END EFFECT N A TRUNCATED
SEMLmFlNlTE WEDGE AND CONE

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
TOMMlE R. THOMPSON

1968

—‘w‘
'

. «ELEM

”ILIBRARY “

Michigan State
University

 
     

This is to certify that the
thesis entitled

END EFFECT IN A TRUNCATED SEMI-INFINITE
WEDGE AND CONE

presented by
Tommie R. Thompson

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Engr. MeCho

[\i\\b‘ K: ) EL
\\ - \ \ 5. "
@393 \}_\é~n C 91% ‘

Major professor

 

Date Aug. 5, 1968

 

0-169

 

ABSTRACT

END EFFECT IN A TRUNCATED SEMI-INFINITE
WEDGE AND CONE

by Tommie R. Thompson

The purpose of this research is to deve10p the stress
distribution in a two-dimensional truncated semi-infinite
wedge and in a three-dimensional truncated semi-infinite cone.

Using a complex valued eigenfunction expansion for
an Airy stress function formulation of the wedge problem, the
stress distribution within the St. Venant boundary region is
determined for several "typical" loadings.

The solution for the cone problem is formulated in
terms of Papkovich-Neuber functions and the resulting stress
distribution in the cone is also determined.

Eigenvalues for both problems are presented for sev-

eral wedge and cone angles.

END EFFECT IN A TRUNCATED SEMI-INFINITE

WEDGE AND CONE

BY

. (#3
Tommie R. Thompson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and
Materials Science

ACKNOWLEDGMENTS

In completing this portion of his graduate study,
the author wishes to express his indebtedness and apprecia-
tion to the following individuals and organizations:

To the National Science Foundation and to the
Division of Engineering Research for awarding the
fellowship which made a portion of this study
possible;

To Dr. Robert Wm. Little, his major advisor,
for his guidance, encouragement, and assistance
in the course of this research and throughout the
author's graduate program;

To Dr. W. A. Bradley and Dr. J. L. Lubkin for
serving on his guidance committee;

To his wife, Lynda, for her understanding and

encouragement throughout his graduate work.

The research reported in this thesis was supported

by the National Science Foundation under Contract 71-1623.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS O O O O O O O O O O O O O O 0
LI ST OF TABLES O O O O O O O O O O O O O O 0

LIST OF FIGURES O O O O O O O O O O O O O O 0

CHAPTER

I. INTRODUCTION AND HISTORICAL DEVELOPMENT.

II. THE WEDGE PROBLEM . . . . . . . . . .

.l Formulation of the Wedge Problem
.2 Results and Conclusions (Wedge
PrOblem) O O O O O O O O O O 0

III. THE CONE PROBLEM . . . . . . . . . . .
3.1 Formulation of the Cone Problem .

3.2 Results and Conclusions (Cone
PrOblem) o o o o o o o o o o o
BIBLIOGRAPHY O O O O I O O O O O O O O O O O
APPENDICES O O O O O O O O O O O I O O O O 0

APPENDIX A. Development of the Asymptotic
Eigenvalues of the Wedge Problem . . .

APPENDIX B. Orthogonality Conditions for
the Eigenfunctions of the Wedge Problem

APPENDIX C. Development of the Legendre
Functions Used in the Cone Problem . .

iii

Page

ii

iv

15
28
28
41
50

54

55

60

62

TABLE

LIST OF TABLES

2.1 Roots of transcendental Eqs. (2.23) and
(2.24)(roots in left half plane) . .

2.2 Roots of transcendental Eqs. (2.23) and
(2.24)(roots in left half plane) . .

2.3 Roots of transcendental Eqs. (2.23) and
(2.24)(roots in left half plane) . .

2.4 Convergence
(loading

2.5 Convergence
(loading

3.1 Roots of transcendental Eq.

in right

3.2 Convergence
(loading

3.3 Convergence
(loading

of eigenfunction expansions
case (a)) . . . . . . . . .

of eigenfunction expansions
case (c)) . . . . . . . . .
half plane) . . . . . . . .

of eigenfunction expansions
case (a)) . . . . . . . . .

of eigenfunction expansions
case (b)) . . . . . . . . .

iv

(3.4l)(roots

Page

19

20

21

22

23

43

44

45

LIST OF FIGURES

FIGURE Page

2.1 Truncated semi-infinite two-dimensional
wedge C O O O O O I I O O O O O O O O O O 6

2.2 Wedge loading case (a) . . . . . . . . . . . '24
.2.3 Wedge loading case (c) . . . . . . . . . . . 25

2.4 Decay properties of Orr at 6 = 30° and 066

at 9 = 0° for loading case (a) . . . . . 26

2.5 Decay prOperties of Orr at 6 = 30° and 066

at 6 = 0° for loading case (0) . . . . . 27

3.1 Truncated semi-infinite three-dimensional
wedge O O O O O O O O O O O O O O O O O O 2 9

3.2 Cone loading case (a) . . . . . . . . . . . 46

3.3 Cone loading case (b) . . . . . . . . . . . 47

3.4 Decay prOperties of ORR at ¢ 60° and 0

at ¢ = 0° for loading case (a) . . . . . 48

3.5 Decay properties of ORR at o = 60° and 0

at ¢ = 0° for loading case (b) . . . . . 49

I. INTRODUCTION AND HISTORICAL

DEVELOPMENT

In 1853, Barre de Saint-Venant published his
"Memoire sur la Torsion des Prismes" [l] in which he solved
the problem of torsion in long prismatic bars of various
shapes of cross section. In a footnote he states that the
influence of forces in equilibrium acting on a small por—
tion of the surface of a body extends very little beyond
the parts upon which they act. This has been the basis for
the more familiar form of St. Venant's principle, the es-
sence of which can be stated as follows: If a system of
forces acting on a small portion of the surface of an elas-
tic body is replaced by another statically equivalent system
of forces acting on the same portion of the surface, the
same stress distribution and deformation are produced inside
the body except in the immediate neighborhood of the region
where the surface forces are applied. "Statically equiva-
lent systems" are those which have the same resultant force
and moment.

This principle is of great practical importance.
Often the exact distribution of boundary stresses is unknown
but the statically equivalent loading can be easily deter-
mined. For these cases, the problem may be solved with the

l

statically equivalent system of boundary stresses and from
St. Venant's principle, the solution can be taken as ac-
curate except in the vicinity of the loading.

On the other hand, if the boundary conditions are
specified according to the exact distribution of the
stresses, the problem may become too complicated to solve
mathematically. Frequently, by modifying the boundary con—
ditions slightly, the solution becomes possible and gives
essentially the same stress distribution in a large part
of the body as does the actual loading. By means of St.
Venant's principle, the solution of the problem may be sim-
plified by altering the boundary conditions as long as the
systems of applied forces are statically equivalent.

The principle agrees very well with reality as can
be illustrated by simple examples but its formal mathemat-
ical proof is rather difficult in the general case.

Early applications of St. Venant's principle to
justify approximations of boundary conditions include prob-
lems investigated by Thomson and Tait [2], Levy [3], Bous-
sinesq [4,5], and Clebsch [6].

For the elastic half space, bounds have been es-
tablished on the decay rates for stresses (rates at which
the stresses approach zero) by Boussinesq [5] and von Mises
[7]. Both investigations show that the stresses decay, as
they must, but that the decay rate is a function of the

type of loading applied on the surface of the body. von Mises

introduced the concept of astatic equilibrium which requires
surface forces to remain in equilibrium even when turned
through an arbitrary angle and prOposed a modification to
the principle. For a more detailed discussion of astatic
equilibrium, see Section 2.2. He concluded that static equi-
librium was not enough to insure the maximum decay rate since
astatic equilibrium may generate decay rates much faster than
simple static equilibrium. In 1954, E. Sternberg [8] pre-
sented a mathematical proof for the bounds on the decay rates
for the modified principle.
Recent research in this area generally follows one
of the two methods:
(a) that which attempts to establish bounds on the
width of the St. Venant boundary region (as done by von Mises),
(b) that which attempts to establish "exact" solutions
within this boundary region for limited classes of geometry.
The geometries investigated thus far include the semi—infinite
strip [9,10] and the semi-infinite circular cylinder [11-14].
The purpose of this research is to further the classes
of geometry for which the "exact" solution is known by deter-
mining the stress distribution in a semi—infinite two-dimen-
sional wedge and in a semi-infinite three-dimensional cone.
The cone solution should then approach that of the cylinder
as the cone angle approaches zero and should approach that of

the half space as the cone angle approaches n/2.

For the wedge, the solution was formulated using an
Airy stress function expressed in terms of a complex valued
eigenfunction expansion. All boundary conditions were taken
in terms of stresses applied on the surfaces.

