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Masters Theses Student Theses and Dissertations

1965

Investigation of the use of optics in the analysis of structures Investigation of the use of optics in the analysis of structures

Jimmy D. Hahs

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INVESTIGATION OF TifE USE OF OPTICS

IN THE ANALYSIS OF STRUCTURES

by

JilvU1Y DEAN HAllS

A

THESIS

submitted to the faculty of the

ID~IVERSITY OF NISSOURI AT ROLLA

in partial fulfillment or the requirements for the

Degree of

HASTER OF SCIENCE IN CIVIL ENGINEERING

Rolla, Missouri

1965

APPROVED BY

ABSTRACT

This study \vas made to determine the feasibility of

the use of optics in the analysis of structures. Since

the curvature of a beam due to a loading condition is a

function of the moment, a measure of this curvature is a

measurement of the moment.

This study used the relationship between object size

and image size formed by a concave or convex reflective

surface to determine the radius of curvature and the moment

at the point.

Overhanging simply supported beams of plexiglass with

concentrated loads at various positions \vere used to det:on­

strate the developed optical theory. Optical measurements

i.vere taken on the beams and the data plotted to obtain

moment diagrams and inflection points which were compared

with the results obtained by the use of sta tics. 1ne com­

parisons \vere within tolerable limits and shm·Ted that the

optical method is feasible.

ii

iii

A CI'JT 0\'JLEDGl<:ET·TT

The author ·~:~ishes to express his sincere than1w to

Dr. J. H. Senne, Jr., Chairman of the Civil Eneineering

Department, for introducing the subject and giving constant

encouragement and advice.

The author is also appreciative of the help given to

him by N. L. Brown, Instructor of Civil Engineering, in

assembling the instrumentation. The fine craftsmanship of

J. Smith is also appreciated.

TABLE OF CONTENTS

ABS'fiti\.CT •••••••••••••••••••••••••••••••••••••••••••

A CKN 0 ~oJL .EDGHENT •••••••••••••••••••••••••••••••••••••

LIST OF FI GURES ••••••••••••••••••••••••••••••••••••

LIST OF TABLES ••••••••••••••••••••• . •••••••••••••••

LIST OF SYHBOLS ••••••••••••••••••••••••••••••••••••

I. INTRODUCTION ••••••••••••••••••••••••••••••

II. REVIEvl OF LITE:HA1\JHE ••••••••••••••••••••••

III. DIS CUSS ION

3-1

3-2

3-3

General Theory ••••••••••••••••••••••

Derivation of General Relationship

for Location of Inflection Points •••

Derivation of General Relationships

Required to Dra1.v the N Diagram •••••• EI

3-4 Derivation of General Relationships

Required to Draw the Homent Diagram.

IV. DEHIVATION OF EQUATION FOR EXPEHHIENTAL

INSTHUl·IEUTATION ••••••••••••••••••••••••••

v. TEST APPA~~AT : ~ •••••••••••••••••••••••••••

VI. TES'l1 Pl\OCEIJ .... :J.\E •••••••••••••••••••••••••••

VII. DISClJSSIOl~ OF'I : ~~ :~S ~JL1.lS ••••••••••••••••••••

VIII. COI'·J CIJ TJS I 0 l~ S ••••••••••••••••••••••••••••••

IX. HECOI-.. il·lENDAT I ONS ••••••••••••••••••••••••••

APPENDICES

I • • • • • • • • • • • • . • • • . • • • • • . • • • • .. • .. • . • . • .. . . . . . . II ••............•..............•...•...•.•....•

iv

Page

ii

iii

vi

ix

X

1

2

4

8

9

9

17

21

26

?7

58

60

'I'ADLE OF COli'I'K!'i'S (continued.)

BIBLIOGHAPIIY. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

VI TA •• •••••••.•••••••••••••••••••••• 41 ••••••••••••••

v

Page

69

70

LIST OF li'IGUHES

Figure

1 • Concave curvatu.re ••••••••••••••••••••••••••••

2. Convex curvature •••••••••••••••••••••••••••••

3. Ellipsoidal segment of a beam ••••••••••••.•••

4. Elements of a cylindrical mirror •••••••••••••

5. Image formed of an object by a concave

mirror when sis less than f •••••••••••••••••

6. Image formed of an object by a convex

mirror • ....•.....•...••.......•..........••..

7. Formation of an inflection point •••••••••••••

8. Image formed of an object by a thin lens •••••

9. Images formed by instrumentation •••••••••••••

1 o. r··licrometer' lens and 0 bj ect ••••••••••••••••••

1 1 •

12.

13.

14.

15.

Image as seen through the micrometer •••••••••

Supporting frame and carriage ••••••••••••••••

Adjusting screws of the carriage •••••••••••••

Instrumentation for runs 1 '

2 and 3 ••••. •..••

Instrumentation for rlllls 4, 5 and 6 • •..••.•.•

16. Image measurement ••••••••••••••••••••••••••••

16(a). Distorted images •••••••••••••••••••••••••••••

17. Channel support for zero readings for

rlUls 4, 5 and 6 •••••••••••••••••••••••••••••••

18. y 0' 1 y 11 2 _..;;:.._..,..,...._ x 10 versus moment - run 1 •••••••••• yli

19. Location of the images formed by the lens

for flat, concave and convex beruas •••••••••••

vi

Page

11

11

12

13

14

1 5

16

19

20

23

23

24

25 29

30

31

32

33

4o

41

vj_i

LIST Uti' F IG!.JHES (continued)

FigLJ.re

20. Y~ - Y11 x 102 versus moment - run 2....... 42

y''

21. Yo- Y" x 102 versus moment- r t.m 3....... 43

y"

22. Loading condition, moment diagram and

elastic curve- run 4..................... 44 23. Loading condition, moment di <tgrarl and

ela stic c urve - rw1 5..................... 45 24. Loading condition, Dornent diagrau and

elastic cur ve - run 6. • • • • • • • • • • • • • . • • • . • • ~-6

25. Initial and loaded readings versus distance-

rur1 4.............. . . . . . . . . . . . . . . . . . . . . . . . lr9

26 . Experim(.;ntal and theoretical nonents versus

distance - rt.m 5.......................... 49

27. Initial and loaded readings vers us distnnce-

rur1 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

28 . Initial a nd loaded readings ver sus distance-

rtm 6.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

29. Expe:cir:1e.::1t a l and t heorc t j_ cal ElOJ:lent s v e;_--s us

dista.nce- rLUl 5..... ................. .... 54-30. Exps r j_u en. tal and t h eor e tj_cal mor:en ts ve r s us

dis tt1 ~:1ce - run 6.......................... 55

31 • Cantilevered beaJJl . • • • • • • • • • • • • • • • • • • • • • • • • 62

32. Def l ection versus time.................... 64

33. Loading conditions- cases 1, 2 C:l.J.'1d 3 •••.• 65

viJi

LIST OF FIG UHES (continued)

Figure Page

34. Deflection versus load- cases 1, 2 and

average................................... 67

35. Deflection versus load- case 3........... 68

ix

LIST OF TABLES

Table Page

I. Loaded Readings- Run 1................. 38

II. Loaded Readings- Run 2................. 38

III.

IV.

v. VI.

VII.

VIII.

IX.

x.

