"'I'IH' -r~ "I” -

 

BOUNDARY METHOD FOR THE

DETERMINATION OF STRESS COMPONENTS
IN SOLID CIRCULAR PLATES

Thesis for the Degree of M. S.
MlCHIGAN STATE COLLEGE

Freédrich G. K. Grohé
W4?

THESIS
O

This is to certifu that the

l thesis entitled

BOUNDARY METHOD FOR THE
DETERMINATION OF STRESS COMPONENTS

IN SOLID CIRCULAR PLATES

presented In]

Freidrich G. K. Grohe'

has been accepted tummls fulfillment

of tlu- requirements for

M.S. degree in CiVil Engineering

MOW

Majm' prolcssur

 

Date 12/16/49

BOUNDARY METHOD
FOR THE DETERMINATION OF STRESS COMPONENTS

IN SOLID CIRCULAR PLATES

By
Friedrich G. K. ggohe'

A THESIS
Submitted to the Schoel of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil Engineering
19%9

THESIS

ACKNOWLEDGMENT

I would like to express my sincere gratitude to Dr. C. 0. Harris,
Head of the Department of Civil Engineering, and to Dr. Richard Pian,
Associate Professor of Civil Engineering, who have given me every
support in carrying out this study. Dr. Harris in particular made val-
uable suggestions for the improvement of some derivations. His critical
readings of the manuscript gave me great help in writing the final form
of this thesis.

I am particularly indebted to Dr. J. Sutherland.Frame, Head of the
Department of Mathematics, who spent many hours of his valuable time in
discussing the development of the boundary method. His mathematical
suggestions were of decisive influence on the solution of the problem.

I also take the opportunity to express my indebtedness to Dr. Ing.
K. Kloeppel, Professor of Advanced Structural Theory and Steel Structures,
at Dormstadt Institute of Technology, Germany. In his outstanding lec-
tures, he made me acquainted with the determination of boundary values
of a stress function, a knowledge which actually inspired me to under-
take this study.

flaws 1' q
-» Q ' "
flit) £va 1 c) 11". ( I

ham)

1 = (-0-)

3

II

NOTATIONS
Rectangular coordinates.
Polar coordinates
Single concentrated load.
Intensity of a continuously distributed load.

Resultant of all external forces applied on the boundary
between a starting point 0 and a point of reference 1:.

Components of R in the direction of the coordinate axes x, y.
Component of R parallel to the tangent at a boundary point 1:.
Radius of the boundary of a circular plate.

Numbering of particular points of a grid lying on concentric
circles.

Numbering of particular points of a grid lying on rays from
the center to the boundary.

Components of a distributed boundary force per unit length
of the boundary.

Normal components of stress parallel to x- and y-axes.
Radial and tangential normal stresses in polar coordinates.
Shearing stress component in rectangular coordinates.
Shearing stress in polar coordinates.

Airy stress function in rectangular coordinates.

Airy stress function in polar coordinates.

Boundary value of the stress function at point 1:.

Boundary value of G== ‘V g;- at point 1:.

Value of F “3&3 at a point of a grid as determined by k
and 10

Value of F039) at the center point of a circular plate.

Extrapolated value of F (’39) at a point outside the plate.

mamas
Acknowledgment
Notations
1.) Synopsis

2.) The Problem

3.) Justification of the Study

4.) Preliminary Description of the Boundary Method
5.) The Airy Stress Function

6.) Boundary Values of a Stress Function

7.) Trigonometric Stress Function with Fourier Coefficients
Obtained by Integration

8.) Trigonometric Stress Function with Fourier Coefficients
Obtained by Finite Summation

9.) Numerical Hethod for an Approximate Determination of
Stress Components

10.) Approximate Determination of Stress Components by
Finite Differences

ll») Numerical Example

a.) Stress Components by Using the First
lodification, Section 7

b.) Stress Components by'Using the Second
Modification, Section 8

c.) Stress Components by'Using the Numerical
Method, Section 9

12.)Discussion of the Results
Bibliography

Tables and Diagrams

Page

28

34

48
52

52

55

III

l.

1.) smopsrs

 

This thesis gives a method for the determination of stress compon-
ents in solid circular plates under any kind of boundary forces which
lie entirely in the plane of the plate. There are no forces applied in-
side the plate. The boundary values of the stress function are deterb‘
mined.from the external forces and then used for the evaluation of the
Fourier coefficients of a trigonometric stress function.

The same principle is used for an approximate method leading to
approximate numerical values of the stress function at certain points in-
side the plate. The stress components at those points are then deter—
mined from these numerical values using finite differences instead of
differentials.

2.) THE PROBLEI

The object of this thesis is the determination of the stresses in
a solid circular plate subjected to arbitrary boundary loads in the
plane of the plate. No loads act inside the plate; body forces are con—
sidered to be absent.

The thickness of the plate is taken as unity. The restrictions for
the thickness are the same as in other two-dimensional problems of elas-
ticity. For the case of single concentrated loads on the boundary, the
results are true only if the plate is thin and the loads lie entirely
in the centerplane of the plate. For the case of line loads uniformly
distributed over the entire thickness, the thickness of the plate is not

restricted.

2.

3.

2.) JUSTIFICATION OF THE STUDY

In the present time principles of higher mechanics,which were con-
sidered merely "academic cases" until recently, are going to be more and
more introduced into practical design. In many countries, postawar
shortage of structural materials obliges the designer to determine stresses
and deformations more exactly in order to create the most efficient struc—
ture with a minimum of material.

Stress functions play an important role in this development. While
there are a large number of functions which satisfy the compatibility equa-
tion, the problem is to bring those functions in agreement with the boun-
dary conditions which are, of course, different for every individual case,
depending upon the shape of the body or plate and the load conditions.

In recent years much work has been done in giving solutions for
problems in rectangular coordinates. However, there seems to be a lack
of general methods in polar coordinates which could enable the non-expert
on elasticity to find stress functions for any loading condition. For a
few special cases stress functions are given, for other cases only for-
mulas for the determination of stress components have been devived. It
is significant, for instance, that neither Timoshenko nor Frocht give a
stress function for the case of two single concentrated loads acting on
the diameter of a circular plate, but restrict themselves to formulas
for the stress components gained by superposition of three different
cases of loading. The stress function itself obtained in the same way
by superposition and coordinate transformation would be so complicated
that it is practically no more differentiable.

In order to have a method of general applicability, a simple rela-
tion between boundary conditions and stress functions should be found
which would allow us to evaluate certain unknown coefficients of the
stress function under any kind of loading.

4.

5.

A.) PRELIMINARY DESCRIPTION OF THE BOUNDARY METHOD

The boundary method is intended to be a method of general applica-
bility. Boundary values of the stress function are obtained from the
external forces as shown in Section 6. This gives the desired relation
between boundary conditions and stress function as mentioned in the fore-
going section. The further procedure leading to the stress components
inside the plate is developed in three different modifications.

FirstȤodification:
The boundary values of the stress function are represented by a con-

tinuous function around the boundary which is expanded in a Fourier series.
The Fourier coefficients are determined in the usual way by integration
around the boundary. A second biharmonic trigonometric series is assumed
as stress function whose Fourier coefficients are obtained through com-
parison with the known coefficients of the expansion for the boundary -
values. The trigonometric stress function is then differentiated as us-

ual for the determination of the stress components.

Second godification:
The boundary values of the stress function cannot be expressed in

one or two functions with continuous derivatives around the boundary.

In order to obtain the Fourier coefficients, it is then useful to substi—
tute the integration around the boundary by a finite summation using nuns
erical boundary values. If certain formulas are used, which will be
found in Section 8, the approximate coefficients of the trigonometric

stress function are immediately obtained.

6.

Third Modification:

This modification is an attempt to establish a purely algebraic
method, avoiding stress functions, integrations, and partial differentia-
tions in its practical application. A grid is laid over the circular
plate. It is advisable to use a standard grid for which constant coeffi—
cients have been already evaluated. Using the boundary values of the
stress function and a formula given in Section 9, one obtains numerical
values of the stress function at the grid points inside the plate. From
these values, stress components are determined by taking finite differ-
ences instead of differentials.

5Q THE AIR; STRESS FUNCTION

As an introduction to the mathematical part, the conditions for the
existence of the Airy stress function may be mentioned briefly.

From elasticity it is known that a function F( r34"; ) (called the
Airy stress function), which satisfies certain conditions, enables us to
determine the stress components at any point of a body which is under
external loads. Assuming that body forces are absent, these conditions
are for two dimensional problems as follows:

a) Differentigl Equations 2;: mailibrium:

Eggs—a—aao

3.23 a. 3:2- = O
a! 33‘

in which chem and tr! are, respectively, the normal compments of
stress parallel to x and y axes and the shearing stress component in rect-
angular coordinates (Fig. 1).

' b; Compatibility Eguation:

(%+-§§:)(6,+o3\=o ’ (2)

c) Boundary Conditions:

at “S CK?) t’ tr: COS Cufl) =3 x

(3)

n, cos (me) + “3 cos (“3) = Y
in which (our) and (“3) are, respectively, the angles of the norml to the
boundary with the x - and y - axis (Fig. 2).
The stress function F(w‘3 ) is defined in such a way that the stress

components are determined by the following equations:

 

9‘F 3'? as?
Uxaaxli ass-m; Tx3=--3-;5§

(4)

8.

Eqs . (4) satisfy the differential equations of equilibrium (1) . Substi—
tuting Eqs. (4) in the compatibility equations (2), it is seen that the

stress function F( my ) must also satisfy the two dimensional bipotential

 

equation:

8“F + as: an: __

a» dxlaat ‘" 03:, " 0 (5)
or briefly: AA F = 0

As a result, a two dimensional problem is solved if a stress function
F029?) can be found which satisfies AA F = O and the boundary conditions.

In pglar coordinates, the stress components are obtained from F(~r.9 )

 

1V:
.L 8F .1. 3'1":
0.3 v7‘ *La g
3‘1:
0": 0*!- (6)
.. 1. 3E. _ A. 3“
“v0= w- a «r 3*89

in which who“ and Two are, respectively, the radial and tangential
normal stresses and the shearing stress in polar coordinates (Fig. 3).

The compatibility equation AA F = O is as follows:

_. ‘8 ~19}. a: iii LEE _.
(.vnwsaaHoswauam)” <7)

 

 

 

! T3” 36;”- d3
( __._ 3f:
: V:.‘7~=1,wsd
I 0'2- -| /‘ Ai 30»:
* ~—~»— ', r- » <7» we
‘ T" L... / 4’._j___ ma. ..__—.-

FIG.“ EQU|LIBR|UM CF F. POI“! C...Fi-'.E£.’T \i\ kECTRC-q. CQJRE.

