The development of an experimental method of obtaining an influence diagram for stress in structural frames

y,

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II~TRODU CTI ON

THE 1.113RARY

CANTERBURY UNIVERSITY COLLEG!i!

N.Z.

The function of an engineering structure is to

:recei,ve

exte~l

and gravitational forces whioll must be

transmitted and distributed to some ex.ternal medium, and

the structural effioiency is a measure of the utilization

of the force

resis~ing

capacity of' the structure •

.structu:r~l

ef'f'iciancy is, however, no overall C:t'iterion

of structural suitability,

for

aesth~tic,

economic and

const:t'uctional qualifications must also be considered.

Originally the stroctu:t'al efficiency was limited severely

by

these latter three cons.iderationah

With the

develop-men.t of high grade

mate:~:oials

a:n.d manufacturing prc:messes

₁

with closely controlled quality and precisely predictable

strength, and stren.gth deforr11a tion characteristics, .it

become economically and constructionally desirable to use

'.fue

advent or arc welding and reinforced

concrete has enabled these basically more efficient structures

This increased structural

efficiency introduce$ greater degrees of indeterminancy

in

the structure a:n.d requires more precision

in

the design

which 1n their tur.n have resulted

in

a necessary examu1ation

of load prediction, factors of safety and c1es1grl

metlJod~t

The external forces to be transmitted by a structure

will generally be contirtuously variable and not precisely

predictable, although in some instances the load may be

quite specifically defined for the whole of the life of

2~

for those expo. sed. to natural forces such as flood; ea.rth•

quake

or

wind

loads, the

design load.s

specified 1n the

governing

code

of

pra.otice

are

based upon the

phenotnenon

of

certain severities

bei:ng definitely

cyclic, the longer

the cycle

the

greater the severity.

If' these

loadings

are then accepted

a probable life periot;l is also accepted.

Similarly, the live

loadbAg values

for

various

type

structures

laid

dovm 1n these

codes

are values which

have been found. by e~porience

to

be satisfactory or safe

for the no~mal

life period or

bu1ldtngs

u1

the

area

concerned,

and

which may

or may

not

be

proved

by

1nvo1m1tary full scale

a.

Factors

of sa£e~y

ruld/or load factors should be

obtai11ed frq:m statistical enquiry into

prev:tous

accident

experience, economic eff'ic:tenoy, cles::tgn

u:n.certainty

and

constructional variationlll.

the analysis of structures

becomafll

n1ore

:rational and empirical then

load

factors

may

be properly

reduced, resulttng in a higher

structural,

efficiency and

greater

safety.

If then$ a structure

withstands the

load.ing

applied

f.or its entire life period

it does not mean necessarilyj that it

is

an

efficient or

even a satisfactory engirJ.eori:ng structure~

With the

values of

the

forces

to

be transmitted decided,

it is

necessary to

relate tben1 to the

associated

forces in

individual

parts

of

tbe structure.

These forces

will be

resisted by the electrical

and

magnetic forces

responsible

P:und.QJrtenta:Lly then, shcmlr1 the ones

with wl.dch an of the loe.~.d. capacity is

Evett

of the

~lter-crystalline

force

to illtrot:.luce the r~implifying concex)t of stress

a:tld

its corollary,

the araorphous

isotropic

mabel~ial ..

th~t

range which

fully

But outside this

1 t does l'lot rt l!n.1ff'ieient explanation.

I

i

I"'or material

_,. s:illl;l.l'c:i.:t" to

concrete,

\'~ihich

idealisec1

t:tcm.shipt a

simplicity

the lack

or

vniformi ty of the

· lU.om1:tally

as

there ·

of comparable · acot:traoy

:no

praotictr.l

s:tmplieity,

stress concept

n1ust be accepted.

it

should be possible to detenaine

the distribution of forces

and deflection within the

their resulting deformation

specified,

although

no

theory of' failu:r•e based

upon consideration of

cally

this inter~relationship of

equations whose solution should give a group of functions

which determine, for the conditions

obtainiltg

at any point,

the new position of that point and the stress and strain

tn

any desired

direction~ These differential equations

genE~n:>ally

hold only within a continuous range of :material·

resolved

into finding

the partial

or

approximate

solution

which

will

solve, with :more or less accuracy, the particu•

This

is

a satisfactory expedient only in

relatively simple eases, as

f'or

nort1a1 analysis it

is

:much

too

laborious a method and sirapl1fied procedure.uJ

'

must be used.

!!'he

qualitative effects

ot

these

simplifi•

cation$ are

very

d.ifficn:tlt to assess, and so recourse raust

be had

to

either voluntary or irtvoluntary tests on actual

111

general,

the

basis for acceptance

of

these simplifications and assu.mptions is

empirical.

With the growing use of indeterminate mo:rmli thic structures_

the

e~perience

upon 'mioh justification of. many of these

simplifications has been

based,

is

not satisfac.tory

and

Qssessment of their effect has become much more difficult.

The Steel structures Research Committee of Great

26

Britain••

Department of Scientific and Industrial Hesearch

undertook an investigation into the development

of

a

rational

design

raethod for steel

frame

st::~mctures

which

involved a large number of test

progran~nes

upon full seale

structures.

The method evolved

fr~~ ttds

prograrame was

but there arose from it the investigation into the

behaviour

of

steel fraraes at failure" It is rnore than

probable that from the imraense amom1t. of research there

will

be developed

a simple but

rational method

for the

are

very

expensive, there have been produced a series of

tests upon inexpetuiJi ve snnall scale raodels of the prototype

which can provide i:n:tormat:ton, impractical. to obtain from

f.ull scale tests.

Tests

upon a

prototype structure

are

generally made either to check analysis by oomparisan. with

actual 'behaviour under know.n load oondi

tions

₁

or

to

estab-lish the behaviour

under

normal service conditions. The

purpose

of :Model tests is eith$X> to provide information for

a

change 1n

design and

to

show the e;fficaey of these altera ....

tions,

or to aid.- avoid

or

check theoretical ::uutlys:l.s by

experi~ent.

Tests upon full

scale

structurf!)s

normally

permit

only simple loadings,. whereas model t«Jst; ·loadings may

There

are

many diffiQulties i:n'V'olved. Ul th(l; use of models however,

' . ' '

the major one being common to

all ....

namely the develOplllent

of techniqu«Js and devic~HJ to el'lable the difficulties in

similarity to be ove:t-.comee

A ruajor advm1.oe in the use of structural models wa•

.

