H
ILL INO
S
UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
PRODUCTION NOTE
University of Illinois at
UrbanaChampaign Library
Largescale Digitization Project, 2007.
No. 493 is missing
ABSTRACT
THE PRINCIPAL OBJECT OF THE STUDY
REPORTED HERE WAS TO ANALYZE THE EFFECTS
OF THE VARIOUS PARAMETERS THAT DEFINE
THE EXTERNAL GEOMETRY OF PROJECTING
NOTCHES ON THE ASSOCIATED STRESS CON
CENTRATION FACTOR. THE STUDY DEALS
PARTICULARLY WITH NOTCHES OCCURRING
SYMMETRICALLY IN RECTANGULAR BARS SUB
JECTED TO AXIAL LOADING.
THE METHOD OF ANALYSIS USED AC
COMPLISHED THE ABOVE OBJECTIVE IN
DIRECTLY BY CONSIDERING THE EFFECTS OF
VARIOUS GEOMETRICAL PARAMETERS ON THE
MAGNITUDE AND DISTRIBUTION OF THE
SHEARING STRESSES ALONG THE BASE SEC
TION OF A PROJECTING NOTCH. AS A PRE
LIMINARY STEP, THE STRESSES ALONG SE
LECTED SECTIONS IN A PROJECTING NOTCH
IN A HALFPLANE WERE OBTAINED USING A
SOLUTION GIVEN BY H. NEUBER. USING
THESE DATA AND A SOLUTION BY L. N. G.
FILON FOR THE STRESSES IN A RECTAN
GULAR BAR RESULTING FROM SURFACE STRESS
LOADS IT WAS SHOWN THAT THE MAXIMUM
LONGITUDINAL STRESS ALONG THE BASE SEC
TION OF A PROJECTING NOTCH IS PRODUCED
PRIMARILY BY THE SHEARING STRESS COM
PONENT ALONG THE BASE SECTION. THE
EFFECT OF THE NORMAL STRESS COMPONENT
ALONG THE SAME SECTION WAS SHOWN TO BE
NEGLIGIBLE. THIS HAS LED TO THE OB
SERVATION THAT THE MAXIMUM STRESS IN A
PROJECTING NOTCH MAY BE APPROXIMATED
REASONABLY BY THE LONGITUDINAL STRESS
IN A PLAIN RECTANGULAR BAR SUBJECTED TO
AN AXIAL LOAD AND A SURFACESHEAR LOAD
ING OF THE PROPER MAGNITUDE AND DIS
TRIBUTION.
THE ABOVE OBSERVATION SERVED AS THE
BASIS FOR EMPLOYING AN EQUIVALENT SUR
FACESHEAR LOADING TO APPROXIMATE THE
STRESSRAISING ACTION OF A PROJECTING
NOTCH. THE RESULTS OF THE SURFACE
SHEARLOAD METHOD WERE USED TO DEVELOP
A PROCEDURE FOR ESTABLISHING A COR
RELATION BETWEEN THE STRESS CONCENTRATION
FACTOR AND THE GEOMETRICAL PARAMETERS
CHARACTERIZING THE PROFILE OF PROJECT
ING NOTCHES.
ACKNOWLEDGMENTS
The studies described herein were conducted as a part of the research under
the Illinois Cooperative Highway Research Program Project IHR64, "Behavior of
Welded Highway Structures." The research was conducted by the Department of
Civil Engineering of the University of Illinois in cooperation with the State of
Illinois, Division of Highways, and the U.S. Department of Transportation,
Federal Highway Administration, Bureau of Public Roads.
On the part of the University of Illinois, the work covered by this report
was carried out under the general administrative supervision of W. L. Everitt,
Dean of the College of Engineering; Ross J. Martin, Director of the Engineering
Experiment Station; N. M. Newmark, Head of the Department of Civil Engineering;
and Ellis Danner, Director of the Illinois Cooperative Highway Research Program.
On the part of the Division of Highways of the State of Illinois, the work
was under the administrative direction of Virden E. Staff, Chief Highway Engineer;
Theodore F. Morf, Deputy Chief Highway Engineer; and John E. Burke, Engineer of
Research and Development.
The program was planned in cooperation with a Project Advisory Committee
consisting of the following members:
Representing the Illinois Division of Highways:
James Branton, Field Engineer, Bureau
of Construction
John E. Burke, Engineer of Research
and Development
George T. Fischer, Engineer of Shop
Plans and Inspection
Representing the Bureau of Public Roads:
J. L. Hirsch, Bridge Engineer
Representing the University of Illinois:
William J. Hall, Professor of
Civil Engineering
John D. Haltiwanger, Professor
of Civil Engineering
The direct supervision of this study was provided by W. H. Munse, Project Super
visor, and W. W. Sanders, Jr., and John Cannon, formerly Project Investigators.
This report is based on a doctoral dissertation written by Arnaldo T. Derecho
under the direction of Professor W. H. Munse. The authors are grateful to
Professors S. Fenves, A. R. Robinson, and F. S. Shaw for their kind assistance
and suggestions in connection with various phases of the program.
Acknowledgment is also extended to the reviewers of this Bulletin: G. T.
Fischer of the Illinois Division of Highways and J. D. Haltiwanger of the Univer
sity of Illinois.
CONTENTS
NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
A. General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 1
B. Objectives of Study . . . . . . . . . . . . . . . . . . . . . . . 2
II. THE PROBLEM AND THE METHOD OF ANALYSIS . . . . . . . . . . . . . . . . 3
A. The Idealized Projecting Notch; Parameters Considered . . . . . . 3
B. Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . 3
C. General Description of Method of Analysis . . . . . . . . . . . . 3
III. ANALYSIS OF STRESSES AND RELATIONSHIPS FOR NEUBER'S NOTCHES . . . . . 5
A. Neuber's Solution . . . . . . . . . . . . . . . . . . . . . . . . 5
B. The SurfaceShear Load Method; Effect of Normal Component
of Stress Along Base Section of Notch . . . . . . . . . . . . . . 10
IV. ANALYSIS OF STRESSES AND RELATIONSHIPS FOR NEUBER'S NOTCHES . . . . . 18
A. FiniteDifference Solution . . . . . . . . . . . . . . . . . . . . 18
B. Correlation of Stress and Geometrical Parameters . . . . . . . . . 20
V. EXPERIMENTAL INVESTIGATION AND COMPARISONS WITH THEORETICAL ANALYSIS . 24
A. Experimental Investigation . . . . . . . . . . . . . . . . . . . . 24
B. Comparison of Results of Theoretical Analysis with
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 25
VI. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 27
A. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
VII. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
APPENDIX A. DEVELOPMENT OF NEUBER'S SOLUTION FOR A PROJECTING
NOTCH IN A HALFPLANE UNDER TENSILE LOADING . . . . . . . . . . . . . 31
A. General Relationships in Rectangular Coordinates . . . . . . . . . 31
B. The Problem in Curvilinear Coordinates . . . . . . . . . . . . . . 32
C. Solution for the Stresses in a Projecting Notch in a HalfPlane
Under Tensile Loading . . . . . . . . . . . . . . . . . . . . . . 37
APPENDIX B. DEVELOPMENT OF EXPRESSIONS FOR THE STRESSES IN A
RECTANGULAR BAR DUE TO SURFACE STRESS LOADING . . . . . . . . . . . . 42
A. Solution for the Case of a SurfaceShear Loading Distributed
AntiSymmetrically About the y Axis . . . . . . . . . . . . . . . 43
B. Solution for the Case of a SurfaceNormal Stress Loading
Distributed Symmetrically About the y Axis . . . . . . . . . . . . 45
APPENDIX C. BOUNDARY VALUES FOR THE STRESS FUNCTION USED IN THE
FINITE DIFFERENCE SOLUTION . . . . . . . . . . . . . . . . . . . . . . 48
FIGURES
la. Symmetrically Disposed Projecting Notches in a Bar of
Rectangular Cross Section
lb. Typical Distributions of Stresses Along Centerline and
Base Sections
2. Profiles of Neuber's Projecting Notches
3. Distributions of Normal Stresses Along the Bases of
Neuber's Notches
4. Distributions of Shearing Stresses Along the Bases of
Neuber's Notches
5. Distributions of Longitudinal Stresses Along the Bases
of Neuber's Notches
6. Distributions of Longitudinal Stresses Along the Center
line of Neuber's Notches
7 Variation of Stress Concentration Factor with Parameters
h/w, R min. /w, and e for Neuber's Notches
8. Longitudinal Stress Along Surface of Rectangular Bar Due
to Surface Stress Loadings
9. Maximum Stress in a Rectangular Bar Due to SurfaceShear
Load and Uniform Stress Across Ends
10. Approximate Variation of Shearing Stress Parameter,
F /pw, with R . /w Neuber's Notches
11. Shearing Stress Parameter F /pw as a Function of R . /w
and h/w for Neuber's Notches
12. Correlation of R . /w of Neuber's Notches with c/w of
Triangular SurfaceShear Load
13. The Stress Concentration Factor as a Function of h/w and
R . /w for Neuber's Notches
mmn
14. Stress Concentration Factors for Neuber's Notches
15. Distributions of Shearing Stresses Along the Bases of
Idealized Projecting Notches
16. Graphical Estimation of Maximum Stresses
17. Shearing Stress Parameter, F /pw, as a Function of h/w and
R/w, for Idealized Projecting Notches with 6 = 90 Degrees
18. Shearing Stress Parameter, F /pw, as a Function of h/w and
R/w, for Idealized Projecting Notches with 6 = 60 Degrees
19. Shearing Stress Parameter, F /pw, as a Function of h/w and
R/w, for Idealized Projecting Notches with 6 = 30 Degrees
20. Extrapolation Curves for F /pw  h/w Relationships
21. Correlation of R/w of Idealized Projecting Notch with c/w
of Triangular SurfaceShear Load
22. The Stress Concentration Factor as a Function of h/w and
R/w for Idealized Projecting Notches with 6 = 90 Degrees
22a. Figure 22 with R/w Scale Expanded and Linearized
23. The Stress Concentration Factor as a Function of h/w and
R/w for Idealized Projecting Notches with 6 = 60 Degrees
23a. Figure 23 with R/w Scale Expanded and Linearized
24. The Stress Concentration Factor as a Function of h/w and
R/w for Idealized Projecting Notches with 6 = 30 Degrees
24a. Figure 24 with R/w Scale Expanded and Linearized
25. Stress Concentration Factors for Idealized Projecting
Notches with e = 90 Degrees
26. Stress Concentration Factors for Idealized Projecting
Notches with 0 = 60 Degrees
27. Stress Concentration Factors for Idealized Projecting
Notches with 6 = 30 Degrees
28. Details of Test Specimens
29. Strain Versus Total Load Plots for Some Gages on
Specimen PS4
30. Variation of Stress Concentration Factor with w/h for
Different 9 and R/h Idealized Projecting Notches
A1. An Element on a Curved Boundary
B1. Triangular SurfaceShear Load
B2. Linearly Varying SurfaceNormal Load
C1 Typical Section Considered in Finite Difference Solution
C2. Finite Difference Biharmonic Operator
C3. Notation Used in Interpolation Equations
TABLES
1a. Maximum Stresses and Characteristic Dimensions for Neuber's
Notches with Base Sections Defined by k=0.125 in Equation (8)
1b. Maximum Stresses and Characteristic Dimensions for Neuber's
Notches with Base Sections Defined by k=0.250 in Equation (8)
2. Stress Parameters F /pw and c/w, and Efficiency of Center
line Section of Neuber's Notches with Base Sections Defined by
k=0.125 and k=0.250
3. Variation of (a ) /p Along Surface of Rectangular Bar
x
due to Triangular murfaceShear Load, with the Depth 2b
of the Bar
4. Comparison of Values of the Stress Concentration Factor
for Neuber's Notches as Obtained from Figure 14 with
Calculated Values
5. Summary of Results of FiniteDifference Solution for the
Idealized Projecting Notch
6. Summary of Strain Gage Measurements on Projecting Notch
Specimens
7. Comparison of Values Obtained from Figures 25, 26, and 27
for the Limiting Case of a Bar with Shoulder Fillet, with
Values from Reference 10
NOTATION
The following notation used in the text and the appendices are listed below
for convenient reference:
In connection with H. Neuber's solution:
x, y, z = rectangular coordinate
a, 8, y = orthogonal curvilinear coordinates
o = the 6 = constant curvilinear coordinate line denoting the
profile of a notch
he, hV, h = scale factors along a, 8, and y directions
u ,
n
E
G, G
F
v, w
U, V, W
e
e
y~8
Tai
(ay
a max
a , a , a
x y z
= a constant
= Young's modulus
= Poisson's ratio
= Lame's constants
= stress function
= harmonic functions
= displacement components along the x, y, and z directions,
respectively
= displacement components along the a, 8, and y directions,
respectively
= dilatation
= dilatation in generalized plane stress
= strain components along the a, 8, and y directions, respectively
= shearing strain in the a 8 plane
= normal stresses acting in the direction of the a, 8, and y
coordinate axes, respectively
= shearing stress in the a 6 plane
= maximum stress in the direction of the a coordinate line
(occurring at the surface of the notch)
= normal stresses acting in the direction of the x, y, and z
coordinate axes, respectively
ix
X,
R
m
aR
m
k.
xt5
In connectio
xy = shearing stress in the xy plane
Y = x and y components, respectively, of the external stresses
acting on the boundary
s = arc length along boundary
y = coordinates of the point of maximum stress
n = minimum radius of curvature of the notch profile
in
in = a coordinate of the point of maximum stress (along profile
n = o )
d = constants
t = coordinates of point on notch profile denoting arbitrarily
defined 'toe' of notch
w = onehalf the width of the notch, measured from centerline to toe
h = height of notch, measured from the base section y = yt to the
highest point on the notch profile
= the angle which the tangent to the notch profile at any point
(a, 8) makes with the x axis
9 = flank angle  the angle which the tangent to the notch
profile makes with the x axis at the point of inflection
p = intensity of uniform tensile stress acting across a section
x = constant sufficiently removed from the notch
F = total force due to the stress 0 acting across the centerline
section of the notch  for thai part of the section above
y = Yt
F (= F ) total force due to the shearing stressT acting along
the base of the notch, from centerline to toe xy
c = distance from the point of maximum shearing stress along the
base to the toe of the notch
n with the surfaceshear load method:
a = onehalf the total length of the bar considered
b = onehalf the depth of the bar
t = onehalf the thickness of the bar
w = onehalf the loaded length, i.e., onehalf the total length
acted on by the surfacestress loading (corresponding to
the halfwidth of the notch)
a = 2rn
= Fourier coefficient for the surfaceshear stress intensity curve
C =
n
F =
Fourier coefficient for the surfacenormal stress intensity curve
Airy's stress function
x
Parameters defining the surfaceshear stress intensity curve:
F /p = total area of stress intensity triangle
T /p = height of stress intensity triangle
c = distance from the apex of the stress intensity triangle to the
end of the loaded length
Parameters defining the surfacenormal stress intensity curve:
s = distance from the center of the loaded length to the point
of zero stress
g = distance from the apex of the positive stress triangle to the
end of the loaded length
2
r = s  s  (w  s  g) (w  s)
(+), = height of positive stress triangle
n
/ /p
n
F n/p
= constant height of negative stress trapezoid
= total area of positive stress triangle
Most of the above
y + +C
S, a , T =
x y xy
quantities are illustrated in Figures B1 and B2.
stress components due to surfaceshear stress loading on
rectangular bar
+ + +
C 0y , T = stress components due to surfacenormal stress loading
(a ) = maximum stress on surface of bar due to combined action of
surface stress loading and the uniform tensile stress p
acting at the ends of the bar
In connection with the finite difference solutions:
F = Airy's stress function
X, Y = x and y components, respectively, of the external stresses
acting on the boundary
s = arc length along boundary
n = normal to boundary
p = uniform tensile stress acting at ends of bar
A = mesh interval
a = ratio of the distance from first interior point to boundary
point, to A
F., Fe F b, Fa = values of stress function F at points i, e, b, and a, respec
tively, adjacent to boundary
I. INTRODUCTION
A. GENERAL REMARKS
The localized increase in stress produced
by a discontinuity in the shape or geometry
of a loadcarrying member has long been
recognized. Such stressraising discontinu
ities are broadly referred to as notches,
whether they be geometrical, metallurgical, or
other, in nature.