 

 

 

 

 

Edouabaws OF R B-ZIJ:-.;>i-IT<Y EustaErf

- - _.._ _._. ._..._.. .__.........»._

 

 

 

 

—-—_

 

 

 

L...“ .l .-_._-_.__-- fl... _ .. A .n -_ _.____.___.._..-. ._ _.___._.___.__--..__. .. a _

 

 

I
/
I / ’
I c
t +
‘/ /./
/ /
d6 ’_,/ ,v"
a 0‘s
:///"1 x.
a: 1 ——D

 

:1, .—' '. E ‘J:L1?.R:o:vx o: 'n Feb“! ELENEAT '.:~.' voazaR :mr«r-.:.‘RT

~- ~,, g,

 

 

 

 

10.

6.1 sounnm VALUES or THE STRESS FUNCTION

As already mentioned in the acknowledgment, the relation between ex-
ternal forces and the boundary values of a stress function was not dis-
covered by the writer but the idea was given to him by Professor Dr. Ing.
K. Kloeppel. Although the idea is therefore not original, a complete
derivation may be given since the relation does not seem to be generally

known.

Fguilibrium pf 55 ficundag: Element
Fig. 2 shows a boundary element of any plate under the action of ex-
ternal forces with the resultant components I and I parallel to the co-
ordinate me. I and I are forces per unit of length.
The conditions of equilibrium are:
site-o: 0-, d4 cos. (as) +- it". old to; (a!) = X-de
iY=O; “trade 665 Car) 4- 0'3 do cos CR3) "3 Yd”
£H=O: Txg =‘ 7:3»
These are essentially the boundary conditions, Eqs. (3). With,
costar): 95-”;- 1 ‘05 (e3)‘="’%:'

we may also write,
6rd! - tra.dx- == X'dd
(8)
r3! dg '— 6" d? '4': Y 'd-4 ,
Introduction 2;: _th_e_ Stress motion

The relation between stress components and stress function is,

 

B‘F 3H: . ,_ __ a“
6": 3*t ; 63: 3x2.) (“'3' 51:83 0")

By inserting Eqs. (4) in Eqs. (8) , the boundary conditions can be
written in terms of the stress function F( v.3 ) as follows:

 

 

 

an: 31;.-
_ a“: .. a“: _ .
31:63 3 ax‘ dX’ -- Y d6

The total differential of F( xxx ) is,

85'
dF=-a-;dr-+—g-E—ol3 ,and,
a: 3!:
F = = --—- --—- 10
(dF amalr'+ra‘3 d3 ()

Eq. (10) may be partially differentiated with respect to x and y:

 

 

3: __ a‘F dx- 9‘:

3r - a“ g 61:03

2: 31F d + I B‘F d (11)
33 2 I 6363 k 03‘ 3

A comparison of the Eqs. (9) and (ll) leads to the following relation:

%E=[Xdo; %=-(Yd4

If the integration of these line integrals is performed along the

boundary from a certain starting point 0 to a point of reference ‘k , we

obtain: aF "
as .. “ .. _ (12)
bit-k := [e Yd”) -' Q3

where RI and By are the components, parallel to the x- and y- axis, re-
spectively, of the resultant R of all external forces acting on the
boundary between the points 0 and 1< (Fig. A).

@9351 [alue 93 the Stress Motion ELI: Point 1: :
If Eq. (10) is integrated byparts, we obtain,

 

'u h:
F“: “-gS-dx-4r L.%§-d3
if. "‘ a‘F 8F “' an:
. a,“ (13:43” L. 3:: rd» + ashfllk-xo) -- Lo 3;; 3 d3

Considering Eqs. (12) and (4) we can write,

”a

he
FR -.-_. - Q3 (wk—Xa- Localtdr *’ 2r (Hk'xo) - (I. a?! d!

The two integrals represent moments of the resultth I?y and Ex
about Point 0. If the point on the boundary where By and Ex act is des-

ignated by r, with coordinates Ir and yr, we may write:

F‘k == - R3 (rk'dro) * Q3(1¢",‘¥’,)+ Qy- (ER-Ado) " p'r (Xv—Xe)

Considering that point r lies between 0 and 1c on the boundary there

results:
:‘k = Rx- (31"!v)- Q3(rk-)e1,.)
F‘k = R’P'ynv.‘ Qh‘x'wv- = RN" (‘3)

where rx is the moment arm of the resultant R, acting at point r, with
respect to 1:. .Fig. 4' shows that the moment is positive if taken in

counterclockwise direction.

ficmetric Interpretation 9_f_ __t_h__e_ Stress Motion
Geometrically Ford ) is the equation of a surface. If we erect

ordinates at every point of the plate, normal to its plans, the length of
which designates the value of the stress function at this very point, then
the endpoints of all these ordinates will form an imaginary A_i_11 M
surface. Hence, we may say that the boundary moment as defined by Eq.(13)
gives the boundary ordinate of the stress surface at point ‘k.

In order to determine the Fourier coefficients of a trigonometric
stress function, a second relation between boundary conditions and stress

function is required. We use the first partial derivative of the stress

aFOny)

function with respect to the normal to the boundary, e.g. 3"“

.In

terms of the geometric interpretation, this means the slope of the stress
surface normal to the boundary.

13..

Bounngz Derivative 3f _t_h_e_ §_t_r_e_s_s_ Function 22?. m )5:

Figure 5 shows the introduction of a new, movable coordinate system
n ,t whose origin moves along the boundary such that the n-axis is always
normal to the boundary. The partial differentiation of F(x,y), with respect

to the normal, gives:

Qf_ a: d.- as 1!.
a“ :3 T; d“ + a! d“ (14)
-d» .43

From Fig. 5: j; = cos (34) J; :1: = C0501)

where (ya) and (xs) are, respectively, the angle of the tangent at the
boundary pointkwith they-andx-axis.

Inserted in Eq. (14), we obtain,

as __ 35_ as
a“ - 8r ocean-+93 cos (*4)

and with Eqs. (12):

OF

3:" _ ‘- 2: C05 (3¢)+ Q» COS(*4)

It is seen from Figure 6 that this equation represents the difference
between those components of Rx and P? which are parallel to the tangent
at the boundary point k. But this is the negative component B, of the
resultant R parallel to the tangent at point k. It can therefore be

written:

3F R

on = o... m) - Q: “s (w = - Q4 (15)
Arbim Selection 9}; _t_l_19_ Starting Point 9_

The starting point 0 on the boundary can be chosen arbitrarily since
its position has no influence on the magnitude of the stress components
although we obtain boundary values of the stress function differing by a

 

 

 

(is. 11" |[\‘

 

 

C
K.

'e F‘h‘J

 

{t‘rf

 

(“At
r.-

 

)1 Z

 

 

 

t
b

_.-___. .._.

 

 

 

!: .illi-- .4

 

lb.

linear function in x and y for different starting points.

To prove this, we may consider the case when the starting point in
Figure 7 is moved along the'boundary to the left for the curve element ds.
Then a new additional resultant of the boundary forces acting along ds
originates whose components are:

out, = dx-do -_- cough ‘- dax a aw.“ s can“.
The boundary value of the stress function at any point k is increased by
the statical moment of these constant forces about point ‘k. The incre-
ment is linear and of the form
chV -' Rut + 3-3
It is seen immediately that these terms will vaniSh if F(x, y) is differ-

entiated according to Eqs. (4) to obtain the stress components.

MEI
The boundary value of the stress function at any boundary point k

is given by the statical moment of the resultant R with respect to that
point (Eq. (13)). R is defined as the resultant of all external loads
on the boundary between a starting point 0 and the point of reference, k,
and acts at the boundary point r.

The starting point 0 may be chosen arbitrarily since a variation in
its position leaves the magnitude of the stress components unaltered.

If the resultant B, acting at point r, is divided into two compon-
ents so that the one is parallel to the boundary tangent at point 1: (R4),
while the other is parallel to the boundary normal at the same point k,
then the negative former component, R4 , is equal to the first partial
derivative of F(x, y) with respect to the normal to the boundary at point

1‘: 9°80 '= ‘ £24 - ' See Figure '7.

 

“w

'6.

 

31L

 

\1 RJ'W‘AHJ '0
é.

 

 

 

 

 

 

3H}

 

 

 

‘II- t; ILIllllui

 

 

 

 

 

 

 

lll’lllwll~v I‘v‘lll

 

r"
Q

‘.\‘\|

'.;\‘V'f"i i. lit...

a... I r‘ 'I‘ .

' Q

I.) \‘

r1.

' ..
K‘.

uh--.

u '-

 

_._'_ _.

 

 

L.___-.____.. _

17.

W) MOONOMETRICW STRESS FUNCTION WITH FOURIER
COEFFICIENTS OBTAINED BY INTEGRATION

For the two—dimensional problem of a circular plate, we assume a

 

nest general stress function in the form of a trigonometric series,

For-‘93 = 9i- i- 2.0. v" cosn9+ 2b”? Sinn94'
as. as! (16)
+—-v'~+£ cr"”’co; “9+2d v“ sinn9
“B. «I;

This function is known to be biharmnic and hence satisfies the compati-
bility equation (7 ) . By differentiating equation (16) partially with

respect to r and multiplying this emression by r, we obtain:

an
Gfr'9)=+§§-= in nv" cosn9 +2 bhnvr"sin n9+
“3' In: (17)
ea
+6.? tH‘Ec (n+L)v-” .20an +£d“ (laid—)V‘” S'm n9
“3! “It

To obtain the stress function for a certain case of loading, the unknown
coefficients a0, an, hm co, on, and tin must be determined; e. g., the
assumed expression for the stress function must be brought in agreement
with the boundary conditions. This shall be done by means of the boun-
dary values of the stress function. The application of the results of

section 6 to this problem may be studied in two examples.

sauna 2.-
Given a solid circular plate with two single concentrated loads P
acting on the vertical diameter as shown in Figure 8. The radius of the
Plate is ’o

.30QO Values 2; __t_l_1§_ Stress Function

We recall from section 6 that the boundary value of the stress

18.

function at any point k is obtained by taking the statical moment of the
resultant R = 3(9 ) with respect to that point 1:.

The point ( ,o ' — {- ~) may be chosen as starting point 0 (Fig. 8).
The load P applied at that point is then split into two forces, each of
magnitude ; , which are considered to act in an infinitely small dis-
tance to the left and right, respectively, of point ( P' ~f- ). This is
done in order to obtain boundary values which are mtric to both, the
r - and y - axis.

The resultant a is then equal to '5' and constant at every boundary
point. The moment of R about any boundary point k is clockwise and there-.-
fore negative according to our sign convention. Noting that the moment
arm is F Ices 9| , the boundary values of the stress function are repre-
sented by the following equation:

P
“P.“ a - {— teos 9| == No-(lCO) , (18)

P
where l = - --E- = coast. and {(9) a: Icos 9l

gounggg ngrivativeg _o_f_ _t_h_e_ Stress motion

From section 6, it is known that the slope of the stress surface,

95‘3- , is given by the tangential component a, of the resultant a. In

our problem:

as P
”9.6:: SV‘ :3"- ‘_ lees-9|

as seen from Figure 8. The expression G( no ), corresponding to Eq. (17),

is then:

Gt(f>|9) 2 Pi:- =“:£— (cas0lz FfP'O) (19)

l9.

