Effect of plastic flow on flexural stresses in reinforced concrete beams
by Louis A Divras
A THESIS Submitted to the Graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Civil Engineering
Montana State University
© Copyright by Louis A Divras (1948)
Abstract:
In this thesis, the fact is discussed that the stress variation In the compression side of the beam is not
actually a straight line as it is assumed by most designers, but it has a shape of a curve, which is
somewhat close to a parabola.
Most designers, while designing certain beams for different loads, base their design on. the straight line
theory, which in turn considers only the instantaneous unit deformation produced by the application of
a load and which is knownas elastic strain.
Concrete, however, is aplastic material and additional deformations occur immediately after the
application of a load. Bven though the load remains constant, these additional strains increase as time
passes. These strains, which are usually celled plastic strains, vary with the amount of load and with the
time. Considering this, it is shown that modulus of elasticity, as far as concrete design is concerned, is
meaningless# In the experimental work# two beams were tested,. One beam was designed b,> the
plastic flow method and ohe was designed by the straight line stress variation method, in which the
modulus of elasticity of concrete must be used.
' ..
The test results demonstrate that the strains do not vary as a straight line from the neutral axis to the
extreme fiber; consequently# the stress does not vary as a straight line from the neutral axis to the
extreme fiber. BBTEOT OF PLASTIC F&OW OB P&BXKRA& STRESSES
. . IBUBGiaFGBOgD OOKOBBTB BSAKB
'
V
imD&A* JKCKRAB
A TmaiB .
, Submitted to the Graduate Gomittee
in
p a r tia l fulfillm ent of the requirements
. fo r the degree of
■■Master of Science in G ivil Engineering
at
Montana State Gollege
Approved*
In Charge of Major Work
X e
Ghairman3 Examining Committee
September, 194#
-j b t ^
2
o f "2^
Table of Contents
Page
T itle
Table of Contents
2
A bstract
3
Symbols Used
4
In tro d u ctio n
6
H isto ric a l Development
8
P la s tic Flow of Concrete
12
P la s tic Flow and Volume Changes of Concrete
14
E ffe ct of Shrinkage
21
S tre sse s Developed Due to Shrinkage
23
The P l a s tic ity R atio
26
P la s tic Flow Design of Reinforced Concrete Members Subjected
to Flexure
29
Experimental Work and Design of Beams
35
Tabulated Data
Tabulated R esults
47
D iscussion and Conclusion
49
L ite ra tu re Cited and Consulted
53
Abstract
In th is thesis#, the fa c t is discussed, th a t the stre ss variation i n '
the compression side of the beam-is not .autuaiiy a straig h t lin e as i t is
assumed by most designers* but i t has &. shape of a curve* which i s some-*
what close to a parabola*
Most designers* while designing certain beams for d ifferen t loads*
base th e ir design on the straig h t lin e theory* which in turn considers only
the instantaneous unit deformation produced by the application of a load
and which i s knownas e la stic ©train*
■Concrete* however*- is a, p la stic m aterial and additional Cteformatione
Occur immediately after the application of a load*. Bven though the load
■remains constant*,' these additional strain s increase as time passes* These
strains* which are usually called p la stic strains* vary with th e amount
of load and with the. time*
Considering th is# i t i s shown that modulus of
. e la s tic ity 5- as f a r as concrete design i s Qoacemeds i s meaningless*
In the experimental work# two beams' were tested*. One beam was designed
h,y the p la stic flow method and ea'e was designed by the straig h t line stress
variation method* in which the modulus of e la s tic ity of concrete must be
used*'
■
•
■
' ..
■The te s t re su lts demonstrate that th e strain s do not vary as a straig ht
' ' -■
lin e from the neutral axis to the extreme fib erco n seq u en tly * the -stress
does not vary as. a stra ig h t lin e from the neutral axis to the extreme fiber®
Symbols Used
Ae - Effective concrete area,
a
Pepth of equivalent stress diagram.
A st Overall area of concrete in a column,
As st Area of te n sile steel.,
b * Width of beam*
0 st Total compressive force*
T ar to ta l te n sile force*
c * Ara of resistin g couple*
f 0 % te n sile stress in concrete*
fg 2 Compression stress in steel*
■i
f c s Oompressive stress of .a concrete cylinder*
.fy « Yield point Stress of steel*
d
Effective depth of beam, .
M - Resisting moment*
■'
;
.
•
’
U = Modular ra tio (which is equal to |-~)».
Bs a Modulus of' e la s tic ity of steel*
A
-
Be s Modulus of e la s tic ity of concrete*
p s Steel ra tio (As/bd),
S n Shortening of a non-reiaiorced concrete block due to shrinkage
Sa sr Shortening of reinforced concrete beam due to shrinkage, ■
f ls s Allowable stress in steel*
f"e ~ Allowable stress in concrete.*
jd % Moment arm of the in tern al couple,
kd s Position of the neutral asds*
'
5»
■ .
■
If 9 Total shear force at the section,
v = Intensity of vertical, shearing, stre ss,
~ Total concrete stra in represented in the trapezoidal diagram at fin a l
rupture,
’ ■
~ P lastic s tra in represented by the horizontal portion of the trape­
zoidal diagram.
^ # P la stic ity r a tio ,
G-Ot3 Total concrete stra in minus p la stic stra in ,
Tstt « Computed theoretical stress of ste e l,
Tdi 1’ » Computed th eo retical stress of concrete, .
Sg. = Computed actual stre ss of ste e l, .
S0 - Computed actual stress of concrete,
<r3 s Actual strain of steel-,
6c “ Actual stra in of concrete.
6,
Introduction
Experience has shorn th a t the actual stre ss d istrib u tio n in a rein*
forced concrete beam does not agree with the accepted theory*
I t i s also
well Imom th a t th is usual theory, when used with properly selected unit
stresses gives re su lts th a t are safe in sp ite of the fact th at the actual
unit stresses may be e n tire ly tilfferente
It has long been recognised th at concrete changes in volume# when
subjected to stress (e la stic deformation), sustained load (p la stic flow),
temperature changes, and changes in moisture content (shrinkage), and many
investigators have worked upon the determination of the magnitude of these
volume changes»
In recent years, th e criticism s of the ordinary straight lin e theory
of reinforced concrete have given ris e to proposals fo r the abandonment
of the modular ra tio SB, in favor of formulas based on various theories
of p la stic actiono In a l l these th eo ries, emphasis i s placed upon the
load, at fin a l rupture of the beam. While the wisdom of such emphasis is
open to question, there can be no doubt th a t i t is desirable to be able
to product, within reasonable lim its, the ultimate capacity of a beam.
Since concrete i s not tru ly e la stic , the stre ss variation across the beam
section usually i s to be assumed a parabola,
sa tisfie d with such an assumption.
the w riter does not fe e l
Since i t s deformations under load in ­
creases with time and because of shrinkage and p lastic flow, the e lastic
or straig h t lin e theory has long been recognised as approximate. Since
shrinkage and flow are impossible of prediction with any accuracy for- a
given structure, i t appears to be a hopeless task to determine exactly the
7,
stresses m der a given load,
Xi is possible, however* to derive a simple
relationship which w ill predict with considerable accuracy the ultimate
strength of a reinforced concrete beam* a value l i t t l e affected by shrink­
age and flow since there is a redistribution of stress w ith'the large
strain s previous to fa ilu re .
I t thus becomes possible to design on the
basis of ultimate strength applying a definite factor of safety suitable
to the conditions,
The writer, in the experiments conducted in the Oivil Engineering labor
story of Montana State College* attempted to find the actual stress d is tr i­
bution in two f u l l size concrete beams.
The undersigner wants to express hi© deep gratitude, to Professor E-, CL
DeHart for his kind assistance in carrying on the. work, He also wants to
thank Mr, J , E, Arboe and Mr, Drummond for th e ir kind assistance ■in offer­
ing the f a c i liti e s of the- Mechanical Department necessary fo r the comple­
tion of the work,
"
8,
',Historical Development
■
I t ie probable that a am ber of observers had' noticed the plastic
flow pbeaommoa in oonerete before igoy, when professor % K. Hatt ^ pab^
.liehed the reaelts of a series of defleetlc# tea ts on beams, Thla appear*
.to be the f i r s t publication on the subject*
According to' Professor H attjr
tMee TrOBnilts taken together show * sort of p la sticity in concrete by which
i t yields -under the action of a load applied fo r a long time or applied -a
number of tim es,'
.
