l
Then equation (34) can be written
(KiM*n.-KCf.)-K]
(**•)
It is desirable to eliminate n in equation (34a).
This can be done by expressing n in terms of k and p
by means of the equation
k 9 - L £ p * + f r * + / - ^ ( p * + n * - if + 4 n e J
(see 35b)
solving for n gives
n *t)j
(+\)
where the plus sign is to be used when z is positive
and the minus sign when z is negative.
The normal stress (J“L can then finally be written
<5 . :
♦ i z]
where the plus sign is to be used for positive z and
«>
(6b)
the minus sign for negative z.
Equation (42) is good for all external points
except the z axis (p=0)*
For internal points it is
EotT
only neoessary to subtract
from the right hand
side of equation (42) to obtain 0J .
Along the z axis (see appendix)
(43)
where n is positive when z is positive and negative
when z is negative, and (43) is good for both external
and internal points*
Radial and tangential stresses*
The radial and tangential stresses can also be found
by means of Gaussf theorem*
The attraction of the semi-infinite cylinder in
the radial direction is the same as the attraction in
the x direction for a point in the xz plane, and is
equal to the potential of a cylindrical shell with a
surface distribution of matter of - p c o s 0 (see sketch)*
This result could also have been obtained by considering
the difference between the potentials of two semi-
(66)
Infinite homogeneous circular cylinders a dlstanoe dz
appart, just as was done for the attraction in the z
direotion.
In order to find the radial
to find the potential of the
above cylindrical shell9 but
f
^
v<\
— IILyc-k— k— I • i*
-
stress it is not neoessary
■— s.
V
/
J
only its attraction in the x
direction upon a point in
■
the zz plane*
This is done by splitting the surface up into semiinfinite rods of width add and mass per unit of length
p cos* add •
*'
cost a d *
x
where
dtl/i^fa2- 2axoosd
(67)
Substituting r for x gives
277
^„ f
r~ 4 Cos6
/, .
‘" ’I ' - . S - t . r c * .
d*V
7P
z
\
< **>
where r* in general, is equal to x^* y*.
A*yj
it is only necessary to turn the
91
To find
-777
cylindrical shell 90
about its axis and find its
attraction in the y direction upon a point in the xz
plane*
This gives (r.31,p.4)
27T
Jp s~
(f+Si»c()sin'faJ*
/ft
where sin ^
■
Substituting the previous values of d and sin of
gives
271
j7 =
Jt*
a
f
t
f in 0 —
(l*
Jo r +a -2arco»0 \
1
\lr*+z
1
— \d s
.2 *r c» » f
Now put z * na and for r larger than a put r* pa,
but for r smaller than a put a*p*r.
(44) and (45) will reduce to the form
Then equations
(45)
(68)
iV
£ ±
- o
+ *
f f e,J f z £ £ £ * l
Jt p + 1 -2 pcose
/ / . ______
(
H
\ ^
{ P .n ^ - e p c ^ r
in
A_
r
* * '
y)t p * + w > * \ !
/-£*s*0
n
\
/„
(46)
Zpc* 9 r °
when r is larger than a
and to the form
&ir
c p ' p f cosf-pjcos**
J
P Z+ I- 2 p'cosi
/,
(
»P‘
) ^
'}p 't+ » >P % h 2 p c o s » )
(■ » )
in
4 jt - -
b
'* p I
* • -
p rl
!-*< > ***
( u
P '^ - ip c o s * V
______ 2£|______
iFZmZpZZl
when r Is smaller than a ,
Integrating (46) an (47) and proceeding in the
same manner as with the equation for <rz $ (see appendix),
the following results are obtained
<rr m- —
- r —* i v *
n
\( ( p * i) * * » * ) ( * ' £ )
(*•)
Hr'**M)k} * fp.'frtotf-epr.glj
(69)
(49 )
tor r larger than a
and, by putting
pci-(
P
(4 S a )
~ (p **2 p
and
(49‘)
*<p- 0 ‘k ] - # [ * ( ( « ) - ■ &
' &
] }
for r smaller than a.
These are the stresses at exterior points«
(70)
For Interior points the stresses are
(4 8 b )
-(r**ip-l)k\-Zpr[KCM')-*|j.Jj
and
(*n)
■ & [*" < *)-■ *?
* & ] j
In formulas (48) to (49b) the elliptic integrals are
determined by
(p+t)*+n*
k.*. !- k *a < £ 0 f * «f
(p *0 *+ n *
£
«n (T,k«)« sin*/'«
n
The above stress formulas do not hold when p equals
zero, or for points along the z axis.
In this oase,
however, it is quite simple to find the attractions
of the cylindrical shell at points along Its axis*
This gives (see appendix)
for external points (n negative), and
W
-
7
&
(
for internal points (n positive)*
From symmetry it is Immediately seen that Tr ^ - 0 .
The most Interesting stresses are those near the
•ltd of the oyllnder, and particularly at the sharp
(*
‘>
(72)
olroular edge.
Here there la a discontinuity in the
normal stresses and the shear stress becomes infinite,
as shown on figures 2 to 5.
