TORSION 0F AXiALLY NON-UNIFORM
CIRCULAR SHAFTS-iTERATiVE Fl-NiTE DIFFERENCE
SOLUTIONS

Thesis for 1515 Degree of Ph. D:
MICHIGAN STATE UNIVERSITY
Dipak Kumar Baza]
1964

 

   
  
 

{Tl-4531!

LIBRARY

Michigan State

University ‘

This is to certify that the
thesis entitled

TORSION 0F AXIALLY NON-UNIFORM
CIRCULAR SHAFTS-ITERATIVE FINITE DIFFERENCE
SOLUTIONS

presented by

DIPAK KUMAR BAZAJ

has been accepted towards fulfillment
of the requirements for

Engineering

7% i” I ’
Ci ’< (I [55‘ ”(a U (7 \Q (Just/2,.“

Major professor

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ABSTRACT
TORSION OF AXIALLY NON—UNIFORM
CIRCULAR SHAFTS - ITERATIVE
FINITE DIFFERENCE SOLUTIONS

by Dipak Kumar Bazaj

A method is developed to apply the iterative technique
for solving finite difference equations to the torsion problem
for axially non-uniform circular shaftss The method consists
of making a general program which could be used for obtaining
the shear stress distribution on an axial section for a variety
of axial non-uniformities of the shafto

A successive overarelaxation iterative method is used
in theprogram and is found to converge rapidly enough so
that one can solve this problem on a digital computers A
fairly fine mesh size can be used in the program, which can
be varied according to the capacity of the computer and avail-
able times

To demonstrate the use of the program we have considered
some examples of collared, filleted and grooved shafts for
the computation_of shear stresses and stress concentration
factorsO The values of the stress functions for the collared
shaft are compared with those obtained earlier by Do No deGa
Allen using a relaxation methods The stress concentration
factors obtained for the filleted shafts are compared with the
experimental values of Lo 80 Jacobson (electrical analogy

l

method) an

method) and of Ao Weigand (precision strain gages)o

l‘x)

§

n

TORSION OF AXIALLY NON-UNIFORM
CIRCULAR SHAFTS - ITERATIVE
FINITE DIFFERENCE SOLUTIONS

BY

Dipak Kumar Bazaj

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

196%

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ducted.
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guidance
My
Euidance
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this Wor?

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financia
Fi

aPPT‘9cia

ACKNOWLEDGMENTS

I wish to express my sincere thanks to Dr. Lawrence E.
Malvern who suggested the problem and under whose pleasant
counsel, lofty inspiration and supervision this work was con-
ducted.

I am also grateful to Dr. G. H. Martin, chairman of the
guidance committee, for helpful suggestions and interest.

My thanks are also due to the other members of the
guidance committee: Dr. H. L. Womochel and Dr. D. W. Hall
Who have contributed to the academic background necessary for
this work, as well as to the staff of the computer center for
their programming help.

I am highly indebted to Dr. C. R. St.-Clair Jr., chair-
man of the Mechanical Engineering Department, for providing
financial support in the form of graduate assistantship.

Finally, I dedicate this work to my wife, Manju, in

appreciation of her patience, understanding and moral support.

ii

gamma
2m 0'.“ '1‘;
LIST or 1
us: or A

Chapter
I. I

IV .

TABLE OF CONTENTS

ACKNOWLEDGMENTS O O O O O 0 O O O O O O O 0

LIST OF TABLES. O O 0 O O O 0 O O O O O O 0

LIST OF FIGURES. o o o o o o o o o o o o 0

LIST OF APPENDICES o O O 0 O 0 0 O O 0 O O 0

Chapter

I.
II.
III.

IV.

VI.

INTRODUCTIONO O O O 0 O O 0 O O O 0
REVIEW OF LITERATURE. . o . o . o .
FUNDAMENTALS O O O O O O O O O O O O
3.1 Elasticity Equations . . . .
3.2 Finite Difference Equations.
3.3 Irregular Star . . . . . . .

METHOD OF ITERATION . o . o o o o o

u.1 Solution of Linear Algebraic
EquationSooooooooooo

O O O O

“.2 Gauss-Seidel Iterative Process . . .

u.3 Application of the Iterative
“a“ Flow Chart 0 o o o o o o o o

PROGRAMMING O O 0 O O 0 O O 0 O 0 0

.1 Program Requirements . .
.2 Program Techniques . . .
.3 Flow Chart for MATGN . .
.H .

5
5
5
5 Flow Chart for BOUGEN.

EXAMPLES OF SOME NON-UNIFORM SHAFTS

6.1 Examples . . . . .
6.2 Collared ShaftS. .
6.3 Filleted Shafts. .
6.u Grooved Shafts . .

O O 0 O
0 O O O
O O O O
O 0 O 0

iii

MathOd o

O O O O

O O O O
O O O O
0 O O 0
O O O 0

0 O O O
O O O O
O O O 0
O O 0 0

0 O O O 0 0 O O

O 0 O O

Page

ii

vii

viii

\10'301

11

11
12
1M
15

18

18
19
22
28

30

30
31
32
3k

Chapter
VII. C3

REFEREA CBS

‘n \Vfi? T‘C
n:?£.lu&CL-

Chapter
VII.

CONCLUSION.

REFERENCES. . . . .

APPENDICES. . . . .

O

O

0

iv

Page
#7

Q9
51

Table
l.

2.

3A.

33.

34A,

“B,

5A.

SB, o

m

as:

7A.

73:

She

St}

(I)

LIST OF TABLES
Table Page
1. Running Time and Number of Iterations. . . . . 39

2. Stress Concentration Factors for Grooved
and Filleted ShaftS. o o a o o o o o o o o o “5

3A. Calculated Stress Function Values for the
COllared Shaft o o o o -o., o o o o o o a o o o 52

38. Stress Function Values for the Collared
Shaft by Allen 0 O O 0 O O O O O O 0 O O O O 52

3C. Calculated Shear Stress Values for the
callared Shaft O O O O 0 O O O O O O o O O 0 53

uA. Stress Function Values for Filleted Shaft,
2r/830025000oooooooooooooo 5”

“B. Shear Stress Values for Filleted Shaft,
zr/S = 00250 O O O O O 0 O O 0 O O O O O O O 55

5A. Stress Function Values for Filleted Shaft,
2r/S=0.5................. 56

SB. Shear Stress Values for Filleted Shaft,
2r/83005o00000000000000oo 57

6A. Stress Function Values for Filleted Shaft,
2r/S=0.75................. 58

SB. Shear Stress Values for Filleted Shaft,
zr/S = 00750 o O O 0 O O O O O O O 0 O O O o 59

7A. Stress Function Values for the Filleted
Shaft, ZIP/S = 1000 o o o o F 0 C 0 o o o o o 60

7B. Shear Stress Values for the Filleted
Shaft, 2r/S = 1000 O O O O O O O O O O O 0 O 61

Table

8A.

83.

11A.

118.

Table Page

8A. Stress Function Values for the Grooved
Shaft. 2r/S = 00250 a o o o o o o o o o o o 62

8B. Shear Stress Values for the Grooved Shaft,
zr/S 3 0025 O O O O O O O O O O O O O O O O 63

9A. Stress Function Values for the Grooved
Shaft, zr/S = 005 0 O O O O O O O O O O O O 6“

QB. Shear Stress Values for the Grooved Shaft,
2r/S 3 0056 O O o O O O O O O o O O O O O O 65

10A. Stress Function Values for the Grooved
Shaft, 2r/S = 00750 a o o o o o o o o o o o 66

108. Shear Stress Values for the Grooved Shaft,
ZP/S = 0075 o o o o o o o o o a o o o a o o 67

11A. Stress Function Values for the Grooved
Shaft, 2r/S = 1.0 a o o o o o o o o o o o o 68

116. Shear Stress Values for the Grooved Shaft,
2P/S = 1000 O o O O O O O O O O O O O O O O 69

vi

 

‘5
‘U.

11

‘oo

Figure

LIST OF FIGURES

Regular Star . . . . . . . . . . . . . . . .
Example of Irregular Star. . . . . . . . . .
Flow Chart for Gauss-Seidel Iteration Method
Flow Chart for Subroutine MATGN. . . . . . .
Alternative Method of Obtaining ER . . . . .
Flow Chart for Subroutine BOUGEN . . . . . .