4

. made by l?rofe&EH>r G.& Beggs with a method of obtaining an

influence line for

forces at any section

of a model. This

1927

·~0~ • • ~ . . . 8

1 the

14,13

The Gottsoehalk Cont1nostat, 18

M.I.T.

Def'o~eter,.

24

1v1oment Indicator,

obtained f'rora

have been published • including

This

pul"pose that mod.el

Ma:n.y other type1 or rnodel

which are those

for

asses

of wind loading

upon

flexible

the

full

s

of very

the

Ol1

the

the

the field of'

there

method

ot

:more or

....

1. Mlic:rcuaeopio . ~.t

selected points

on

either a

proto ...

type

or model.

2.

surface

strain ctesc:ription by the

une

of brittle coatings •.

3. Microscopic surface strain measlU'ementa

by X"'ray ..

4. overall st:t"'ain :raeasu:r:>eraent b'Y IJhoto•

elastic methods with ei the:r

ttlo

or

three d:bnens:i.o:nal

s.

Each of these include very lucid and pov,ro:rful of

advanced very rapidly over the last decade and this is

I>ri:ncipally 1 to .the :neeessi ty for effioie.ney 1n

the phe:noman.al strid.es forward

Of tb.ese four 1:>asic

types₁ the firs.t three d.o not come within the scope

;J,t.~.vestiga tion., and no further l"ef erenoe to

will be mad.fh

'J?he photo-elastic effect was first observed by

34

Brewster and the major initial development the engirleer•

32

in.g applications was made by E.o. Coker:,. of the University

of London~ resins • with

photo ... elust:tc propertie~,. which tnay be readily

together with

sheets has

popular.

polariscope requires a large and accurate

to provide an adequate field of light fOl"'

a structural model. For• two dirne:r.ts :tonal analysis the

'

does readily provide an ove:r.all stre.s~3 analysis

of

Uilloaded bottnclaries

tor one

specific loao :at a

time. '+':i:l.ree. dimensional a:nv.lysis is more dif'ficml

t.

most useful. metl1od O.epe:nds upon stress fixation or

Maxwell, re ... d$IIlonstl"'ated by Solakian and explained by

46

Kl'lske ro1d l:Iet«nyi. A tered. light method has

been.-applied ·to 1.1 very restr.icted model and. stress type with

34 35

s.

'l'he photo-elastic

method anablE~s

a precise st:ress

analysis

to

be

rnade

for 2'r1Y specific

loadi:ng~

The purpose of' this thesis is to develop a simplified

and

inexpEm.si

ve

polariscope

and

to

develop and

examine a

method of ootatning experimentally, a direct load/stress

relationship

tor

art'f! load at eny point in a stJ?Uoture ...

i,e. an

influe:ru;e diagram for st:tless directly

by combinip.g

Beggts

Deformeter allalysia with photo•elastic stress analysis.

This will enfltble those assumptions

regarding

force

distttibu-tion

over

sections

to

btl' el.1m:ina

ted. The

finance a:vailable

ro:r

this

project

was very

li~ted and it was

necessary

to

keep the whole

:method as simple and inexpensive as possible,

so

that it could be readily and inexpensively repeated 1n a

normal

design

af ficEh It

would

have been preferable to

develop the method under precise conditions

and

to have

adapted it to a sinwler procedure~ but

this

was

Ullfortunately

not

possible•

IJ.lhe

possibility or

obtaining a

stress

influence

line

is first theoretically

establiahed,

equipn1e:nt atld techniques

developed

are discussed,

the results

obtained

given

m1d

discussed

and

in

conclusion, an

assessmerrq

or

the

:method

{;' l ( } 11 { .l \1 ; J .: '

The most simple and ge[1era1 method of consideration

of

any g~eral

theory concerntng structural m1a1yeis, is to

discuss first its application to structures of some ideal

elastic

material,

and than to

consider the

probable

variation

in

result when

it is applied. to actual structures.

Theref'ore the theoretical discussion of this experimental

method of obtaining

Influ~ce

diagrams

for _ is

considered initially

1n

te:rrr1s

of

structu:r:-e

of

mt ideal

ms.terial;

and the transfer of results from model to

prototype, in

order to define the neces

conditions

of

similarity;) is

discussed on

the same basis.

'l'b.e

assu:rnptions involved are reviewed.

The

application

of

to models, and the

ttJansfer

of these results

to prototype structures are

with regard to

relationship

of

stress

to

st:t~a1n

cha:racteristi{l

of

actual

structural

'11_HEORY

ANALYSIS

use of

detlecti011

diagrams

of structures, or

theil.'

moclels

₁

under

speci:t:ic displaoetnents ~s

influence

d1agrmas

tor various forces, is a method of

long

stancttng,

In order

to find the governing conditions,

the

derivation

of the

method

will.

be fully

discuss&d.

The

law on

which

this

method

is based,

was first 38

forrnule.ted by l;;iaxwell, 1864₁ although

later,

both Betti

38

It 1s knOY;\ll as l\UtXW$ll* s Law

ot

Reciprocal Deflections •

. This states:•

t

.

'

'

"It

s

I l

s , •••

S are the

corresponding

d1splace;rllents 2

•

•

•

at points l 2 .u. n for the force systell. l;) , 11 , ... p

l

a

n

~ppJ.ied at the same points and i.n the sante directions as

the

torce

system.

P P •••

:P

·whose

corresponding

displact:~w

l S

n

mEmts

are

s

s

....

8

thon

•

2 n

I

•

•

'

•

*

.p _{S +P}

s

p

_s

p _{8 +p}

s

p

_s

til

•••

~

1 l 2

₂

n n

11

a a

_•••

nn

ox-

·stated

by

Betti: ....

"The first f.orce system acting through the cor~E)s ...

pending displacements of the second f'orce system does

s~e amount of' woPk as the second force system actin$

·through

the corresponding displacements of the first

force

system".

'!his law

applies

to all structures from the

simplest statically determinate to

the

most complex.

indeterminate

structure. As a simple yet represm1.tative

exarr1ple, oonsider a two b&y f'l"~JlH., with one single storey

bay and one

double storey

bay •· ... two

or

the feet being

subject

to

complete f1.xity1 s,x>.d the other being tre~ to

rotate although fixed in position, under two oond~tions

of :Loading, 'l"he first (Fig. 1) being a point load :P

Tl

ll~

removed,

and

the root A moved

tn

a purely horizontal

direction, without be1ng permitted to rotate, a

dist~ce

H

a

H , - - - t l (

Ef+C . ~~~

+vc .