Figure 2 shows the normal stresses along a radial
line in the xy plane (just below the base of the oylln*
der nsO-, and just above the base n=0+), along a
radial line at a distance of 0.1a below the base ( m -0.1 )
and along a radial line at a distance of 0,1a above
the base (n« 0.1 ).
Figure
s gives the normal stresses along a generator
on the outside of the cylindrical surface (p*l+) and
along a line parallel to the generator at a distance
of 1.1a from the axis of the oyllnder (p»l.l).
Figure 4 gives the normal stresses along a generator
on the Inside of the cylindrical surfaoe (p»l-) and
along the axis of the cylinder.
Figure 5 gives the shear stress along a radial
line in the xy plane and along a generator.
With reference to figures 2 to 5, a is the axis
of the cylinder, the longitudinal distance z from the
base of the oyllnder is put equal to na and the radial
distance r from the axis of the cylinder is put equal
to pa#
Appendix to Problem 2
A«
Evaluation of the elliptic Integrals»
To evaluate the elliptic Integrals entering Into
equations (32), page 59, consider first the Integral
40
I
( &)
i
't w + J X * - * # * - * . )
In vhioh A 24 04 * , 4 ^ 4
Nov put
(Si)
a *+ *
vhere
0 < W4 I
Then
dw.
or» dV'= - 2
^ (o *Hr)
also, from (52),
dw*
w
(74)
, *
**Mi
where k * ■ r--
(34 -)
and 0 4 k 4 1 •
Expression (51) oan then be written
w
cfw____
/
( V M
fa**A, J.
NOW put Is sinf , whioh gives dw« cos / d/ and
If the Integral In (Sla) is put equal to ut then
fw
Z I
- 4 ”
,
'9 i(h w * )(l-k * w*)
f
r*
d *
K fl-k *s m *Y
there
*
» ' i m f • snu • l/
■
»
• • • f a f * </ a
f b $ H * U
m »/ 2 3
f **♦»,
m f/y**)
(V$)
<7b|
•aft
A t*fhk*sm*t * th v •\fl-k‘sn*u = | / p ^
F ( / » k ) I s tlM •lliptio Integral of the first kind.
Then
00
du
i*
(ss)
+ *,
and the first Integral in equations (32) becomes
’X
” <***•)* I * * udtt
^
a* u^, Is obtained by putting V' = />i in (54), making
S
0 s S and
</. =
f
-
= ft
share K Is the eomplete elliptic integral of the first
kind*
From (53) and (54q) it is immediately seen that
an. d l
4f
(? 7 )
(76)
and
an
This gives
f*
I sn*u du s
1
)
where E s f ^l-k*sin*]f d ^
o
and is called the complete elliptic integral of the
second kind*
Then (56) gives
f ,('t* * * W ('t* 4 * )(* -U )(+ -X,)
<a *
+
h
)
\
^
and the shear stresses (55) follow immediately from
(52) and (58), and the formulas (d)9 page 12*
From (52) and (54)
a ,, a ***, - ^ s n ' u
sn *u
(77)
Using (55) and the above equation, the expression
under the Integral sign in the last of equations (52)
oan be written
sn*u
n,
Putting
a **2 ,-a *s n *u
0 ^
— 7-
2
2
s k sn of
gives
(S9J
a**?,
*>
r H
/ _________
_
2
I
k s n ^ s n ^ u d tt
(6o)
2
sn«
0*Ha*+1kt CHd dHot
7 T (« ,< *)
in Jaoobl's notation, where //(k 9ei) is the eomplete
elliptic Integral of the third kind (r«339 P.420)
and (r«349 pages 15 and 142)•
7Toy,A) *
J,
In general
/ - k sn « sn u
where u Is defined by (53)*
A1so9 see (r«339 p.421)
f(K,«)* K2(«)
d
(78)
S
(6 /)
where Z(u)*E(u) - — uf
see (r*349 p«15)«
From equations (35) and (59) is obtained
(6Z)
a *+ *a
and it is easily seen that
sn* * 4
JH
(43)
2
as sn 0< is larger than l f oi must be a complex quantity,
and due to the periodicity of sn u there are Infinitely
many values of or that satisfy (62)•
The perlodloity of sn(u9k) is (4K, 21K*)» see (r*329
p.35), and along a line K+iv the function sn(K+lv)
varies from 1 to 4- when v varies from zero to K99
k
(r#32, P«36)•
But 9 from (63), 1 ^ sn «c
and one particular value of e€ that satisfies (62)
oan be obtained by putting o( s K* iv9 where O ^ v ^ K 9,
and v is always real*
Then, (r#33, p*422)9
(79)
Z(K + l v ) * - [ Z(v ♦ K ’, lc*) + ^ K K 7
J
and, by the addition theorem,
Z( v * K ’f k*) « Z(v, k')f Z|K», M M - k ^ s n f v . k M s n f K S k M s n f v f
But, from (r.34, p.15), Z(Kv,k')s O«
Also, sn(K* ,k*) = sin
= 1, and, see (r*32, p,33)
sntv^KSk*) = £ hSv»k ')
ctn(V'k')
This gives
Zlv+K1,**)- Z(v#k f)-k,2sn(v,k*)
and finally, by means of (61),
*j£]‘
Row, see (r.32, p.32),
in*
c»*4n*
m
sn(K+iv)
dnfckj
cn(K*'tv)dn(K*iv) ~ V * snW.V)cn(v,k')
and equation (60) beoomes
.