Loaded Readings- Run 3 •••••••••••••••••

Initial Readings -Runs 4, 5 and 6 ••••••

Loaded Readings - Huns 4, 5 and 6 •••••••

Experimental Homent Run 4 •••••••••••••

Experimental Homent- Run 5 ••••.•••••••• Experimental Noment RWl. 6 •••••••••••••

Creep Deflection ••••••••••••••••••••••••

Deflection of Cantilever Beams ••••••••••

39

47

48

51

52

53 63

66

X

LIST OF SYHBOLS

The symbols are defined '\·!here they first occur in the

text and are listed here in alphabetical orc,er for convenience.

c Hatio of ll.s to s 2 E Modulus of elasticity (gm/cm )

F Focal point

f Focal distance (em)

I Homent of inertia (cm4 )

L Span length (em)

H Homent (gm-cm)

m Magnification of the beam

m_ l·lagnification of the lens L

P Concentrated load (gm)

R Radius of curvature of the elastic curve (em)

Rm Radius of curvature of the mirror (em)

s Distance from the object to the beam (em)

s 1 Distance from the beam to the image it formed (em)

s1 Distance from the lens to the image formed by the

beam (em)

sL Distance from the lens to the image it formed (en )

As Distance from the object to the lens (em)

w Uniform load (gm/cm)

y Object size (em)

y 1 Size of the image of the beam (em)

y 11 Size of the image of the lens for any load (YJicro­

meter divisions)

xi

LIST OF S:Ll-lBOLS (continued)

y11 Size of the image of the lens vlhen the moment is 0

zero (micrometer divisions)

z Distance from the top of the 1Jeam to the neutral

axis (em)

a Deflection (em)

1

I. INTHODUCTION

There are many methods of analyzing structures available

to the engineer. One of the most common methods is by the

use of equations derived from statics and mechanics of ma­

terials '"hich are especially convenient '\vhen the structure

is statically determinate. The solution of these equations

for highly statically indeterminate structures becomes diffi­

cult and in some cases almost impossible.

A second raethod of analysis is by the use of models

which, one might sa:;~, is a field of its mvn. There are

many types of models and many approaches which can be used

in model analysis. In general, the use of models is of par­

ticular value ,.,hen the structure is either highly indeter­

minate or has many variables.

Since the curvature of a beara due to a loading system

is a function of the moment, a determination of the curva­

ture is a means of determining the moment in the beam.

The shape of the image of an object formed by a reflective

surface is a fLmction of the shape (curvature) of the su1·face.

Thus, the relationship of thG shape of the image :Ls a function

of the moment if the curvature is due to the applicatj_on or

a moment to a reflective beam.

This study -vms made to determine the relationship bet\·Jeen

the moment and the shape of the imac; e and to nH.'1.ke an experi­

mental investigation to check the feasibility of this re­

lationship.

2

Optics are used in conjw1ction ·vri th mechanical levers

as strain ga ges. These gages are very precise and sensitive.

The general procedure using mechanical-optical stra in gE~t;es

is to secure a reflect i ve pivoted member or lever in the

gage. As the structure is deformed, t he lever i s r o tateC

due to a change bet1veen the gaging points causing the attached

reflec t ive surface to rota te. A collimated beau of li t;:ht is

reflected from the gage to a scale; the motion of the beam

indicates t he def ormation of the gage length. To increa s e

the magnification of the movement, a telescope can be used(1).

'!The only p urely optical strain gages are of the in­

terferometer type" (1). This type of gage uses t\w parallel

flat ontica l plates f asten ed sec u.rely to t h e structure with ­

out ,,rarping. A monochromatic J.ight beam is passeC. through

a half-silvered mirr or normal to t he o:ptica l plat es. The

interference fringes are observed with a telescope through

the back side o f the mirror.

As the deformation of the structure occurs, t he inter-

f erence rinr;s are counted <: s the:l Jr;ove pc:::.st a n es t a bl i shed

point on one of the optica l f lats. dith each move~ent of

one compl ete f ringe past t he referenc e malce, the op ti cal

f l a ts have moved one-half of one v.J.~;_ v e length o.f t h e l i :_,ht

used. With proper adjustment,tlds ga ge can be very sensitive.

"The use of optical fla ts, or t:!:1e i n t e r fermome t cr as a strain

gage, is the closest to t he absolut e meas ure1!Ien t, of stra :i.n

that we are able to attain,(1).

A..t1.other method of measuring deformations of a nodel

is the Beggs Deformeter \'lhich uses a micrometer r.nicroscope

to r;leasurc the movement at one point due to a l:nmvn change

in the conditions at an.other point. The la'\-T of reciprocal

deflections can be applied to analyze the model (2).

One useful method of solving certain structural

problems is the use of ::1embrane analogies. This method

3

uses the contours of the shape of a thin membrane under

pressure on one side to be analogous to a certain structural

phenomenon. One method of measuring these contours is of

interest in the follm.;ing study. The method is an application

of optical and geometrical principles. A light source is

located at a known elevation above the memb:bane boun.6ary

or some convenient reference plane. The light is po~;i tioned

so that the reflection fror:1 the mem"L:rane is observed along

a line that intersects the vertical line through the light

source. By measuring certain distances and angles, the

slope of the membrane at the point of reflection and the

elevation of the membrane at that point may be cor!lputed ( 1).

III. DISCUSSION

3-1 General Theory

The curvature at any point in a beam is a function of

the moment, the modulus of elasticity, a~d the moment of

inertia of the cross-section. For small deflections, this

well-knm·m relationship is given by the follm·ring:

1 = 1·1 'R EY

i.vhere

R = Radius of curvatu.re of the beam,

H = Homent,

E = Nodulus of elasticity, and

I = Homent of inertia.

(Eq. 1)

The loading and elastic curve for the longitudinal

dimension of t1.vo beams are shm·m in Figure 1 and Figure 2.

Using a beam that has a reflective surface and taking

a small element of this surface as shovm in Figure 3, an

ellipsoidial mirror is formed. This mirror has a radius

1\n for a segment along the longitudinal axis ·of the beam

and a radius Rro1 for the segment along the transverse axis.

When the beam is loaded so that no torsion is obtained

and the deflection is small, the curvature of the segnent

along the transverse axis approaches zero or the radius

approaches infinity.

This. asswnption allovrs the surface element to be con-

sidered one of cylindrical shape as shown by Figure 4,

'WiLth the radius of the mil"rco.r· equal to the radius o.f' the

top surfa ce of the beam and the height of the cylj_nder equal

to the \ITid th of the beam. Although the c urv<:?.t ~u·e o i' the

beam is constant only if E, I, and H are constant, the

curvature for a small ele::nent cc=> .... n. be assurned to be constant.

F'igure 4- shm·rs both concave curvature and convex: curva-

ture.

Figures 5 and 6 show the construction for finding the

image formed by a cylindrical concave mirror and convex

mirror, respectively. The procedure for locating lines 1,

2, and . 3 is e~)lained in Appendix I.

The concave mirror represented is of la:q;e radius ~·

The focal distance (f) is given by the follmving:

f = -Rm 2

(Eq. 2)

5

Since 1\n approaches infinity as the curv<,~ ture approaches

zero, the focal distance also approaches infinity as the

curvature approaches zero. When the object y is located

at some distance s between the focal point and the surface

of the mirror, the image yr \dll be at some distance s r on

the opposite side of the surface.

The relation bet,, ... een s, s 1 , and I\n is given by the

follmving:

1 - 1 = ~ (Eq. 3) s s• -m.

The magnification m is given by the follm·ring:

ID : zt: ~I y s (Eq. 4)

6

\.Jl1en the following sign convention commonly used by the

optical industry is adopted, the relationship for the focal

distance, radius and the mat;nification can be a~)plied to

convex mirrors C'~S·O ( 3).