COllared Shaft o o o o o a o o o o o o o o o

Filleted Shaft, 2r/S = 0.25. . . . . . . . .
Filleted Shaft, 2r/S = 0.5 . . . . . . . . .
Filleted Shaft, 2r/S = 0.75. . . . . . . . .
Filleted Shaft, 2r/S = 1.0 . . . . . . . . .
Grooved Shaft, 2r/S = 0.25 . . . . . . . . .
Grooved Shaft, 2r/S a 0.5. . . . . . . . . .
Grooved Shaft, 2r/S = 0.75 . . . . . . . . .

Grooved Shaft, 2r/S

1000 O O O G O O O O 0

Stress Concentration Factors for Grooved
and Filleted Shafts.) o o o o o o o o o o 0

vii

Page

17
26
27
29
31
35
36
37
38
no
1+1
1.2

#3

M6

 

H3

LIST OF APPENDICES
Appendix Page

A. Tables of the Calculated Values of the
Shear Stress and the Stress Function . . . . . 51

B. Program in FORTRAN Language. . . . . . . . . . . 70

viii

 

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elastic;
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uniform
first fo
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ability 0

The

53853 in

CHAPTER I
INTRODUCTION

Considerable progress has been made in solving plane
elasticity problems, both by analytic methods and by finite
difference methods. The elastic torsion problem for a non-
uniform shaft has drawn much less attention, although the
first formulation of such a problem was given as early as 1899.
This is because of the difficulty of obtaining an analytic
solution for such a problem and the absence until recent years
of iterative schemes which could solve linear algebraic equa-
tions involved in the solution by a finite difference method.
The present work applies recently developed iterative tech-
niques to the finite difference solution of torsion problems
for axially non-uniform shafts, and studies the practic-
ability of the methods.

The study of the stress distribution and the maximum
stress in shafts with non-uniform axial cross-section, subjected
to torsional couple, is of considerable importance in the field
of stress analysis, as it provides the stress concentration
factors and the points of maximum and minimum stresses and
thus is helpful in choosing appropriate factors of safety in
the design criteria of machine parts. It can also be helpful

in optimizing designs.

 

Becaus«
solution of *
either used .
ence methods
ever, these 2
deficient in
in particula:
develop the i

In rem
speed digits
dizensional 1
the finite d
this work a 1

torsion prob

 

2

Because of the difficulty of obtaining an analytic
solution of this problem, all the work done until now has
either used analog-experimental solutions or finite differ~
ence methods using graphical or relaxation techniques. How»
ever, these solutions always either lack generality or are
deficient in accuracy or convenience. The relaxation method,
in particular, requires practice, experience and patience to
develop the skill required.

In recent years, the use of matrix iteration with high
speed digital computers has made it possible to solve two-
dimensional elasticity problems to the desired accuracy by
the finite difference methods using a very fine grid. In
this work a general program is prepared which can solve the

torsion problem for a variety of axially nonouniform shafts.

 

 

fined-inn!“ sun-

CHAPTER II
REVIEW OF LITERATURE

Michell (8, 1899) was the first to formulate equations
defining the torsion problem for a solid of revolution.

Foppl (3, 1905) undertook this problem independently in order
to determine the maximum stress in filleted shafts. He
suggested an indirect method of obtaining a displacement
function v and then finding a contour s for which v is a solu“
tion by assuming a velocity potential function o.

FOppl also used a hydrodynamic analogy assuming that
most of the twisting moment is concentrated in a thin layer
at the surface. This also was not very successful because of
the nature of his assumption.

Willers (22, 1907) was the first to put the torsion
problem in a form involving thesxress function and the twist
function. He used Runge's method of numerical integration
to find the distribution of the shearing stress in fillets of
a specifically dimensioned collared shaft.

Between 1912 and 1933 many papers* were published, most
of which extended the mathematical theory and obtained the
stress distribution in the mathematical solids of revolu-

tion such as ellipsoid, paraboloid, etc. In this connection

 

*For further reference on these see Higgins (5, 19u5).

3

only one w'
in a shaft
process of
obtain a ‘2
difference
did this 1
mesh poin‘
of each 5<

Hort
applied r
and obtai
a SPecifi
flicker,
mical ::
Very time

Pelaxatic

u
the work of Timpe (17, 1912), Melan (9, 1920) and Neuber
(10, 1933) are the basis of later development.

Of all the authors mentioned above, Willers was the
only one who furnished a means of determining the stresses
in a shaft of an arbitrary contour, using a time-consuming
process of numerical integration of limited accuracy. To
obtain a better method Thom and Orr (13, 1930) used a finite
difference procedure involving the stress function. They
did this first by estimating the stress function values at
mesh points and then by calculating the value at the center
of each square and repeating this back and forth.

More recently Southwell (13, l9u6) and Allen (1, 19u6)
applied relaxation techniques to the finite difference method
and obtained almost the same results as Thom and Orr did for
a specifically dimensioned collared shaft. Their method is
quicker, better formulated and much less vulnerable to arith-
matical mistakes, than that of Thom and Orr, but still
very time consuming, especially for one not an expert in
relaxation techniques.

In addition to the analytical method of solution of
the torsion problem, there have been attempts to solve it
by analog-experimental methods. In this connection Wyszo-
mirski's (23, 191a) work using fluid flow through a thin
slit and Jacobson's work (6, 1925) and later Thum and
Bautz's (15, 193M) work using an electrical analogy are
important. Timoshenko and Goodier (16, 1951) give results

of Jacobson's electrical analogy.

3.1 Elastic:

 

The net

 

for circular
texts on elas
:cso tibilit'
equation 5 or
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N Goodier

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1‘6 ‘
T62

CHAPTER III
FUNDAMENTALS

3.1 Elasticity Equations

The mathematical formulation of the torsion problem
for circular shafts of variable diameter is given in most
texts on elasticity. From the equations of equilibrium and
compatibility for isotropic materials, a partial differential
equation for a torsion stress function and another equation
for a displacement function are derived.

Following the notations and derivation of Timoshenko
and Goodier (16) the final form of the partial differential
equation for the stress function o in cylindrical coordinates

with z axis along the axis of the shaft is:

a 1 a. a 1 a. :
ss‘;357)*s'z(;stz’ 0‘“
°Pfi§-§.l£+32¢=o (2)
Pr 5
3r as

The only non—zero stress components are Inaandtez, and these

are related to the stress fun;tion ¢ by

T = - 1 3¢
re ~ -—
r2 32
- (3)
T67. : l” 13
r2 3r

A second equation, similar to equation (1) can be de-

rived for the displacement function o, where V is the angle

of rota:

of the s

F
4r
V-
‘e
as o
Lu 5 .3
q
|
‘
q
.

#4"?

GI“

from the

that 511;

of the
m
1

shaft 1

Where e

end at

CORdit
c, ,
Men
5°: nt
the he

filth 1

6
of rotation of an elemental ring of radius rlin a cross~section

of the shaft. The equation for e is

8 3
5r (r3 ar) 82 (r3 a ) = o, (a)

13¢_a

6;? “F ' 5g and
_1 91,335 (5)
Gfiaz—Sr‘.

The boundary conditions for the function ¢. obtained
from the requirement that the boundary be stress free, require
that function ¢ be constant on the boundary of an axial section
of the shaft.

The magnitude of the torsional moment applied to the
shaft is given by

T = 21! (Ma)- #0)), (6)
where «a)and «9)are the values of the function o at the boundary
and at the axis, respectively.

Equations (2) and (6), with the help of the boundary
condition, are sufficient to determine completely the stress
function ¢ at every point in the shaft for a given twisting
moment T, and the shear stresses can then be calculated with
the help of equations (3) by differentiating the function ¢

with respect to r and z.

3.2 Finite Difference Equations
The partial differential equations (2) or (M) have been
solved in closed form only for a few simple cases. For most

practical shapes of the shafts we must resort to approximate

 

sethods. A <
on a digital
the equation
divided into
algebraic eq
ential equat
gives the va

Follow
Timoshenko a
ence equatio

01+e

where O is t
2| 3’ '4 are

‘3 1the radiu

3:3

 

“£311
The f'

no: .