'
In 1915j Professor P* 8». MacMillan ® reported te s ts on beams and slabs
Iii which he measured fiber deformation changes as well ag deflections under
conditions where the effect of shrinkage due to drying out were -separated '
from the effect of p la stic flow* Zn some of h is te s ts on Slabsjr though
th e deflection continued to Increases the fib er stress"measurements on the
steel SiadjLaatWBd compression after one year*
Apparently the loading m s
not great enough to form any cracks in the region of the steel and the tot a l shortening di the beam due: to SJirinkage was gyeat enough to cause com*, ;
preseion in the steel and the combined effect of shrinkage and plastic
flow on the CompreeBion side were sufficient to allow tbs increased deflection*^ the concrete on the tension side was, apparently taking the f u ll ■.
bending plus the Compressions which the shrinkage caused in the ste e l,
fhes© te a ts illu s tra te d the effect of shrlrdcage b etter than that, of p la stic
flow,
3& 1916* two other important papers* one by
B, Amlth ^ and another
I* Superior figures throughout th is th e sis re fe r to bibliography.
%
by A9 Tc Goldbeek and. B9 B9 Sidbh
M adtilaa
and a major d lsous s i on by F9 R9
uers published® The f i r s t of these papers brought out the
fast that the plastlo rebovery after a load that bad been sustained for
some time was removed with considerably lose than the p la stic flow under .
the sustained Ioad9 I t 'also showed th at concretes under water flow less
than these in air*
The second paper showed that the p la stic flow is com­
paratively rapid at x rrst and appears t o .undergo & progressive slowing down
as time goes on* , Tests on beams and slabs in which fib er deformations
were measured brought out the information th a t the deformations at six
months were roughly three times the i n i t i a l deformations and a t two years3. .
four t IjTiesa MacMillan made calculations and reported th e p la stic flow in
terms of a reduced modulus of e la s tic ity fo r the concrete or increased
modular r a tio of (modulus of e la s tic ity of ste e l to modulus of e la s tic ity
of concrete)*
Xn I 9I 7f two other important papers appeared in the Proceedings of
the American Ooacrete In stitu te, One* a report by A, B. lord ^ of measureHcnts made during a year on a f la t slab panel* confirmed previous reports
and SBphaBiaed the effect of shrinkage la reducing the ten sile stresses
xxi steel*
Lord reported th a t the p la stic flow and shrinkage tended to
reduce the negative beading resistance* thus producing greater deflections
in regions of positive moment* He also found th a t the “straig h t line
theory* of bending fo r reinforced concrete did not apply fo r the conditions
of the te s t oven with low stresses in the concrete and the steel *
So 'Be Smith 7, in 1917* found th a t the law of p la stic flow of con­
crete tended toward a lim it and th a t the amount of p la stic flow depends
3.0»
somewhat upon the kind, of aggregate Used6 He found that the coarser the
aggregate, the greater the plastic flow* He also found that the action
of p la stic flow is to lower the neutral axis of a beam, which happens
when the modulus of e la s tic ity i s reduced*
I t follows then that the oout**
press ion, area i s increased, thus reducing the resultant couple arm and con,** •
sequently, increasing the u n it stress in steel*
A
la 1927, Oscar Faber ° 5 m authority on mechanics, published in Eng­
land the re su lts of te s ts he had made on beans*
The effect of shrinkage
and p lastic flow was brought into the theory of mechanics by way of change
of the modular ra tio with time*
Froxessor R* B6 Davxs of the University of California, was commissioned
to make te s ts so th at proper adjustments and corrections fo r p la s tic flow
could be made* The re su lts of these te a ts are the most comprehensive that
have so fa r been made and they are published in the Proceedings of the
American Concrete In stitu te, f i r s t in 1926
and la te r in 1931 10s as
well as in the Proceedings of the American Society for Testing Materials
in 1930
and 1934
A number of papers have la tely appeared ia WdLoh theoretical aspects
of p la stic flow have been discussed,
Among them is one by O= S6 Whitney
in which ho shows how the p la stic flow property of concrete .eliminates many
of the secondary stresses in archee, a problem that has concerned designers
considerably and has been the cause of very special procedures and devices
introduced into design and construction.
The effect of p la stic flow figures
largely in the conclusions drawn from the- comprehensive series of tests on
columns la te ly carried out by the University of I l l i n o i s , ^
11
Loren* G. Straub ^
made aome tests and report# that a repeated appli­
cation load* produce effects similar to a sustained load because of the
difference between the plastic flow and plastic recovery.
Result# in a
eerie* of rigid frames constructed and tasted at the University of Illinole
16
serve to cheek the theoretical calculations.
P lastic Flow of Concrete ^
Ordinary calculations fo r stresses and stre ss d istrib u tio n are based
on.the th eory th a t the material i s e la s tic ^ th at the deformation is pro*
p o rtio m l to stress* and no term is introduced fo r time.
It is now known
th a t some stru ctu ra l m aterials' are affected by time and th a t stra in s or
deformations continue to change a fte r th e stresses have become constant,
th is property i s called "time yield" or "creep" or "p lastic flow". Hone
of these terms apply to s te e l'a t low temperatures so long as the stresses
are within the e la s tic lim it, fhe designer o f ste e l parts for stress at
high temperatures where the e la s tic properties are not so d efin ite is eon*
eerned with the property usually known as "creep" and rath er good studies
have been made of th is property of metals.
Other m aterials such as timber,
concrete and resinous substances as celluloid have th is p la stic property,
at. ordinary temperatures and a t any degree of stre ss,
property is different fo r different m aterials,
f c nature of th is
fhe term "plastic flow"
&$ used in th is th e sis i s th a t property of a m aterial which i s evidenced
by the long time continuation of a varying increase of deformation or
s tra in under sustained load or decrease of stress during sustained stra in ,
A plain concrete specimen in compression under a constant load w ill
continue to shorten a f te r loading.
a fte r the load i s applied.
This shortening begins immediately
The change in length at constant stre ss is
quite rapid at f i r s t and slows down as time goes on.
The shape of the
curve of time plotted against deformation i s th a t of a power or exponential
function not unlike th a t of a cubic parabola,
llben ste e l and concrete are -
both present, as In a building column, the p lastic flow is lessened because
of the presence of ste e l which i s not p la s tic ,
increasing proportion of
the
$‘he ste e l must take oh Sn ■
load a@ time goes on* Ba beams, the defies*
ile a continues after, the load: h&s Ibeeih applied, aa& tke stress i s Wreased
in the tension s te e l 'because of the reduction of the resu ltan t internal
couple arm,
The straig h t lime representing r e la tio n s of the. f i b e r -deforma^
tim e continues to change and an increasing depth of the beam i s brought
in to compression,
'
%be ccwept of plastic flow ae Wligated by the term i t s e l f i s readily
applied to conditions of sustained load or stress but the action of p la stic
flow under sustained deformation or stra in i s .somewhat more d iffic u lt to
understand, although i t !La lit t le ,, i f any, lees important. I f a plain eon*
orete specimen is. placed in a testin g machine and a certain deformation i s
imposed and held, the scale beam w ill drop. I f an. attempt i s mad# to keep
the beam poised, the indicated load w ill have to be reduced, rapidly at
f i r s t and then progressively more slowly,
i f a beam-is loaded to a eer*
tain e la stic deformation ,and thia deformtion I* held, the load w ill have
to be reduced in the' Same way as i t would have to be reduced fo r the specie
men on the te stin g machine. I f a. beam is subjected to secondary stress,
the p la stic flow in the concrete will, cause th e fib er stresses in com*
pression in the concrete to be reduced and as a consequence, the stresses
in. the s te e l w ill have to reduce 'also, ' Thus th e action of plastic-..flow
on secondary stresses Is generally beneficial,-
14,
: P la etia i fXom and Volmaa Changes: of 'Generate ;
The best picture o f what goes on in sid e a piece of concrete when
shrinkage or plastic, flow takes.place see^e.to be th a t 'given b y -Etf
Davis4, H, E* Davis aM Hamilton of the University of California, ^
In
te s ts which th e y made miser mass curing conditions, the .stresses developed
due to thermal changes in large concrete cylinders under complete ,axial
restrain t, were measured. I t was found th a t the degree to which flow takes
'
place has an important ■influence upon the stresses 'developed,- They have
also found th a t in a period of 10 years* 95 percent of the p la stic flow
'-
occurring in 10 years todk place in th e f i r s t 5 years and the remaining
5 percent in th e la s t five years* Tests to determine the effect of water .
cement ra tio and of aggregate cement ra tio upon plastic- flow indicated t h a t '
the more the cement used, per u n it volume of concrete* the less the flow and
th a t th is variable was very important* Mithin the range of normal concretes
tested* i t was- bbserwd th a t with pastes' o f the same water cement ratio* the
flow was p ractica lly proportional to the amount" of the paste in the concrete*
In te s ts to compare th e flaw of concretes in Csial-tension -and com*
pressaon* i t was observed th a t a t le a s t "during, the early periods*, the flow
under te n sile stress was greater than th a t under compressive;, stress* 'In the la te r stages* the rate of flow was le s s under te n sile than, under com*pressive stre ss,
.