In the design of staticallyloaded
members, the increase in stress occurring in
the immediate vicinity of a discontinuity is
generally neglected. Here, reliance is placed
on the beneficial effect of the redistribution
of stresses that occurs when the stresses in
the critical portions of the material reach
the yield point. However, in members subjected
to repeated loadings, in which failure, if it
occurs, is generally nonductile, the localized
stress concentrations produced by notches
become significant. This is particularly true
in the case of members subjected to relatively
low (compared to the yield point of the
material) stresses which result in failure,
but at long lives. (1,2,3)*
A common type of discontinuity in welded
members is represented by the weld reinforce
ment in a transverse buttwelded joint. Other
members which are buttwelded into a main
member, with the axes perpendicular to each
other, provide the same general type of
Numbers in parentheses refer to entries in
References Chapter VII.
discontinuity. Experimental studies on
buttwelded joints subjected to axial fatigue
loading(1,2,3,4,5,6) have shown definite and
often significant reductions in the fatigue
lives of the members as a result of the weld
reinforcement. Reductions in fatigue strength
of aswelded specimens have been found to be
as high as 45 per cent (relative to the
fatigue resistance of the corresponding plain
plate specimens).
Experiments by Newman and Gurney,(5)
J. E. Tomlinson and J. L. Wood, 6)and W. 0.
Dinsdale verify results of tests conducted
at the University of Illinois aimed at
isolating and studying the effects of the
external geometry of buttwelded joints
They indicate that a reduction in fatigue
strength increases with the severity of the
notch geometry up to a certain point, beyond
which the effect of the notch appears to
remain more or less constant. This observa
tion has led to the conclusion that for a
given material and loading condition, particu
larly for longlife fatigue, the fatigue
behavior of a member is, phenomenologically,
at least, a direct function of the local
stress and strain condition in the critical
portions of the member, and hence of its
geometry. However, the relationship between
stress and/or strain and fatigue life may be
far from simple.
Using specimens with the notch simulating
the weld reinforcement machined from
base metal.
This study is an outgrowth of the above
mentioned investigation on the effects of
external geometry on the fatigue behavior of
buttwelded joints under axial loading.
Because of the dominant influence of high,
localized stresses on the fatigue behavior of
buttwelded and other similar joints, it was
felt necessary to study in detail the effects
of the various parameters determining the
external notch geometry on the stress concen
tration factor.
B. OBJECTIVES OF STUDY
The primary objective of this study is
to analyze the effects of the various geomet
rical parameters that characterize the profile
of a projecting notch on the associated
stress concentration factor. The study deals
particularly with rectangular bars which have
symmetrically disposed projecting notches and
which are subject to axial loading. The
ultimate aim of the analysis is the development
of a method of predicting the elastic stress
concentration factors for projecting notches
of varying characteristic dimensions.
The term projecting notch is used here
to denote the geometrical discontinuity
characterized by the projection of a part of
a body above an otherwise plane surface. As
mentioned previously, the weld reinforcement
in a buttwelded plate as well as attachments
or other members buttwelded into a main
member fall under this category.
0* * *
Used here in its broader sense, i.e., any
stress raiser.
II. THE PROBLEM AND THE METHOD OF ANALYSIS
A. THE IDEALIZED PROJECTING NOTCH; PARAMETERS
CONSIDERED
The idealized projecting notch for which
this study seeks to develop quantitative rela
tionships between the elastic stress concen
tration factor and the significant geometrical
parameters is shown in Figure la. This
represents a twodimensional problem with the
projecting notches disposed symmetrically
about the axis of a rectangular bar subjected
to axial loading. The following geometrical
parameters describe the notch adequately:
R = the radius of the circular
transition curve
h = height of notch, measured from the
base of the notch to the highest
point on the notch profile
w = onehalf the total width of the
notch, measured from centerline to
toe or point of tangency with the
base
e = the "flank angld', i.e., the angle
which the tangent to the notch
profile, taken at the point of
tangency of the circular arc and
the straight flank of the notch,
makes with the surface of the bar
or base of the notch.
A bar of unit thickness will be assumed
throughout the following discussion.
B. PREVIOUS STUDIES
To the author's knowledge, little has
been done to develop a relationship between
the elastic stress concentration factor and
the geometrical parameters that determine the
profiles of projecting notches in bars sub
jected to axial loading.
H. Neuber presents a solution for a
projecting notch with a variable transition
radius in a halfplane and gives a plot of the
variation of the stress concentration factor
with the minimum radius of curvature of the
notch profile. However, no mention is made of
the effects of the other parameters that
determine the notch profile.
Values of the stress concentration factor
for projecting notches in flat bars subjected
to bending, based on photoelastic studies by
M. M. Leven and A. J. Hartmann, are given by
R. E. Peterson.(10) Peterson,(10) as well as
R. B. Heywood, ll) indicate that no similar
data are available for the case of projecting
notches in bars subjected to axial loading.
C. GENERAL DESCRIPTION OF METHOD OF ANALYSIS
An approximate method of analysis has
been used since it is difficult to find a
coordinate system or a set of transformation
relationships in closed form which gives a
profile having the general shape of the
notched bar under consideration and which
allows a controlled variation of the geometri
cal parameters that characterize the notch.
As a preliminary step, Neuber's solution()
for a projecting notch in a halfplane was
considered (see Figure 2). The distributions
T. A. McCreery(12) gives an approximate
solution for the stresses in a notched
bar using conformal mapping  the
mapping function being given by a power
series expansion.
of stresses along selected sections in a
number of Neuber's notches were plotted and
from these plots qualitative relationships
were deduced. However, because the series of
notches associated with Neuber's solution
represented a very limited number of combina
tions of the geometrical parameters character
izing the profile of this type of notch, a
method had to be developed which would permit
the extension of the available data to a wider
range of values of the parameters considered.
This was necessary not only to give a better
picture of the influence of the different
parameters involved but also to provide a
method by which the desired correlation for
the case of the idealized projecting notch
could be obtained on the basis of a relatively
few calculated values.
The method developed to extend the results
to a wider range of values of the geometrical
parameters is based on the socalled "equivalent
surfaceshear load method. 13 5) The
effect of the notch is approximated by imposing
on a flat bar an equivalent shear stress load
ing, acting over the full width of the notch.
The maximum longitudinal stress along the
surface of the bar produced by such a loading
is then assumed to be equal to the actual
maximum stress produced by the corresponding
notch. In effect, it is assumed that the
projection on a notched bar is removed by
passing a section through the base of the
notch and the remaining plain bar loaded with
the equivalent stresses acting on the cut
section. The stresses along the base section
of the notch consist of (1) a shearing stress
component of varying intensity and (2) a
normal stress component. The normal stress
component may reasonably be neglected since it
is generally of a smaller magnitude and con
tributes an insignificant part to the maximum
stress along the section, as will be shown in
the subsequent development. As in previous
papers, (14,15) the shearing stress distribu
tion curve has been approximated by a broken
line surfaceshear load curve and the resulting
stresses in the plain rectangular bar have
been obtained using L. N. G. Filon's solution
for this case. However, unlike previous work
based on the surfaceshear load method, the
method of analysis developed in this study
does not use directly the maximum stress
produced by the "triangular" surfaceshear
loading to approximate the maximum stress in a
notch. Rather it utilizes a series of curves
obtained by the surfaceshear load method,
mainly as a framework of reference and as a
means of extending the results of calculations
for a limited number of cases to a wider range
of values of the parameters.
Using a plot of the maximum stresses
derived by the surfaceshear load method
together with data from Neuber's solution, a
correlation was obtained between what appeared
to be the most significant geometrical param
eters and the stress concentration factor for
Neuber's notches.
In order to apply the method developed to
obtain a correlation for Neuber's notches to
the case of the idealized projecting notch of
Figure la, a finitedifference solution for
the stresses in a rectangular bar with
symmetrically disposed projecting notches was
applied in a limited number of cases. The
.method of finite differences permitted the
geometrical parameters defining the notch
profile to be varied independently of each
other so that their separate effects on the
stress concentration factor could be deter
mined.
To obtain an experimental check of the
resulting relationships for idealized project
ing notches, a limited number of tests were
made on notched specimens. These test results,
as well as available data on bars with
shoulder fillets subjected to axial loading,
have been compared with values from the
theoretical analysis.
III. ANALYSIS OF STRESSES AND RELATIONSHIPS FOR NEUBER'S NOTCHES
A. NEUBER'S SOLUTION(8)
Neuber considered a projecting notch in
a halfplane with a profile coincident with
a "3 = constant" line of the coordinate system
defined by the following transformation rela
tionships:
xa+ a2 2
(1)
2 2
a + P2
Equations (1) determine an orthogonal,
isometric curvilinear coordinate system
(a, P). By assigning to P a fixed value, P 0'
and allowing a to vary, corresponding values
of (x, y) are obtained defining the notch
profile. Examples of such profiles, with
values of 3 ranging from 0.25 to 1.25, are
0
shown in Figure 2. The characteristic
dimensions of the notch profiles are listed in
Tables 1a and Ib. The above series of
notches are referred to as Neuber's notches.
In the twodimensional case considered
here, Neuber gives the following expressions
for the stresses along the a and P coordinate
axes: (The development of Equations (2), as
well as other relationships concerning
Neuber's notches, is given in Appendix A).
See Appendix A.
I )F 1 f3h )F )h )F\
a h2 2 h3 a 7a 7p (2)
1 )2F 1 ( /h )F ) h )F (
S 2 h 2 3 \ (2)
h h h
In the above equations, h is the common scale
factor or modulus of the assumed isometric
coordinate system; F is a stress function.
For the particular coordinate system defined
by Equations (1),
h2 = 2 1 + 22  22 +1
(h2 + 12)2
and for the problem of the projecting notch
with a profile given by P = Po'
F = (P  Bo)2 [1 1
2 (20 2 + 1)(a2+ 2)
(2b)
In Equation (2b), p is the applied tensile
stress acting across a section parallel to the
axis of the notch and sufficiently removed
from it to allow for uniform stress. Differ
entiation of Equations (2a) and (2b) and
substitution into Equations (2) give the
stress components along the curvilinear
coordinate lines a = constant and P = constant.
The corresponding x and y components of stress
0
are obtained by using standard transformation
relationships.
The maximum stress (along the profile
PB = P) is given by
max) = 1 +
max [
(1 + 4P02) (/_  Po)]
4D (1 + 2P0) JP
0 0 0 ý 1
and occurs at a point defined by
( = 1 + 0 + v 2 + 4o (4)
max
The rectangular coordinates of the point of
maximum stress (x , yo) are readily obtained
by substituting the value of a given by
Equation (4) and the corresponding 1 into
Equations (1).
The minimum radius of curvature of the
notch profile is given by
R. = 2 2 2 (5)
R 2P +P (5)
and occurs at a point determined by
/R 1 + 2 P (6)
mmin
The corresponding x and y coordinates are again
obtained from Equations (1).
The angle which the tangent to the notch
profile makes with the x axis at the point
(a, P ) is given by
0
 20 o
CP  tan  22 "
(a2 + P 2 ) +(P 0 2 )
The angle 0 which the tangent to the notch
profile makes with the x axis at the point of
inflection is defined as the "flank angle."
Definition of Base of Notch
The reader will note that in Figure 2 the
curve defining the notch profile does not
become horizontal (i.e., parallel to the
x axis) at any point except at x = 0. It
tapers off gradually, becoming almost hori
zontal for large values of a and x. Equations
(1) indicate that a approaches x as a becomes
larger, just as P approaches y for large
values of P. Because of the absence of a
definite horizontal line in the immediate
vicinity of the notch, the "base of the notch"
is not clearly defined and an appropriate
horizontal section y = yt must be selected
which can be taken as the base of the notch.
This then determines the width of the notch
as well as its height. The following were
considered in choosing the section.
(1) Previous experimentaA measure
ments on filleted bars(l6,17,18) with
circular arcs, as well as the results of
"Photostress" measurements carried out in
connection with this sudy, have shown
that the maximum stress occurs at a point
slightly above the toe of the notch  at
a point where the tangent makes an angle
of approximately 10 degrees with the axis
of the bar.
(2) On the basis of the above
observation, the horizontal section to be
used in defining the limits of Neuber's
notch should intersect the notch profile
P = Po a short distance beyond the point
of maximum stress. Furthermore, this
distance should be a function of the
minimum radius of curvature of the notch
profile.
In view of the above, base sections
determined by the relationship
x = x + k * R .
t o mimn
were tried, assuming values of 0.125 and 0.250
for the constant k (corresponding to angles of
about 7 degrees and 15 degrees, respectively,
for a circular arc fillet). As will be seen
later, a variation in the values of k within
the above range does not produce significant
changes in the results obtained. In the sub
sequent discussion, the point on the notch
profile defined by the coordinates (xt, yt)
will be referred to as the toe of the notch.
The relative magnitudes and distributions
of the stress components along the base and
centerline sections of the notches were
calculated using Equations (2) and are plotted
in Figures 3 to 6. A value of k = 0.250 was
used in Equation (8) to define the base
section y = y t. In all of the abovementioned
figures, as well as in the following figures
and discussion, the uniform tensile stress p
has been taken as unity. With this assumed
value of p, the magnitude of the maximum
stress (o ) is equal to the stress concen
max
tration factor.
Discussion of Stress Distributions Along Base
and Centerline Sections
A comparison of the distribution curves
for the stress components along the base of
the notch indicates that the peak of the
normal stress intensity curve of Figure 3
occurs slightly to the left of the peak of the
corresponding shear stress intensity curve of
Figure 4. As will be shown in connection with
the surfaceshearload method, this is signif
icant since the longitudinal stress on the
surface of a bar produced by a surfacenormal
stress loading has a distribution similar to
that of the load itself. On the other hand,
the maximum longitudinal stress on the surface
of the bar produced by a surfaceshear load
lies slightly to the right of the peak of the
shear load intensity curve. This relative
displacement between the points of maximum
longitudinal stress produced by surfacestress
loadings corresponding to the normal and shear
stress components along the base of the notch,
together with the comparatively small magnitude
of the longitudinal stress created by the
normal component, combine to make the effect
of the normal stress component on the maximum
longitudinal stress negligible. Neglecting
the normal stress component implies that the
stressconcentrating effect of the notch is
approximated reasonably well by the effect of
the shearing stress component.
In the following discussion, therefore,
the main emphasis will be placed on the
shearing stress component along the base
section y = y t. Since this shearing stress
component is the predominant factor, a study
of the effects of the different shearing
stress distributions and the variation of
these distributions with the geometrical
parameters that characterize the notch profile
should lead to a reasonably good correlation
between the notch parameters and the theoreti
cal stress concentration factor. This concept
underlies the socalled equivalent surface
shear load method of calculating stress
concentration factors.
The following additional points are worth
noting in connection with the distributions of
stresses along the base and centerline sections.
An examination of the shearing stress
distribution curves corresponding to the
different notch profiles in Figure 4 reveals
that these differ principally in two respects,
(1) the distance c from the peak of each curve
to the toe of the notch (indicated by x/w = 1.0
in Figure 4), and (2) the total area under
each curve F /p. The above two quantities,
expressed in terms of the dimensionless ratios
c/w and F /pw which describe the essential
features of the shear distribution curve along
the base of the notch, shall be referred to
briefly as stress parameters.
Table 2 indicates that the stress
parameter F /pw increases with increasing
height of notch and decreasing radius of
curvature of the notch profile, while the
parameter c/w decreases with decreasing radius
of curvature of the notch profile.
It will be noted that in Figure 4 the
shearing stress is not equal to zero at the
toe of the notch because of the peculiar shape
of Neuber's notches and the arbitrary manner
in which the base of the notch had to be
defined. Thus, the toe of the notch does not
mark the point of tangency of a curve and a
horizontal line  as in the idealized notch
of Figure la. In the idealized projecting
notch, the shearing stress is equal to zero at
the toe.
Table 2 also lists the dimensionless
ratio F /pw, shown equal to F /pw, correspond
ing to various values of (3 (F being the total
force due to x along the centerline section
x
for that part of the section above the base of
the notch). For equilibrium, the total force
F must be equal to the corresponding total
force F due to the shearing stress acting
along half the width of the base section
y = yt. Values of F /pw have been calculated
for notches whose widths were determined by
using Equation (8) with k = 0.125 and k =
0.250.