For the cone, the solution was formulated in terms
of the Papkovich-Neuber functions with boundary conditions
again being taken in terms of stresses alone.

Previous investigations of various wedge and cone
problems are presented in [18] to [32].

The transcendental equations which will be developed
for the wedgelEqs. (2.23) and (2.24] agree with those developed
by Williams [41] in his investigation of stress singularities
resulting from extension of angular plates with free edges.
However, Williams presents only the real part of the minimum
root of these equations. Williams is interested in the be-
havior of the solution near r = 0 and is concerned with
bounded displacements at the origin, whereas the problem
outlined in Chapter II does not contain the point r = 0 but

is concerned with solutions which are bounded as r + w.

II. THE WEDGE PROBLEM

2.1. Formulation of the Wedge Problem

Consider the wedge shown in Fig. 2.1. Formulating
the problem in terms of an Airy stress function, w, the

stresses in polar coordinates can be expressed as:

_ 1 1

0rr - ? w,r + ;2 w,68 (2'1)

088 = w,rr (2'2)
_ 1 _ 1

Tre ‘ :7 w,e E w,r8 (2'3)

where w r denotes the partial derivative of U with respect

I
to r. Assuming plane stress conditions, the equilibrium
equations of elasticity are satisfied and the defining equa-

tion for w becomes

V V w = 0 (2.4)
where

2 32 2

1 1 a
V = +——+ .
8r r 3r :7 382

O)

The boundary conditions to be satisfied are:
Solution + 0 as r + w (2.5)

066(r' i B) = 0 (2.6)

Fig.

 

2.1.

 

Truncated

semi-infinite two-dimensional wedge.

Tre(r, _ B) = 0 (2.7)
o (r e) = 0° (6) (2 8)
rr 0' rr °
r (r e) = 1° (6) (2 9)
r0 0' r6 '

0

where ogr(6) and Tre(9) are the specified loading functions.

Assume the solution for w in Eq. (2.4) to be of the

form:

a
w = Z r nfn(8) + Cre sin 0 + Dre cos 6 (2.10)
n

where the last two terms are included in order to incorporate
that portion of the solution corresponding to an = 1. The
necessity of these terms will be more apparent after the next
few steps. Substituting the assumed expression for w into
the biharmonic equation yields the defining equation for

fn(9) as:

f:V(0) + [oi + (an - 2)2]f;'(0) + a§(an - 2)2 fn(6) = o

(2.11)
where the primes denote differentiation with respect to 6.

Taking the solution for fn(6) in the form

the characteristic equation for allowable values of m is:

4 [2 _ 2]2 2 _ 2_
m + an + (an 2) m + an(an 2) — 0 (2.12)

[m2 + 01:][m2 + (an - 2)2] = 0

Thus,

m = iia (2.13)
n

m = ii(an - 2) . (2.14)

Thus, the general solution for fn(0) can be written as:

I I I
fn(6) = An srn one + Bn cos anB + Cn s1n (an - 2)0
I
+ Dn cos (an - 2)0 . (2.15)
If an = l in Eq. (2.12) above, then repeated roots exist and

the solution corresponding to these roots satisfies all the
boundary conditions. These are the terms included in w as
Cr0 sin 0 and Dre cos 0.

In terms of the stress function assumed initially,

the non-zero stresses are:

OLn.-2 " 2 2 .
Orr = g r [onfn(0) + fn (8% + E'C cos 0 - f D s1n 0
(2.16)
an-Z
089 = g r [anmn - 1)fn(e)] (2.17)
an-Z ,
Ire =—g r [(dn - 1)fn(e)] . (2.18)
For stress free boundary conditions at 8 = :8:
“66' _ = 0 => £413) = o
0 — :8
I
Trel =0 => f,‘(:B)=0 .

6 = i8

Thus, the boundary conditions at 6 = :8 reduce to

specifying:

f(iB) = 0 (2.19)

n

fn(:B) = o . (2.20)

However, since any linear elasticity problem can be
solved by the method of superposition, separate fn(6) into
its even and odd parts in 6. This will facilitate the solu—
tion for the eigenvalues, an, and in general will make the

problem more tractable. From Eq. (2.15),

féo)(0) = A; sin ago)0 + c; sin‘aéo) - 2)0 (2.21)

fée)(e) = Bn cos aée)0 + D; cos (use) - 2)0 . (2.22)

Reference to an even problem implies one in which
the stress function, 0, and the Orr and 008 stresses are
even functions of 0 and reference to an odd problem implies
one in which these are odd functions of 6. The even problem
will be indicated with a superscript (e); the odd problem
will be indicated with a superscript (o).

The boundary conditions at 6:: B can then be ex-

pressed as:

fé°) (e) = o fée) (B) = o
for an odd for an even
, problem , problem
féO) (e) = o fée) (B) = o

10

Applying the above boundary conditions to the odd

problem yields the following transcendental equation for the

odd eigenvalues:

age) sin 28 - 2 sin (4:8) cosl}aéo) - 2)8] = 0 . (2.23)

For the even problem, the corresponding transcen-

dental equation for the even eigenvalues is:

aée) sin 28 + 2 cos (aée)8) sin [(aée) - 2)8] = 0 .

 

(2.24)
The eigenfunctions can then be expressed as:
. . (o) )
Sln — 2 8
féo) = An sin (an (o) - 2’0 - ‘% (0) sin a;0)0
sin on B
(2.25)

(e)
cos - 2 B
f(e) = Cn cos {an (e) - 2)9 - (an (e) ) cos crime .
cos an 8

 

(2.26)

Thus, for the odd problem, the stress expressions

become:
“(0)” () () ()"] 1
_ n o o o _ — .
orr - g AnY [an fn + fn 2BY Sln e (2.27)
(o)
_ o‘n '2 (0) am) (0)]
096 _ g A Y [a n - 1)fn (2.28)

a -2 '
Tre =‘E AnY n {(930) - 1)fé°)'] (2.29)

11

where y = r/r0 and féo) is the bracketed portion of Eq.

(2.25).
For the even problem, the stress expressions become:
“(e)’2[() () ()"J 1
_ n e e e -
Orr — E Cny an fn + fn + 20v cos 6
(2.30)
(e) . ‘
a -2
_ n (e) (e) __ (e)
Gee — :21 cm)! [on (an 1)::n ] (2.31)
(e)
a -2 '
__ n (e) _ (e)
Ire — r21cm) [‘an 1):?n I (2.32)

where, again Y = r/r0 and fée) is the bracketed portion of
Eq. (2.26).

The transcendental equations, Eqs. (2.23) and (2.24),
were solved on the CDC 3600 digital computer using a Newton-
Raphson iteration technique in the complex plane since the
eigenvalues will be complex numbers. However, the Newton-
Raphson technique is quite sensitive to the initial guess
for the root of the equation. Care must be taken so that
the initial guess is in the neighborhood of the particular
root being sought or roots may be skipped; i.e., the method
may converge to a root other than the one being sought. To
avoid this skipping of roots, asymptotic expressions for the
roots were developed and these asymptotic values were used
as the initial guess for each eigenvalue. For a detailed
discussion of the development of these asymptotic values,

see Appendix A.

12

For this specific wedge problem, only those eigen-
values which have negative real parts will be used to insure
the solution goes to zero for large r.

For a number b = c + id which is a solution to the
transcendental equation, note that its complex conjugate
B = c - id is also a solution. This is the condition which
must exist if the stresses, which are, of course, real quan—
tities, are to be expressed in terms of complex eigenfunction
expansions.

Having now satisfied the boundary conditions at in-
finity and at the wedge angle, 8, the remaining boundary
conditions at r = r0 will be satisfied.

Using a generalized approach to orthogonality as
outlined by P. F. Papkovich [l7], orthogonality conditions
were established for the eigenfunctions. For details of
the method as applied to this particular problem, see Appen-
dix B. The orthogonality condition for the eigenfunctions

is:

8 II II II
j:8|:am(am - 2)an(an - 2)fmfn + 4fmfn - fm fn ]d6 = O .

(m # n) . (2.33)
However, it was not possible to interpret this condition
physically in terms of stress, displacement, or mixed
boundary conditions on the end of the wedge.

Therefore, the last boundary conditions, those at

r = r0, were satisfied numerically by truncating the series

13

expressions and determining the constants in a least squares

sense.