.
.
In technical books fo r about a decade* there was frequent reference ■ '
to the strength and e la s tic ity of concrete but only an occasional -reference
to shrinkage due to .drying and almost no reference to the gradual deform-:
tlo n udder the action of sustained load* which- has been called tiCreepit*
V
"time y i e l d a , n d "p lastic flow’s
As.
^ however^ of the research
of recent years* there has been developed, a general conception of the
effect of shrinkage and p la stic flow upon the behavior'of concrete strncturds.
I t is now believed th a t shrinkage .and p la stic flow are closely re­
la ted phenomena* each being prim arily due. to changes.in the amount of ab­
sorbed ra te r in the cement g el and being but l i t t l e d irec tly Influenced by
the free water occupying the pore spaces within the concrete mass* ■•■
Many researches were made to find the magnitude of p la s tic flow and
shrinkage under the conditions surrounding a given concrete structure. Bowyerji they were not able to succeed.
The property of shrinkage. Cf concrete due to drying i s altogether un- ,
desirable*
Shrinkage .can, never be e n tire ly eliminated* yet there seems to- ' ■
. be th e .possibility through the proper selection of materials' and methods, of ■
. reducing shrinkage to a point where it- w ill not be a facto r f o r serious
consideration*
On the whole* p la stic Sow does not seem to be ah unde­
sirable property.
In certain reinforced, concrete imbers* i t tends, to
make possible more e ffic ie n t use of .steel*- such as th in structures sub­
jected to drying* as well as in mass stru ctu re s■subjected to thermal
changes due to the hydration Of cement,,'
•
P la e tic .Flow Under long lim e lo a d in g ::
The p la stic flow of plain and reinforced concrete cylinder a* which
have been under load f o r 10 years* are shown in figure. I*
■ -
One load of
specimens was loaded at the age of &8 day* and the other a t the age of
3 months. Prior to loading* 'th e concrete was moist cured and a fte r lo a d -.
log, i t da* stored in a ir at 70 percent relative humidity and
. .
'
'
- - . '
'
/f',/,
J
■
*
'fi
-
16
E
i
i
csss
*s%*
tu t
SSS
B
4>
-B
-§
£
5
.s
I
- H T J__ u__ _____Ui___ ll
%ors
20 30
SO 70 KX)
200 300' 5007001000 2000 3000 5000
I
2
5
id
•Time Since Application of Stress to Specimen, log scale
Tm 1 .—Flow U nder I-onR Sustained Stress
15 20
30 40 5060 80 100
150 200
300400 500600
Time Since Application of Stress to Specimen, days, log scale
F ir.
Z
—Flffect of A c e re e a te C tm e n t R a tio a n d W a t e r C e m e n t R a tio U pon Flow.
17.
sustained, compressive stresses were 300, 600, 900, and 1200 pounds per
square inch. For the concrete loaded a t the age of 20 days to a stress
of 900 pounds per square Inchj, about 00 percent of the flow took place
within the fir s t year and about. 9? percent within g years.
An approximate
proportionality of stress to deformation appears to have been maintained
over the loading period. For the same sustained stre ss, the age at Ioadiag does not appear to have a large effect upon the rate of change in
length a fte r the f i r s t few months under load.
As the flow, however, m s
progressing, more and more load was transferred, from th e concrete to the
ste e l and the rate of flow decreased in much greater proportion than would
be the case fo r plain concrete sustaining the same in i t i a l stre ss.
^pf,.Iggregate-Gement Ratio and Water Cement Batio upon P lastic Flow.
In order to determine the effects of these factors,, a series of te sts
were mad® in which concretes of three d ifferen t po&oat contents and two d if­
ferent water cement ra tio s were used.
All specimens were subjected to a
sustained compressive stre ss of 800 pounds per square inch at th e age of
28 days.
The concrete was cured under standard moist conditions u n til time
of loading, a fte r which the te s ts were carried out in a ir a t 50 percent
re la tiv e humidity and 70° F„ The re su lts of these tests are show on Fig­
ure 2.
At the end of these te s ts , i t was found that concretes having the
earn® water cement r a tio , th e ric h e r mixes have more flow than the lean
mixes. This i s what might be expected i f the major portion of the flow is
considered to be due to the movement in the hardened paste.
Statements
also have been made th a t $ (I) lean concretes flow more than ric h concretes,
(2) weak concretes flow more than strong concretes.
For the same consistency.
18«
a Ieaa concrete requires a Iiigher water cement ra tio than a rich concrete*
Xt appears reasonable to suppose that the more open the pore structure#
the le ss tr ill be the areas of contact between neighboring hydrated cement
grains and so under a given load# the higher w ill be the stresses over
such areas of contact,
There appears the Interesting p o ssib ility th a t the
p a rtic le sic© d istrib u tio n of cement#, through i t s effect upon the structure
of the hardened cement paste# m y have an 'important influence upon the rate
and magnitude of volume changes in general, .
Flow ila Aasinl Tension and. Qompressiont
The re su lts of a series of te s ts to study the flow of concrete under
sustained tension and compression are shown in figure 3,
low heat Portland and a normal Portland are compared,
Two cements# a
Prior to the time
of loading#, the specimen© were maintained under.mss curing conditions
from a sta rtin g temperature of 40° F,
The load was applied, a t the age of
28 days. During the period of te s t under sustained load# the temperature
was SQ0 F»
ThC tension, specimens were subjected, to constant sustained stresses
of 50 and 100 pounds per square inch and the compression specimens ware
subjected to constant sustained stresses of 200 and 400 pounds per square in c h ,„ At the end of these tests# i t was observed th at at the early ages#
the flow under te n sile stress was greater than that under compressive
stress*
However# a fte r a few weeks under load# the ra te of te n sile flow ■
.
was considerably le ss than the ra te of compressive flow# ■While th is pro­
perty of flowing more in tension than in compression may not be character- ■
Xntlc of a l l concretes# i t obviously is a desirable property, since i t tends
19
15
20
30 40 50 60 60100
150 200
300 400 500 600
Time Since Applicotion of Stress to Specimen, doys, log scale
Ku. 4
KtTevt of Composition and Kimness of Cement upon I low
Flow for Stress of I lb. per sq in
040
W J a n d - 1J * ? ! * *
0.30
I
k
^
/
020
A
--------------------------
T
Normal PorHond - Tension
'
" ^ n d A c o jn o r e jS S ^ 7 Ww - I S - J - « - -
-
I
.ComprosvoJL------------ --N o r r ^ l ^
E 010
I
20
40
60
60
100
120
Time Since Application o f Stress to Specimen,days
Ku; 3
Flow in Compression and Tension
Mass Cured Concret -
go,
to reduce cracking e ith e r due to drying or thermal changes„
Fiber Strains Under Sustained Flexural load,
Almost the same re s u lts as those in determining the flow in axial
tension and compression ware found in te s ts to determine the change of
stra in with time in both, plain and reinforced concrete beams under con-*
s tant sustained bending moment. Obviously, whether the stru ctu ral element
be plain or reinforced, i t is desirable that the concrete of which i t is
composed be of quality* which, under drying conditions, w ill exhibit maxi­
mum flexural strength under sustained moment-* In other words, i t is de­
sirable th a t the concrete be me in which the effects of p la stic flow and
shrinkage due to drying, one tending to o ffse t the other, would combine to
produce the most favorable d istrib u tio n of stress*
E ffect of Fineness and Oomposition of Qement booh.Plastic Flow*
At the end of these te s ts , i t was observed th a t th e coarsely ground
low heat cement exhibited much the greater p la stic flow a t a l l ages, and
th a t i t s ra te of flow was greatest not only a t tho early ages, but also
at the la te ages* Boferring to figure 4, i t i s seen th at in the ease of
tho low heat portland cement, the flow i s greater fo r the coarser cement*
In the case of the normal port land cement,, the reverse is true*. The reason
fo r the reversal is not evident, but i t seems possible th at i t may be due
to variations in the p artic le Bias d istrib u tio n of the cements* Results
of la te r series of te s ts covering a wider range in specific surface of ce­
ments exhibit sim ilar p e c u lia rities again pointing to the p o ssib ility th a t
the p a rtic le else d istrib u tio n of a cement may exert an influence Upon the
flow ch aracteristic of cement,
.
