Post on 03-Feb-2022
STRESS CONCENTRATION FACTORS FOR CIRCULARNOTCHES
Lawrence Jerome Smith
DUDLEY KNOX LIBRARY
NAVAL POSTGRADUATE SCHOOC
MONTEREY, CALIFORNIA 9394Q
i as \ Hay o i o
ft 0:0
1
Monterey, California
STRESS CONCENTRATION FACTORSFOR CIRCULAR NOTCHES
byLawrence Jerome Smith
June 19 7 1*
^«KLSi
l Thesis Advisor: J.E. Brock
Approved for public release; distribution unlimited.
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Stress Concentration Factors forCircular Notches
5. TYPE OF REPORT & PERIOD COVERED
Engineer's Thesis;June 197^
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORfa.)
Lawrence Jerome Smith
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Naval Postgraduate SchoolMonterey, California 939^0
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16. SUPPLEMENTARY NOTES
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Stress ConcentrationFinite Element
20. ABSTRACT (Continue on raverae eldo It ncceaaery end Identity by block number)
Existing finite element programs PLISOP and PLIMEG aremodified and adapted to the successful determination of stressconcentration factors for thin elastic bars containing V-shapedround bottom notches and subjected to tensile or bending loads.Accuracy and reliabilty are demonstrated. Comparisons aremade with previous analytical and experimental determinations.The effects of finite bar thickness are discussed.1IIK
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DUDLEY K
NAVAL FOS
wONTEREY
Stress Concentration Factorsfor Circular Notches
by
Lawrence Jerome SmithLieutenant Commander, United States Navy
B.E., Villanova University, I960
Submitted in partial fulfillment of therequirements for the degrees of
MECHANICAL ENGINEER
and
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
June 197^
DUC
NAV
DUDLEY KNOXI
NAVAL POSTGRADUATE SCHOOLMONTEREY, CALIFORNIA 93940
ABSTRACT
Existing finite element programs PLISOP and PLIMEG are
modified and adapted to the successful determination of
stress concentration factors for thin elastic bars containing
V-shaped round bottom notches and subjected to tensile or
bending loads. Accuracy and reliability are demonstrated.
Comparisons are made with previous analytical and experimental
determinations. The effects of finite bar thickness are
discussed.
DUD
NA\|
r/iOl
TABLE OF CONTENTS
I. INTRODUCTION 9
II. BASIC METHODS OF OBTAINING SCF'S 11
A. CLASSICAL ANALYSIS 11
1. Neuber's Notch Analysis 11
2. Limitations of Neuber's Analysis 1^
3. Availability of Neuber's Solution I 1!
B. EXPERIMENTAL INVESTIGATION 15
1. Strain Gage Measurements 15
2. Limitations of Strain Gage Measurements - 16
3. Photoelastic Evaluations 16
k , Limitations of Photoelastic Evaluation — 19
C. NUMERICAL TREATMENT OF ELASTIC FIELDEQUATIONS 20
III. DETAILED PROBLEM SPECIFICATION 22
A. GENERAL 22
B. FOUR PROBLEMS 22
C. ONE STRUCTURE 26
D. FOUR SETS OF BOUNDARY CONDITIONS 27
IV. SOFTWARE SELECTION 31
A. BASIC PROGRAM SELECTION 31
B. MESH GENERATION — — 32
C. MODIFICATIONS TO PLIMEG 3^
1. Stress Gradient Considered 3^
2. Curved Boundary Node Relocation 36
3. Boundary Condition Calculations ^°
D." SUBROUTINE PROBLM 40
E. SUBROUTINE ANS 42
V. PROGRAM OPERATION ^6
A. INPUT ^6
B. OUTPUT ^6
VI. PROGRAM OUTPUT ANALYSIS ^9
A. EQUILIBRIUM ^9
B. MAXIMUM STRESS LOCATION 4 9
C. CONVERGENCE ^9
D. CORRELATIONS 5^
VII. DISCUSSION OF RESULTS 55
A. RESULTS PRESENTED 55
B. EFFECT OF NOTCH DEPTH 56
C. EFFECT OF ANGLE 6 58
D. SINGLE NOTCH GEOMETRY 62
VIII. CONCLUSIONS AND RECOMMENDATIONS 65
A. CONCLUSIONS 65
B. RECOMMENDATIONS 66
APPENDIX A. CURVES OF RESULTS 67
APPENDIX B. NEUBER'S NOTCH ANALYSIS 73
APPENDIX C. THREE-DIMENSIONAL EFFECT 78
1. PREAMBLE 78
2. A SPECIFIC PROBLEM 79
3. PROCEDURES TO SOLVE THE PROBLEM 8l
4. NUMERICAL DIFFERENTIATION FROM 2D FEM — 90
5. DIRECT COMPARISON FROM 3D FEM 91
6. CONCLUSION 93
APPENDIX D. SUGGESTIONS FOR PRODUCING ADDITIONALNUMERICAL RESULTS 95
APPENDIX E. PROGRAM USER'S MANUAL 97
1. TO EXECUTE THE PROGRAM 97
2. TO CHANGE BOUNDARY CONDITIONS 10 1\
3. TO CHANGE THE SHAPE OF THE NOTCH 105
i|. TO CHANGE DIMENSIONING 108
5. TO MAKE A MAJOR MODIFICATION 109
APPENDIX F. COMPUTER PROGRAM LISTING — 113
BIBLIOGRAPHY 164
INITIAL DISTRIBUTION LIST 166
ACKNOWLEDGEMENT
The invaluable advice and counsel of Professor John E.
Brock is gratefully acknowledged. A special thank you,
also, to the computer operating personnel, especially those
on the swing shift.
The work was made easier by the encouragement and
sacrifices of my wife, Polly, and the understanding of our
boys, Kevin and Kurt.
I. INTRODUCTION
Stress intensifiers , In the form of holes, grooves, key-
ways, threads, etc., result In local modification of simple
stress patterns and give rise to stress concentrations. The
designer or analyst is interested in determining the locally
maximum, governing or limiting value of stress, designated by
the symbol a , which is calculated as the product of amax
nominal stress, a , and a stress concentration factor,' nom J 3
designated by the abbreviation SCP, thus:
o = SCP x a nvnmax nom
An elementary formula is used to compute a , the termsJ nom'
of which must be precisely defined; this accounts for the
gross geometry of the member and the loading which is applied
The SCF is intended to provide a corrective multiplying
factor; its value depends on the details of local geometry
and, usually, on the particular type of loading that is
applied.
The designer employs available tabulations of SCF's,
together with the appropriate and easily calculated a ,
employing the formula given above, to estimate the maximum
stress which he compares with an allowable value; the latter
being established by experience, by code, or by contract.
The task of the person who supplies values of SCP is to
determine values of cj and of a ^ and their quotients, themax nom ^
SCF's. To make the values of SCF most widely useful, the
stress analyst selects an appropriate set of parameters to
describe the geometry of a class of stress intensification
problems. A number of ' individual problems are undertaken,
taking care to vary the parameters so as to cover the useful
range of each and thus to permit making reasonable inter-
polations for intermediate values.
The SCF's obtained in this manner are usually presented
in the form of parametric curves drawn through the actual
data points established. The designer then must interpolate
between the curves to match the actual parameters of his
problem.
The particular classes of problems selected for this
study, because of their prominence and importance, are
characterized by a thin, flat bar under tension and/or
bending. We consider two forms of stress intensifiers
:
1. Two notches on opposite edges of the bar, locatedsymmetrically about the long axis of the bar andshaped symmetrically with respect to a plane normalto the long axis of the bar.
2. A notch on only one edge of the bar.
In general, the notches are circular at the bottom. Details
of the notches, with sketches, are presented in Section III.
Much of what follows, however, is applicable to other
geometries
.
We confine attention to the two-dimensional case, i.e.,
the case of plane stress. Realizing that a condition of
plane stress is never actually found In a real world
application, the effect of three dimensionality on SCF's
was also Investigated, see Appendix C.
10
II. BASIC METHODS OF OBTAINING SCF'S
A. CLASSICAL ANALYSIS
Neuber [11] has made the most extensive analytical
investigation of SCF's. His book contains treatments of a
number of geometries and loadings and his methods include
a few cases of exact analytical solutions together with a
number of ingenious approximations and interpolations. The
best known of Neuber' s analyses is that of the deep hyperbolic
notch.
1 . Neuber's Notch Analysis
Using curvilinear coordinates and complex stress
functions, Neuber presented a solution for axial tensile
loading (without bending) of a geometry consisting of
symmetrically located, hyperbolic shaped notches in an
infinite sheet of unit thickness (Fig. 1).
y
Fig. 1. Symmetric hyperbolic notches in an infinitesheet.
1
1
He arrived at the following solution for the stress
concentration factor for this notch:
t\ cot n nSCF =
7T - 2nQ
+ sih(2n )
where n nis a parameter defining the particular notch
geometry, namely, the angle between asymptote and notch
centerline . The nominal stress is given by the axial force
divided by the minimum width of the sheet at the notch. The
hyperbolic boundary is described by the equation:
2, r2
x y = 12 2 2.2
a cos ru a sin n
The half width at the bottom of the notch is D. The
minimum radius of curvature is
2p = D tan (r\
Q)
The width of the sheet approaches infinity on each side,
unlike cases of practical interest in which the notch is an
indentation in a bar of finite width. Accordingly, for
Neuber's deep hyperbolic notch, it is not possible to specify
a gross width or the ratio of gross width to net width as
is usually done for practical cases.
Neuber further presents an approximation for shallow
notches in a bar of finite width with notch depth specified.
12
The analysis involves employing a solution for an elliptic
hole in a bar. Somewhat arbitrarily the bar is
cut through the center of the hole parallel to the edges of
the bar and reassembled to form the notched bar (Fig. 2).
WD
* tradius = p ^
Jl_
Fig. 2. Neuber's shallow notch approximation
The boundaries of the notched bar are not unloaded as they
should be and there are unaccounted for dislocations along
the axial center line.
Taking the half width of the bar as W and using D,
p, and c as defined for the hyperbolic notch analysis,
Neuber arrived at the following approximation for stress
concentration factor for the shallow notch:
SCF = 1 + 2[(W - D)/p]^
Noting that nominal stress was defined the same way
in both cases, i.e., on the basis of net area, Neuber devised
13 » \
a nomographic scheme to obtain SCF for a wide range of the
parameters
:
p = minimum radius of curvature
D = half width of the bar at the notch (net half width)
w~ = half width of the undisturbed bar (gross half width)
The nominal stress is o' ^_ = (axial force per unit thickness)/2D.nom
A summary of Neuber's solution for the hyperbolic
notch is shown in Appendix B.
2
.
Limitations of Neuber's Analysis
The hyperbolic notch analyzed has a definite shape,
namely a hyperbola having the asymptotes y = ± x tan n Q. The
values presented by Neuber are based on two extremes and an
interpolation between them. It must be recognized that other
geometries only approximate these notch shapes and therefore
the data must be judiciously applied.