On the end boundary, for the even problem, consider

0 O

the expansion of the real functions Orr and Tre in terms of

the complex eigenfunctions. Take

"
oo = Z A e + Z A e + 20 cos a (2.34)
rr nn nn
n n
o '—
rre — Z Anwn + Z Anwn (2.35)
n n
where
II
¢ = a f + f
n nn n
I
Wn — -(an - 1)fn

(e)

and where an refers to an and fn refers to fée). The two

conditions, Eqs. (2.34) and (2.35), are then sufficient to
' —
imply An = An. Thus, the boundary stresses can be written

as

crr = Z Anon + Z Anon + 20 cos 9 (2.36)
n n

T0 = Z A w + X X‘F (2 37)

r6 n n n n n n '

Satisfying both the specified stresses in a least

squares sense by using:

N N
00 + 110 = Z A (e + 1w ) + 2 A'($ + 1? ) + 2D cos 9
rr r0 n n n n n n n n

(2.38)

14

leads to minimizing the integral:

0 N N -| — — -o
r g - Z An(<1>n + iwn) - Z An(<1>n + iwn) - 20 cos 6 g
n n

N _’ _ _ N
- 2 A (e - 1w ) - 2 A (e - iv ) - 20 cos 6 dr = minimum

n n n n n n n n

(2.39)

where g0 = 00 + 1T0 Note that the integrand is the

rr re'
modulus squared of the error in the series approximations.

After minimizing the integral with respect to the

jth constant, the following equations were developed:

8 N 5 N
e ‘; “ ‘1‘ ' “
1:8 E An( j n + w w ) + g An(¢J¢n + ijn)

_ B _ _
+2D<I>. cosede=%f [(90+g—0)<I>. -i(go-§O)‘¥-de
3 '8 J J

(j = 1,2,...,N) (2.40)

B n n n j n j n
1 B O —0 O —O
+ 2D¢. COS 6 d6 = — (g + g )T. - i(g — g )W. d6
3 2 _B J J
(j = 1,2,...,N) (2.41)
8 . N N _
2 cos 6 Z A 0 + 2 cos 6 Z A 5
-8 n n n n n n
B
+ 4D cos2 9 d6 = I? (go + 3°) cos 6 d6 . (2.42)

The last equation is determined by minimizing Eq. (2.39)

with respect to D.

15

In matrix form these equations generate a Hermitian
matrix; i.e., Aji = Kij‘ The maximum number of terms, N, was
chosen so that the series representations for the specified
boundary stresses converged within some e error term. For
all loading cases which were used, N = 15 yielded a < 0.5%.
For all loading cases, the integrals were evaluated

analytically.

The same method is applicable to the odd problem.

2.2. Results and Conclusions
(Wedge Problem)

 

The roots of the transcendental equations, Eqs.
(2.23) and (2.24), were determined by the Newton-Raphson
method as discussed briefly in the previous section. The
results are shown for several wedge angles in Tables (2.1)-
(2.3) .

The system of equations generated by Eqs. (2.40)-

(2.42) was solved for the following loading cases:

 

00 T0 Principal Decay
rr r0 for Stresses
a -2
(a) A + B02 0 r 1
n0 3 2 0‘1"2
(b) 0 A sin '—8— + 3(9 — B 6) r
(c) A 3(93 4 826) r-1
(d) 1.0 0 r"1

{8) since these are even problems.

where a1 refers to 0

Loading cases (a) and (c) are shown in Figs. (2.2) and (2.3),

16

respectively. The constants A and B were chosen so the load-

ing system itself would be in static equilibrium where the

required conditions for static equilibrium are
if

W? X F") = 0 ; (2.44)

II
o

(2.43)

i.e., the resultant force and the resultant moment must be
zero.

In 1945, von Mises [7] introduced a stronger condi-
tion of equilibrium denoted as astatic equilibrium and de-

fined by the expressions
ZF = o (2.45)
[If = o . (2.46)

Note that astatic equilibrium implies static equilibrium but

that the converse is not true.

Extending the above definition to include the distri-

buted forces on the end boundary used for this particular

wedge problem, the conditions for astatic equilibrium become:

 

 

B70 0 _
{F = 0: j, Lorr cos - r sin 0 d6 = 0 (2.47)
-B
B—
Jls Logr sin cos 0 d8 (2.48)
__ Bo
ZrF‘= 0:f-B Orr d0 = (2.49)
8o
J. T d6 = (2.50)
-8 r0

17

B
0 O . _
j;8[orr cos 0 - Tre Sln 0] cos 0 d8 — 0 (2.51)
B o o
0 cos 0 - T sin 0 sin 0 d6 = 0 . (2.52)
-8 rr r8

von Mises shows that for the half plane, loadings in astatic
equilibrium may generate faster decay rates for stresses
than those loadings which are in simple static equilibrium.

However, the conditions of astatic equilibrium can-
not be used to justify the faster decay rates in cases (a)
and (b) since neither satisfies the required conditions. As
was expected, case (a) did yield a faster decay rate than
case (d) since (a) is in static equilibrium. The interac-
tion of Ogr and T39 for case (c) results in a slower decay
rate than for cases (a) or (b).

Case (c) presents another particularly interesting
result. Within the range r0 < r < 1.4r0,
of stress at 0 = 8 increases to a value approximately 500%

the orr component

its corresponding value on the boundary before it begins to
decay. This can be seen in Fig. (2.5). Physically, this is
fairly easy to justify since the large shear stress on the
boundary near 0 = 8 results in large radial stresses in the
vicinity of the corner at 0 = 8. For all the loading func-
tions used, 8 was taken as 30°.

Tables (2.4) and (2.5) show how well the truncated

series represent the specified loading functions for different

18

values of N where N represents the number of pairs of eigen-
values used in the truncation.

Decay properties of the stresses are shown in
Figures (2.4) and (2.5).

The solutions of Equations (2.23) and (2.24) which
lie in the right half plane can be obtained from the left

half solutions by the relations:

(even-right) = -a (even—left)

+ 2
n n

(odd-right) = -a (odd-left)

+ 2
n n

For the stresses to be bounded at infinity, the
real part of on < 2. Therefore, no eigenvalues in the

right half plane enter into the series summations.

Table 2.1--Roots of transcendental Eqs.
(roots in left half plane).

19

(2.23) and (2.24)

 

 

 

 

 

n age) ago)
8 = 10°
1 - 11.0795 -i 6.3844 - 20.4864 -i 7.8711
2 29.6943 8.8302 38.8245 9.5441
3 47.9149 10.1141 56.9819 10.5889
4 66.0338 10.9959 75.0753 11.3522
5 84.1094 11.6690 93.1380 11.9542
6 102.1623 12.2135 111.1832 12.4514
7 120.2015 12.6710 129.2176 12.8750
8 138.2319 13.0654 147.2447 13.2439
9 156.2562 13.4120 165.2667 13.5708
10 174.2762 13.7212 183.2849 13.8642
8 - 20°
1 - 5.0578 -i 3.0954 - 9.7541 -i 3.8431
2 14.3550 4.3241 18.9184 4.6817
3 23.4625 4.9670 27.9952 5.2047
4 32.5206 5.4083 37.0410 5.5865
5 41.5577 5.7450 46.0717 5.8877
6 50.5836 6.0174 55.0939 6.1364
7 59.6028 6.2462 64.1107 6.3482
8 68.6178 6.4434 73.1241 6.5327
9 77.6297 6.6168 82.1349 6.6962
10 86.6395 6.7714 91.1438 6.8429
Asymptotic expressions are given by Eqs. (A.18), (A.19),

(A.22),

(A.23).

Table 2.2—-Roots of transcendental Eqs.
(roots in left half plane).

20

(2.23) and (2.24)

 

 

 

 

 

n aée) ago)
8 - 30°
1 - 3.0593 —i 1.9520 - 6.1820 -i 2.4557
2 9.2457 2.7780 12.2860 3.0171
3 15.3142 3.2078 18.3351 3.3665
4 21.3514 3.5024 24.3644 3.6213
5 27.3752 3.7271 30.3842 3.8222
6 33.3919 3.9088 36.3985 3.9881
7 39.4043 4.0614 42.4094 4.1294
8 45.4139 4.1929 48.4180 4.2525
9 51.4217 4.3085 54.4250 4.3615
10 57.4280 4.4116 60.4308 4.4593
8 - 45°
1 - 1.7396 -i 1.1190 - 3.8083 —i 1.4639
2 5.8451 1.6816 7.8688 1.8424
3 9.8856 1.9702 11.8981 2.0764
4 13.9079 2.1673 15.9158 2.2468
5 17.9223 2.3175 19.9278 2.3810
6 21.9325 2.4388 23.9365 2.4918
7 25.9401 2.5407 27.9432 2.5861
8 29.9460 2.6284 31.9485 2.6682
9 33.9508 2.7056 35.9528 2.7409
10 37.9547 2.7743 39.9564 2.8061
Asymptotic expressions are given by Eqs. (A.18), (A.19),

(A.22),

(A.23).

21

Table 2.3—-Roots of transcendental Eqs. (2.23) and (2.24)

(roots in left half plane).

