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Masters Theses Student Theses and Dissertations

1960

Stress distribution in splined shafts in torsion by the membrane Stress distribution in splined shafts in torsion by the membrane

analogy analogy

Charles L. Edwards

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Recommended Citation Recommended Citation Edwards, Charles L., "Stress distribution in splined shafts in torsion by the membrane analogy" (1960). Masters Theses. 5576. https://scholarsmine.mst.edu/masters_theses/5576

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i

T~ t 2\tc·~ c. i . \ .

STRESS DISTRIBUTION IN SPLINED SHAFTS

IN TORSION BY THE ME1'1BRANE ANALOGY

BY

CHARLES L. EDWARDS

A

THESIS

submitted to the faculty of the

SCHOOL OF MINES.AND METALLURGY OF THE UNIVERSITY OF MISSOURI

in partial fulfillment of the work requi"±ed for the

Degree of

1960

Approved by

ACKNOWLEDGEMENTS

The author wishes to thank Dr. A. J«> Miles for his assist ..

anoe and for suggesting the thesis subjecto The author also

wishes to thank Professor Go L. Scofield for his.:suggestions

on experimental procedure.

ii

PREFACE

The purpose of this investigation is to detennine torsional

shearing stress distribution in various shapes of shaftso

The project will include shafts with circular boundarieso

No attempt will be made to show the relation between stress

and applied torqueR

iii

iv

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS •••••••••••••••••••••••••••••••••••• a•••••••o••••V

PA.~T I: Derivation of the Equation for Torsional Shearing Stress and Methods. of Solution ••••••••••••••• o o ••••• 1

PART II: The Membrane Analogy And its Applications •••••••••••••••••••• 6

PART III: Procedure and Experimental Da:ta•••••••••••••••••••••••••••••8

PART IV: Conclusions ..... · .............................................. 34

BIBLIOGRAPiff o • o o o o ••a• o o • • o • o • o o a o o o a• • .o • a o ••a o • • • o o o • • • • • • o o a o • • a••• 36

VITA•••••••••••••••••••••••••••••••••••••••••••••••••••••••o••••••••o37

v

Lisr OF ILLUSTRATIONS

FIGURE Page

1 Sketch of infinitesimal body in pure torsion ••••••••.••••••••• !

2 Stress distribution in a non-circular shaft ............. u •••• l

3 Angular deflection of a plane•••••••••••··~··••••••••••••••••2

4 Sketch of non-circular prismatic shaft ••• o ••••••••••••••••••• 2·

5 Sketch of an element turning about a center without distortion ••••••••••••••••• a••·······················2

6 View of element A in the ZX plane••••••••••••••••••••••••••••3

7 Free body diagram of three dimensional element in-pure torsion••••••••••••••••••••••••••••••••••••···~··••••3

8 Geometrical representation of phi ••••••••.•••••••••••••••••• o. 4

9 Homogeneous membrane blown up over a. given cross-section ••••• 6

lOA Templates used in experiment••••••••••••••••••••oao•••••••••a8

lOB Templates used in investigation to simulate circular and splined shafts {Photographs) •••••••••••••••••••• 9

11 Component parts of.apparatus used in the investigation {Photograph} •••••••••••••••••••••••••• 110

12 Membrane apparatus with templates in place and glass plate removed {Photograph) .............. o•••o•o•••oll

13 Membrane apparatus with templates in place and glass plate in place (Photograph)ooo••o•••••••••••••••••ll

14 Membrane apparatus with templates and glass plate in place and the target in the marking position ·(Photograpijol2

15 Illustration of blowing up of films with small pressure from underneath shell (Photograph) ••••• o••••o••••••l4

16 Sketch of membrane on a circular cross-section •••••••••••••• 15

17 Membrane over circular template {Curve)•••••••o•••••••••ooo21

FIGURE

18

19

LIST OF ILLUSTRATIONS (Continued}

Lines of constant phi,circular

vi

Page

template (Theoretical), (Plate).••••••••••••••••••••••••••••22

Lines of constant elevation,oircular template (Experimental), (Plate) •••••• booQODoo••············23