(80)
,
f
*—-_ [
f -
\
* * i <»*+*,
d n (*k ')
[tfp /v L )
ft'*Mfoh')cnW)l
■j
(6
]-Kj
From (5)t (32) and (64) equation (34) oan be written
down at once.
Evaluation of the Integrals appearing in equations (46)
and (47)♦
Consider first
r rw
I
tfcose- e.cos*e
JB
0 p * + l- 2 p c o s e
__/_r *ir
I
dcoso- * c a s 9 ^
~ P *H J
/-c ,c a s fi
where
c =
* P
p *+ l
and is always smaller than 1 for p4 ^t
for p equal to 1 #
The above Integral oan be written
oqual to 1
(81)
and is equal to
2W
For r larger than a d * p and e * 1, and for r smaller
than a d « 1 and e * p *•
In the xy plane, then,
^-7
7 = r*
jr* = ^p*
(f
\>or r>*)•
Us)
(forr<a)
Which is just half of the corresponding seoond deriva­
tives of the logarithmic potential for an infinite
eireular cylinder, see (r*30, p*72)»
The radial stress at points in the xy plane will then be
<Tr
-f—
r* - 47^-j
p * /-v
(rya)
(66)
J-
(r<a)
(82)
These stresses should be just half of those for
z
«
whloh oheeks vlth what is later found for this ease*
Now oonslder
I
(d c o s *
-
eco s*e)d *
_
2
J0 (p*H -2pco$6)fp*+nz+l-2pcos*
f
JJ^p*+rf+^2pm 0
7T
(6 7)
/
(hC,cos*)Jp**ff+t^2pcot0
where
P *HL 0
& * - £». = -C—
Ci
Zp
Cf \ C»
'
e a - b and f *
&P
*p
p *+ /
J
' f - </
*P
lb* right hand side of equation (67) can now be written
a (0
j f [(*£ ***
gpeest
]
(63)
To integrate (68) put
<Ps
■■
£
Then
cos0 * 2sin
-1
d0= - 2d*
and the limits are 0 « O f
and 0 * 7 T»
0.
The first integral in (68) then becomes
ft
~ 4 =
—
r
^
ftp + ! ) * + » * j
where
k*s
(p+l)‘+ n e .
r<#;
'
(70)
The integral in (69) is easily seen to be
which is equal to
j y [ ( ( P * 0 * + » * ) e ( K ‘ F ) + ((f>-0ee - 2 P < t ) K \
Using the same transformation on the last of the
integrals in (68) gives
(7/)
(84)
Og s
with
ZC'
=
!*c,
* p
(pH)
this beoomes
2
£
_____ a ft
e f(P*+l)
f
(P*0‘*)!(ph )*+ n* J,
£
'
*
/-.»
(t- c&sin*</)ff-/'*sin*v
The integral in (72) can be written
Jr
J
K
du
J
f-Cg£tra ti
share
fr *
_
sn** =
£ V ______
I- k * s n * « t $tr‘i
= !£+?)*+»*
k*
(p + ,)*
This last integral is easily Identified with Jaoobi9s
elliptic integral of the third kind, and as
14
^
ka
it oan be solved in exactly the same manner as the
Integral of the same type in equation (60).
This gives
( du_
[l-Wu
,
dm*')
fun../.•)-](£!!'i
.ILL 1
M m W c W n
*'
**'*
(85)
With the values of the elllptlo functions given on
pages 70 and 71, this becomes
I
^
(»>
By oombining (69), (71), (72) and (73) the expression
(6 8 ) oan be written
\(74)
-
[<A
- H f c * * ^ ♦#]
When r is larger than a d • pn and e= n.
Then, by
combining (65) and (74),
0 = - p { ir'
-)
Whenever a double sign ooours in any of these equations
the upper sign is to be used for positive z, or n,
end the lower sign for negative z9 or n*
(7f)
(06)
When r is smaller than a dcn'p9 and a * n ,p ^ l vher*
n*« np*#
Then,
(74)
v TTv
ZK
In equation (76)
4 p -
k*=
P
but, on putting p f« -L , this is seen to be identioal
vlth equation (70), and (76) takes the font
£
Jr*
. r
; &
I
k c m i
•
- * &
* &
-i)
] )
Equation (77) is only good for r smaller than a*
From equations (75) and (77) it is now easy to
arrive at the expressions for radial stress, by means
(77)
of (Xa)9 when it is reoalled that T» 0 for exterior points.
The tangential stresses are derived in exaotly the
same manner from the lower equations in (46) and (47).
Thus equations (48) to (49b) are obtained.
B» Special oases.
At an infinite distance above the plane of the
base of the semi-infinite cylinder the stress distri­
bution will be the same as for an infinite oylinder.
This stress distribution oan easily be oaloulated by
means of the ordinary theory of elasticity, as it oan
be considered one of plane strain (r.36).
The boundary
conditions at infinity will then only require that
uniform pressure be applied to the ends of the infinite
oylinder.
The effect of removing this local pressure
is negligible in the interior of the infinite solid.