 

 

(e) (o)

 

 

 

(A.22) ,

(A.23).

n n n
B - 60°
1 - 1.0941 -i 0.6046 - 2.6307 -i 0.8812
2 4.1517 1.0493 5.6657 1.1720
3 7.1758 1.2690 8.6834 1.3493
4 10.1895 1.4179 11.6944 1.4779
5 13.1985 1.5311 14.7020 1.5789
6 16.2050 1.6223 17.7076 1.6622
7 19.2099 1.6989 20.7119 1.7330
8 22.2137 1.7649 23.7154 1.7947
9 25.2168 1.8228 26.7182 1.8493
10 28.2194 1.8744 29.7205 1.8983
8 - 75°
1 - 0.9130 -i 0.0000 - 1.9367 -i 0.3637
2 3.1455 0.5232 4.3518 0.6299
3 5.5567 0.7117 6.7605 0.7783
4 7.9636 0.8347 9.1662 0.8836
5 10.3684 0.9268 11.5703 0.9656
6 12.7720 1.0008 13.9735 1.0329
7 15.1748 1.0626 16.3759 1.0901
8 17.5770 1.1157 18.7779 1.1397
9 19.9788 1.1623 21.1796 1.1836
10 22.3803 1.2037 23.5809 1.2229
Asymptotic expressions are given by Eqs. (A.18), (A.19),

22

Table 2.4--Convergence of eigenfunction expansions (loading
case (a)).

 

 

 

 

 

 

 

 

 

8 = 30°
Specified Function No. of Paired Eigenvalues
0 _ 2

e Orr ' A + Be N = 6 N = 10 N = 15

0° 1.0000 0.9994 1.0006 1.0000

5° 0.9134 0.9143 0.9136 0.9133
10° 0.6537 0.6522 0.6534 0.6538
15° 0.2208 0.2230 0.2209 0.2209
20° -0.3853 -0.3876 -0.3846 -0.3854
25° -1.1646 -1.1646 -1.1652 —l.l647
30° -2.1170 -2.1341 -2.1215 -2.1170

Specified Function No. of Paired Eigenvalues
0 _

9 Tre ' 0 N = 6 N = 10 N = 15

0° 0.0000 0 0 0

5° -2><10‘4 2x10‘4 1><10‘4
10° 6x10"5 —3><10’4 -4><10‘5
15° 7><10'4 -8><10"4 5x10”5
20° -3><10‘3 -7><10‘5 -3><10"4
25° 5x10'3 6><10'5 -2><10’4
30° 0 0 0
B = _ A sin 8

28 cos 8 + (82 - 2) sin 8
1.0

W
ll

23

Table 2.5--Convergence of eigenfunction expansions (loading
case (c)).

 

 

 

 

 

 

 

8 = 30°
Specified Function No. of Paired Eigenvalues

0 Ogr = A N = 6 N = 10 N = 15

0° 1.0000 0.9845 0.9938 0.9982

5° 1.0107 0.9993 1.0009
10° 1.0035 1.0072 0.9995
15° 0.9732 1.0033 0.9986
20° 1.0549 0.9925 1.0043
25° 0.9483 1.0076 1.0017
30° 1.0646 0.9733 0.9582

Specified Function No. of Paired Eigenvalues

0 Tie = 3‘93 ' 829’ N = 6 N = 10 N = 15

0° 0 0 0 0

5° 2.2603 2.2745 2.2551 2.2588
10° 4.1332 4.1093 4.1336 4.1355
15° 5.2310 5.2552 5.2408 5.2308
20° 5.1665 5.1685 5.1690 5.1693
25° 3.5519 3.4850 3.5407 3.5529
30° 0 0 0 0
B = A sin 8

2(82 - 3) sin 8 + 68 cos 8 ’

A = 1.0

24

 

 

 

Figure 2.2. Wedge loading case (a).

25

 

 

 

o T0
r’ r0
4) .——-\
// \\
5°01 / \
/ \
/
/
/
4.0.. /
/ \
/ \
/' 0 \
rr
3.0" // 0 \
————T
/ re
/
/ 0
2.0" / Orr - A
0 _ 3_ 2
/ Ire B(0 8 0)
/
l. f
/
I/
\
I . \‘ e
0 5° 10° 15° 20° 25° 30°:
Fig. 2.3. Wedge loading case (c).'

26

a
II

) 0 at 30°

‘ —— rr

0.54

r/r

 

-1.01.

-l.5+-

 

 

Fig. 2.4. Decay properties of Orr at 0 = 30° and

Gee at 0 = 0° for loading case (a).

at 9 = 30°

00

 

-15.

 

Fig. 2.

5.

 

Decay properties of Orr at 0 = 30° and

at e = 0° for loading case (0).

O

86

III. THE CONE PROBLEM

3.1. Formulation of the
Cone Problem

Consider the cone shown in Fig. 3.1. The boundary

conditions to be specified are:

Solution + 0 as R + w (3.1)
o¢¢(R,B) = 0 (3.2)
TR¢(R,B) - 0 (3.3)
0RR(R0,¢) = ogR(¢) (3.4)

TR¢(RO,¢) = rg¢(¢) (3.5)

where 03R(¢) and Tg¢(¢) are the specified loading functions.
The problem will be solved using Papkovich-Neuber
functions. The displacement field may be expressed in the

form [15]

H = E + Vx (3.6)

where E is a vector function to be determined and x is a
scalar function to also be determined. Substituting Eq.

(3.6) into Navier's equation, it can be shown that

1 _

x=‘4(1- 0) (R°B+B (3.7)

 

O)

28

29

 

 

 

Fig. 3.1. Truncated semi-infinite three-dimensional cone.

30

where E and B0 are harmonic functions. R is the position

vector .

Thus, the displacement equation becomes

_ 1
4(1- v1

V(RB + B

 

u = E (3.8)

0)
where
2— 2

V B = 0 and V B0 = 0 .

Formulating the problem for the cone in terms of the
spherical coordinates R, 0, 0 it is necessary to determine
the functions E and B0 so that

v B = 0 (3.9)
and

V B = 0 (3.10)

where, for an axisymmetric problem; i.e., no 0 dependence,

 

V2 = :§-§%(R2fi 3 1+ —7——————§$(sin ¢ §¢) (3.11)
s1n

E = eR 8R(R 6) + e¢B¢(R,¢) (3.12)

B0 = B0(R,¢) (3.13)

F = Rafi . (3.14)

The non-zero displacement components for the cone
become

_ _ 1 8
uR - BR 471—3—3) Ffi (Bo + RBR) (3'15)

u =B — .142—
¢ ¢ 3(1 - v) R 80

It should be noted that neither BR
function. However, it is possible

ponents of E such that

2 _
V (Bz) - 0
and

V2(Bpeie) = 0

where Bp and B2 are the components
ordinates. See Appendix C for the

equations. Then the components of

(80 + RBR) . (3.16)

nor B is a harmonic

¢
[15] to find the com-

(3.17)

(3.18)

of E’in cylindrical co-
solution of these

E in spherical coordi-

nates can be determined by using the transformation equations:

B

R Bp s1n ¢ + B2

B4»

Bp cos 0 - Bz

cos 0 (3.19)

sin 0 . (3.20)

For a solution which approaches zero as R + w Eqs.

(3.17) and (3.18) yield

-Ol.n-1 dpan (u)

130 = E AnR ——a$——— (3.21)
. -an-l
32 = g AnR Pan(u) (3.22)
where
u = cos 0 (3.23)

32

 

(-a ) (a + l) k
p (u) = n k “I k (1 g“) (3.24)
0‘n k=0 (k!)
(y)k = Y(Y + 1)(y + 2)...(Y + k - l) , k 2 l (3.25)
(7)0 = 1 . (3.26)

Equation (3.24) is one of the possible hypergeometric series
representations for Legendre functions [16] and is conver-
gent for

ll - pl < 2 . (3.27)

For the cone problem, the range of u will be 0 < u s 1.
Equations (3.19) and (3.20) give the components of

E in spherical coordinates as:

-a -1

— n 2 ' ' 3 28
BR — g R An(u - 1)Pa + Anupafi ( . )

n
X -an-l ¢[ ' ' ] (3 29)
B =- R sin A uP + A P .
0 n n an n on

where

It is now desired to findB0 so that

v B = 0 (3.30)

and so that E and B0 both contribute the same power of R

to the displacements. This will then allow the boundary con-

ditions at 0 = 8 to be satisfied in a tractable manner.

33

If B is of the form

0

’0.
_ n
BO — Z BnR Pa _1(u) , (3.31)
n n

these conditions are satisfied.