§erie§ M g Boundag Values:
The expression for the boundary values, Eq. (17), may be expanded

in a trigonometric series. Figure 9 shows the function

{(9) -' \cos9\

which is an even function with the period 1' . Hence there will be only

even cosine terms in the Fourier expansion which is expressed by intro-

ducing Zn instead of n,

a. so
{(9) = T + 5‘ g; -cos an (20)

M

The coefficients 1‘; and a“. are determined as usual:

q” ‘1. £17 2. 7h.

"i‘ = 1W L .(.(O)d0 == fitL c059 d9
“9 2.

‘Z a “i"

' ll?
“2. = :Lfm cosine d0

l'l‘ '
:— L c059 cos Zh9 do

_ _'-_ QUI‘hs-UO * flu name "a.
It 1K~| an“ 9

R =: 3; '16-th
3... ‘1' kn‘-!

Inserting the evaluated coefficients in Eq. (20), we have,

605 Zing]

 

Q I _ 00 ("In
With Eq. (18) , the trigonometric expansion for the boundary values of the

stress function can be written:

2.9 °° I‘LL"
scoot = N--GC°) = " 1fiti’é.wt—t c“ “9] (21)
From Eq. (19) there follows:
9 °° c-I)“
69:9) ._._. F(r|°) c “ if“? g nut-i (as 24.9] (22)

 

an... ‘fs- f

 

'r.

AU

9‘:

m
r.
.._.
.tM
2
-..
..
_
\m
L
C»
5.
.~
_
. .
. _
_

'.‘i\..'.

 

 

 

+(9)“\L03 6|

”T

.L .
i
9

_
“a :1

 

 

e
D

.unisf

no...) '- .

 

 

21.

The unlmown ,ooefficients of Eq. (16) can now be determined from the
condition that for all boundary points, Eqs. (16) and (17) must be identi-
cal with Eqs. (21) and (22), respectively. For the boundary, r =f> ,
Eqs. (16) and (17) read as follows,

W so
F(f.9) a 25- + f. a“ f" Cosn9 *‘ E. be. e“ 5;“ “9 " (23)

It:

Co I. we MN. 2 K“ .
4.1-? 4.2a“? cognG-i- cl“? smn9

“at “8!

00

N
G(?\&) = E Gan?“ co; n9» 4. E. bun?“ sin '59 8-

“Cl

90
+c.f’- + Z Cu (nu-z.) faces u9 *- §.dh(“‘L) F": sin :49- (24)

Hi.

If we compare the coefficients of Eq. (21) with those of Eq. (23)
and the coefficients of Eq. (22) with those of Eq. (24) , there results:

:2 P? a. pl. {-L)"
— -—--— . 1.: __ . .__...._———_
qo n" I I.“ q‘ P2.“ (ZM‘I)

P P 0‘

I‘ " _-——- . Ail——

co - W I c '3 "' J— ' a“ l

f L“ tr P "‘ (ZWH)
are = CM '5‘ 0 (or an odd n
b“ a: d“ --"= 0 {mr- all M

With these values of the Fourier coefficients, Eq. (16) can be written,

PP as '9 {—l)'" he»
Fr9=*J-*‘+Z-‘MV’C& —
< \ ‘ *- fi‘ ee. ‘T (“tan-u °‘ "‘9

| P L__ es l,____£fl:___ W‘L‘L
-3... fur-c- Z w figurafl') COSZnB

Hun

22.

 

 

from which,
P
F(Vx9) 1: - 'F' «L (+.(£ l +
1 {it 3)] W)
+ :éf-U“ ‘- ‘JH (if)..- zu'—\](%)Mws 2‘9 )
or:
1 v.
F6138) = "‘ 3;{ LE!“ ‘
" (25b)

 

so 2. ,_ p. _
.. iC-I)“ P (can) V (2.. n) (fru‘casanO)

as: uh‘—l

The stress components, in polar coordinates, are then obtained by

differentiation according to Eqs. (6) .

23.

Example _2_.
Next, the case shown in Figure 10 may be considered. Two distributed

loads of intensity q and radial direction are applied on the boundary over
an arc length Zpfii where P is taken to the right and left of the vertical

diameter.

_S________eries meiong. fauna dag!” Values:
The series expansion for this case can be obtained directly from the

expansion in Example 1. . We consider the single concentrated load P of
Example 1 as a load element of the distributed load q. If this single
load element is moved along the boundary of the circular plate for an

. angle wt (in a counterclockwise direction), the boundary values of the
stress function are altered by this angle due to the new position of q,
(Fig. 11). Eq. (21) must then be written in the form,

2 0° (. V‘
dF(P'9) '= '- "¥"["§'_ "' EH Tfii‘r ‘05 Zn (9+oLJ] (26)

where dF( [9.9 ) is the element of the stress function on the boundary due
to the load element q. It is seen that Eq. (26) becomes identical with
Eq. (21) if d. approaches zero.

The total stress function F( “G ) for the boundary values is then
obtained by integrating Eq. (26) over the angle on which q is applied:

 

 

4»
mph) a [ an”) e - “9 [kt—2"” W. €052u(94-4)] om
-F

- .- iii: {P _ 2 {-U" F[siuzu(&u)]‘P}

Ii keg “at“ 3M -P

u '- '. °° r-u“ ___2__e
F(?\0) = _. Jat'l-‘E— [ 2 -— é. Kl COS 1H9 in? ] (27)

q

r".'" '7

, (Winn;

---?_.

 

 

/
/
."

§ ’ 7'“ um‘f' "
' LLLL'J U

j ‘1

ti . . . ' .’ _ _ :. - ~ _ , ,g . a. . , -a . I - . ,- -.
I ‘<‘ “ "’“"'L):- -'-~‘- 3 "W" '.,‘..‘r:..-.t'.t.( Fu.\.kiu‘ Tic Lufio C

 

.. gm...

 

 

 

 

I . . 0-. - ‘ * ~00 . r '
F ' '- ' n . t. ‘1 ‘v‘ ;‘ ' ‘ ‘ - .«v‘ .' .' ' .’
‘ I A | . ..5o(.[. -5. _( ‘ ‘Qq u.‘ |‘-|.“.( so‘ b t-HVh‘1‘\

 

Vila}; i7.

 

 

 

zq .

25'.

If we denote the total force exerted by the distributed load by P, then,

P3 Zq?%
Substituting this expression in Eq. (27) , we obtain,

 

Pp ' 0° (4)" ‘ 53V. 39.2 ]
F(f‘9)=—-{:’[:~él MAL-l C—Os ZMQ a"?

which is the same as Eq. (21) with the exception of a new constant factor,

s‘m 3“ B
M
Z»?

If P approaches zero, then,

I‘m _____Esiu3u z: I
P90 - up

and the expression becomes identical with Eq. (21).
It is seen that in this case again, G( “8' ) is identical with F( PI 9 ),

 

so a .
8 “32$ _ (.4) sun 2:43
6%,” = 7:" = " w [15 E. nut-4 “5 “9' up 1 ‘ (28)

Fourier Coefficients 21; the §tress mnetioggwfi l
The Fourier coefficients of F( T. 9 ) are determined in the same way
as those of Example 1. The general expressions for F( «39 ) and G( r, 9 )

on the boundary are , respectively,

on 60
FfP‘O) =.- Ef‘iv Z. qM (“can v19" 2 b“ ‘5“ Sin «9 *-

uwu KB]

oo (23)

4: Ff P" + g 6“ fun“; :49 +— EOd“ raw—Si“ M9

GCP‘9\ = i'qnfl?“ COS I19 ‘5 é.b“n F“ 5;“),‘9 (-
' (24)

BO . a d ‘
+ co f‘t g c“ (us-2.) ru+Lcei u9 + éfldu (net) f“ rm m9

26.

The comparison of the coefficients of Eqs. (27) and (23) as well as

of Eqs. (28) and (21.) leads to the following result:

 

949%

a, i. '- a.

a“ e C“ a: 0 If- VI )3 Odd. (29)
L z - H)“ -s' 2.

at“: rcpt“ n(zu-—\) W‘ “P

bu=d“=0 {cumin

3 _. 3-11
c, 1‘.
q (‘0m .
= _ .-—-—--' . ———-—-'-' o s
‘2... Writ. n(Zvu-\) M 2'“);

If these evaluated coefficients are inserted in Eq. (16), we obtain the

following expression for the stress function F( V'.9 ):

 

 

 

He‘s) = .. 3.155{ (A [H- gr] ~
‘ £533- [ .2.-. - 1;” <f1‘](-}f)‘f‘esess.eu up) 0°“)
ov:
RV.” =' ‘ 3: i F ”1”“ ' (30b)
_. 1| _—[—§‘:.— - 2:1“. 1 (we: as an}

It is seen that Eqs. (30a) and (30b) can be also obtained directly after
the relation between example 1 and 2 has been established with Eq. (27).

Then in Eqs. (25a) and (25b) there is substituted:

“air?

sin in

and the summation term is multiplied by 1‘9 .

27.

i___trees 9%
If Eq. (13) is differentiated according to Eq. ( .6), the following

expressions for the stress components are obtained:

06 u-L ”
at, = C. - { é‘ugnmq)? cos n9+ u£ bkn (u-t)v-""iiun9- *-
N a N M
t- f. “(m-L) (usuwr cosug .- “i dn(M-rz,) (”4) v- 53“ "9}
ea a-L '° sq. (31)
0'9 =- Cet inunfie-Uv cosu9 +£ bunk-0v- sin n9+~
“3| Isa!
‘O “ W
V Z c“(u+l) (wt-t) v- cos “9 + i d“ (hi-L)(u+1)v~"siu m9-
n- N 0° '-
%a = é.O“W(M-\) v““_§m e9 - if bgn (ml) wr'" casn9 *-
we " M»
+ 5. CM is (nu-l) 4-“ suns -- 2 d“ n (eel) v- c-os m9
“3' “B:
For our particular example where only even cosine terms appear in the
trigonometric stress function, these equations become:
4- -- Ce “‘th VI (Zuni) v- 60: 2a 4-
“ a
+ f Ghana-2.) (zen) v~ “€652M9}
(32)

a “-8.
0‘s = Co + i “all“ (zu-mr Co: as i-

“I!

a
t 2. a,M c zeta) chins.“ cos ass

“II

so _ on
' “C33: 6. Q2“ZM(3M-\)V'zu Leia ““9 *' 5— Ca in as“) warm 1:49

“8: In.