1. DravJ all diagrams 1.vi th the incident light traveling

from the left to the right.

2. Object distances s are positive if the object point

lies to the left.

3. Image distances s' are positive if the image lies

to the right of the surface.

4. Radii of curvature 1\n are positive if the center

of curvature lies to the riBht of the surface.

5. Transverse dimensions y and y 1 are positive if

above the axis.

Substituting Eq. 4 into Eq. 5 and rearranging, the

following is obtained:

1 = =1 (y' - ~l I\n 2s y' ( Eq. 5)

Since H is the radius of curvature of the elastic curve

of the bear1 and the elastic curve is located some distance z

below the top surface of the bean (Figure 4-), H is equal

to the radius of the top s urface plus z for a concave surfac e.

Since the radius of the top surf ace is equal to 1:\n' the radius

of the mirror, R is equal to R + z. m

Substituting Rm + z for R in Eq. 1 and rearranging,

the follo'\lling is obtained:

1 R m

= =_;.;M.......,..., EI - zH

(Eq.6)

7

EOt:lat; :..n.::..-· T:~q~ • r; ~and 6 rr·i VeS t11e follO''. ng 1 t. . . -- ... _ s - ~ ./ ~ 6~- · ,·.l . re a ~onsrnp;

1 : +1 (y - y I) : J.vl TI 2s y' EI - zH m

(Eq. 7)

Solving Eq. 7 for H, the follouing is obtained:

H = +1 (y- z')EI 1 (Eq. 8) 2s yt . 1-2CY-y1 )

2s y

Since z is a small quantity for raost cases 2s and y - y' Y'

is small, the product \vill be very small. vJi th this assumption,

Eq. 8, reduces to the follm·Ting:

H = + EI ( y - y I )

2s . y1 --(Eq. 9)

From the adopted sign convention and l'igure 5, y, y 1 ,

s an.d s 1 are positive for a concave surface. Also, y' is

larger than y. Thus, for a concave surface the raoment 'Hill

be a negatj_v e ·Hhich establishes the sign convention of com-

pression in the top ·fibers as a result of negative monent.

This sign convention for moment V-Till be used in the remainder

o r the text.

\men Fir,ure 6 is considered, the surface is convex. ,,

The radius of curvature of the convex reflective s u.rf<'l.cc is

H + z (Figure 4). 11hen :;_:~ - z is substituted into 3q. 1

aJ.'ld similar steps are ta!;:en as for conc c.vc su.rf'e.ces, t r.te

follovTing i s obta:Lned:

M = +EI (y- y') 1 y I T1-+-:--z~( ~-r ---y-=-r )

2s y'

(Eq. 10)

\vith the assumption that z (y- y 1 ) is very small, 2s yi

.E:q. 9 is obtained for convex, as well as concave s urfaces.

For convex surfaces, y, y', sands' are all positive; but

Figure 6 diagramrn.atically shmvs tha t y' will be less than

8

y, wLich gives the quantity (y- y') a positive sign and

makes the moment obtained from Eq. 9 positive. Thus, the

moment required to obtain convex curvature is positive, vrhich

agrees '\-Ii th t he adopted sign convention that compression in

the top is due to negative moment. Conversely, tension in

the top fibers is a result of positive moment.

Eq. 9 shm,rs that if E, I, s, y and y' are knovm or

measured, the moment at that point can be computed.

3-2 Derivation of General Relationship f or Location of

Inflection Points

Figure 7 illustrates a general loading condition the. t

will produce a..'l inflection point. An infleetion point is

the point at vrhich convex curva tm~e changes to conc .::lve

curvature or vice versa (4). Using the fi rst ca:.::e as sho\.vn

in Figure 7, to the left of the inflection point (convex

curvature) the moment is seen to be positive and to the

right of t he inflection point the moment is negative , or

to the left y' is less than y and to the right y' is grea ter

than y. '£he moment is zero at the inf l e cLLo.n poi:lt.

When Eq. 9 is set equal to zero, it is seen t ha t the

only quantity that can ph,ysically be zero is y - y'. Hence,

y = y 1 at t he inflection point.

Therefore, the inflection point can be found by locating

the point on the beam where y = y 1 •

3-3 Derivation oi' General Relationships Heoui:ced to Draw

the H Diagram EI

One convenient method of f inding the deflecti on and

the a'1.gular rot ~ ;. tion is to apply the principles of moment

area. It is desirable in this application to obtain the

l .. I diagram. Er

Referring to Eq. 9 and rearranging, N is given by the EY

follovTing:

H = -1 (y' - y) - - - I EI 2s y . . (Eq. 11)

9

When the object distance is held constant and the above

equation is r ci:lri tten, the f ollovJing i s obtained~

H = K1 (y' - y) liT yl

\vhere K1 = -1 2s

(Eq. 12)

Since H is proporti onal to (y' - y_), a plot of y' - y EI yf yf'

versus distance gives a diagr am tha t is proportional to the

H diagram. EI

3-4 Derivati on of the General Relationship Heg,uiyed to

Draw the Homent Diagram

From Eq. 9 the moment at any point is r; i ven as a ft:mction

of E, I, y', y and s. I f t hese are measured and the f1JJ1ction

plotted versus t he dis t a.nce, the moment diagra:rJ is obtained.

For a beam of constant I and E, Eq. 9 can be re\Iri tten

in the f ollowing f orm:

10

H = K2 (1) (;z I - :z) s yf (Eq. 13)

vJhere K2 = -EI ~

If the object is moved parallel to the longitudinal

axis of the beam, s vrill remain constant for small deflections.

Hev1ri ting the above equation, the f ollo\'Ting is obtai ned:

M = K3 ( y t . i y) y (Eq. 14)

Thus, a plot of y' - y versus distance will f~~ ive a yi

diagram whose ordinates are proportional to the moment dia-

gram.

Loading condition

Elastic curve

Loacling condition

Elastic curve:

11

figure 1.. Concave curvature

Figure 2. •. Convex curvature:

12

Figure 3· Ellipsoidal segment of a beam

I· b ~ A-A -

z t

Concave reflective: surface;

Convex refle:cti:ve. surface

0

Figure ~. Elements of a cylindrical rn.irJJor

13

( 2)

Reflective ~ surface

-- --- -~.:;::::.. ,.._ - - - -=" ::::--f----- :_-- ---?- =:::>1-- I

I

~~ ~ -----11·-----c

I I

• I

R

Object (y) I I

t I . 1 •• J ~-~'~

~ f ~

,.,,

Image (y')

Figure 5. Image formed of an object by a concave mirror when s is less than f ~

+

Reflective surface

, ) .. -=-=-,.. =-------· ~ E t

--Object (y)

7' I

~ s ~--Y

- -- -- -==~-=.---=::--::_-c F - --Image (y')

f

R J I

Figure 6. Image formed of an object by a convex mirror _. V\

Loading condition

Homent diagram (gm-cm)

Elastic curve

lt::p=1===;=~ _r_=· =========:::-,

RL t- RR t a .1.. b .j. c ~

Figure 7. Formation of an inflection p~int

16

17

IV. DEHIVA'I'ION OF EQ -. JA'.U01'·I FOH EXPEHii.li.!:N'l'Ai. INSThUtiEN'l'ATION

Consider the beam to be a segraent of a cylindr:ical

mirror '\vi th the magnification c;iven by the follmdng:

y I S I

m - - = y s (Eq. 4)

The image formed by the mirror is the object of the

lens. Figure 8 shovrs the formation of an image by a thin

lens. The procedure for the locating lines 1, 2 and 3 is

explained in Appendix I. The magnification of the lens is

where

~ = :Hagnification of the lens,

sL = Image distance of the lens,

s1 = Object diste.nce of the lens,

yu = Image size of the lens, and

y' = Image size of the mirror.