Curved boun ‘
c

at .
a distanc

7
methods. A convenient procedure for solving such problems
on a digital computer is the finite difference method in which
the equation is discretized. The section of the shaft is
divided into a grid, and for each node in the grid a linear
algebraic equation is derived from the original partial differ-
ential equation. Solution of these linear algebraic equations
gives the value of the required function at each node point.
Following once again the notations and derivation of
Timoshenko and Goodier (page #91) for a square grid, the differ-
ence equation corresponding to (2) is given by

01+¢2+¢3+¢e-“¢o-§_11<¢1-¢3)=o, (7)
Po

where o is the point for which the equation is formed. 1,
2, 3, u are the neighboring points to right, top, left and
bottom of point 0 Figure (1), h is the mesh interval and r0

is the radius at point 0.

 

 

 

N

4
Fig. 1. Regular Star

3.3 Irregular Star

The finite difference equation (7) is suitable only for
nodes with a constant h, i.e. for a regular star. Near a
curved boundary there will be one or more neighboring nodes

at a distance less than h. Such points have irregular stars,

 

and for 1
cooplica'

in Fig.

ah and t

before,

0)

Q)

idare a.
7'
.2

05 the

cf -o:

0) lo)

I
“M 1
“MI-$52“

3
3
5:50
3
5
or;

8
and for them the resulting difference equation becomes more
complicated. For example, consider the irregular star shown

in Fig. (2). Points 1 and u lie on the boundary at distances

2 a and b i 1

3 h’ cab—41 - ‘I‘
bb

 

Fig. 2. Example of Irregular Starz

ah and bh respectively from 0. Using the same notations as

 

before,
a. I a 21’¢o, 31 a ¢o’4’3
35' 01 ah ’ 3r 30 h

where 31 is the approximation of the derivative at the center
3r ol .

 

of the interval o-l and so is the approximation at the center

 

 

 

3? 30
0f 3'00
2
3 1 9 ‘¢o _ 90'93 (a)
at? “ err—7w (4T. *5")
o 2'
Similarly
2
3 ¢ ~ 1 9 '9 90-92 (b)
3'27 O‘W)('%fi‘a"T
Also
. ~1 ¢-¢ eo-o3)
5% o '_ 2' ( ah +
or Be 1 (o ~a¢ - (l-a) o ).
s; . “2's. 1 3 °

 

Substituting these in equation (2) we get the finite differ»

ence equation for an irregular star corresponding to equation (7).

2 _ 3 _g . + 2 . 2 3 n_ .
(a(I+a) 2' roa) l I+5 2 + (ha + 2' r ) 3

o
+ 2 e, g 2 + 2 - 3 (l~a)h ¢ _ o. (8)
5(I+5) '5 '5 2 roa ° '

If a=b=1, then equation (8) reduces to equation (7) for regular
star.

An alternative method of obtaining the second partial
derivative of o with respect to r or z, i.e. equation (a) and
(b), is by expanding the function o (r, 2) into a power series
in the neighborhood of the point 0. (See, for example, Wang,

(20), page 138.) Thus, considering o as origin,

_ 2 2
d (r, z) - do + a1 r + azz + a3r + auz
+ asrz+ - - -
At r = o, z = -h we have ¢ = ¢2 and at r = 0,2 = bh, ¢ = o“.

Neglecting higher powers of r and 2, it follows

2
¢2 . ¢O "' 32h + auh

2
and in 2 oo + a2 (bh) + a1+ (bh) .
Solving these equations simultaneously we get

3 _ 2 .-
32 <¢t+ ¢o) + b (¢o ¢2)
.bh(l+b)

 

and
an ”(¢u‘¢o)+b (¢2‘¢o)
bh2(l+b)

 

Similarly using point 1, o and 3,

a1 = (cl-do) + a2 (do-ea)

 

ah (1+a)
and

a3 a (cl-do) - a (do-d3)

 

ah2 (1+a)

At the point 0 (r = o, z = o)

 

2 _
a o - 2a = 2 ¢ -¢ ¢ -¢
3‘7 3 H<1+a) l 1 9-- -9-3
P o ah h

and

10

and

32 g 2a1+ . 2 ¢u'¢o ‘ ¢o’¢2
WTT,

32 0

These are the same equations as (a) and (b).

 

t.l g:

1
obtain.
a part:

to use

invers.
linear
PPopor

Pmsen

CHAPTER IV

METHOD OF ITERATION

“.1 Solution of Linear Algebraic Equations

Little is known concerning the extent of approximation
obtained by solving the difference equations corresponding to
a partial differential equation. It is therefore desirable
to use an extremely fine mesh size in order to get a good
approximation. However, since the number of nodes increases
inversely as the square of the mesh interval, the number of
linear algebraic equations to be solved also increases in that
proportion. Thus the problem of solving difference equation
presents a serious practical difficulty.

Methods for solving linear algebraic equations can be
divided into two classes: direct and iterative. Direct
methods such as Cramer's rule and Gaussian elimination methods
are impracticable because of the size of the system. On the
other hand the iterative method, which begins by assuming at
each point an arbitrary value of the variable and then succes-
sively improves the values, yields the answer only as a limit
of a sequence of calculations, each extending over the entire
field, and therefore becomes time-consuming. In addition to
the fact that the iterative methods can solve a large number
of equations, they can usually take full advantage of numer-
ous zeroes in the storage of matrix A, of the matrix equation

11

 

the sc
tions.
by seI
variai
differ
pre-de
provec
ally c

i.
59.8 ‘1:

Eraph
05 on:
Cirec'

ch‘o

0|.

12
Ad = B, obtained from the set of linear algebraic equations.
The iterative method also tends to minimize the roundaoff

error, because of its selfacorrecting nature.

H.2 Gauss-Seidel Iterative Process

A well-known linear iterative process for approximating
the solution of a set of simultaneous linear algebraic equa-
tions, is the method of Gauss-Seidel. This method is effected
by selecting first arbitrary trial values for the set of
variables and then improving these values gradually until the
difference between the two successive values is less than a
pre-determined number at every point. This method can be
proved to be convergent for a strictly or irreducibly diagon-
ally dominant matrix (for a statement and proof of the theorem
see Varga (19), page 73).

A matrix A is irreducible if and only if its directed
graph is strongly connected. (Varga, page 20.) As the nature
of our equation (7) and (8) of Chapter III is such that a
directed graph for any ordered pair of points is always strongly
connected, our matrix A is always an irreducible matrix.

Also an irreducibly diagonally dominant matrix A 3 aij
is defined to be one in which

n
'aiilz 2 'aij'with strict inequality for at least

3°31
ji

~19-

one i (Varga, page 23). The strict inequality always holds in
our case for all the points adjacent to the boundary.
The set of linear equations is

n
2 a.. X. = b- (la)

. = 1 l] 3 ‘ ‘
J 1:11279091‘)

 

or in ma

AX

saccessi

"€830? x

equation
by the v

(Jacobi
uses in
:revious
his has

does not

Vic's-d b'

a, an u.

t; ‘
...aL
A
A
“Tits

\
b

3‘
Av- .

13
or in matrix notation
AX = B (lb)
. . . (d5 2%m .
Starting with a trial vectorlxj or we improve these

(1) [21

successively to X , X etc., which converge to the solution
vector X. The improvement is effected by cycling through the
equations, replacing only the ith component of the trial vector
by the value necessary to satisfy the ith equation.

The difference between the ordinary iterative scheme
(Jacobi Method) and Gauss-Seidel method is that the latter
uses in the process the improved values available of (i-l)
previous components to improve the values of the ith component.
This has an advantage, when working with the computer, that it
does not require simultaneous storage of two sets of approxi-
mation £K+D and MKiin the course of computation. It can also
be shown that the rate of convergence of the Gauss-Seidel method
for a symmetric matrix is greater than that of the Jacobi method
(See, for example, Todd (18), page 90H).