Numerical values of the stress components at any point (v.9 ) for the case
of loading shown in Fig. 10 are then obtained if the Fourier coefficients
are evaluated according to Eqs. (29) and the coordinates of (wt-.9 ) are

inserted in Eqs. (32).

8o) TRIGONOMETRIC STRESS FUNCTION WITH FOURIER
COEFFICIENTS OBTAINED BY FINITE SUMMATIOH

Under general loading conditions, it is often not possible to find
a function 119.9 ) (expressing the boundary values of the stress function)
whose derivative with respect to 9 is continuous from zero to 1"" . The
integration for obtaining the Fourier coefficients must then be split into
several intervals, the endpoints of which are determined by the points of
discontinuity of 99%- . In such a case, it may be useful to euploy
the following approximate method which replaces the integration by a fin-
ite summation around the boundary.

wression £91; £13 Boungpgz 33.1% 9.; £13 £29.29; Function

As seen in section 6, the numerical value of the stress function at
arm boundary point can be found from the external loads. In the following
derivation, we shall consider those numerical values at selected points k
which are spaced at equal (intervals 49 around the boundary, 1r being an
integral multiple of; 49 . The total number of points 1: taken on a semi-
circle is denoted by m . The polar coordinates of such a point 1: are
than (9.33" . Anemple withm=6, 159:? is showninFig. 12.

We may further define F]: as the mmerical value of F( r.3’ ) at k.
Correspondingly, Gk is the numerical value of G( n9 ) = V’ '95? at
point 1:.

Knowing the values Fk and Gk for all points 1:, k = l, 2, 3 "0-2m-l,
2m, we may express those values in the form of two finite trigonometric ‘

series,

-I
F(p.3;- _)=:*2nu‘° s“:JtE:COS1flT-t~

.-
=0

+ Z. Ba 53M 13": (33)

2.9.

 

 

 

/
v /
x , i'fi
‘ _
Q. m
1 I. A
//
I;
I. . .
set
i
\ p
\x
\
\
/ \.\ x
.4 . x s
I
\
/
’l/
v/
. P
s ,
=
v

 

Mad:

 

 

6 :o

s a
n

———..’__-

(49

.f"'.¢“i|

Ll T‘I‘§ ‘I

 

\‘n/
k=lo

iii-J

\
'a
‘-

Fu {2.

I

4‘

"\
.r a...

f'r

n
=8
2: .< a

 

 

 

 

 

 

JU-

‘H? d- “H “1“? d“- - u . u‘k"
Gwen? = T‘étcuwsvr‘Twflw *‘éPes'n—5 (31.)

Since we break the series up after a finite number of terms, we must

take the last terms, An and Ca, only half. The reason for this is the
same as for taking the half of the first term, A0. If we take Am and 0].,-
instead of E“: and sf- , the point ( p.16? ) would have a double weight.

The proof may be found by writing the series

-i
R- ” n'ku‘r a...
Ti-é‘flhcos‘a- r-z'cos’ku

-ind.

interns of cm“L and e

Since we do not know the funtions F( 9‘9 ) and 6( Pp ) but only the numeri-
cal values Fk and Gk at a finite number of points, we cannot use the
usual method of integrating around the boundary in order to determine the
Fourier coefficients. As an approximation, we take a finite summation of
the values F]: and 6r The formulas for the determination of the Fourier

coefficients are then as follows,

R. ‘tgfu ,

“a g 5;:ng “51a: '(35)
B“ = 3': E‘Fk-sinl-EF

C = ‘L :5-

c w. 1:!“ 'k
C a = 5:. :5: Gy‘ Cos 3-3: (362
D“ =tgG‘k-SME'E‘E

31.

Fourier Coefficients of the General Stress Motion F159;)

 

We assume again a most general stress function F(V'\9 ), this time
in the form of a finite trigonometric series,

\M-I

a.
Farm =- 25- + 2 qu'“ ca; “9 4.. 9? v“coxm9 + Z bnv-"siu n9 4-
In; MCI
‘*~‘ a“ ‘9‘ , (37)
+ 1:- v‘wé cult-“Loos uS v 1:?” (as “9 *' gdnfu.‘siun9
ue. . H=|
0F .
The corresponding expression for G( V} ) = V’ 37,: is ,
“A n a“! “A M
6:038) =-" 2 <3an cos m9 *- TWW" “5W9 " "é b“mr“su'u «9' :-
ue. CI
“d ea. Cu Inn-L (38)
+co'r‘t 2 cm (ueL) V“ “sag:- -: (“4152.) V’ C05 m9 +-
\u m-L
+ Z d“ (u+?.)"" sau v.9
“BI
For the boundary points 1:, these equations read as follows,
Fcl‘i=°-L.'§ n «1::- at» _% s.uu-
r. .. a as? as t:- +7? «m + "M T *
(39)
c, L w". My; Hkfi' c... at“. M an. “1";-
+-;_-pt- _Cuf aim—*7? cas'krré‘dhr sin-a:-
th E)‘2‘ "‘ Elsi-3L“ “ca 16" :4: "-3151"
f... “‘ovmpcos w. tun? s “+ue.b“"P g...“ +-
Hp| L
*‘efz'i- 1 Cu. Cut-Z.) mice: 135 ‘- 41! (WWI) fa“ CORR? 4- (40)
"3.

m —
+ g (1“ (Ma) 9“”- shay

KC!

Eqs. (33) and (31.) must be identical with Eqs. (39) and (40), respectively.

By comparing the coefficients of the individual terms, the following

relations between the Fourier

g, =

 

O a
cadet) a“-ch
0-“ = ——_;—_
if
b“ g (“*t) But-Du
if“

32.

coefficients are obtained,

 

c = ¢° (41)

O aft

—n (1.4- c‘“
zfn‘f’,

 

_n 3“+D“
zfuva.

I)

 

du

The coefficients of the general stress function are here expressed in

terms of the coefficients of the special stress function for the boundary

values and its first derivative, Eqs. (33) and (34). But the latter co-

efficients can be evaluated by using the known boundary values F1‘ and Gk

as seen from.Eqs. (35) and (36). Hence, the Fourier coefficients of the

general stress function, Eq. (37) are also/known. If we substitute Eqs.

‘

(35) and (36) in Eqs. (4;), we obtain,

 

 

 

 

 

ha
|
e,-_- 3;- {-‘EZF-‘r-Gk]
. A“ ..
a“: 1;?“ I‘m: [tutqu-G‘t] cos 15-1
I a... . uku-
bh = 1“?“ £.[(VI+L) Fk’Gk] 3m 7-: (42)
I 2 G
co "' a“ r‘ “' k
I a“ In}?
c“ = inf“; é,[G“-“ Fklt'os T
a... ..
du‘ ‘ £[Gk'an]$)“r—EL

zu fK~L y‘.

It is seen that the Fourier coefficients an and cIn in Eqs. (37) and (39)

are merely a special case of

the coefficients an and cm“ They follow

immediately from the expressions for an and on, Eqs. (42) by setting n=m.

As a result, we have obtained a trigonometric stress function, the

33.

Fourier coefficients of which are approximately known. The determina-
tion of the stress components follows the same procedures as previous-

ly described in section 7 (see Eqs. (31).

34-

2.) NUMERICAL METHOD FOR AN APPROXIMATE DETERMINATION
OF STRESS COMPONENTS

As mentioned already in the preliminary description, Section 1,,
this third modification of the boundary method is an attempt to establish
a purely numerical method in_which neither stress functions nor differ-
entiations nor integration appear in the practical application. Approxi-

mate stress components are to be obtained by die algebraic means.

Expression for the General Stress Function F(139 ):

The derivation of this method is based on the same assumptions as

 

made in Section 8. The general stress function is again assumed to be,

as “4 QM m .
F0139) =3 "i‘* éogv“cagu9+-:v-“cos “.9*- é‘b“\r“smh9+
|

 

 

 

(3'7)
fi-l. In
* ff v'L *- 2 c“v-“H‘cosu9 *- 5:- V'hucsst I- is ”(I-”‘33“ “9‘
In. H3. K
The Fourier coefficients of Eq. (37) are determined by,
( M
co = :3: é‘CzF‘t'G’k]
‘ 3“ II'k‘
a“ 3 1w- ?“ é [(MtL)F\(-Gk] Co; fi
‘ uh - u'kIT (42)
by = W fi‘tfiuh) Fw’éu] Sm 7
‘ '2
Co g am 9‘ u‘. GR
c. ‘ a '-
V‘ _ We §.LGu‘“Fu] cos :3;
l h- . \K"
d.“ = Lu rm“ 5.[G'k’“F‘t] SIM FT“:

35.

in which F]: and Gk are, respectively, the known numerical values of the
stress function and the product of r times the first partial derivative
of the stress function in a direction normal to the boundary at any

boundary point k. In contrast to the development of section 8, however,

Eqs. (42) may not be evaluated but inserted in Eq. (37):

.. h I.-
Ftv93= 3‘;{£_F + ide- $0»; (I-1)]«39m% +kil—Fr(‘}:J L‘*§(“%:)]'

Hun ft

.(0519 cc; .E3,+ . . , + Fr(.r:)“‘[|+ %(I-%:)]cosm9€oskr +-

+ ZLFk(!'-)C‘+" 0" (1’)] SI'H9 SEMI; I £16104 [l+% (‘-;:}]

RC:

' SI“ 28 SIMEEI“ 'L ‘ ' "' ‘ -.+sz(F-Ju.['+.=:1(‘-F:)15;“(w'|)gsih£t:uL-)h.}l .-

.. h- I.
- :{:—[I--1+£a~$-'—1<I[Iunseagat-‘grfi—hl-

G a.
-¢osZGoog——§ t------‘-+ —£*(: M[l-‘i-hlflosm9 Cash's- v-

+Z‘RG ( :WD s‘“9‘;n‘£;+£_Gk(‘ ) 0-?! :lsiul853Mz-EL‘I-
* ..-........ {-GR (:5: ) Wr‘[|-% :]$Ih(m~ ”9 SI“ CW-xkr}

 

If we note that,

"' - ' -E
C05n9w55£1+53un9 mar-33- = case“ a.)

and if we simplify the resulting expression, the equation for F( v.9 )

becomes,

=~.m= zt-{Eatua'u rrIu-i 0-; ERcosn("§)‘({)"f”50":‘)i

NC!

~<osvu(9- -—)J - (43)

~£Gu‘(t--)[‘+£Z(r coin(9"")“(!) ‘05“‘(9 353])

36.

Reduction _o_f_ E: (52) to a Single Summation:
The summation over n in Eq. (1.3) shall now be replaced by a simpler
expression. For this purpose, we introduce two new functions 9‘ and Ya .