20 Stress distribution in a circular shaft •••••••••••••••••••• D24

21 Circular shaft with semi-circular spline •••••••••••••••••••• 16

22 Lines of· constant elevation, one-spline template, (Experimental), (Plate) ••••••••••••••••••••••••••• 25

23. Lines o.f constant phi, one-spline template, (Theoretical)~ (Plate) ••••• o •• o ••••• o •••••••••••• a 2.6

24 Graphical -solution for fourth order equations, (Curve} ••••••• a••••••••••o•••••••••••••••••••••o•27

25 Membrame over circular one-spline 1:ernplate8 (Curve) o o •••••••••••••••••• .•••••••••••••• o •••••••• a 28

26 Stress distribution in a Gircular one-spline shaft I (Curve) a O O II O O • 0 ••• 0 a O • 0 a O ••• 0 ••• a a O II O a a • . a O 2 9

27 Circular shaft with four semi-circular splinesa•••••••••··~·20

28 Lines of constant elevation1 four-spline template, (Experimental), {Plate).0000••••••••••••••••••0•0~30

29 Membrane over circular four-spline teinpla te, {Curve) •••••••••• a •• ~ ....... o ••••.•••••••••••••••••• 31.

30 Stress distribution in a circular four-spline shaft1 (Curve)•••••••••o•••••••oooo•••••••••••o~••••••••••••32

Sl Stress distribution in· a · circular, one-splin~, and four ... spline shaft, (Curve} D ~ 0 0 0 o_o O O O O O O O ~ 0 0. 0. 0 0 0 0. 0 0 0 a O 33

32 Effect of an additional spline on a membrane ••••• o • ·• •••• a o o o 35

PART I

DERIVATION OF TiiE EC.UATION FOR

TORSIONAL SHEARING STRESS AND METHODS

OF SOLUTION

The stress distribution for a shaft with a circular cross-

section in pure torsion is a linear function of the radial distance

from the geometric center {s = T,.) . The stress distribution in a s 7T

non-circular shaft is not this simple. Since many- shafts encountered

in Mechanical Engineering are.of ~he non-circular sort, and since

these shafts tr3nsmit a relatively large torque, the ability to

predict torsional shearing stres·s distribution becomes important.

Den Hartog describes the derivation of Saint Venants compat-

ibility equation for a non-circular prismatic shaft in pure torsion(l).

In this discussion Den Hartog points out:

1. Shearing stresses on sections normal to the axis of the

shaft must act parallel to the surface of the shafto

l Ssn

Fig. 1. Sketch of e;p T

infinitesimal

Does Not Exist .: Ss =Sst

body in pure torsion.

2. Plane cross-sections do not remain plane since the shearing

stress is not distributed in a circular manner about the

center of rotation. el:> r

Fig. 2. Stress in a non-cfrcular shafto

(1) All references are in bibliography.

1

3. The stresses acting as shown in Fig~ 3 will rotate and

warp the cross-section. The projected shape of the section reIDnins

unchanged. ~ y

x x

Fig. 3. Angular deflection of a plane.

Saint Venant assumes the displacements of an element (A Fig. 5) in a

shaft to be: ~

G{=- 9,r. y V=-~lX

W= f fr,!/}

Fig. 4. Sketch of non-circular prismatic shaft.

Fig. S. S"~etch of an element turning about a center without distortion.

2

(1)

;-/here u is the displacement in the X directio7 v the displacement in the

y direction, w the displacement or warpage in the Z direction and ~,

the unit angular displacement of the element after the torque is applied.

The displacements now must be changed to strains. The total shear­

ing strain ~of element A is the . sum of . angles C and D, (fig. 6).

Fig. 6. View.of element A in the ZX planeo

.Thus:

Within the elastic limit the shearing strain angle is small so that

Yx z= = e.u -1- £Yx o~ ax The same· method when applied to the ZY plane yields:

Yy~ - eY -1- .ow ~ oy·

Substitution of equation {l) produces Yx:2. == &/y t- ow Yyc = -~Xt-Jw BX ay

From Hooke's Law V 55 =.BI

Where G is the shearing modulus of elasticity and Ss is the

shearing stress. Putting/this into e~ations {4) gives .(Ss)xr= 6,9t~.1-. ~J 1t fl/- ax .

... 15sJu:c. ~ l7A .. ::-:-~.,( r.QW I _J o_j('/. .

A relation between the str~sses in equation {5} can be found by

equilibrium conditions. The fact that stre~s changes from one

side of an element to the parallel side a small distance away is

taken into account. Thus the stress in the ZY plane changes from

Fig. 7 Free body diagram of three dimensional element in pure torsion.

