The influence of plastic flow upon the stresses developed in short span reinforced concrete beams
by Dimitri Lambropulos
A THESIS Submited 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 Dimitri Lambropulos (1948)
Abstract:
In this thesis, it has been attempted to determine the actual stress variation In the concrete on the
compression -side of short-span reinforced concrete beams* and to prove that the "Straight-Line
Theory”, adopted up to now, gives only a portion of the stresses developed,since, in the deri-vation of
its formulae, the effects of plastic flow and shrinkage on the final stresses set up in the concrete are not
considered.
Data and results were obtained from' tests conducted on two short beams; one with web reinforcement
designed by the plastic flow method, and one assuming straight-line stress-strain variation.
The actual stress distribution on the compression side of tho reinforced concrete beams tested-, as
obtained from the results shorn in Tables IV and V, are shorn in Figures 15 and 16. 4 line representing
the distri-button of stress in the concrete is curved, somewhat close to the shape' of a parabola, but far
from a straight line.
The capacity of the test machine was not sufficient to break tho beams; therefore, a comparison
between the two web reinforcement design methods was not obtained. SMKREWWSEBS DETBtOFBD W SBORt 8PM
CmQOStm ,BBAHB
W
D m m i w m opm os
I THESIS
SW aitied td the Graduate Oowittee
' . ,
' ia
p a r tia l fulfillm ent of the requirements
fo r the degree of
Eaeter of Seienee in C ivil Engineering ■
at
Montana State College
Approved;
f i - Z .
In Charge of Major Work
g-
Oe P A /
Septemberj, I f 40
J
/
/
u
4
2.
TABLE OF CONTENTS
A b stra ct
Nom enclature
In tr o d u c tio n
P l a s t i c Flow and Volume Changes o f C oncrete
S t r e s s e s D eveloped Due t o Shrinkage and P l a s t i c Flow
The P l a s t i c i t y R a tio
a f f e c t o f P l a s t i c Flow and Volume Ciianges on D esign
P l a s t i c Theory o f R ein fo rced C oncrete D esign
E xperim ental Work
D u scu ssion and C onclu sion
L ite r a tu r e C ited and C onsu lted
X
In th is thesis,*, i t has been attempted to. determine the actual stress
variation In the concrete on the compression side of short-span reinforced
concrete beams*, and to prove th a t th e *Btraight«&ln@ Theory” $ adopted np
•to now* gives only a portion of th e stresses developed* since* in the' d e ri­
vation of i t s formulae*, the effects of p la stic flow and shrinkage- on the
f in a l stresses set up in th e concrete are not considered*
- P a ta ' and,resu lts were obtained from' te s ts conducted on two short '
beamsj one with web reinforcement designed by the p la stic flow method* and
one assuming stra ig h t-lin e s tre s s-s tra in variation*
The actual stre ss d istrib u tio n on the compression/ aide of tho rein ­
forced concrete beams tested* a s obtained from the re su lts shorn in Tables
I? and If* are shot# in Figures 15 and 16* 4 lin e representing th e d i s t r i - ' ■
but!on of stress in the concrete i s curved* somewhat close to the shape'of
a parabola * but far from a straight lino*. '
'
: '
The capacity of the te s t machine was not sufficient to break the beams;
therefore * a comparison between, the two web reinforcement .design methods
was not obtained* ■
;' ;
4*
NCWKOLATtm
# Gospressive '^fcrees.4e steely
s * The ehorteplng of the donorete block due to siirjjikage i f the block was
not reinforced*
I'
fc % Tentile s tre e t in concrete* ■
■.
Se.* Modulus of e la s tic ity o f oteel*
.
Be sf Modulus o f"'e la s tic ity of concrete,
Ac sr Effective area of concrete,,
p "® S teel ra tio
(As/bd),
A » Overall area of concrete in a column '
a = S ■
■
.
a © ■» E la stic shortening of column
"
:
^
.
.
Sp - The p la s tic flow of a. unit length of plain concrete under unit stre ss
from th e time, o f loading to m y given time under consideration-*
dd % The p la stic flow of % unit length o f plain concrete fo r a given in«,'
f in ite sintal time in terv al d t due, to a u n it /stress, intensity* ’
die t The change of stre ss in concrete,.
die 5 The change of stre ss in steel*
i&L * Actual u n it ste e l stress combining e la s tic and p la s tic effects*
A fc $ Total change in concrete -,stress due to p la stic flow*
A fs -3 Total change in ste e l strength due to p lastic flow*
f% ~ The in te n sity of --stress in the concrete a t the time in terv al d t due
both to- the applied load and. the e ffe c ts of plastic flow up to th a t
tim e, and assumed constant through th e small in te rv a l dt*
fee ■- The original concrete stress due to elastic-actio n only*
fee * The o rig in al s te e l: ,stress due to e la s tic ,action only.
JkU
■ ■
' 6.
y * fhe p la stie flow in milliontiis of m. ineti p er. inch fo r s nnit otryss
of one pound per square' Ineh6x & fime a fte r Ioadingji In days*"
Sj, and
.■
Constants varying with the m aterial. -
Bex = The o ffOetive modulus of e la s tic ity .
, .
Be * Instantaneous modulus o f elasticity*
7
,
.
f u 1* * Oompregsiv© strength of a concrete cylinders
O % The to ta l compression, ,
,
•
a -a Depth o f equivalent 'stress diagram*, ’
■M * The depth from th e compressive side of -a -hem to the neu tral axis*
b » Width of beam*
c » Arm of re sistin g couple (p lastic theory method) *
H m Eoment of resistan ce»' ^
'i
T 'p Total tension*
7
■
,
,
' .'