Eq. (43) can then be written in the form,

‘ he In; L
a { 5-. “actual - 0—5:.) 4 w.)
in which,
<1)“ (v1 9-1‘é) = | + '2 2_ (3%)“:9; "(Q-33‘) +- (f-‘f‘m w. (9-3?) (45)

and

I!"

‘Pk‘fl 9-1‘5) - g. 7- (ff 1:;- (!- fives n (ea—133+ (ff-:5- (t-fij) Cos M9355) (46)

If we differentiate Eq. (45) with respect to r, and multiply it by r, we

obtain,
8:? “M ‘A ..
v. .333 = “i2.“ (f) Costa (93%] NM (7‘)“‘ws w (94.5)

Comparing this result with Eq. (46), we may write,

9::
v‘ _‘t
J2:- ('- Fl) 9' 3V“

-6
,3?
?
i”?
u

Hence we can express Eq. (1.4) as a function of' q)k only,

' In. . a l“ I V“
Form = if ‘— pk [?k*i'(“%)*'5?]-E.Grz(h7‘) 91¢;

u.‘

or in the form,

._ _'. l“ - I v" a a.“ ' ’L
FN‘O) ‘ 2M { £‘*u[‘*'£(“ft)*37 ?k’k£ Gk: (ll-F") ?"'}

Consider now Eq. (45). In order to simplify writing, we may introduce

the function,
H-!
Nka-M) = H- Z ant-“cos HoL +v~"“ co; Met
Kat

which is of the same type as Eq. (45).
By using the familiar relations,

'5 ind

1' e = v-“(cosnat +£s‘m not)

I ';“‘ ‘ e
V’ c =-. v-“(Gosnd—cs'uud.)

we find that,

n ‘ ..'u
2.v- Cosud .—_- vs“ (e¢M&~_e_‘ “J

and hence ,

*4 :4 {Mat —'m¢ V'u dud -€wu
Nk(v-.cl\=l+£‘r(e +9. )+--£-[e +c )

h...

Consider now the series,

\i-V‘
Z.

 

['+"*""*"g+..........+ I‘M-q]

3'7.

(4'7)

(48)

(49)

(50)

(51)

38.

The sum of the series on the right side is known to be,

S 3 2.. ‘ h.» (52)

If we substitute in Eq. (51) first x = re“ and then secondly

a

X’ = 'H- ‘ and add both series, we obtain,

 

 

 

 

I . ' -‘ w"‘ ' - -' .
I+ v-(c‘d+¢“‘)+v-'- (e‘u+c‘u)r """ ‘ *V' (e‘('“')‘+c‘(”')“) «-
+ :2 incl. e-i'MX)
z. e 1-
WM ‘ L‘ V" Jud "'“M
d. .—
= |+ Z¢“(em‘+em ”'1'“ ‘0‘ )" Nkw'fl‘)
“3|
Using Eq. (52), the sum of these two series is,
.d. . _\ -0
S 4- S' = “‘""“ . l-v-W‘e‘wd H-Wc ”t n-v‘“e“""
'- l-V‘e
= NR ('14)
This can also be written in the form,
N ( g [ \*V'£ “ |§+e‘£‘ _’ I: 4.H‘ 3+V‘C “ - e-‘.w‘ ‘*V’¢.“
R *3d)’ 2' . we‘d + —I'.¢ L - *id -¢:K
" l-v-c "’"‘¢- """9

Substituting Eq. (49) and simplifying the resulting expression, we
‘amm,

(can. (FY-“cos lad ““4 Zsluat s'u
Nk'("u°‘) = J )+ 'r 'Wd
l-Zv-Cosqt+v~|- .. lrcosotq-v-L

n-v-l) 0-?“ cos mat) +v~mH chu'd aim and
Nk (V14) = <53)
t~ 1v- ¢b$ol + we.

39.

Recalling that Nk(r,u) represents essentially the function (pk , we

may write,

(I- g)[\-(}t)ucosm(9'§” " (EQMHLSM (9' g) 8"“ “A ("la")

 

(Pk (V‘O-E; (54)

I-L(%-) cos (0—?) + (3%)"

With Eqs. (1.7) and (54), we have arrived at a simpler expression for
F(r,9 ) than Eq. (43) was. The double summation over k and n has been
reduced to a single summation over 1:. However, the resulting expressions
are still too complicated for practical purposes and may therefore be
simplified further.

Bractical Aspects:

The purpose of this numerical method is to find the numerical values
of the general stress function F(r, 9 ) at certain selected points in-
side the circular plate. A grid may be laid over the plate as shown in
Figure 13. We are then interested in the numerical values at the grid

points, e. g. in the values 1" (£5. 33"] where {is}: 3%) indicates
the position of the grid points on the plate. The notations l, n, k, and
m may be understood from Figure 13. Later we shall also use the short nota-
tion 1'; to give the position of the point on the plate.

If we use a grid with constant angle A9 between the rays, the
equation (54) for (pk can be reduced to the ease in which 9 = 0. Then,

in Eq. (54) '

(05 M C -1“;" = 605 MT = (-‘)k
H .. -
(f)... 2.53“ (9--\§) sin w(9-%)= 0
*0?

cos (9-%) = CO: V

 

 

 

 

 

 

 

 

‘k=3,
1a“ V-l
X. , fi___\\ ./
.\ _
x} I .1 \--\‘ ‘ \ 1‘5
ms ‘ us:
I [x y/ \ l/ ‘
‘\ f ‘N‘ \ ,xu'i‘“
\ /"q1‘3
.\ ‘ \‘ “
the - ' " l X . . “‘0‘”.
t s h; A a 3
i ‘\ u f ;
. ' /
\\ \ / ’
0 I, \ 'l V"‘
Vq ‘ ‘
. c ‘ 0 A9 = i
u“ ‘k I 6
. 0.035; n = 5
K13 . .
(wuss: m = 6
mo. .3: exec/mu. Fca R out!) at: R caRCULRR PuRTE
X A 3.4. F ti
tan d t — ltd»--
3}: 3‘+2.Awtund F:=F'-'+2AV‘
\
a.) ; b.)
H
a j 3
. l/
3‘ 2 / .LFI“
I It
) l _L

 

 

 

 

Fm. us: "JRLUES as THE

45V

 

 

 

lw—Ar —--’i<~- Av- —

STRESS. Fun/Jaw (Latinas THE PuRTE

 

 

 

4-1 .

Eq. (54) can then be written in the form,

(‘-—:k)[l-(~u (fl r]
"‘_(+)L (55)

 

E‘s,

m-‘E-

|-L(pJ¢OS-E:7

This is strictly valid only for the values F (g .0) ‘, e.g., for the
values at the points lying on the ray 9= 0. However, since we have a
summation around the boundary, according to Eq. (1.7), we can use the same
set of values 4’1: , Eq. (55), for all F (3- L“ 7:) if we "rotate" the
values q‘‘ in clockwise direction for the same number of angles 49 as
k in F (5": ,‘f—é) amounts to. An example, using Eq. (1.7) may illustrate

the procedure :

 

'k-o; F(li".°) = t{:2Ft['*i(l‘%:)*g'J-] (Pk-
.. ...
—getz(a-; fink}
if. ”f | 2m
1.... :(“.-;-)=;{é‘u[ ------ 1nk_-:é:e~w<r.,-.}

 

R's-S: F(t g?)=‘L{ZFk[- """" Jchhs—za“ ---- (Pk-5'}

9““ 1w» k“

 

and so forth. By observing this rule, the simple expression for Qt:
Eq. (55), can be used for determination of all 1:- ( Elf-‘13)

Further gigglificatign 2;: Es. (57) and (55):
Eq. (55) can also be written in the form,
tot-~51)
{hf-+9 E5 (56)

= Cl-f-kaflq 'Ut

 

(pk = Ct-(-l)‘({')‘“]'

42.

where ,

(57)

Considering now the fits'l: partial derivative of 9’1: with respect to r

which appears in Eq. (47), we obtain from Eq. (55),

9%:
v- .__.

8v- i—Z(%)‘°$1g-+(-;—')‘ [l-l(§)cosk7:: ? (if-VJ"

 

: {eewpflae‘ + [wt-0‘13“](t—fl')°?-(?)fwsi:-— fl] __

M {-l)k (flufl-S)
l- lat)“: lg +(;)'-

This equation can be brought by algebraic means to the following form,

(58)

 

39.. ‘k V'M 2. i(%*{) “‘(“)x(_i’t)~
*T=[""“ (1‘)] Uk- . p - v: r... ' k
:(r; "“0 (7)

where Uk is the same as defined by Eq. (57). In Eq. (47), ~r 3-5 is

multiplied by in - {1). Using Eq. (58), this product gives,

. 3 .
207'? ”'3? = (I-c—n“(f-]”‘]l 37(“5’”: ‘ . P
*+

 

-.L(t *" “('nk (HM U
1 ft ‘_( K V’ M k
'4) (T)

and by contracting and rearranging,
\ ,n. 9
-("§3)v~—1‘- [I-(~I)k(§)M]'Uk.

1‘ 3v- =
[aha {Ut- me» (E) }_%(«+%:)] (59)

 

Considering now the term,

\ '3. .
E\+i(l-%:)f‘5‘;] (PR ‘“ “far”

we can write,

1

" . rm I ,Q 9
[nus-swim = [new-,3 1am (“Fangs

If we substitute Eq. (59) for the last term on the right—hand side of
this equation, and contract the resulting expression, we obtain,

 

L k w; ( vL MC“! V’ m
(”Hui-J'wathl’kr'D-C-U (a 1.207..) MUN — {ff} ]
‘-(-|) (7)“ (60)

‘Using Eqs. (60) and (56), Eq. (47) for the stress function at the grid

points becomes,

 

\ | inf-U" I“
H‘s'nfi -- E E‘iiFuU “'3“ {19“} ‘0'?“ U“ (”*H’ {9} ]‘

poi)“ (%)‘*
(61)
\ *l w.
_. G.k 1(l—F1)[l-(—|)“(_"CJ ]U‘}
We now introfluce a new notation and can write Eq. (61) in the form,
if I 2‘“ .
spa-‘0) = $1? 2: {Fst-Vvék‘vk} (62)

 

 

in which,
I g__:
UR ‘ ’ ' " -cosE
267” T) “
\Ik -..= { (:— :91) [I-c-u“ (-F-)"‘].uk (63)
sod-0‘ I a.
\A/k = Uk+l- (F) (64)
i—GI)“ (355)“
If we set,
2* = WW" (65)

we arrive at the following final expression for the stress function at

the point (3.1" so) 3

I la.
PH} .0) = 3'; 3“ Fez, five} (66)
The stress function for other points than those lying on the ray 9= O
are obtained by rotating the values of 2* and Vk in clockwise direc-

tion as mentioned above in connection with qk . For instance,

 

 

" 3‘ - 3—2 F 2 - v
1"“ F(T|:) _' 2w; knf 1‘. k—g Gk. k4}
I.»
‘0'?“ F(EE.E'~:') = 3.}: §.{Fk'Zk-s'&k'Vh-3§ (66a)

and so forth.