Thus, the displacement expressions for the cone

become
—dn-l ,

uR = Z R Anon + An)(l + kdn)uPa

n n

_ gn[(1 + kan)An - an]Pdn-l} (3.32)
11¢ = E R [(1 + kan)An - an - k(Anan + An) T

I
”(Anan + An)[1 - k(0(n + 1)]Pan s1n 0}) (3.33)

where k = 4(Ier5) . However, without any loss of gen-
erality, the two equations above can be expressed in terms

of the two arbitrary complex constants CA and D; as

-d. '1 I I
— n -
uR - Z R <Cn(l + kan)uPan anDnPan-L} (3.34)

dP

~an-l , , an-l
11¢ = R {[Dn'- kCn:\ T

- Cg[1 - k(0(n + 1)]P0‘n sin 0} (3.35)

34

where

C = A a + A
n n

D
II

(1 + ka )A - kB
n n n

As will be shown later, this is not the complete solution
for the displacements.

Applying the strain-disPlacement relations and the
constitutive equations, the non-zero stresses can be shown

to be:

-a -2
_ n _ 2 _ ]
ORR - Xv [ [on + San + 2(2 0) uPa + ZvanPan_l]kCn

n n
-2 _ 2(2 - v)
+ an(dn + 1)Pan_1Dn} + COY [l + COS B W COS¢:|
(3.36)
2 -dn-2 [
0 = Y (1 - 2v)(l + 20 )uP + 20a P _
00 n n on n on 1

I I
+ pP _ kC - [a P _ + pP _ ]D
an 1] n n an 1 an 1 n

 

-2 cos 0 - cos B _

+ Coy [ 1 + cos ¢ + cos ¢ 1] (3.37)
-an-2 2

o¢¢ = E3, “on — an + (1 - 20)]upan + 2(1 - v)onpan_l

I 2 I
- uP _ kC .- [a P _ - uP _ ]D
an 1] n n an 1 an 1 n}

 

-2 cos ¢ - cos 8
+ C0Y [ 1 + cos 6 ] (3'38)

35

n

—0Ln-2 2
TR¢ = g y sin 0 -[[(0n + l) - 2(1 - 0)]Pa

+ [(Otn + 1) + 2(1 - 0)] Pan_l]kCn + (an + 1)Pan-1Dn}

 

-2 . cos 0 - cos B
+ Coy s1n ¢[ 1 + cos ¢ ] (3.39)

where Y = R/RO. The C0 term is included in order to in-
corporate that portion of the solution corresponding to
on = 0. That portion of the solution was determined in-

dependently using the Navier equations. The necessity of

this term will be apparent after the next few steps.

The boundary conditions at 0 = 8 are:
0 = 0
.1)“.
T = 0
R¢|¢= 8

for all R 2 R0; i.e., for y 3 1. Applying these conditions
to Eqs. (3.38) and (3.39) results in the transcendental
equation for the determination of allowable an. After much
simplification, the transcendental equation can be written

as:

36

2

an uo[2k(ug - l)d§ + 2k(u3 - l)dn + pg]P:n(uo)

+ pop2k(ug - l)a: + 2k(ug - 1)on + 1]p§n_l(uo)

D

- 2kL2ug(ug - 1’0:

 

2 2
+ (“0 - 1)(3u0 - l)an
+ [U4 + 2(1 - 2V)H2 + l] P (n )P (u ) = 0 (3 40)
0 0 a 0 a -1 0 °
n n
where “0 = cos 8. Thus, it is obvious thatcx.n = 0 is a

solution to this equation and must be included for a com-

plete solution. For an # 0, the transcendental equation is

u. [463 - 183+ 463 - m. + shim

+

Ll0 n

 

“)3

F2k(ug - 1)ai + 2k(ug - l)a a _l(uo)
. n

 

- 244303,- 1).; + (.3 - 1168 4a.

+ [u3 + 2(1 - 2V)“; + 1]]Pa (00)P

(u)=0 .
n 0

dn-l
_(3.41)
The above equation was solved on the CDC 3600 digital
computer by calculating the value of the equation at points
of a grid system in the complex plane and plotting curves
along which either the real or imaginary part of the equa-
tion was zero. The intersection points of the curves then
correspond to the roots of the equation. Using this inter-

section point of the kth set of curves as an initial guess

37

for the kth eigenvalue, the modulus of the equation was

determined at that point and at the four neighboring points

H-

a h

n

H-

a ih
n

where h is the spacing of a new grid. The smallest modulus
of the five calculated was then taken as the improved solu-
tion and the process repeated until the center point rep-
resented the smallest modulus. Then h was reduced by a
factor of ten and the process repeated until the modulus
was less than 81. In all the numerical work, 81 = 10_5.

As in the wedge problem, for a number b = c + id
which is a solution to the transcendental equation, its
complex conjugate 3’: c - id is also a solution.

Having satisfied the boundary conditions at 0 = B,

the non-zero stresses can be expressed as:

 

-a —2
_ n 2
ORR — X CnY {c [on + San + 2(2 - vflkuPa + ZkvanPa -1
n n n
—2 _ 2(2 - v)
+ wndnmn + l)PC1n-l + COY [l + C03 8 W C05 (1)]
(3.42)
-dn-2 .
066 = Z Cny (l - 20)(1 + 20n)kuPa + 2kvanPa -1+ kuPa _1
n n n n
' -2 cos ¢ - cos 8
wn OLnPOLn-l + uPan-1) + C0Y [ l + cos ¢ + COS¢ l

 

(3.43)

38

 

 

-a -2
o¢¢ — X Cny [an - on + (1 Zvflikupa +-2k(1 v)anPa -1
n np n
' 2 ' -2 cos 0 — cos B
- kuP _ - w (a P _ - uP _ ) + C Y [ 1
an 1 n n an 1 an 1 O 1 + cos 0
(3.44)
-dn—2 2 ]
TR¢ = X CnY Sln ¢ - [(an + l) - 2(1 - V) kPa
n n
I I
- [0%14-1J + 2(1 - vikPa -1 + wn(an + 1)Pa -1
n n
-2 . cos 0 — cos B]
+ COY 51n ¢ [ 1 + cos ¢ (3.45)

where

__[0%{+1)2-2(1-v)]kPan(u0)-+[(an+'l)+-2(l-»v)]kP;n_l(po)

w-
n

 

(an + 1)Pan_l(uo)

and ak is the kth root of Eq. (3.41). The C0

sent the portion of the stress field contributed by on = 0.

terms repre-

0

The displacement components corresponding to on
which must be added to Eqs. (3.34) and (3.35) for the com-

plete displacement field are:

 

c .
_0(l—2\)) _4(1-v)
11R - E—[W (l + C05 8) W C08 (1)] (3.46)
_ C0 _ (1 - 20) (1 + 8) sin 9 + . ¢
“0 ' fi‘ - v C05 1 + cos 6 51“ (3.47)

The last boundary conditions, those at R = R0, will

now be satisfied. These conditions had to be satisfied

39

numerically as was the case for the wedge problem. The
method of least squares was again used to determine the
constants for the truncated series.

Using a similar argument as in the wedge problem,

the stresses on the end of the cone can be expressed as

 

 

ORR = Z cnen + cn n + con (3.48)
n n
0 _ _ _
TR¢ — Z cnwn + Z ann + cox (3.49)
n n
where
2
0n — -[0n + San + 2(2 - 0)]kuPa + 2kvanPa -1
n n
+ wnan(an + 1)Pa -1
n
’ 2
‘1’ =- (0!. +1) -2(1-\)):|kP sind)
n n a
L n
I
- gun + 1) + 2(1 - 0)]kPan_l s1n 0
I
+ wn(an + 1)Pa -1 s1n ¢
n
_ _2(2-v)
n - l + cos 8 1 _ 20) cos 0
A = sin ¢(cos 0 - cos 8)
(l + cos 0) °

Following the same procedure as outlined in Section
2.1 for the wedge problem, the following equations were

develoPed for the determination of the constants:

40

- 1 B 0 —0 — 0 ‘
+ W.1) sin 0 d¢ = --J[ (g + g )9.- i(g - g )V. sin¢>d¢
J 2 0 J J ,
(j = 1,2,...,N) (3.50)
0 E cn(<1>j<1>n + ijn) + E on (ej Eh + ijn) + c0(¢jn

B
+ 113.1) sin (1) do = 1'- f [(90 + EOHDj - i(go - EOHIj] sincb dcb
0

N

(j = 1,2,...,N) (3.51)

B N N - - 2 2
.Ih 2 Cn(n¢n + an) + g Cn(n¢n+-1Vh) + C0(n + A ) sin ¢ d¢

n
B 0 0 0 0
=%J’ [(9 + 9 )n ‘ i(g - E )A] sin ¢ d¢ (3.52)
0
where
0 _ 0 0

In matrix form, these equations also generate a Her-
mitian matrix. Again, as for the wedge, N was chosen so
that the series representations for the Specified loading
functions converged within some 82 error term. For the

loading cases which were used, N = 5 yielded 5 < 16% where

2
the maximum error occurred only in the neighborhood of
¢ = B. The integration required for each matrix element

was performed numerically using the Newton-Cotes method.