Sla
Tgffeot of -Shrinkage ^9
■ The shrinkage of Ooncrate i s a .very complex action and is effected
by many factors^ enqb as temperature, moisture OoMitlons, amount of rein*
xoreemsnt and size and shape of the concrete m ss as well as fey the charge*
t eristics of .'the 'concrete it s e l f .
• ". '
Concrete-exposed to the atmosphere. Continues tc shrink fo r a long time., .
rapidly a t .firs t- a n d 'a t-a -deereasing ra te u n til a fte r several years^ i t praoticffilly ©eases»- laboratory inirestigations .indicate th a t the ©mve o f . shrin*
kage strain plotted against time i s sim ilar ho th a t, of p la stic flotn .' The •
amount of shrinkage is greatly affected by temperature and humidity stor*
age.
In saturated a ir or in water,, there i s "a slight, increase- Sn length,
which continues. over a- long period*. The .lower'the- humidity of a ir ,' the- greater the shrinkage of exposed concrete »• , I t -is apparent th a t large •
mas-seaJdo not shrink- as rap id ly or as much as small masses exposed to the
.
drying effects of the air* - I t is also apparent that the concrete near t h e '
outside of large- masses w ill shrink sore 'than the ,interior# settin g up nn- ■ ■-■-.
equal stresses which a re, a t le ast to some degree, released; through plas*ti© flow. -Xf th e 1shrinkage i s too rapid, cracks-may'form due to the re*
strain in g effect of reinforcing ste el or a damp; core- of concrete. Paris ^
has measured'the .shrinkage,of laboratory specimens of concrete stored fo r
about 850 days under various-.controlled humidity conditions.
These were
4 x 14 inch cylinders ■of I ,part cement to. 5,6? p arts gravel# watterr-cement
ra tio of 0.89# stored in fog. fo r 28 days and found the re su lts shown on .
'!‘able I .
Storage Condition
Total Change in Length (in inches)
50 Percent B eiatite Humiditgr
0=006707 shrinkage
70 Percent Beiative Hnmidity
0 &000706 shrinkage
100 Percent Belative H m idity
O>OQ0176 expansion
Water
O9000142 expansion
Tahle I ,
Effect of Hoistnre on Change•in Length o f. 4" x 14» Concrete Cylinders
88 ,
'
Stresses Developed due to Shritiftage
Ihe effect o f shrinkage upon the stresses in a reinforced concrete
member may be understood by considering 'figure 5> which shows a member of
unit eroea ae&t&onal area- A, eym etrldally reinforced with eteel In amount
P&* I f the piece were without reinforcementj. th e shortening or shrinkage
would be B but th e -action of steel reduces the actual, change of length to
Sa*.
'
,
'
:
'
'
I t is obvious from the figure th a t th e in teractio n of the two materials
re s u lts in a compressive stress in the ste e l' and a te n sile stress is the •
concrete, the two stresses being equal in magnitude fo r equilibrium*
A
plain concrete specimen t a l l shrink a s 'i t .dries out without producing any
e%r*ee in the member due to shrinkage+ Steal, on the other hand* does not
shrink as the Concrete dries out so i t s re s tra in t tends to reduce the vol­
ume change of. reinforced concrete..
Gdnsideriag figure 5»-8% caa.de* that.the fihal length of the steal
and concrete must be the same so equating the strain s, we have
■
dr
S sg - I '
S ^
fc
^^
w
■
(S)
where
tensile stress in concrete
i s - compression stress. $n steel •
As the stresses -at a given section must be in equilibrium,, equating the
ceKKpareeeiare fo r# , in the s te e l to Iihxa tension stress in this concrete* we
have
■
.£sPA 4' fe le x fe(l^P)A *
( 3)
Solving equation ■$ fo r f s ,
fa sr
.>?
'
PA
P
.
Siifostltutiag tfoia valtie of fs in SqtiatifoB ( I ) 3,
M W
riSc
Solving for £®s
f e '*
'
% S. ,1»
: Ee
tP '
RPtW J
W
Zn the same way2
f s «■
...
R (W )P
( 3)
As we don't know exaetly the value -of.the- shrinkage: e o e ffie ie n t Sjl the
*ae of these equation Ie at beet acrnde. approalaablon* P&aetlo flow aote
t© tfoclttoe the. stress, set up in th e eonorete hy shrinkage .,and th is my foe
allowed by using an lnorsased value of a 1# the oomputatians as V m be
demonstrated font* to cover the effect of flow in reducing shrinkage, CBanV llle ' state* that a la generally about 1# and w ill usually l i e between
10 and 0O4 Other investigators have used BBitxeh larger value-#
-
'
As '< l 3l* hut, Ac & A ^ Asy so Ac s? a * PA $. (l*p)A«
** Quoted by 0, St Whltoey in Journal
Voly 28* 19)2* p, 500,
*** Por eaample, Ztraeaure end Bbmrer in "Principles Cf Reinforced Oonorete
Construction* 4 th ed** p* 3 5 1 * * 1 # taken as 4 $ for th is oaaa*
* In columns,, i « M
%5
/
1 (O)
“ t o o l . z e d 'I '
S t r e s s -S l r o i n
“ Oiflgroms I ------
O
0.000«
a 0016
'f
(b ) I
Approx,motion
to Stress • Stro.n
■*“ D*cgr o r s
00024
00032
0 0006
Unil S t r a i n , f ,
•----
-►-T- - w—■——- - -
-• -T-
C 0016 " 0 0024
Unit S i r g in,. I c
R g . 6 & 4 d e a l i i e d i t r e n -s tra in c u rv e s l®» c o n c r e te o f d iff e r e n t s lie n g th s
S
£
/< V
. u>iit Len
Fm. r52
file P la stic ity Batio
-For the purpose of. explaining the behavior of concrete in' a flexural ■
memberj a ty p ical set Of stress strain-curves extended to .rupture fo r con­
crete i s assumed, -as shorn in figure 6a$ these curves are then further
idealised in, that- they are represented approximately by & series- of traps*soids as shown in ,figure 6b, ' This s e rie s'o f curves is su fficien tly broad
in scope to cover a ll grades of concretes from the highest to the lowest
' strength.
The. proportions of the trapezoidal diagram fo r -a given 'em*
pressive strength.of concrete5 fe , depend on two'coefficients> namelyv Se,
the in i t i a l modulus -of ela sticity of, the concrete,, and jS, the ra tio of the
p la s tic deformation to th e to ta l deformation of the concrete at rupture,
defined as the "p la stic ity ra tio " , ' Although a ..wide scatter is shown, by
the te s t data rela tin g to th e in i t i a l modulus of e la s tic ity of concretev
1
.
.made with gravel or crushed stone, a satisfacto ry approximation may.be
made from the following empirical formula fo r the modular ratio* ^
*1
- '
where T-lO is the numerical value of the -compressive ■strength-, expressed
in pounds per square inch,. The formula i s simple and easy to use.
The
modulus of -elasticity of steel,. Be, may be taken as' 30 x IO0R
The existence"of a- property of concrete measured quantitatively by
the p la s tic ity ra tio may be recognised in the behavior of concrete com­
pression specimens, a t loads near rupture.
The sudden rupture of the spec!-*
men of high strength almost 'immediately'after1the maximum lo ad -is reached,
gives proof th a t the horizontal portion of the idealised stress strain '
-diagram i s very small fo r concretes of Iaigti strengths
The gradual breakdown of the specimen of low strength^ accompanied
by large stra in s and the maintenance of a considerable proportion of i t s
msMmia resistance gives evidence th a t the horizontal portion of the trape­
zoidal diagram i s .relativ ely large fo r concretes of low strength*
If £ y
i s the to ta l concrete stra in represented in the trapezoidal diagram at fin a l
rapture5 and f$£, is th e p la stic •'s tra in represented, by th e horizontal por*
tlon of the diagram* then^ i s the p la s tic ity ra tio which.my have a numeri­
cal value within the .extreme range between aero and one*
For the purpose of Calculating the ultim ate resistance of the beams>
i t i s proposed th a t @be determined from the.- empirical equation ^
( 2)
where £fc i s again the numerical value of the compressive strength of ooivcrete expressed in.pounds p e r■square inch..
The essen tial feature which i s being brought out here i s th at fo r a
given strength of concrete^ there are two .characteristics necessary to
describe'the stre ss strain relationship a l l the. way to rupture„ These are
the. modular ra tio defined empirically ■by equation (I) and the .plasticity
ra tio defined empirically by equation (2)*
The complete trapezoidal stre ss
diagram is thus, determined by f }c since n5 Bc ' and ^ arc then known from equa»
tio n s I and S5 and since the strain s defined .is figure. 6b are then obtains
• able from equations ,
.