Figure 6 is a plot of the distribution of
the longitudinal stress 0. acting along the
centerline section. It shows a decrease in
the "efficiency" of the section with an
increase in the height of the notch. Effi
ciency is used here to denote the ratio of the
average tensile stress along the centerline
section to the maximum tensile stress occurring
in the section, for that part of the section
above the base of the notch. Values of the
efficiency corresponding to various values of
p. are given in Table 2. It is interesting to
note that in Figure 6 the stress a becomes
x
compressive in the upper portion of the center
line section for the higher notches.
In order to make a comparison of the
total forces (F_ and F ) corresponding to the
different notches, all the notches considered
were reduced to a "standard width." The data
listed in Table 2 correspond to a standard
halfwidth of 4.0 units.
Qualitative Correlation of Stress and
Geometrical Parameters
Having decided to eliminate from further
consideration the effect of the normal
stresses along the base of the notch, the
next step was to establish the relationship
between the geometrical parameters R . , h, w,
and 0 and the stress parameters F /pw and c/w.
Once such a relationship is established, the
stress concentration factor corresponding to
a notch of given dimensions may be determined
by loading a plain bar with the appropriate
shear stress distribution acting over a length
equal to the width of the notch. In the
subsequent discussion, the geometrical
parameters Rmin, h, and w will be considered
in terms of the dimensionless ratios R . /w
min
and h/w.
It must be pointed out that the geometri
cal parameters characterizing Neuber's notches
have fixed relationships so that the parameters
are not independent of each other. Each
profile in the series is uniquely determined
by the value of o , which in turn determines
the values of the above geometrical parameters.
Because of this, a particular notch profile
can be described by any single geometrical
parameter. In spite of this fixity in the
relationships between geometrical parameters,
an attempt was made to establish a correlation
on the basis of the limited data available and
to develop a method by which these results
might be extended to a wider range of values
of the parameters. In this case, the other
combinations of the geometrical parameters
would represent notches of approximately the
same general shape as the notches in the
series considered by Neuber. This was done
mainly as a preliminary step until more
definite indications of the relationships
between the stress and geometrical parameters
could be obtained using the results of a
finitedifference solution.
In the following, qualitative relation
ships between the stress and geometrical
parameters are postulated on the basis of
observations made in connection with Figure 4
and related data on Neuber's notches. After
the qualitative relationships have been
established, consideration is given to the
problems of determining quantitative relation
ships between the stress and geometrical
parameters and developing a method for extend
ing the results to a wider range of values of
the parameters.
The following were considered in
determining the geometrical parameter with
which to relate the stress parameter c/w.
(1) The parameter c/w exhibits the same
variation as the geometrical param
eter Rmin/w, i.e., c/w decreases
with decreasing values of Rmin/w.
(2) As noted previously, the distance
from the point of maximum stress
(on the surface of the notch) to
the toe of the notch is a function
of the radius of curvature of the
notch profile, the maximum stress
occurring at a point where the
tangent to the profile makes an
angle of approximately 10 degrees
with the axis of the bar for the case
of notches with circular arcs. The
maximum longitudinal stress along the
base section of the notch, which may
be considered as produced predomi
nantly by the shearing stress
component along the same section,
occurs at a point only slightly below
the point of maximum stress. The
location of the point of maximum
longitudinal stress along the base
section is in turn a function of the
position of the peak of the shear
stress intensity curve. Hence, the
latter, as defined by c/w, must be a
function of the radius of curvature
of notch profile.
Although the available data do not
indicate an exclusive relationship between
c/w and R . /w, it was decided to relate c/w
mmn
to the geometrical parameter R . /w alone,
since the latter appears to be the parameter
most intimately connected with it.
No clear indication was obtained from the
limited data available as to which of the
geometrical parameters has the greatest influ
ence on the stress parameter F /pw. However,
it was known that for the limiting case when
R . /w approaches zero, the maximum stress in
mln
the notch theoretically becomes very large,
which means that the shearing stresses along
the base section also become very large. Thus,
F /pw must be a function of R . /w. The
geometrical parameters h/w and 9 must also
influence the parameter F /pw, both of these
quantities being measures of the relative
magnitude of the discontinuity represented by
the notch. Tables 1 and 2 show that F /pw
increases with increasing values of h/w and 6.
In the limiting case when both h/w and 0 are
equal to zero (no notch), F /pw is equal to
zero. Therefore, the stress parameter F /pw
is a function of the geometrical parameters
R . /w, h/w, and 6.
mm
The investigation thus far has led to the
following qualitative correlation between the
stress and geometrical parameters: (1) the
relative distance c/w from the point of maxi
mum shearing stress along the base section of
the notch to the toe of the notch is directly
related to the minimum radius of curvature of
the notch profile, as given by the parameter
R . /w; and (2) the total shearing force along
mmn
the halfwidth of the base section of the
notch, as given by the parameter F /pw, is a
function of the geometrical parameters R min/w,
h/w, and 6.
The next step was to determine quantita
tive relationships between the stress and
geometrical parameters using the data for
Neuber's notches. To accomplish this and to
develop the method for extending the results
to a wider range of values of the parameters,
Filon's solution(19) for the case of a bar of
rectangular cross section under an arbitrary
surface stress loading has been used.
B. THE SURFACESHEAR LOAD METHOD; EFFECT OF
NORMAL COMPONENT OF STRESS ALONG BASE
SECTION OF NOTCH
Filon's Solution: Expressions for Stresses
In 1902 Filon gave a solution(19) for the
stresses and displacements in a bar of rec
tangular cross section in a state of general
ized plane stress subjected to a surfacestress
loading of arbitrary distribution. Using
Filon's solution for the special case of a
surfaceshear loading distributed anti
symmetrically with respect to both x and y
axes, the following expressions are obtained
for the resulting stresses:
S  A cos a a
x n ab
n1
+, 2A 2 cos hab ab sin hab
+ 2n sinh 2ab+2ab coshay cosax
nl
+ 2A cos h aob  a y sinbhayos ax.
nl
r X2A ab sin hab
y 2n sinh2ab+22ab cos ha y cos a x
n=
OD
Scos h a b
2An sin h2ab+2aby sin hy cosx.
n=l
A derivation of the expressions which follow
is given in Appendix B.
2 cosh ab ab sin hab s
\xy L 2An sin h2ab+ 2b shaysinax
n1
+ 2An sin h2ahb+b2aby cosh ay sina x.
n=l
In the above expressions, A represents
the coefficients in the Fourier series
expansion of the surfaceshear load distribu
tion curve. Also a  nif/a, 2a being the total
length of the bar considered. The depth of the
bar is equal to 2b.
The expressions for the stresses in the
bar produced by a surfacenormal stress
loading distributed symmetrically with respect
to both x and y axes follow.
S2 sin habab cos hab
2x C 2n sin h 2a b + 2a b cos hay cosax
00
+Z 2C
n1
T 2Cn
n1
 2Cn
n1
nxy  2Cn
n1
sin h a b
sin h2ab+2ab y sin h y cos ax.
sin h a b + a b cosh ab
sin h 2a b + 2a b cos hay cosax
sin h a b
sin h a2 b + 2a b ay sin hay cos ax.
a b cos h a by sin
sinh2ab2ab inhay sin ax
. , _sin hab
+ 2 Cnsin h2ab 2 a by cos hay sin ha x.
n=l
In the above expressions, C represents
the coefficients in the Fourier series
expansion of the surfacenormal load distribu
tion curve.
In the analysis which follows, the smooth
distribution curves associated with the stress
components along the base of the notch are
replaced by brokenline approximations.
These approximating curves are intended to
incorporate the major features of the actual
curves, at the same time allowing relatively
simple expressions to define them. Thus,
instead of the shear distribution curves of
Figure 4, the general brokenline approximation
of Figure B1 (Appendix B) is used. The
Fourier coefficient A corresponding to this
approximate distribution curve is given by
2 T
A ac o sin aw  sin a (wc) (ll)
n ac 2p w c
where the terms involved above have the
significance indicated in Figure B1 (Appendix
B).
Similarly, the normal stress distribution
curves of Figure 3 are replaced by the general
brokenline approximation of Figure B2b
(Appendix B). The corresponding Fourier
coefficient, which appears in Equations (10)
is given by
C =2
n ap
 H() 0(+)
n sin a (sr)+ n r sina(sr)
S s a(wsg)
+ cos a(wg)  cos a (sr)
  cos aw  cos a (wg)
ag
(12)
Effect of Normal Stress Component Along Base
Section of Notch
The longitudinal stresses a along the
surface of a rectangular bar due to each
component have been calculated, in order to
show that the effect of the normal component
of stress along the base of the notch on the
maximum stress is negligible compared to that
of the shear component. The results are shown
in Figure 8. The curves of Figure 8 corre
spond to surfacestress loadings which
approximate the stress distribution along the
base of Neuber's notch with Po = 0.60. In the
case of the surfacenormal stress loading, the
total positive force (i.e., the positive area
under the curve) was made equal to that of
the corresponding curve of Neuber's notch.
(The positive area under the curve must be
equal to the negative area under the curve.)
In addition, the locations of the point of
zero stress and the point of maximum positive
stress, with respect to the end of the loaded
length (corresponding to the toe of the notch),
were made equal to the corresponding quantities
for the Neuber notch.
In the case of the surfaceshear load,
the total area under the approximating shear
load "triangle" and the distance c from the
point of maximum shearing stress (the apex of
the triangle) to the end of the loaded length
were made equal to the corresponding quantities
for Neuber's notch. Thus, the plot shown in
Figure 8 has been based on the following
values:
for
curve 
the normal stress distribution
F +
n 0.041
pw
s/w  0.60
g/w = 0.180
for
curve 
the shear stress distribution
F
1  0.136
pw
c/w  0.124
with w = 4.0.
Values of a  4.0 and b  2.0 were used in
Equations (9) and (10) to calculate the
longitudinal stress a along the surface of the
bar due to the above surface stress loadings.
As mentioned previously, the maximum
longitudinal stress along the surface of the
bar due to the combined action of the surface
shear load and the uniform tensile stress p
lies to the right of the peak of the shear load
triangle. In Figure 8, the distance of the
peak of the surfaceshear load triangle from
the centerline of the bar is given by x/w =
0.876 (corresponding to a c/w = 0.124), while
the maximum stress (a ) along the surface of
x max
the bar due to the surfaceshear load and the
uniform tensile stress p lies at approximately
x/w = 0.950. On the other hand, the longitudi
nal stress along the surface of the bar due to
the surfacenormal stress loading has the same
general pattern of distribution as that of the
normal stress load producing it. Thus, the
maximum longitudinal stresses produced by the
surfaceshear load and the surfacenormal load
generally do not occur at the same point and
are, therefore, not directly additive.
For the case shown in Figure 8, the normal
stress component contributes 0.086 to the total
maximum stress of ox/p = 1.55 (occurring at
x/w  0.925). This contribution represents
5.6 per cent of the total maximum stress, or an
excess of 5.1 per cent over the maximum stress
occurring at x/w  0.950 due to the shearing
stress component alone. A similar calculation
for the case of Neuber's notch with o = 0.30
shows that the contribution of the normal
stress component is only 0.50 per cent of the
total maximum stress; for the flatter notch with
P " 0.90, the normal stress contribution is
1.6 per cent.
Although based on approximations to the
actual distribution curves of the stress
components along the base of the notch, the
preceding discussion shows that the effect of
the normal component of stress may be neglected
when calculating the stress concentration
factor for a projecting notch.
Comparison of Maximum Stresses for Neuber's
Notches and Those Obtained by the SurfaceShear
Load Approximation
It will be noted that in Figure 8 the
maximum stress produced by the surfaceshear
and normal loads, together with the uniform
stress p, is 16 per cent greater than the
maximum stress obtained for a Neuber notch with
P =. 0.60. This discrepancy is due primarily
to the difference in slopes of the stress
intensity curves near the toe of the notch,
particularly of the shear stress intensity
curve, rather than to the replacement of the
smooth distribution curves by brokenline
approximations. Thus, in the surfaceshear
load triangle, the shear stress starts from
zero at the end of the loaded length (corre
sponding to the toe of the notch) and increases
to its maximum value at a distance c from this
end. In Neuber's notch, on the other hand, the
shear stress along the base section is
generally not zero at x/w = 1.0 and increases
more gradually to its maximum value at a dis
tance c from the toe. Had the base sections
in Neuber's notches been taken slightly below
those used in Figure 4, the ordinates to the
shear stress intensity curves would have
shown a further decrease beyond the point of
maximum stress, becoming almost zero as the
slope of the profile at the "toe" approached
the horizontal. This means that the slope of
the shear stress intensity curve at and near
the toe of the notch is relatively small,
increasing gradually from a value close to
zero to a maximum near the peak of the curve
as one moves toward the centerline of the
notch. This slow increase of the shearing
stress near the toe of the notch is a direct
reflection of the geometry of the Neuber notch,
in which the profile does not become horizon
tal, i.e., the radius of curvature of the
profile decreases gradually from a large value
away from the main projection to a minimum
near the point of maximum stress.
In contrast to the Neuber notch, the
idealized projecting notch, such as is shown
in Figure la, has a welldefined base section
such that the shear stress along this section
becomes zero at the toe of the notch. At the
toe of an idealized notch, the radius of
curvature of the profile changes abruptly from
infinity (for the straight surface of the bar)
to the finite (constant) radius of curvature
of the circular transition curve. In this
case, the shear stress intensity curve would
be expected to have a slope at and near the
toe of the notch greater than that of a Neuber
notch of comparable dimensions (with the base
section defined by either k  0.125 or k =
0.250). In this respect the surfaceshear
load triangle used here would be a closer
approximation to the shear stress intensity
curve along the base section of an idealized
projecting notch than to that of a Neuber
notch.
As will be seen in the next section, the
maximum stress produced by a surfaceshear
loading increases with an increase in the
slope of the shear stress intensity curve near
the toe of the notch. This difference in
geometry between the Neuber notch and the
idealized projecting notch  as reflected in
their respective shear stress distribution
curves along the base sections  would lead
one to expect a higher stress concentration
factor for the idealized projecting notch than
for the Neuber notch with the same minimum
radius of curvature and the same h/w and e
values.
It must be pointed out that although the
maximum stress along the surface of a rectan
gular bar due to the surface stress loads
discussed above is not the same as the maximum
stress in the actual notch, the value obtained
for a properly distributed surface stress
loading should be very close to the actual
maximum stress. (As mentioned earlier, the
maximum stress in the actual notch occurs
along the surface of the notch, at a small
distance above the base section). This fact
makes the surfaceshear load method quite
useful as an approximate method of calculating
stress concentration factors for projecting
notches. For Neuber's notches with the base
section defined by k = 0.250 (see Equation 8),
the ratio of the actual maximum stress to the
maximum longitudinal stress along the base
section varies from 1.099 for 0o = 0.40 to
1.003 for P = 1.50. The ratio decreases with
decreasing notch severity.
However, this difference between the
maximum stress along the surface of a notch
and the maximum longitudinal stress along the
base section of the notch need not cause any
difficulty when applying the surfaceshear
load method. This difference can be allowed
for by defining the equivalent surfaceshear
load as a triangularly distributed surface
shear loading having the same value of F /pw
as in the corresponding notch and a value of
c/w such that the maximum longitudinal stress
along the surface of a plain bar produced by
such a loading is equal to the actual maximum
stress in the notch. With the above defini
tion of the equivalent surfaceshear load, the
added effect of the normal stress component
along the base section of the notch on the
maximum stress is indirectly accounted for.
This will become clear in the next section.
Quantitative Relationships Between the Stress
and Geometrical Parameters
In order to complete the correlation of
the stress and geometrical parameters for
Neuber's notches, it was necessary to develop
a procedure for determining the essential
quantitative relationships which would allow
its extension (to the case of idealized
projecting notches with appropriate modifica
tions where indicated). The procedure utilizes
a series of curves which give the maximum
longitudinal stress in a plain rectangular bar
as a function of the parameters F /pw and c/w,
characterizing the triangular surfaceshear
loading. This series of curves is shown in
Figure 9. The values of the maximum stress
plotted in Figure 9 were obtained by consider
ing 400 terms in the series expression for a
(see Equation B16 and the paragraph immedi
ately following) and by taking w = 4.0,
a  4w, and b = 2w.