The basis of constructing an iterative scheme is pro-
vided by dividing the matrix A into a lower triangular matrix
L, an upper triangular matrix R and a diagonal matrix D such
that

A=L+D+R. (2)

Assuming matrix A has no zero entries on its diagonal, we
write

DX = B - (L + R) X (3)

from which we derive

(K) i'1 (K) “ (K-l) +
00 e = - °' 0 ' - .' 0 e e u
an x1 sail xJ 2a13 xJ b1, ( )
3:1 j=i+l
where i=1,2,---n , K 3. l

 

a":
.315

OVEF

n:

O
f
,1)

1“

(K)

and finally obtain X. by dividing by aii°

J
(K) 1’1 (K) “ (K=l)
X. = - z a... X. + r. a... X. - bo (Ha...)
3- °-1 1] J 1'3 J ' 1 11-
3‘ j=i+l

O

This is the Gauss-Seidel iterative scheme. In this, now, we
can introduce a relaxation factor w to obtain a successive

over or under relaxation iterative scheme.

ial
x.(K) = (1-.) x (“'1’ + <./a..) - z a . x (K)
i 1 ii 3:1 1] J
n
- 2 ai° X.<K-1) + b;] (5)
j=i+1 3 _.|

This latter iterative scheme with w>l is found to give a
better rate of convergence in the present problem (See also

Forsythe and Wasow (u), page 260).

4.3 Application of the Iterative Method

The importance of the iterative method of solving linear
algebraic equations comes from the fact that it can take ad»
vantage of some of the special properties of thg_ggefficient

matrix A and the constant matrix B. These properties are

- m MMMWM m “’m

 

common in matrices derived from most elliptic partial differ~
ential equations. The properties are

(a) Matrices A and B are usually of large order, but
are sparse i.e. the non-zero elements are much less in number
than the zero elements.

(b) The non-zero elements of A and B are easy to gener»
ate and therefore the coefficient matrices themselves do not
require any storage place in the memory.

These properties can be well utilized in the iterative
method, since for a Gauss~Seidel iterative scheme only one

equation is required at a time, which can be generated just

 

befoz
e an:

ble 1

Hi

U58 C

of t?

in Fi

15
before its use. Thus the storage of the coefficient matrix
A and the matrix B is completely eliminated and it is possi-

ble to solve a large number of linear algebraic equations.

.— .~.-.m_W—..h--—..~mn~ “as;

u.u"Flow Chart

Ralston and Wilf.(12) give a detailed analysis of the
use of the Gauss-Seidel method in a computer. On the basis
of their summary of the calculation procedure, the flow chart
in Fig. (3) and a description of the flow chart follows.
‘ Box 1: K is the counter which counts number of cycles of
iterations. W is over-relaxation factor.
Box 2: i identifies the equation. 1 5 i 5 n where n is the
number of linear algebraic equations. ER is the error esti-
mate summed over all point for the kth cycle of iteration.

Thus,
n -1)
ER‘K) = r. x1153 XKKD
i=1 1

Box 3: Qi is the value of the two summations under the square
bracket in equation (5) of section 9.2.

Box H: P1 is the final calculated valte of X‘K)
Box 5: D is the difference value of X1 in Kth iteration and
(K-l)th iteration.

Box 6: D is added to ER to get summation of box 2.

Box 7: X1 takes its value after Kth iteration i. e. X1 K): Pi°
Circle A: If i=n which means all the n equations have gone
through a cycle, then flow proceeds to box 9, otherwise the
flow is directed to box 8.

Box 8: i is increased by l for iterating the next equation.

Flow is directed back to box 3.

 

16
Circle B: If ER 5 EC, then the X93 have reached the desired
accuracy and the output can be printed or called. However,
if ER> EC then the flow is directed to Box ‘9 and another
iteration cycle starts.
Box 9: K is advanced by one to start another cycle of iterac

tion and the flow is back to box 2.

 

SWFURT'

 

 

 

* a m-nr w
_ .32. f9 E ii“: +3»?er _

_ ' K-
[E =: c b. paw/m, +(1-W)’X.;

 

 

 

. i
[3: Pii- X‘Efi‘

I at"; at) IDlj

 

 

-1 1
M=oi

 

 

 

 

 

 

 

 

 

Fig. 3.~=Flow chart for GauSSaSeidel iteration method

 

 

 

 

_ 2-
'._.1 _4'-1 .. tee-1311 . Qt)
Vi newt-.0 I f ’lqi" E104! 9‘; +£35.99 1]

7

[a = c to. new... +c1-w)'x{

|3= — «a

6

I Ed”); es“) mi]

 

 

 

 

 

 

apfl__ '2
lie—F2]

 

 

 

 

 

 

   

 

 

 

Fig. 3.~-Flow chart for Gauss~Seidel iteration method

CHAPTER V
PROGRAMMING

5.1 Program Requirements

One of the objectives of this work is to make a general
program which could solve for the stress distribution in any
axially non-uniform shaft. To fulfill this object, this prom
gram must do the following things:

(1) Locate the mesh points and number them.

(2) Identify the mesh points having irregular stars

and get the values of the factors a and b in equation

8 of Section 3.3.

(3) Generate and label the non-zero coefficients of

matrix A as often as is needed.

(u) Modify the non-zero coefficients whenever an

irregular star occurs.

(5) Solve the matrix equation AX=B by an iterative

scheme of the type in Section H.M.

(6) Generate the boundary values at ends of the shaft.

(7) Perform differentiation both in z and r direction

and calculate the stress at each node.

(8) Print the results thus far calculated in proper

order and place.

In addition, the program must have access to data pro-
viding the radius of the shaft at each section, values of the

18

19
stress function at the center line and on the surface, and the

mesh size.

5.2 grggram Technique

To incorporate all the requirements of the program in
Section 5.1, it is divided into three parts; two subroutines
and a main program.

The first subroutine (named MATGN), which forms the
main part of the program fulfills the requirements (1) to (6)
of Section 5.1. This has three overlapping loops. The outer-

most loop is for the iteration cycle and this corresponds to

v—— —._r_~-—

 

the loop of the iteration scheme in Section u.u.

The center loop generates the mesh by adding a mesh
length to row (I-l) to get row I; then after obtaining the
radius at the section where the I-th row occurs it generates
the number of mesh points in this row.

The innermost loop is for points belonging to the same
row. It performs the following functions:

(1) It selects the mesh points one at a time, starting

from the center line,

(2) numbers them in succession,

(3) calculates the radial coordinate at the mesh point,

(H) determines the non-zero coefficients of the matrix A,

(5) determines the non-zero coefficient of the matrix B,

(6) if the mesh point is adjacent to the boundary curve

and has an irregular star, then it modifies coefficients

of the matrix A,

(7) it labels the neighboring four points, and

(8) iterates the Ith row according to the iteration scheme.

20

After all the mesh points in one row are considered, the
row is advanced by one and the inner loop is repeated until
the whole section is covered. This completes one iteration
cycle. This is continued until iteration is completed as in
Section “.“.

The stress function is constant on the center line and
“constant also on the surface of the shaft. The difference
between these constants is proportional to the torsional moment
applied to the shaft (equation (6) of Section 3.1). In this
work a constant value of moment, T, is chosen which gives the
difference in the stress function at the center line and at
the boundary as T/Zw. Although it does not in principle make
any difference whether we assume zero value of the stress
function on the surface or on the center line, it does simplify
the program if the zero value is chosen at the surface, be-
cause it facilitates the calculations with irregular stars.

It is also possible to divide subroutine MATGN into two
separate subroutines. The first one generates the non-zero
elements of matrices A and B and then stores them. Since there
are at most five non-zero elements in each row of matrix A,
it is possible to store them in five different unidimensional
arrays. The row number of the element in these arrays remains
the same as in the original matrix, and the names of the
arrays indicate for which of the points 0, l, 2, 3 and “

(Fig. 1) the element is generated. The matrix B also needs,
in this procedure, a storage place. The second subroutine
picks up row by row one element from each array and iterates

according to the scheme of section “.“.

21

This procedure of dividing subroutine MATGN is found to
take about 15% less computer time for approximately 1200 alge-
braic equations, although it sharply cuts down the capacity
of the program because of the storage space required.