*
where
Ir
- -. f 'c
^ i = ( I - >Ec
and trhere
to ta l concrete s tra in represented in the trapezoidal dia­
gram at fin a l rupture=
. .
p la stic stra in represented h j the horizontal portion of the
• trapezoidal diagram*
(? # p la s tic ity ra tio ,
» t o t a l concrete strain minus p la stic strain .
'.
'
89,
'
'
/
P lastic Flow Design of BeinWced Oonerete Members
Subjected to Plexure 1 *■
The present Method of designing' members under bedding^ combined
Whdihg and direct stress i s m ectisfactcry for several ,reac<ms»
•The usual formulas based oh assumption of a cracked section and
straig h t lin e variation of stress are f a r from correct both under working
loads when the concrete i s not, M aterially cracked' and under, ultimate loads
when the stress, in the, concrete i s not even approximately proportional to
-
the distance from the neutral ax$.s«.
The usual flexure formulas are complicated by th e use of the. value of
the modular ratio, (the modulus of elasticity' of steel to the modulus of
'
e la stic ity of concrete),, which ie quite unpredictable under high loads
. and a c tu a lly has l i t t l e effect on the ultim ate strength of the beam.
effective value of n 1$ tddel^ different for dead and liv e loads.
The
In the
,case of arch rib s , th e deed, and liv e load stresses, cannot be computed sep­
arately with difference- values of n and added together, because dead load
-may produce compression only while the 'liv e load moment alone would require
M sm ptim of & cracked aeqtloa*
complicated with two values of n*
The. Calcul&tiCA in th is case would bo very
' -: '
The present method makes no prediction of. .!the loading which w ill causa
cracking' and does not give accurate control over the' factors- of safety
under dead loads and liv e loads=
I t has been pointed out recently by several investigators that while'
the- -stress v a ria tio n -in the concrete 'is approximately lin e a r under very'
lig h t loads, and parabolic- under intermediate loads., as' the ultimate load
•is approachedj,. i t assumes a shape about as shown in figure ?a„ The stress
in crease very rapidly soar the n eu tral -acds.and i s nearly uniform fo r the
greater part -bf the depth o f the compression section* probably decreasing
slig h tly toward the .edge of the beam# The usual parabolic formulas have
been based on. the theory th a t the stress variation in the beam followed theshape of. a parabola^
•
■
This evidently la not tr u e because of the greater ultimate- strains
■®ad the differen t behavior of the concrete Im the beam=
I t i s , therefore* proposed th a t a rectangular block of uniform stress*
as indicated in figure rIhi be used to represent, whatever s tre s s ■My ex ist in
the concrete* Whatever- i t actually is*, i t must have m average intensity*
f c ? and an effective depth* a* The resu ltan t i s a t the middle ■of the re**
tangle. Pridar ultim ate load* Hookers law and the. theory of e la s tic ity have
no significance as f a r as the internal stresses are concerned* Ho further
theoretical, ju stific a tio n of the assumption a rectangular compressive stre ss
block i s necessary i f the formulas derived therefrom accurately predict th e .
ultimate strength of the member*
51
rig. 7 «.
A ctual S treea D is tr ib u tio n
P;
AnsuEed D esign S tr e ss
D is tr ib u tio n
HrtnjorCtti I 'oncirU Sfrtttlx-rt I tuUT HUrurr
C m<3 & f c
>
Assa. ed D esign S tr e s s D is tr ib u tio n
. Sj-mple Fleamye
fhe assumed relatio n s fo r a rectangular beam under simple flexure are
shown In figure 8*
. •■
•
I t 1® .ash'tffitsd th a t in m under^relhforead,
.
that I s 5 one wfaieii
■r
w ill f a i l in the te n s ile steel# the concrete w ill crack as the ste e l s tr e t­
ches and the depth of the beam in .compression# a % w ill ,be reduced u n til the concrete unit stress-reaches tfae ,ultimate.' and 'as T. s?1'$
a ■«
'-■
(i)
bfo
in which$' 'As, = Area p f te n sile steel#
fy - Yield point stress in steel* '
• ,■
'
b w Width of beam.
•
.
<
,
fc = Ultimate strength of concrete,
This determines' the lever arm of the ste e l reinforcement since C ® 4 I t w ill be assumed th a t the ultimate compressive strength of the con­
crete in the beam i s etguai to 85 percent ■of the ultimate compressive strength
Of a tested cylinder
The values' of a and a are derived as follows for any p articu lar bending
moment. Bee figure 9„
- ' '
' ; ' M% %
Ge ^r, W
3 f o ( ^ ' (2)'
fo r which
a 3 'd **^4^ or
(3)
* The depth* a# i s le ss than the distance to the neutral axis# kd.
"
53
.
and so® sequent,
Z~
~^ ~
)
(4)
Th*@« axpraoslooe are lmu*p#uU*nt of th e are* of e te e l and the value of
4%.
TM required s t e e l a re a w ill hex
K = Te = fy A * x C
A* = M
(5)
Sine* th e assumed oompreeslve s t r e s s distrlb u t.'lo n ha* no exact th e o r e tic a l
b a s is , th e I i a itiU i value of tu e depth of eompreseion, a , fo r exual con-'
Crete
s t e e l stre n g th * in fle x u re «**L be determined a x fo rlr.o n tally .
I f th e beam has a t l e a s t s u f f ic ie n t s te e l t o f u l ly develop th e stre n g th '
o: th e concrete, a d d itio n a l s te e l does not a e t e r i a l l y increase th e s tr o n ^ h
of the beam*
The l i i i t l n g v alue of a
as computed from t e s t s I* reported as
#
0*53?d . The value of fg sssumsd in equation (3) I e 85 percent of th e
corresponding cy lin d er stre p i l o*' fg = O .igf^c.
The fle x u ra l stre n g th of a f u l ly rein fo rce d re c ta n g u la r be<ci with ten ­
s i l e s te e l only i s th en , fro * equation (2 ),
R = (d - #)&bf* = (d - C.2685d)0.537dbfe = 0.393hd^fc.
D iic I* a simple equation by which to detsn&ine the se ctio n olweueioae.
* The lim itln ? value o f a as computed from t e s t s rep o rted by S la te r and iy ee
*
.
■ ,
34. '
Uo determine the ste e l area, we say th at
H » Ue , but T » UylfrAs and 0 = (d «
a 0.73Sd
therefore$
K = W S <0. 73M) ^
A s- - J - _
Uhls balanced ste e l area is-' considerably more than has been used generally
The p la stic theory tends to gilre a smaller beam section izith more steel
than the working^atres@ straig h t lin e theory.
^ fy is th e.y ield stress of s te e l.
' SG,
Experimental Vfork
As mentioned in the introduction,, the purpose of th is -work was to
find the actual stress v ariatio n in the compression side o f the beams.
For th is purpose, two & feet long beams were designed*
One of the
'beams m s designed by the straig h t lin e method-and the other by the plas­
ti c method.
The straight lin e beam was designed fo r a load of IiOOO pounds
and the other fo r a load of 5000 pounds. ■ The strength of the concrete was
taken as 2000 pounds per square: inch, with a water cement ra tio of 8 gallons
per sack of cement
and a cement-fine aggregate-coarse aggregate of 2-2i:-3£=
in order to find the actual strength of the, concrete,, fiv e standard '
cylinders were made to be te ste d . I l l of these cylinders were made from
the sum© paste as the one used fo r the beams.
.
,^ .niililmimm I......................................... I
& Reinforced Concrete Design, by Sutherland and Reese, Second Edition,
John Wiley and Sons, Bew York.
Design of Beams
Btralfgfafe lin e Method*
Load 4 k ip s ; f*s » 80000 pounds per square inch
f"c 5 2000 x 0.45 =' 900 pounds per square inch
Span « S fe e t
& ~ IBt £■ 22j£JSf,. s 15 ’
Be
2 x IO6
, H
2000 x 4 '* BOOO pomd^foofe
s SOOO x 12 X: 96000 pouad-inch
Ka
I
I T S : '
nf»c
*
°-w
K si 2. *
j s l -§
3
0*866
Bj a 2 8 ^ x 0.366 x 0d403 S 1$8
M a BM2
96000
d5 s Z S . * ^ 158 x 6 % 10 inches
S teel Area requiredi
Ma
a fs A9 jd
96000 * BQOOO * 0,866 %10 z &a
Ose 3a 0” bars
A* *
4*8
Area provided ,O.*6o I n ^ 0*555
= *525
o*k«.