The reasoning used to arrive at this solution is
ingenious but experimental evidence described later indicates
a systematic difference between Neuber's analysis and the
behavior of actual test sections.
3. Availability of Neuber's Solution
A designer may use Neuber's equations and perform the
indicated interpolation to determine a particular SCF based
on the parameters of interest. However, much of that work
has already been done
.
The most extensive, readily available tabulation of
SCF for this case is presented by Peterson [12]. He presents
a family of curves using dimensionless ratios of the
Ik
parameters as arguments. Other authors, such as Faires [6]
present the same data over a narrower range of parameters.
B. EXPERIMENTAL INVESTIGATION
Values of maximum stress may be determined by experimental
stress analysis including strain gage measurements and photo-
elastic evaluations.
1. Strain Gage Measurements
Kikukawa [9] used resistance strain gages to study
the stress in a bar with two symmetrically located U-shaped
notches subjected to an axial tensile load.
Fig. 3. Symmetric U-shaped notches
15
The geometry of the test section (Fig. 3) is specified by:
R = radius at the bottom of the notch
D = net half width
W = gross half width
Only the gross width of the bar was varied during the tests.
2. Limitations of Strain Gage Measurements
In using strain gages, difficulty arises from the
highly localized character of the stress distribution at the
place in question. A very small gage length is necessary
to obtain satisfactory results. Accordingly, the physical
size of available strain gages limits their usefulness to
the investigation of relatively large notches. Furthermore,
models must be very carefully machined to obtain dimensions
within satisfactory tolerances. Good adhesion of the gage to
the model and correct placement and orientation on the specimen
are difficult to obtain.
Data scatter dictates that some sort of smoothing
process must be used before the results are useful to a
designer. Large expenditures of time and other valuable
resources are required to obtain even a few benchmarks which
can reasonably be used to infer SCF's over some useful range
of the parameters employed.
3. Photoelastic Evaluations
Flynn and Roll [7] used photoelastic techniques to
study the stress in a bar subjected to an axial tensile load
for both U-shaped and hyperbolic-shaped symmetrically located
notches. The geometry of the U-notched test section (Fig. 3)
is specified by:
16
R = radius at the bottom of the notch
D - net half width
W = gross half width
Fig. 4. Symmetric hyperbolic notches
The geometry of the hyperbolic-notched test section (Fig. 4)
is specified by:
p = minimum radius of curvature
D = net half width
W = gross half width
The gross width of the bar was the only parameter which was
varied in these tests, also.
Appl and Koerner [2] used photoelastic techniques to
study the stress in a notched bar subjected to an axial load
The notches were symmetrically located, V-shaped with various
angles and had round bottoms.
17
Tw
y D
I
S
Fig. 5. Symmetric V-shaped notches
The geometry of the test section (Fig. 5) is specified by:
R = radius at the bottom of the notch
D = net half width
W = gross half width
6 = half included angle
They present SCF's for several notch angles using the ratio
D/W as a parameter.
Cole and Brown [5] used photoelastic techniques to
study the stress in a bar with a U-shaped notch on only one
side of the bar. An axial load was applied such that its
line of application was through the center of the net cross
section
.
18
w RwD
<>
Fig. 6. U-shaped notches, one side only
The geometry of the test section (Fig. 6) is specified by:
R = radius at the bottom of the notch
D = net width
W = gross width
They present SCF's for several combinations of the parameters.
l\. Limitations of Photoelastic Evauluation
Models, no matter how thin, do have a third dimension.
These experimenters have chosen thin models purposely so that
their results could be reported as plane stress values.
Appendix C discusses the effect of th.e third dimension, showing
that the stress at the base of the notch actually varies through
the thickness. Photoelastic evaluation reveals the stress
19
averaged through the model thickness. The difference between
the averaged and true maximum is small, but the three
dimensional effect should be recognized.
Errors due to several edge effects including those
caused by residual machining stresses and time dependent
material changes must be recognized.
The same limitations regarding model preparation,
data scatter and resource expenditure that were mentioned as
limitations to strain gage measurements apply also to
photoelastic evaluations.
C. NUMERICAL TREATMENT OP ELASTIC FIELD EQUATIONS
Several numerical techniques, notably the Finite Difference
method, are available and have been used to solve partial
differential equations. However, useful results to the
elastic field equations are difficult to obtain, especially
if the geometry of the problem is not very simple.
Recently a new numerical method called the Finite Element
Method (FEM) has been extensively developed and used. In many
ways it is similar to earlier numerical methods in that the
partial differential equations are replaced by a finite system
of algebraic equations. There is, however, a big difference
in the detail of this replacement and this difference in
detail permits the user to deal with complex geometric
shapes
.
The method has been well proved by the solution of
problems which have classically known solutions. Thus it
20
seems to provide a new and' powerful way to determine stress
concentration factors.
Considering all of the above, the Finite Element Method
(FEM) was chosen for use in this study.
21
III. DETAILED PROBLEM SPECIFICATION
A. GENERAL
Before proceeding with FEM program selection, it will be
convenient to have the problem specified in full detail.
This is done so that further reference to computer program
selection and modification can be with the requirements of
this problem specifically established.
In Section I we limited the study to a family of thin,
flat bars with general end loading. The stress intensifiers
were limited to two forms: a single notch and two symmetrically
located notches. Now specifically we deal with that general
loading and a detailed description of notch shape. A single
notch shape is considered: the bottom of the notch is
circular, characterized by its radius, and the sides are
V-shaped. The included angle of the V may be between 0°
(U-shaped notch) and 180° (unnotched bar)
.
This problem family was chosen because of its practical
importance and because it includes much of the available
experimental data.
B. FOUR PROBLEMS
Division of the problem family into two general classes
based on the number of notches has already been indicated.
The two classes are shown in Fig. 7.
We speak of general loading at the ends of the bar-
containing the stress intensifier. However, by virtue of the
22
to
<D
wwcd
rHO
o
.QO
PL.
O
bO•H
23
principle of Saint Venant, if the bar is several times longer
than its width, as we assume and as is the case in practical
applications, then only the static properties of the load,
i.e.;, the net axial force and the moment (about, say, the
central axis of the bar) are of importance. Thus what we
have called a general loading can be treated by the super-
position, in proper proportion, of a case with an axial force
and no moment and a case with moment and no net axial load.
Based on this, the four specific problems shown in Fig. 8
and described in Table I were selected for study.
TABLE I
Problems Studied
Problem Number Notches Loading
1 Two Axial Force
2 Two Moment
3 One Axial Force
4 One Moment
The general loading of the class of problems with two
notches can be obtained by an appropriate superposition of
problems 1 and 2. The maximum stress (a ) always occurs^ max
at the base of one of the two notches and since they are
symmetric, it is apparent that the maximum stress can always
be calculated.
24
t
y i i i y
CO
6(U
rH
O
Oh
E<L)
rH
O£h
m A A A A A A A A
to
6(D
rH.ao
P-.
U
oPh
OD
•H
M I t t Y
e<L)H,0O
Ph
The general loading of the class of problems with one
notch can similarly always be obtained by an appropriate
superposition of problems 3 and 4. However the maximum
stress does not always occur at the base of the notch.
Consider first the bending of problem 4. Maximum tensile
stress occurs on the straight boundary opposite the notch and
maximum compressive stress at the bottom of the notch.
Superimposing problem 3 can bring the stress at the bottom
of the notch to any value, say zero, producing maximum stress
on the boundary opposite the notch.
Accordingly, for this class, data should be generated and
made available both for the point at the bottom of the notch
and also for the opposite point oh the straight boundary.
C. ONE STRUCTURE
By taking advantage of symmetry, the four problems may
be reduced to four cases each of which is completely described
by the single geometry of Fig. 9 and one of four different
sets of boundary conditions. Face B is a circular arc of
radius R. Face C is a straight line inclined at angle 6.
The other four faces, A, D, E and F, are straight lines
parallel to the x or y axis. The length of face A is
designated D, the length of face E is designated W, and the
length of face F is designated L. We take L = 5W so as to
assure, by St. Venant's principle, that fine details of
end loading do not influence the stress distribution in the
notched portion of the bar.
26
Face C
Face B
Face D
Face E
Face F
-c
W
-<>•
Y-H
Fig. 9. Structure face:
The geometry of the structure can now be adequately described
R Dby the dimensionless ratios g , rj and .
D. FOUR SETS OF BOUNDARY CONDITIONS
The four cases to be described here and summarized in
Tables II and III are equivalent to the four problems
described earlier. Cases c and d are problems 3 and k
directly. Appropriate superposing of two cases a and of
two cases b along face F yield problems 1 and 2 respectively.
At this point we delineate the boundary conditions
corresponding to each of the four cases. It is evident that
we have to specify values of the displacements u and v along
the faces A, B, C, D, E and F and/or the values of the forces
27
P and P applied along these faces by external agencies,x y
Along faces B, C and D there are no externally applied forces
and the displacements u and v are not specified (that is,
they are unknowns which will be solved for by the PEM
program). All internal points of the structure, likewise,
have no specified values of u or v and are not externally
loaded.
Because of the symmetry in all four cases, we specify
that u = all along face A. Along face F, for cases c and
d, there are no boundary forces and values of u and v are
unspecified. For case a, face F is along a line of symmetry
and we take v = all along face F. For case b, the condition
of antisymmetry is satisfied by taking u = at every point
along face F. In neither of these cases are there any
prescribed boundary forces along face F. In all four cases,
the point which marks the intersection of faces A and F is
restrained so that u = v = 0.
Loadings are applied on face E. The displacements along
face E are unspecified. All forces F are zero. Values of
F along this face are such that they produce the required
loadings as shown in Table II.
A summary of boundary conditions for faces A, B, C, D
and F is given in Table III. The conditions are given as
triples (u,v,Fj. All values of F are zero. An asterisk (*)x y
indicates that no value is specified.
28
TABLE II
EXTERNAL LOADING ALONG FACE E
Loading CaseNumber
Static Effect
M
PW2
3P
2- PW
- PW
The displacements along face E are unspecifiedF are zero.
All forces
29
TABLE III
BOUNDARY CONDITION SPECIFICATION
Pace Case 1 Case 2 Case 3 Case 4
A (0,*,*) (0,*,*) (0,*,*) (0,*,*)
B,C,D (*,*,0) (*,*,0) (*,*,0) (*,*,0)
F (*,0,0) (o,*,*) (*,*,0) (-*,*9 o)
30
IV. SOFTWARE SELECTION
A. BASIC PROGRAM SELECTION
The purpose of the study was not only to obtain some useful
data (i.e., SCF's) to supplement and refine that already
available through other sources, but also to perfect computer
software for generating such data and to establish its
reliability. Reliable, efficient programs capable of solving
the two-dimensional plane stress problem and three-dimensional
stress problem have been written and are available. Some of
them are proprietary but many appear in the literature and
portions of others have been published. Unfortunately, mainly
due to the state of the art in the field Of computer science,
most programs are machine dependent and therefore could not
be considered for use in this investigation.