!1-'---.K

F' G

r,I ~·n ~~

+v•H

" +·v•c

ovinu; tlle Foot

Now

~der

the first system of loading, the external

forces or loads., acting on the structures with their

corresponding displacements are:•

At A At B At

a

Force · HA VA. )!A

HB

VB 'MB HO VO

Oorrespondi:ne;

Displacement

0 0 0 0 0 0 0 0

lS~

and und&~ the

second system

of

loadinga-At A

• t •

HA VA

MA

At B

•

•

•

HB VB liB

0 0 ()

At C At E

'

.

HC VO 0 0

•

0 0

:M

c

a

_l

•

'With thea• two

systems,

then by the Law of

Reoiprocal-Def:t,ectionm:•

.

'

HA,H -t VA,O

+-a

-+-lllhO 1-VB.O 7:l1m .. o + HC.O +VC,O + 0 0 ·r-J?~S

t t t

.

'

'

'

'

HA.Ot- VA, OrMA,O +

.o

·rV:B.O -t-MB.O +-liO.O +-VC,O rO~M l-O.S

•

whence

lU ... H + 1?.-S ~

o

a.

If

the rnora

en.

t a:!:; 8, point M,

requi:r:•ed for

f'irst

system of

will

be

be

cut

a

rota.tion

applied

l

rotation

-liii applied to U: (h 2

tvA

n .K

fvn

HC

~ G

t

'.fG

on

F'i u;n.v:· 4 ;

the

F,G

a

different

the

:&".G

M

to

a

l,

H r----+---.Y

G

fv'T

t

V1_C

e ni rJf.O:r'"LE:-\1

c

In

order that the structure

in two

separate sections, the meJnber H K

·;:auDt

be cut by a

theoretical

section

~d

the

actual

stress system

in

the replaced by the

equ'ipollent

force

system on

!l.l.his will 11ot

of Considering :n.c:>w

two

force

systems ond

systems

at t;he feet

will be neglected as they will not tt.ffect the :r~esultl

being removed from the eqw:t

tio:n.s

as 'they were in the

cases of'

points

B and 0 above.

VJitll the

of the section L

,' I

\l.Oaditrlg

At :L

HVM

J, lj L

corresp~tlng H,V.M,

J)isplaee:mtm.t L L L

H,V, M.

mmm

ax1.d Ul'lder the s~co:nd loadirl.g cond1 tio:n. ""'

F'orce

Cot'responding

Displacement

At L

• ' t

BVM

I~ L L

' t •

H,

v,

la,

L L L

A·t },~

• t t

Hv V.l~

Then by the Law of Rec.tprocal Deflections •

.

.

.

.

.

'

.

U

:W, _,_

V V,

+

M M, +l't H,+ V V,-t M lv1, +l

LL :t.L

L

L

mm :mm mm

• t ' • .• '

H₁

+

V V, + J,'I M, I! H,+ V

qr,+

M 14,+0~8

L·X..

Io~L LL

mm

mm m.m

of tho structure on

At

L

At

Fo:ttce

•H

-v

~M

-H

.. v'

L L L m .m m

Oorreepel':td:1ng

H,

v,

!11,

L.

v,

I

:Displacement

L L L m ;m

_m

At L At

' • • ' t '

·-n : . ,.,

-11

-v ....

L

L

L

m m

m

Oorr•.spon~

' ' •

l):'lSpl6\(H~:ment · L, V, M , . L L L

~en by the Law

•

'

• t

, V, .. • M

:m .•ttl 2

.,.H H₁

:LL

-v v, ·

•

:t,

L

'

'

•li H, •V V,

m m

m

;m

•l~ ( ... m )

m

2

'

... fl

n,

LL

•

-v ·v

_LL

""H H, ' t', .

-

-t • ' •

o.H, .,.

o.v,

+O•M, +O

:a:,

+0 v, M (M

L

L

L

m

m

o.H,

o.v.

o.:M,

-r

o.H, o.v,

o.l!I

L L :m

m

m

t

l

p

s

2

'

V1henoe li (lli+M ) ::::

...

1 2

'

o:t• M ~ ... P

s

il:;l71

1 2

in dir~otian ~d

location

to the fo~oe

pointfJ on the structure

e magnitude ifJ

value are applied, the value or tlle initial force rr1ay

Th1ts was stated by Mull&r ,... Breslau:•

''If' any tunot:ton is aJ,lowed to produce :t'reely

a

small

corresponding

displaoeraent

t a • the

load

line of the

structure

will be deflected

by the f:lmou.:nt of the Inf'luen.oe ordinate for

that point multiplied by •a• "

This method may be applied to any point in a

structure, provided that at tho points of support or

interaction with the surl"OUllding

media,

the st!*ucture

is eii;her completely restru:tn.ed f;;;,om movement or di&lplaca•

ment corresponding to any force, or is not restra:ined in

any way by that force. f'mere there exist cases of partial

16.

oan

be

replaced

by

a structure or structures, continuous

with the one

Ullder

cor1siderat1on

₁ and

whose supports fulfil

the

above conditions, W:i.tllout

changing in

any

way

the loG.d

deflectio:p.

relationships at the points

or

interaction, from

those

obtaining in the

original case,

thel'n !Ul

analysis

of'

the composite structure will provide the true

variat:i.on

ot

the respective

forces with variation in position

of

applied load~

At Ulterior sections. in the structure,

it

is

necessary that the forces

acting

upon both faces

of

the

cut section after the movement has been applied be the

same, and that the mechanism

which

applies

the

deformation

does not provide fillY external restraints to either part

of'

the

member,

but imposes

purely

a relative

xuovement

between the

two

portior.Ls.

As

Ma~well's

Law of Reciprocal Deflections depends

upon

the Principle

of

superposition,

and the accuracy

of

application of this principle becomes .less, the greater

the

deflections,

i t is

necessary

that

the

displaeetflOilts

be kept

as small as possible.

To tltls and a method

of

anaJ.ysis

w~s origi!utted by J:lrofeesor Chi1;.

Beggs.

This

in.volves special mechanisms

for

applying small accurately

repeatable movements of the point

u:n.der

consH\era.tion

₁

and the measurement of the de!leetions or displacements

of

targets, loea ted stra

tegioally

on

the 1noclel1 by

micro•

meter microscope.