The first step in the procedure was to
relate the stress parameter F /pw to the
geometrical parameters influencing it. Because
each of Neuber's notches is characterized by
different values of R min /w, h/w, and e, the
available data is insufficient to determine
accurately the variation of F /pw with the
above geometrical parameters. However, an
attempt was made to obtain an approximation
of the variation of F /pw with the parameters
Rmin/w and h/w, on the basis of previous
observations. This approximation of the
relationship between F /pw and the geometrical
parameters R mi /w and h/w is shown in Figure 11.
The parameter e was not included in Figure 11
since e varies in the same manner as h/w, and
its effect on F /pw may be considered as
already included in h/w. This similarity in
variation between h/w and 6, which is common
to projecting notches with rounded tops, is
clearly shown in Figure 7 for the case of
Neuber's notches.
Figure 11 was obtained by first preparing
a preliminary plot (not shown) which was
similar to Figure 11 but whose curves passed
through points representing average values of
the parameters corresponding to the two values
of the constant k used in the foregoing.
These F /pw versus h/w curves, corresponding
to particular values of R min/w, all pass
through the origin (for the limiting case
when h/w = 0, i.e., no notch) and become
steeper with increasing values of h/w. One
such curve corresponding to the notch defined
by Po 0 0.50, with average values F /pw =
0.1428, h/w = 0.824, and R . /w = 0.250, (refer
to Tables I and 2) is shown as a dashed curve
marked R . /w = 0.25 in Figure 11.
From the abovementioned preliminary plot
the curve of Figure 10, showing the variation
of F /pw with R . /w, was obtained. The
T mi n
values plotted in Figure 10 were based on
F /pw values corresponding to h/w = 0.80. The
ordinate in Figure 10 is expressed in terms of
the ratio of the F /pw value corresponding to
a particular value of R . /w to that corre
mln
sponding to Rmin/w = 0.250. By assuming that
the variation of F /pw with R . /w shown in
Figure 10 applied to all values of h/w, the
F /pw versus h/w curves shown in Figure 11
were obtained. These corresponded to values
of Rmin/w in multiples of 0.10. In this
figure, the curves were cut off at values of
h/w approximately equal to twice the h/w value
of the corresponding Neuber notch. This, in
effect, limits the applicability of the
resulting correlation to projecting notches
with h/w ratios no greater than twice the
corresponding value of the Neuber notch with
the same R . /w value. In connection with
mIn
Figure 11 it is worth noting that the stress
parameter F /pw increases rapidly for the
smaller values of h/w and changes much more
slowly as h/w increases. This is consistent
with the observation made earlier concerning
the decrease in the efficiency of the center
line section with increasing height of notch.
The next step involved determining the
relationship between the ratio R . /w of
min
See page 8 for the special connotation
attached to this term.
Neuber's notches and the ratio c/w of the
equivalent surfaceshear load triangle. This
relationship, which is represented by the
straight line of Figure 12, was obtained by
using the series of curves of Figure 9. The
same two values of k used earlier were used to
prepare Figure 12.
Using Figure 9 in determining the value
of c/w of the equivalent triangular surface
shear load corresponding to Rmi . /w of a
particular notch, a value of (or ) /p equal
max
to the associated value of (o ) /p was
a max
assumed. (As mentioned previously, this step
obviated the necessity of correcting for the
difference between the actual maximum stress
and the maximum longitudinal stress along the
base section of the notch. At the same time
it accounted for the added effect on the maxi
mum stress of the normal stress component along
the notch base.) The desired value of c/w was
then taken as that value corresponding to the
intersection of the appropriate F /pw curve
with the line representing the above value of
(r ) /p. As an illustration, Tables I and 2
give the following for a Neuber notch defined
by p = 0.60, with the base section determined
by k = 0.250:
('d
max
p
R .
min
w
 1.335
= 0.349
h/w = 0.632
 0.136.
Using Figure 9,
from a value of
a horizontal line is drawn
( a) (o1)
max max 1.335
P p
until it intersects the curve corresponding to
F /pw = 0.136. A vertical line dropped from
this intersection to the c/w axis gives a
value of c/w = 0.30. This value of c/w, when
plotted against the corresponding value of
Rmin/w (0.349), determines a point in Figure
12. The same procedure has been followed for
a number of Neuber's notches in preparing
Figure 12.
The points relating the stress parameter
c/w to the geometrical parameter R . /w in
Figure 12, for both values of k, lie very
close to a straight line given by the equation
R .
min = 1.25(c)
W (
Equation (13), which has been taken to
represent the relationship between the above
parameters, indicates that a triangular
surfaceshear load with c/w equal to 0.100
produces a maximum longitudinal stress (ax)max
along the surface of a plain rectangular bar
which is equal to the maximum stress ()max
in a Neuber notch with R . /w = 0.125 and
having the same value of F /pw. Thus, as
developed in the foregoing, the equivalent
surfaceshear load consists in a triangularly
distributed surfaceshear stress loading having
a value of F /pw equal to that along the base
section of the corresponding projecting notch.
Its value of c/w is such that the maximum
longitudinal stress along the surface of a
plain rectangular bar produced by such a
loading  acting together with an axial
tensile loading of intensity p at the ends of
the bar  is equal to the actual maximum
stress occurring in the corresponding
notch.
The use of results of the surfaceshear
load method (which is based on a bar of finite
depth with symmetrically disposed projecting
notches) in conjunction with Neuber's notches
(which are notches in a halfplane) is consid
ered permissible since the final correlation
is independent of the assumed bar depth 2b.
This neglect of the effect of the bar depth on
the maximum stress is not only demanded by the
fact that Neuber's notches have an infinite
depth associated with them but is also justi
fied by the fact that the maximum longitudinal
stress obtained by the surfaceshear load
method is affected very little by the depth of
the bar. This latter fact is clearly indicated
in Table 3, which shows values of (a ) /p
X max
corresponding to values of b/w ranging from
1.0 to 10.0, F /pw  0.075, and F /pw = 0.150.
The maximum difference between the various
values of the maximum stress and those
corresponding to b/w  2.0, the ratio which
was used in obtaining the results plotted in
Figure 9, is 1.09 per cent.
The final step in obtaining the desired
relationship between the stress concentration
factor and the geometrical parameters charac
terizing the profile of Neuber's notches was
carried out using Figures 9, 11, and 12. The
consolidated results are shown in Figure 13.
Thus, the relationship between F /pw, R . /w,
¶ mm
and h/w of Figure 11 was replaced by the
curves marked A and B in Figure 13, the
B curves serving essentially as a curved
F /pw scale. It should be noted that a pro
gressively decreasing scale was used for the
h/w axis, since the variation of h/w was
assumed to be linear between the marked points
on the top scale. The relationship between
c/w and Rmi . n/w of Figure 11 appears in Figure
13 as a change from the c/w scale of Figure 9
to the R . /w scale at the bottom of the
mmn
figure.
In using Figure 13 to determine the
stress concentration factor for a projecting
notch of given characteristic dimensions, the
procedure is as follows.
(1) From a point representing the
given h/w value on the top scale, a
vertical line is dropped until it
intersects the curve marked A corre
sponding to the given value of R . /w
m n
(estimating intermediate values if
necessary).
(2) One then moves from this point
in a direction parallel to the curves
marked B (again estimating intermediate
values when necessary) to the given
value of R . /w as indicated on the
min
bottom scale.
(3) A horizontal line drawn from
this point to the (ay)max/p axis gives
the stress concentration factor.
A more convenient form of the relationship
between the stress concentration factor and the
geometrical parameters characterizing the
profile of the Neuber notch is shown in Figure
14. This figure, based on Figure 13, repre
sents the final results of the correlation for
Neuber's notches. The curve corresponding to
h/w = 0.20 in Figure 14 appears as a dashed
curve in Figure 13. In both Figures 13 and 14,
the dashed lines indicate the use of the
graphs for the case of a Neuber notch defined
by po = 0.60 and k = 0.250.
The values of the stress concentration
factor for Neuber's notches (obtained using
Figure 14) are compared with the calculated
values in Table 4. This table includes only
those notches with characteristic dimensions
which fall within the range of the respective
parameters in Figure 14. A maximum error of
6.0 per cent occurs for the higher notches,
with smaller errors corresponding to the
flatter notches.
The larger errors in the values of the
stress concentration factor obtained from
Figure 14 occur for the notches defined by
Bo  0.25 and po 0 0.30. The profiles of
these notches have flank angles exceeding
90 degrees, i.e., the notch profile in each
case flares out above the point of minimum
radius of curvature so that a vertical line
intersects the profile at three distinct
points (see Figure 2). Had the points
corresponding to the above two notches been
plotted in the (a ) /p versus e graph of
Figure 7, (these points would fall beyond the
limits of Figure 7) they would lie farther
from the straight line representing the
variation of the maximum stress with the flank
angle e than the points on the plot corre
sponding to the other notch profiles. In
contrast, the points corresponding to these
same two notch profiles in the (a )max/p versus
h/w plot of Figure 7 lie along the straight
line representing the variation of the maximum
stress with the ratio h/w. This indicates
that for these two particular profiles, the
flank angle does not vary in the same manner
as does the ratio h/w for the other notch
profiles, and that the effect of the flank
angle on the maximum stress cannot be
considered as being directly reflected in the
ratio h/w. Thus, the difference between the
calculated values of the stress concentration
factor and the corresponding values obtained
from Figure 14 for notch profiles with flank
angles exceeding 90 degrees is the result of
an increased effect of the flank angle.
0 0
IV. ANALYSIS OF STRESSES FOR IDEALIZED PROJECTING NOTCHES
A. FINITEDIFFERENCE SOLUTION
In order to arrive at a correlation
between the stress concentration factor and
the geometrical parameters for the case of
the idealized projecting notch of Figure la,
using the same method employed in connection
with Neuber's notches, a finitedifference
solution was employed. The finitedifference
method, although approximate, allows a
controlled variation of the geometrical
parameters defining the notch profile, a
requirement essential in this study. Thus,
although the geometrical parameters charac
terizing the profiles of Neuber's notches have
fixed relationships, the finitedifference
method made it possible to specify the dimen
sions of the notch profile and to vary the
parameters independently of each other. This
flexibility allowed the effect of each
parameter to be ascertained more fully.
A brief description of the method of
finite differences and a derivation of the
boundary conditions in terms of Airy's stress
function for the particular case considered
here is given in Appendix C. Because of the
twofold symmetry of the notched bar section,
only onefourth of the entire section was
considered in the solution. This is shown in
Figure CI of Appendix C. A value of w equal
to 4.0 units has been used.
Using the finitedifference method,
values of the stress concentration factor
were obtained as well as the stress parameters
F /pw and c/w, corresponding to a number of
notch profiles. All calculations made in this
study were carried out on the University of
Illinois' IBM 7094 electronic digital computer.
The limited storage space in the computer is a
major problem which must be considered in
using the method of finite differences. The
storage space available in the computer
memory limits the number of equations which
may be solved and hence the fineness of the
grid which may be used for a given area. This
is particularly important since accuracy is
heavily dependent upon the fineness of the
grid used.
By solving the resulting set of simul
taneous equations by iteration rather than by
a direct method and by taking advantage of the
sparseness of the associated matrix of
coefficients, it was possible to write a
computer program capable of handling 1000
points on a 40 x 45 mesh grid.
The complete solution involved three
steps. The first employed a relatively coarse
grid, the starting values of the stress func
tion for the iterative solution being taken to
equal zero. The values of the function
obtained from this coarsegrid solution (with
values at intermediate points calculated by
linear interpolation) were then used as
starting values for the iterative process of
the second step. In the second step, a grid
spacing equal to onehalf that of the first
step was used. By thus obtaining rough
approximations of the values of the function
with the coarsegrid solution of the first
step and using these as starting values for the
second step, the convergence of the iterative
process in the latter was hastened considerably.
A further refinement of the grid spacing
was carried out in the third step of the solu
tion, which used a mesh interval equal to
onehalf that of the second step and included
only a portion of the area considered in the
first two steps. For this final step, the
area considered was bounded by a horizontal
line a few mesh intervals below the base of
the notch and a vertical line a few mesh
intervals beyond the toe of the notch. This
new boundary is indicated by the dashed line
marked S in Figure C1 of Appendix C. Values
of the stress function along this new boundary
as obtained in the second step  with inter
mediate values calculated by parabolic
interpolation  were used as the new boundary
values for the third step.
In addition to the values of the stress
function obtained in the preceding coarsermesh
solution as starting values for the second
and thirdstep iterations, systematic over
relaxation (20,21) was used to improve the
convergence of the GaussSeidel method. The
optimum value of the overrelaxation factor
(corresponding to the maximum rate of
convergence) was determined by trial and
ranged from 1.65 to 1.75, decreasing with the
closeness of the starting values to the true
solution and increasing with the number of
equations involved (the factor can vary
theoretically from 1.0 to 2.0). The conver
gence criterion used required that the absolute
value of the residual in 99 per cent of the
equations involved be equal to or less than a
specified value. The results given below were
obtained using a maximum allowable residual of
0.05 for the first step and 0.01 for the second
and third steps. The magnitudes of these
residuals were selected with the aim of
obtaining a solution requiring only a reasona
ble amount of computer time and are believed
to have resulted in relatively small errors in
the calculated values of the stress function.
With the above criteria, convergence was
achieved after an average of 340 cycles of
iteration for the first step, 80 cycles for
the second step, and 40 cycles for the third
step. These correspond to an average of 150
points (or unknowns) for the first step, 550
points for the second step, and 700 points for
the final step. In all cases, there was a
slight increase in F /pw and a comparatively
greater increase in a max/p with each refinement
of the grid spacing, corresponding to each step
of the solution.
Because of the limited capacity of the
program, the total length a of the section
considered was limited to a value slightly
greater than twice the halfwidth of the notch
w; the depth of the section below the base of
the notch b was limited to a value slightly
greater than w. The results of the calcula
tions listed in Table 5 were obtained using
a/w = 2.10 and b/w  1.05 for the first two
steps of the solution, and al/w = 1.425 and
b'/w = 0.375 for the final step. The
restriction on the number of unknowns capable
of being handled by the program eliminated the
possibility of studying the effect of the
depth of the bar 2b on the stress concentra
tion factor. However, the results of a
limited number of twostep solutions carried
out to check this effect indicate a slight
decrease in both F /pw and a max/p with an
increase in b/w. A solution using a value of
a/w  3.00 did not show any significant
difference from the stress values correspond
ing to a/w  2.10.
B. CORRELATION OF STRESS AND GEOMETRICAL
PARAMETERS
To carry out the desired correlation, the
stresses at points along the surface of the
notch as well as those along the base and
centerline sections were calculated for notches
with flank angles of 30 degrees, 60 degrees,
and 90 degrees. The ratio R/w was given
values of 0.30, 0.40, 0.50, and 0.60, while
h/w was varied from 0.30 to a maximum of 0.90.
The results of the calculations are given in
Table 5.
Using Simpson's rule, the values of F /pw
and F/ pw in Table 5 were calculated from the
values of the stresses along the respective
sections. It is worth noting the close agree
ment between the values of these two quantities
for the different notches considered. The
values of the ratio c/w, as well as those of
max /p, were obtained graphically, i.e., by
max
passing smooth curves through points repre
senting calculated values. Examples of such
plots are shown in Figures 15 and 16. In all
cases considered, the maximum stress occurred
at a point along the circular arc where the
tangent made an angle of between 5 and 15
degrees with the axis of the bar.
Although approximate, the results of the
calculations are in general reasonably con
sistent and provide a good indication of the
relative effects of the different parameters
considered. The calculated values along the
boundary, where interpolation equations had to
be used, are likely to be in greater error
than those corresponding to interior points.
Notches with e  90 Degrees
Stresses for notches with flank angles
equal to 90 degrees were calculated for values
of h/w equal to 0.30, 0.45, 0.60, 0.75, and
0.90. These values of h/w were chosen to
maintain the same grid spacing throughout.
Thus, for the notch with h/w  0.60, the
height of the notch h was divided into four
equal parts in the first step of the solution;
for the notch with h/w  0.75, it was divided
into five equal parts, etc. The program was
written so that the mesh interval for the
firststep solution was determined by dividing
the height of the notch into a number of equal
parts, the minimum number being three.