The second subroutine of the program (named BOUGEN) is
for generating boundary values. On the basis of Saint-
Venant's principle it is assumed that sufficiently far from
the non-uniformity along z-direction, where the shaft is uni-
form, the stress function is independent of 2. From our
computations we noted that the value of the stress function
is fairly independent of 2 at a distance greater than 0.75
the diameter on either side of the non-uniformity. The end
boundaries for the solutions are therefore chosen beyond this
distance and the boundary values at the mesh points on the
ends are obtained from a mathematical solution for a uniform
circular shaft.

An alternative method of obtaining the boundary values
of the stress function is by a numerical solution. In this
the boundary values at the mesh points are obtained by per-
forming iterations in the r direction only. Since the differ-

ential equation here becomes z-independent, the terms ¢2 and

2“ do not appear in equation (7) of section 3.2. This equation

can now b wr’tt 3h -2 + 3h = .
e 1 en as ¢1(1-Tr7) o0 ¢3 (1+2?) 0
o o

Southwell (13) and Allen (1) in use of relaxation method prefer to

use the numerical solution for the boundary values. We have noted

in this work that for the mesh size used, there is very little

 

‘5,

.. ‘ _--'x. "'15ij
V

\J

22

difference in the values of the stress functions by the two
methods. For numerical solution, the subroutine BOUGEN has
the same basic form as subroutine MATGN. It has, however,
much fewer points and they all lie on a straight line.

The main program performs differentiation in the r
and 2 directions. A three-point center derivative formula
is used to obtain the derivative at each point lying inside
the boundary curve. A one-sided three-point derivative for-
mula is used to obtain derivatives on the surface. The latter
is also used for points with irregular stars. From these
derivatives, stresses are obtained by equation (3) of section
3.1, and the resultant shear stress is obtained at each point
by a vector sum of the stresses in r and 2 directions. This
part of the program also contains print statements to print
the stress function and the shear stress in the same order

and place as the mesh point on the section of the shaft.

5.3 Flow Chart for MATGN

As the subroutine MATGN forms the main part of the pro-
gram, a step by step description of its flow chart (Fig. “)
is given below.
Box 1: In this an over-relaxation factor, W=l.3, and a
counter K which counts the number of iterations performed,
are introduced.
Box 2: ER is the error estimate and has the same meaning as
ER in section “.“. IM is the counter which identifies the
boundary values supplied by subroutine BOUGEN.
Box 3: I and IR count and label the rows on the grid and the

node points on the grid respectively.

23

Box “: Z and ZB are the distances of rows I and I + 1 re-

spectively from the top end of the section. Y and YB are

the

radii of the shaft at rows I and I + 1 respectively. These

radii can either be read from data, or an equation of the

boundary curve with respect to some origin on the center line

of the shaft can be supplied to compute them.
Box 5: JN(I) is the number of mesh points on the Ith row
Box 6: J is a counter which labels points lying in a row

starting from the center line.

0

Box 7: R is the radius at any point J. AA, AB, AC, AD and

AE are the non-zero coefficients of matrix A for points 0

l, 2, 3 and “ (Fig. 1) respectively for mesh point IR. For

points which lie adjacent to the boundaries these coefficients

will be modified in boxes 8 to 1“.
Circle A: This selects all the points lying in row 1 and

directs their flow via box 8.

Box 8: IM is advanced by one each time this box is encountered.

AB is set equal to zero and B takes the value BR(IM) supp
by subroutine BOUGEN. B, if not modified, later becomes

non-zero element of matrix B for point IR.

Circle B: This selects all the points lying in the last

and directs their flow via box 9.

lied

a

POW

Box 9: IM is once again advanced by one in this box every

time it is encountered. AB is made zero and B=BR(IM) is
supplied by BOUGEN as in box 8.

Circle C: This selects all the points lying adjacent to
the center line and directs their flow to box 10.

Box 10: AD is made zero. A constant value of 0.398T is

added

2a

to B. This is obtained from equation (7) of Section 3.3 and
from the assumption that the stress function has a constant
value of T/2n on the center line.
Circle D: This selects all the points which lie adjacent to
the boundary curve and also are the last points in the row and
directs their flow to box 11.
Box 11: AB is set equal to zero. ‘1
Circle E: If the last point in a row has an irregular star,
i.e., if Y-th then the flow is directed to box 12.
Box 12: In this the values of factors a and b are obtained
as follows: J

a = Y-R and
from the equation z=f(y) of the boundary the value ZL is
determined where Y=R. Then

b = ZL-Z.
If b is found greater or equal to one, then its value is taken
as one. Values of AA, AB, AC, AD and AE are modified accord-
ing to equation (8) of section 3.3.
Circle F: If the number of points in row I is greater than
that in row I+l and if b is less than or equal to one then
the flow is directed to box 13.
Box 13: AB is set equal to zero.
Circle G: If any of the internal points, where J = JN(I)
are adjacent to the boundary, then J > JN(I + l) and the
flow is directed to box 1“.
Box 1“: AE is set equal to zero.
Box 15: This identifies the number of the neighboring points

1, 2, 3, “ of IR by giving them number KR, LR, MR and NR

25
respectively.
Box 16: From this box onwards the iteration process starts
and the steps are very similar to that of Section “.“. In
this box :

QsQi: X.

1 Aij i
i

CquM:
43-"

Since all the Aij's except those corresponding to AB, AC, AD,
and AE are zero we get
Q = AB-X(KR) + AC°X(LR) + AD'X(MR) + AE'X(NR).
Box 17: In this n
P = P5 = (B - Q) W/AA + (l-W) X(IR)
which follows directly from section “.“.
Box 18: D is the error estimate for point IR and since P
is the latest value of X(IR) from equation (5) of section “.“
D = X(IR) - P.
Box 19: ER is the sum of the error estimate for all the points
ER = ER +113]- I
Box 20: Here iteration of point IR is over. J is advanced
by one to the next point in the row. X(IR) takes its latest
Value P. B is set equal to zero and IR is advanced by one.
Circle R: If J > JN(I) i.e. if the last point in the row is
already considered then the flow is directed to box 21, other-
wise the flow is directed to box 7.
Box 21: I is advanced by one to change over to next row.
Circle K: If I is greater than the total number of rows by
one, which means that all the section has been considered,

then flow is directed to box 22. Otherwise the flow is to box

“.

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.5 u u i 33.? L .Efih
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27

Box 22: K is advanced by one, indicating completion of one
more iterative cycle. BC is the error criterion, which in
this case is assumed to be 0.001(IR).
Circle L: If ER 5 EC i.e. if the desired accuracy of X's
has been attained, the flow proceeds to box 23. If ER > EC,
the flow proceeds to box 2 where IM and ER are set to zero and
another complete cycle of iteration beginso
Box 23: The values of X can be printed or called by some
other subroutine. K also may be printed to get the total
number of iterative cycleso

Later in the work an alternative method of obtaining

the error estimate ER is also triedo In this ER is defined

38

EfiK)= Max {K'1'- Km)
1 ix)

for Kth iteration: (For previous definition see section “on,

 

box 2). Boxes 18 and 19 are modified to make these changes

as shown in Pig. (5). The explanation of the figure followso

 

IS M

Box17——>~ D: lxgéhg‘ 13>? fiBOX 20

 

 

 

 

 

 

 

EF2=IJ

 

 

 

Figo 50 Alternative method of obtaining ERo

Box 18: Here D is the ratio of the difference between the
two successive values of X and the latest value of X at a
point.

Circle M: In this if D > ER then in box 19 ER is set equal

28
to D. In this way after all the points are considered, ER
has the maximum value of D.
The maximum value of ER over all the points is compared
with BC the error criterion as before in circle L, Fig. (4).

However, here EC is given a constant value 0.0001.

5.“ Flow Chart of Subroutine BOUGEN

As stated earlier, subroutine BOUGEN, Fig. (6), is
basically the same as MATGN and, therefore, we do not give
here a step by step description of its flow chart, which is
self explanatory once the flow chart of MATGN is understood.
However, some of the variables which do not appear in the sub-
routine MATGN are described here. Also, since two sets of
values of the stress functions are to be generated in this
subroutine, one for each end of the shaft, there are state-
ments which make a shift to the next set after the first is
calculated. The calculated values of the two sets are stored
under the same array name in the order of their calculation
and in the order they are required by subroutine MATGN. The
variable ID in box 1 takes the value of IR in box 17 when the
first set of calculations is over and then in the second set
the labeling of points starts from ID + 1 in box u and 6.
Circles A and F determine the shift from one set to another.
L is the value of large radius and S is that of small radius
at the ends. T corresponds to variable B in subroutine MATGN.
IG determines number of points considered in one set for the
purpose of calculating the error criterion. If the alternative
definition of ER is used, a modification similar to subroutine

MATGN (Fig. 5) is required in boxes 13 and 1H.