45.

Value 31; the sgess Function at the Center 9_f_ _t_h_g flats:
The numerical value of the stress function at the center of the plate

is given by,

k8: L (67)

This follows immediately from the Eqs. (57), (63) to (66) by setting r==0.

In this case,

ESQ-3.13% 2;, Qt; _S_‘§_r__9_8'_§ Function g‘tside the 213319:

In order to determine the stress components on the boundary by means of
finite differences, we need the numerical values of the stress function
on a circle outside the plate which is drawn at the same distance A r
from the boundary circle as the other concentric circles of the grid in-
side the plate are spaced.

If we know the value of the stress function at a grid point on the
circle next to the boundary as well as the slope of the stress surface
on the corresponding boundary point, the value of the stress function on
the circle outside the plate, which lies on the same ray, can be deter-
mined approximately by extrapolation.

From Figure 14a, it is seen that \the tangent of the angle at is given

 

by, ‘hin at = 33-3.
2.4?

It follows for the ordinate y, ,

I1

\53 y‘bzort’and

46.

The tangent 0‘ represents the slope of the function at a certain point.
For the case of a stress surface representing the stress function F019 ),
the tan at corresponds with the slope of the stress surface in a direc-
tion normal to the boundary. We have seen in section 6 that this slope
can be obtained from the external forces for all boundary points. Hence,
for points lying on the same ray of the grid, the value of the stress

function at a grid point on the circle outside the plate is given by,

3:
F“ = F“ 2A ~—
k 'k + '- ar“ (68)

in which F12" is the value of the stress function on the grid circle
inside the plate and next to its boundary and E; is the slope of the
stress surface at the corresponding boundary point 1:. Figure 14b shows

a cross section of the imaginary stress surface on the boundary.

Noting that
9.5.. _ 53:. I P
or“ P 0 AV“: 71—

where n is the number of concentric circles of a certain grid chosen,

Eq. (68) becomes,

Fo" -..= F“... 4» 3%- Gk (69)

Practical Application _9_i_‘_ the Derived Formulas:

It is seen that Uk’ Vk, Wk, and Z k are constant for a certain grid
and independent of the individual problem as far as the size of the
circular plate and the loading conditions are concerned. Using Eqs. (57) ,
(63), (61.), and (65), these constants may be evaluated for one, two, or

47.

more standard grids taking a different number of points fer each of the
standard grids. In solving an individual problem, it is advisable to
use only standard grids for which the coefficients have already been eval-
uated and tabulated. In this case, it is only necessary to determine the
boundary values of the stress function, It and Gk, from the external forces
(Section 6) for the corresponding grid points, and to perform the multi-
plication and summation according to Eq. (66).

The approximate determination of the stress components from these

numerical values of the stress function is shown in the following section.

1 . APPROXIMATE DETERMINATION 01" STRESS COMPONENTS
BY FINITE DIFFERENCES

Using the numerical method, Section 9, only numerical values of the
stress function at selected points are obtained. Hence, we cannot deter-

mine the stress components by differentiation according to the following

equatiorq
' 6F .+.L _EE:
°'v- = 33 37 ~r‘ can
at:
0’9 = eq-L (6)
. 0F \ at:

11"" = tuft—36‘- ; 9+39

The differentials in Eqs. (6) must rather be replaced by finite differen-

35;)

1?
ces. To determine the stress components at the point ( T. m , we

7 use the numerical values of the stress function at that point and in its
neighborhood. ‘

The first derivative of a function at a point ( ‘1'}. 13’) , or briefly
(1"), can be expressed approximately by the difference of the values of
that function at two neighboring points, divided by the distance between
those two points. Thus we obtain the following relations, (See Figure 15) s

 

 

 

 

tM I-t
Be a F“ - F,‘
6v- 2.4+
at: 2 FL“- 2. F; .- Pg"
av" Art
F1. __ 1 (70
E ,5 RH FR-t )
69 LAB '
‘ 1 1
at; ~ Fkfl- 2. ('1'k + [:7‘M
39‘- A9?-
!” 1" lot _ ..
6"? Ft" ’ Fkh '. Fir-t ‘P h:-“

 

ll

 

3w33 hAwAG

 

 

 

3-1:..-z

«(rat

L

r’II-l”
_//
f/
lw
FL“ L;
F1
kn 9
\‘x
Fu’i

FJu‘TJ :‘yer. ”1!:

in

At? 13E:

\

.1'

:1.

L)

c. ““

In
k-I

' .n. . ,. .‘.-"_~. ,. A .0. ‘-~
Kt'.\§“n I 0;!“ 3:" .. t n:---| ~O|'.p--N f:“\‘ ...

 

 

v‘u\|

l .
\V" ‘r .J‘o-.r\_‘ r|‘\"-E 8“:':':,I\F_\‘~F:.

 

 

 

 

 

 

SO.

51.

If we replace the differentials in Eqs. (6) by the expressions (70) , we

 

 

 

 

obtain,
-1 l-
o ‘ — Pk“. Fu + .l. FJM-IF‘: +F‘:_
V“ V" ZAV‘ T'- 493.
'- ~|
Pk“...}_ C:+ F: (71)
0’9 =--" A ‘-
*
1 to! l-| l I l-l
- 3 .1— Fk'bl FR I ‘ Fk+c-F-k|-| 'I:-I+F'|f—‘
*9 ’4“ 149 * nov- 49
Noting that,
Av="'P—" urn—Lt ' A9=i (72)
“ I h I M
Eqs. (71) become,
In" [ l“ H Zen" 1 -1 L
6* = 119‘- .“ - F‘k + -'-1 kn k+ Fk-c)]
v"- [ 1H _‘l' in
0'83? F‘k-Lk‘k+:k]
(73)

"" —- fl;[ 1' .. I -12. Ft”- 1"F1¢| I-l
cw ‘ ‘. F F ' z( 11"»; Fk:‘-F. ‘9‘.“de )]

Stress goggonents £9; a Particular Standard Grid:
, For a standard grid with m = 12, n == 10 (Figure 1'7)J Eqs. ('73) became,

2.. 1H ‘ I-i l9? 1 _ I
0'..- = «the “F. “Ff”... 1‘ “in.”

 

e, [Fri— * PIE] ‘7"

'1

l ,
TT3 = W Flu-i— Fk-\ - T ( FR“- F‘kM-F F‘k-t “)1

52.

ll. NUMERICAL EXAMPLE
The case of a circular plate, shown in Figure 10, may now be inves-

tigated. Two uniformly distributed loads of the intensity q act on an
arch length of its on the boundary. Let the angle {3. be no": 7% .
Using the boundary method, the variation of the radial and tangential
stress on the horizontal and vertical diameter, respectively, is to be
determined. Since the loading is symmetric with respect to the :9 axis

as well as with respect to the y-axis, it is sufficient to find the stress

variation for one-half of each diameter.

(a) Stress Components by Using the First Modification
ngeloped in Section 7. .

In example 2, Section 7, the radial and tangential stress components were
found to be,

'° --2.
O". = Co“ {2: a Zn (and) van cos Zn9 {-

1“: M

°° a.
+ Z ('2... (Zia-L) (2M H) V' as: 2mg} (
h‘. 32)

be “_2-
0'9 = Co + 5.. Q 2u(ZM-\)Y' cos 2n9 +-
M: a“

90

+ Z a,“ (bowl) (UH-i) 13'“ cos Zn9'
“an

 

in which,
__ ‘43
c, = w
(29)
amp
= ('4 u q a“
a2“ 3 “(la-i) rfk-L
C1“ = ‘64)“ q $3M 2mg

 

fl (lat!) ‘fi' Pa“

Inserting EqS. (29) in Eqs. (32), we obtain,

Zu-L

0'? = - 3-3- f 9.4- if—U“ Sin Zine (L) cos 1M9 "

-£ C-I)" —;‘-‘-sm2n(l(% )2. «an9?

._. (73)
(To = " '21 {‘1‘ 2 0“)“ SIM 1“? ( F)Z“’LCOS 2M9 4"
Ht . Z.“
+ €2.04)“ -—:‘- Sm in? (it?) cos 2M9}
For the right half of the horizontal diameter ( 9:9 ), this reduces to,
00
6v = " if?" { [5+ éflC—U“ Sin 2kg (%)u-L - 2 ( JO 7;" SIM Zn? 6?)?“ }
H~l

(74)

0’9 3-1;;{g‘ gC—i)“ SIM 2h? (9") new“ +§|(-|)n£§ 5;“ 2"? (Tr—)2“?

“3|

In calculating the stress variation on one-half of the vertical diameter,

we note that for 9'= ‘E‘ ,

cos ZuO = 603 MTV == (4)“

Eq. (73) then becomes,

¢v=—%{F*2$IMZMP(“)h-.L-;aisma“F(%-2M) f

he.

(75)

=I (3'

SF— 25in 2"P(T’ ")ul+ :.-“-? sin 2a? (Pt)h)

The results of the evaluation of Eqs. (74) and (75) are given in the
first columns of Tables 6 to 9 and are shown in Diagrams l and 2.
The stress components are determined for the following points on the

horizontal and vertical diameter, respectively,
T=°“f'o'1f'o'gf‘ ------ 0.9?

For this computation, only the first ten terms of the infinite series in
Eqs. (74) and (75) were used. It is seen that the convergence of these
series depends primarily on the ratios (%)“’L and ( PL) l“ . While

the convergence is very rapid for points near the origin, it becomes

slower and slower towards the boundary. On the boundary itself, the series
are divergent.

The divergence of the series on the boundary can be avoided by con-
tracting the two summation expressions in Eq. (73). Then, for each stress
component, two new series originate, one of which converges also on the
boundary while the other can be replaced by a respective formula for the
sum of'aninfinite and convergent series.

Although this is not of practical importance, the computation of the
stress components has been carried outto the fifth decimal in order to
supply a good comparison with the values obtained by using the other modi-
fications of the boundary method.

(b) Stress Components by Using the Second Modifi-
cation Developed in Section 8

Figure 16 shows the loading and the boundary points at which the
boundary values of the stress function are used for the determination of
the Fourier coefficients. Thirty-six points k are spaced at equal inter-
vals A9 = 10° around the boundary.