Ofaeek for shear:
v
—
-g ,
3 38*4 pounds per square inch'
Being special anchorage (deformed bars)
■ v a Go03fc
0.03 x 2000 * 80, pounds, per square inch'
60>38*4
Bb stirru p s required.
Sbeok fo r bonds
2000
u st
Allomble v,
37i"d x 0.866 x 10
- 42»5 pounds per square inch
OtiOUtc- = O0Ok x 2000 a SO pounds per square inch
80>42,$ o+k.
Masdatam D eflections
4000 x (8 & 12^
A = J i!*
40131 48' F ^ a c lO ^ K S ''
a- 0=0737 inches
12
' Ihe cross section of th is beam is show in figure 9<
Chock of Design fo r the F irs t-Boama .
» - A = 2^
r
= 0-01
K « / 2 p T f (pn)s - pn S' fz(0~01) x 15 / (0,01 x 15)® * O0Ol a I?
a =f / o 3 o - 0=0225' - . 0,15.. s P »52? - 0.15 » .377
K * 0.377
j = I - ^ » I ^
^ 0.87&
$ * 3 L , _. 96poo..
» 11000
jd 0.875% 10
0 a -I** -s 96000
#
0 , 87^ % 10
11000 pounds
96000.
, fg .s
. ■
B a Gjds§S(Kd)(,b)jd
5*25
fc
fe a
s
18300 sounds per square inch ,
,
3
192000
0,377 x 6 %IOO x .875
Si ,970 pounds per square inch
38
Fig.Io
Cross S e c tio n o f th e Beam Designed by th e P la s tic Flow Method
....... ZsJ1 —
Fig .9
Cross S e c tio n o f th e Beam Designed by th e S tra ig h t Line Method
P lastic Method?
Load 5 k ip s.
Span S feet
f ?c ~ 2000 pounds per square Inch
fc ~ 900 pounds per square inch
fg # 20000 pounds per square, inch
fy s, 50000 pounds per square Inch
H = O z c = O.egf'c (0*537)d z 0„732bd
Assuming b * 6 inches
120000 % O.eg Z 0.4^ Z 2000 % 0 J37d % 6 z 0»?32d
d2 » l / ^ S 222, a 8,1$ inches
y 1800
Use d = 10 Inches
Steel Area required;
M 2 Tc 2 TyAsC
120000 S 20000 z IQ z 0 o732As
A* =0*823 ino&2
Vee
0 inch bars which furnish IoOO'?0,823 in*^' o,k, .
•
Check for shear;
tf
-OtirtA
v =—
___ 3 ____ 2500____ _ 2 $8 pounds per square inch
b Z 0,732d
6 Z 0.732 % 10
Allowable
, v & 0,03 f *0 s 60 pounds per square inch
60>£8
o,k»
Check fo r Bonds .
u .=
£& x 0,732d
?
5r | z 0.732 x 10
3 43»5 pounds per square in
Allowable t i 0,04fc - 80 pounds per square inch
80>43,5 o,k.
Maadmum Deflections
^
5000(8 x 12)3 z 1 2 % 0,0921 inches
ZL s PI:
48B1 ** 4 8 z 2 3 c l O ^ ^ 6 ^ 1 0 ^ '
Cross section of th is beam is shown in figure 10,
40.
For the same design loading p la stic theory gives smaller sections hut
uses more steel»; Therefore * fo r Countries where there is an abundance of
ste e l j, p la stic theory i s much more economical than the straig h t line, met hod. 0
Oh the other hand, countries as Turkey, where there is no ste e l and an
abundance of concrete, th e straight lin e method is very economical because
the imported ste e l i s very expensive„
The factor of safety should obviously depend on the nature of the
structure and i t s loading, - I t may be desirable to provide d ifferen t f a c - ,
to rs for different kinds of loads in some' cases,, As a. general .proposition,.
i t appears th at allowable- dead plus- liv e loads could be four tenths of the •
ultim ate loads, and for extreme combinations of dead and liv e loads, and
wind, temperature and shrinkage effects , one h alf of the ultimate might
' not be excessive.
41
Sample Calculation of Calculated Stresses at Failure
Straight Line Method;
/ / / / / / / / / / / ,
To find the neutral axis of the above section
l^x ( i ) ~ nAe(lO-x)
3x2 e 90 - 9x
3%2 / 9x - 90 = 0
d iv id in g by 3
x 2 / 3x - 30 = 0
x = -3 - i / T / A x I x 30 -
3tI
= r l .r,
2
- -2 / 11.5
X2
therefore,
“
2
= -7.2
= A* 25
kd = 4.25 inches
jd = 10 - 4.25 / |(4 .2 5 ) =* 8.6 inches
To fin d the s te e l stress at fa ilu r e :
M = Tju = TgA8Jd
"_ K
A ijI
f S —
but M S Pl _ P
X 8
x 12
4:£o
#*24P
Substituting back we ,get ■fg * &ML ?s
■ Aajd
fg s r ^ x 13000 - 505OO pounds per sq0 inch
3 x 0^2Q x 8.6
To find th e concrete stress a t fa ilu re
a Ojd » = # d x b x j&
fo ? "T — ------ *
■ M x b -x jd
fH' «»
b o b '# = 24?
68P
_ „ U B x 13000
3 2047 pounds per
M x b x jd 4»25 x 6 x 8,6
square1Inch
P lastic Plow Methods
To find the s te e l stress a t failu re
M - Tc, ^ fgAsc
f" a JL ZUP5 AsC
5 x 0.20 x 0.732 x 10
fs ~
1.0 x 7 .3 2
* '656OO pounds per square, inch
To find the concrete stre ss a t failu re
,M s
Co -
O sSSfcf it a x b x c
fC«" s _: / ..... 24?________ .
5,37 % 7.32 x 6 x 0. 8g
24 x 20000 - 2390 pounds par
201
.square inch
To find, the unit stra in in steely we m ultiply the exbensoraeter- reading by
I .
1822
43«, •
■Sample. Galeulation o f Actual Stresses in S te e l
Bg=S&
6S
Eg C SO xiO ^
Ec = modulus of e la s t ic it y of concrete (pounds per square inch)
Bg =' modulus of e la s t ic it y of s t e e l (pounds per square inch)
Ss = actual str e ss in s t e e l (pounds per square inch)
s' actual strain in s te e l (inch per inch)
JSc * actu al str e ss in concrete (pounds per square inch)
~ actual strain in concrete (inch per inch)
• B5 =S 30 3£ IO^ x O0GOI38 Sr 4140G pounds per square inch
Sample Calculation o f Actual Btresseo in Concrete
E0 = &
/
<©
B0 = 1 .4 x IO^ x c00112 % 1570 pounds per square inch
The r esu lts of straight lin e and p la s tic flow methods are shown on Tables
IV and V,
Data fo r th e Beaa with 3 Longitudinal Bars - Straight Line Method
Load in
Pounds
S tress in
S te e l
Ifos/ia^
I
2
0.00011
0 .1 2
0.000098
3000
0.26
WOO
k
•
StL froa top
O
0
Q
0.000016 '.
0,00015
0 ,00014'
0 .0 0 0 2 2
0 .0 0 0 2 6
0 .0 0 0 2 2
0 .0 0 0 2 1
0.00035
0.00040
0,00035
O.L3
0.00033
0,00050
0.00032
0.00048
. 0.72
.. 0.00039
0,00063
0 .0 0 0 6 3
0.00059
1 .0 0
0.00082
0 ,0 0 0 7 4
0.00076
O.OOO7I
7000
1*25
0.00104
0,00090
0.00089 ~
0.00081
8000
1*30
0.00123
0.00104
0.00101
0.00095
9000
1*69
0.00138
0,00112
0.00112
0.00106
—
-™—»
0.00130
0 .0 0 1 2 5
0.00148
0.00138
12000
0.00180
0.00149
13000
0 .0 0 1 7 0
0.00162
1000
0 .0 2
2000
3000
3
1« from top • 3lt from top.
0
0
0
Ocnerete Strain in in /in
Strain in
S te e l
' in /in
6000
.10000
HOOO
.
.
---------
Table I l
• 0,00118
0,00130
.
0,00138
Daia fo r the Beam x-Ath 5 LoBgitudinal Bars - P lastic Plow Hethod
Load, in
Pounds
, . ' v V
I
:
2
3
Estensomster
Beading
0 /
Strain in
S teel
in /in
I 11 from top
3U from top
Stt from top
O
0
0
0
Strain in Concrete
^zdo
0.06
0.00005
0,008138
Q. 000125
0.000113
‘3000
0.33
■ 0.00016
0 .0 0 0 2 5 0
0 .00033S
0.000225
4000
0 . 2$
0 .0 0 0 4 0 0
0.000475
0.000350
3000
0.36
0 .0 0 0 3
0.000500
0^)00587
0.000500
6000
0.49
0 .0 0 0 4
0.000600
0.000700
0.000587
7000 .