Summation of forces in the Z direction finds

d~O)'d2= le) (ss)x~-;- 2) t$.s~..c7-== 0 L ~x o_y J ·

3

(2)

(3)

{5}

0 (ss)x~ ox

The next operation in Saint Venant' s derivation of the compatibi_li ty

equation is. :to choose a ·.unique function£ such that

This f function so chosen leads to the condition of continuity • .

Substitution of equations (7} into (6) shows the continuitv. ~ o:z.l _ 2>;l.l ..

ox c;y "cJyax

.!/ Fig. 8. Geometrical representation o·f phi 0

Phi i can be plotted above an 1..'Y plane forming a curved surfacea

From equations (7) it can be seen that the slope of this surface

in the X direction is equal to the stress in ~he Y directiono

The X stress is the negative of the slope in the Y direction.

It can be shown that the slope in any direction is equal to the

stress perpendicular to that direction of slope.

Now all the equations can be written in terms of this function.

Equations (5) become

- o~ - (]{fJ,y.,. ~) c,!I - ax +~= G/-&tX-f-0:Y}

2>X . "?)y Taking a of the first of equations (9) and () of the second ~ i5x

equation and subtracting gives

4

{6)

{7}

(8)

(9)

(10)

This equation is known as the equation of compatibility because

it unites stress and strain conditionso

Equation (10) is the partial differential equa.tion for a shaft

in pure torsion. Solution of torsional stress problems thus becomes

a matter of solving equation (10)o

Ftnding the phi(f) function of equation {10) is very difficulta

Saint Venant working !°n 1855 . solved this equation for such shapes

as rectangular bars, ellipses, triangles, and semicircleso ( 2)

Many shapes such as shafts with squa·re splines cannot be readily

reduced to mathematical formulae. 'Inis fact precipitates the

use of analogies to equation (lO)o

Less than 20 years after Sta Venants work Sir William Thomson

(Lo:rd Kelvin) .devi$ed the fluid-flow analogy. {3) He pictured

the lines of . constant I {Fig. 8) as streamlines of vortical flow

within i;he boundaries of a given shape. The partial differential

equation for this flow is similar to St. Venant's compatibility

equation a The str.esses become proportional to the linear velocity

of the fluido

Lines of constant~ can also be interpreted as constant

electrical potential lineso The current flowing between these

lines is proportional to the stressa

5

P&itT II

THE MEMBRANE ANALOGY

AND ITS APPLICATIONS

A homog~neous membrane is stretched over. a hole _of given

shape with an initial tension T per unit length of periphery.

A small lateral pressure P defl.ect-$ the membrane. The tension

Twill not change appreciably if Pis i=. . .

lr-arrrrnw-T ~ p . ,·

Fig a 9. Homogeneous membrane expanded- over a

The vertical component of force Bis (4)

·-T £E. dy. ol(

The vertical component of force C is

· T~ d!J + 2-_(TOr. dy)dx The vertical ~lponents fof'l andqfare

-T oiE.. dx and atj

r li- dx + ~ (r az dx) dt respectiv~-lf. oy oy.

cross-section.

Adding the four components and equating them to the force of

the pressure - Pclx dy gives

then

0 2z + · 3-:..z - _ F ol/-:L. cJ"A::z. - T .

In 1903 15"randtl showed that equai;ions (10) and (11) were

similar and developed the Membrane Analogy for torsion. Taylor

and Griffith (5} were the first to use ~his analogy, applying

it to cylindrical bars. Since then others have used thi~ analogue

6

)(

(11)

for various odd-shaped shafts. Among these were Newbauer and Boston (6)

who in 1947 detennined the shear~ng stress distribution in twist ...

drill sections"

7

Christopherson and Southwell (7) proved.that equation (11) can

apply to any plane-potential problem, since all the partial differential

equations of this type .of problem take on this form. This fact

was previously sunnised since ¥..iles and Stephenson (8) had successfully

used equation (11) as an analogy for pressure distribution around

an oil or gas well.