‘Boun [alues:
Taking the starting point at (P’°)’ the boundary values are given by

(See section 6 for method),

Fk'Gkao u- oétas
‘w-Ge=-qM‘sm5°l it use
pk "' C'u3-Lqpp‘ice39\ N- Iaékfizc
F“ 3" Gk=‘qp?‘s‘w\$' H- ‘k-L‘l
FR 3 Gkgo if Legit-:36

The evaluation leads to the following results,

 

 

 

 

1‘ Fk" Gk 1‘ F1!= Gk

o o 19 -0.98 1.81 c
l O 20 -0.93 969 C
2 o 21 -o.86 603 c
3 O 22 -O.76 601. C
4 o 23 -o.64 279 c
5 o 24 -o.5o 000 c
6 o 25 -o.34 202 c
7 0 26 -0.17 365 C
8 o 27 .0.04 358 c
9 -0.04 358 C 28 O

10 -O.l7 365 C ‘ 29 0

11 4.34. 202 c 30 o

12 -o.50 000 c 31 o

13 -0.64 279 c 32 o

11. -o.76 60/+ c . 33 0

l5 ~0.86 603 C 34 O

16 -—0.93 969 C 35 0

17 -o.98 1.81 c 36 0

18 .1.00 000 c

 

 

 

 

 

 

 

 

in which C = fag-q . It is seen that the boundary values are symmet-
trical with respect to the x - axis.

 

 

 

der. ‘6'. PaéTE WlTn SET .3? SQUJ‘ORRY 'POMTS FOR THE NUMEQK‘

m. ammné gaseous Habonfifmm)

 

__.-.._.Jb

 

 

5'6.

57.

Fourier Coefficients of the Stress Motion:

The Fourier coefficients of the stress function, Eq. (37),are then
determined according to Eqs. (42). It is seen immediately that all sine
terms drop out since the sine is antimetric with respect to the x - axis,

while the boundary values are symmetric with respect to the same axis.

 

Hence,
0“ == 0 I. dM :2 0
Noting further that Fk = Gk and m = 18, Eqs. (42) reduce to,
‘ 3‘ ' 'x nkq
a° = 37 1.2.. F“ a“ = 3.69" EIO'H.” F“ c” «s
‘ 3‘ ' 3‘ 0‘ch (76)

The evaluation of equations (76) leads to the following results,

 

 

 

 

n 0w?" n cn-f"”‘

0 —0.31 992 C 0 -O.31 992 C
1 0.50 000 C 1 0

2 -0.31 349 C 2 0.10 450 C
A 0.09 812 C 1+ -0.05 887 C
6 -0.05 270 C 6 0.03 763 C
8 0.03 173 C 8 -0.02 468 C
10 —0.01 917 C 10 0.01 568 C
12 0.01 075 C 12 ~0.00 910 C
14 ~0.00 502 C 14 0.00 435 C
16 0.00 11.6 c 16 .0.00 129 c
18 —0.00 016 C 18 0.00 014 C

 

 

 

 

 

 

 

 

All add an’a and cn'o except a1 vanish.
_Sjress Components:
From Eqs. (31) which given an expression for the stress components,

it is seen that the term with the coefficient a1 vanishes. Hence, we

have again only even cosine terms for the determination of the stresses,

58.

and the equations become as follows,

{We

%- a ..
O’V.= C,’{ “2 qu‘Z-M(ZM~\) V" 2'60: 2449 *’ ‘5' “(W-l) V'w 16011449 *-
=1

. 1011.4.)
I.
+ Z. C2... Z-(M-l) [au+t)v-ucos 2M9 4- 22. (wt-UCM“) 1"“ to: m9 }

“=1

('77)
{he-L) Zu-L 0. -
org = c, + 2 gal“ (Lu~l)* cos 21.8 . ~53 w (us-1M3“ has M9 .
K31
ihhl)

cm
*' (“1(u+l)(au.-t)v-h‘coc 2.1.8 +- 7: (M+L)(m¢-l) v-u‘cosmg'

K8!

In order to obtain a convenient expression, we introduce a new notation,

 

 

 

Z, d _. R“
C, :3 ‘7‘ 1: co -§-q
.. :32: d _ — \ 5:.
an. "' Pu. ‘ a,“ elm-2.. 3‘]
\ 3..., .. \ r
a‘“ = u d d. an. u-z. 3 (1
P P
am _. ‘- _‘ E.
CLM = “.1. d - CL... Lu 3 (1
F 9
an _ "‘" .3.— E;
can g ‘13: " cw [NM 3 q

where C = fag: q . The new coefficients 3. ‘ a,“ , etc. represent

merely the numerical part of the coefficients co. a,“ , etc. Taking the

constant factor %q in front, we can now write Eqs. ('77) in the form,

.. i Chm-z.)
u _ ‘- __ ' e- lu-L
i co“ at“ Zn (la-I) (7,") C03 2H9 "

"v =°~ ’51
a, —L
_T whet-l) ('F—‘ju (0.: “$—
{fa-L)
' 2 2,“ Names“) 0%.)“ “s 21.9 ‘ (78)

“6|

-— fféu—L)(w+l) (ff‘cog m9)

°'o = 36") c. ‘- 2 a}... butler-v1) (T) ca: 2143' c-
me.
3 a.
* 3: W (”‘0 (f) tcosmg *-

ifm-L)_ w in
+ fi‘ cl“ Z(u4-l)CZu§-\) ('F) Co; 2.49 g

1- % (m+L)(uu-l) (fjwcaswgi

 

.The equations for the stress variation in the right half of the horizontal

diameter (9: 0) are then (with m == 18),

- _, 8 __ - "' ‘
— 22 2a.- 2: a“ it. I "
._-. u. on...” (P) .. ‘ sou. (f) f
(79)
\T - g .- “u - V"
a. - .a i W) (51‘ 4 .0. (7)"

WW1 (§)“~ —- m":

8
«- Z.
“O.

The corresponding equations for the stresses in the upper half of the

vertical diameter (9=% ) are as follows,

IT _ ‘ n- - a
0' = S-q { co“ Zf—t) ahauau-u (1%)“ L--—;-_! 306 ('92:)‘6‘

a _ ..
"' Z I")h chaun) (ZuH) (_‘P’IJ‘k_ $3 .30“ (7":ng

he.

(80)

0‘!
ll
0 la

qg 20 ‘- 82(4)“ a&“Z-K (244'” (%)‘“-&+ §f8.30‘ (%)“+

" — E. v- a
Q- E‘(-\)“ Chlfiaupl) (auaIJ (36:)2“... —-E «380 {-F’)‘ }
Eqs. (79) and (80) are used for the evaluation of the stress components
or‘" and 01, on the horizontal and vertical diameter, respectively. The
results are given in the second columns of Tables 6 to 9 and are shown

in Diagrams l and 2.

Lo) Stress Commnents by UsLnLthe Numerical Method
Developed in Section 2

Figure 17 shows the loading conditions and the grid used for the
numerical example. The grid consists of 24 rays spaced at equal inter—-

vals ae-m and of'lO concentric circles. Hence,

W‘ll" “2‘0; 49=|g°=€E
In addition, we have an eleventh circle outside the boundary of the plate.

Boundigz Values:
The boundary values are obtained in the same way as in Section 11, b.

If (9,9) is the starting point, these values are given by,

Fk=4k=0 if-OékéS’

a .= e. = wpe‘w 5° :4 «=6

6|.

 

 

 

 

 

H?

 

49-150

 

'3. IO°

 

 

naiO

HUN SR. ~15“.

0‘. “1?.

v
|

F.)

r
LL

9.;

STRnCR¢r<D (I

{4

WITH

an

FLRT

4;}. l7:

M a". .t

 

has)

.‘

3. -Fh

“ I \~ 0

5

0
§

AHMVL

‘
a
Q

‘.

 

 

 

ch GK :3 ~ZqPr" (COSG' 3(- 7ék§|1
FR=GK =2 ‘qpf’. 86:45" 1‘ k=¢8

Fk=Gk= o if "3;“:Z-‘4'

This gives the following mmerical values,

 

 

 

 

0 0 13 -0-%59 2583 C
1 0 14 -0.8660 2540 C
2 0 15 4.70“. 0673 C
3 0 16 -0.5000 0000 C
4 0 17 —0.2588 1905 C
5 0 18 -0.0435 7737 C
6 4.0435 7737 C 19 0
7 -0.2588 1905 C 20 0
8 -0o5000 0000 C 21 0
9 -0.7071 0678 C 22 0
10 -—0.8660 2540 C 23 0
11 4.9659 2583 C 24 0
12 ~1o0000 0000 C

 

 

 

 

 

 

 

 

where C=f g-q

Evaluation 9_f_ _t_h_e_ Grid Constants:

The grid constants Uk, 71:: Wk, and 1]: are evaluated according to
Eqs. (57), (63), (64), and (65). The results are found in Tables 1 to 4.

3.21.1122 2.: 12.11:. was _____nmction inside. in: gate

The nnnerical values of the stress function at the grid points in-
side the plate are determined by means of Eqs. (66) and(66a). Since we
are only interested in the stress variations on the horizontal and verti-
cal diameters, the values of the stress function have not been calculated
for all grid points but only for those which we need in the difference
equations, (74). These values are given in Table 5.

The value of the stress function at the center of the plate is

62.

63. '

determined from Eq. (67). We obtain,

F, a: — a. lace 6062. (1

111393 9_f_ th _S_t__1_‘_§_s_s_ Function Outside the 21312:

The numerical values of the stress function on the additional circle
outside the plate are obtained from Eq. (69) which becomes, for our ex-
ample with n = 10,

s
F = Fk+a.z.6\-k

u
“ (81)
The result is also found in Table 5.

It should be mentioned that these values obtained from Eq. (81) are
only rough approximations since they are gained from linear extrapolation
as described in Section 9. However, they are only needed for the differ-
ence equations (74) if those stress components on the boundary are re-
quired which cannot be seen immediately from the boundary conditions. In
our example, for instance, this would be the case with the tangential
stress do at the point ( f}; ). However, one should remember that

those stresses obtained by using values F; are likely to be inaccurate.

Sgess Components:
The stress components at the grid points are determined according

to the difference equations, (74) . The results are given in the third

column of Tables 6 to 9 and are shown in Diagrams l and 2.

64.

g. DISCUSSION OF THE RESULTS
Stress Components fl Using Frocht's Formulas:

In order to obtain an estimate of the accuracy of the boundary method,
it is desirable to compare the results obtained in the numerical example,
Section 11, with the results of a similar problem gained by a well estab-
lished and generally accepted method.

Frocht has investigated the case of a circular plate loaded with two
single concentrated loads acting on the vertical diameter (See Figure 8).
He arrives at the following formulas for the stress components on the

horizontal and vertical diameter (See Photoelasticity, Vol. II, p. 127):

 

 

 

 

 

 

 

Horizontal Diameter: ‘19 d1-ux‘
03': wtd [dH-ux‘]
(82)
c. ___ __ 2.9 [ ltd“ _i]
3 Wtd (d‘+u¥‘)‘—
Vertical Diameter: 0' 3 2.9 = Co“...
" fitd. ‘
.. 3-9 [ e- L l (83)
" d" 7-! d+ag d

in which 1'. is the thickness. of the plate, d the diameter, and x, y the
coordinates of the points on the horizontal and vertical diameter, respec-
tively.