41

Because of the lengthy computer time required for
N > 5, the solution was not determined for larger values
of N. Therefore, the convergence near 0 = 8 for the cone
as shown in Tables (3.2) and (3.3) is not quite as accu-
rate as one may desire. However, over 70% of the boundary,
the agreement was within 5%.

3.2 Results and Conclusions
(Cone Problem)

The roots of the transcendental equation, Eq. (3.41),
were determined by the method as outlined in the previous
section. The results for several cone angles are shown in
Table 3.1.

The system of equations generated by Eqs. (3.50)-

(3.52) was solved for the following loading cases:

 

00 T0 Principal Decay
RR R9 for Stresses
-a -2
(a) 1.0 + A62 + B93 0 R 1
-d -2
(b) A B(¢3 - 82¢) R 1
(c) 1.0 0 R"2

Loading cases (a) and (b) are shown in Figs. (3.2) and (3.3),
respectively. The constants A and B were chosen so the load-
ing system would be in static equilibrium.

For this axisymmetric cone, the conditions for astatic

equilibrium become

42
B
2F = 0 : Jr [pchos ¢ - 13¢ sin 0] sin 0 do = 0 (3.53)
0

8
25:0: f ogRsin¢d¢=0
0

8
.j; [ORR cos 0 - 13¢ sin 0] sin 0 cos 0 d¢== 0 .
(3.54)

Similar to the results of the wedge problem, loading
case (a) did yield a faster decay rate than case (c) since
(a) is in static equilibrium. However, for the cone, case
(b) yields a faster decay than for the similar loading
function on the wedge. None of the loading cases is in
astatic equilibrium.

Similar to the decay in the wedge, loading case (b)
for the cone results in an increase of the ORR component of
stress within a small region before it begins to decay.
Again, the interpretation of this result is quite similar
to that of the wedge.

Tables (3.2) and (3.3) show how well the truncated
series represent the specified loading functions.

Decay properties of the stresses are shown in Figs.

(3.4) and (3.5).

43

Table 3.1--Roots of transcendental Eq. (3.41)(roots in
right half plane).

 

 

 

n (1n n (In

B = 15°: 8 = 30°:

1 9.9170 + i 5.0850 1 4.7409 + 1 2.3589

2 22.6618 6.1604 2 11.1025 2.9327

3 34.9063 6.8925 3 17.2181 3.3041

4 47.0343 7.4239 4 23.2783 3.5723
5 29.3163 3.7814

8 = 45°: 8 = 60°:

1 3.0371 + 1 1.3520 1 2.2189 + 1 0.7425

2 7.2610 1.7771 2 5.3518 1.1135

3 11.3298 2.0309 3 8.3933 1.3111

4 15.3657 : 2.2118 4 11.4151 1.4493

5 19.3883 2.3525 5 14.4289 1.5560

8 = 75°:

1 1.7530 + 1 0.0460

2 4.2211 0.5707

3 6.6413 0.7429

4 9.0523 0.8582

5 11.4593 0.9457

 

44

Table 3.2—-Convergence of eigenfunction expansions
(loading case (a)).

 

 

 

 

 

 

 

8 = 60°
77 3 No. of Paired
Specified Function Eigenvalues
¢ Og¢=l.0+A¢2+B¢3 N=5
0° 1.0000 0.9033
10° 0.8418 0.7914
20° 0.4716 0.4597
30° 0.0458 0.0681
40° -0.2791 -0.2654
50° -0.3465 -0.3412
60° 0 -2x10'2
No. of Paired
Specified Function Eigenvalues

¢ 13¢ = 0 N = 5

0° 0 0
10° 4><10‘2
20° 5X10"2
30° 2><10‘2
40° -1x10"3
50° -1><10’2
60° 0

 

483 - 3(282 - 1) sin 28 - 68 cos 28
284 sin 28 + 483 cos 28 - 382 sin 28 + 283

B _ 282 - 28 sin 28 + l - cos 28

-84 sin 28 - 283 cos 28 + 1.582 sin 28 - 83

45

Table 3.3--Convergence of eigenfunction expansions
(loading case (b))

 

 

 

 

 

 

 

 

8=60°
No. of Paired
Specified Function Eigenvalues
4’ °g¢=A N=5
0° 1.0000 1.1230
10° 1.1033
20° 1.0478
30° 0.9415
40° 0.9686
50° 0.9953
60° 1.1621
No. of Paired
Specified Function Eigenvalues
9 13¢ = 8(43 - 82¢) N = 5
0° 0 0
10° 0.7627 0.7125
20° 1.3947 1.2788
30° 1.7652 1.6995
40° 1.7434 1.7588
50° 1.1986 1.2062
60° 0 0
B = - 8A sin 28

 

(284 + 282 + 3) - 68 sin 28 + (482 - 3) cos 28

A = 1.0

46

 

 

10° 20° 3 ° 40°

-0.50
0 2 3
O =
RR 1.0 + A9 + B¢>
0

-1. 1b =

0 TR¢ 0

 

Fig. 3.2. Cone loading case (a).

47

 

 

 

00 I0
I R
RR 6 0
I RR
2.0.)- -_-_ 0
Rd)
//’—\\\
/ \
// \
/ \
‘/ \
/ \
/ \
1.0 ,1 a
/
/
/ \
’ \
- / o _
/ ORR ’ A \
/
/ 13¢ = B(¢3 - 82¢) \
/
/ \
0 10° 20° 30° 40° 50° 60°

Fig. 3.3. Cone loading case (b).

48

 

RR' (M)
0.4
|
0.3..)
\
\
\ O _ o
\ RR at CD - 60
0.2..\
\ ____ OM) at (1’ = 0°
\
\
\
0.11
\
\
\
\\
0 1.0 \\‘~1___23_..(.)___.___——?i10 'R/RO
-0.1.
03R _1.0 + A¢2 + Bd>3
0 _
TR<1> “'0
-0.2..

 

Fig. 3.4. Decay properties of ORR at 0 = 60° and

0¢¢ at ¢ = 0° for loading case (a).

49

at ¢

0' 0
RR 60

c at 0 0°

¢¢

 

 

 

 

3.0 0
/ 0 _
/ ORR - A
I, )rg¢ = 3(43 - 82¢)
I i
-2.0«I
I
I
I
-3.0.1
I
J
-4.0 _

 

Fig. 3.5. Decay properties of ORR at o = 60° and

o¢¢ at ¢ = 0° for loading caSe (b).

10.

BIBLIOGRAPHY

Todhunter, I. and Pearson, K., A History of the Theory
of Elasticity and of the Strength of Materials,
Vol. 2, pt. 1, Cambridge Univ. Press, London,
1893, Chapter 10.

 

Todhunter, I. and Pearson,K., A History of the Theory
of Elasticity and of the Strength of Materials,
Vol. 2, pt. 2, Cambridge Univ. Press, London,
1893, pp. 401-416.

 

 

Todhunter, I. and Pearson,K., A History of the Theory
of Elasticity and of the Strength of Materials,
Vol. 2, pt. 2, Cambridge Univ. Press, London,
1893, pp. 206-207.

 

 

Todhunter, I. and Pearson, K., A History of the Theory
of Elasticity and of the Strength of Materials,
Vol. 2, pt. 2, Cambridge Univ. Press, London,
1893, Chapter 13.

 

 

Fung, Y. C., Foundations of SoligMechanics, Prentice—
Hall, Englewood Cliffs, N.J., 1965, pp. 301-309.

 

Todhunter, I. and Pearson, K., A History of the Theory
of Elasticity and of the Strength of Materials,
Vol. 2, pt.*2, Cambridge Univ. Press, London,
1893, pp. 154-161.

 

 

von Mises, R., "On Saint-Venant's Principle," Bull.
Amer. Math. Soc., vol. 51, 1945, pp. 555-562.

 

Sternberg, E., "On Saint-Venant's Principle," Quart.
Appl. Math., vol. 11, 1954, pp. 393-402.

 

Johnson, M. W., Jr. and Little, R. W., "The Semi—
Infinite Elastic Strip," Quart. Appl. Math.,
vol. 23, 1965, pp. 335-344.

 

Papkovich, P. F., "On One Form of Solution of the Plane
Problem of the Theory of Elasticity for the
Rectangular Strip," Dokl. Akad. Nauk.SSSR, vol.
27, 1940.