Because of this limitation on the number of
parts into which the height of the notch could
be subdivided and in order to maintain the
same grid spacing in the final step of the
solution for all the notches considered, the
stresses for notches with h/w = 0.30 were
obtained by using only the first and final
steps of the complete solution. For this
particular value of h/w, the height of the
notch was divided into four equal parts and a
maximum allowable residual of 0.01 was used
for the iterative solution in the first step.
With the height of the notch divided as
indicated above, a grid spacing of 0.150 units
was obtained for the thirdstep solution.
This gave values of the ratio of the mesh
interval to the radius of curvature A/R of
0.125, 0.094, 0.075, and 0.062 corresponding
to R/w values of 0.30, 0.40, 0.50, and 0.60.
Essentially the same procedure as that
used for the case of Neuber's notches was
employed to obtain the desired relationship
between the stress concentration factor and
the geometrical parameters defining the notch
profile.
A plot of the variation of the shearing
stress parameter F /pw with h/w for different
values of R/w is shown in Figure 17. The
values of F /pw used in Figure 17 have been
taken as the average of the calculated values
of F /pw and F a/pw. Since, for equilibrium,
these two values should be equal, their
average appears to provide the most represen
tative value. The curves through the plotted
points corresponding to different values of
h
a
FIGURE la. SYMMETRICALLY DISPOSED PROJECTING NOTCHES
IN A BAR OF RECTANGULAR CROSS SECTION
notch
xy
FIGURE Ib. TYPICAL DISTRIBTUIONS OF STRESSES
ALONG CENTERLINE AND BASE SECTIONS
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1.0 1.2
FIGURE 6. DISTRIBUTIONS OF LONGITUDINAL STRESSES ALONG
THE CENTERLINE OF NEUBER'S NOTCHES
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SURFACESHEAR LOAD AND UNIFORM STRESS ACROSS ENDS
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S k = .125
S k = .250
w = 4.o
ttttt 0 r'
.025
.050
.075
.100
S/pw
FIGURE 11. SHEARING STRESS PARAMETER FT/pw AS A FUNCTION OF
Rmin/w AND h/w FOR NEUBER' S NOTCHES
1.6
1.2
C
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na4
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FIGURE 12.
CORRELATION OF Rmin/w OF NEUBER'S NOTCHES
WITH c/w OF TRIANGULAR SURFACESHEAR LOAD
ul
BASE SECTION OF NOTCH DEFINED BY: Xt = Xo* kRmin
S k = .125
e  k = .250
if IZ
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.1136
.1220
.1241
.1254
.1294
.1062
.1135
.1165
.1187
.1196
.1028
.1105
.1125
.1138
.1147
.0984
.1071
.1094
.1100
.1110
.1101
.1189
.1216
.1247
.1253
c/w
a
max
1.66
1.71
1.71
1.72
1.74
1.55
1.60
1.61
1.61
1.62
1.44
1.47
1.48
1.49
1.50
1.39
1.42
1.43
1.43
1.43
1.59
1.63
1.63
1.64
1.64
TABLE 5. SUMMARY OF RESULTS OF FINITEDIFFERENCE SOLUTION FOR
THE IDEALIZED PROJECTING NOTCH
pw
Fp
1
I
Flank
R/w
IFO
pw
.40
TABLE 5. (Continued)
h/w
.30
.45
.60
.75
.90
.30
.45
.60
.75
.90
.30
.45
.60
.75
.90
.30
.45
.30
.45
.30
.45
.30
.45
 .1048
 .1128
 . 1156
 .1177
 . 1191
 . 1011
 . 1095
 .1108
 . 1130
 .1141
 .0975
 .1063
 .1087
 .1100
 .1106
 .0988
 .1088
 .0972
 .1048
 .0950
 .1023
 .0930
 .1006
.1055
.1136
.1165
.1185
.1202
.1024
.1101
.1118
.1135
.1144
.0981
.1068
.1089
.1101
.1110
.1015
.1098
.0990
.1056
.0963
.1034
.0943
.1002
c/w
.14
.14
.14
.14
.14
.16
.16
.16
.16
.16
.18
.18
.18
.18
.18
.12
.12
.14
.14
.16
.16
.18
.18
0
max
P
1.50
1.54
1.55
1.55
1.56
1.44
1.47
1.48
1.48
1.49
1.39
1.41
1.42
1.43
1.43
1.54
1.59
1.46
1.50
1.42
1.46
1.37
1.41
F I
pw
8S
U
ii
II
as
88
11
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. P4 .4 V C4 v
g ^ iG S E; 0% 10i i
S4 Ca4 C w0 4 C4
0 0 0 %4
B 9 %0 S 9n
ja N^ ' < 4 N 4 4 Nt >4
N A N N S tS N NM N C
4 .
o1 o 0o
I
)0
I 5
i
LU
L 0
U
Of
U
LU
LU
«0
C .
D
CA
O S
LU 
ILU
(D
CD
Ln =!
LU LU
O I
H 
LU
C). cc
II
i s
I_i
0
I
R/w clearly indicate that the stress parameter
F /pw is a function of the ratio R/w, as
indeed it should be (F /pw being theoretically
infinite when R = 0). Although no values less
than 0.30 are available for h/w,the F /pw
versus h/w curves of Figure 17 have been
extended below this line since in this region
the curves are reasonably well determined by
the condition that when h/w = 0 (no notch),
F /pw = 0, i.e., the curves must pass through
the origin.
In order to extend the data obtained from
the calculation to a wider range of values of
the ratio R/w, an extrapolating curve was
obtained graphically, i.e., by passing a
smooth curve through points representing
calculated values in the F /pw versus h/w plot
of Figure 17. This curve was then prolonged
on both sides in what appear to be reasonable
extensions. This extrapolation is shown in
Figure 20. The F /pw values used in preparing
Figure 20 were those corresponding to h/w =
0.75. The ordinates to the curve in Figure 20
are expressed in terms of the ratio of the
value of F /pw for a particular R/w to the
value of F /pw for R/w = 0.40. In extending
the curve of Figure 20 beyond the plotted
points, the following limiting conditions were
considered.
(1) As R/w approaches zero, the
value of F./pw becomes very
large.
(2) As R/w becomes very large
(limiting case: no notch),
F /pw approaches zero.
By assuming that the variation of F /pw with
R/w given by the above extrapolation curve
applies for all values of h/w (an assumption
which is reasonable in view of the approxi
mately parallel directions which the calcu
lated F /pw versus h/w curves take for the
r
different R/w values). F /pw versus h/w curves
r
for R/w  0.05, 0.10, 0.20, 0.70, and 0.80 are
shown in Figure 17.
Figure 17 clearly shows that for a given
value of 9 the shearing stress parameter
F /pw is a function of the parameters R/w and
h/w. It is worth noting in connection with
Figure 17 that for a given value of R/w, the
stress parameter F /pw increases very rapidly
for small values of h/w, the rate of increase
diminishing with increasing h/w. For h/w
greater than about 0.60, the increase in F /pw
with h/w becomes very small, a trend which
may be more clearly understood in terms of the
diminishing efficiency of the centerline
section with increasing height of notch, i.e.,
as a greater portion of the section comes
under compressive stress.
With the relationship between the stress
parameter F /pw and the geometrical parameters
e, h/w, and R/w known, the next step was to
determine the relationship between c/w and R/w
using Figure 9.
Table 5 clearly indicates that the ratio
c/w is a function of R/w only and is indepen
dent of h/w and e. This fact confirms the
assumption made in this regard in connection
with Neuber's notches. The procedure used in
determining the relationship between c/w and
R/w follows that used for Neuber's notches.
A plot of the c/w versus R/w relationship is
shown in Figure 21. In Figure 21 the tendency
of the curve to flatten as R/w increases is
due primarily to the fact that the triangular
surfaceshear load approximates the actual
shearing stress distribution curve along the
base section of the notch more closely for
notches with large R/w values than for notches
with small radii. This is apparent from a
comparison of the shearing stress distribution
curves corresponding to different values of
R/w shown in Figure 15.
As in the case of Neuber's notches, the
graph giving the stress concentration factor
See page 8 for the special connotation
attached to this term.
as a function of the geometrical parameters
was obtained by superposing the relationships
given in Figures 17 and 21 onto Figure 9. The
composite plot for this case (e  90 degrees)
is shown in Figure 22. In this figure a
maximum value of 1.0 is shown for the ratio
R/w. This value represents the limiting case
above which the flank angle becomes less than
90 degrees.
The procedure to be followed in using
Figure 22 is similar to that outlined for the
analogous figure on Neuber's notches. Thus,
for a notch of given dimensions, one proceeds
as follows.
(1) From a point representing the
given h/w value on the top scale, a
vertical line is dropped until it
intersects the curve marked A corre
sponding to the given value of R/w
(estimating intermediate values, if
necessary).
(2) One then moves from this point in
a direction parallel to the curves
marked B (again estimating intermediate
values when necessary) to the given
value of R/w as indicated on the bottom
scale.
(3) A horizontal line drawn from this
last point to the amax /p axis gives
the stress concentration factor.
It is significant to note that the
parameter R/w enters twice in the plot of
Figure 22. The dependence of both stress
parameters, F /pw and c/w, on the radius of
curvature of the notch profile clearly points
to the marked influence of this parameter on
the stressraising action of the notch.
A more convenient form of the relation
ship given in Figure 22 is shown in Figure 25,
which represents the final correlation for the
case of notches with e  90 degrees. Figure
25 was prepared by first expanding and
linearizing the R/w scale of Figure 22. This
intermediate step is shown in Figure 22a, in
which the final curve corresponding to h/w 
0.20 is presented as a dashed curve. (It will
be noted that the expansion of the R/w scale
of Figure 22 was carried out with respect only
to the curves marked B, which are actually
F /pw curves; the curves marked A were
superposed directly from Figure 17 after the
h/w scale at the top of Figure 22a had been
set.) Because the B curves of Figure 22 are
not well defined for values of R/w less than
0.20, the corresponding curves in Figure 22a
have been extended to cover this region. The
shape of the extensions are indicated by the
slope of the curves in the region R/w > 0.20
and by the fact that the curves tend to large
values of max /p as R/w approaches zero.
However, the analysis has not been adequate
in verifying the relationships in this low
R/w region.
In Figure 25, the dashed line inter
secting the h/w curves marks the boundary
between notch profiles having R/w less than
h/w and those having R/w greater than h/w. It
will be noted that for the latter case the
flank angle is actually less than 90 degrees,
and is given in this case by the angle which
the tangent to the circular arc at the top of
the notch makes with the axis of the bar.
Notches with 6 = 60 Degrees and e = 30 Degrees
The same procedure as described above was
used in analyzing notches with e  60 degrees
and 6  30 degrees. Stresses were calculated
for values of h/w  0.30, 0.45, 0.60, 0.75,
and 0.90 in notches with e  60 degrees; and
for h/w = 0.30 and 0.45 in notches with
e = 30 degrees.
The curves relating the stress parameter
F /pw and the geometrical parameters h/w and
R/w for the above cases are shown in Figures
18 and 19. The F /pw versus h/w curves in
these figures were cut off at points
representing the maximum possible values of
h/w associated with the respective R/w values.
(The maximum value of the ratio h/w in each
case corresponds to R  0.) The extrapolation
curves used to extend the range of the F /pw
versus h/w plots to other values of R/w, as
well as the c/w versus R/w relationship for
both of the above cases, are shown with the
corresponding relationships for 9 = 90 degrees
(Figures 20 and 21, respectively). For notches
with e  30 degrees the plotted points in
Figure 20, representing calculated values,
were based on F /pw values for h/w = 0.45.
A comparison of the F /pw versus h/w
curves of Figures 17, 18, and 19 corresponding
to the three values of 0 under consideration
shows that the curves associated with the
larger values of R/w are almost coincident for
notches with 9e 60 degrees and e = 90 degrees,
while the curves for e  30 degrees lie close
to the corresponding curves for the above two
values of e. This indicates a diminishing
effect of the flank angle on F /pw as R/w
increases. This decrease in the effect of e
with increasing R/w is also indicated in the
c/w versus R/w relationships shown in Figure
21, where the single curve representing 0 
60 degrees and 9  30 degrees joins the curve
for e  90 degrees at the larger values of
R/w. The abovenoted tendency immediately
becomes evident from a comparison of the
calculated values of F /pw and max /p for the
different cases given in Table 5.
The composite plots giving the stress
concentration factor as a function of the
geometrical parameters for notches with 0 
60 degrees and 0  30 degrees are shown in
Figures 23 and 24, respectively. As in
Figure 22 (for e  90 degrees), a maximum
value of 1.16 is shown for the ratio R/w in
Figure 23, this value representing the
limiting case above which the flank angle
becomes less than 60 degrees. (The corre
sponding limiting value for notches with 6 =
30 degrees is 1.99.)
Plots of Figures 23 and 24 in the same
form as Figure 25 for e = 90 degrees are
given in Figures 26 and 27. The curves for
the higher values of h/w in the latter figures
were cut off at points corresponding to values
of R/w for which they represent the maximum
possible h/w values. As in the case of
Figure 25, the dashed line cutting the h/w
curves in Figure 26 (for notches with 0 =
60 degrees) separates notch profiles with
flank angles equal to 60 degrees from those in
which the flank angle is less than 60 degrees,
i.e., when R/w is greater than 0.50 h/w.
V. EXPERIMENTAL INVESTIGATION AND COMPARISONS WITH THEORETICAL ANALYSIS
A. EXPERIMENTAL INVESTIGATION
Even though small in number, the speci
mens tested in this study were intended to
provide a basis for comparison with values
obtainable from the theoretical analysis of
the preceding chapter. Specifically, it was
the author's intention to determine experi
mentally the maximum stresses and the total
tensile force across the centerline section of
the notch F in each of the specimens.
As originally planned, the stresses were
to be determined by means of strain gage as
well as "Photostress" measurements. The
latter method makes use of a sheet of
birefringent plastic cut into the same shape
and bonded onto the face of the specimen, and
a reflection polariscope operating on essen
tially the same principle as the standard
photoelastic apparatus. However, the strains
at the centerline section of the notches were
of such small magnitude that it was impossible
to obtain accurate readings for the stresses
along this section without loading the speci
mens to a point where the critical sections
would have yielded. In view of this limita
tion, the Photostress measurements were
abandoned and only strain gage readings were
taken.
Test Specimens and Procedure
The six specimens tested were fabricated
from 1/4inchthick plate conforming to
American Society for Testing and Materials
specifications for A7 steel. The general
dimensions of the specimens are shown in
Figure 28a, while the general shapes of the
notch profiles are shown in Figure 28b. The
measured values of the geometrical parameters
defining the notch profiles are listed in
Table 6, which also includes the results of
the tests.
An average of ten metalfilm strain gages,
having gage lengths of 1/16 of an inch and
1/32 of an inch, were used on each specimen.
In two of the specimens, where slight varia
tions in the measured values of the radii
were observed, it was found advisable to
install additional gages on the second notch.
The 1/32inch gages were used along a short
length on the shaped edge of each specimen
where the maximum strain was expected to occur;
the 1/16inch gages were used to measure the
nominal strain and the strains along the
centerline section of the notch.
The specimens were tested in a 120,000
pound Universal testing machine. Because
some eccentricity in the loading was noted
early in the tests (due to a slight curvature
in the specimens as well as misalignment in
the bolt holes), an effort was made to reduce
the effect of this eccentricity by taking
three series of readings with the gaged side
of the specimen facing one direction and
another three series of readings with the
specimen turned 180 degrees. In order to
minimize the effect of a nonuniform seating
of the bolts which fastened the specimen to
the pullheads, the bolts were loosened at the
end of each series of readings and retightened
before the start of the next. Each series of
readings was started from zero load, and then
brought up to the maximum load in increments
of 5000 and then 10,000 pounds. The maximum
load was determined by estimating, on the basis
of the smallload readings, the load which
would just cause the material at the critical
sections to yield. The maximum load for each
case was then set slightly below this
estimated value.
Test Results
Strain versus totalload plots for several
of the gages on specimen PS4 are shown in
Figure 29. Similar plots were prepared for
each of the gages used. It will be noted that
in Figure 29 the lines tend to converge and
form narrower bands as the load increases.