When a uniform pressure p aots on the surfaoe of an in­
finite cylindrical hole9 of radius a9 cut through an
infinite solid, the stresses in the solid are (r#28,p.57),
(88)
Tbo elongation of the radius due to the pressure p
will then be
which for r s a becomes
For a solid infinite oyllnder of radius a and outside
pressure p the stresses are (r«£8, p# 56)
(j't = <rt = - p .
The radial dlsplaoement at the outside of the oyllnder
then Is
**)) = - f j M f ♦ •'‘S) .
The axial strain Is
j - f c - u ( < r r t <rt ) J = y
(<rz +2>>p).
If the pressure p is solely due to the temperature
expansion of the hot cylinder
(89)
or
The condition of plane strain In the oyllnder gives
Summing
from
the last two equations gives
» - 2p,
whloh
Then, in the infinite solid, far above the plane of the
base of the semi-infinite circular oyllnder, the stresses
are
(76 )
<rt*o
(80)
outside the hot cylinder, and
<T -*c er
- —
ISLL
Qr
r t -- - J
(79)
1
/-I/
inside the hot oyllnder.
From the symmetry of two seal*
infinite oyllnders put end to end
(see sketch) it is easily seen
that for r > a the stresses <Tr
and CTt are just one half as
large for the semi- Infinite
oyllnder as for the Infinite
oyllnder, when nsO.
For r < a the stresses fTr
and <Tt for the infinite oyllnder
will obviously be equal to the
stun of the corresponding stresses just above and just
below the xy plane for the semi-Infinite cylinder, as
there is a discontinuity of stress in this case.
The stress <JJ however, is continuous across the xy
plane (except for p= 1 ), and it is easily seen to be
(91)
just one half as large for the semi-infinite ae for the
Infinite cylinder everywhere in the xy plane9 exeept
for p*l#
As the stress for the infinite oyllnder is the same as
that for the semi-infinite oyllnder at n * e »9 this
consideration will serve as a oheok on the formulas
developed#
The stresses trz
and <T» #
From (35a) it is easily seen that k becomes zero when
n becomes infinite#
Also9 k 9»l and K a E s ^ #
From (38) it is seen that it is neoessary to evaluate
the expression
K- X
&
of whloh both the numerator and denominator are zero#
To do this put
From this equation it is immediately seen that
(so)
K-E
Also Llm —
•0
k -*0 k
(si)
(92)
K-E
and
Llm
k-*0
k« '
4
•
(3 2 )
From (80) It follows that
TrL = 0
for n:tf,
which it should be.
For the stress C \ equation (42) will be used.
From page 63
sn*(v,k»)« sin*JP« 1, whloh gives
E(v,kf)s Ef* 1.
v
Also tsK's co %and — *1.
Kf
Then (42) yields <T2 s 0
whloh cheeks with (78).
<TZ -
fP« 90# and
for exterior points,
For interior points
, whloh oheekswlth (79).
In the xy plane n s O and, from (35b)
k «
*P
( p *+i - )/(p*-0e ).
In using this equation for k it must be remembered
(93)
Then,
fop p>l, k » ^ and for p<l, k*p*
Equation (38) then gives
T
- +*7
* by
For p * 1,
K -E
k ir
(S3)
k * 1 and Ztz « e© .
For p * 0, k * 0 , and equation (61)Immediately gives
Tt l s o, whloh Itshould be*
As p-ksO for p <l and 1-pkaO for p > l 9 equation
yields
From page 63 it Is then seen that
s i n * ^ ^ 0 for p > l and sin*J^#= 1 for p< 1 *
.Then p is larger than 1, then,
E(vfk*)a va 0 and 0 * * 0 in the xy plane*
.
’/hen p Is smaller than 1 , however,
E(v,k,) * E f and v » K f, which gives
v
Strik1) * -* E*« 0, and
K*
(48)
(94)
(84 )
just below the base of the oyllnder (n«0-).
Just above the base (n * 0+) It Is easily seen that
the axial stress has the same value as riven by (94),
whloh It should, as <J^ Is continuous*
These results oheok with (78) and (79) In oonneotlon
with the dlsousslon of these formulas*
Along the axis of the oyllnder (peO) It Is obvious
that
r 1-2* 0. From (35a)
k =
Using this, and letting p-*0, gives
(P -h )U -P k )
m
{n *
Also, it is immediately seen that k * 0*
p In terms of k and n gives
p . l+k*
ik
-
i ( * £ - ) * - ( i , n l
solving for
(95)
where the minus sign Is taken because p is smaller
1.
When k-tO, and for finite values of n9 this oan
be written
„ - h i l ~ [ u jl . i
V ~ £k
•***
Z £*$[•
I tk
.
,+ -*)L
~ " J1 ~. r\ t4,t
I* .
i*
k-fO
Then pk • (1 ♦ n*)k*and equations (81) and (38) immediately
give T/-23 09 which it should be*
From page 63 it is seen that sin* / « l 9 whloh
v*K'«*>
v
, — • I,
Kf
IT
and 3(v9kf)l Kvv
«
Kay
gives
1*
Then (42) gives
<rz = 1
1
1*1
•-*
* 1
(43)
fiZ p !
both for positive and negative values of n*
Equation (43) oan easily be oheoked by finding
y v
“JJj
direotly, for points along the axis of the semi*
infinite circular oyllnder*
Along any line p a l there is no difficulty in applying
formulas (38) and (42), except for the case of n » 0*
Then k a 1 and K * ae 9 whloh immediately gives t r t s d?