The two programs selected were developed at the Naval
Postgraduate School. PLISOP [^] is a two-dimensional finite
element program capable of solving the plane stress problem.
It utilizes isoparametric elements which means that curved
boundaries can be represented. All data handling in this
program is confined to the main core storage of the computer.
The program has internal documentation which includes instruc-
tions for its use. TRISOP [10] is a three-dimensional finite
element program utilizing isoparametric elements. Because
of the large amount of data implicit in the three-dimensional
analysis, much of the program coding deals exclusively with
31
handling the data. Most of the storage is outside the
computer main core on direct-access devices which provides for
the storage of large volumes of data. The price paid for
this expanded storage is the large computational time needed.
Internal documentation includes instructions for its use.
Prior to the start of this investigation both of these
programs had been debugged on the IBM-36O computer available
for use in this study and solutions to a number of problems
had been compared to the classically obtained solutions to
demonstrate that the programs are capable of producing very
good approximations when used correctly. The remainder of
this section deals specifically with the two-dimensional
program. Cf. Appendix C for remarks concerning the use of
TRISOP
.
B. MESH GENERATION
The first step in employing a FEM program for the solution
of a specific problem is to provide appropriate input data,
describing the problem, in the form called for by the
specifications of the particular FEM program employed. This
requires the preparation of a very large volume of data.
Material properties and program options must be selected,
of course, but the great preponderance of the input has to
do with geometry. The user must discreti^e the structure
into elements and describe these elements and their inter-
connection to the program. Coordinates for each node used
(specific points on the element boundaries) must be provided
along with load boundary conditions. ' The displacement boundary
conditions constitute the third large block of information
32
required. The elements and nodes are numbered for identi-
fication using numbering conventions dictated by the coding
of the FEM program. Specified data card format must be
adhered to rigidly. The discretization ultimately selected
for most of this study would require over 600 data input
cards.
Manual preparation of such a volume of data is unthinkable.
However this problem is not new and others have dealt with it
before and have provided programs to accomplish the task.
Such a program is called a mesh generator. It is written to
accept a gross description of the structure under consideration
(Fig. 12) and its basic function is to superimpose a grid,
establishing the elements (Fig. 11) required. Coordinates
of all nodes are calculated. Elements and nodes are numbered
according to the criteria of the FEM program which the mesh
generator serves. The program output is in the form of a
deck of punched data cards. The deck, compatible with the
FEM program, is in two parts: element descriptions (element
connectivity cards) and nodal coordinates (element node cards).
PLIMEG is a computer program originally written by J.R.
Adamek as part of his NPS thesis [1] for the purpose of
simplifying the preparation of input to PLISOP. Modifications
to Adamek' s program were made as a part of the present study
in order to make it more useful for the purpose of inputting
data for the notch stress intensification problem and making
optimal use of the computer facilities in solving the problem.
33
The basic input to PLIMEG is a description of the geometry
of the problem, divided, by the author, into several large
subdivisions, referred to as superelements, or, briefly, as
SEL's. The coordinates of the SEL corner points are specified,
the connectivity is indicated, and, in an appropriate fashion,
the geometry between corner points is specified. Other inputs
to PLIMEG are m and n, the number of subdivisions required
along two adjacent SEL boundaries. The numbers m and n must
be consistent on any boundary common to two SEL's.
Internally, the program PLIMEG now establishes additional
nodal points within each SEL, dividing each SEL into m x n
smaller elements — the finite elements themselves. The output
data deck is compatible with PLISOP . A program option may
be used to obtain a graphical output depicting the completed
grid.
The subdividing done in PLIMEG is actually accomplished
in a non-dimensional coordinate system (£,n) and the grid
established there is mapped into the SEL by a coordinate
transformation
.
C. MODIFICATIONS TO PLIMEG
1 . Stress Gradient Considered
The structure considered in the study (Fig. 9) is of
interest because of the stress concentration. That concentra-
tion implies a large gradient which, in turn, demands small
elements in the finite element discretization. The use of a
uniformly fine mesh is undesirable from the standpoint of the
number of equations that must ultimately be solved. Not only
3*1
is the solution more time consuming but the maximum number
of elements is directly dependent on machine storage capacity.
The IBM-36O, using PLISOP, imposes an upper limit of slightly
over 180 quadratic elements, and even that size of problem can
only be considered by special treatment.
Noting that the high stress gradient is very localized,
it is possible however, to use small elements in that
immediate area and larger elements where the gradient is
smaller. By extension, bands of different size elements
may be used. This technique has been successfully employed
by many users. Pfeifer [13 3for example, uses the three-
dimensional stress analysis program (TRISOP) in this way to
solve the problem of a concentrated force on the boundary of
a semi-infinite body. His results compare well with Boussinesq's
classical solution.
Accepting the usefulness of graded element size, a
logical next step is to devise a way to vary element size
progressively as the stress gradient changes. This was
accomplished by modifying PLIMEG . Entering the program at
the point where the nondimensional grid has been established,
the element corner nodes, represented by (£.,,n.) can be
systematically moved by use of the relationships:
exp[ar U, - €_ 1n )] - 1
r = (r _ r )*> 1 mm _ .
s i vsmax smirr r (r _ Nn _
exp[a^max-Wn )] ^exp[a (n .
- n , ) ] - 1n = (r) _ n . ) - n _1_mln _i max 'mirr r , N1 _
35
A wide range of element size variation is provided for by
allowing the user to specify the exponential coefficients
a,, and a as PLIMEG input values. The intermediate nodes are
inserted on the new grid lines thus established, equally
spaced between corner nodes (one if quadratic elements are
used and two if cubic elements are used) . From this point
the coordinate transformation and other details of PLIMEG
are as originally programmed. Mesh generation as originally
programmed and as modified is illustrated in Pig. 10 for
a representative general SEL. Note that PLIMEG, as modified,
provides the capability of dividing a SEL into any specified
number of increments with any specified exponential coefficient
(positive, negative, or zero) in both the E, and n directions,
each completely independent from the other.
2 . Curved Boundary Node Relocation
Isoparametric elements are used both for the SEL
description and the elements in the completed mesh. Three
element types are used. Each is capable of representing a
different degree of variation along an element boundary.
The element types, with their common names and nodal description
are shown in Table IV. The maximum degree of polynomial which
can be exactly represented by each element is given. If
straight line, parabolic, and/or cubic boundaries determine
the geometric shape of the structure, they may be exactly
represented by proper choice of SEL type. If more complex curves
are needed to describe the structure, or if there are
discontinuities, additional SEL ' s can be used to meet the
requirements of the particular use.
36
A y A
[^p^>
BWwiaw^BMMMhMWMMMwiwwwaaatAi I I—a^o X
Nodes repositionedexponentially
Coordinate transfer withequal increments
n4 A
pnap~^>
<r- x
Coordinate transfer withunequal increments
Fig. 10. Mesh generation with and without equal element size
37
TABLE IV
ISOPARAMETRIC ELEMENT TYPES USED
CommonName
Number ofNodes
ExactlyRepresents
Linear
Quadratic
Cubic
Four
Eight
Twelve
Straight Line
Parabola
Cubic Polynomial
It is important to note that SEL boundaries which
cannot be exactly described by the element type chosen (i.e.,
are of higher order than the element used to describe them)
become slightly distorted in the mapping procedure. Except
for those boundaries which coincide with the boundaries of
the structure, this is of no consequence. The structure
boundary nodes, however, must be corrected to locate them
properly in the real coordinate system. A provision to
accomplish this for the curved boundary of Face B (Fig. 9)
has been included in PLIMEG.
A representative mesh, as used in this study, is
shown in Fig. 11. Quadratic elements are used througout
this study.
33
T3
to
w
.cCO
&
•H
3 . Boundary Condition ' Calculations
Boundary conditions for the structure are in two
categories — force and displacement. As established in
Section III D , there are two sets of force conditions and
four sets of deflection conditions. These are easily coded
into PLIMEG where the appropriate input data for the FEM
program is prepared. A particular set of boundary conditions
is imposed by a code number which exactly corresponds to one
of the four problems outlined in Section III B.
D. SUBROUTINE PROBLM
The test structure, divided into the SEL arrangement
utilized, is illustrated in Fig. 12. SEL numbers and SEL node
numbers are indicated. The SEL arrangement was selected
after considerable experimentation with other arrangements,
based on the following considerations:
1. Small elements are required at the base of the notchbecause of the high stress gradient there.
2. Large elements are adequate away from the notch,specifically at the end of the structure where loadsare applied.
3. Storage capacity of the computer limits maximum numberof elements to not more than 180.
t\. Extreme element corner angles (i.e. near 0° or 180°
)
must be avoided.
5. Each SEL must be a four-sided figure.
6. Relying on the principle of Saint-Venant , a singleelement boundary may be used to represent the loadedend.
The coordinates of most of the nodes are obvious after
certain key nodes are located. The origin is at node 27.
Nodes 1, ^4, 7, 20, and 33 are established by the parameters
*I0
to
Jh
O•HPcdo•H<M•H-P
CD
•H
vlWCO
W<Drdo£
cd
'Ci
c
o.o
'd'Gcd
CD
Hcdoto
o
o
co co- rHv-3 Cd
W >CO O
(L) £k -H
P Uo cd
p aco cd
c\j
r-i
60•H
11
directly (R/W, D/W, 0), when L = 5W is provided. The x
coordinate of node 30 was established at W + R and the y
coordinate of node 1*1 at D/6 . Boundary A is an arc of the
parabola passing through nodes 1*1 and 20 with zero slope at
node lH . Boundary B is an arc of another parabola through
nodes 4 and 17 perpendicular to boundary A at node 17.
Boundaries C and D are straight lines. Intermediate nodes,
where indicated, are located at the 1/3, 2/3 points along
the curve establishing the boundary. Table V is a summary
of the SEL node coordinates.
The program PROBLM, written to calculate SEL node
coordinates uses the above criteria and requires a single
card input: parameters R/W, D/W and 0.
E. SUBROUTINE ANS
The answer required for each problem is the stress
concentration factor which is calculated in subroutine ANS.
This program produces a single card reporting problem
parameters and the stress concentration factor associated
with them. The stress concentration factors reported are
based on the following definitions of anom
Case 1, uniform loading:
c = P/Anom
where
:
P = total load applied *
A = 2D = net width of the bar
^Actually here and elsewnere, loads are per unit thicicness.
42
TABLE V
SEL Node Coordinates
Node SEL's x coordinate y coordinate
1 1 D
2 1 R sin(30° - 6/3) Note (1)
3 1 R sin(60° - 26/3) Note (1)
k 1,5 R cos 6 (D+R) - R sin 6
7 5 x4+ (W-yjj) cot W
9 1,5 x^ + (x17
-xi])/3 Note (3)
12 1,5 x^ + 2(x1^-x
lj)/3 Note (3)
Ik 1,2 D/6
17 1,2,5,6 R + W Note (2)
18 5,6 x1?