(Eq. 1 5)

Figure 9 shows that s1 = s +As + s 1 = (1 + c + m)s.

\>!hen this is substituted into Eq. 15, the follm1ing is

obtained:

Y'

-si m = c:

1 -c-c-+ 1 + m") s

Y'' (Eq. 16)

\Vhen ym is substitu.ted into Eq. 16 for y' ru1.d rearranc;ed,

the following is obtained:

-(c + 1) m = st + 1

yus (Eq. 17)

Assuming that H = R and substituting Eq. 17 into m

-~ =! (1 - ~), rearranging ru1d equating the resultan t to

H the following is obtained: - ' EI

1 = -1 f1 + H 2s L 1 . {U.Z + J =

c + 1 · SY11 .YJ (Eq. 18)

In order to obtain a \vorkable equation, consider Eq. 18

\vhen H = 0. It can be seen that ( 1 + ( 1 ) (§1..:1. + 1 )) c + 1 SoY'~

must be equal to zero.

vrhere

s0 = Object distance \·Then N = o,

y~ = Hicrometer reading when f.l = 0.

Solving f or y and substituting into Eq. 18 and simpli-

fing, the f ollmring is obtained:

£L = 1 = 1 <c + 2) ~o <to) -11 EI R 2s c + 1 [ syu J (Eq. 19)

If the object is moved parallel to the beam and the

deflection is small compared to s0 , the follO\ving equation

is obtained:

H = 1 ( c + 2) ( y" 1 ) EI 2s c+1 ~-

Solving for the moment,

H = K3 (yu - y") 9

where

= 1 (c + 2)(EI) 2s c + 1

(Eq. 20)

( Eq. 21)

18

Object (y)

s

I ~hin lens I '\ '"Y-::: · t I d >/ '-...._ -....._ F 1- z:_~ \ - '*......_. I

~ 'IT;' - ---~-- · '.......... I Image Cy•J r -- --····

---~~, -­~ . ---::---

+-----f _____ ....c,:.l~r-----~ ~' --~~ s'

~r~ ~-'1

Figure 8. Ima.ge formed of an object by a thin lens ~

'-0

Image formed by the lens

Lens

1

j_ ----~

Object (light source)

Reflective

s

surface of~------------~----------~~--­the beam

Image formed by the beam

Figure 9. Images formed by ins trumenta:I;ion

20

V. TES'l' APPAHATUS

Figure 10 shows a detail of the micrometer, lens and

the object used in this study. The micror:1eter is the

measuring device used in the Beggs Deformeter. It can

be read directly to the nearest division and vras estimated

to the nearest .5 division for the first group and to the

nearest .2 division for the last group of ru..'1.s. The units

of divisions were substituted directly into Eq. 20 as is

explained in the Test Procedure of this study.

The lens used was the object lens of a solar attach­

ment to a transit.

The object used was a 224 light bulb which had its

energy supplied by flashlight batteries. The design pro­

duced by this obj ect is shm.m in Figure 11.

The frame and the supporting carriage for the microme­

ter are shm.·m in Figure 12. 'l'he frame and carriage \vere

adapted from photogrammetry equipment.

The frame \vas equipped with adjusting scre\vS -vrhich

allowed vertical adjustment and leveling of the track of

the carriage. Shims were placed bet\veen the track and the

frame to establish a vertical plane for the rollers of the

carriage.

21

The carriage \vas equipped with several adjusting scre\·Js

that facilitated the opera tions considerably. These screws

are labeled in Ii''igure 13 and are descrj_ bed as follov.rs.

22

Sere\-! one allO\ved the carriage to be moved parallel

to the longi tudional axis of the specimen and in a horizont:ll

plane.

Scre\v two allowed vertical adjustment of the measuring

system.

Screw three allm.ved rotation about an axis parallel to

the transverse axis of the specimen.

Screw four allowed the rotation about an axis parallel

to t he longitudional axis of the specimen.

The specimens used in this stu6.y were made of plexi­

glass. The bottoms of each were coated with a black dye to

improve the image formed on the top surface of the specimen

and to eliminate the image formed by the bottom surface of

the specimen.

The specimen used in rtlll 1 was 1/8" X .4n X 15" and

in run 2 was 1/8" X .ti-n X 19. 6" and in rtlll 3 was 1/411 X

.4" X 15".

·The specimen for the second gro up \vas 1/4" x .411 x

14.211 • The EI of this specimen \'las found as explained in

Appendix II.

The minimurn \•Tidth of the specimens ·vms contr ollec3. by

the Hid th of the object. It \·TaS found tha.t the i nc:,ge vJaS

distorted "~l'lhen it fell '"i thin approximately 1/811 of t he

edges of the s pecimen.

23

gure 10. M!cro ter, lens and object

; jd

F1 u.r 11. Im s s n through e micrometer

24

.·' Figur 12. Supporting ._rr , carri

25

,u ting crw .·of . th carriag

. VI. TEST PROCEDUHE

The tests ;..;ere made in ti•TO general groups, the first

group consisted of three rw1s and vras to shmv the general

yu _ yu proportionality of the moment at a point to ~0--­

y" •

The second group vlas a set of three runs and '!,vas to

26

sho\oT the relationship of Eq. 21 along the beam and to locate

the inflection points.

Figure 14 shows the general instrumentation of the fj_rst

three runs and Figure 15 shO\·ls the general ins t rumentation

for the second group of runs.

A baseline was established on the table and the fra":le

was adjusted to allO\v the carriage to move horizontally

and parallel to the baseline.

The supports for the first three runs i·lere placed on

lines that were perpendicular to the baseline and the

speci.rnen positioned on the supports so that they 'i•rere para-

llel to the baseline.

The support frame for the second group \.,ras placed so

that the outside angle \vas parallel to the baseline, leveled

and \•Teighted to eliminate r:10ving (Figure 1 :·). The knife edge

supports \vere positioned on the support frane so the.t the

edges ·Here perpendicula r to tho outside angle 'l,·lhich also

made them perpendicular to the motion oi' the carria&:e. The

edges \vere leveled in the earlier rtms by shims placed bet1r1een

the \vooden blocks and the alwninwn support frame an d by set

screws in later runs.

27

A line parallel to the baseline vras established on the

knife edge supports to allovl the positioning of the specimen

so that it had proper rela tionsr1ip to the sup ports and the

carriage.

By adjustment of the control screi·rs of the carria ge,

the image viaS positioned in approxima tely the center of the

micrometer. In order to locate the object in relation to

the specimen, a moveable pointer ,,.ras placed on a meter stick.

Since the measuring system vms focused on the ir::age forraed

by the beam which vias approxima tely t'.vice the distance from

the object to the beam, the pointer \vas not in focus, but a

shadov-r through the image was used to determine the proper

positioning. The left reaction was used as the reference

point for the ::;easurements of each of tbe rWls.

