Stress concentration factors for circular notches

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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4. TITLE (end Subtitle)

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

8. CONTRACT OR GRANT NUMBERfiJ

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Naval Postgraduate SchoolMonterey, California 939^0

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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


D - net 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


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



D = net 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



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 discretiê 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 )] êxp[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—aô 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

Âctually 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

>TjS

3Pto

CD

o •

C hO<D CbO-Hk ?H

 &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° (SCPÔ 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 =

û = _ 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.Û-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

oQ'J>a: oo<r-oo

inH--4-xmLUiMzr-<o o o

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-

I'- -Hi-1—

«

sô 3LL -—-» t •--—

—

3iT> (M^CNJtM O-_» ^•O'-M'-l -J-4- —« i—(—

»

z:—» II ooo •> OrO •-H

—.Q. *<r\00 OiMHOJMOOH r-t UJ*-O0'«-*" »W ^-* *-^ QO

<<X>

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2:wH- ÔO-J-3<ixcoco> Oh- <-JOh-OOO vO 00>-oo -. -^. —••

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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

rH

OuP-.

> u A A A A A A A A

Y Y Y Y Y Y

ECDH.OOuPL.

E0)

rH

O

P-.

C\J

ECD

rH&O

CO

ECD

rH

OP.

P-.

u

o

CD

&En

onW

bO•H

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

^WCO

•

CO

CU

d(1)

C rrj

>>^ CO

cd

'Oc -pP" p.Q OS

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•

to to«- iH.J ctf

W >CO O

tu C£h -H3-P ?H

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k P,•P D,CO cd

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hO•Hfe

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

NOî 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

.

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

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No. Copies

1. Defense Documentation Center 2

Cameron StationAlexandria, Virginia 22314

2. Library, Code 0212 2


3. Chairman, Department of Mechanical Engineering 2


4. Prof. John E. Brock, Code 59BC 2


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

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