+ (x20
-x1?
)/3 Note (2)
19 ' 5,6 x17
+ 2(x20
-x17
)/3 Note (2)
20 5,6 5W W
27 2
30 2,6 R + W
33 6 5V7
Notes
:
(1) Determined by circular arc of radius R through 1,4.
(2) Determined by parabola A through Ik3 20 with zero slope
at Ik.
(3) Determined by parabola B through k3 17 perpendicular to
parabola A at 17.
(k) Points not listed are on straight line boundaries equallyspaced between corner nodes.
4 3
Case 2 , bending moment:
a = Mc/Inom
where
:
M = total moment applied
c = D = distance from neutral axis
I = 2DV3 = moment of inertia
Case 3 ? uniform loading:
a = P/Anom
where
:
P = total load applied
A = D = net width of the bar
Case k3bending moment
a = Mc/Inom
where
:
M = total moment applied
c = D/2 = distance from neutral axis
I = D /12 = moment of inertia
F. PROGRAM MAIN
The four programs: PROBLM (establishes SEL node coor-
dinates based on parameters), FLIMEG (discretises the struc-
ture and computes boundary conditons), PLISOP (solves the
M
stress problem) and ANS (calculates the stress concentration
factor) are joined together by MAIN. All card output, except
the single card from subroutine ANS, is suppressed. Input
data and intermediate results are passed within the program
in common blocks and via computer direct-access devices.
The functions of the main program are to read and store the
input data, call the various subroutines in turn, provide
output of the solution cards and prints, and to increment
the problem parameters so as to run through a related series
of evaluations
.
45
V. PROGRAM OPERATION
A. INPUT
The problem input data deck consists of nine cards, a
typical example of which is shown in Fig. 13. Card 1 lists
the problem parameters: R/W, D/W, and 6. Cards 2 and 3
are title cards which may contain any message to identify
the run. Card H lists the mesh parameters: number of SEL's,
number of nodes in each element (not SEL) , problem identifi-
cation number and an indicator for plotting the mesh. Cards
5 through 8, are SEL connectivity cards. Each card lists
the number of increments to be used in the £ and n directions,
their respective exponential coefficients, and a listing of
the nodes, appropriately ordered, defining the corners of
the SEL. Card 9 lists the solution parameters: number of
materials, an indicator for the plane stress solution, and
a code for the numerical integration technique.
The job control language cards (JCL) needed to make the
program operational, details of card format, and program
options are discussed in Appendix D.
B. OUTPUT
Although the output required in this particular applica-
tion is a single number (SCF) , a great deal more is available.
All of the input data is echo checked and the. following
important data passed between subroutines is printed:
46
oOf CO
oomxmUJCM
o oiucr> • •
xco iH •-•
l-N I 1
zm
LUr-l
i-o oo<cr* •
OCX) f*H
coUJCMac—
i
<ol>0 00
SO3lA C\M-C\JC\]
org
QO •--* i—*—t ^—
I
<coor-
vO1 U\
^c\jo-<
OHIO
O CO»h-r-
H-»C0
CM!XJ-—
i
0_JOCM--<>t<lT-00.0 f-« *-»
OSTQ0
»l/)vO
m
O OCOOO'-tCO^H.qc/oc^
OhOo m^^rMmo
<f r- 1>-o r\i
-< CM
h-oonoi-smcMrO
4-l^r-O.-HCM-HCO
OUJa
3
LLi
to
o
47
SEL node coordinates.'
Element node coordinates with the load boundary conditionsas imposed.
Element connectivity.
Displacement boundary conditions.
Material properties
.
Displacements u and v for each node.
Reactions at the nodes where displacement boundaryconditions were imposed.
A force equilibrium check which is the result of summingall x-direction and all y-direction reactions.
Average stress components at each node.
Average strain components at each node.
Stress concentration factor.
Although the stress concentration factor is independent
of Young's modulus and Poisson's ratio, the values 30 x 10
and 0.3 respectively were coded into the program as represen-
tative of common engineering materials. The average stress
and strain components are computed and printed in two ways
:
the cartesian coordinate components (a .a , and t „) and thex * y * xy
principal values (c_.,c . and t „ ) with the inclination of1* 2 max
the principal axes (<$>).
Other output in the form of data cards and the plotted
mesh has already been mentioned.
W
VI. PROGRAM OUTPUT ANALYSIS
A. EQUILIBRIUM
Static equilibrium is maintained in the finite element
solution of all problems studied. As indicated earlier, the
equilibrium of forces is quickly checked from program output
by comparing the imposed loading with the reaction summations
printed by PLISOP as "equilibrium check".
The check on moment summation is slightly less convenient
Program output, including loads and reactions at all nodes,
and the coordinates of nodes, is available but the actual
arithmetic must be done by hand. These external calculations
were made for several representative problems. The results
were as required: moments and forces sum to zero.
B. MAXIMUM STRESS LOCATION
Using the program output, the location of the maximum
principal stress was determined for several parameter combina-
tions. In every case the maximum occurred tangent to the
edge at the bottom of the notch.
C. CONVERGENCE
To investigate convergence behavior, we first identify
the rows and columns of the SEL divisions used. The columns
are referred to as band A and band B; the rows as band C and
band D (Fig. l'l). In what follows, band D always consists
of one row of elements. This limit was imposed because
^9
"> V " V _JL_ •
o
<u
<c\j <A\ \ 1
CM
iH
to
*H
CD
.Q63s
>>TjS
3Pto
CD
o •
C hO<D CbO-Hk ?H
<U Q)
> &C Eo 3o r.
m TJo cCm cd
JOto
TJ CD
C -Pcti ri
.o o•rl
h-3 "3
W cCO •H
•
=riH
bC•HP^
50
of core storage availability in the computer used.
The refinements were accomplished by first assigning a base
number to each of the other bands, then multiplying by a
successively larger integer for each new mesh. Table VI
shows the number of elements per band and total elements in
the mesh. The mesh ultimately used is included for comparison.
TABLE VI
Convergence Subdivisions
"total=WW (Refer t0 FlE
-1 * )
FinalBand First Second Third Fourth DataDivision Run Run Run Run Runs
(1) (2) (2) (2) (3)
NA 3 6 9 12 12
NB
2 H 6 8 3
NC
2 ll 6 8 8
ND
1 1 1 1 1
TotalElements 15 50 105 180 135
(1) This division was used primarily to verify correctperformance of program.
(2) These divisions were used to verify and evaluate convergence(3) This division was used for volume data generation as an
optimal compromise giving good accuracy and satifactorycomputer turnaround.
51
The refinement was done for several combinations of the
problem parameters. Figure 15 is a representative plot of
the stress concentration factor vs. (reciprocal of) total
elements in the mesh for two combinations of the parameters
as shown . In both cases only the three finest meshes used
in the convergence study (50, 105 and 180 elements)are
shown. Fitting a straight line through the points and
extrapolating back to the axis of ordinates, an approximation
for the converged result can be obtained. The value obtained
using 135 elements (see below) has been added. Note that it
plots on this same straight line.
Consider the upper curve shown which represents the case
of a shallow notch. The extrapolated value for stress
concentration factor is SCF = 1.773. The 180 element result
is less than 0.8$ high while the 135 element mesh used is
less than 1% high. Results from studies using other parameter
combinations, including ¥ 0, showed the same convergence
pattern.
The mesh of 135 elements was chosen for data runs because
that size mesh requires approximately 3/^ of the computer core
storage space available. From a practical standpoint data
collection proceeded without undue special arrangements in
the computer center. The larger number of elements required
elaborate special consideration by operating personnel which
was not warranted by the 0.2$ difference in the results.
52
SCF
r.85
Key
SymbolParameter
D/W R/W
o 0.95 0.25
0.05 0.25
1.8 •
1.75
1.2 •
?\ »iXy(
w >
*
1.15 -
1.1 •
<>
200 100_1_
50
Reciprocal number of elements
Pig. 15. Convergence Plot
-£*-
53
D. CORRELATIONS
The data collected is discussed in the next section. As
will be seen, experimental data from the literature correlates
very well with values of SCF calculated. The trends established
are verified and the experimental values are sometimes
identical to three significant figures.
5^
VII. DISCUSSION OF RESULTS
A. RESULTS PRESENTED
The SCF's presented in Appendix A document a represen-
tative variation of the parameters for problem 1. It is
obvious that either 6 = 90° or D/W = 1 constitutes a bar
with no notch and results in stress concentration factor = 1.
Several computer runs with D/W very small showed that as D/W
approaches zero, SCF indeed did approach unity.
To establish the curves, D/W = 0.1 was taken as the
initial value and incremented by adding 0.2 through
D/W = 0.9. Likewise, R/W was incremented by steps of 0.05
from 0.1 through 0.5. 6 was incremented by steps of 15°
from 0° through 60°
.
The data are presented as several families of curves.
Each family has a specified 6 associated with it. The curves
are plotted as SCF vs. D/W with R/W as a parameter for each
family. Linear interpolation between the curves may be used
without appreciable error. Interpolation between the families
for 6 = 60° and 6 = 90°, where stress concentration factor
is one, should not be attempted. Data points actually
calculated are indicated by a cross. The curves through
these points were generated by fitting a sixth order poly-
nomial through them.
Only one run was made to prove the programing for problem
2; no data for this case are presented. One run was also made
for each of problem 3 and k. The data collected are presented
55
in Section VII D in the form of a correlation with experimental
data from the literature.
Time was not available for more extensive data collection.
However, now that the software has been perfected and proved,
it is a perfectly straightforward task to obtain additional
data as computer time becomes available. Appendix D suggests
the parameter ranges which should be investigated.
B. EFFECT OF NOTCH DEPTH
Selected data from the studies of problem 1 are plotted
in Fig. 16. The parameter D/R is constant. Experimental
data plotted is from Flynn and Roll; however Kikukawa's data
shows the same trends . The data in the experiments was for
the symmetrically located U-shaped notches as in problem 1,
with 6=0. .Both Neuber's results and Flynn and Roll's
experimental data are included for the hyperbolic shaped
symmetric notches with the same parameters.
The general close agreement between the FEM and experimental
values for the U-shaped notch is easily seen. The largest-
difference noted is for W/D ~ ^.33 where the difference is
less than 2%.
We might examine the difference by utilizing an aspect
ratio defined as effective length from the notch to point of
load application divided by notch depth. Table ' /compares
this value. As can be seen, the calculations al ' a much
more favorable (i.e., larger, and thus more realistic) ratio
than that provided in the experiment.
56
1.9 •
SCF
1.8 .