0.61
. 0.0005
0.000725
0.000730 \
0 .0 0 0 7 1 2
8000
0«74
0.0006
0.000888
0.000900
9000
0.86
0.000?
0.00304
0.00101
0.000975
• IOOOQ
0.99
0.0008
O.OG&16
0 .0 0 1 1 4
0.00110
UOOO
1.12
0 .0 0 0 9
0.00330
0 .0 0 1 2 5
0.00121
12000
1 .2 6
0.001
0.03143
0 .0 0 1 3 8
0.00127
13000
1.38
o .o o u
0.00156
0 .0 0 1 5 0
0.00137
14000
1.52
0.0013'
0.00170
0.08160
0.00150
0 .0 0 0 2
.
.
Tabls U I
. 0.000875
load. InPounds
IgOOD
Sztensemetsr
Beading
1 .4 6
16000
S train im Soncrets
S train In
Steel
in /in
I ft from fop
3H from top
5Wfrom fop
0 ,0015..
.0.00180
0.00168
O.0M63
0*00191
0.00180
0.00175
0,00208
0^00280
0.0G087
<W30212
0.06200
M*'m*m**»*a«a*
17000
ISOOO
«•
0.00230
19000
——
0.00238
20000
fa ile d
*=.«. =**.**
1JEaMe I I I
0.00306
ecmtinne 4
Straight Line Hethod
■v
BaaB: with 3 Longitudiaal Bars
Bg * 30 % ^
Load in
Pounds
Sg ^
% IO^
O
.
0
0
0
0
0
480
154
210
196
3000
308
364
308
6300
490
560
490
10500
700
7^
882
1065
672
1000
WO
asm
3000
9300
13990
219
438
657
4000
16600
876
3000
6000
53250
57900 .
1095
1314
17700 ■
24600
7000
$000
32550
37500
1533
1752
31200
36900
9000
41850
46500
1971
41400
12000
51150
55800
5409
2628
13000
60450
1000D?r"11000
,
.
2190
.
§
O
8
SH
A
Computed theo- Computed th e Computed
• Computed actual stre s s of
re t le a l stre ss o re tic a l stress ' Actual
concrete in Ib s/in 2^
of concrete in stress of —------ .— —
of ste e l in
.....- 1■■
I b /iA
Ib/in -«•
ste e l in
lh / inlj’-x3 " from top
5 Mfrom top
—*——
2847
T a b le I ?
# For method of computation^ see page Al,
-is* For method of computation^ see page 43,
:
882
1035
1260
1460
1245
1415
. 1570
1820
1570
1750
2075
1935
1485
1652
1820
2240
2090
1930
2380
2270
825
1000
.
1135
1330
,
.
f l s s t i c Floi-x Kethod
Beam with 5 Longitudinal Bans
% =
Load in
Founds
O
Computed the­
oretical
stress of
ste e l in
Ib/to* "
0
.■
2200
7200
3000
4000
9823
13120
3000
16400 ■
I968O
.
G
263
358
10000
32800 ■
HOGG '
3 6 0 8 0 ...
39360
42640
43920
49200.
32&80
$3760
39040
62320.
63600
I670
.1790
I 9l 0
2030
2130
.2270
#90
7000
8000
22960
9000
29320
12000
13000
14000
13000
16000
17000
10W0
19000
20000
26240
x
Be =
Computed the­ Computed ac­
oretical
tual stress
stress of con- of stee l in
lb /in 2"^
Crete in „
lb/in^ ‘
.477
396
Tl?
935
954
1093
1194
1312
1430
1330
6000
30
* For method of computation, see.page 42..
*5^ For method of computation^, see page 43»
1 .4
* IO^
Computed Actual Stress of Concrete
in Ibs/in^ ^
I ff from top
311 from top
'5” from top
0
1300
4800
6000
Q
193
G
. Q
' 138
9000
700
12000
840
1013
1242
1435
1623
330.
360
13000
18000 .
21000
24000
27000
:
:
1820
2000
2182
30000
33000
39000
45000
— —
\
2380
2320
2670
2910
3080
3330
173
474
' 665
. 820
980
313
490
700
820
IOgO
1000
1260
1225
1412
1364
1340
1695
1778
1595 .
1750
1930
' 2100
2240
2330
2320
2800
2970
3020
1918
2100
2280
2430
2620
2800
2880
. ;
... / -
'
. -
4 a ,.
. ; .
.
.
.- ,
• Discussion and Ooaclusion'
The following discussion i s based m Tables IV and f and Figures I l 5
12 and. %3.
'
'
At low loads there i s a -great difference between the actual and the
computed stress in ste e l fo r both design.methods* This i s due to the fact
th a t at low loads, most of th e tension i s taken by the concrete and a very
small amount by the steel*
The neutral mils a t t hat time is fa r below the
actual one and is very -near to the ste e l bars*- Consequently,, the moment
arm of the re sistin g moment i s very small as shown in figures 11 and 12«,
However, as the load is increased, i t is seen th at the actual stress is
catching up with the computed one* This i s due to the fact t h a t .at high
loads ste e l i s taking a l l the tension and none is l e f t fo r the concrete.
The neutral axis goes up and the moment arm of the re sistin g moment is in­
creased.
At low stresses, there i s a small difference between the computed, and
i-
"
actual concrete stress* ■This difference Is probably the e ffe c t of sh rin k -.
age.
The evaporation of water causes shrinkage in the concrete, which in
turn produces some tension in th e concrete and compression in the ste e l.
Therefore, a t low loads, we can -say th at the concrete has very small stress
due to the fact th at the- compressive. s tr e s s ' produced by the -application of
a small load is overcome by the tension in the concrete produced by shrink­
age.
See figure 14«,
,
50.
53.
(a) Shrink 'e Str sf>
Lr lbution
(b) Compress Stress D istri­
bution dui to Loads
Fig. 14
Stress Distribution
At high loads, however, i f we compare Table IV with Table V, we see
that Table IV shows the measured concrete str e ss to be somewhat le s s than
the computed concrete stress by the straight lin e method.
#
This indicates
that the beam has too much concrete, as was expected.
Table V, on the other hand, shows the measured concrete str ess to be
greater than the computed stress
by the p la stic method.
This i s due to
the e ffe c t of p la stic flow, which produces a redistribution of str esse s.
This redistribution of stresses raises the neutral axis by a small amount.
On the other hand, however, i t lowers considerably the point of application
of the compressive force (see Figure 12) thus decreasing the moment arm of
the r e sis tiv e moment and increasing the str e ss in the s t e e l.
Also, the
strain s at high loads are increasing rapidly while the str esse s are almost
constant.
This, of course, occurs somewhere in the v ic in ity of fa ilu r e . *
# See page 41 for computation of stress in concrete by straigh t lin e method
** See page 42 for computation of str ess in concrete by p la s tic method.
where the stre s s-s tra in curve becobs s f l a t „
. I f we examine ta b le V 'and cosipar© the concrete stress corresponding
.to the ste e l stress a t the yield, point with the concrete stress corres­
ponding to the ste el stress a t the yield point in table I ? ? we see th at in
. table y, when the -steel stre ss is at .its yield points the concrete stress
i s very near i t s ultimate cylinder stre s s# which -indicates th at the p la stic
flow theory gives a balanced design.
On th e other hand, however, i f we
look at table I? , we see that although the stress in ste e l has reached i t s
yield p o in ty the corresponding concrete stress' is very low, which proves
that, the stra ig h t lin e method does not give as balanced a design as the'
p la stic method does,
Another important case which is observed is the fact th at the compressive stress is always greater than the te n sile one,
Ih is i s most probably
due to the measured concrete strain s which were, somewhat inaccurate.
How­
ever, the greatest mistake-in th is case i s where we assume the modulus of
e la s tic ity of concrete cylinder to be equal to the modulus of e la s tic ity
of the beams te ste d .
I t has been shown th a t the modulus of e la s tic ity of
■concrete is something very inconsistent and that we should never re ly on i t ,
■■ I t i s assumed that-Hooke’s Law applies to concrete also to some d efin ite
lim it, but th is is not true as the stress stra in curve of concrete is not
a straight lin e as shown in figure 13», Therefore, i t i s concluded that a
'
small error in reading the strain s multiplied- by a. wrong modulus of e la s ti­
c ity w ill give re su lts th a t a re 'in e rro r„
- - Comparing the external .applied moment- with the in te rn a l resistin g moment
we see th a t in every ease the external moment is greater than the internal
SG.
o$i©s ixi ths most favorable ease the ■e rro r'heltig 13 percent,
Figure 16 shows the cracks which formed in the beam a t fa ilu re due to
excess deformation iii s te e l,
.