It is probably the best of many methods

17.

point :U1 6. struct'l.l.re with variation in unit load

structLt:t".~s, subj.ect to

load:!.ng

in that pl$e

onl;r.

the

f() system at a po1l1t raay be completely specified" if

the vt1lU~s of two

ruutuall'Y

perpendicular resolutes

an.d

In a three dimens:tonal

ayst~₄₁• the force resolutes in. each

or

three mutually

·~ '

a th:l:*ee .dir.lensional rolalyeis, six separate diS~plaeeraents

ar1d related deflection measurements mttst be maclEh

lliOfWJ~.

Hov1ever₁ once the ef'.f'eet variation is found, mtd

i l l Of tWC)

and a couple (considering a plane structure) o:r

additicm found. The effect at a point

1n a . structure will be detel">!Idned by tbree individual

f'orc.es, provided that they are different 1n direction ttnd

involve,

at J.east

once,

thE: two resolutes snd the couple•

This ~varaiat:ton may be found in the following

manner

-V a:hd li representing vert:i.cal and horizo:ntal resolutes,

If' the force at the section be mad.a up frott

the forces under consideration are

v

H

v

,.

v

v

H )1

ll h h

and

v

II l!ii

m m m

the values

ot

v,

H₁ t~n.n M~

being

knovm for

all

required

load.

positions~

Let the multiple of the first force be A

the .second

and

the

thii'd

thf.U.t

A(V

tH

)

(V ~H ) -rC(V

Jl

)

vvv

hhh

lllll'llll

v

lr!f

9 • •

and.

also A V B V

V

v

h

·m

All B:ti li

v

h

m

and AM B 0 M

v

h

m·

From these

equations

•' B

c

v

H M

.

.

•

"'

.

•

6

.

•

V (H M ... H ) H (V M . •M V ) (Ii V· • V H )

A = • m h -· 11...m • h 1'11 · h m ' h r m · h m ·

v

U1

)A

".!.

l11l ')

-r! ·

CV

!l ·

-V·

I )

+

M (V

li ·. ·

..

-v

:rr

J

~ hv vh m vh hv m hv vh

V ·

(M H • H M ) H (V M .

.,.;,v

M ) rK ( H V • H V ) B :::. · ,• ·m. v m v ' m y . v m . • m v v m

v

(U

Y ..

H'

I )

-r

H

(V

I

•V

M )

+

M (V

lt ...

v

H

'J

m hv vh m vh hv · m: hv vn

V (H M •H ~~ ) H (V H ) (li V ... H V )

c

~ t hv vh • vh vh ' vh hv

v·~tH

I ·

~

It

I

1

.;.)1

(V M · •V M ) El" (U V . •

H.

V

l

m hv v h . m vh hv +'m vh hv

. l

. 2

1

l'hus A8 B and 0. a:t .. e lil1ear functions involving

const&t.t multiples of V H and M as the other ractoros

• •

•

depend only up~1 the inittal selection

of

torces, and

A B and

c

may be found for any values

ot

V H and M

t ' '

which will etlable the surn of the thl:>a& .force syli!tems to

be equipollent to th.e force system acting at the section.

'J,1J.1eill \Talue a.nd va.rin t:l.on might readily be found experi"'

men tally in the same mru:u1er that tile value srtcl variation

of

v

.

H and M are

round,

but the displacements would be

'

.

more difficult to

apply.

This involves purely a cbatlge

in convention.

3. STRESS li:,~UIVAI,IiliOR OF LOADING

If a systerr1. of exterxlal loading, \Vl1ich does not

cause either elastic instability

or

ru1y departure from

elastic behaviour, is applied to a structure, there will

exist at

any section

of an1

mmabers

oertain

stress

distribution which represents upon sun1mation across the

section, a ;particul,ar force and couple., whose rne.g:nit'Ude

and direction are depend~.:mt upon, tl:}e structure and the

loading system applied.

If the

member is cut at this

section, ano. replaced by ar:t. exher!lal force and couple,

equipollent to, e.nd dist:6ibuted i:n the srune maxu1~r as

the stress force across the uncut sect:lon., the stress

1n, and behaviou.r

of

the structul?e w::tll 'be unchanged.

If however, these forces are distributed across

20.

the

section will

no

longer be the

E.Uime,

although

as the

stress

at

poi!1ts

further and further

retnovE;~d

from the

section is considered, the value of the force applied

will

become more important 1n determining the

stress

values

than

the actual

distribution

of' 'the.force

across

the

seQtion.

'lherefore,

the stress will tan1d to vary leQ'

and less from the value obtaining when ·bhe section was

u:n.cut. Frocht has t:!Jhown that in a comp;.:-ession specittlen

loaded

centrally

by a point load, the stress becom.es

uniform across

the

section at

a cliatanee

from

the

point

of loading approximately equal to the maXimum width Cif

·the section.

This dependen.ce

or

stress

values,

at

poin.ts

removed

from the

area ir1

which

the load

is

a:pplied, 1.1pon

the

value of the

load

rather t11an

its

distribution, is

know.n as Saint ... venant• s principle

of

elastic

equivalence

of

statically

equipollent forces.

It

is derived frOill

the more

genera~, p~ir!ciple

of

Minimum Stored Energy

which

may be

stated

as follows:•

and associated force system

are in

equilibrium then the

total stored energy is a minimumn •

. By

the principle of superposition, the stress

in

the

structure may be found 1Jy sunn:uation

of the stress

systems produced individually

by

forces acting

at

the

section, which are

in

total.- equipollent

to the forces

existing

in

the stress system

across

the

section,

provided that the behaviour of' the structure is elastic

21.

1

ts

o:t~iginal condition and also that there no

instability. order that the pr1Iloiple Superposition

be apJ.::>licable, the forces which are applied must be

sv.ch as will Ollly cause small deflections of the structu:t"e•

Consider the atru.ctux•.e when free of exte!"'tlal load.i:ng.

If a force systern is applied at a section, then t(!e force

in every part of the str:uctu.re whicla is singly co:rmected

with regard to the section under con.sideratio:n will be

uniquely Q.eterxnined by the fol"Ce syst.em at the section

If an exter:r1al load is llO'~N applied to some

point

of

the structure wldch is

singly oon:nected

with

to th~tt section, the:tl the fOl"~CH9 system will o:nly

be uniquely detert.1:t:n.ed. between that section

Therefore, the stress the

s

trt:te

ture

Wlli.ch the f'o:roe

atld exter:n.al J.o~d or nn.tl tiply connected portions.