These formulas are obtained from three different superpositions. Each
one of the forces P, acting on the edge of a semi-infinite plate, pro- 1
' duczes; a radial stress distribution with respect to the point of appli-

cation of the respective load. The. superposition of these two cases

65.

results in uniform compression on the boundary of a circular plate.
The third superposition, uniform tension of the same intensity, removes
the uniform compression and thus satisfies the boundary conditions.

The formulas gained from superposition of these well known cases give
accepted values for the stress components and will be used for a compari—
son with the boundary method. The loading condition in our numerical axe
ample (Figure 10) is slightly different. However, the principle of
Saint Venant states that if forces acting on a small portion of the sur-
face of an elastic body are replaced by another statically equivalent
system of forces on the same portion of the surface, this redistribution
of loading has a negligible effect on the stresses at distances which are
large as compared.with the linear dimensions of the surface on which the
forces are changed. Hence, we may expect essentially the same stress
distribution on the horizontal diameter as well as on the portion of the
vertical diameter close to the center.

.If we take Frocht's load P as of the same magnitude as the resultant
of a uniformly distributed load, '

P a ’- PM
and if we further assume the thickness t as unity, we obtain the stress
variation on the horizontal and vertical diameter for Frocht's case of
loading (Figure 8) from Eqs. (82) and (83). The results are given in the

fourth column of Tables 6 to 9 and are shown in Diagrams l and 2.

Comparison 2;,Egocht's Ialues with Those Obtained fromuthg_Bounga£z Method:
Tables 6 to 9 and Diagrams l and 2 allow a comparison between Frocht's

values and the values obtained by using the three modifications of the

where

‘boundary method. It is seen that everywherevthe principle of Saint Venant

66.

holds true, e.g. on the horizontal diameter as well as on the vertical
diameter close to the center of the plate, all four values are in close
agreement. On the horizontal diameter, the four different sets of values
could not be plotted as four distinct curves since deviations mostly ap—
pear in the third decimal only. Even if these numerical values have
still to be multiplied by the load intensity q, the difference is im~
material.

On the vertical diameter, the boundary method leads to finite values
for the radial stress on the boundary while Frocht's value approaches in-
finity. But this is consistent with the different loading conditions.
Frocht uses a single concentrated load, and the radial stress at its point
of application is equal to infinity, whereas in our example, using a uni-
formly distributed load, the radial stress at the boundary point of the
vertical diameter equals the load intensity q. Modifications I and II
furnish values which are too large. The error in modification II amounts
to 14% at that point. Modification I would probably lead to a correct -
value if ten or twenty more terms of the series had been considered in
the computation. It was already mentioned that the corresponding series '
shows a slow convergence for points close to the boundary. At the same
point, the third modification gives a value which is smaller than the
correct value by 11%. The inaccuracy of modifications II and III on the
boundary is due to the fact that the number of boundary points which were
considered for a representation of the boundary values of the stress
function was not sufficient. If a higher degree of accuracy is required,
it is necessary to take one boundary point on each side of the yaaxis
so that this point lies under the distributed load applied to that part

of the boundary;

In our example, for instance, we would have to space the boundary
points at a constant angle of A9: S° . -

On the vertical diameter, the boundary method shows a change in sign
for the tangential stress near the boundary, while Frocht gives constant
tension. This may again be understood from the different boundary con-
ditions. It seems natural that aluniformly distributed load in radial di-
rection produces a compressive tangential stress at the point of appli-
cation of its resultant and in the neighborhood of this point.

Comments 93 _t_h_g first Modification _o_f _th_e_ Bounds}: Method;
The practical application of the first modification of the boundary

method seems to be restricted to some special cases where the function
expressing the boundary values of the stress function has a continuous
first derivative with respect to 9 . In those cases, this modification
leads to correct values of the stress components provided the computa-
tion is carried to a sufficient number of decimals and considers a
sufficient number of terms of the infinite series. The disadvantage
is that this series is likely to converge very slowly at points close to
the boundary. Although ten terms were taken into consideration in our
numerical example, the radial stress on the vertical diameter at point
.1 = 3 is obtained as -l.02 168 q while in reality it should be below
-1.00 000 q, since --l.00 000 q is the value on the boundary.

In order to avoid the disadvantage of infinite series with their
slow convergence, or even divergence on the boundary, Dr. Frame has

developed the following formula which should be mentioned in this connection.
Its derivation is not given.

it? UT
a

| N- ' _
F(W‘Q) = 1"? I (F‘U) d9 ‘1‘?— ("’"LJ Pa?— 33;: lo (F‘U) d9 (84)
O

68.

where F is the function representing the boundary values of the stress
function, and.U is as follows,

l-({‘

U = 8
l—-L(%)Ces(9-o)+(%y‘ ( 5)

 

Eq. (84) can be obtained from our former Eqs. (57), (63), (61.), (65), and
(66) by letting m approach infinity and by proving that U is a harmonic
function, while 7, Eq. (63), and 2 , Eq. (65), are both biharmonic.

Using Eq. (84), Dr. Frame has shown that the stress function for the
case of two single concentrated loads acting on the vertical diameter

(Figure 8) is as follows,

9 a
F(~r‘9)=- %[R-arc+nu‘§*f] (86)
anhich R :2 «$059-
and s = i(£:“f>t)

However, considerable difficulties are encountered in integrating Eq.
(86) to obtain the case of a uniformdy distributed load.(Figure 10.)

In summary, the first modification of the boundary method leads to
correct values for the stress components provided the loading conditions
are such that the resulting expressions are practically integrable and
differentiable. However, this modification does not satisfy the too
basic demands we have made for this study: it is not a procedure of
general applicability to any kind of loading, and its performance is
likely to require some special mathematical knowledge as to integration
and differentiation of complicated expressions and to the theory of

infinite series.

69. '

Qomments 9g _thg §_e_c_9_n_d Modification 3;; the w W:

‘ The second modification gives a good approximation to the correct
values if a sufficient number of boundary points k are taken. The results
will then be accurate enough for all practical purposes. As compared with
the third modification, the advantage of this method is that the stress
components can be calculated at any point of the plate, not just at cer-
tain.selected grid points. loreover, we obtain an approximate stress
function in the form of a finite trigonometric series which can also be
used for the determination of deformations of the plate under the applied
loads.

The calculation of the Fourier coefficients according to Eqs. (42)
and of the stress components accordingxto Eqs. (73), consists essentially
of summing a finite number of products. In order to cut down the time of
computation to a reasonable amount, it is desirable to use a modern elec-
tric calculator which can perform positive and negative accumulative mul-
tiplications. The use of a slide rule would not only increase the amount
of time required, but also lead to useless results since the computation
should be carried at least to the fourth decimal. This is necessary
because in the course of this computation, small quantities are some-
times multiplied by large factors, and numbers differing only by a very
small quantity are subtracted from each other. If, however, a calculator
is used, the consideration of several decimals scarcely increases the
amount of work.

In summary, the second.modification satisfies both-of our demands.
It is a method of general applicability to any kind of loading, and its
procedure is such that every engineer should be able to perform the

computations.

70.

Cements 93; the, M Modification _o_f_ th Boundary W:

The third modification of the boundary method also gives approxima-
tions close enough to the correct values to satisfy all requirements of
practical engineering, provided a grid with a sufficient number of points
is used. This is necessary not only to obtain a good representation of the
stress function on the boundary, but also to keep the inaccuracy arising
from the use of finite differences inside the plate within acceptable lim-
its.

The larger amount of work involved in this modification is required
for the evaluation of the grid constants 01‘, Wk, Wk, and 2: b However,
this work has to be done only once for a certain grid, and the same con-
stants can be used for any case of loading provided the same grid is al-
ways chosen. For practical application, it is therefore suggested that
for all computations only two or three or four ”standard grids" be used,
for which the grid constants have been determined once and collected in
tables. An example for such a standard grid with m = 12, n = 10,
as; 3E .—. \S" is shown in Figure 17. This grid was used in the numerical
example, Section 11, c, and its grid constants are given in Tables 1 to 1..
Other possible standard grids could be:
= 10°
= 5°
Using one of those standard grids, the grid constants of which are

3:18, 11:15, A9:

klo ale

m=36, n=25, A0-=

already known, the work to be done reduces to the determination of the
boundary values and boundary derivations of the stress function according
to Section 6, the determination of the numerical values of the stress
function inside and outside the plate according to Eqs. (66), (66a) , and

(69) , and to the calculation of the stress components by means of the

'71;

difference equations(73). Here also an electric calculator with provision
for positive and.negative accumlative multiplication is desirable. Using
such a calculator as well as prepared charts which contain the constants
of the standard grid to be used, the computation can be performed within
a reasonable period of time.

The main feature of this numerical method is the elimination of all
mathematical operation except algebraic ones. After having determined
the boundary values and boundary derivatives, almost all the work can be
done by any person who knows how to use an electric calculator, if this
person is given.simple instructions, and if prepared charts are used
which lead the computer automatically from step to step.

A further advantage of this method should be emphasized.l The writer
knows about approximate methods for rectangular coordinates where the
numerical values of the stress function at the points of a rectangular
grid are obtained in the form of a system of 5 linear equations with p:

' unknowns which may be solved by the Gauss algorithmus, by different
matrix methods, or by relaxation methods. However, the solution of such
a system.of linear equations is always troublesome, and the number of
grid points taken is restricted by the possible number of equations in
this system. If there are more than, say 20 or 30,equations, it is not
feasible to solve them. At least, the effort would not be justified by
the result. This is a severe restriction on the accuracy of those num-
erical methods. Moreover, if one is only interested in the stress com-
ponents at certain cross-sections as'in certain portions of the plate,
one has nevertheless to solve the whole system.of linear equations, thus
obtaining also the numerical values of the stress function of those
points which are of no interest. The third modification of the boundary’

method, however, permits the calculation of the value of the stress func-
tion at any point inside the plate independent of the values at other
points. Unnecessary work is therefore avoided in this method.

In summary, the third modification of the boundary method satisfies
our two basic demands. It is not only a method of general applicability
within the scope of this study, but is also of reasonable simplicity in
its practical application.

Further Aggcts 9_1_1_d_ Possibilities:

The writer has considered this study as an interesting example of
the power of practical mathematical methods in engineering. However,
this investigation represents only a small step in the direction indica-
ted. At this point, many questions relative to extended applications of
the boundary method arise which could not be studies due to lack of time.
The most important ones may be the possible extension of the boundary
method to ring problems and the determination of deformations from
numerical values of stress function obtained by using the third modification.

BIBLIOGRAPHY

Theory of Elasticity, S. Timoshenko, McGraw-Hill Book Company, Inc.,

New York, 1934.

Photoelasticity II, M. II. Frocht, John Wiley and Sons, Inc., New York,

1948.

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