The final strainversustotalload line for
each gage was taken as a line connecting the
origin with the point representing the average
of the six measured strain values at the maxi
mum load. The values of the stresses and
their ratios to the calculated nominal stress
were determined using the values of the
strains obtained in this way. The total force
across the centerline of the notch F (which
is equal to the total shearing force along
onehalf the base width of the notch F ) was
determined graphically.
B. COMPARISON OF RESULTS OF THEORETICAL
ANALYSIS WITH EXPERIMENTAL RESULTS
Column 11 of Table 6 lists the predicted
stress concentration factors corresponding to
the notches in Specimens PS3, 4, and 5 as
obtained from Figure 27. The per cent
difference between the predicted values and
the respective measured values are recorded in
column 12. Except for the case of PS5 which
represents a notch with a relatively large
radius of curvature, the values obtained from
Figure 27 are considerably lower than the
measured values.
In an effort to further check the values
obtained by the finitedifference solution
and the subsequent analysis, the case of
projecting notches with low h/w or large w/h
values was considered, i.e., notches approach
ing the case of a bar with shoulder fillets.
For this purpose, Figure 30 was prepared
showing the variation of the stress concentra
tion factor with the ratio w/h for notches
with e = 90 degrees, 60 degrees, and 30
degrees, and R/h values of 0.50, 1.0, and 2.0.
In this figure, the various curves were drawn
as fine lines in regions where one or more of
the defining parameters would no longer be
realized. For instance, the entire curve for
e = 90 degrees and R/h = 2.0 was drawn as a
fine line because e is always less than 90
degrees for this value of R/h.
Figure 30 shows that for a particular
value of e and R/h, a max/p increases with
increasing w/h until a value of the latter of
about 5.0 is reached beyond which max /p
becomes approximately constant. This constant
value of the stress concentration factor
corresponds to the case of a projecting notch
with relatively large width or a bar with a
shoulder fillet and may, therefore, be com
pared with the corresponding empiricallybased
values given by Peterson(10) or Heywood.(ll)
Such a comparison is presented in Table 7.
Since Figures 25, 26, and 27 were based
on a solution using a halfdepth of bar b
approximately equal to the halfwidth of the
notch w, the value of b used in column 4 of
Table 7 was made equal to the value of w
corresponding to the w/h value at the point
where the appropriate curve in Figure 30
becomes horizontal. The points marking these
w/h values are indicated by small crosses in
Figure 30. The values of h and R used in
Table 7 were chosen mainly to give the desired
ratios of R/h = 0.50, 1.0, and 2.0. (The
ratios of these quantities, rather than their
absolute values, are the significant
parameters.)
The values listed in column 7 of Table 7
were obtained from Figure 57 of Reference 10.
Since the figure gives values corresponding to
the case of e = 90 degrees, the values taken
from this figure were corrected for the cases
where e  60 degrees and e = 30 degrees by
using the empirical relationship given in
Figure B3 of Reference 26. The last column
in Table 7 gives the per cent difference
between the stress concentration factors
obtained from Reference 10 and those from
Figure 30.
The values of the stress concentration
factor as obtained from the analysis of the
preceding chapter are found to be consistently
lower than the corresponding values given by
Peterson. The largest differences listed in
column 9 of Table 7 occur for notches with
small radii and flank angles of 60 degrees and
90 degrees. The values listed in columns 7
and 8 of Table 7, however, compare reasonably
well. It is particularly worth noting the
close agreement between the values for notches
with e  30 degrees. This is in contrast to
the large differences listed in Table 6
between the values from Figure 27 and the
experimental results for specimens PS3 and
PS4. This would seem to indicate that the
measured values of amax /p listed in Table 6 for
PS3 and PS4 are too high for the recorded
R/w values and may have been caused by small
irregularities at the critical sections of the
profiles produced by grinding in the prepara
tion of the specimens.
The above comparisons suggest that the
values given by Figures 25, 26, and 27 may be
slightly lower than what they should be. It
may be noted here that had a further refine
ment of the grid spacing been possible in the
previouslydiscussed finitedifference
solution, slightly higher values of the stress
concentration factor could have been obtained.
(This could be expected on the basis of the
observed increase in the calculated maximum
stress accompanying each refinement of the
grid spacing.)
*0e
VI. SUMMARY AND CONCLUSIONS
A. SUMMARY
The principal object of this study was
the determination of the effects of the
various geometrical parameters characterizing
the profiles of projecting notches on the
associated stress concentration factors. In
particular, consideration was given to notches
with circular transition curves (referred to
here as idealized projecting notches) occurring
symmetrically in a rectangular bar subjected
to axial loading.
The method of analysis used here accom
plished the object of the study indirectly by
considering the effects of the different
geometrical parameters on the magnitude and
distribution of the shearing stresses along
the base section of a projecting notch. As
a preliminary step, the stresses along selec
ted sections in a projecting notch in a
halfplane were obtained using a solution
given by H. Neuber. By combining these data
with a solution by L. N. G. Filon for the
stresses in a rectangular bar resulting from
surface stress loads it was shown that the
maximum longitudinal stress along the base
section of a projecting notch is produced
primarily by the shearing stress component
along the base section. The effect of the
normal stress component along the same section
was shown to be negligible. This has led to
the observation that the maximum stress in a
projecting notch may be reasonably approximated
by the longitudinal stress in a plain rectan
gular bar subjected to an axial load and a
surfaceshear stress loading of the proper
magnitude and distribution.
The above observation was used as the
basis for employing an equivalent surface
shear loading to approximate the stress
raising action of a projecting notch. As used
in this study, the equivalent surfaceshear
load consists of a triangularlydistributed
surfaceshear loading having a total area
under the distribution curve (from centerline
to toe of notch) F /p equal to that along the
base section of the corresponding projecting
notch; the apex of the triangular distribution
curve (defined by the ratio c/w) is located
such that the maximum longitudinal stress along
the surface of a plain rectangular bar pro
duced by such a loading  acting together
with an axial tensile load of intensity p at
the ends of the bar  is equal to the actual
maximum stress occurring in the corresponding
projecting notch.
Using the results of the equivalent
surfaceshear load method, a procedure was
developed for establishing a correlation
between the stress concentration factor and
the geometrical parameters characterizing the
profile of projecting notches. The procedure
developed allows the extension of results
of calculations for a relatively few cases
to a wider range of values of the geometrical
parameters. The resulting relationships,
obtained for the case of Neuber's notches and
also for idealized projecting notches (in
which the stresses were obtained using a
finitedifference solution), provide a clear
indication of the effects of the various
parameters considered.
B. CONCLUSIONS
Although the values of the stress concen
tration factor obtained for the idealized
projecting notch are approximate, owing to the
approximations inherent in the finite
difference solution used, the following conclu
sions may be drawn validly from the above
study.
(1) The stressraising action of a
projecting notch is a function of the
magnitude and distribution of and may be
reasonably well approximated by the effect
of the shearing stress component along its
base section. The effect of the normal
stress component along the same section
is negligible.
(2) The distribution of the shearing
stress component along the base section
of the notch is characterized principally
by the stress parameters F /pw and c/w
(F being the total shearing force along
the halfwidth of the notch w, p being
the nominal stress, and c being the dis
tance from the peak of the shear stress
intensity curve to the toe of the notch.
The stress concentration factor increases
with increasing values of F /pw and
decreasing c/w).
(3) The shearing stress parameter
F /pw is a function of the geometrical
parameters R/w, h/w, and e.
(a) The value of the stress
parameter F /pw increases with
decreasing values of R/w, becoming
theoretically infinite when R = 0.
For a given value of h/w, the effect
of R/w on F /pw diminishes with
decreasing values of the flank angle
0. (See Figures 17, 18, and 19.)
(b) For particular values of
R/w and e, F /pw increases rapidly
for small values of h/w, the rate of
increase diminishing markedly and
progressively becoming smaller as
h/w exceeds a value of about 0.50.
(See Figures 17, 18, and 19.)
(c) For given values of R/w
and h/w, F /pw decreases with
decreasing values of the flank
angle 6, the decrease being small
for large values of R/w and
increasing with decreasing values
of R/w. (Compare Figures 17, 18,
and 19.)
(4) The stress parameter c/w is a
function only of the geometrical param
eter R/w and is independent of h/w and 0.
(See Table 5 and Figure 15.) It decreases
with decreasing values of R/w.
(5) The stress concentration factor
for idealized projecting notches is thus
a function of all three geometrical
parameters defining the notch profile,
i.e., R/w, h/w, and 6, the most signifi
cant of these being R/w. The relative
importance of each of these parameters
varies with the values of the other
parameters, as discussed above.
VII. REFERENCES
1. R. K. Sahgal and W. H. Munse, "Fatigue
Behavior of Axially Loaded Weldments in
HY80 Steel," University of Illinois
SRS 204, September,1960.
2. A. J. Hartmann and W. H. Munse, "Fatigue
Behavior of Welded Joints and Weldments
in HY80 Steel Subjected to Axial
Loadings," University of Illinois SRS 250,
July, 1962.
3. W. W. Sanders, Jr., A. T. Derecho, and
W. H. Munse, "Effect of External Geometry
on Fatigue Behavior of Welded Joints,"
Welding Journal, February, 1965.
4. H. Kihara, T. Yoshiaki, M. Watanabe, and
Y. Ishii, "Effect of Flaws in Welds on
Their Strength," Nondestructive Testing
of Welds and Their Strength (The Society
of Naval Architects of Japan [Tokyo,
1960]).
5. Newman and Gurney "Fatigue Tests in Plain
Plate Specimens and Transverse Butt Welds,"
British Welding Journal, VI, No. 12
(December, 1959) 569594.
6. J. E. Tomlinson and J. L. Wood, "Factors
Affecting the Fatigue Behavior of Welded
Aluminum," British Welding Journal, VII,
No. 4 (April, 1960) 250264.
7. W. 0. Dinsdale, "Effect of Reinforcement
Shape on Fatigue Behavior of Butt Welds
in NPS/6," British Welding Journal, XI,
No. 5 (May, 1964) 233238.
8. H. Neuber, "Theory of Notch Stresses:
Principles for Exact Stress Calculation,"
(English Translation by F. A. Raven,
Translation 74, David Taylor Model Basin,
Navy Department, Washington, D.C.).
9. M. M. Leven and A. J. Hartmann, "Factors
of Stress Concentration for Flat Bars and
Shafts with Centrally Enlarged Section,"
Proceedings SESA, IX, No. 1 (1951).
10. R. E. Peterson, Stress Concentration
Design Factors, (John Wiley, New York,
1953).
11. R. B. Heywood, Designing by Photoelasticity,
(Chapman & Hall, London, 1951).
12. T. A. McCreery, "A Computer Method for
the Approximate Solution of Two
Dimensional Elastostatic Problems by
Conformal Mapping," (University of
Illinois Ph.D. Thesis, 1961).
13. Ernst Weinel, "Ueber die Spannungserhoehung
in Kerbstaeben," (Proceedings of the Fifth
International Congress for Applied
Mechanics, Cambridge, Massachusetts, 1938)
5153.
14. M. Hetenyi and T. D. Liu, "A Method for
Calculating Stress Concentration Factors,"
Journal of Applied Mechanics, XXIII,
No. 3 (September, 1956) 451457.
15. A. Mathews, "Calculation of Stress
Concentration Factors in Filleted Members
by the Equivalent SurfaceShear Load
Method," (University of Illinois Ph.D.
Thesis, 1959).
16. M. M. Frocht, "Factors of Stress
Concentration Photoelastically Determined,"
Transactions ASME, LVII (1935) A67.
17. E. G. Coker and L. N. G. Filon, "A
Treatise on Photoelasticity," (Cambridge
University Press, 1931).
18. S. Timoshenko and Dietz, "Stress
Concentration Produced by Holes and
Fillets," Transactions ASME, XLVII (1925)
199237.
19. L. N. G. Filon, "On an Approximate
Solution for the Bending of a Beam of
Rectangular CrossSection Under Any System
of Load, with Special Reference to Points
of Concentrated of Discontinuous Loadings,'
Phil. Transactions of the Royal Society,
CCI, Series A (London, 1902) 63155.
20. D. Young, "Iterative Methods for Solving
Partial Difference Equations of the
Elliptic Type," Transactions American
Mathematics Society, LXXVI (1954) 92111.
21. F. G. Lehman, "Simultaneous Equations
Solved by OverRelaxation," (American
Society of Civil Engineers Second
Conference on Electronic Computation
[September, 1960] 503512.
22. S. Timoshenko and J. N. Goodier, Theory
of Elasticity, (McGrawHill, New York,
T95).
23. M. Salvadori and M. Baron, Numerical
Methods in Engineering, (PrenticeHall,
24. S. H. Crandall, Engineering Analysis,
(McGrawHill, 1956).
25. D. N. de G. Allen, Relaxation Methods in
Engineering and Science, (McGrawHill,
1954).
26. K. R. Wichman, A. G. Hopper, and J. L.
Mershon, "Local Stresses in Spherical and
Cylindrical Shells Due to External
Loadings," (Bulletin No. 107, Welding
Research Council, August, 1965).
* * *
APPENDIX A. DEVELOPMENT OF NEUBER'S SOLUTION FOR A PROJECTING NOTCH
IN A HALFPLANE UNDER TENSILE LOADING(8'
A. GENERAL RELATIONSHIPS IN RECTANGULAR
COORDINATES
The Threedimensional Problem
Neuber approached the elasticity problem
of satisfying the equilibrium equations in
terms of displacements,
(% + G) e + G V2u 0
ýx
( + G) e + G 72v = 0
(A1)
(X + G) + G 72w = 0,
and the associated boundary conditions, by
assuming displacement functions of the form
2Gu  2r1P 
2Cv = 2rP2 
F
2Gw  2q3 F
where
F  (P + xpPI + yp2 + zc3
(A
and where u, v, w are the displacement
components along the rectangular x,y,x axes,
respectively, and
ýu + v + w
e  xy z
G E
G 2(1 + v)
E = Young's modulus
v = Poisson's ratio.
(A3)
When Equations (A2) are substituted into
Equations (Ai), it is seen that if the func
tions T0,' T 2' and p3 are harmonic, i.e.,
2 2ri 2ý(pi ý2pi
p = + y2 + 0,
x2 5y 4z
(A4)
i = 0,1,2,3,
the Equations (Al) are satisfied provided the
constant n has a value
q = 2(1  v).
(A5)
Generally, any one of the harmonic
2) functions c. (i = 0,1,2,3) may be dropped
without loss of generality. Thus, the problem
is reduced to finding harmonic functions which
satisfy the prescribed boundary conditions.
The Twodimensional Problem
Similar relations hold for the two
dimensional case, where the stresses and
deformations are functions of two coordinate
variables (e.g., x and y in rectangular
coordinates).
(A3)
SvE
(1 + v)(I  2v)
(a) Plane Strain. The equilibrium
equations in terms of displacements are
(X + G) e + G 2u 0
(X + G) I + G Vv  0
Ty
(A6)
where
bu bv
Equations (A6) are satisfied if the displace
ment functions are taken as
ýF
2Gu = 21P,  Fx
(A 7)
2Gv ly
t/2 t/2
e  = f x dz + dz,
t/2 t/2
(All)
e, u, and v indicating mean values across the
thickness t in the z direction.
If the displacement functions are again
given by Equations (A7), with u and v taken
as mean values, Equations (A10) are satisfied
 provided T0 and qI are harmonic and
2
+' T V;
(A12)
where
F (P0 +Pl'
provided the functions TO and P
i.e.,
2 2 T i 2 i
V (.P  +   , 0
' ox oy
n = 2(1  v).