In the field of Heat Transfer1 Wilson (9) in 1948 used the ·

mem1::>rane analogy to solve two-dimensional steady-state heat-conduction

problems. for this problem the height of the membrane represents

the temperature distribution in the member being analyzedo

Comparing equation (10) for torsion to equation (11) for

the membrane, it is seen that~ must equal Z and P in~st . be equal T

to z (J 'Jo This plus the fact that the membrane must be stretched

over-a hole geometrically similar to the shaft in torsion leads

to impossible technological problems. No attempt is made to meet

these conditions except for geometric similarity. Thus in the

membrane {usually a soap bubble) P is equal to some function of;?.Go}u T

The height Z is equal to a function of p. _e= K :z.G~ iE= = K, f T

This leads to a relation between shearing .stress and the slope of

the buhbleo

(S5 ) y2 = !<:i. oz= -. "?))( .

(S5))(i! == K3 aJE oy

PART III

PRCCEDURE.AND EXPERIMENTAL DATA

The experimental apparatus used was the same as -that used by

Wilson (10)~ Templates were machined to the specifications of Fig.

,----l-'-3,34.5 '1--t+-­

C1~c~/qr

Fig 10 Ao

o. D ..

O a e-s p//oe F o LI r-Sp //o (!T.

Templates used in experiment.

4s0

-:zz;,_ -~

AA 3, .3 4-5 /I Q.l).,

The component parts of the apparatus are shown in Fig. (ll)o

These parts consist of an aluminum frame with a removable circular

shell, a circular target with clamps, a glass plate with a micrometer

head projected through the center, and the necessary test templates·o

The micrometer has a sharp point so that it may be screwed down to

touch th~ soap film on the template. The target has a- piece of paper

clamped on it so that as the target is brought down a point on top of

the m_icrometer head makes an impression on the paper. The micrometer

reading can then be recorded next to this mark. This procedure is

illustrated in Figso {12), (13), and (14).

The soap solution used was the same as that used by Wilson {11);

2 grams of sodium oleate.and 30 c.c. of glycerin in one liter of dis-

·tilled water. Different solutions are recommended by other invest-

igators such as Neubauer and Boston (12) who used 6 c.co of glycerin

and .s gram of sodium oleate per liter of distilled wate!• The soap

films produced by Wil~9n's solution proved to be sufficiently tough ?--:. •

and durable. A qµanti i:y. of soap solution is poured into the shell.

8

With a piece of celluloid o~ plastic a film is swept over the template.

A small wire is then used to remove bubbles and most of the excess

solution from the filmo

Figa 10 Ba Templates used in investigation to simulate circular and splined shaftso

9

Figa 11. Component parts of apparatus used in the investigation.

10

FigD 12e1 Membrane apparatus with templates in place and glass plate removed.

Figo 13. Membrane appa-ratus with templates in place., and glass plate in place.

11

Fig. 14. Xembrane apparatus with templates and glass plate in place, and the target in the marking position.

12:

The glass plate is innnediately placed over the.shell to prevent.

contamination of the film by dust and carbon dioxide. The excess

soap solution in the shell maintains a moist atmosphere to prolong

the film life.

Den Hartog (13) recommends that the film on the template be

eXpanded by a small pressure underneath the shell (Fig. 15}. With

this method the film is stretched farther in the expanding process1

thus shortening the film life. Also, when the micrometer point is

brought down on the film the chances of bursting the expanded film

are ·very high. Another disadvantage to this method is that leaks

in the shell and pressure regulation -device are very difficult to

stop, causing the height of the film to vary constantly.

For these reasons the soap films in this experiment were not

expanded. They were allowed to sag downward so that the only

pressure acting on them was that caused by the weight of the film

plus the excess moisture distributed throughout the film.

With the glass plate in place contour.lines (constant phi)

can now be plotted on the target sheet. The micrometer is left

at any· desired reading qnd the glass plate moved until contact

is made with the film. By placing a light so that its reflection

is seen in the film the point ·of contact may be clearly seen~

This is possible because the reflection of the micromet~r point

in the film will come into contact with the actual tip. When

contact is made the target is brought down on the micrometer top,

marking the paper •. Numerous points are so plotted then lines

13

drawn to connect these points,. These lines thus become contour lines.

Fiq. 15. Illustration of blowing up of films with small pressure from underneath shell.

14

15

The micrometer tip is peri~dically moistened with soap solution

to prevent film breakage when contact is made.

In order to compare experimental results with theory, two

problems for which the solutions are known were investigated.

The first was a solid circular shaft. A membrane stretched over

a circular hole of radius R is blown up { or allowed to sag) Fig. (16).

·IZ7

.. ~ r I . Lt'~\,.