Conclusions
The p la stic theory proposed for the design of concrete members is
•different from.;the straig h t lin e method in th a t it- recognises the p la stic
action of concrete instead o f attempting the predict the stresses at work* '
ing lo a d s,- The■p la stic ,flow method 'does not involve the modulus of elas­
t i c i t y of concrete,,
'
.
The application of the p lastic theory i s much simpler than the straig h t
■line stre ss v ariation method and agrees-better'w ith te s t results*
I t can
be taught more easily than the present method and w ill lead designers to a b e tte r understanding of the actual stru ctu ral action.
The design of p la stic theory gives smaller sections, with greater amount
of s te e l.
This shows th a t the p lastic flow method i s much more economical
than the straig h t lin e method for countries which have an abundance of s te e l,
■For a, given ■story height 3' the plastic design- method w ill reduce the required'
building height since shallower beams are needed.
The present method has been useful during-a. time Vnen. knowledge and
- experience with reinforced concrete were- being accumulated.
I t provides
-
in some cases what id now an excessively high factor of safety against.' con­
crete fa ilu re .
I t leads to uneconomical design and an inconsistent facto r -
of safety,, - I t i s based on an. empirical value o f
which Be in turn assumes
the stress strain curve-of concrete to be a straight lin e t i l l a definite
point5 which is not tru e as Eooke1S Law does not apply to concrete at a l l .
■
56, ■
. , - . ■
Suggestions
!Phe apparatus used to te s t the two beams is shorn ia figure 15,
As
i t is seen from th is figure# to measure■the concrete strain an Annexe d ia l
reading to 0,0001 inches i s used, . In order, to measure the ste e l strain#
however#' an extensoiseter is used, The exbeiisometer# as was expected# gave
very good re s u lts .
On the other hand# i t was very d iffic u lt to measure
the concrete strain s because as the instrument was very sensitive# i t was
impossible to count the revolutions which tends to lead to. mistakes,
Therefore# I recommend' th a t i f any experiments'are to be conducted on the
same theory# a l l apparatus should be used in the "same way except in the case
'=
1
of measuring the concrete str a in s. Instead of using one AmetS d ia l and
moving i t up and .down# three should be used and'they should be fixed somehow
on the p lates so that the readings w ill be more accurate.
57
Fig. 16
Cracks Fonaod at F a ilu re
SG.
•L iteratw e Cited and Consulted
I . Effect of Tiae Element in loading Concrete, tgr # . g. B att, Proceedings*
A, 8 , T» 3L, 1907, page 421.
2» Shrinkage and Time E ffects in Reinforced Concrete, by P k R, IfcEiillan,
University of Minnesota^ Studios in Engineering, B ulletin 1%
3» The Plow of Concrete Under'Sustained load, by 2, B6 Siaiths Proceedings,
A .C .l., Vol6 12, 1916, page 31?•
4» Tests of large lieinforeed Concrete Slabs, by A. T6 Goldbeck and B6 B6
Smith, Proceedings, A6C6I 8, Vol6 12, 191b, page 324.
5» Design of Reinforced Concrete Slabs, Transactions, A6S6C6B6, Discussion
by P6 R6 m m ila n , Vol. 80, 1916, page 1?&3*
6, Bxfcensometer Measurements in a Reinforced Concrete Building over a
.Period of One Year} by A6 Re lo rd , Froc6 A .C.I., Vol6. 13, I 9I 7,
page 45»
7 . The Plow of Concrete Under Sustained loads, by Ec B6 Smith, Proc6 A6G .!.,
vol, 1 3 , 1 9 1 7 ».page 99,
0. P lastic Flow, Shrinkage, and Other Problems of Concrete and Their Effect
on Design, by Oscar Faber, Min6 of Proc6 In st, of C iv il'Engineers
(Great Britain)* Vol. 22$, 1927-1928, Part I .
9» Flow of Concrete Under Sustained Compressive S tress, by R6 B6 Bavis5.
Proc* A6C6I 6, Vol6 24, I 9P83 page 303, with Discussion by J 6 R6
Shank, page 327.
10. Plow of Concrete Under the Action of Sustained loads, by R6 S6 Davis
ahd H. G6 Davie, Pros, A6O,!+, Vol, 27, 1931, page 837, with Dis­
cussions by 0+ A, Money and M6 B. Garnet* J, R6 Shank, Iara Jorgensen, and convention discussion. Discussion by Hardy Cross appeared
In Proc6 A6C8I 8, Vol, 28* 1932, page 26$,
I I . Flow of Concrete Under Sustained Compressive S tress, by R6 E6 Davis,
Pros. A,8.T.M6, Vol. 3 0 , 1930, page TO?.
12. P lastic Flow of Concrete. Under Sustained Stress, by R, B6 Davis, II. E6
Davis* and J. 8+ Kamilton, Proc6 A6S6T6M,* Vol. 3 0 , 1934* Part I I ,
page 354,
13, Plain and Reinforced Concrete Arches* by Charles 8, Whitney* Pros, A8C6I,*
Vol. 28* 1928* page 479* td th discussions by A* L» Gemeny5 Clyde T. '
Morris, M, K, G leaville, and F. G6 Thomas, Bernard I , Weiner, Gilbert
C6 Staehlo, and authors closure, Proc6 A6O8I 66 Vol. 29, 1933, pp.
. 87 to 97*
14# Sttitdles lit H eiaioreed Concrete I l I 3t The Creep o r Flow o f Concrete IInder
Io ad 3 by W. H3 G la n v ille jl Dept* o f S e le n tifie and X ndnstrial He**
search , Building Research, T echnical Paper So6 12, 1930«
19» P la s tic Flow in-C oncrete Arches, by lo re n z G# Straub, Proc*,
. T ol. 9$ , 1931, page 6X3 »
16* The B ffee t of P la s tic Flow in R igid Frames of Reinforced Concrete, by
F , SI* R ich art3 E* L* Broma3 and T# G* Taylor3 Proe* A6G6X63 F o l6
BG3 1934; page IS l#
17* The P la s tic Flow of Concrete by J* R*: Shank3 Ohio State- U n iv e rsity ,'
S tudies Engineering S eries B u lle tin Ho. 91, September, 1935*
IS# P la s tic Flow and Volume Changes of Concrete, by Raymond EU Davig3
Harmer B* Bavis3 and Bldwod EU Brown, A6S6T9H*. Proceediags3
Vol, 37, 1937, page 317.
19* E ffe c t o f Shrinkage3 by C harles VJhitney3 AflG6I d Jo u rn a l, Proe6 28?
September, 1931 - June3 1932,.page 49®*
20» Summary o f the R esu lts a n d ' In v e stig a tio n s Having to do w ith Volumetric
Changes i n Geaonts3 H ortars and Concretes due to Causes Other Than.
S tr e s s 3 by R# B9 Davis3 A6C6X9 Jo u rn a l, February, 1930, Proo63 Vol6
26, page 4O7 *
21* Reinforced Concrete Design, by Sutherland and Beeae3 Hew Io rk 3 JoJm
Miley and Sons3 In o 93 Second e d itio n , page 109#
22» nBrachzustand und S ic h erh e it im ' Bisenbet Onbalkentf? by Rudolf S a llg e r3
Boton und B isen3 Ve 35, Ko9,19, October 5S 1936, pp» 317-3BO3 and
Ho9 20, October 20, 1936* pp* 339-346»
23# “Her Beiwert uUir3 by F r i t z V# Brfmerger3. Baton und E isen 3 V* 35, Ho* 19,
October 5 , 1936* pp* 324-332#
,24» Design o f Reinforced Concrete Members Under Flexure or Combined Flexure
and Direct- Compression, by C harles $:» Mhitney3 A0G6I 0 Journal3
Proc6 33, September 1936 - Jxme3 1937a page 485»
25* Compressive S trength of Concrete in Flaxureu3 by S la te r and Iy se 3
Jo u rn a l American Concrete I n s t i t u t e , Jm e 3 1930, Proc 63 Vol9
P * 6 3 l.
2 6 3,
MONTANA STATE UNIVERSITY LIBRARIES
N37
37122
op.
Di ^ffo?.f. nf pl a s t i c fliat-on -
f l e x u r a _l s t r e s___
s e_
s 4i nvi r eAi4nM-—
f n r n e d eoncrete_Jbggm§=:
4.
67122
Do4*
cop. 2