1.7
; DR
W I
t i
1
-*
2 3
Fig. 16. Effect of notch depth on SCF for R/W =1.5
W/D
O-
D-
-O Numerical U-notch (Finite Element)
-Q Experimental U-notch (Flynn & Roll)
Experimental Hyperbolic notch (Flynn & Roll)
Analytical Hyperbolic notch (Neuber)
57
TABLE VII
Aspect Ratio L/(W-D)
Source W/D = 2 W/D = 3 W/D = 4.33
PEMCalculations
Experiments
10
8.6
7.5
4.33
6.5
2.6
The maximum point on the curve in Fig. 16 occurs at
W/D = 2, a relatively shallow notch. Experimental data
matches at this point exactly. Note, the error that would be
made if Neuber's hyperbolic notch result was used for such a
notch is more than 7%. Experimental data presented by Flynn
and Roll point to a systematic error in Neuber's results.
C. EFFECT OF ANGLE 6
The data plotted in Fig. 17 compares the stress concentra-
tion factor for G = 60° (SCP^O to the stress concentration
factor for 6=0° (SCFQ
) . The parameter used is W/D. Two
experimentally determined curves show that increasing the
angle of the V from 0° to 60° decreases the SCF.q . This
effect is more profound as the W/D ratio is increased (i.e.,
for deeper notches).
Replotting the data as in Fig. 18 shows the variation of
SCFg with changing W/D for constant SCFQ
. Note the close
correlation of the experimental data and FEM data.
5,8
A
SCF60
2.6
2.4
2.4
2.2
2.0"
1.8
Key
a W/D =1.43 FEM
A W/D = 1.66 EXPTL
W/D = 2.0 PEM
o W/D = 3.0 EXPTL___ _
= SCP for notch having angle
l.i 2.0 2.2 2.4 2.6
SCF
2.8 3.0
Fig. 17. Effect of V angle on SCF. The dot-dash line(SCFgQ = SCFq) is included for reference to emphasize that
SCPgQ
is always less than SCPQ
for any value of the other
parameters. Similar data could be plotted for other valuesof 6.
59
SCF,60
2.6-
2.5-
2.k •
2.3
2.2
2.1
Key
(h FEM
A EXPTL
FEM
O EXPTL
<!
SCFQ=2.6
SCFQ=2.3
W/D
Fig. 18. Effect of W/D on SCFg These lines
2.6) of the
data shown in Fig. 17. The same plot symbols are used as
in Fig. 17.
represent profiles (for SCFQ
= 2.3 and SCFQ
60
Figure 19 demonstrates that increasing the angle 6
reduces the stress intensification and this effect is more
profound for smaller radius notches than for notches where
the bottom radius is large. That is, the SCF for a given
notch may be reduced by increasing either R or G while the
other parameters are unchanged.
1.0
SCF,60
SCF,
0.9
"^_
s
<~o
JbxY
a w/D=1.43
o W/D=2.0
8 10
W/R
.+—{,-—
Fig. 19. Effect of angle on SCF as function of radius
61
D. SINGLE NOTCH GEOMETRY
The only experimental data available dealing with the
single notch geometry is that presented by Cole and Brown [53.
Their data is for a particular case of loading such that the
line of action of the applied force is through the midpoint
of the net width of the bar. By an appropriate combination of
the SCF's calculated for problems 3 and 4 this loading can be
reproduced as shown in Pig. 20. The defining relationship
for SCF as used by Cole and Brown is
:
aQPT7 - maxb0 ** - T7d"
where
a = maximum stressmax
P = total load applied
D = net width
The defining relationships for problem 3 (SCFO and problem
4 (SCFh) are contained in Section IV E.
For the parameter values D/W - 0.5, R/W =0.1 and 6 = 0°,
Cole and Brown report SCF^ = 1.95. The computed results are
SCF~ = 7.85, SCF, = I.98. Making the appropriate substitutions
and solving for maximum stress:
62
w
yD «
D/2
T" Point load (P)
r^p-
/
>1
Uniform load (a = P/W)
-r-4-
"i*"
-H
-—
^
-—->s»-
-;•
Bending moment [M = (W - D)P/2]
Pig. 20. Bar with one notch only. A point load decomposedinto constant and linearly varying components. St. Venant'sprinciple accounts for the absence of higher ordercomponents
.
63
%ax = SCP3
(I ) "SCVx }
(SCF3
- 3 SCP1|)(|)
1.9K§) = scpC^)
Thus the FEM analysis gives SCF = 1.91 compared to Cole and
Brown's 1.95. There is approximately 2% difference in the
results. A special computer run with this specific loading
gave SCF = 1.91 also. This calculation served to verify
superposability of the two cases and the correctness and
internal consistency of the programming and the inputs.
6 'I
VIII. CONCLUSIONS AND RECOMMENDATIONS
A. CONCLUSIONS
The FEM program as presented in this thesis can be used
to calculate the stress concentration factor for the specific
problems listed, over a wide range of the parameters used.
Further, by appropriate superposing of these four specific
problems, the SCF can be calculated for bars which are notched
either on one side only or symmetrically on both sides and
are subjected to general tensile and bending loads.
One of the conclusions from an analysis of the data
obtained for problem 1, the case of the symmetrically notched
bar subjected to a tensile load only, and a comparison with
tabulations and recommendations in the literature is that
the three parameters (minimum radius of curvature, gross
width of the bar, and net width of the bar) do a rather good
job of characterizing the geometry of the notch, in the sense
that interpolated values of the SCF obtained from available
data agree rather well with the more accurate values obtained
by FEM analysis in this study.
However, the analysis also shows that there are some
small but systematic deviations that are assignable to the
fact that these three parameters are not sufficient to
characterize the geometry of the notch completely. The data
presented in Appendix A accordingly provides a more complete
and accurate representation of SCF's in this case than was
previously available.
65
The stress concentration factors presented in the litera-
ture for several specific cases of U-shaped and V-shaped
notches are essentially correct. However, the data presented
is limited, mainly because of the difficulties associated
with experimental investigations. This program overcomes
those difficulties and can solve for a single SCF in less
than three minutes of computer time as compared to several
hours by accepted experimental methods.
B. RECOMMENDATIONS
Accordingly, it is recommended that the program described
herein be used to determine SCF's for problems 1 through 4
and that the results be published for general use. Appendix
D presents a specific plan for obtaining the necessary results
Further, if interest in other notch shapes warrants it,
this program should be appropriately modified and the SCF's
calculated and published.
An axi-symmetric stress analysis program has been written
by Bahkshandehpour [3] as part of his NPS thesis. That
program could be similarly adapted to calculate SCF's for
general axisymmetric geometries having notches.
66
APPENDIX A
CURVES OF RESULTS
The stress concentrations presented in this appendix are
for the problem of symmetrically located, round bottom,
V-shaped notches subjected to uniform tensile loading. The
data collected is plotted as crosses. A sixth order
polynomial is shown fitted through these points.
1 The curves are plotted as SCP vs. D/W using R/W as a
parameter with each family representing a single value,
where
R = radius of the bottom of the notch
D = net half width of the bar
W = gross half width of the bar
6 = half included angle of the V
67
6CO
o
S•H
1O
COCO
B-pco
0.2 0.4 0.6D/W
1.0
Plane stress values of SCF for 6=0°.
x Indicates FEM data, a = SCF(^r)max 2D
68
PL,o00
b
g•H
CDO6oCOto
K-p00
Plane stress values of SCF for 8
x indicates FEM data.
15°
a = SCF("4r)irax 2D
bCO
o
co
S
(L>
8ootoCO
pCO
1.0
Plane stress values of SCP for = 30c
x indicates FEM data, a ov = SCF(^-)
70
in 2
o•poctf
P^
S•H-Pctf
pC<DOSotoCO
KpCO
0.2 Q.k 0.6
DA/
Plane stress values of SCF for 6 = ^5°
.
x indicates FEM data, cj = SCF(~)ITlcLX. C.U
1.0
71
D/W
Plane stress values of SCF for 6 = 60°
.
x indicates FEM data, a ov = SCF(~)max cu
12
APPENDIX B
NEUBER'S NOTCH ANALYSIS
This appendix is included since we have been unable to
find a reference which quotes these useful and important
results in a brief and clear fashion.
V/e are concerned with the two-dimensional case of an
axial load transmitted in the y-direction through the net
section of a thin bar (plane stress case) having a hyperbolic
boundary. All quantities which follow are for unit thickness.
To describe the geometry and to provide a basis for the
solution for elastic stress distribution, Neuber employed
the mapping
z = c cosh Cj C~ cosh (z/c) = ln[ ( z/c) + (z /c - 1)
2
]
(Bla,b)
where z and r, are complex variables
z=x+iy , c=C+in (B2a,b)
and i is the imaginary unit . In real terms this results in
x = c cosh £ cos r\ ; y = c sinh E, sin n (BSa^b)
£ = h cosh" 1(G+H) , r, = h cos~ 1 [G-H] (B4a,b)
where
G = (x2+y
2)/c
2; H
2= G
2+ 2(y
2-x
2)/c
2+ 1 (B5a,b)
73
This mapping provides a correspondence between the xy
cartesian system of coordinates and a £n system useful for
defining the geometry of the notch. Lines of constant £ are
ellipses
( -)2
+ (•
c cosh £ c sinh £)d - 1 (B6)
while lines of constant n are the hyperbolas
(—* )
2- (—^— )
2= 1
c cos n c siniy(B7)
Figure B-l shows the pattern of the £n network.
n = constant
tfinsm
Fig. Bl. Neuber's hyperbolic, symmetric notches in aninfinite plate.
74
A complex Airy stress function is used to solve the
elasticity problem so that
O E T r HO (B8)
along the notch boundary which is the hyperbola
x - y cot n Q- c cos n Q
(B9)
corresponding to the value n = n - The half width at the
base of the notch is the value of x when y = 0. We denote
this by a, so that we are able to evaluate
c = a sec ri • (BIO)
This choice of c selects a definite shape, namely that
which has net width 2a and whose hyperbolic boundary has
asymptotes
y = ± x tan nQ
. (Bll)
It is significant to note that except for a change of overall
scale, the value of n nis the single parameter which defines
•the geometry of the notch. The two "halves" of the hyperbolic
boundary are fixed relative to each other by the fact that
their mutual asymptotes intersect at the origin x = y = 0.
One of the most significant derived quantities is the
minimum radius of curvature of the hyperbola, p, which occurs
at the base of the notch.
75
p = a tan 2nQ
. (B12)
The solution, not given in detail here, permits finding
the stress state (o„,o„ } a„„) or (o r ,o ,i r ) at any point ofx y xy c, ri ^n
the field. A constant, b, is involved in this solution. For
our purposes we need only the stresses along the line
y = 0, viz:
o o 2a = a = 2b esc n (sin n - sin n n )
o 2 2°v
=°E ~ 2b csc J
n (sin n + sin nQ
) (B13,a,b,c)
T = T r =xy Cn
One of the results we will use in Appendix C is the sum
ax
+ ay
= ^ b csc n•
(Bl4)
along y = 0.