It has shovm that in a s·bructure, · variation

of f'Ol~ce at a.TlY poirtt, wi t:h the vv.ria.t:ton i:n pos:i. tion of

a load specified in both magr1itud~ and direction,

may

be

found by deflection. atudies of the structure, ~td the

value of the stress 1Il specific portions may be tU:liquoly

deterrailled by smmnation of the stretis valuea for ind.iviclual

forces which a:r.•e in t;otal, equipollent with the :r•equired

force existilJ.g at the section l..'lncler cor:u:tideratio11,.

Therefore, if the var•:tat:ton of each of these forces

a:r•e f'ound, then the stress

at a

po:t:ntt w:'l.trdn

those

at:vuo ...

ttn1al

parts

a.t which the force system 'Ul:liquel:r deto:t"mines

the stress, may be

rou:nd.

by

•.rhat

:ts.

if

for u:nit

force apvlied. at :poiJ.tt A t~he valur:is

of

th.ree

f'oz•.ces, wh:t'ch

a1~e

such as w;tll detol"!'t1ine specif:t ...

cti:tlly the t(>tal force,

u"o

poitlt B~ a:t:•e h v a:u.d n1, thel1

st:r.eas at poil:tt 0₁ 0 is on r.t J,>a:r.t of t;l1e

uniquely deterro.illec1

by th.e f'o:t:>ce a.yste:u~ applied to the section 1:1.'1:; A,

for

u.:nit i:nclividual VfJlues of the three :forces e.-1::; B are

respectively

h

v

th«n

where

oe

is the stress at C for unit load at A and

if the load at A it;~J .P ... in 'the same sense as unit load, then

the stress at 0

If

the stress

value required at 0 is a

boundary

stress,

then the stress will be in the

same direction

in

each

case,

and the

swm:;atio:n

of' stress is

quite straightforward.

If

the

stress

is at an interior point and

the

directions

of' principal

stress

vary in each

case, then

there is

required

the

vector

sum af'

the three stress eondi tiona

obtaining~

With a throe dimensional stress distribution,

the principal stresses in the

plane

of the boundary surface

23.

For interior stresses in a three dimensional

only.the boundary

stress.

In a plane

structure, by plotti:ug the stress due to un1 t value of

of rut ects a.t; suf'fic1en.t seet10l1S

to permit the whole of' the structure to Ul.liquely

deternined

with regard to one of the sections, and the

vax•iation of each of these effects with. -unit load position:,

:ts obta:tn.ed a series of Influence for ::;;tress,

whi~h the stress £\ t any point for any loading

eol:t.di tion; to be fou:nd.

varlatioxl of

foroe

or effect at; any section,

with unit load

the:v of the actual struotu:t'e, or n1o:t•e generally,

and the stress d.:l.stl"ibution fl"Oli1 :t>ho:to-elastio

of

H

model --

to

• applied three separat;e

P:t'ovided these lu.tte:r

of Which

forces or

effects~

are sufficient

·!Jhe section,

their Vt:,tlue is 1rn.m>£:.tl~l'i&.l₁ t:l:r.td t11.eir VU:l1iat:tor.1. 1nay be

fottn.d fro:m the displacell1{.1t~t ₁ provided again that the di.splar.:ement Hl)J:llyses a:r•t! also

to U);l.ique:ty dete:t"tui:ne

cOJ.1Si(le:t'a·h:ton.

f'or :::ru.ffieient

111. this discussion the :t"ollow:Ulg Syfllbols wlll be used:• E -=You:ng1 s Modulus

u :: J:oiason• s Hatio

l

.:::Length

of'

one

Cm'Lt:~;e

Line Dime:nsion

'

r ,r • • ,:;:.Ratio of other Centre Line Dimensions to l,

• 2

• t

r

,r

•••=Ratio

of section depths

to l,

t . 2

.

~ • t

r

,r

••=Hatio

ot

section

brf}adths _{to l,}

'

2

,s .,. • • :::Ratto of other loads to P

' • 2

'

s ,s •••=Directional cos~1es

or

res~eotive load

• 2

::::stress

d = DeflectiorJ. of

ce:n.tre

line from unloaded position

e =Strain

a :: Depth of' section

I :::-Seotional tnoment

or

Inertia

Subscripts

p ::: Prototype

m

=lAodel

A structure :~:nay be completely elaeticaJ. ly

25.

t

"

_{s .,.,a}

•

_~'~;x6y,s)

'

.

I

-fnf"

- "'

1~, u, l, r

...

,

r

...

,

l"

...

,

P,

•

_'

..

t tt t

and d-=-P.n(E, u, l, :t:' ••• , ~ ••• , r ... , P, s ... ,s ... ,x,y,z)

• t • • '

where fn and Fn represent

functions

involving

the

bracketed

' •'· ,,., . vr;~.r:i.ables. '

These relationships may be expressed

in ~

non

dimensional form,

•

•

••• r

•

r •••r

'

.

t • '

EU1d. therefore, these no ftmctions must the

same for both prototy:pe., to

ly terrn,s tnus t be the same

for botll

that

I

""

::

B

zm

I

1m

~

um

::::

u

p

•••

..

..

r

m

:::: _r p

'

t

t

..

..

.

•

....

r Ii1 -- r p

t

•

..

•

•

• :::

"

•••

r m r p

'

I

and

.,.

sm

'

t

s m

•

•

•

•

- $ p

'

:::. s

•

'

.

.

.

p

and th1s be:tng so then

2

In

order that these relationships hold, the model and

proto-type must be strictly geometr!cally similar.

The value of

Youngts :Modulus

is

not material, as the te:rom involving it

may be adjusted

by

alterhtg the model loading

in

magnitude;

but Poisson t s

ra·tio

must

be the same fol' both.

As

the

variation

in

Poisson•s ratio is reasonably small, it is

generally only an important factor

in

the stress relation ..

11hips of self stra:t:ned bodies or bodies which contain holes

with loaded

bounr~ries.