The general procedure of taJ::ing a reading vras to move

the cross hair of the micrometer by turning the micrometer

screw until it v:as tangent to the bisector of the outside

ring (li'igure 16). A series of three settings was tal'Cen

at this position and then the cross hair '\vas :·Jovec'. to t he

opposite side of t he i ma ge and a s eries of three readings

tal-dng at this position (Figure 16 ). The d ifference betl·reen

the averages of these readings \vas a measure of the line

AA' of Figure 16 . The diameter of the i uag e along the lon­

gitudinal axis via~ (1.1 Heading), but the relationship used

in this study was a ra t io of the dianeter of the image Hhen

the beam Has loaded and ;_mloaded so that th~ cancelled.

Also, the tmi ts cancelled so that the micrometer read:i.ngs

were used directly into' Eq. 20.

The loaded readings ·v1ere taken in the same general order

of locating the image in the proper position and then setting

the cross hair on the right and the left sides of the imc.ce,

respectively, as shown in Figm'e 16. Figure 16(a) shows an

enlargement of the distorted images as seen through the

micrometer.

The initial reading for Run 1 ·vms taken ·~:~i th the dead

load acting. Loaded readings \•Tere tal~en at the point in

question for different load increments.

The initial reading for Bun 2 was taken on a flat plate

of plexiglass of the same t hickness as the specimen. The

moment of inertia \vas larger at the midspan, lx1.t the curva­

ture of the plate vras neglected. Loaded readint;s '.·rere ta1:en

at the point in question for different load incrernonts.

The initial reading for Hun 3 vias taken on a plate of

plexiglass of the same thickness a s the specimen. The

curvature of the plate \vas neglected in the computation.

Loaded readings \·Tere ta1ren at the point in question for

different load increments.

The initial readings for the second group \vere tal:en

on the specimen in questj_on vrhile resting in a metal chc: .. nnel

as sho~1 in Figure 17. The curvatur e due to the dead load

of the channel and the specimex~ v.rn. s negl ected in the con-

putations. The carriage Has moved hori.zontally and parallel

to the longi t u.dinal axis of the specimen and initial rea1..iings

taken at intervals along the specimens. The s pe c imens \-!ere

then loaded, and the loaded readings vrere taken at intervals

along the beam.

29

F1 ur 11t-. I:m.strum .n·t'at"16n ;or runs 1, 2 and 3

30

F1 ur 15. I tr t ti

Outside ~'---...... _

ring

Inner core--

A

A

(3}

Zero curvature

Concave, curvature

2)