To determine the constant b, we note that the tensile
force (per unit thickness) transmitted in the y-direction is
aP = 2 / a dx (B15)
y
Substituting Eq . (B13b) and solving for b, we get
b = (P cos n )/2a(TT - 2n + sin 2n ) (Bl6)
76
Substituting Eq . (Bl6) into (B13b), and evaluating at
n = rig, we get the maximum tensile stress (at the base of
the notch) to be
amax= (2P cot n
o)/a(7T " 2n + sin 2n ) (B17)
The nominal tensile stress is
"nom " Ti < Bl8 >
and the stress concentration factor is
SCF = o /o m = (1\ cot r) n )/(-i\ - 2^ n + sin 2n n ) (B19)max nom '0
In terms of the ratio of minimum radius of curvature p and
net half-width a, we have
SCF = VkO-2 arctan k + 2k/(l+k 2)] (B20)
where
k = p/a (B21)
This shows the SCF in terms of the ratio of two evidently
significant and measurable geometric dimensions.
77
APPENDIX C
THREE DIMENSIONAL EFFECT
1. PREAMBLE
Before proceeding with this discussion, the following
points should be made clear. First, there is every reason
to place full reliance upon the accuracy of the determina-
tions made by the two-dimensional FEM program, that is, to
assert that these calculations accurately represent the
theoretical two dimensional problem (plane stress, as
presented here). From this standpoint, the study of three
dimensionality is intended only to assess the valididty of
photoelastic determinations which were compared to the FEM
determinations, and we in no way depend upon three dimensional
behavior in presenting, with great confidence, our results
for the two dimensional case. However, one cannot evade
the responsibility of making some sort of evaluation
of the effect of finite thickness upon determinations valid,
strictly speaking, only for cases of zero thickness or
infinite thickenss.
78
2. A SPECIFIC PROBLEM
In addition to other difficulties of photoelastic evalu-
ations, there is an inherent error which is due to the fact
that all photoelastic models have finite thickness so that
the conditions of plane stress are violated, at least to some
extent
.
To a first approximation, as shown by Timoshenko and
Goodier [15], the variation of a two-dimensional stress field,
due to finite thickness, may be obtained by employing,
instead of the usual Airy stress function<f>
(x,y), satisfying
2 2 2
V\= ,_u =
^_u =
^u = _ x (cia,b,c,d)
8x^ y dyx 3x8y xy
2
<j)(x,y;z) =<J)
(x,y) - ^Q+y) (x,y) (C2)
where? 2
G (x,y) = V2
<f>= —J- + —
I
(C3)8x^ 8y^
The (modified) stresses
2 2 2
x3y
2' ^ 3x
2'
x* 8 x3y1"'
obtained from this modified stress function, thus have a
parabolic dependence upon the third dimension, and, in general,
79
agree with the elementary (strictly two-dimensional) solution
only on the plane of symmetry, z = 0. -
Note thatQ
is the first tensor invariant of the
elementary solution , that is
% " <°x>0+ (Vo (C5)
where we have added the subscript to indicate that these are
the elementary or unmodified stresses.
In photoelastic experiments, the light beam is affected by
the distance-average stress-state and this differs somewhat
from the stress-state at mid thickness. It is the latter,
however, which corresponds to the theoretical plane stress
distribution which the photoelastic evaluation is intended
to reveal.
To determine the correction it is necessary to have,
locally, second derivatives of the first tensor invariant.
This involves two-fold differentiation of a stress field
which itself has been determined, usually, by graphical or
numerical integration procedures. The problem is somewhat
easier to deal with at points on the free edge. Here there
is only one stress, namely that in a direction parallel to
the free edge, with which we are concerned, and all that is
required is two-fold differentiation of Q~ in a direction
normal to the edge. Nevertheless, it .is quite unlikely that
sufficiently accurate stress determinations will have been
made on the interior of the model area to permit a reliable
evaluation of the second derivative needed.
80
In the particular case with which we are concerned in
this thesis, namely the matter of stress concentrations at
the bottom of notches, we can construct a plausible theoretical
means of obtaining the desired information. This procedure,
and experimentally accomplished equivalent procedures are,
however, of doubtful accuracy and probably do no more than
give a very rough estimate of the proper correction which,
fortunately, usually is not large.
d26The problem, specifically, is to determine
dn2
where 6 is the sum of principal stresses and n is the
direction normal to the boundary at the point at which the
correction is to be made. The evaluation is at the boundary.
3. PROCEDURES TO SOLVE THE PROBLEM
The following procedures suggest themselves. First, one
can use the photoelastic stress information, properly integrated
so as to obtain the stress-sum, and perform two-fold
differentiation. Or, still proceeding experimentally, one can
perform another experiment, using an interferometric procedure,
to get the isopachic lines, which provide a direct experimental
evaluation of the (average) stress- sum, and then differentiate
the field twice in the normal direction. Third, one can
analytically determine the desired second derivatives by
studying an available theoretically exact solution to a
geometry which "matches" that of the case under investigation.
Fourth, one can construct a photoelastic model of such a
geometry for which an analytic solution is available and infer
81
the desired correction by comparing photoelastic results
with the theoretical results and then apply the same correction
to the case under investigation.
Although superficially these procedures seem reasonable,
each has flaws of fatal or near-fatal magnitude. The first
two methods described do indeed relate to the geometry of
the case at hand rather than to some substitute geometry.
Thus, in theory, they are capable of providing the desired
information. However, the fact that a stress-sum field must
be constructed with sufficient accuracy that a second
derivative can be relied upon indicates the need for a degree
of precision considerably beyond what might be regarded as
normally achievable
.
The third and fourth methods are essentially the same —
one is dealing with a substitute geometry for which a
theoretical result is available. In the third method one
needs a more detailed mathematical representation of the
solution; in the fourth method all that is required is the
theoretical value of the edge stress — and a second photo-
elastic determination. In each of these methods however,
the reliance is upon the assumption that the second derivative
of the stress- sum is the same for the substitute geometry as
for the case of interest, or at least that the ratio of this
second derivative to the edge stress itself is the same for
the two cases.
One is quite disinclined to believe that there is any real
justification for such reliance. If the substitute geometry
82
(with known analytic solution) is sufficiently close to the
actual geometry that one may rely upon the coincidence of
the second derivative of the stress sum, then indeed there
should be even closer coincidence between the edge stresses
themselves so that a photoelastic investigation is not called
for at all. It is the very fact that the geometry of interest
does not coincide with a geometry for which there is an
available analytic solution that causes us to make the
photelastic study in the first place.
However, even though we may have deep and quite reasonable
doubts concerning the accuracy of our evaluation or estimate
of the second derivative of the stress-sum (or its equivalence
in other terms), nevertheless, for thin photoelastic models
there is reason to believe that the corresponding variation
in edge stress through the thickness of the model is "small"
so that a greater degree of error may be tolerated in this
estimate than would be tolerable in the basic evaluation of
the edge stress.
Our own study of this matter has involved (1) analytical
differentiation of the stress-sum field for a known geometry
"approximating" that of the case under study, (2) numerical
differentiation of the stress-sum field obtained by use of
a two-dimensional FEM program, and (3) actual use of a three-
dimensional PEM program.
83
In the following paragraphs we concern ourselves with a
U-shaped notch. The radius of the bottom of the notch is
R = 0.2 inch, the gross half width of the bar is W = 1.3
inches, the net half width is D = 0.3 inch and the half
thickness is 0.125 inch. The angle 0, as used in the text,
is 6 = 0°.
a. Analytical Differentiation
1. Neuber's Deep Notch
The geometry of Neuber's hyperbolic notch in
an infinite plate is shown in Fig. Bl . Equations B13 and
Bl4 in Appendix B give the following expressions for
stress :
(a ) n = 2b csc 3n (sin 2
n + sin2ru
)
(C6)y'o
6 = (ax
+ ay
) = 4b esc n (C7)
Letting esc ri = (1-u )" and designating -r— by ' ,dn
we get
eg = 4b (l+2u2)(l-u
2)
5/2csc
2n /D 2
(C8)
.Along y = 0, a varies both with respect to x and z; letting
c = v/2(l+v)
84
we get from eq. (Ck) :
ay
= (ay
) 2D- c G" z
2(CIO)
Taking 2X = t , the thickness, and evaluating the constant,
b:
D X
p = *J / / c dz dx (Cll)y
b = p cos n AXD[G - 3H/2] (C12)
where G = it - 2n Q+ sin 2n Q
(C13)
H = X2c cot 3
n /D2
(Cl'O
From eq. (CIO)
'Vmax = i|b csc ^0 (C15)
(O . = (o) - c e " x2
(ci6)y edge y max
Letting
Ac = (a ) - (a ) . (C17)y y'max y edge
Aa/ J— = H csc n (1 + 2 cos n ) (C18)
y max
8
Pig. CI. Stress profile through the thickness
Figure CI shows the stress profile through the model thickness
The photoelastician sees a averaged through the thickness
2X:
ho(cr ) = (a ) ^v y'ave v y'max 3
(C19)
Consider the case: D = 0.3, p= 0.2, A = 0.125,
p = D tan 2nQ
. We find
86
e " = 18.85
(a ) = 0.982 (a )y ave * v
y max (C.20)
(Here and later F " denotes the evaluation of 6Q" at the
base of the notch.)
That is to say the photoelastic evaluation of stress provides
an uncorrected value for (cr ) that is nearly 2% low.y max *
2. Circular Hole in a Plate
One substitute geometry which suggests itself
is that of a circular hole in a plate under uniaxial tension.
Figure C2 shows a round hole in a large plate.
-v.^U-UWIH 1 **
Pig. C2 . Circular hole in an infinite plate
87
If the plate is large in comparison with the size of a
circular hole the following relations hold:
ar
= S[(l-R2) + (l+3R
i|
- i4R2)cos 2a]/2 (C21)
aa
= S[(l+R2) - (l+3R
Z|
)cos 2a]/2 (C22)
'0=
°r+
°a= S(1 " 2r2 cos 2a) (C23)
where R = rQ/r (C24)
Introducing the three-dimensional behavior, we
have
2 2
(a) = ^4 = ^ + cz2 iii (C25)
a8r
28r
28r
2
and evaluating at the edge (r = rn ) , we get
(a ) , = a n + Aa na edge
= [S-2S cos 2a] + [6S(z/r )
2v cos 2a/(l+v)] (C26)
Table CI shows evaluations of this formula
at various points around the hole, in terms of the ratio
X/rQwhere A is the half thickness.
88
TABLE CI
EDGE STRESS AROUND HOLE-
ot a /SA
15
30
45
60
75
90
-1
0.732
1
2
2.732
3
1.38
1.20
0.69
-0.69
-1.20
-1,38
-1.38
-1.64
oo
-0.35
-0.44
-0.46
Inserting the numerical data as previously used: rfi
= 0.2,
A = 0.125; and solving at a = 90°:
(a ) , = 0.82 a na edge(C27)
Proceeding as before:
6 " = 99.79
(a ) = 0.94 (a )a ave J K crmax
(C28)
89
A more precise analysis using Howland's [8]
formulation yields essentially the same result. It Is apparent
that if a circular hole is used to Infer the needed correction,
as suggested in Section 3 above, SCF's reported for the
U-shaped notch will be slightly high. This is apparently the
procedure used by Flynn and Roll in their experiments.
i|. NUMERICAL DIFFERENTIATION FROM 2D FEM
For this evaluation, we deal directly with the U-shaped
notch. The same numerical data was used to facilitate direct
comparisons. Our objective is to take the FEM computed values
of a + a at the nodes along the line normal to the notch,x y
fit a polynomial through the values obtained and compute the
second derivative in the normal direction. Figure C3 shows
the notch.