Even than it is not a critical

factor, and lack of strict adherence to equivalence of

a, lo; 34, 38

Poisson's ratio will

not involve

~Y

appreciable error.

consider stress

cono~ltrat1one

under conditione

of full geometric similarity, taking as an example the

stress under the point

of S.J_;pl:tcat:ton of'

concentrated

load.

then

lf P

be

the load per unit

tl1iclmese

b

27~

of' interaction of' load and material the origin of the polar

co ..

ordinates

thus for ·the model

Fp ::

~ip

Whence li1n. xw.m bm

Pm

ru.td

so with true geometr,ie similarity, the stress

valuea

at

corresponding points

bear

the same relationship

itt

this

area

the load

will

o:nly hold at points

Which are

removed

from

' '

the point

of

interaction, as

it

is.

improbable that

in

practice

the actual loaded surfaces would

be

identical

1n

both the

model

~d

the prototype.

vVhere the

stress concentration

is caused

by

corners or niches

having a definite radius of'

curvature,

the

value of stress is,

1n

general, a

function

of both the load and the

radius of curvature.

Therefo :re,

the stress

valut.lS should

bear

the same

relationship.

However, where

the concentration

is caused

by

a

sharp

change

in

direction such as the conjunction

or

two planes or

curved

surfaces whioh contain an acute

angle

at

the

line

of

contact,

and v1he:re this exists in an area of

approximately

l.U:liform strEuJs.

the

valtH~

of tbe

stress concentration

will be

i:ndepende:nt

or

the dimensions

of

the structure.

It

will be

dependent solely upon the load value

obtaining,

and therefore,

28.

With true geometric simila~ity between both th&

stl'"Ucture and the loading systGrn of the model an.d the p~ototype ~

the stress values obtaining :1:r1 the model, will bear a eons.tQJ:lt

:relationship to the p:rototype stress valutu'l, except in some

specific cases of stress concentrating features wllioh are

JJlentical w:lth those ill the protot1pe, where the

relation-ship will be different~ Consideration

or

a particular

feature should enab-le·a·deciaion

to

be·made rega.I~d.ing

the

relationship

which will e;x".ist. between t':he xn(ldel the

prototype to be

deciaed•

+n many eases, there are great practical, difficulties

ii"J.vol ved in producing a geometrically similar model of. a

structure, and also in a:.i;Lalys:U'lg the resul stre~rs patt

obtained pllotoelastieally., In these eases, ird'ormati.on

the t;!tz>ess conditions obta::tn:tng i:n si::;l..,_lcture,

tla of .suf.fieiar:tt accuracy for the lJ"Ll:t ... poses of

. $t:ruetural analysis, any. be. found from stu(lies of nlodels,

rn.ucb s~..mpler in f'omn, which are

onlr

partially similar to

the pro~otyp,th 'l1l.ese will be called models of specific

similarity, as opposed to 1nodels of geometric sim.il,ar1.ty.

Genera ll;r, in stl.:>Uc.rtures ~ the f"oroe at atly section,

mt:ly be considered as being made up of three crJrtlpO!lents in

a ple.ne struoi;ure, and six, ill a three d1m.e-.nsional structure ....

nru:nely a direct thrust, two shearing forces, two perpendicular

ruoments and a torsion or twisting r.noment~ The stress and

deflection values obtaining the strtlctm:"e may be rega1~ded

the

forces

individually.

In most cases, it will be f'ound that the stress SJld

deflection g~1erall7 over the structure, will be a;ppro:xima tely

equivalent to the

stress

c1.ef'lect1on caused by one of the

above effect~J, especially ir1 ax~eas

of

maximum

stress and

of

maximu:n:t deflection~ In a two dimensional system, the

eJtress tzJ.d def'le<)tion as socia terl with the benciing t.'loment

a:::oe approxirAately

equive.l.eJl:t to

.the stresses and deflections

in the structure, ~speei~l1y whe:n the· str•ucture :has a

pre-dordnance

o:e

slender ru01.11bers ... that is, t~he cent1•e line

dimensions are large

con1pared

~with the aoct:Lc)n d.im~sions,

therefore,.

a model of specific

similarity

to the

pl'oto•

type, as the bendir.tg raoment stJ:?ess and deflections

are concerned VJtlll allow the atre~w and cleflect:ton valu,es

b1 ~1e prototype

to be predicted with reasQnable

degree

of

acwu:NlOY•

Where

some

other

effect OJ? combi:n.ution

of'

effects predorninate in dotel'!lli:n:ing the stress &l.d deflection

values in the

structure~ the

specific similarity

should

be

liltli ted

to

COl1.SidO:t"atiOll

Of

these effects. 1'h1S may tl.lSO

tnvolve

a specific

similarity

tor other effects,

but

ditf'ererit similarities

in

each case, and

therefor•e,

the

error

involved

by

neglecting these other effects,

will

be

reduced.

Examine the necass~:lcy

conditions

involved in a

specific

similarity

for bending moment.

F

and. therefore

and the ratios

Im

fP

,

must be constant for all sections of tl1e structure.

For considerations of deflection

d:::Fn(MIE)

!n(:Pli:E)

and the specific similarity does not include the depth of

section ..

t

a •

The .following muf.d; be constant fol' all equivalent

sections:""

For conditions of stress concentration similarity,

it is necessary that there be a geometric similarity between

the model and prototype over the area of' concentration, and

the previous considerations of' stress concE>..:ntration will

again apply.

The specific similar! ty between model and

prototype w!uch is considered necessary, will in all cases,

depend upo:n the prototype shape and section shape, the degree

of accuracy required in the transfer of results and the

facilities available for model makil1.ge

must be considered separately, a:nd the similarity necessary

3lo

0 • ASdUMPTIONS.

The assumptions made about the structure and

material,

in

order

that it may be

treated

tn

a calculable s~lse in

the

theoretical

discussion of the

method, are primarily,

that the following

laws

and

principles hold: ...

Hookes Law: my

system

of loading comprising any number

of forces in any direction,

will

produce

a

definite

and

proportionate

deflection

of every other

part.

The structure

is elastic:

It will recover its

origtnal

for.ra

after

any system of loading has been applied and

then

removed.

Saint - Venant•s

princi~le: The principle of

elastic

equivalence of statically equipollent loading systen1so

Forces

applied at

one

point ~l a

structure

~T.lll

jnvolve

stresses, which, except 111 ·the immediate locality of the

forces,

will d.epend &lraotit •=~ntirely

upon

the resultant

action

of

the forces

e.:r1d very

little

upon their

distribution.

F:r:ineiile or.su;pe:rpos~tioP;:

The value of'

any effect (stress,

def'leotio:r.a., moment of

:resiste,nce

etc., )

produced

by

a

loading system,

oomprisilJ.g

several loads,

will

be

indepen-der.t.t of' the order of appliot:1tion of the loads and the sum

of

the

values of the

effect produced

by the

individual

forces

a.otillg alone.