In establishing Equations (A7)
Equations (A2) has been set eq
and 9 is necessarily equal to
deformation is to be independen
(b) Generalized Plane Str
equilibrium equations in terms
ments for this case have a formr
that of Equations (A6),
( + G) T + G V u 0
( + G) + G V2~ 0
where
 2 XG vE
X + 2G 2
1 v
B. THE PROBLEM IN CURVILINEAR COORDINATES
In the following, only orthogonal curvi
linear coordinate systems are considered.
l are harmonic,
are harmonic, As stated earlier, assuming the
displacement functions to be of the form given
by either of Equations (A2) or (A7) with the
i = 0,1 (A8)
constant 1i assigned the required value, the
problem reduces to one of finding harmonic
functions c1 satisfying the boundary condi
(A9) tions. When the boundary is other than
straight, the boundary conditions are
' 2 in generally difficult to satisfy unless the
ual to zero boundary coincides with a coordinate line.
zero if the Hence, the use of an appropriate curvilinear
t of . coordinate system simplifies this part of the
ess. The problem considerably. However, the problem
of displace of finding harmonic functions, i.e., functions
similar to satisfying Laplace's equation, become compara
tively complicated for the general case by the
use of curvilinear coordinates. Thus, for an
orthogonal curvilinear coordinate system
(A10) defined by
y = y(oa, B, 7),
x = x(a, 13, 7);
(A11) z = z(a, P, 7),
the Laplacian (in three dimensions) is given
by
a /
h ahh y + hah,\
+ b h V (A13)
where h , h , h are the scale factors corre
sponding to the a, p, 7 directions. For the
twodimensional case,
2 hlh h ( a (A14)
An important special case of orthogonal
curvilinear coordinates is the isometric (or
conformal) system in which the scale factors
are equal, i.e.,
h = h = h,
and for which Equation (A15) reduces to
(A15)
Thus, for an isometric system, the condi
tion that the functions (0 and P1 be harmonic
assumes the same simple form it takes in
ractangular coordinates
 +  0, i  0, 1. (A16)
Before going into the conditions which
the displacement functions must satisfy along
For a curvilinear coordinate system defined
by x  f(a, p); y = g(a, p) to be isometric,
the following conditions must be satisfied:
a free boundary, the expressions for the
stresses in curvilinear coordinates will first
be developed.
Expressions for Stresses in Curvilinear
Coordinates
(a) The Threedimensional Case. The
displacement components along the a, 7, 7 axes,
U, V, and W, respectively, may be obtained
from the expressions for the displacement
components in the x, y, z directions by noting
that the cosines of the angles between the
x axis and the a, p, 7 axes are given by
cos (x,a)
so that
h2
a
I x
= a ; cos(x,p)
x cos(x, ) 1 x
7 (A17)
tx 2 + 2 + 2
with similar expressions for the y and z axes.
Using the above expressions, one obtains
1 ( x , 8y , z
U *  u T + v + w z
V I (u 6 + v + w;
h  p ( dp f13 7
1 I u. x + v + w \
h 7 77^w^
. (A18)
Substitution in the above equations of
the expressions for u, v, and w from Equations
(A2) gives the displacement components along
the curvilinear axes in terms of the harmonic
functions P.,
2GU a F1 F 2n(
h I 2 1 T
a I C3
+ 2 ýa + 3 F
(A19)
2 1 + 2
h 2 2 ,2
v h2 2 4) *p
dx + ýy
.7.  ibb'
X *
2GV  
h 0
S F
[~3
+ 2n (&
( 2 a +z cp
+ 2 i +'3 To/J
2G  F + 2n x1
(P +2 T7
where
(A19)
From geometrical considerations the
following expressions are obtained for the
strain components in the a, p, 7 directions in
terms of the displacement components
a h\da ha Wp h 2/
(A20)
ch a / \V
a ho a /h()) h ' TJ
similar relations being obtainable for e , e ,
y , and 7 . By substituting Equations (A19)
into the above equations, the following
expressions are obtained for the strains in
terms of the harmonic functions (.:
I
2 G c Z 2 nL" a x 2 +
ain2 h
a i
G7A
a2F
an _ n
h h 3 5
aB
(A21)
2
a
h h 2 7
a7
2 h /
+ 2h
2 7P)
so that
n2 a2 i2
2 n 2 an 2 n 2
a P Y
(A22)
Similar expressions may be written for e , e ,
7 , and 7 .
Substitution of the above expressions for
strain into the stressstrain relationships,
oa" = e + 2GEa
o = Xe + 2GE
T =X e + 2Ge
7 7
where
e  e + e + e
a p 7
O7 = G 7 0
T = G7t
07 " 7
. (A23)
yields the following expressions for the
stresses in terms of the harmonic functions:
hh 6
+ 21 ax + 2 a 2 8
+ Fp  +0 +
+P Zz + 3 ax
"ClE 7 '
. 2F + 2_, . x 2 ,
" n2 hrF + 1 "x
1 5x P2 2 2
+ + T ;,
+ +
+,3 (TX ii z
S(A24)
etc.
(b) Stress Components for the Two
dimensional Case. Since for the twodimensional
case the stress distribution is independent of
Poisson's ratio,* an arbitrary value may be
assigned to this ratio without affecting the
resulting expressions for the stresses. If a
value of unity is assigned to Poisson's ratio,
the constant ,T (see Equation A5) becomes
equal to zero so that the bracketed terms
appearing in Equations (A24) drop out. By
noting that
ý2 F 2
V2F F 2F
ýn2 ýn2
a P
we obtain from Equations (A24)
expressions for the stresses:
a . 1 = 1 ; I F
a n Pn a a
+_I a ýF
h h2"7 T
a
the following
(A25)
+"1h = 13
h 2 h N \
ce 5nXn _ýh Tci (2F,
S 5 22
a
2 h h"
2h a h2
P
(A25)
where F is given by Equations (A7). It will
be noted that the function F has the same
character as Airy's stress function.
For an isometric system, Equations (A25)
reduce to
lP I D + I , h F
=1 +  F\ , T (A26)
C h
or, performing the indicated differentiation,
29 /  
F + F I ;h dF dh ýF
a h2 )132 h30(aa /
1 ý2F I lI' ýF )h ýF
h " 2 +0: h 3 Tp7  Z T
T 2F 1 + bh F . h 5F
ap  3 *a73 0T)
> (A26&
)
When body forces are neglected.
Boundary Conditions Corresponding
to a Free Boundary
The conditions on the function F corre
sponding to a free boundary for a two
dimensional curvilinear coordinate system will
be derived by first considering the conditions
for the case of rectangular coordinates.
First, it should be noted that for the
twodimensional case, the function F appearing
in the foregoing equations can be made to
correspond to Airy's stress function. Thus if
we consider F as given in Equations (A7),
i.e.,
F M cp0 + x(1 '
and take the displacement components in the
x and y directions as
2Gu   (F TIC)
2Gv =  (F +ylp1)
(A27)*
By next considering the equilibrium of an
element located on a curved boundary S,
bounded in the interior by sides parallel to
the x and y axes (see Figure Ai), and
denoting by X and Y the components of the
external load per unit length of arc acting
on the element, the following relationship is
obtained:
Y d AF
ds F
Y = " d (F
If the boundary is free of external loads,
then
where
} (A29)
pl  If dx ,
the equilibrium equations in terms of dis
placements are satisfied, provided the
constant n is given by
n  2(1  v)
for the case of plane strain, and
2
= (1 + v)
for the case of generalized plane stress.
With the displacement components assumed
in the form of Equations (A27), the stress
components are given by
s 2 F s 2F F
x y2 ' y x2 '
as in the case of Airy's
SWF
xy "xy '
stress function.
Obtained from Equations (A7) by introducing
the following transformations [the
starred quantities refer to those appearing
in Equations (A7)]:
F*  F +nl
S 1cp
0 0 1" '
along boundary
fJalong boundary
" constant, c1
(A30)
= constant, c2
The above conditions on F may be
simplified for the case of a problem involving
a single continuous boundary by introducing a
new stress function,
F = F  (c x + c2y)
(A31)
With the above form of the stress function,
the expressions for the stresses given by
Equations (A28) remain unaffected, while the
displacement components given by Equations
(A29) are each augmented by constant terms.
8) For a problem involving only a single
continuous boundary, these additional constants
represent a rigid body displacement and hence
do not alter the problem. (However, for a
problem involving two disconnected boundaries,
the constant terms appearing in the expressions
for the displacement components of the bounda
ries represent a relative displacement of the
boundaries with respect to each other.) If F
See, for example, pp. 100101 of Reference 22,
is now substituted for F in Equations (A30),
one obtains
along boundary
Jalong boundary
(A32)
Solving for ýF/)a and ýF/6p, one obtains
a Dh2 P
dF .1 (TyF dx _ dF
where
D 1 h x y "x  X
Since one may write
F F  clx  c2y
= 0 + xl  clx  c2y
= (P0  c2y) + (  c )x
F 0 + x P1
where
C0 = P0  c2y,
,l Il  cl
and subject are new harmonic functions, the
conditions corresponding to a single free
boundary may be written as
ýF
= 0
along the boundary. (A33
ýF
E 0O
y
Equations (A33) may now be used to
derive the corresponding conditions for a
curvilinear coordinate system. This is done
by writing dF/dx and dF/dy in terms of their
respective components along the a and p axes.
Thus,
dF 1 dF dx 1 I F dx
hh
F I F _F
a hB
Along a free boundary, dF/dx = 0 and
dF/dy = 0. Hence, the conditions correspond
ing to a free boundary, for the case of a
problem involving only a single continuous
boundary, are given by
=0
along the boundary.
g= J
(A35)
(The above relations could have been deduced
directly from Equations (A33) by noting that
the two components of the derivatives of F in
two nonparallel directions  in this case x
and y  are both equal to zero.)
) Thus, the above solution of a problem in
plane elasticity involving a single free
boundary consists of finding harmonic
functions cp0 and Cp such that the function
F = P0 + xcpl satisfies Equations (A35) in
addition to other prescribed boundary condi
tions (usually concerned with the type of
loading). Once this is done, the stresses
and displacements may be calculated from F
using Equations (A28) and (A27), respectively.
C. SOLUTION FOR THE STRESSES IN A PROJECTING
NOTCH IN A HALFPLANE UNDER TENSILE LOADING
The projecting notch considered in the
following is defined by the equations
(kA34)
[A34)
x a + 92 2 and y  2
a + a + B
(A36)
with p = p (a constant), and is assumed
subjected to a uniform tensile stress directed
parallel to the x axis. The profile of the
notch, p = o, represents the lower boundary of
a halfplane which extends indefinitely in
both positive and negative x directions as well
as in the positive y direction. Profiles
corresponding to different values of the con
stant are shown in Figure 2. It will be noted
from Equations (A36) that a approaches x and
p approaches y for large values of a and/or B.
The (a, p) coordinate system described by
Equations (A36) can be shown to be orthogonal,
and also isometric, i.e.,
h2 = h2 h2 = 1 + 2(2 2 )+1 (A37)
a (a (2 + p2)2
For a loading consisting of a uniform
tensile stress of intensity p directed parallel
to the x axis and acting at infinity, the
boundary conditions to be satisfied are:
stress in the x direction and then modifying
it to account for the disturbance of the uni
form stress field due to the presence of the
notch. Thus, corresponding to the initial
condition of a uniform tensile stress a = p
throughout the region considered, one readily
obtains
F =.2 .
(A39)
The function F as given by Equation (A39) may
be obtained by taking the harmonic functions
C0 and cp as follows:
PO = (2  x2)
cp1 =2 x
(A40)
Then,
F = p0 + xp y2
The next step is to transform F into
curvilinear coordinates using Equations (A36).
F  L2 2 _2 22 2+
(A41)
(a) at x = a = I.o:
X =  p.
x y2
(b) along p = o:
ýF = 0
ýF
=0°
The solution is obtained by first
considering a condition of uniform tensile
That is the relations defining the system,
given by Equations (A36), satisfy the
orthogonality condition:
.T.T ' 4 0.
(A3J
>
The corresponding expressions for the harmonic
functions c0 and cp are:
8) 1 0 . p2 2 2(x2+ 2) + 2
0 2 2 2 ( 2 2)
a + 2 (a+P )÷
(A42)
1 2\ 1p2
The expression for F given by Equation
(A41) now satisfies the first boundary
condition. To satisfy the second boundary
condition and to account for the disturbance
of the uniform stress field due to the
presence of the notch in the vicinity of
x = a  0, additional terms must be appended
to p0 and cpI as given by Equations (A42).
These auxiliary functions must be such that
their effect vanishes for large values of x
or a so that the satisfaction of the first
boundary condition is unaffected by their
addition. Since the final function F has to
satisfy the boundary conditions along the
surface of the notch, expressions having a
form similar to the relations defining the
notch profile become logical choices for these
auxiliary terms. Among the harmonic functions
in this category are:
a, p, 2 2' 2 2
a +p a + p
(A43)
( 2 2 2
The functions C0 and P1 may now be written as
0~ *
0 2 L 2)2
2 2a +P
+ 2  2 + Aa
(a + 22
+ + C(B2 2) + D 2a 2
a + P
+ E P + G P2 2 12
a 2+ P (a + P )
2 2 + a + H 2)2
1 a +P t+ 2
The stress function F then becomes
F = 90 + xP I 0 +(a+ 2._2)
a + P
. p2 _ 23 + p2
2 2 2 2 2 2
i2 a + B (a +2)2
+ Aa+BP+C(pa52) +D 20 2
a +p
2 2
+ E 2 + 2 + G 2
a + (a 2+ 2 )
+ H 2 + (2 a2 2 (A45)
(a +2)P
. (A44
The constants appearing in Equation (A45)
are determined by making F satisfy the condi
tion along the free boundary
ýFj
S0 and II
o]
= 0. (A46)
This gives the following:
A = C = D = 0
B =  2P
E = 2
2P2 + 1 (A47)
3P2 + 1 4p2 + 1
G = and H =
2p2 + 1 2 2 + 1
Substituting the values of the above constants
into Equation (A45) gives
F =  [2  2pP
2 _ _ _2 02+2
(p +l a2 2) 2 2+1
(2 P2 +,)(a2+ 0 2 202 +
(A48)
which may be reduced to a simpler form by
introducing a constant term (which does not
affect the stresses)
P _ +
2 2+o
2 o 2
2 o
(A49)
The final expression for the stress function
then becomes
F =  (pp )2 [1  (2 )(2+
2( 2°+ )L(a 2+ 2
(A50)
The above form of F makes it readily apparent
that along the boundary p = p , F equals zero,
so that the second boundary condition is
indeed satisfied.
)
With the function F now known, the stress
components cr , , and T may be calculated
using Equations (A26a). The derivatives of h
and F appearing in Equations (A26a) have been
calculated and are given below:
h 2c(a 2 _3  1)
(a22 )2 [(2+2 2 2+2 2 2c2+ 1/2
)h 21(30O2 0  0)
(a2 )2 [(a2P 2 2+2 2+1 /2
[ °v2
ýF 1 a(pp0) 2
ýF a + pp
TPF  [(()2 3a2)
sa2 [22 23] p
^2F F a4+a2(6PPO 2ý2  3a2)
.62 _(2P2 (a 2 + P2 p
V2Ii a (   3 0 0
2 (2p 2 +1)( 2 2 3
o0
As a check on the satisfaction of the
first boundary condition, it will be noted
that for large values of x and a, h  1,
T  r , and all of the terms entering into
the expressions for the stress components,
except 2F/rp2 (which becomes equal to p),
tend to zero. Thus, as x  ± o, a = p, and
T= = 0. Also, for large values of p,
y xy
 = p and a = T = 0, i.e., the stresses
x y xy
are finite as y  + 0.
Along the surface of the notch,
= 0, so that
_o h2 =
Taj ý= 0h 2 )P _
 (2 p2+1) (2+2)]
2 2p p (A52)
+ 2  2a +1
((2 + 2)2 J
The maximum stress is obtained by differenti
ating the above expression for cr with respect
of a and equating the result to zero. This
gives
S1 + 2 1/2
1 1 + 02 0 2p+ 4J
S (l+4 P2) 2 o
max 4po(1 +20o
(A53)
(A54)
The stress concentration factor is then given
(u
by the ratio max
The curvature of the notch profile is
given by
2po(3c 2 p2 1)
\= 0 0
3 p P O 3/2 '
S 0 [(a22)2+22 2 2+]
a 5O +13 2 +o
(A55)
Differentiation of the above expression for
XK , to obtain the minimum radius of curvatume
of ?he notch profile, yields
2
aR = + o
Rmin 0
See Equation (A37).
2 I+ 2
R . = 2 + +f
mmn o o 0
max
50
(A56)
In transforming the stress components
along the a and p directions to components
parallel to the x and y directions, the angle
between the a coordinate line (or p  constant
line) and the x coordinate line at the point
is required. This angle is given by the
following:
P  tan 2 2( C 0 2 (A57)
(l+ 2) 2+(P3 a J2)
APPENDIX B. DEVELOPMENT OF EXPRESSIONS FOR THE STRESSES
IN A RECTANGULAR BAR DUE TO SURFACE STRESS LOADINGS*
In the following, consideration is given
to a rectangular bar of length 2a, depth 2b,
and thickness 2t, subjected to a uniform
tensile stress p acting at its ends. In addi
tion to the axial load p, a surface stress
loading  either shear or normal stress 
acting over a length 2w is imposed at the
center on both top and bottom surfaces of the
bar (see Figure B1).