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T +mnnmx Exommm<n0 m 35mm...
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CHAPTER VI

EXAMPLES OF SOKE NON-UNIFORM SHAPTS

6.1 Examples
In this chapter, we shall apply the programming tech-

nique of Chapter V to prepare a program in the Fortran language
(Appendix B) to be run on a CDC 3600 digital computer to

solve some specific torsional problems. Three varieties of
shafts having axial non-uniformities are considered:

(1) Collared shafts,

(2) Filleted shafts, and

(3) Grooved shafts.

Although only the three most usual types of non-uni-
formities encountered in practice are considered, the program
of Appendix B is very general and could be used for any other
types of non-uniformities, such as non-circular fillets, grooves,
etc. With a memory capacity of 32000 (capacity of the M.S.U.
CDC 3600 computer), as many as 18000 points can be considered.
This, however, will not leave any place in the memory for stor-
ing the stress values. But, since for a central three point
differentiation only three rows of values at mesh points are
required at a time, it is possible to store the stresses at
the same place in the memory as the stress function. This can

be done either by use of a slow speed memory (magnetic tape)

30

31
or by substituting the stress values in places where the
values of the stress function have no further use. This is
not necessary in any of the examples solved in the present
work, since the maximum number of points considered is not
more than 1500.

A mesh size of L/ZR, where L is the maximum radius, is
used in all the examples except the collared shaft where the
size used is 8/16, where S is the radius of the shaft. The
different mesh size for the collared shaft is used for compar-
ing the result of this work with that of Allen (1), who used
a relaxation technique. The relaxation solution is given by

Southwell (13, page 153).

6.2 Collared shaft

 

 

 

 

a,

 

 

 

 

 

Fig. 7. The collared shaft

The specifically dimensioned example used by Allen is
shown in Fig. 7. He used a value of the stress function on
the boundary as 0096* and zero value at the center line. The
program in Appendix B was therefore modified to suit this re-
quirement. The values of the stress function obtained by the

two methods are tabulated in Tables 3A and 3B. Only the values

 

*Allen used this value to compare his results with that of

Thom and Orr (1n).

32
for points corresponding to mesh size 8/8 are tabulated due
to space limitations. The maximum disagreement between two
sets at any point is not more than u%. The third Table 3C
gives the calculated values of the stress. Neither Allen nor
Southwell give these values in their publications.

Since the collar is symmetrical about the plane perpendic-
ular to the center line and passing through the middle of the
collar, ¢2 = *0 in equation (7) of section 3.2 for all the
points lying on the plane of symmetry. Thus in all the cases
where such a symmetry occurs, one needs to compute the values
of the stress function only for the points which lie on one

side of the plane of symmetry.

6.3 Filleted Shafts.

Filleted shafts are those which have a step and a fillet
of a specified radius to avoid sharp corners and consequently
high stress concentration.

The extent of the stress concentration is directly re-
lated to the diameter at the two ends and the radius of the
fillet. It is possible to see the effect of the fillet radius
on the general distribution of the shear stress, since we have
now a program which gives the shear stresses at each point.

As an example, a set of four shafts are considered with
the ratio of their large radius L and small radiusES equal to
1.5 in all four shafts. The only dimension varied is r, the
radius of the fillet. The value 2r/S is taken as, .25, .S,
.75 and 1.0. In doing the calculations L was chosen 3 units,
8 was chosen 2 units and thevalue of torsional moment T was

taken as 162n. The values of the stress and the stress function

33
were obtained by substituting these values in the program and
are tabulated in Tables u through 7 for each of the shafts.

Although the values of the stress and the stress function
tabulated are based on the arbitrary values of S, L, and T,
these numbers can also be used for any other set of values
of 82, L2 and T2 to give the shear stress and the stress
function, so long as L/S and 2r/S remain constant. This can
be done by use of what may be called the stress factor and
the stress function factor.

The maximum shear stress in a uniform shaft of the same
radius as the minimum radius of the shaft under consideration
can be calculated from the mathematical solution and is given
by

T = 2T (1)

From this the value of the maximum stress for T = 162w and

S1 = 2.053 00.5. Now we define the stress factor k to be

the ratio of the tabulated.stress at any one point to that of
the maximum stress 1max of equation (1). This factor remains
constant for a particular set of L/S and 2r/S. From this it
is possible to determine the shear stress at any point for

any other set of values of T, L, and S. Thus,

k = T2 = 11
1m2 EU°5

or 12 = Tm2 . T1 (2)
55.5

where 12 is the required value of the shear stress at any point
for 82, L2 and T2, Tmz is the maximum value of the shear stress
from equation (1) for 82, L2 and T2 and t1 is the tabulated

value of the shear stress at that point.

39
A similar conversion formula can be obtained for the
stress function by defining a stress function factor m. This
factor is defined to be the ratio of the stress function at a
point to the maximum value of the stress function on the sec-
tion of the shaft. Thus,

m = £__ = constant, (3)
emax

where emax = T + ¢minimum (Equation (6) of section 3.1). If
I

T = 1621 then emax = Bl for e minimum = 0. From this we get

$2 = 11;? 91 (4)

where 02 is the required value of the stress function at any
point for 82, L2 and T2, ¢1 is the tabulated value of the stress
function at the same point and ¢max is the maximum value of the
stress function for 82, L2 and T2.

Figures 8, 9, 10 and 11 give a general distribution of
the shear stress and the stress function over a cross section
of the shafts on the basis of the tabulated results. In these
figures level lines of constant stress and stress functions

are plotted.

6.0 Grooved Shafts

In this case four shafts with grooves having semi-
Circular bottoms are considered. The same values are chosen
for L/S (1.5) and 2r/S (.25, .5, .75 and 1.0), where r, this
time, is the radius of semi-circular groove bottom. For conven-
ience of calculation the values of S, L and T, again, are taken
to be 2 units, 3 units and l62n respectively. The stress and

the stress functions corresponding to these values are tabulated

35

STRESS STRESS FUNCTION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 8.--Fillgted shaft“, 2r/s = 0.25

 

r ‘—
. . (w. .v .mh.

 

STRESS FUNCTWON

8 ejggfi

//

“' 005

STRESS

 

 

 

 

 

 

 

 

 

 

 

 

 

40
so
as
29
15
i%:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 9.--Filleted shaft, 2r/S

 

STRESS

()3

“J

STRESS FUNCTTON

 

 

‘\

1O

 

4'6 .0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. lO.-uFilleted shaft, 2r/S a 0.75

1.0

 

 

 

 

 

 

 

 

STRESS STRESS FUNCTION

1” w ~ , a W

/%

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

39
in tables 6 through 11. Since all the dimensions of the
grooved shaft are the same as those of the filleted shaft,
the same equations <1-u) apply here for getting the shear
stress and the stress function for different sets of values
of S, L and T. Figures 12 through 15 give the constant
stress and the constant stress function level lines corres-

ponding to the tabulated values.

6.5 Running Time and Number of Iterations

The running time on the CDC 3600 digital computer for
the programs and the number of iterations, for the examples
considered here varied with the type of shaft. There is only
a slight variation for the shafts of the same type. Table l
gives the actual running time and number of iterations for
each of the filleted and grooved shafts considered. For the
example of the collared shaft, the time taken is 5 minutes,

29 seconds and the number of iterations 1u2.