Prrt~?iple of

o,onservation

of enersz:

The

same atnount

of-energy will be

stored

in a

structure

clul:'ing the appliCc1tion

of' a loading system as is

:t•eleased.

when the

system is

l\s the method is concerned primHrily with models,

and the tr.a:nsfer of the stress values Obtained f'rom. the

InflU~lce diagr~s for stress to the prototype structure

is the final step, the assumptions involved :in the model

analysis will first be discussed, and those it1voJ.ved ir1

the transfer of' the results to the prototype structures

of normal type, will then be considered.

1111 MODE.L IAA'J.'l~RIJU,S

The model materials available for photo-elastic

work were all plastics, coxmnercially availabl.e, e.lthough

lll some cases specially produced ~1d developed for the

purpose. They all exhibited in varying degl .. ees, creep

under load. - that is a variation of st:r.'ail:J. wj. th the time

The relationship governing these variables

is such,. tha.t with any constant tirae of a:pplication of

load, Youngt s Modulus has a constar.Lt value, and the strain

becomes arJproximately constant with any

load

after

a

certain time a Thezlef'o:~.~e, the

zua

terial obeys Hooke a •

Law,

in effect, provided either suff'icient ti:tne is allowed to

elapse

to

enable const~t

conditions

to be reached, or any

~eadmgs are taken a constant time a.f'te:t~ application or

tho loan. SimiJ,arly, a:eter the load is removed., the

material reverts to its original. shape eventually ...

the titne taken, deper>.ding upon the length

or

time the

load. had been applied.

The models for use in the deformeter wo:t?k were also made

33.

As there is a definite strain apJ)lied. to the model, tho creep

under load is evidenced by a var•i<Jtion :l:n ·t;he load required

to :maintain the d.ispl~ccme:rlt₁ and thus t1oes not eft\aQ.t the

resUlt in a'tl"if way,. rr'ho displacement ia posi t;ively l"6l'llOV6d,'

and the:t~efore ·the mod~l

mu.Ert

r•otum1. to ita Ol'l~tginal unloaded

position. These xnater:1.als behave effeetlvely in a :UltU:lne:r

perrnitt:tng 1nterin•t:rt;ation on the aostunpt;ion o:r purely elastic

behaviour, provided that f'or s:ny testa where :lrl:t'ol.,mation

concerning lo~d fJt:Pasa and load atx1u:tn N~lationahip _iEJ

required, all :meaaurmne:n.ts u:._-.e truten afte:t• auf~f':toitlxrt tirae

has elapsed to allow the creep rate ·to become VSI"Y small,

or the:t•e is a system involving H very good titne control.

As

the

photo.-elastic pl:umomenon is strain-de.pencle:nt,

i:n:f'orma.,..

tion oota.med under OOl1.stant strai:n. co:nditions of loading (su.ch

as the applict:.:t tion of specific displacements),. nmy related

to stress through in!'o:t'lllation :r.•egs.rd:tng the strain load tin1e

relationship fo1~ the moc1el material obtainecl from other tests.

Saint Vtmcm.t•s principle does :not admit of a11.y rigid px•oof.

It is a

specific statement

of the more general

theorem of

minimum stored m:1e~•gy configura tiona of loaded struc'L"'Ul"ll\HJ.

It has be~ demonstrated by

many

investigators.

consideration

or

this, it 1:3 evident tllat ·vilwx•evei• pos;Jible,

the sections to which the fo:pces a;~?e ar)plied should :not be

in posi tiona where tbe stz•ess oondi tiona are likely to be

critical, i f this is possible. However, as it is generally

34.

structure is attached to the supp<n"ting media, und as

these sections are likely to be critical for

maximum

st:reso condi tiona, some care should be taken to ensure•

where possible, the stress c.'tistribution across these

sections

approximates the

actual

diSJtrioution

possible

in practice. Therefore, at hinged ooru1.ections, the

hinge

should

be an accurate model of the

actual

hinge

connection to be used in the prototype, u.l th.ough a study

of the hinge as a

separate

entity would

probably

be

necessary in ord.er

to

obtain accurate values for the

stress around the hinge ... the hinge on the 1nocl.el being,

in all probabillty1 too sme.ll to permtt this to be done.

For fixed end concli tiona, the model was carr:ted on past

the section considered, but clamped as rigidly as possible

to the body of the defomning mechanism. In this way,

the stiffness increases at the t:~ection to a very high

value, and yet the model is continuous over the section;

thus the stress distr:tbut:l.on across 1~h.e section is reasonably

unifo:rra, ru:td the fixed encl condi t:ton is rigorously applied.

'.rhe exactness with wh:toh this simule,tes the o.ctual stress

condi tiona obte,in1.:ng in the pro:botype, can only be c:lecided

by cons1derat1.on af the proto1-;ype attacbrnent to, a:nc1 the

properties of

1 the SUiJJ)ort:tng Jnedia. At :t:nte:~r:l.or sections,

the member is clamped to tbe etefomning mechen:tsr11 over parts

external to the section, so the stress co:nrlitions around

35.

Therefore, i t is 1n this cu:use advisable to have this

section at points well removed from the location of areas

of ma;d,mum stress. Provided enough variation in stress

is evident, the force analysis of the member, when. sorae

o.ther portion is load.ed, rnay provide sufficient

int"orma--ti~, ana this is probably the bast way for the force

analysis at interior sections; but in most cases the

stress variation is very small and is not sufficient

to give an accurate picture.

~he Principle of superposition arises from the two

initial laws, but it is necessary for the deflections to

be small, as if they are large, the 'latter loads are

applied to different shaped structures m1d this will

produce a different final shape and force distribution.

Il1 order that .this Principle raay be applicable, the loads

applied must be such as will produce a :maximum stress

value with a mini:mttra deflection, although the errOl""

involved is not excessive 1<'Vithin the of deflection

possible in stiff models before the proportional lllnit

of the material is reached.

The Principle of Conservation of ].lnergy is

necessarily true if the model is elafJtic.

2. PROTOTYFE MN.L'Ji;RIALS

Therefore, with attention to the dtltails necessary

to ensure a reasonably accurate fulfillment of these

assumptions, the stress values obtained fJ."Om interpre•

tation of the InflUe:i'lce Diagraras for strt:lss, \Vill be

36.

isotropic elastic material. lf the ~rototype is also

ae:nsi'bly elastic, then tl1e stress values obtained will

'be immediately transferable vdth rega:~.~d to the conditions

governing the simila:t"i ty of the model to the prototype,

and the majority of steel structures loaded within the

elastic :r•ange will. conf'o:m:n to these COJ:lclitione.h For

plastic hinges and ultimate load.il!.g, however, the model

results .will be applicable only for the location of the

initial Plastic Hinges and not 1n any further tmalysis.