(96)
This oheoks with what was found in the xy plane.
Also, sin
J - , whloh gives
EfT,**)* v s
and S ’s K's £ .
9 - 45*, and
Equation (42) then gives
Writing
it is seen that each term in this series is finite,
and if the series therefore is multiplied by either
Ef
EjVfk1) - - v
or
1 -k,
each of whioh becomes zero
K*
when ksi , the terms all become zero.
The series thus
obtained will satisfy the conditions of convergence
and approach zero as k approaches 1 ,
Then it is easily seen that
(97)
and there la a discontinuity lr <rs of ^
£ £
aa the xy plane la orossed*
The atreaaea Or and Oj >
in equations (48) to (49b) theModulus k oftheelllptlo
integrals la determined by theequation
.*
ka
4P
—- t
(P+0 * + if *
When n Is infinite the tern
examination*
((p+l)*+ n*)(K«S) requires
In this oaae k » 0 and, from (8b),
m 4p
(p 4 1)a♦ n*»-^
, which gives
k
((P ♦ 1
K*5
n4 )(K-E)* 4p — 7 • This, by
k*
virtueof(88),
is seen to approaoh piT when k approaches zero*
n
Also, - p s s s s s n approaches 1 as n approaches infinity
y(P 4 l )*4 n*
and, from page 70, sin^ / s !•
K*y ,
E(v,kf)S
1,
With k** 1 this gives
v* K*« ee and —
* 1,
(98)
and aquation (48) baeonaa, for eztarior points,
<Tt * -
I
2p*
CttT
/-*
,
whloh oheoks with (78).
For Interior points equation (48b) glwes
t 1 £* T
* r - - J —
whloh oheoks with (79)•
n
For n » - «o t
;
— T" y f
V(p*l)«* n*
1 and both equations
(48) and (48a) give <7J.a 0.
In the same manner <Jf Is seen to oheok with (78) and
(79) for n • so and to be zero for n* -ee f whloh are
the proper boundary conditions in these two oases*
In the xy plane (n«0) / * 0, see page 70.
It is
then Immediately seen that equations (48) to (49b)
take the form
(99)
01s-
I
E«T
4p*~t~v
(p/i)
cr.- 4 pi *
But
/-*
to;
oj.o-.^fsr
r
f 4 /-«/
(p< 1, » » 0»1
(p<l» n « 0+)
This oheoks with (66) and also with (70) and (79) In
oonneotlon with the disoussion of thoaa formulas*
Along the axis of the cylinder (p*0) <7> and dj are
equal everywhere*
Equations (46a) to (49h) then beeoas
very difficult to apply on aeoount of the term
*/p* ,
However, in this case a staple expression for the stress
perpendicular to the axis of the cylinder can be derived
very easily, by considering the potential at points
along its axis of the semi-infinite cylindrical shell
previously used, see pages 65, 66 and 71.
(100)
Then» for the solid semiinfInite oiroular cylinder
of uniform mass density
p
lit
j p
= - J
(/- sin*)cos 6 ad&
where sin et s
-Z
ll a**,
This gives
fj =**
7 T p (l+ -= = )
1 fh n *' i
which is good for both positive
and negative values of n.
For n < 0 (external points)
equation (50) is obtained
and for n >0 (internal points)
equation (50a) Is obtained.
2
2 * P
For p* 1 it is convenient to put (p+1) ♦ n * --y f
from (86)f in equations (48) t (49) and (49b).
n
Then
V ( P + 1 ) S n*
(101)
and with p «l this becomes
♦ \/l-k* = Ik*, plus or
minus aocording to whether n is positive or negative*
Also, sin*J^=l, which gives
v a K 1 and
E(v,kf)sE*,
It is then easily seen that for n > 0
—
£ « rf
/- u L <
(61)
and for n < 0
* r - f S
4 ( ¥ - i « )
(87 a)
«?■
Whan n a t c c , k* 0 and kf* 1*
By means of (82) it
if than immediately seen that (87) checks with (78)
and (79)* for n a * oo f and that (87a) gives
a o for n ■ • oo ,
(102)
NUmerloal Calculations for Problem 2 ,
A.
The stresses X r z and (Xj .
n *0.