Fig. C3. Symmetric U-shaped notches (not to scale)
90
Examination of the stress-sum profile reveals that a
smooth curve is not obtained even with the finest mesh used.
(This is primarily because of a minor systematic inherent
discrepancy between the FEM evaluation of stress at a midside
node and at a corner node.) Choosing the element corner nodes
only, a fourth degree, least square polynomial can be calculated
Examination of the second derivative of this polynomial reveals
that it is slowly converging to some value with each
successive refinement of the mesh. Viewing each of these
results as a term in a convergent sequence, a convergence-
accelerating procedure due to Shanks [1^1 was used to estimate
" at the base of the notch. The numerical result was:
e " 2» 10
(a ) = 0.99^ (a ) (C29)x ave x max
$. DIRECT COMPARISON FROM 3D FEM
The FEM program used in this study was TRISOP [10].
Quadratic elements were used for the solution. The mesh was
generated by TRIMEG [1], modified in the same way as the two
dimension mesh generator (PLIMEG) , to progressively vary the
element size. The mesh in the x-y plane was the same as that
used in the 2D solutions. Two equal size elements were used
for the half thickness in the z direction. Because of symmetry
about the midthickness plane, it was sufficient to consider
only half the plate. The half thickness of the bar (previously
designated X) was 0.125. Figure C'l shows the notch.
91
Fig. C^ . U-shaped notches. (xy, yz, zx planes are planesof symmetry. The planes z = ±X are outside faces,the' planes y = ±w are the unnotched boundaries)
Each computational run involved nearly three and one-half
hours of computer time. Because of this only a few runs were
made to insure proper program operation and gather this data.
Since a large data gathering task was not contemplated here,
the mesh generator and FEM program were not coupled together.
All data input to the mesh generator was manually prepared.
All calculations subsequent to determination of nodal stresses
was done outside the computer.
A parabola was fitted through the values a at the corner
nodes along the line y = a = 0.3. The result of this procedure,
92
cast again in familiar terms was
e " = 14.98
K } ave = °'991 (*x
) max (C30)
The former value was inferred from the coefficients in the
parabolic fit.
6. CONCLUSION
We are now in a position to compare the various results
and estimate the importance of the three-dimensional effect
The aspect ratio of the bar considered here is
AR = thickness _ 0.25 ~276" ~ 0.1width
The results obtained are summarized in Table C2
(C3D
TABLE C2
Results of Three-Dimensional Analyses
Method o /oave max"5" "eo
Remarks
Neuber notch 0.982 18.85 Two derivatives ofanalytical stress-sum.
Hole in plate 0.9^ 99.79 Two derivatives ofanalytical stress-sum.
2D FEM 0.99^ 10 Two derivatives of thecurve fit throughstress-sum fromnumerical data.
3D FEM 0.991 14.98 Parabola fit throughnumerical computationsof a .
X
93
As can be seen from the results presented here, the
effect of three-dimensionality is indeed small. The two-
dimensional and three-dimensional FEM studies compare
closely enough to permit extracting the third dimension
effect from PLISOP. The extraction is not easy. However,
the procedure outlined is readily accomplished with the
computer and requires much less computational time than does
the execution of TRISOP.
We see also from these results that an indiscriminate
adjustment of photoelastically obtained values of SCF based
on the SCF for a known geometry may cause overcorrection.
The overcorrection is small, however, and the error causes
conservative estimates, that is, one overestimates the
ratio (a /a ) , .max ave notch
9^
APPENDIX D
SUGGESTIONS FOR PRODUCING ADDITIONAL NUMERICAL RESULTS
Stress concentration factors for the four problems
presented in this thesis can easily be calculated by using
the program in Appendix F. For all four problems the useful
range of the parameter D/W is from zero to one and the range
of 6 is from 0° to 90°. Note that either D/W = 1 or e = 90°
describes a bar with no notch and results in SCF = 1.
D/W may be incremented in steps of 0.2 (program statements
AA 013^0 and AA 01200) and 6 in steps of 15° (program
statements AA 01380 and AA01180) for all problems to provide
sufficient data. The incrementing step selected in the
program for R/W is 0.05 (program statements AA OI36O and
AA 01190)
.
The SCF's for problem 1 over the full range of D/W, all
8 up to 60° and R/W from 0.1 to 0.5 are presented in Appendix
A. To duplicate this information for the other three
problems, all that is needed is three additional computer
runs. The sample input deck of figure 13 of the text may
be duplicated and used with a single change for each new
problem. The entry on card h in columns 11-15 will be
changed to 2, 3 and ;4 for each successive run.
Each computer run will require 12.5 hours and will
provide the raw data (a deck of punched cards, each listing
the problem parameters and corresponding SCF) paralleling
95
the data plotted in Appendix A. A simple plotting routine
will be needed to plot the data.
In Section III C of the text, we pointed out that the
SCF at two points are of interest for problem 4, namely the
point at the bottom of the notch and also the opposite point
on the straight boundary. As presented, this program
calculates only the first of these for each new problem.
To make the information more complete, an investigator
should add the card "Y(15) = SSJNT(19,D" following PP 0930.
Following SS 00090 add the card "SCF2 = SCF * Y (15 )/Y (1*1 )"
;
change present card SS 00130 by adding ",SCF2"; change
present card SS 001^0 to "FORMAT (6D15.8)".
96
APPENDIX E
PROGRAM USER'S MANUAL
This user's manual is intended to be used with the program
listed in Appendix F.
A user familiar with neither computer coding or the Finite
Element Method (FEM) of numerical solution can, with the aid
of this manual
:
1. execute the program listed to solve the four specificproblems presented in the body of this thesis.
A user familiar with both computer coding and FEM, in
addition to the above, can:
2. change the boundary conditions applied to the structure.
3. change the shape of the notch in a minor way.
k. change the program dimensions to suit the user's needs.
5. modify the program to solve a different, parameterizedproblem.
1. TO EXECUTE THE PROGRAM
The job control language cards needed are listed in
Figure El. No explanation of their make-up will be attempted
in this manual other than to comment that some of the cards
listed beginning with //FTXXF001 provide the direct access
storage needed by the program. Others provide linkages
needed to print, plot, and punch cards.
A sample input deck is included in Figure El. The entries
on the cards are listed in Table El in detail. Figure E2
(identical with Figure 9 in the text) is a sketch of the
structure considered.
97
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uj<7> • •
ICO f—< .—
i
l-N i t
>««« •oLULU 2ftnh-H-— o<t-
~* UJLULU ro—
»
_J_JI- Ol-vli-H LULULU LU—Io T~*)_l HO ooo . ».UJ <.CT> • •
u_ J2Q oco .-tt-t
>o UJLU *• —«r-o Z-Z-Jt QOY- •—» wuu zir»LL. ii ii z: 1-4 <f•* O-O-^ m
r-H OOl/O II LUC\J idO Ml-lQ. CC^ oO CJCD00 <o LUu_ • »*»-H cr> oin — »-»o oOuOo »-,»-. •- Zh H-
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«
s^o 3LL -—-» t •--—
—
3iT> (M^CNJtM O-_» ^•O'-M'-l -J-4- —« i—(—
»
z:—» II ooo •> OrO •-H
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<<X>
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w*^ t—
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LCLL00 00LL ULLLoOOi—
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- r-< (\| UA vQOOUOUODC
98
Fig. E2. Structure Faces
Figure E3 (identical with Figure 8 in the text) shows the
four problems solved. The user must make some selections
and can change other numbers; however some entries on these
cards should not be changed. They are indicated by:
User must choose -( )-
User may choose ( )
User may not choose *( )*
Some additional explanatory material may be found in the
thesis by Adamek [1].
Notes to Accompany Figure El and Table El.
a. Card k: If NPLOT= 1 or 2, only a single problem is solved
This is done to avoid unnecessarily drawing the same mesh
99V \
t T t t V
mE
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100
with only minor modification 225 times. This also
provides the capability to stack as many problems as
desired in one data run. If NPLOT ¥ 0, at the end of
calculations for each problem^ control returns to the
reading section of MAIN to begin a new problem.
b. Program PLIMEG has an ordering of SEL's with which the
present use conforms. This means that here we use only
SEL's numbered 1, 2, 5, and 6. Cards 5, 6, 7, and 8
relate to SEL's numbered 1, 2, 5, and 6, in that order.
c. Cards 5-8: Element size increases with coordinate
direction for positive EXPX and EXPY . For the smallest
element to lie at the base of the notch EXPX should be
positive and EXPY should be negative. If the values
shown are not used the stresses calculated may be farther
from the converged result than 1% shown in the body of
this thesis.
d. Cards 5-8: ROWS must be the same for SEL's 1 and 5 and
for SEL's 2 and 6. Likewise for EXPY.
e. Cards 5-8. CLMNS must be the same for SEL's 1 and 2 and
SEL's 5 and 6. Likewise for EXPX.
f. Cards 5-8: Dimension statements in the program limit
the ROWS and CLMNS as follows; ROWS: SEL1 + SEL2 < 9;
CLMNS: SEL1 + SEL5 £ 15.
g. Card 9: Card columns not listed are not used and should
be left blank.
h. Card 9: For a discussion of NGP see reference [16],
Chapter 8. For this formulation NGP = 2 is adequate.
101
TABLE El
LIST AND DESCRIPTION OF INPUT DECK
r
CardNumber
CardColumns
VariableName Explanation
problem parameters1
1-10 (RW) radius/gross width ratio
11-20 (DW) net/gross width ratio
21-30 (TH) Notch semi-angle (degrees)
2 identification
1-48 -(TITLE)- as desired by user
3 additional identification
1-48 - (TITLE )- as desired by user
k mesh parameters
1-5 *(NSEL)* number of SEL'
s
6-10 *(NPT)* number nodes per element
11-15 -(NPUNCH)- 1 indicator for problem 1
2 indicator for problem 2
3 indicator for problem 3
k indicator for problem 4
16-20 -(NPLOT)- indicator for no plots
1 indicator for printer plot
2 indicator for CALCOM PLOT
5-8 super element deck
1-5 *(SELNR)* SEL identification number
6-10 (ROWS) number of columns (divisionsin y direction) in this SEL
11-15 (CLMNS) number of columns (divisionsin x direction) in this SEL
102
TABLE El (Continued)
CardNumber
CardColumns
16-20
21-25
26-30
31-35
36-40
46-50
51-60
61-70
VariableName
*(Node A)*
*(Node B)*
*(Node O*
*(Node D)*
*(TYPE)*
*(NPTSB)*
*(C0N)*
(EXPX)
(EXPY)
Explanation
upper left corner node number
lower left corner node number
lower right corner node number
upper right corner node number
material identification number
number of boundary nodes in thisSEL
code for ' interconnection betweenSEL — for use see thesis byAdamek [H]
exponential coefficientXI direction
exponential coefficientETA direction
9
11-15
26-30
31-35
*(NMAT)*
(NSTRES)
(NGP)
FEM parameters
number of materials in thestructure
2 dimension assumption made
:
for plane stress
1 for plane strain
number of Gauss points used inthe integration
103
Computer time required may be estimated by knowing that
the compile and link steps require approximately two minutes
total and each problem executes in three minutes. Core
requirement with this configuration is 69OK. The core
requirement may be reduced to 6l5K if cards with serial
numbers GG 00530 , GG 007^0 and GG 00750 are removed from the
deck. This removes the plotting capabilities from the
program. If NPL0T= 1 or 2 is used, the program terminates
by "end of file on unit 5" (the card reader is empty) after
the last problem is solved.