The applicnti()n of the results to a prototype

structure of reil1.forced concrete is tnuoh less

straight-forward, as the rela.tionship betwee:n. stress and strain

for concrete is by 110 means lirwar and is dependent

upon the tit11e of' tlpplicu tion of tlle load, also the

is not hol110geneotuh !J.'herefox•e., the rotation

pel~ unit length tu1i t

morrleJ;J.t •

which will in future

be called tl1e basic roti:t tio;n_. two tinct values,

both of' which vary with both the magn.i tude of' loading

and the t1lue it has bEllan appliedJ the f'irst applies

before the concrete fails in tension and the second

after this has occurred. · Therefore the behaviour of

the structure will depend upo11 its prevlous loading

history and the value of the loading system considered,

When this load approaches the maximum, thos a parts of

the structure where the rnembera have t:oo.cracked sect:i.ons

are small compared vd th the purts where the member has

the rotation.

-t,h~ QSS"mpl!ao"

Thus for . the o.f simplicity,

or ct completely craclred seotton as the basil!! of 9J1&lysis

of b&h#l'Viour Of St:P.UCtl.U'aS Ul106l? load S,.s justitiable.

As the results o,:re obtained for elastic behaviour of

the

stmu~ture 1 t is

necessary

to assv.me n correspo:n.di:ng

elasticity in the

prototype. As thE!~ concrete wilJ.

normally be stressed to only approxir!lately one halt' the

ultir(late, r~nd th.e

assl.lmption

o.f a const$nt modulus itJ

r~asona.bly

consistent

within thitJ

range,.

this does not

h1volve

any

serious

error,

although

for loads

higher

th~ this the

assumption beeomee

untenable.

eases, depandine; upon the analysis required₁ results

may be

t;r:-ansrerred

in terms

or

$train and the loe.d

distribution obtei

ned

:fron1 these results, as the strain

va:r:i.a tion acrcu~a the concrete section remain~ linear to

loe-d valu~s ~ppro:x1mat:1ng the

ultin:late

in the concrete

prototype a11d. tn the :model~r

The

major.diffieulty

arises

from the

non-homogenuity

of the ool.lerete men:ibE>:r, unct to overcome this the steel

mu~t~t

be considered

to be translated

into tension concrete

wlaieh has the ss.rae properties in tension as the concrete

Tltis must be done in such a

man.ner

as to ensure tha.t the

section

at1ll has the same

strain-load properties, and so that this theoretical member has

The

strain

values obtained from the model m.talysis,

horlfever,

wlll

theor·etical

stra:t:n

to

tra tiona will\ lJ.b.le to

the model arJ.alyaia to

is all cases aruall.

U COllS tall t

ati•uctu:ral is valitl.,

should

of

of

o.ccn.:trv.cy fo:r•

1 lll

~ ... nalysis.

For

a

:Modulus for the ccncrete

1n lo C<J. tiona

; as

conc1i tio:rts

strain

concan-with the same

for the l:lp:plica tion

st:ru.ctl:l.l?e to

aatisfucto:ry

the

the

the purpose

the

deter-to tht~ pr•ototype

wlll the sru:11e as the ratio of t;hat of the plastic

to t of its S:i..lJ.g With loads which

uro sufficient to cause the atr•ength of th.e concrete

to be structure 'Hill accurately

39.

or

greater

ma~11tude

the reproduction of the conditions

obtaining in the structure

in

the

rilodel

becomes more

difficult, and the simplest way is probably to

intro-duce artificial tension cracks

1n

the plastic, but

1n

general,

tJ::u~se

would be dif:f'icul t to looate,

3. 'l'IIREE Dil.!IImSIOllAL SY8'11

l!XS

In three 41mensional structures involving large

surfaces and point loads, the general analysis of the

t~tresa

values beco1nes more difficult.

The

rnajor

problel'll

is the distribution

or

force intensity across the section

considered, for, as this would vary considerably with

load position, the application of the force system to

the section becomes

difficult~ In

these structures

the major section dime:nsion is large compared with the

oent!"e line dimensions of the model, and the area within

which the stress is

dep~dent

upan the load distribution

rather

tlWl

load value becomes a large portion of the

structure.

Thus these structures do :not comply with

the conditions neoesse,ry for the validity of the

method~

Therefore, even although it is possible to arrive at

general relationships between load position and force

va:ri~ltion

across the section, the number of analyses

which would nave to be made would be

L~ch in e~oess

of' the number required if' direct photo-elastic an.alyses

were made of :raodels under several probable worst condition$

of loadtngJ

these loadings

bei~ng

decided from

o~1er

considerations.

The criterion of' application of this

obtained frorn consideration of Saint-Vmumt.• s principle,

for if the maxinrum. section dimension is large 1n con1patti•

son with the centre line dimensions, then the nuri1ber of'

analyses required is prohibitive a.r.1.d the general raethod

is impractical.

4. SIDt!MAR'Y

In summar~ ₁ this method of determining influence

diagrams

for

stress in

structures

by deformeter

m1d

photo-elastic

studies

of

model:i will

enable

accurate

a:malyses to

oe

made where the model is geometrically

similar

to the prototype, and

bo~1

are

elastic

structures

in which the section dimensions are relatively small in

comparison with the

centre

lilLe diJ:nens:tons.

specific oases of

cone en

tra tio11

the

model

results ·will

Xl.ot

be transferable in the same ratio

because the stress cor.~.cent:t'*ation is Pl"'oduced by con:t'igu ...

ratious in the n1odel wbi,oh are ide:n.tical with those in

the prototype a:o.d not s.calar replt'!Odl.lct:lons,. 'Wl::H~re

, ' I l l ' ' . ,

stress in the prototype is gove~1ed by one particula~

effect1 a model of specific sind.larity will enable stress

analyses of suff1Ci~t accuracy for mm.1y design purposes

to be obtained from a model simpler, both to construct

and to analyse, although, with only specific similarity

obtainin~h caution :nlUst be used ill tlle

transfei.'

of' stress

concentration values to the prototy9e•

Vft1~1

results are

applied to

reinforced cmtcrete