1
3
4
*'(<)
0- sm'(t)
K
£
1.571
1.596
1.571
1.54 6
1.468
1.320
1.171
1.029
n
2
0
•25
.50
.75
•90
.99
1.00
0
.785
1.57
2.35
2.82
3.11
3.14
14° °30f
30*
48# 30*
64* 10*
81* 50*
90*
r>a
r <a
/-* T
T
TZt tUL
it
-M
“M
0
•064
IT
0
4.0U0
2.000
.016
.069
•139
•251
1.333
.188
.380
•746
1.111
.353
1.010
.740
CO
1.000
1.686
1.910
2.280
3.346
0O
5
1.000
6
0
.050
.218
.590
1.109
2.317
oo
(102)
1
3
2
4
5
6
(2)
n
n 2
0
.01
.10
.20
•50
T
+l
0
.0001
.01
1.00
1 .00
•04
•25
1.01
1.0025
1.00
2.00
1.00
1.0625
1.25
4.00
2.00
7
8
(S) 4 (6)
1.00
1.01
1.105
1.221
1.641
2.618
5.828
13
9
\I(T)
1.00
1.005
1.050
1.105
1.282
1.617
2.41
IT
A =
15
16
.516
1.118
2.828
•990
•905
.820
.610
.382
.172
CO
.719
.382
.271
.140
•060
.018
' m
77
? * ■
l.Ou
1.00
1.005
1.020
1.125
1.500
3.000
12
&K
90 0
82°
65 •
55#
37# 30*
22* 30*
9* 50*
17
▼
ILL
* (MU)
•122
0
.01
.100
.201
tt)-(s)
£
•024
0)^(4)
11
1.00
/-*
2.242
1.145
.77
.343
1.005
1.033
1.118
1.414
.318
.320
.334
.352
.408
.515
.767
14
CO
1.00
1.00
1.001
10
18)
MOT
1.00
1.028
1.164
1.26
1.413
1.512
1.559
\fl3)
eo
3.370
2.309
2.03
1.756
1.634
1.583
18
/
l+ (l°)
CO
2 .00
1.074
.735
.647
.558
•520
.504
1.990
1.905
1.820
1.610
1.382
1.171
Ul)
.500
.503
.525
.549
.621
.723
.854
(104)
19
20
21
22
23
2A
e'*
if(755
$##T/09)
.707
.709
.785
.742
.788
.855
•925
25
E l* k ')
.7854
.7852
.797
.809
•836
.876
•934
45*
45*
46°
48°
52 #
58 #
67°
0*)*
1.00
10*
30*
10*
40*
26
.998
.820
.672
.372
.142
.029
27
cc '
1.571
1.570
1.498
1.432
1.280
1.137
1.040
/-te/J
0
.002
.180
.328
.628
.858
.971
28
29
.7854
.7856
.826
.868
.990
1.198
1.600
p« 1+
31
(ishtoht*)
0
•0012
•048
•089
•182
•307
•422
32
(K )* (U )
0
•0013
•035
•058
.1017
•1596
•213
33
/ (* * )
i'zr
•25
•2499
•250
•251
•249
•248
•254
0
.0447
.424
.573
.793
.926
.985
V
1.571
1.572
1.649
1.731
1.983
2.415
3.153
/(ID
34
L i <Tm
**rz L
(27j
.7854
.7854
.750
.718
•638
•565
•528
p« 1 35
U S i*
s(S2)+Ctt) 04)-/
•250
•2486
•215
•193
•147
•088
•041
-•750
-.751
-•785
-•807
-•853
-•912
-•959
sin'7*5)
0
2° 34*
25*
35*
52* 30*
67* 50*
80*
30
/-(/*)
0
.001
.095
•180
•380
•618
•828
(105)
nt io.l
1
p
.500
.800
•950
1.000
1.050
1.200
1.500
2.000
7
2
p
5
*
•25
•64
•9025
1.00
1.1025
1.44
2.25
4.00
/.OI+(2)
1.26
1.65
1.9125
2.01
2.1125
2.45
5.26
5.01
6
9
(3 )* (7)
.766
•403
•218
•2005
•229
•493
1.276
3.02
13
2.026
2.053
2.130
2.210
2.341
2.943
4.536
8.03
14
5
ff
(*h
•74
•55
•0875
•01
•1125
•45
1.26
3.01
10
d (S)
.493
.780
•893
•906
•898
•815
•662
•498
15
sin"1(9)
29°
51*
63*
65 #
63*
54*
41*
29*
30f
20*
20f
50*
40'
30*
50*
16
(J3)
/- ( i * )
•753
•376
•152
•094
•057
•022
•007
•004
w *
•243
•608
•797
•820
•807
•665
•438
•248
------■
6
(4 )*
•547
•1225
•0076
•0001
•0126
•205
1.588
9.07
11
•587
•1625
•0476
•0401
•0526
•245
1.628
9«11
12
05
A*
m
4
Os)
•757
•392
•203
•180
•193
•535
•562
•752
•995
•959
.749
•522
•295
•066
•0125
.0053
K
1.6818
1.9602
2.2541
2.3088
2.2702
2.0209
1.7057
1.6847
17
jW )
•998
.978
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(118)
Normal Stresses.
p» I ♦.
(119)
Normal Stresses.
£«T .
w-i4
|-tf
P* • " •
AjciSOfCyl
-.3
Axis ofCul
(120)
Shear Stress Tr z .
p» i.