T A-A T ~

Fig. 16. Sketch of membrane on a .circular. cross-section.

Outting a concentric circle out of the membrane and summing forces

in the Z direction gives:

T dz: 2. rrr = Prrr;z. Where 'T' is the~~rce per unit length of periphery, 'P' is the

pressure, and 'r' is the radius from the centera

Solving for z gives: ,!!: = - .J:i:r dr := :.... Pr-:,.'+ c :::t:r 4T

Z=-O wh~n r.:::. R. .. -", Z =- P {R;;;!.-;-;;e.) . 4-T

Or for a given membrane Pis constant so: T

~ -= c1

( R.-::i..- r-:2.) Where c1 is a constant.

If the membrane is adjusted so that equations (10) and {11) are

equal then Z becomes equal to phi(_p) and E.. equals .:<G~thus:

i= ~Ge, l~~r1 4-

T

e, is a function of geometry and torque so that for a given_

ahape and torque:

f - C (R~-;r-:i.) - ~

Where c2 is a constant.

(12)

(13)

16

Equation (13) is the theoretical shape for the phi~)function and

equation (12) is the theoretical shape for a membrane over a

circular templ~te Fig. (17) shown both the theore~ical curve (equation 13)

and the ~xperimental curve (equation 12). For comparative purposes

the experimental values were-multiplied by a constant. This gives

the same effect as ·equating the constants iii equations (12} and (13).

The lines of constant phi(f}for the theoretical and experiment~l

circular shaft are plotted in Figs. (18) and (19) respectively.

Fig. (20) is the stress distribution in the circular shaft. This

curve was found by graphical differentiation of Fig. (17). Fig. 20

can also be found by differentiation of equation {13).

o~ = Cz_(o-zr)=Ss

This shows the linear relation between the radius and the stress.

The second problem with a known solution is given by Den Hartog (14).

The shaft is circular with one semi-circular spline. Fig. (21).

Fig. 21. Circular shaft. with semi-circular spline.

The phi (j) function is given a·s: ·

(14)

Where C is a constant, a is the radius o~ the shaft, and b ·is the

radius of the spline.

17

This function must meet the conditions. of continuity ( equation_ 18.}

md the boundary condi tionso

Equation (a) shows the continuity of equation (14):

• * •

To meet the boundary conditions phi(~) must be _equa~ to zero

at t~e boundary:

X=-b j y=o i= c (!/-;i.qb + :!:;/-H2·):::c{-:Z.q/;+jqb).::o

x = -x. o. J ,ro 'I?= c (-1t1fo--::zq/aq)+z bQ::i.q _ H'-) -?q-:a.+o

~- C (tq~D""--4q~f 4a~-l')· = C (b~b) =o -?a-:z_

y A . .

-~ x ·polnt A Fj,,ZI _

x-= b 0d-fi <J = b ~/3

p:::c{b-·-c~=z,e,J+ b~,BJ-.zqh~t~91~~-h::)

1=c~ ( Coq~-J-~,8)-2qb<W,8+ ~t?'aC(J'l./3 -bi b~~~~t-~ff/ .

J = c [b:i.(r) -2ab Co<:1-{3 + 2-b0a eoo-/3 _ g]

. . . b::z£i]

. ~:::: C L - Z ab Cdl.f!,_ f- Zq b Co<J,8] o

18

19

By in_spection it can be seen that the maximtun . shearing stress

occurs at Y equal zero and X equal b. This can also be shown by

/':.. {x'J-J/1):zB"°a -:2h~CIX2

)

the phia) function. o~ = re (J.;<-.za i- rx2-;y-z J 2..

-ox.

o~ =-- fssllrJqt.,--=- c &-b-7..a rab~tab)-= GB; (;?. o, -b) c15i C:,7'

Taking the slope at any other point on the boundary will give

stresses of smaller valu~ than equ~tion {15).