 

50

 

 

 

ll.

12.

l3.

14.

15.

l6.

17.

18.

19.

20.

21.

22.

51

Little, R. Wm. and Childs, S. B., "Elastostatic
Boundary Region Problem in Solid Cylinders,"
Quart. Appl. Math., vol. 25, Oct. 1967.

 

Horvay, G. and Mirabal, J. A., "The End Problem of
Cylinders," J. Appl. Mech., vol. 25, 1958,
pp. 561-570.

 

Hodgkins, W. R., "A Numerical Solution of the End
Deformation Problem of a Cylinder," U.K. Atomic
Energy Authority TRG Report 294.

Warren, W. E., Roark, A. L., and Bickford, W. B.,
"End Effect in Semi-Infinite Transversely Iso-
tropic Cylinders," AIAA J., vol. 5, Aug. 1967,
pp. 1448-1455.

Lur'e, A.I., Three-Dimensional Problems of the Theory
of Elasticity, John Wiley and Sons, New York,
1964.

 

 

Lebedev, N. N., Special Functions and Their Applica-
tions, Prentice-Hall, Englewood Cliffs, N.J.,
1965.

 

ProkOpov, V. K., "On the Relation of the Generalized
Orthogonality of P.F. Papkovich for Rectangular
Plates," J. Appl. Math. and Mech., vol. 28,
1964, pp. 428-433.

 

Alblas, J. B. and Kuypers, W. J. J., "On the Diffusion
of Load from a Stiffener into an Infinite Wedge-
Shaped Plate," Appl. Scientific Research (A),
vol. 15, 1965/66, pp. 429-439.

 

Baker, B. R., "Closed Forms for the Stresses in a
Class of OrthotroPic Wedges," J. Appl. Mech.,
vol. 32, March 1965, pp. 26-30.

 

Chen, W. T., "Stresses in a Transversely Isotropic
Elastic Cone Under an Asymmetric Force at its
Vertex," ZAMP, vol. 16, 1965, pp. 337-344.

Christensen, R. M., "Deformation of an Elastic Spheri-
cal Wedge," J. Appl. Mech., vol. 33, March 1966,
pp. 52-56.

 

Conway, H. D., "The Stresses in Infinite Wedges Linearly
Tapered in Width and Thickness," J. Appl. Mech.,
vol. 26, Sept. 1959, pp. 458-460.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

52

Godfrey, D. E. R., "Generalized Plane Stress in an
Elastic Wedge Under Isolated Loads," Quart. J.
Mech. Appl. Math., vol. 8, 1955, pp. 226-236.

 

Horvay, G. and Hanson, K. L., "The Sector Problem,"
J. Appl. Mech., vol. 24, Dec. 1957, pp. 574-581.

 

Knowles, J. K. and Sternberg, E., "On Saint-Venant's
Principle and the Torsion of Solids of Revolu-
tion," Arch. Rational Mech. Anal., vol. 22, 1966,
pp. 100-120.

 

Low, R. D., "On the Torsion of an Elastic Cone as a
Mixed Boundary Value Problem," Quart. J. Mech.
Appl. Math., vol. 19, pt. 1, Feb. 1966, pp. 57-64.

 

 

Morgan, A. J. A., "Stress Distributions in Semi-
Infinite Solids of Revolution," ZAMP, vol. 5,
1954, pp. 330-341.

Silverman, I. K., "Approximate Stress Functions for
Triangular Wedges," J. Appl. Mech., vol. 22,
March 1955, pp. 123-128.

 

Sneddon, I. N., "Boussinesq's Problem for a Rigid Cone,"
Proc. Camb. Phil. Soc., vol. 44, Oct. 1948,
pp. 192-507.

 

Srivastav, R. P. and Narain, P., "Certain Two-Dimen-
sional Problems of Stress Distribution in Wedge-
Shaped Elastic Solids Under Discontinuous Load,"
Proc. Camb. Phil. Soc., vol. 61, 1965, pp. 945-
954.

 

Sternberg, E. and Koiter, W. T., "The Wedge Under a
Concentrated Couple: A Paradox in the Two-
Dimensional Theory of Elasticity," J. Appl.
Mech., vol. 25, Dec. 1958, pp. 575-58 .

Tranter, C. J., "The Use of the Mellin Transform in
Finding the Stress Distribution in an Infinite
Wedge," Quart. J. Mech. Appl. Math., vol. 1,
June 1948, pp. 125-130.

 

Wang, C. T., Applied Elasticity, McGraw-Hill, New York,
1953.

 

Love, A. E. H., A Treatise on the Mathematical Theory
of Elasticity, Dover, New York, 1944.

 

 

35.

36.

37.

38.

39.

40.

41.

53

Sokolnikoff, I. 8., Mathematical Theory of Elasticity,
McGraw-Hill, New York, 1956.

 

Long, R. R., Mechanics of Solids and Fluids, Prentice-
Hall, Englewood Cliffs, N. J., 1961.

 

Hobson, E. W., The Theory of Spherical and Ellipsoidal
Harmonics, Cambridge Univ. Press, London, 1931.

 

 

Hildebrand, F. B., Advanced Calculus for Applications,
Prentice-Hall, Englewood Cliffs, N. J., 1962.

 

Novozhilov, V. V., Theory of Elasticity, Pergamon Press,
London, 1961.

 

Coddington, E. A. and Levinson, N., Theory of Ordinary
Differential Equations, McGraw-Hill, New York,
1955.

 

 

Williams, M. L., "Stress Singularities Resulting from
Various Boundary Conditions in Angular Corners
of Plates in Extension," J. Appl. Mech., vol. 19,
1952, pp. 526-528.

 

APPENDICES

APPENDIX A

DEVELOPMENT OF THE ASYMPTOTIC EIGENVALUES

OF THE WEDGE PROBLEM

Consider the transcendental equation for the even

eigenvalues, Eq. (2.24), in the form:
anB Sln (28) + 28 cos (anB) s1n [(dn'2)8]= o (A.l)
For an complex, seek solutions in the first quadrant
of the complex plane in the form:

an = xn + iyn (A.2)

Substituting this expression foran into Eq. (A.l)

 

yields:
(xn + iyn)B sin (28) + 28 cos [(xn + iyn)8] sin [ (xn - 2)
+ iyn)8]= O (A.3)

Making use of the elementary trigonometric relations:

sin (a i b) = sin a cos b t cos a sin b
cos (a i b) = cos a cos b 1 sin a sin b
sin (iy) = i sinh y
cos (iY) = cosh y

yields the following coupled algebraic equations for the
real and imaginary parts of an:

55

56

xn8 sin (28) + 28<cos (xn8) sin [(xn - 2)8]cosh2 (ynB)

. . 2 _
+ Sin (xn8) cos [(xn - 2)8] Sinh (yn8)} — 0 (A.4)
and,
ynB Sln (28) + 28 cos[2(xn - 1)B] Slnh (ynB) cosh (ynB) = 0
(A.5)

It is desired to develop asymptotic solutions for
these equations: i.e., solutions which will be accurate for
the higher eigenvalues. Substituting the exponential ex-
pressions for the hyperbolic functions and neglecting higher

order terms, Eqs. (A.4) and (A.5) reduce to:

¢
$1 sin 28 + Be 2 sin ¢l = 0 (A.6)
and
C"2
$2 sin 28 + Be cos ¢1 = 0 (A.7)
where
$2 = 2yn8 .

These, then are the equations which hold forlarge ¢1 and ¢2.
Since solutions are being sought in the first quad-
rant, xn and yn are restricted to be positive numbers.
However, for Eqs. (A.6) and (A.7) to be satisfied, sin $1
and cos ¢1 must be negative.
For Eq. (A.7) to be satisfied, cos $1 must be a

¢

small negative number since e 2 >> ¢2. Thus, take

¢l = (4n - l) g - En , n = 1,2,3,... (A.8)

57

where an is a small correction term to be determined shortly.

From Eq. (A.6)

 

- ¢
_ _ Sln 28 1
¢2 - ’6” [ 8 Sin cpl] o (A.9)

But from Eq. (A.8), sin $12 —1 and for an approximation of

¢2, use only the first term of ¢1 in Eq. (A.8); i.e., take

<b2 = £n[s—in7§§ (4n - 1) g] , n = 1,2,3,... (21.10)

Thus, ¢l and $2 become:

(bl = (4n - 1) 121- en (A.11)

(b — in Si“ 28 (4n - 1) 1’1 (A 12)

2 ‘ —E— 2 °
for n = 1,2,3,...