Convex curvature

(1) Centering position

(2.) Neasuring right side

(3) Measuring lei't s:tda

Figure 16. Image measurement

31

~~~ \

•'

Image formed by . dead load curvature

tii!aa~ . ro;rmed by >:, c~~~x durvat~e

:1 ,,,

Image formed by . con:c·ave curvature

Figure 16(a). Distorted images

Figur 1?.

f

Channel support for zero readings for runs 4, ; and 6

33

VII. DISCUSSION 01? RESULTS

Tables 1 , 2 at'ld 3 shoi1 the data obtained for HLms 1,

2 and 3 respectively. :i?igure 18 sho1,rs the loading condition

Yn _ ..,rtl and a plot of o J versus moment for Hun 1. 1'he figure

yl'

shol<'lS that for a ra..11.ge up to vThere t6 Y11 is approximately yfi

-7 x 102 the plot is a straight line but after this range

it is curved. 1'his curvature of the line is due to the

image not falling on the plane of tJ:1e cross hairs.

Figure 10 shm.vs the forn1a tion of the ir:1af:e by tl1e beam

and the lens of the mi·:~rometer for a plc,..'1e, concave and

convex curved beam. Each of these cases shm·rs the position

of the object vrith respect to the ce.nter :Line of the bee.Ll

and 1rL th respect to the lens to 'be const<:-.nt. 3ach o:f' the

cases shm·IS the distance from the lens to the screen of

the micrometer to be a const.::l.nt.

Only the imae;e of the pl<:me bear.'l 1 s image 1vas actually

located on the plane oi' the micrometer screen. In order

to keep the j_nstrLU}lentation as sinple as pos~:ible, no

adjustment \..ras provided to readjust for this condi tj_on.

Since it vrould have been necessary to oal:;:e an adjustmer:t

on each side of the imc~ge due to the radius of curv<.;,ture

changing along the segm.ent of the beam, it \vas decided to

lceep the change in curvature small so that the effect of

the image of the lens not being exactly on the plane of

the micrometer screen vrould not be significant.

Figure 19 aloo shm·rs the object to be located synmetrical

35

vri th the center line of the lens. For Huns 1 through 6 the

object \vas located slightly off center to allovr the lens to

function. The amount of the object being off center vras

very small vlhen compared with the object distance of the

lens and 1-1as neglected in this study.

Figure 18 gives a larger ~ - Y'' value for a moment yt• -

than would be expected if the straight line portion is

extended. This difference agrees \•li th the case for conc :_.ve

curvature as shown by Figure 19.

Figure 20 shows the loading condition and the plot of

yg - 7' versus moment for Run 2. This type loading condition yh

causes convex curvature when the concentrated load becomes

large enough to compensate for the dead load. Convex

curvature is a result of tension in the top fibers or

positive moment. The plot is a straight line u~' to a

point, at "IIThich time it becomes curved. If the straight

line portion is extended and the YB - Y" value compared yfl

for a moment larger than 60 gm-cm, the straight line

t · ·11 · 1 1 _.oor the Y'o' - Y" than ex enslon Wl glve a arger va ue , y"

plotted data shm-1s. This is due to the image not falling

on the plane of the cross hairs, as explained above.

Figure 19 shO\vS that for convex curvature the yg - y' ytt

will be smaller than it would be if the plane of the image

and the cr~ss hairs are at the same position.

Figure 21 shovlS the loading c ondition a..lJ.d the -plot of

You - .,.u t "' ~ · 3 .Y versus momen i or ..:•un • The plot of Y3 - Y"

:t'

versus moment plotted a straight line in this ca se since

36

the curvature of the specimen ·vias small enough to elimina te

the image being out of focus.

The loading conditions, moment diagrams and elastic

curves are shovrn in Figures 22, 23 and 24 for 1-\uns lt, 5 and

6, respectively.

Table 4 shmvs the initial readings for Huns 4, 5 and

6. Table 5 shows the loaded readings for Runs 4, 5 and 6.

Figures 25, 27 and 28 shows a plot of the initial

readings (yg) and the loaded readings (y") versus distc:~nce

for nuns 4, 5 and 6, respectively.

Tables 6, 7 and 8 shovJ yg - yn , y'' and the experinen tal

moment computed from Eq. 20 for Runs 4, 5 and 6, respectively.

Figure 26 sho"'.vs a plot of the experinental monent and

the theoretical moment f or .i:iun 4. It can be seen that the

curves a cree Vli thin experimental accuracy along the full

length of the s pecimen.

F'igure 29 shovTS a plot of the experimen tal and the

theoretical moment s vers :J.s distru-1ce for itt..m 5. The moments

agree \d thin experir1ental accuracy along the length of the

beams.

Figure 30 sho•ds a plot of the experimental and the

theoretical moments versus distance for Hun 6 . The theo-

retical moment is seen to be considerably larger for the

37

specimen frora appro:;.:ir:1a tely -7cm to +5en. j_'his derivation

is due prin:::.rily to the inage not being on t h e scuJe plane

as the :::Tos s hairs in this region. '.J..'he curves acree 1'l'i th

the ·:·n·ed ·ictio:l o-r Fi ~,·~ rr·e 10 ·ror convex :··q rva•- ,, ,,e and ,.,..: t h • --~ _., .}_ - (~; '- ·- / ..;,., ... ._, 'v ·• • V '-'•• •• ,.1 • . ~ j _ •

the results of Run 2 ;:,:.s shmm in Figure 20.

1,he t vJO curves of l."igure 30 a gree \-Ti thin expcriuental

accuracy for the re:·nainder of the length •

.According to the developed theory, the inflection -ooints

should be located 1;lhere y8 ; yn. This point is the inter-

section of the plots of y~ a..rJ.d y 11 versus distance.

The distances from ths left support to the experimentally

determined inflectj_on points \vere 6.4, 3.2 and 3. 8 em for

Runs 4, 5 and 6, respectively.

The theoretic2.l distanc e from the left suppol..,t to the

inflection point as determined by stat:l.cs is 6.5, 3.05 and

8.8 em for .J:\uns 4, 5 and 6, respectively.

TABLE I.

Loaded Readings

Load Noment Y" (gm) (gm-cm) (Div)

11-.2 -12.7

8.4 -25.6

12.6 -38 •. 4

16 •. 8 -51 •. 4

21.0 -64.0

* Y" = -11 2. 5 Di v 0

-115.0

-117 • .o

-119.0

-121.0

-127.0

TABLE II.

- B.tm 1

y" - )''* ~Div

+ .2.5

+ 4.5

+ 6.5

+ 8.5

+14.5

Loaded Readings - Run 2

Load (gm)

Dead Load

4.2

8.2

12.6

16.8

20.7

24.9

29.1

1:-loment (gm-cm)

- 25.6

~.3

+ 11.0

+ 38.4

+ 59.6

+105.0

+126.8

+148.0

** y" = -101.1 Div 0

yu (Div)

-101.5

- -100.1

- 99.6

- 97.2

- 96.2

95.5

- 95.3

- 95.5

y'' - Y" ** ~Div)

+0.4

-1.0

-1.5

-3.9

-4.9

-5.6

-5.8

-5.6

38

Y8 II 2 - y X ·10 yu

- 2.2

-- 3.8

- 5.5

- 7.0

-11.4

Y3 - yu 2 __.;;... __ X 10 yn

-0.4

+1.0

+1.5

+l.J-.o

+6.1

+5 0 . /

TABLE III.

Loaded Headings

Load Homent (gm)

8.4

16.8

25 .• 2

64.3

80.8

97.3

127.8

144.3

160.8

190.8

207.3

223.8

*Y11 = -101.7 0

(gm-cm)

- 30.7

- 61.5

- 91.9

-233.5

-294.5

-355.5

-465.0

-526.0

-?86.0

-695.0

-756.0

-815.0

Y'' (Div)

-102.4

-102.6

-102.5

-104.0

-1o4.1

-104.3

-105.8

-105.8

-106.4

-107.3

-107.5

-108.5

39

- ltWl 3

~· - yu* Yt] · - Y"x 10 2

~Div) yil

+0.7 -0.7

+0.9 -0.9

+0.8 -0.8

+2:.3 -2.2

+2.4 -2.3

+2.6 -2.5

+4 •. 1 -3.9

+4.1 -3.9

+4.7 -4.4

+5.6 -5.2

+5.8 -5.4

+6.8 -6.3

4o

Y"-Y" 0 x102

y t I

-10

-15 ~--------'-------'1----...J -100 -50 -25 0

Moment (gm-cm)

Figure . 18. ~1x1o2 versu~ moment - run 1

B

0~11 \ A '\

I I

\ \

I I \

: \

A'

I I

. I ,.