Fig. 22 shows the experimental plot of constant elevation

for the shaft of Figo 21. Fig. 23 shows the theoretical plot

whi.ch represents lines of constant phi ~ ). Fig. 24 shows the

graphical method used to determine lines of constant phi~ for

the theoretical solution in Fig. 23. Points were chosen at Y

equals zero between X equal band X equals aQ At these points

the values of phi(~}were found from equation (14). These values

of phi{i)were then used at Y equals b to solve for the values of

Xo Solving for X involved fourth order equations such as follows:

:J ::= Oj X=~P . :2t:SQ ,:}... r- t?_ b'' f = .;.- c ( ·il?--?.c(2bf 2 b-b ) =- C '-3b - 3a u u =.b, x-= 7\ :r ==- c C3.b-:i..-3qb] 7 } -~ ~ 6:i..]

c [) 61 - :i ab] = c: (x"-...:,._q-x + if"' +- ~-,_,, ~ ~b"-) ~c J?-1qxf-::,,ab\ . . x -t'j I i-

2 O. 1--3 - X4 -t ( 2-b -3 o.) bx 2- 3 b5 (a-b) 4-= /,b72~' j b=,4S75''

-'X<I +3,31-SX 3 -/..Z/Xa== .. 3J/

Y~ e,=>ll ( Y=- -x1-+3.31-5X3-1,B1x

2

The line on which the maximum stress occurs CY-equal zero,

Xis between band a) is plotted for the experimental and theoretical

solution in Fig. 25.

20

Fig. 26 shows the stress distribution along the line of maximum

stress for both the theoretical nnd experimental solution. Since

the stress is -proportional to the slope of the phi (or Z) curve the

stress becomes:

Ss'~ ~ =-- c r:z X-;;?..a - zh~) (16) ~x . -x~

Fig. 26 was found by graphical differentiation of Fig. 25. Equation

{16} if plotted would agree with Fig. 26.

This plot is therefore a qualitative result. The actual

numerical value of the stress depends on c. This constant is

a ~u~ction of a1 ·and therefore a function of torque.

The last problem worked was a shaft as shown in Fi'J• 27.

:8 Fig. 27. Circular shaft with four semi-circular splineso

The theoretical solution to this problem is not knowno

-:-i_·, .. The experimental plot of constant elevations is eho't'm in

Fig. 280 The point at which the maximum stres·s occurs was seen to

be at Y equals zero and X equals b, or at the corresponding point

on auy of the other splines. Fig. 29.is the plot of the membrane

along the line Y equals zero and X lying between b and a. This

curve (Fig. 29) was graphically differentiated to show the stress

distribution in a qualitative m~nner {Fig. 30}o

For comparative purposes the stress distribution in the circular

shaft, the one-spline shaft, and the four spline shaft are- shown

together in Fig. 31.

. ··i·· ·.·· .

.. ,.1.

·····i· -. . . . .:!::

.. . t ..... - ••• -+. -+ ~ ••

... , I• • •- •

.. ~r-•• • 1 • .. .. : ! : .

::::t· :::.!:::. I . . +-·--sl----+~--+<~-+-~+---,1---+~-i-.~

.::l:: .... ::·:;:··· ::::1: . : : : .1: :::.:t·::.

.1 ..

'.! I

··---l-I·

j. : i

. .. ! , I

I

·I · !' I ------

i ... ~.:.+:::

. t. \ ..

. ...

·~-+-~+----if---+~-+-~-+----+----1f----+·~--+<~-+

.. ·:1<

...... !. ;., ' I

:1

~ . .. .

. . ... .

. .· .. -.. . ··:;·::: ..

:\)

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'

1 ' .. · :: . .. ---- -; _ _____ ___ ...

' .. I

. , . . . . : t ~ . . .

·-r-; '. ;; t' ; ·_ '. : .i....t,- -t- · -·· ···

::·: :.: r.: 1:

: ! ~: 1:: i: . . _ .. _. l ' - . -

, . !• •

• I •..

: ::1~:: :

PA.t.~T W

CCHCLUSIONS

34

The results obtained from the investigation were consistent in

several ways. The plot of _the membrane was always_ just below the plot

of the phi(f}function1 and repeated testing yielded consistent curves

for the investigations in which the solutions were known. It was con­

cluded from th~s that the re~ults for the four-spline shaft were slight­

ly under the theor~tical (unknown) solution.

Comparison of the figures for the contour lines shows that the

theoretical plots are slightly smoother-than the experimental

plots. This may be due to the fact that some of the points were

plotted using the color ·bands in the film as guides (Fig. 15).

These bands changed with membrane age and were also shifted by

the micrometer tip, since· the tip seemed to attract the film at

a distance of approximately .001 of an incho

A. slightly moist membrane was not a deterent factor in this in­

vestigation. Wilson (15) found that with 'age' the membrane dried

out and shrank upwards towards th~ desired theoretical curve. How~

ever in this problem the theoretical curve was ·slightly below ev~n a

moist membrane. Therefo~e immediately after the excess solution was

removed measurements were taken. Although this eliminated the

variable of membrane age, it.gave best results.