Now substitute Eqs. (A.11) and (A.12) into Eqs. (A.6)
and (A.7) to solve for en. Equation (A.6) is satisfied ap-

proximately and Eq. (A.7) yields, for small En

. sin 28 n n _
Sln 28<fn [———§—— (4n - 1) 7] - (4n - 1) 5 an} — 0 .

(A.13)
Thus, for sin 28 75 0,
' 28 N
in iiEg—— (4n - l) -]
E = [ 2 I n = 1,2,3,... (A014)

 

n n
(4n - 1)?

Therefore, the asymptotic eigenvalues in the right

half plane for the even problem are:

58

 

Kn sin 28 (4n _ l) 1
Xn = 12’1- (4n - 1) 721+ 28 - [ 1T 2] (A.15)
(4n - l) -2'
_ 1 sin 28 n
Yn ‘ jg £n[-_—E—— (4n - l) E] (A.16)
where age) = xn + iyn (even solution in right half plane).

A similar development for the decaying solution re-
quires seeking solutions to Eq. (2.24) in the left half
plane. Taking the solution for an in the form

an = -un - ivn , (A.17)

the asymptotic values in the left half plane for the even
problem can be determined in a manner very consistent with

the previous deve10pment as:

1 fl Kn [sin 28 (4n _ 1) %

 

u =‘—— (4n - l) - - 28 - (A.18)
n 23 2 (4n - 1) 321
_ 1 sin 28 _ n
Vn - IB- £77. [T (4n 1) 7] (A.19)
where use) = -un — ivn . (even solution in left half plane)

For the odd problem, working with the odd transcen-
dental equation, Eq. (2.23), the asymptotic values become:

in sin 28 (4n _ 3) n
@n - 3) 11+ 28 - [ E 7 (A.20)

2 (4n - 3)

 

'o
u
MP
m
an:

qn = %g fin [$32—8- (4n - 3) 2’1] (A.21)

59

(o)

where an = pn + iqn (odd solution in right half plane).

The decaying solution is:

in [§£E_Z§.(4n - 3) %]

 

l n B
r = —— (4n - 3) - 28 - .- (A.22)
n 8 7 (4n-”12‘-
_ 1 sin 28 n
sn - IE in [———§—— (4n - 3) 2] (A.23)
where ago) = -rn - isn (odd solution in left half plane).

These values represent the approximate solutions to
the appropriate transcendental equations for the higher
eigenvalues; however, they are also quite accurate for the
lower eigenvalues as may be observed using Tables (2.1)-

(2.3).

 

APPENDIX B

ORTHOGONALITY CONDITIONS FOR THE EIGEN-

FUNCTIONS OF THE WEDGE PROBLEM

Using the method of generalized orthogonality by

P. F. Papkovich [17], the defining equation for the mth

eigenfunction of the wedge problem is the fourth order

differential equation:

2

' 2 2 . _
m (8) + am(am - 2) fm(0) - 0 .

2
+ (am - 2) ]f

IV
fm (8) + [on m

(3.1)

th

Similarly, the n eigenfunction is given by:

2

£37m) + [1131+ (an - 2)2]f;1'(e) + 0131mm - 2) fnm) = o .

(B.2)

Multiply Eq. (B.l) by an(an - 2)fn' Eq. (B.2) by

am(am - 2)fm, subtracting the equations, and integrating

over the boundary yields:
8 (1(OL - 2)f fIV - 01(01 - 2)f fIV + 01(0) - 2) a2
-B m m m n n n n m m m n
2 " 2 2 IL“
+ (an — 2) ]fmfn — an(an - 246m + (am — 2) ]fnfm

+ [dm(dm - 2)a:(an - 2)2 - dn(dn - 2)ai(dm - 2)2]fmfn d6 = 0 .
(B.3)

60

61

The boundary conditions for the particular wedge
problem formulated in the earlier chapters require fm(i8),
fg(18), fn(i8), f;(18) to vanish. Applying these condi-
tions after integrating by parts twice for the first two

terms and once for the next two, Eq. (B.3) reduces to

8 II II 2
J; |§m<am - 2) - dn(an - 2)]fm fn + -am(dm - 2) an
8
2 2 2 ' '
+ (an 2) )+ dn(an - 2)(am + (am - 2) )]fmfn
- dm(am-2)an(dn-2)[am(dm - 2) - dn(an - 2)]fmfn d6 = 0
(3.4)
II
Expanding and simplifying the coefficient of fmfn
and factoring out[a (a - 2) - a (a - 2)] from each term
. m m n n
yields:
8
[dm(am - 2) - dn(dn - 2)]J:B[hm(dm - 2)ocn(o:n - 2)fmfn
I :4 II II ’
+ 4 fmfn - fm fn d9 = 0 . (B.5)

Thus, the orthogonality condition is:

B U U I | I
.jLB [am(am - 2)an(an - 2)fmfn + 4 fmfn - fm fn ] d6 = 0,
(m # n) . (B.6)

APPENDIX C
DEVELOPMENT OF THE LEGENDRE FUNCTIONS USED
IN THE CONE PROBLEM

Solving the cone problem by the method outlined in

Chapter III requires solving the equations

V2(Bz) = o (c.1)
V2(Bpele) = o (c.2)
where B2 = Bz(R,¢) and BD = Bp(R,¢).
Assume a solution of the form
-an-l
13.2 = R fn(¢) (C.3)
and substitute into Eq. (c.1). Then
2 .0Ln-l
V R fn(¢) = 0 (C.4)

where V2 is the Laplace operator in spherical coordinates.

This leads to the following equation for fn:

2 I l l
(1 - x )fn (x) - 2x fn(x) + an(dn + 1)fn(x) - 0 (C.5)
where x = cos ¢. This is Legendre's equation.

Change variables [16] by letting t = %(1 - x). Eq.

(C.5) becomes
62

 

63

t(l - t)f;'(t) + (1 - 2t)f;(t) + an(an + 1)fn(t) = o
(C.6)

which has a solution

 

f = (f ) = F -d , a + 1:1; 1 - x (C.7)

n n 1 n n 2
where F(a,8;y;z) is the hypergeometric series defined by

(a)k(8)k 2k

F(G,B;Y;z) = (C.8)
kEO Elmk

(A)k 1(1 + l)(l + 2)...(l + k — 1) , k 2 l (C.9)

(A) = l . (C.10)
A second independent solution is generated [16] by

letting t = x-2 in Eq. (C.5). This leads to an equation

whose solution is

 

/? P(a + l) a a
f = (f ) = 3 n a +1 F‘EE + 1' 22 + 2‘“ + 27;?
n 1“2 T‘an+§)(2x) n n x
(c.11)

The solutions to Eq. (C.5) are called the Legendre

functions of degree an of the first and second kind, denoted

by Pa (x) and Qa (x), respectively; i.e.,

 

 

n n
pa (x) = F -an,an + 1;1;l—§—§ , Ix - 1| < 2 (c.12)
n
/F F(a + ]J a a
_ n n 1. 2.1_
Qan(X) ' 3 an+l F‘E‘ + 1' 2"+ 2'an + 2'x2 '
r(an + 7)(2x)

le > 1 . (c.13)

64

The general solution for Eq. (C.5) can then be written as

fn = APa (x) + 30a (x) . (C.14)
n n

The cone problem outlined in Chapter III requires
B = 0 for a finite solution along the cone axis. The a1-
1owab1e solution for fn becomes
fn = APan(x) . (C.15)
Thus, take

B = AR n P (x) . (C.16)

Similarly, for the solution of Bp, assume

-an-l
Bp = R gn(¢) (C.17)

and substitute into Eq. (c.2). This leads to the follow-

ing equation for gn:

l

l _ x2] n

(1 - X2)g;'(x) 2xgg<x) + [an(an + 1) -

where x = cos ¢. This is Legendre's associated equation

which has as its solution

gn(x) = C P:n(x) + D Q;n(X) (C.19)

where

65

 

 

 

 

 

a (a + l) _
P1 (x) = - n “2 <1 - X2)F('a + 1.a + 2:2;l X .
an n n 2
ll - x] < 2 (c.20)
1
Q1 (x) = -/?P(dn + 2) (x2 _ 1)? F an + 3 an
a a +1 _ a +2 2' 2'2—
n 2 n I‘(d + 2)x n
n 2
3 l
+ 1;an + §;-7 , lxl > 1 . (c.21)
x
I
Thus for the cone problem, D = 0 and the allowable
solution for gn become
' 1
9n = C Pa (x) (C.22)
n
which can be written as
dP
oLn
9n = C d¢ (C.23)
since
%_dPa (x)
l _ _ 2 n
Pan — (l X ) "'- 3X (C024)
Thus,vtake
-an-1 den
B0 = CR -—d$—- . ((2.25)

The components of B can now be determined in spheri-

cal coordinates using the transformation equations.