 

 

 

 

 

”"93: Table 1. Running Time and Number of Iterations
Grooved Filleted
Time No. of’ Time No. of
2r/d Min. Sec. Iterations Min. Sec. Iterations
.25 6 21 160 6 #3 150
.5 6 2B 163 7 l” 192
.7 6 90 163 7 30 151

1.0 6 40 159 7 1“ 151

 

 

40

STRESS STRESS FUNCHON

 

10
8
4.
2
_. 3°
70
J9

 

 

 

 

‘/\

 

 

 

 

 

 

 

 

 

 

 

Fig. l2.--Grooved shaft, 2r/S = 0.25

141

STRESS STRESS FUNCTWON

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 13.--Grooved shaft, 2r/S 3 0.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

an
The running time and the numbers of iterations, of
course, depends on BC the error criterion (see 5.3). The
smaller is the value the BC the larger will be the number
of iterations and consequently the larger will be the running
time.
'In all the calculations, here, the error estimate ER

as defined in Section u.u is taken to be .1

n
EEK): z XIK)- xix D for kth iteration.
1:

1.

 

 

1

Another definition of ERK 2 Max XéK)- xéK-D was also tried

m
Xi

   

 

'- A N . ‘-"‘-'1
Vim __

with the value of RC = .0001. It was found that the two
methods take nearly the same time and the same number of

iterations.

6.6 Stress Concentration Factors

From the values of the shear stresses obtained for the
filleted and the grooved shafts, the stress concentration
factors are calculated. The stress concentmnflxxfactor K
is defined to be the ratio of maximum stress in the shaft to
the maximum stress in a shaft of uniform radius, the radius
of the uniform shaft being the minimum radius of the non-uniform
shaft under consideration. Thus,

K = 1255

1m

where Enis the same as in equation (1) of Section 6.3.

Table 2 gives the calculated values of the stress con-
centration factors in each case. An additional value of K

corresponding to 2r/S = .125 for both types of the shaft is

also tabulated.

#5

Table 2. Stress Concentration Factors for Grooved and Filleted

 

 

 

Shafts
Grooved Filleted

2r/S Maxm stress K7 Maxm stress K
.125 70.93 1.85 66.0 1.6“
.25 63.20 1.56 51.36 1.28
.5 55.57 1.37 07.76 1.177
.75 52.83 1.30 96.0 1.136
1.0 09.60 1.22 00.93 1.11

 

 

In Figure 15 is plotted the stress concentration factor
K versus 2r/S for the grooved and the filleted shafts. The
dotted line in the figure shows the experimental (methods of
electrical analogy and precision strain gages) values of the
factor K for a filleted shaft based on the work of Jacobson
and of Weigand and given by Peterson (11). Peterson has not
given the stress concentration factor values for the grooved

shafts of the type solved here.

M6

 

    
   
    

 

 

 

"9 l T I i
l-B- -
, GROOVED SHAFTS
'.7 -‘ -—¢
reh- -—
1-3 - —~
‘tFILLETEo SHAFTS
\\(EXPERIMENTAL VALUES
12+— '* .4
H _ FILLETED SHAFTS \+ _
(CALcuLATED VALUES)
0 1 .L 1 1
0 0-25 0-5 0-75 1-0

2?
1;.

Fig. 16.rmStress concentration factors for grooved
and filleted shafts.

 

CHAPTER VII

CONCLUSION

The iterative technique is found useful for solving the
finite difference equations of axially non~uniform circular
shafts subjected to a torsional moment. It is found that the
method of successive over-relaxation converges rapidly enough
so that one can solve this problem on a digital computer.

The results obtained by this method for a collared shaft agree
closely with those obtained by the relaxation method. The
method of iteration, however, is much more convenient and
takes far less time. This method allows for all types of
non-uniformities in circular shafts without any basic changes
in the programming. A fine grid can be used for the solution,
restricted only by the capacity of the computer and the avail-
able time.

From the figures of stress distribution for filleted and
grooved shafts, it can be seen that the maximum stress con-
centration occurs at the place where the cross-section is
minimum. As one goes away from the non-uniformity the stress
levels off to the value for the uniform cross-section shaft,
which is similar to the case of pure tension or compression.
In the case of torsion, however, the actual value of the

stress concentration factor is much less than that in the case

n?

”.8

of tension or compression. The calculated values of the stress
concentration factors obtained by the present method are
slightly smaller (maximum difference is approximately 6%)

than those obtained by the methods of Jacobson (6) based on
electrical analogy and of Weigand (21) based on precision strain
gages for filleted shafts.

We have by example shown that the finite difference
method with the help of the iterative technique furnishes a
feasible and convenient way to determine the stress distribu-
tion for axially non-uniform circular shafts under torsional
couples and that it agrees well in one case with the experi-

mental values.

1.

2.

3.

0.

5.

9.

10.

11.

12.

REFERENCES

Allen, D. N. deG., "Relaxation Methods in Engineeringgand
Sgience," McGraw Hill, New York, 1950.

Feddeeva, V. N., "QQmputational Method of Linear Algebra,"
Dover Publications, New York (translated from Russian
by Benster, C.D.), 1959.

Foppl, A., "Die Beanspruchung auf Verdrehen an einer
uebergangstelle mit scharfer Abrundung," Sitz.-Bayer.
Akad., Wiss. Math. Phys. Klasse, 35, 209-262, 1905.

Forsythe, G. E. and Wasow, W. R., "Einite Difference
Methgds for Partial Differential Equations," John Wiley

and Sons, New York, 1960.

Higgins, T. J., "Stress Analysis of Shafting Exemplified
by Saint—Venant's Torsion Problem," Experimental Stress
Analysis, Vol. 3, 90, 1905.

Jacobson, L. 8.. "Torsional-Stress Concentrations in Shafts
of Circular Cross-Section and Variable Diameter," Trans.
American Soc. of Mech. Eng., 07, 619-601, 1925.

Love, A. E. H., "The Mathematical Theory of Elasticity,"
Dover Publications, New York, 1900.

Michell, J. H.. "The Uniform Torsion and Flexure of In-
complete Tores, with Application to Helical Springs,"
Proc. London Math. Soc. 31, 130-106, 1899.

Melan, E.‘,"Ein Beitrag zur Torsion von RotationskBrpern,"
Tech. Blatt (Prague), 52, 017-019 and 027-029, 1920.

Neuber, H., "Elastischestrenge L63ungen zur Kerbwirkung
bei Scheiben und UmdrehungskSrpern," Zeit f. Angew.
Math. und Mech., 13, 039~002, 1933-30.

Peterson, R. E., "Stress ancentration Design Factors,"
John Wiley and Sons, Inc., New York, 1953.

Ralston, A. and Wilf, H. S..,"Mathematical Methods for

Digital Computers," John Wiley and Sons, Inc., New
York, 1960.

09

13.

10.

15.

16.

17.

18.

19.

20.

21.

22.

23.

50

Southwell, R. V., "Relaxation Methede in Iheenetieel
Physics," Oxford University Press, London, 1906.

Thom, A. and Orr, J., "The Solution of Torsional Problems
for Circular Shafts of Varying Radius," Proc. of the Roy.
Soc. London, 131A, 30-37, 1931.

Thum, A. and Bautz, W., "Die Ermittlung von Spannung-
Spitzen in Verdrehbeanspruchten Wellen Durch ein Elek-
trisches Modell," Zeit-Verein Deut. Ing., 78, 17-19,
1930.

Timoshenko, S. and Goodier, J. N., "Theory of Elasticity,"
McGraw Hill, New York, 1951.

Timpe, A., "Die Torsion von UmdrehungskBrpern," Math.
Ann., 71, 080-509, 1912.

Todd, J., "Survey of Numerical Analysis," McGraw Hill,
New York, 1962.

Varga, R. 8., "Matrix Iterative Analysis," Prentice Hall
Series in Automatic Computation, New Jersey, 1962.

Wang, Chi-Teh., "Applied Elasticity," McGraw-Hill, New
York, 1953.

Weigand, A., "Ermittlung der Formziffer der auf Verdrehung
beanspruchten abgesetzten Welle mit Hilfe von Feindehnungs-
messungen," Luftfahrt-Forsch., Vol. 20 (1903), p. 217
(NACA Translation No. 1179).

Willers, A., "Die Torsion eines RotationskBrpers um
seine achse," Zeit F. Math. and Phys., 55, 225-263, 1907.