For the twodimensional state of stress
assumed here, the elasticity problem may be
formulated in terms of Airy's stress function,
F. When body forces are neglected this
function is related to the stress components
by the equations
2 2F
y x2 '
2F
rx 2 p
a T =T 0
y xy
xy
y
along y= ± b
(B3)
(B4)
for the case of a surfaceshear loading; or
Equation (B3) and
2F
y = g(x)
y x2)2
along y = 2 b
(B5)
T = 0
xy
xy  . (B1)
The equations of equilibrium are identi
cally satisfied by Equations (B1), and in the
absence of body forces the compatibility
equation becomes the biharmonic equation in F,
4F )4F 4F 4
+ 2 x2 V F V 0.
(B2)
For the particular case considered, the
problem becomes one of finding a function F
satisfying Equation (B2) in the plane region
considered as well as the following boundary
conditions:
for the case of a surfacenormal stress
loading. In the above equations, f(x) and
g(x) are the distribution functions associated
with the surfaceshear and the surfacenormal
stress loadings, respectively.
For both of the above cases, the problem
may be considered as divided into two parts.
The first part, representing a condition of
uniform stress a = p along the entire length
of the bar due to the action of the uniform
tensile stress p at its ends, corresponds to a
stress function (as in Appendix A) of the form
1 2
F =  py ,
(B6)
See, for example, pages 4652 of Reference 22.
which satisfies Equation (B2) as well as the
first boundary condition (B3). The second
part represents the stress condition arising
ý2F
F '
Tx ~^2
from the action of the stress loading on the
surface y = . b, the stress function for which
must satisfy condition (B4) in the case of a
surfaceshear loading or condition (B5) in
the case of a surfacenormal stress loading.
Moreover, the stress function for this second
part must satisfy the condition
S= a = T = 0 at x = ± a.
x y xy
(B7)
The stress function for this second part
corresponding to the two cases noted above
will be considered separately.
A. SOLUTION FOR THE CASE OF A SURFACESHEAR
LOADING DISTRIBUTED ANTISYMMETRICALLY
ABOUT THE Y AXIS
For the case of the surfaceshear loading
shown in Figure B1, the resulting state of
stress in the bar will have a distribution
which is symmetrical about the y axis (i.e.,
an even function of x). The solution to
Equation (B2) may then be taken as a series
of the form*
F Y(y) cos ax
(B8)
where
ncr
a 
a
From Equation (B10),
" =c2F
x 2
Zj [cCl 2cosh y + C2a sinh ay
n=1
+C3 a(2sinhcy +c y coshc y)
+C*4c(2cosh oy+acy sinh ay)] cos ax
2)2F
y x2
00
o_Z [CIcoshoy + C2sinhry + C3y coshcay
+ C4y sinh ay] a cos ax
S2F
xy xy
= [c1Ca sinhay + C2a cosh ay
n=l
+ C3(coshcty + cy sinh cy
+ C4(sinhay+ ay coshhay)] asinax. j
When Equation (B8) is substituted into The constants CI to C4 in Equations (Bll)
Equation (B2) a fourthorder ordinary are determined by using the boundary conditions
differential equation in Y(y) results which of Equation (B4). For this purpose, the
when solved, yields function f(x) describing the distribution of
the shearing stresses along y  t. b is
Y(y) = Clcoshacy + C2sin h ay expressed in terms of a Fourier sine series
(B9) (the shearing stress distribution shown in
+ C3ycos hy + C4ysin hcy. Figure BI being antisymmetrical about the
The stress function thus becomes y axis, i.e., an odd function of x). Thus
F X= (Ccos hcay+ C2sin hcy + C3y cosh
n0
+ C4y sin hay) cos ax,
See, for example, pages 4650 of Refere
co
(Txy) + An sin cx
y+nb
nCl
(B10) B
(TXy)yb Bn sin ox
snce 22. n=1
(B11)
(B12)
Substituting the above values into the third
equation of Equations (B11) and solving the
resulting relationships together with the
second of Equations (B11) with o = 0, one
obtains
Cl L b sinh a b C
 L cosh a b ] C4b
(An13 r b sinhab
a sinh2ab+20
C = b cosh a
2 sinh a b
b
C4 =
AnBn [ b cosh ab
s sinh2rb2?b
Noting that, by the usual sign convention foi
shearing stresses, A = B , one immediately
n n
gets C2 = C = 0, Equations (Bll) now
become
00
2 coshab b sinhb coshcos
Tx 2An sinh 2ab +2ab coshoy cosox
n=1
00
7 cosha b
+ n 2An sinh 2ab+2b ysinhaycosax.
nI
00
2A u~ab sinhab cosh
y A n sinh 2b+ 2b cos
n=l
00
S 2A coshab .
 2An sinh 2ab+ 2ab oy sinhy cosox.
nl
xy
n=,1
nI
n=l
2A coshab ob sinhb snhysin
n sinh 2ab+ 2ob sinhsinax
2An sinh 2ab+ 2b y cosh aysin nc.
It will be noted that at the ends of the
bar, x = + a, T = 0; however, x is not
equal to zero and has a distributxyion other
equal to zero and has a distribution other
than uniform. Thus, it is difficult to
satisfy fully the boundary condition specified
by Equation (B7). However, if the dimension
a (i.e., the halflength of the bar) is made
sufficiently large in comparison with b and
w, instead of the stress distribution for a
given by the first of Equations (B14),we may
consider at x = ± a, a statically equivalent
uniform stress distribution without materially
altering the stress condition in the immediate
13) vicinity of the bar centerline, i.e., in the
region of application of the surfacestress
load. To do this, the total forces acting at
the ends of the bar due to the stress a must
x
be calculated.
r If the total tension at either end of
the bar is denoted by T and the bending
moment by M, then
b
(T)x+a = (T) = f (Cx ) dy,
b
or
Sb
(T) ' . 2A 2coshObzabsinhbab f fcoscydy
nx=a b sinh2c +2cb a
n=l b
(E
Sb
n sinh 2ab + 2ab f
2n A 2 bosh co ysinh aydy
n l b
114)
00
(T) Z 2A cos aa
(T)x=a = 2An a
n=l
(B15)
By observing that y coshay and y sinhay are
odd functions of y, it is found that
(M)xt a = x)
b
dy = 0.
Thus, the condition of zero load at the ends
of the bar can be satisfied  insofar as the
effects near the bar centerline are concerned
a sinh2teb 2ab
by introducing uniform compressive stresses at
the ends of the bar with a magnitude equal to
the average stress corresponding to the total
end force (T)x=ta given by Equation (B15).
This corrective stress loading, which is trans
mitted undiminished throughout the length of
the bar, must be added to the expression for
a given by the first equation of Equations
(B14), which now becomes,
Z An cos aa
x . b a
nl
00
" Y2A 2coshababsinhab y
+ 2An sinh 2ab+2h cosh cos
n I
S,2A cosh b
+ 2An sinh 2ob 2ob y sinhaycos ax.
nI
(B16)
The complete expression for the stress
( due to the action of the surfaceshear
x
loading shown in Figure B1, together with a
uniform longitudinal tensile stress p acting
at the ends of the bar, will then be equal to
(p + x). The expressions for a and r as
x y xy
given by Equations (814) are unaffected by
the corrective loading and the stress p at the
ends of the bar.
Expression for the Fourier Coefficient A
Corresponding to a Triangular SurfaceShear
Loading
For the brokenline distribution of the
surfaceshear loading shown in Figure B1,
obtains
¶ (x) o   x when 0< x < (w  c)
r(x) =° (xw) when (w c)< x w .
In Figure B2b, the quantities a(+), w,
n
s,,and g are assumed to be given while the
quantities cr and r are to be calculated
n
from the requirement that the sum of the nega
tive (compressive) stresses must be equal to
the sum of the positive (tensile) stresses.
This condition occurs along the base section
of a projecting notch. It can readily be
verified that
r = (s  1s2  (ws g)(w s)
r(+) r
+(wnsg) nr sin a(sr)
(B26)
+1 [cos a(wg)  cos a(sr)]
(+)
 [cos d  cos o(wg)]
for n = 1, 2, 3,.... oo. A
tion for the first term C
0
separate calcula
gives
(B27)
C = 2 f  (x) dx  0.
o ap  n
o
This result follows from the condition, stated
earlier, that the algebraic sum of the
surfacenormal stresses on each of the
surfaces y  1 b must be equal to zero.
a p (
ap n
I c
APPENDIX C. BOUNDARY VALUES FOR THE STRESS
FUNCTION USED IN THE FINITE DIFFERENCE SOLUTION
In the finitedifferences method, the
problem of finding a continuous function F
that satisfies the compatibility equation as
well as the associated boundary conditions is
replaced by one requiring values of F at a
discrete number of points, which also satisfy
compatibility and the boundary conditions.
Thus, the area under consideration is assumed
to be subdivided by a grid (usually square) or
lattice and the value of the stress function F
is calculated at the intersections of the grid
lines or node points (see Figure C1). Corre
sponding to this discretization, the compati
bility equation (B2) is replaced by the
finitedifference biharmonic equation shown
diagramatically in Figure (C2). The numbers
within the circles in the bracketed biharmonic
operator of Figure (C2) represent the
coefficients of the values of F at these node
points.(23,24) To obtain the finite
difference approximations for the derivatives
in terms of the values of the function at the
node points, the usual procedure is to con
struct an interpolating polynomial (with degree
equal to the order of the derivative) which
takes on the given values of the function at
the node points. The derivatives of the given
function are then considered to be equal to
the corresponding derivative of the interpola
ting polynomial.
For the particular case considered here,
the stress function has to satisfy, in addi
tion to the compatibility condition, the
following boundary conditions (see Figure C1):
Y  0 everywhere
X = p on AB
X = 0 on BCDEF
(a)
(b) (C1)
(c)
where X and Y are the respective x and y
components of the external stresses acting on
the boundary of the area considered. Since
Equations (Cl) are expressed in terms of
stresses, it is necessary to transform them
into the corresponding expressions in terms of
the stress function F. This is done by using
Equations (A29) which relate X and Y with. F,
i.e.,
X = s ý)
Ts; (ty)
(a)
(b).
(A29)
(C2)
In applying the biharmonic operator to
points near the boundary of the area consid
ered, the value of the function F, as well as
its derivatives, DF/rx and cF/ýy, are required.
The latter are used in the interpolation and
extrapolation formulas for determining the
values of F at points adjacent to a boundary.
(a) Values of Derivatives 5F/rx and
6F/7y Along the Boundary. From Equations
(Cla) and (C2b), one obtains
=  Yds = 0 + C1.
Since F is known to be symmetrical about the
vertical axis x  0 (i.e., the line FG in
Figure C1), its derivative ;F/rx must be
antisymmetrical about the same line. Hence,
so that C1 = 0;
x 0,
x0
everywhere.
(C3)
Along the vertical portion of the
boundary AB, ds = dy. Using Equations (Clb)
and (C2a)
 f Xds = pdy = py + C2.
AB AB
Again, since F is known to be symmetrical
about the horizontal axis y = 0, 5F/5y must
be antisymmetrical about the same line,
that is,
yj = 0 hence C2 = 0
y=0
F py
7y = py
on AB.
(C4)
(b) Values of the Function F Along the
Boundary. To calculate the values of the
stress function F along the boundary, it is
convenient to use the following expression
for F:
F f cos (x,n)  sin(x,n) ds
s a (C6a)
where
n = outwarddirected normal to the
boundary
(x,n) = angle between the positive x axis
and the normal n.
Equation (C6a) may be derived from the
expression for the partial derivative of F
with respect to s, i.e.,
5F F dx +F dy
s 7x Tds y ds'
(C6b)
by noting that
d_  sin(x,n) and s = cos(x,n).
ds ds
An alternative expression to Equation (C6a)
is obtained by integrating Equation (C6b) by
parts. This yields
On the portion of the boundary BCDEF,
X = 0, and
F= X ds = 0 + C3.
BF
The above equation indicates that 6F/6y has a
constant value along BCDEF. From Equation
(C4), one obtains for point B,
OF
T bj pyj
yb yb
Hence,
F
y bp
dY
= bp.
F + F 62F a2F
^ y (^^X T ai^ E y
(C6c)
Proceeding now with the calculation of
the values of the function F for the portion
of the boundary AB (with ds = dy and (x,n) 01
one obtains from Equations (C6a), (C3), and
(C4),
F = (py)dy = py2 + C4.
See, for example, page 109 of Reference 25.
(c5) **
See, for example, page 484 of Reference 22.
on BCDEF
Since the addition of an arbitrary constant to
the values of F along AB does not affect the
value of the prescribed stress r a 2F/ry2 = p
on AB, the constant C4 may be chosen such that
F  0 at A. Hence,
1 2
F =  py on AB.
(C7)
On the portion of the boundary BC,
ds = dx and the angle (x, n) = 90 degrees.
Using Equations (C3) and (C6a), one obtains
F  f(bp) (0) dx = 0 + C5,
which indicates that F has a constant value
along BC. From Equation (C7),
F I b b2
c
The two preceding expressions for F] give
C6 =~ 1
6 2
C  2 b P.
Thus,
F = bpy  b p
on CDEF.
(C9)
To summarize, the following boundary
conditions in terms of the stress function F
have been obtained:
(1) Derivatives Along the Boundary
=F
; = 0 everywhere
=  b2p for point B.
B
Thus,
F  b2p
on BC.
2)F
Ty py
ýF
y bp
(C8)
on AB
on BCDEF
(2) Function F Along the Boundary
Along CDEF, Equations (C3) and (C5) give the
following values of the derivatives of F:
ýF 2F
Tx 0 y = bp.
Using Equations (C6a),
F = ybp [y s(bp) ds = bpy + C6.
It should be noted that the quantity (bp)
appearing in the above integrand is a constant,
so that the second term vanishes. For point C,
with y * b, the above equation yields
FJ  b2p + C6.
c
From Equation (C8), also for point C,
1 2
F = 2 py
on AB
F = bpy   b2p on BCDEF.
With the values of the stress function
and its derivatives along the boundary known,
the values of the stress function at node
points adjacent to the boundary may be
obtained by interpolation and extrapolation.
The finitedifference equations may then be
written for the interior points using the
biharmonic operator of Figure (C2). This
yields a system of simultaneous linear,
algebraic equations in F which may be solved
by either direct or iterative procedures.
(c) Interpolation Equations. When a
curve forms part of the boundary of the area
considered, not all the grid points generally
can be made to coincide with the boundary. In
F  py2
such a case, the regular finitedifference
biharmonic operator of Figure (C2) cannot be
applied to interior grid points adjacent to
the boundary. Moreover, in order to be able
to apply the operator of Figure (C2) to the
second interior point in this case, it is
necessary to use an external "fictitious"
grid point. In this study, the value of the
stress function at such points adjacent to
the boundary have been calculated in terms of
the values of the function at and near the
boundary and of its derivatives at the
boundary. The following interpolation and
extrapolation equations, which are given by
D. N. de G. Allen, (25) have been employed:
l +2cx a2
F.= +2 Fb + Fa
e (1 +a) (1 +a)2
e (l+0) (l +a)
aA ,cF
I + a (t
F = value of F at the first exterior
e ("fictitious") grid point e
Fb = value of F at the boundary point b
(intersection of grid line with
boundary)
F = value of F at the second interior
a point a
F
S = value of the derivative of F at the
b boundary point b
A  length of regular mesh interval
a = ratio of the distance from the first
interior point i to the boundary
point b, to A
Equations (C10) are derived from the equation
of a second degree parabola obtained by speci
b fying the values of the function at the second
interior point a and the value of the
(CIO) function and its derivative at the boundary
.F point b.
I+a 'dx
b
where (refer to Figure C3)
F. = value of the stress function F at
the first interior grid point i
For the purpose of interpolating along
grid lines in the y direction, only the last
terms involving the derivatives need be
changed in Equations (C10).
*
(2 l t)A