The four-spline shaft shows a slightly higher maximum stress than

the one-spline and also a sharper drop in stress. At first this seems

unreasonable. ~ig. 32 illustrates the effect an additional spline

could have on .a membrane. . The s .lope is fncreased at the boundary and

decreased elsewhere.

· · ... --. · ~ •o/d 1/Aembrcme ~, 'Shqpe · ·

i-,.-..,1,_z v~ s s s. . 1/ew Me>7lb,z:,"e · · ·

Fig. 320 Effect of an Shqp~

additional spline on a : Membranea

Fig. 31 is a comparison betwa'en the various shaf::ts assuming

the angular deflections ~ to be equal o The stress ratio:"L between ·

the one-spline and dirctilar shafts is 2o54o . The ratio between the

four-spline and circular shafts is 2.7.60 These are not legi:timate

stress concentration factors because the relation between torque

and angular deflection is not the . 1same among the shaftso

This suggests that a technique for finding the relation between

stress and torque in this ·problem should be developedo An accurate

method for measuring the volume under the membrane would :have to be

used since a very small volume is involve:d• The purpose for fin~ing

the volume is that it can be shown {16} .that the torque is proportional

to the volume under the membrane.· Neubauer and Boston (17) used

graphical intergration for.best resultsn

35

BIBLI03ru\PHY

.1. Den J-{artog, J.P • ., M.:_ Strength tl :Materials, McGraw-Hill 1952, pp. 3:..10.

2. Ibid. Page 19.

3. Den Hartog, J.P., 11.1• Strength .2f Materials1 McGraw-Hill 1952, page 21.

4. W'ilson, L.H., Thesis., The Application of the Membrane Analogy to the Solution of Heat Conductidm Problems, Missouri School of Min-;;; 1948, -page7_o __

5. Taylor, Ga I. and Griffith, A.A~Adv. Como for Aeronautics, T.ech. Repto -(British_) 1917-1918, ~ 3M.

6. Neubauer, T.P. a.nd Boston, DaW •. , Transactions .A..S.M.E. Vol. 69, No. 81 November, 1947.

7. · christopherson, B.A., and Southwell, R.V., Proceedings of the Royal Society, S€ries A, Vol. 168, 1938~ pp. 317~350.

s. ~riles, J~.J. and Stephenson,. E.A., A.i.--uer. Inst. of lFi.in. and ~· Engrs. ~!. Pub., No~ 919# May.,_ 19~ --

9o W1lson3 L.H., Thesis, ~ Application of ~- r-1ew.hrane Analogy . to the Solution of Heat Conduction Problems. r!issouri School of llin~ 1948$1 --

10. Ibid. Page 16.

11. Op. Cit .. vlil"son L.H., Page 11.

12. Neubauer1 T.P.,. and Boston1 D.W., Transa·ctions A.S.H.E. Vo., 69 1 no. 8.,. November., 1947, Pc1:ge ~97D

13. Den Hartog, J.P., Ad. Strength of Materials, McGra,.11-Hill · 1952, pcige 13.

14. Ibid~ Page 337.

. . ' 15. Wilson, L.H., Thesis, ~Application £!.~Membrane Analogx:

to the.Solution of Heat Conduction Problems, Missouri School of Mines';' 1948, page2~ .

16. Den Hartog# J.P., M• Strength .2£. Materials:, McGraw-Hill 1952', page 9.

17. Meubauer, T.P., and Boston, D.W., Transactions A.S.M.E. Vol. 69, Noo 8, November 1947. Page 9020

1f7 VITA

The author w2.s born January 21, 1934, at Hiawatha, Kansas.

His high school edudation was completed in 1952 at Lafayette

High School in St. Joseph, Hissouri, at which time he entered the

St. Joseph Junior College at st. Joseph, Missouri and attended there

fn:un 1952 to 1954. In September, 1954 he entered Missouri School

of Hines a.nd Hetallurgy at Rolla, Missouri. In June1 1957 he

received his Bachelor of Science in Mechanical Engineering. He

be came an instructor ·in Mechanical Engineering September, 1957

· and entered graduate school.·