Pl a;:-~c of cross hairs

Lens Object

1.·

I I

---~~~~-~t-\~1-+~ Longitudinal 1 ! \ I axis of bean;. I I

:I \ I II ~ I

I I II I I \ II II

I I ~ 1 ~: \ II L __ j I ,I \

L I _j

Convex image

Flat image

Concave image

L -- ~ L --

Figure 19. Location of the images formeci by -..;:ne l t;..:lS for flat, concave and convex beams

~- 1

n . ....u 2 Y 0 ":' -1 x1 0 · . yu

+8

+6

+2-

'0

-2 -50-·

42

5.08 ·I r 12.24~ 1 A. f-t-5.08 ' 1 I

7.88 i ; 8 :-- ·-11.1 I em

...

v ~

,j .

..

v. I

I · 0 1;50 175

Moment ~gm-cm)

ent - run Z

yn-yu 0 :x:102 . yn

p

A.-- t

0~------~------~--------~------~----

-8--------._------~------~~------~--~ -900 · -700 -$00 -300 -100 0

· Homent (gm-cm)

li'igura a.t. Y&-Y11 .JC1o2 VGrsus moment . - run 3 :y" .

Loading condition

Noment diagram (gm-cm)

Elasti.c curve

P1 = 49r87gm

~~

~.08 ..L

15.24

Inflection poin

~--__......,~- 6.5cm

Figure 22.4 Loading condition, moment di as;ram ar.cl eJ.astic curve - run 4

44

Loading condition

Homent diagram (gm-cm)

Elastic curve

P1 = lr9.87gm

-775gmlcm

p2 = j 3lt. 09 gr.:

l ~ ~:

Inflection iPoint

Figure ~3'• . Loading condition' 'momQnt· OJ.agram d l tic curve - run 5 ·· · ··· ~ ·· . ·

45'

46

p1 p 2

:::: 150 5~---:-• • /,_,. .. J

Loading ····· ·-

"' r·49gm

condition -·

t

Noment diagram (gm-cm)

Elastic curve

f:igur.a: 24.

•

l RL I I Rn

I

16cm I ·15cm I 5cm I I

36cm

Inflec~ion point

8.8cm

Loading condition, moment uia6~~n wlQ clastic curve. - run 6

47

TA1JLE IV.

Initial Readings - Runs 4, 5 and 6

Run Distance Y8 (em) ( Oiv)

4 - 2.5 -105-7

4 6.0 -106.0

4 12.0 -105.0

4 15.5 -105.0

4 22.0 -105.2

4 31.0 - ~106.0

5 - 2.5 -106.3

5 6.0 -106.8

5 12.0 -106.0

5 115-5 -106.0

5 22.0 -106.0

5 31.0 -106.0

6 -15.0 -117.2

6 -10.0 -117.0

6 - 5·0 -116.9

6 0.0 -116.0

6 5.0 -116.0

6 10.0 -116.3

6 15.0 -116.3

6 19.0 -116.3

6 20.0 -117·5

48

TABLE V.

Loaded Readings Runs 4 .-' and 6 - ' :..>

Hun 4 Hun 5 RLm 6

Distance Y" Distance y" Distance ~rll

(em) (Div) (em) (Div) (em) (Div)

-4.6 -105.0 _LI-. 7 -107.1 -15.0 -116.2

o.o -103.5 -2.0 -105.5 . -12.5 -115.7

2.0 -104.6 o.o -104.6 -10.0 -111.1 .• 8

4.0 -105.4 2.0 -105.5 - 7.5 -11Y-.O

6.0 -105.8 4.0 -107.0 - 5.0 -113.8

8.0 -105. 9 6.0 -109.2 - 2.5 -113.0

10.0 -106.7 8.0 -111.3 o.o -112.5.

12.0 -107.5 10.0 -112.3 2.5 -113.2

1 L1-.0 -107.5 12.0 -11 1~· .3 5.0 -113.9

16.0 -10'7.9 1 Lr. 0 -117.5 '7. 5 -115.5

18.0 -107.2 16.0 -116.2 10.0 -116.11-

20.0 -107.4 18.0 -114.2 10.5 -116.7

22.0 -106.6 20.0 -112. 8 12.5 -117.7

2lt.O -107.1 22.0 -110.6 14.0 -119.1

26.0 -106.1 21.1-.0 -109.8 15.0 -11 9.6

28.0 -106.6 26.0 -108.6 17.5 -118.1

30.0 -106.2 28.0 -107.3 19.0 -117.2

30.0 -105.5

-100

-102

Y" (Div) 6 -10

+ •

..,.,

?

Yo" yn ·

-->./1\ v · ~ ~

' -~ I ! I !

l

l

-'- -L ~

fo"" -~ ----- ...._ T

-108

~ ~

~ L?v. ~

~ .. , .

- 110-10 -5 0 5 10 15 20 25 . 30 35

Distance ( CJ:!l)~

Figure 25. Initial and loaded readings vers~s distance - run 4

Mo (gm-cm) :

600

4oo

200

· o

-200

• • • ExperimentaL ---- Theoretical

j \. v ·~ r\.. ~

~ ~ ./

v ~ v-r-"

. -4<>~10 -5 0 5 10 1$ 20 25 30 35

Distance (em) · .. ~

. Figiire 26 •. Experimental and· theoretical mo:4ent_s ...........,,,..., distance - run lt · ·~

-96

-100

-104

... yu -108 (Div}

-112

-116

-120

• +

Y'' yt6

I ~ 7

. •

.

"- ' "<; r- •

"'· ~ y

~

~ I

I I l i ---- . ---

I· I I

\ I - ··· -- ---·

_!. _!_ ~ • L :+

L!_ v

L v

II"

-10 -5 0 5 10 15 20 2.5 30 35

· Figura 27.

-110

-112

-114

yll -116 (Div)

-118

-12.0

-124

Distanca (em):

Initial and loaded readings versus ciista:..1ca - ­run 5

• yll

+ n Yo

L / "t /

v -~ . :,

\ 3.. ' .-1---t v ~ 'l\ !.

,/ -L~

..

\ vI ' I -

-20 -15 -10 -5 0 5 10 1:5 20 25

Distance (c:n)

Figure 28. _ Initial and loaded raadL~gs versus distance -run 6

50

51

TABLE VI.

Experimental Homent - Hun 4

Distance y" - )11 t' Y8 - Y" 2

Experimental (em) ~Div X 10 ( DJ. v) y" Noment

(gm-cm)

-5.08 o.o -105.2 o.o 0

-2.50 -1.3 -104.4 +1.25 +148

-0.00 -2.5 -103.5 +2.42 +286

2.50 -1.4 -104.8 +1.33 +158

5.00 -0.5 -105.6 +0.47 + 56

7.50 +0.4 -106.2 -0.39 - 45

10.00 +1.4 -106.8 -1.31 -155

15.00 +2.6 -107.8 -2.41 -286

20.00 +2.1 -107.3 -1.95 -232

25.00 +1.2 -106.6 -1.13 -133

30.00 +0.1 -106.0 -0.09 - 11

30.50 o.o -105.9 o.o 0

52

TABLE VII.

:E...':cperi:me:1tal Loment - Hun 5

Distance y" - rl tl Y" - yn 2 Experimental 0 X 10 (em) ~Div (D~v) yt1 Hor1ent

(gm-cm)

- 5.08 o.o -107.2 o.o 0

- 2. 50 - 0.8 -106.1 +0.75 + 89

o.oo - 2.1 -104.6 +2.01 + 238

2.50 - 0.7 -106.0 +0.66 + 78

5.00 + 1.7 -108.0 -1.57 - 186

7.50 + 4.9 -111.1 -4.41 - 523

,10.00 + 6.2 -112.3 -5.50 - 651

12.50 + 8.8 -114.8 -7.66 - 910

15.00 +10.7 -116.7 - 9.18 -1088

20.00 + 6.5 -112.5 -5.78 - 685

25.00 + 3.1 -109.1 -2.84 - 336

27.50 + 1. 5 -107.5 -1.34 - 166

30.50 o.o -106.0 o.o 0

53

TABLE VIII.

Experimental l·ioraent - Run 6

Dista11ce vu - y" yll yg Y'' 2 Experimental - X 10 (em) "~Div) (Div) II l·ior.1ent y .

(grJ-cm)

-16.0 o.o -117.2 o.o o.o -15.0 -0.5 -116.7 +O.Y-3 -1 52.4

-1 0.0 -2.1 -11 4. 9 +1.83 +223.0

- 5.0 -3.3 -113.6 +2.91 +354.0

o.o -3.5 -112.5 +3.13 +381.0

5.0 -2.0 -113.9 +1.76 +214.0

10.0 +0.8 -116.6 -0. 69 - 84.0

12.5 +2.2 -118.0 -1.86 -227.0

15.0 +4.1 -119.8 -3.42 _tl-16.0

.17. 5 +2.l.t -118.1 -2.03 -2LJ-7 .0

20.0 o.o -116.5 o.o o.o

+800

+600

+200

l.fo (gm-cm) . 0

-200..-

-ltoo

-600

-800

-1000

-1200

• • • EA~erimental

---- Theoretical

~ v \ I

\ I

I I i

I

I I i

I i

1

:

I -10 -5 0 5 10 20 30

Figure. 29...

Distance (em)

EA~erimental and theoreti cal momonts VGTs~s distance - run 5

54

35

1.200

1050

Experimental ! I

' •••

----- Theoretical I I I

900

. 750

600

·+::·1J.5o

/ \ I

~ .

\ I \ I

\

300

1150

Mo 0 (gm-cm)

-150

-300

I ~ !'\.\ l

/~ ,,

/ v. ,\ \ .

1\, I \ I

\ I .. ,, \ I \ w

-450

-6~20 -15 -10 -5 0 5 10 1:5 20 25

Figure~ 30.

Distance (em)

Experimental and theoretical moments V6J:'SUS distance-~ 6 .

55

VIII. CONCIXSIONS

From this study, the f'ollmiing conclusions l.va.ve been

made.

1. The mornent in a beam is directly proport ional to

Y0" - Y11 when EI is a constant.

y"

2. The inflection points can be located by the use

of the optical relationships developed •

. 3. The moment diagram for a berun can be plotted by

the use of the derived optical relationship.

4. To allov7 great er precision in the measuring a.'1d

greater curvature to be :measured, the lens should have

greater magnification and an internal focusing device should

be used to eliminate the parallax caused by the inage plane

not coinciding "~•Ji th the plane of the cross hairs.

5. To allow the data to be repea ted more readily, an

object that remains a. constant size •.-.rhen the licht source

is replaced should be provided.

6. To facilitate the measuring, the brightness of t he

object should be increased.

57

IX. RECOHhEI'JDATIONS

As a result of the investigation of this thesis, certain

recorrunenda tions follo\IT:

1. The theory should be e:A."":panded to include curved

beams and arches; and tests made to demonstrate its appli­

cation.

2. The theory should be expanded to include plates

and shells. In these applications, the size of the image