Wyszomirski, A. "Stromlinien und Spannungslinien ein
Versuch Problems der Elastizitatslehre mit Hilfe Hydrau-
lischer Analogien Experimentell zu losen," dissertation,
Dresden, 1910.

APPENDIX A

51

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APPENDIX B

70

 

71

PROGRAM 1TSLN4
DIMENSION X(800019JN(500)0AH(100).BH(100)OBR(IOO)Q
128(500)9YB(8000).ER(40)
COMMON X.JN.IR.1.AH.BH.IGR.BR
NH=8
900 FORMAT (3HROW913911(F602))
920 FORMAT(12(F602))
NG=5
DN=100
w=lo3
155:0
NF=O
CALL BOUGEN (NHcNC)
CALL MATGNI (NC.DN.NF9NG.NH.W)
K21
100 1:2
LS=25
L=LS+JN(1)‘2
103 PRINT 9000(10(X(J)9J=LSOL92))
1=1+2
18:1-2
IAzl-l
LS=LS+JN(18)+JN(IA)
L=LS+JN(1)‘2
106 IF(1-K)10301039107
107 1:1
1000 HN=FLOATF(NH)/2.0
1R=JN(I1
KL=K-l
DO 1006 1:2.KL
398 R=0.0
NP=JN(I1
1A=1“l
DO 1006 J=19NP
1R=IR+1
RzFLOATF(J)/FLOATF(NH)
KR=1R+1
LR=1R-JN(1A)
MRle-l
NR=IR+JN(11
TX=(X(MR)‘X(KR))*HN
TY=(X(LR)‘X(NR))*HN
1F(J‘1)1003.100301004
1003 TX=(81.O-X(KR))*HN
1004 1F(J-JN(1))100601005o1005
1005 TX=X(MR)*HN
ZB(1)=(4.O*X(1R)‘X(MR)1*HN/(R+l.0/FLOATF(NH))**2
1006 YB(IR)=(SQRTF(TX**2+TY**2)1/(R**2)
K=K-2
(CONTINUED ON NEXT PAGE

72

200 132
LS=25
L8L$+JN(11‘2
203 pRINT9200(YB(J)9J=LSOL02’QZB(1’
131+2
1331‘2
IA=I-1
LSBLS+JN(1A1+JN(IB)
LRLS+JN(1)-2
206 1F(1-K)20302039207
207 CONTINUE
1017 CONTINUE
END
SUBROUTINE MATGNI (NCQDNQNFQNGQNHOW)
D1MEN510N X18000)QJN(500)9AH(100)OBH(100)QBRCIOO)
COMMON XOJNQIROIOAHOBHQIGROBR
900 FORMAT (18H NO OF ITERATIONs=v131
350 FORMAT(1H093HAA=9F150603HAC3oF150693HAD=9F1506)
250 FORMATCIH003HAH(01292H139F159693HBH(91202H1=9F15061
DO 460 L1=108000
460 X(L113000
KNT=O
CN =FLOATF(NC1
HN=FLOATF(NH)
FNBFLOATF(NF’
GN=FLOATF(NG1
470 ER=000
159:0
1M=O
ZSIOO/HN
131
1R30
300 1F(Z-GN+100/HN)30103129302
301 Y=300
YB=Y
C03000
CURVE‘O.
GO TO 306
312 Y=300
YB=3OO
GO TO 306
302 1F(Z-GN-DN)303v3100304
303 ZA=Z*GN
ZB=ZA+100/HN
CO=OOO
C03001F Y83o ATZ=GN0CO=NON ZERO 1F Y ISNOT 3. AT Z=GN
CURVE=10
GO TO 311
310 ZA=Z-GN
(CONTINUED ON NEXT PAGE

304
305

306

307
308

409

30
31

32
704
702
703
701

73

2832A

CURVano
Y=*(SQRTF(1oo-‘ZA’10)**2)1+300
YB=-(SQRTF(100‘(ZB-1o1**211+300
GO TO 306
1F(Z-GN-DN-FN)305030506
Y=Y

YB=Y

CURVE‘OQ

JN(1130

RzloO/HN
1F(Y-R)40994090308
R=R+100/HN
JN(113JN(1)+1

GO TO 307

NP=JN(I)

pNzFLOATF(Np)

Do 200 J=loNP

1R=1R+1

RRFLOATF1J1/HN

AAZ‘QO
ABB-(100"(3oO/(200*HN*R)1)
ACS-IOO
AD='(100+(300/(200*HN*R1)1
ASS-100

1F(1-1)10192

AC=OOO

IM=IM+1

8389(1M)+B

1F(J‘1)39304

AD:OOO

B=B+8100*205
1F(z-GN.DN-FN)70505
AESOOO

1M=1M+1

ACB-ZQO

IAxl-l

IB=1+1

JN(15180

RB=100/HN
IFCYB-RB)32032931
RB=RB+1.0/HN
JN(1818JN(IB1+1

GO TO 30
1F(J-JN(1B))1097049702
1F‘J-JN(111100703010
1FCJ-JN(I)) 7010703010
1F(CURVE)999022
1F‘CURV517089709‘708

(CONTINUED ON NEXT PAGE

7”

709 A53000
GO TO 10
9 A3800
1F(CO‘2001109710910
710 A53000
GO TO 10

22 IF(JN(1)-JN(IB))33933034

33 AB=OOO
IGR816R+1
BH(IGR)=100
AE=-1.0
GO TO 35

34 AB=OOO

708 AE=000
IGR=IGR+1
399 BH(IGR13-(SQRTF(100’(R-3001**2))+100
BH(IGR)=(BH(IGR)‘ZA)*FLOATF(NH)
35 IF(J-JN(I))70517060706
705 AH(IGR)3100
GO TO 707
706 AH(IGR)=(Y-R)*HN
707 CONTINUE
AA=(2o/AH(IGR)+20/BH(16R1-(3o*(10-AH(16R),1/(J*20
1*AH(IGR))
AC=*(20/(1+BH(IGR)))
AD=‘(2o/(1+AH(IGR)+30/(20*J)1
1F(KNT-1)23024924

23 PRINT 25001GR9AH(IGR)QIGRQBH(1691
PRINT 3509AA9AC0AD

24 CONTINUE

10 1F(1-1)11911912

11 LR=0
GO TO 13

12 LR=IR-JN(IA)

I3 KR=IR+1
IF(J-1)14914015

14 MR=1
GO TO 16

15 MR=IR“1

16 NR=IR+JN(1)

20 03000
Q=AB*X(KR)+AC*X(LR)+AD*X(MR)+AE*X(NR)
P8(B’Q)*(W/AA)+(100‘W)*X(IR)
D=(X(IR)-p)/P
1F(ER-ABSF(D))7IO72Q72

71 ERBABSF(D)

72 X(IR)=P

200 83000
z=Z+100/HN
(CONTINUED ON NEXT pAGE)

21

150

04>

71
72

10
100

75

I=I+1

GO TO 300

KNTzKNT+1
IF(00001-ER)470021021
PRINT 9009KNT

END

SUBROUTINE BOUGEN (NHQNC)
DIMENSION X(BOOO)QJN(500)0AH(100’0BH(IOOIOBR(100)
COMMON XOJNOIRQIQAHQBHQIGRQBR
1080

DO 150 JA=IQIOO
BR(JA13000

”3100

HN=FLOATF(NH)

Y=300

N=23

IR=O

ERBOOO

IF(Y-200)29992

IR=ID

N315

DO 7 J‘ION

R=FLQ‘TF(J)/HN

T300

IR=IR+1

AA=200
AQ=-(I.0*(3oO/(200*HN*R)1)
AC3-(IOO+(3oO/(200*H~*RII)
IF(J*1)49394

AC=OOO

T3810*205

GO TO 6

KR=IR+1

MR=IR*1

Q3000
OBAB*BR(KR)+AC*BR(MR)
P=(T-O)*(U/AA)+(100‘W)*BR(IR)
D=(BR(IR)*p)/P
IF(ER-ABSF(D))71972072
ER=ABSF(D1

CONTINUE

BR(IR)8P

IF(00001-ER)19808

Y3200

ID=1R

IF(N‘15)101091

pRINT 1000(BR(J)9J=191R1
FORMAT(1H0017(F70111

END

END

   

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8812