(121)
Remarks on Problem 2«
On aooount of the discontinuity of stress at the
surfa06 of the semi—InfInite oiroular oylinder separate
formulas had to he derived for the oases p> 1 and p< 1*
Also, for p < 1 separate formulas had to be derived
for the oases n > 0 and n < 0*
Hie shear stress Is oon-
tinuous everywhere exoept at the oorner (p»l, n«0),
where It beoomes infinite, but of the normal stresses
only the radial stress is oontlnuous when the oyllndrloal
part of the surface is orossed and only the axial stress
is oontlnuous when the base of the oylinder is orossed*
The stresses obtained from the different formulas when
p is put equal to one and n is larger than zero, or n
is put equal to zero and p is smaller than one, must
therefore be considered to be either for points im­
mediately inside the cylinder or immediately outside*
When the oorner is approached from different directions
different limiting values of stress are obtained, all
normal stresses being discontinuous and the shear stress
becoming infinite*
This is shown on figures 2-5, and
it is also shown how the stress along a line that
passes near the oorner has a rapid variation as the
oorner is passed, although it is not discontinuous*
The greatest normal stress that occurs anywhere
(122)
la the axial stress Inside the oylinder and far rroa
Its base*
This Is a ooapreaslTa stress of
»•#
The greatest radial and tangential stresses are both
the same and ooeur Inside the oylinder9 but Infinitely
close to its base*
Their value is *75
,and the
stress Is compressive*
Outside the oylinder the largest stresses are at
the surfaoe and drops off rather rapidly for points
farther out in the exterior, in aoeordanoe with saint*
Venant*s principle*
In this case the greatest axial
stress occurs just below the base of the oylinder*
is a compressive stress of magnltute *5 ^
It
* The
greatest radial stress occurs at the oylindrloal surfaoe
and at a distance of about *5a from Its base*
It Is
a compressive stress of magnitude *535 TTJr • Th*
greatest tangential stress is tensile and of magnitude
*5 -y—
. xt ooours at the surfaoe of the oylinder
and far from its base*
_
/ ^ I t e numerical calculations were all checked by
the faot that for exterior points the sum of the three
normal stresses is zero, and for interior points -2 SSI
^ #
/
/ This Is also a oheok on the method of solution, as the
/ axial stress was oaloulated from an entirely different
surfaoe potential than the radial and tangential
stresses*
.....
—
•—
(123)
Notations.
E
Modulus of elastloity
Coefficient of thermal expansion
T
t
x 9y,z
Rectangular coordinates
u,v,w
Components of displacement
p
€z Normal strain components
¥ *it ¥yz
(Tjc
Temperature
Shear strain components
e
Unit volume expansion
G
Modulus of elastloity in shear
\)
Poissonvs ratio
^
Displacement potential
cTz
tty, T T i f Z
Normal stress components
Shear stress components
V
Potential at point (x,y,z)
p
Density of matter
S
surfaoe of discontinuity
Rectangular coordinates tangential to
and perpendicular to the surfaoe s
a fo
semi-axes of generating ellipse
r*
e
Eooentrioity of ellipse
n
Ratio of x to semi-axis a, or z to
semi-axis o
(124)
p
(Tf.f G~t
Badius of curvature
Radial and tangential stresses
r
x
y
a
radius of semi-infinite circular cylinder
K
Complete elliptio integral of the
first kind
E
Complete elliptic integral of the
second kind
k
p
Modulus of elliptic integrals
Ratio of radial distance r to radius
a of semi-infinite cylinder
n
Ratio of axial distance z to radius
a of semi-infinite cylinder
T f0 9z
t
Cylindrical coordinates
tangent to oircle r equal constant
p f Reciprocal of p
TTtKf0*) Jaoobifs complete elliptic integral
of the third kind
Z(u)
Jaoobi*s Zeta funotion
(125)
References*
Parentheses containing the letter r followed by a
number refer to this list.
Papers.
1.
"Memoirs sur le Calcul des Actions Moleculaires
develop pees par les Changements de Temperature dans
les Corps Sol ides"; by j. M. C. Duhamel, Aoademie
Royale des Sciences de L vinstitute de France, vol. 5
1838, p. 440.
2.
"Die Gesetze der Doppelbreohung des Llchts in
oomprimirten Oder ungleichformlg erwarmten unkrystal
linisohen Korpern"; by F#
Neumann, Abhandlungen
der Konlglisohen Akademie der Wissensohaften zu
Berlin, 1841, p. 3.
3.
"TJhtersuchungen iiber die 31asticitat fester iso-
troper Korper unter Beriicksiohtigung der Warme";
by C. W. Borohardt, Monatsberichte der Konigllch
Preussischen Akademie der Wissensohaften zu Berlin,
1873, p. 9*
4*
"On the stresses caused in an Elastic solid by
(126)
Inequalities of Tnetntnrv"; by j# jnpliasu^ z^es*
••near of Uathenatles, vol. e9 1879, p. 16a#
5*
"On the stresses in Solid Bodies due to unequal
Heating, and on the Double Refraction resulting there*
from"; by Lord Rayleigh, Fhilosophloal Megaslne, 1601,
p# 169*
6*
"Spannungen und Foraanderungen elnes Hohlsyllnders
und elner Hohlkugel, die ton lnnen ersarat verden,
unter der Annahme elnes llnearen Toaperaturvertellungs*
geaetzea"; by a, V. Leon, Zeltsohrlft fur lfetheaatlk
und Physlk, vol. 52, 1905, p# 174,
7.
"Temperaturepannung in Hohlzyllnder”; by r # Lorens,
7. D. I., vol. 51, 1907, p# 745.
8.
"The Thermal Stresses in solid and in Hollos
Clroular Cylinders oonoentrloally Heated”; by C. H. toes,
Proceedings Royal Society of London, rol# Id, 1922, p«411*
9.