should change along both axes since the moment along both

axes wilJ be changing.

3. The study should be expanded to include indeter­

minate beams, beams of variable moment of inertia, and

frames.

4. A study should be made to demonstrate the :;;easurment

of rotation of a segment of a structure similar to the method

of measuring the slope of a membrane analogy as noted in

the Review of Literature.

5. To eliminate the effects of' creep and permanent

distortions, other materials should be tested. One might

try a netal bea..'ll that has either a polished surface, or

segments of its surface polished.

APPEHDIX I

The graphiS!al procedure used to shmv the formation of

the image by a curved rllirror shown in :Figures 5 and 6 is

explained as folloHs (3):

58

1. A ray parallel to the axis. After reflection,

this ray passes through the to cal point of a

concave nirror or appears to come f rom the

focal point of a convex: mirror.

2. A ray from (or proceeding to~rard) the focal

point. This ray is reflected parallel to the

axis

3. A ray along the radius (extended if necessary).

'rhis ray intersects the surface normally and

is reflected back along its original path.

The graphical procedure used to show the formation o:f

the i mage by a thin lens of Figure 8 is explained as

follm·rs (3):

1. A ray parallel to the axis. After ref raction

by the lens, t his ray passes througll the second

focal point of a converging lens, or appears

to come froM the second focal point of a di-

verginr; lens.

') '-• A ray througi1 tile center of tJ.1e l ens. This

ray is not apprecL~.bly devi,:..tted, s :i.nce the hw

lens surfaces through \•Thich the centre .. l ray

passes are very nearly parallel i f the l ens

is thin. A ray passing through a plate \fl th

parallel faces is not dev:Lated, but only

displaced. .1.'or a thin lens, the displacenent

may be neglec t ed.

59

3. A ray through (or proceeding toward) the f irst

focal point. This ray emerges parallel to the

axis.

APPKWIX II

The product of EI for the specimen used in Runs 4, 5

and 6 was determined by the use of a cantilevered segment

of t he beam 1vi th a concentrc.l. ted load on its free end.

The general equation used ,,.;as as follovTs:

PL3 ~· = 3EI

where

s = p = E = I =

Deflection,

Load,

Nodulus of elasticity, and

}foment of inertia.

60

·vJhen a plot of deflection versus load is made, the slope

of t he line is eq ual to 13 • Thus, EI can be computed by 3EI

measuring the slope.

Figure 31 shows the general test equipment used to

measure EI. The beam t>Tas clam·:·!ed bet\•leen tvTO alw-11inwn angles

to give a fixed-end condition. A needle was attached perpen­

dicular to the end of the beaY.l , a scale was positioned at

the end of t he needle, and the concen·Gra ted loads \:ere placed.

in the pan.

A specimen t aken from t he same sheet of plc:x:iglass as

t he s peci rJen used in Runs 1.:. , 5 and 6 \vas checl~ed f or cr eeD

under a constan t load. The specimen ,,ra s .1.1 11 ':.ride and nominal

1./411 thick. The loading condition is shown in Figure 32.

The zero reading ·Has taken -vri th the loading pail on the beam.

A 100 gm weight was placed in the pan and readin:;::s ta~:en a s

shm·m in Table 9.

Figw:·e 32 sho\vs a plot of deflection versus time, a.:nd

that the major portion of the creep deflection tal;:es place

in a reasonably short period of tina.

61

To alloH for the effect of the creep o:f the plexiglas.s,

the readings for the runs on the specimen for Runs 4,5 and 6

were taken to minutes after the increment of load had been

placed in the loading pan.

Three sets of data ".vere taken for the speci~:1en. The

three loading conditions are shown in Figure 33 as Case 1, 2

and 3, respectively. The data for the three cases are shmm

in Table 10.

The plots of deflection versus load for Cases 1, 2 and

the average of t and 2 are shmvn in Figure 34-. The respective

EI's ar ':: 5.02 x 105, 4.86 x 105 and 4.93 ?\: 105 gm-cm2 •

The plo.t of deflection versus load for Case 3 is shown

in Figura 35. The EI was computed from the slope to be

lt. 95 x 105 gm-cm2.

This EI compe.red '\'li thin the limits of experimental accu-·

racy to the average of Case 1 and 2, and ·vias used for the

specimen.

62

Figur 31. cantilevered beam

63

TABLE IX.

Creep Def lec tion

Load Reading Def lection Time (gm) (in) (in)

0 lt-.04o

100 4.240 .200 0 min

100 4.255 .215 5 min

100 4.260 • .220 10 min

100 4.270 .230 60 min

100 4.275 .235 130 min

100 4.290 .250 14 hrs

100 4.290 .250 2l.r- hrs

.oo 0 50

I~ . 20:: ~~ --

' --------100 150 200 250 300 350

Time (min)

Figure 32. Deflection versus time

0\ +

65

Case 1

p

17.9cm '>I

r===-==---

Case 2

p

- :::::: - ~·.·-\ -- . j

-- <::" -- c - ..... J.

Figure 33. Loading conditions- cases 1, 2 ru1d 3

66

'J~ABLE X.

Deflectio=l of Cantilever Dearils

Case 1 Case 2 Case 3

Load .., Load ..., Load (gm) (inx10.)) (gu ) (inx10.)) ( gm) ( r::u:; )

0 0 0

10 15 10 15 5 0.7

20 35 20 30 10 1.5

lto 65 ~-o 60 15 2.5

60 95 60 90 20 3.5

70 110 80 120 25 lt-.5

110 1 '70 110 16 5

11t0 220 1ltO 210

67

24o -i • • Case 1

I I I ...., __ .._ Case 2 Average I

! --- i !

200 1

1.60

120

80

4a

Q; 0 40 80 120 160

P (Load in gm.)

Figure 34. Deflection varsus load- cas es 1, 2 and. average

68

4.0

3.0

s {mm) ' 2.0 ~------~~----~------~

1.0

o.o 0 10 20 30

P (Load in gm)

Figure 35. Deflection versus load - case 3

69

DIDLIOGHAPHY

1. LEE, GEORGE HALOR,(1963). An Introduction to Experine.ntal

Stress Analysis, 6th ecl.i tion John ~'Iiley

and Son, Inc.

2. EURPHY, GLENI'-r , ( 1950). Similitude in En.&~ineering 'I'he c .. - '

Ronald Press Company.

and ZE1·iAJ:SEY, H. \1., ( 19 55). University

Physics, 2nd edition Addison-Wesley

Publishing COE1pany, Inc.

4. \'lANG, CIEJo-KIA, and ECKEL, CLAHEIJ CE LEvHS,(1 9 57).

Elementary Theory of Structures, J.icGrav.r­

Hill Boo.k Publishing Company, Inc.

70

VITA

Jitlrny Dean Hahs, the son of Raymond and Till ere (Statler)

Hahs, vras born on Har.ch 13 , 1.91+0 at Sedgewickville, l<issouri.

He attended the·. Cape Girardeau, l-lissouri and t he

Farmington, Hissouri grade schools and the Farmington,

Hissouri high school, from which he graduated in 1957.

He attended Flat River Junior Colleg-e in Flat River,

Hissouri for one year and in 1958 he enrolled at the Hissouri

School of ~'fines: and Hetallurgy and r .ecei ved the degree of

Bachelor of Science in Civil Engineering in Nay, 1961 from

that institution. He also attended Tulsa University,

Tusla, Olclahoma, Joliet Jnnior College, Joliet, Illinois,

and Northern Illinois University, DeKalb, Illinois.

His professional experience has CO !~Sisted of employment

with Texaco Incorporated, Illinois Highway Department, a:n.d

HcDonnell Aircraft Corporation. He received an assistantshi:;:~

to the University of Hissouri School of Nines and l-:Ietallurgy

in 1963, and \•ras appointed an Instructor in Civil Engineering

in 1964 vlhile working tm'lards the degree of Nastor of

Science in Civil Engineering .

In September, 1961 he vTaS married to Hiss Ruth Fope

of i:-ledora, Illinois. They have two children, Daniel and

Janet.