2. TO CHANGE BOUNDARY CONDITIONS
Computations for the load boundary conditions and codes
indicating the displacement specifications are entirely
contained in subroutine COORD. The serial numbers on all
FORTRAN statements used to calculate boundary conditions
begin with the three digits EEX or EEY . These instructions
assume that the user can calculate externally applied forces
F and F to describe the loading desired.
Nodal displacement specifications may be imposed in any
of several ways. Each specification is indicated by a code
assigned to one specific node. The array NBC (45^,2) is
used to transmit the coded information. NBC (1,1) is the
node number at which the I displacement specification
applies and NBC (1,2) is the code for that specification
taken from the following list.
104
1: A roller on an x axis (i.e., v = 0)
2: A roller on a y axis (i.e., u = 0)
3: A fixed node (i.e., u = v = 0)
A cumulative tally of the number of nodes where a
displacement specification has been imposed is contained
in the variable NPBC and the tally appears in the output.
At least one nodal displacement must be specified in
each of the two coordinate directions in order to prevent
rigid body motion which might result in meaningless outputs
The cards which must be modified are serialized EEY
.
The actual changes to be made are left to the user.
3. TO CHANGE THE SHAPE OF THE NOTCH
Only a minor change can be accomplished easily. The
notch chape is modified but no modification of the SEL
arrangement is contemplated. That is, we only wish to
relocate SEL nodes 1 through 7 (Fig. E*l, identical with
Fig. 12 in the text) and subsequently to generate an
appropriate FEM mesh. It is assumed that the same problem
parameters (R/W, D/W, 6) are used.
In subroutine PROBLM, the x and y coordinates of nodes
1 through 7 must be re-coded. The FORTRAN statements that
must be changed can easily be identified by the user. Some
computational short cuts may be useful under certain
conditions. Before introducing the shortcuts, however, the
user must understand the data string output by this
subroutine
.
105
CO
uCD
.O63£
CO•H-PCti
O•H<i-i
•H-P£CD
•d•H
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d(1)
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106
We refer to each data string written on direct access
device 10 as a node card which has the format (315, 2D25.16).
The first three numbers (integers) are the number of the SEL
(variable J), the node number (variable K) and the short-
cutting code (Variable I or L) . The other two numbers
(Real*8) are the x and y coordinates of node K (variables
X(K) and Y(K) respectively). The node cards are grouped by
SEL number and the groups arranged in increasing numerical
order. Within each SEL group the cards are arranged
according to the SEL connectivity. That is, the node in
the upper right hand corner first , followed by one card for
each node used to describe the SEL, ordered in counter-
clockwise fashion. In this program SEL ' s 1, 5, and 6 have
12 nodes each; SEL 2 has k nodes.
In what follows, some familiarity with PLIMEG and PLISOP
is assumed. The coded numbers are described and/or
calculated as shown in Table D2
.
The shortcutting code referred to above may be used if
the SEL boundary between any two corner nodes is a straight
line. L=5 for 12 node elements (or L=4 for 8 node elements;
not used in this program) causes PLIMEG (the mesh generator
itself) to calculate the coordinates of these intermediate
nodes. Therefore the coordinates need not be calculated in
subroutine PROBLM. Card serial BB 00790 is an example of the
use of the shortcutting code. Note that where the short-
cutting code L is not used, 1= must be specified as has
been done in card serial BB 00830.
107
The structure boundary nodes along the circular arc
(points 1 through *l ) are corrected (see page 38 of text)
in subroutine COORD. A user modifying subroutine PROBLM
must also recode the corrections made by cards serialized
EEZ. The variables CORD(N, 1) and CORD(N, 2) are the x and
y coordinates of node N respectively.
*1. TO CHANGE DIMENSIONING
The cards listed in Table E2 include all of the
controllable dimension statements in the program. Each is
an exact reproduction of its parent card in the program
except that the dimension numbers, where appropriate, have
been replaced by a code: NR2 through NR15 . By calculating
the numbers represented by these codes, preparaing a new
dimension deck and replacing the parent cards in the program
with the new cards (on the basis of serial number) a user
may tailor the program size to meet his needs.
The designations included in the name column are variable
names used in PLIMEG and PLISOP. If several different
versions of the dimension deck might be required, the computer
terminal can be advantageously used. This can be done by
reading the coded deck into the system, then using up to
fourteen change commands to change the codes to the previously
calculated numbers, and then ordering a new punched deck.
If several problems requiring different dimensions are
to be solved, they can all be solved using a program with
the largest of the dimensions. From the standpoint of
108
efficient computer usage , • however , it is better to dimension
the program to fit each job.
A rough estimate of core storage required (in K BYTES)
may be arrived at by making the following computation:
CORE =2 * M°V K01 °
+ 180
5. TO MAKE MAJOR MODIFICATION
A major modification to the program involves, primarily,
a complete rewrite of subroutine PROBLM. The guidance of
Section 3» together with a good working knowledge of PLIMEG,
can be extended to accomplish this. The user may be guided
by carefully studying the preceding portions of this appendix
109
TABLE El
CONTROLLABLE DIMENSION STATEMENTS
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111
TABLE El (Continued)
EXPLANATIONS OF
DIMENSIONING CODES
Code Name Explanation
NO 3 MAXMAT The maximum number of materials to beconsidered
NO^i MAXNJT The maximum number of nodes to be considered
N06 NGCOL The maximum number of columns in the basisSEL grid
N07 NGROW The maximum number of rows in the basicSEL grid
NO 8 MAXNEL The maximum number of elements to beconsidered
NOll NJT The maximum number of nodes per element(4, 8, or 12)
N015 NGP The maximum number of Gauss points to be usedin the integrations (2 through 5)
NO 2 NGROW * NGCOL * NO 11
NO 5 (NGCOL + 1) * NGROW
NO 9 2 * MAXNJT
NO 10 NBAND The half bandv;idth of the stiffness matrixcalculated from:
IF N011= 4, NBAND= 2 * (ROWS + 3)
IF N011= 8, NBAND= 2 * (3 * ROWS + 5)
IF N011= 12, NBAND= 2 * (5 * ROWS + 7)
Note: ROV/S is the maximum number of rov/s ofelements in the completed mesh
NO 12 MAXNJT + 1
NO 13 2 * NOll
NO 14 3 * NOll
112
APPENDIX F
COMPUTER PROGRAM LISTING
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BIBLIOGRAPHY
1. Adamek, J.R., An Automatic Mesh Generator Using Two andThree-Dimensional Isoparametric Finite Elements ,
Master's Thesis, Naval Postgraduate School, Monterey,1973.
2. Appl, P.J. and Koerner, D.R. "Stress ConcentrationFactors for U-Shaped, Hyperbolic and Rounded V-ShapedNotches," ASME Paper G9-DE-2, 1969.
3. Bakhshandehpour, M. , Finite Element Solution for Axi-symmetric Transient Thermal Stresses , Master's Thesis,Naval Postgraduate School, Monterey, 1972.
4. Cantin, J.G., PLISOP , an internally documented FORTRANdeck, Naval Postgraduate School, Monterey, March 1972.
5. Cole, A.G., and Brown ,. A.F . C.
, "Photoelastic Determinationof Stress Concentration Factors Caused by a Single U-Notchon One Side of a Plate in Tension," Journal of the RoyalAeronautical Society , v. 62, p. 597, August 1958.
6. Faires, V.M. , Design of Machine Elements , 4th ed.,MacMillan, 1965.
7. Flynn, P.D., and Roll, A. A., "A Comparison of Stress-Concentration Factors in Hyperbolic and U-Shaped Grooves,"Experimental Mechanics , v. 7, P. 272, June 1967.
8. Howland, R.C.J. , "On the Stresses in the Neighbourhoodof a Circular Hole in a Strip Under Tension," Phil
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Trans. Royal Society , v. 229, p. *»9, 1930.
9. Kikukawa, M., "A Note on the Stress Concentration Factorof a Notched Strip," Proceedings 10th InternationalCongress of Applied Mechanics, Elsevier, N.Y., p. 337,1962.
10. Leonidas, E., A General Purpose Three Dimensional StressAnalysis Program , Master's Thesis, Naval PostgraduateSchool, Monterey, 1971.
11. Neuber, H., Kerbspannungslehre , 2nd ed., Springer,Berlin, 1958; translation, Theory -of Notch Stresses ,
Office of Technical Services, Dept . of Commerce,Washington, D.C., 1961.
12. Peterson, R.E., Stress Concentration Factors, Wiley, 197^.
16k
13. Pfeifer, C.G., Evaluation of a Three-Dirnonsional StressAnalysis Program , Master's Thesis, Naval PostgraduateSchool, Monterey, 1972.
Ik. Shanks, D., "Non-Linear Transformations of Divergentand Slowly Convergent Sequences," Journal of Mathematicsand Physics , v. 3^, P. 1, April 1955.
15. Timoshenko, S. and Goodier, J.N., Theory of Elasticity ,
2nd ed., McGraw-Hill, 1951.
16. Zienkiewicz, O.C., The Finite Element Method inEngineering Science, McGraw-Hill, 1971.
165
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 2
Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 939^0
3. Chairman, Department of Mechanical Engineering 2
Naval Postgraduate SchoolMonterey, California 939^0
4. Prof. John E. Brock, Code 59BC 2
Naval Postgraduate SchoolMonterey, California 939^0
5. Lt . Howard L. Crego 1SMC 1195Naval Postgraduate SchoolMonterey, California 939^0
6. R.E. Peterson 1
Westinghouse Research LabsPittsburgh, Pennsylvania 15235
7. LCDR Lav/rence J. Smith 2
21022 HagerstownHuntington Beach, California 92646
166
ThesisS5963c.l
1528S5Smith
Stress concentrationfactors for circularnotches.
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Thesi s
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1E2S96Smith
Stress concentrationfactors for circularnotches.
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thesS5963
Stress concentration factors for circula
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