"The Calculation of Temperature Stresses in Tubes";
ty L# H. Barker, Engineering, rol# 124, 1927, p. 445#
10#
"Thermal stresses in Thin walled Cylinders"; hy
(127)
C. H. Kent, Transactions A.S.IT.E., vol. 55, 1951, p.167.
11.
"Thermal stresses In Spheres and Cylinders Produced
by Temperatures Varying with Time"; by c. H. rent,
Transactions A.S.M.E., 1932.
12.
"Distributed Gravity and Temperature Loading la
Two-Dimensional Elasticity Replaoed by Boundary Pres­
sures and Dislocations"; by r. a . Blot, Transactions
A.S.M.E. , vol. 57, 1935, p. A—41.
13.
"A General Property of Two-Dimensional Thermal
stress Distribution"; by ??. a . niot, Philosophical
Magazine, 1935, p. 540,
14. "Thermal stresses in Plates"; by J. L. raulbetseh.
Transactions A.S.M.E., vol. 57, 1935, p. A-141.
15.
"The Thermal stress in a strip due to Variation
of Temperature Along the Length and Through the Thickness";
by j. n . Goodler, Physics, vol.7, April 1936, p. 156.
16.
"Temperature Stresses in Flat Rectangular Plates
and in Thin Cylindrical Tubes"; by j. p. Den Harteg,
Journal of the Franklin Institute, vol. 222, ffo. 2,
August 1936, p, 149.
(128)
17*
"Thermal Stress"* by j. jy. Goodier, Transactions
A.S.M.E., March 1937, p. A-33.
18.
"Thermal stress in Long Cylindrical Shells due
to Temperature Variation Round the Circumference, and
Through the Wall"; by j.
n
. Goodier, Canadian Journal
of Research, vol. 15, April 1937.
19.
"On the Integration of the Thermo-Elastic Equations”
by J. N. Goodier, Philosophical Magazine, vol. 23,
1937, p. 1017.
20.
"Thermal Stresses in Cylindrical Pipes"; by H.
Poritsky, Philosophical Magazine, vol. 24, p. 209.
21.
"The Stress produced in a Semi-Infinite Solid
by Pressure on Part of the Boundary"; by
a
. E. H.
Love, Philosophical Transactions of the Royal Society
of London, Series A, vol. 228, 1929, p. 398.
Books containing articles on thermal stress.
22.
"Vorlesungen liber die Theorie der Elasticitat";
by Franz Neumann, Leipzig, 1885, p# 107.
23#
"Vorlesungen uber Teohnisohe Mechanik"; by Aug.
(129)
Foppl, Leipzig and Berlin, 1922, 4th ed., vol. 5, p.238.
24.
"Strength of Materials"; by j. cose, New York
and London, 1925, p. 456.
25.
"Applied Elasticity"; by s. Timoshenko and j. r.
Leeeelle, East Pittsburgh, 1925, pp. 146 and 278.
26.
"Strength of Materials"; by
s.
Timoshenko, M i
York, 1930, part 2, pp. 467 and 550.
27.
"A Treatise on the IGathsnatloal Theory of Elasticity";
by A. 15, H. Lore, Cambridge, 1927, 4th ed., p. 106.
28.
"Theory of Elastlolty"; by S. Timoshenko, M r
York and London, 1934, pp. 203, 364, 366 and 276.
Other refereneee.
29.
"spherioal Harmonies"; by T. X. Use Robert,
New York, 1927.
30.
"The Theory of the Potential"; by
*># ?*e
Milan, New York, 1930.
31.
"A Treatise on Analytical Statics"; vol. 2, by
(130)
E. J. south, Cambridge, 1892.
32.
"Elliptic Integrals"; by h . Hancock, New York, 1917.
33. "Theory of Elliptic Functions"; by h . Hancock,
New York, 1910.
34.
"Elllptlo Functions"; by A. Cayley, London, 1895.
35.
"Funotlonentafeln mlt Formeln und Kurven"; by
E. jahnke und F. Emde, Leipzig and Berlin, 1909.
36.
"On Frictional Effects in Shrink Fits"; by
J. N. Goodier, Stephen Timoshenko 60th anniversary
volume, UaoMillan, 1938, p. 51.
I
(131)
Table of Contenta.
Blograpbloal sketch of the author
1
History of the thermal stress problem
2
General theory
9
Discontinuous temperature distribution
14
Problem 1
19
Appendix to problem 1
27
Numerical calculations for problem 1
56
Figure 1 (stresses pertaining to problem 1)
49
Stresses at interior points
60
Remarks on problem 1
65
Problem 2
66
Appendix to problem 2
75
Evaluation of the elliptic Integrals
75
Special oases
07
Numerical calculations for problem 2
102
Figures 2-5 (stresses pertaining to problem 2)
117
Remarks on problem 2
121
Notations
125
References
125
Ac k n o w l e d g emen t s
The author wishes to express his gratefulness to
Dr, J. N. goodier, Professor of Mechanics of Engineering,
Cornell University, for having suggested the two problems
of this thesis and for invaluable aid in their solution*
He also wishes to thank Dr. H* poritsky, Engineering
General Department, General Electric Company, for a
valuable suggestion in the solution of the second problem.
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