; ,.
.
:
' . . . '
-
'
_
d"'- » The distance from the compression ste e l to the tension .steel*
,
■
As" .si Area of the compression steel* . .
,
'
. ". .
As & Area of tension steel*
P - Applied load.*.
'
ds sf & short element of distance*
V “ BhWring force*
.
-
■
-
v a in te n sity of shear*, '
HO sf Increment in Compressive force*
A t * Increment' in tensile- force*.
,
u *i Intensity of bond stress* '
SLo - The sum of the bar perimeters*
t '* Average Shearing .intensity* :
.
.
.
.________________ ____________________ __ ___________ ;______________
:
8.
<£s = Unit cheat* deformation,
G » Shearing modttlns of ela sticity for concrete.
?.
mmoDWTim
Io Bjt&tm&ntaaS/scope off ^robXem
Arom several investigations on the behavior of Oonereteff i t has been
found th a t th e elasfcie strain s due to application of a load give only a
portion of the actual strains a t any time in the concrete and steel,- In
addition to the above stated strains# there are also stra in s produced
oven before the application of the Ioadff due to shrinkage.
A long^eontin-*
uing Ioadff such as dead load# tvill give increasing p la stic strain s as time
passes.
Another facto r th at might give an increase in the above strain s
i s the change in temperature.
In ary case# the rela tio n between s te e l and concrete stra in s m i l be
very differen t from th a t given by the equations derived by the standard
method# due to loads only,
Recently# certain designers have advocated
designing for failure# the method of analysis, being based upon te s ts of
beams to fa ilu re ,
% th is approach# the ultim ate stra in s t-dll include
shrinkage# load and some p la stic flow stra in s.
These additional strain s due to shrinkage and flow in members in
bending# resu lt in producing a non lin e ar stress v ariation in the eon*,
crete on, the compression side.
An assumption of a parabolic variation
is frequently made which the w riter feels ie not ju stifie d .
In th is thesis# the author has attempted to fin d th e actual stress
variation in th e concrete on the compression aide by te s ts conducted on
f u l l s i se beams in the laboratory at Montana State College# and to deter*
mine the shearing stress- at fa ilu re .
8.
2 6. Acknov/ledgesisHt
Grateful acknowledgement as made to Professor I* 0» PeHaart fo r h is
kind,' assistance^ in te re st and suggestions during
before the experi­
mental Xfork0 Acknowledgement i s also duo to the Mechanical Engineering
Pepaartment fo r the use of th e ir equipment*.
9,
2LA3TI0 FLO# AMD TOLDHE OKABQES OP OONGRBBE
I t i s only comparatively recently th a t the attention of the engin­
eering profession has bees, celled to the p la stic action of concrete lander
sustained constant loads* ^ *
In the technical lite ra tu re up to. about twenty years Ugojf there xsas
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 refer­
ence to the gradual deformation under th e action of sustained load, which
has been variously called "creep", "tim e.yield," and "p lastic flow"*
As a re su lt of the research of recent years, there has been developed
a general conception of th e effect of shrinkage and p la stic flow upon the
behavior of concrete structures.
Under Constant load, the deformation of
the concrete increases progressively, due to p la stic flow, which may be of
appreciable magnitude and cause large changes of stress from those set up
in it ia l ly , thus leading to marked changes in design and construction prac­
tic e s *
I t is now generally believed th a t shrinkage' and p la stic flow are
closely related phenomena, each being prim arily due to changes in the
amount of absorbed water in the cement gel and being but l i t t l e directly
influenced by the free water occupying the pore spaces within th e con­
crete KBSS0
Plastic, flow of concrete has been found to depend upon such factors
as the magnitude of stre s s, the strength of th e concrete, th e duration
% Hwhere re fe r to references given in Bibliography,
10.
of the loading periods the humidity of the atmosphere* th e age of the
concretea the ch aracteristics of the aggregates and. the quantity of ee-»
ment paste 6 **
In sp ite of the rath er extensive researches, th at have been made in
th is field* i t i s not yet possible quantitatively to sta te with any degree.
of certainty what i s lik e ly to be the magnitude of eith er p la stic flow or
shrinkage under the conditions which surround any given concrete struc-r
turn* nor is i t possible quantitatively to sta te with accuracy vibat is
the effect' of p la stic flow and shrinkage upon the magnitude and dietribu*
%
tion of stresses
The property of shrinkage of concrete due to drying i s altogether
undesirable* M t while probably shrinkage can never be eliminated* there
seems to be the possibility* through the proper selection of m aterials
and methods* of reducing shrinkage to a point where i t w ill not be a faeto r fo r serious consideration* On the whole* p la stic flow does not seem
to be an undesirable property= In certain reinforced concrete members*
i t tends to maim- possible more e ffic ie n t use of steel* and in th in s tr u c t'
tunes subjected to drying* as well as in mass structures subjected to
thermal changes due to the hydration of cement* i t tends to promote a more
favorable d istrib u tio n of stresses than would otherwise e x is t0
Beoent findings of Investigation© carried cm a t the University of
California are as follows, I
P lastic glow Under lo ngtim e loading
In t e s t s Ufider long-time loading* i t was found that even a fte r 10
years some p la stic flow was s t i l l discernible in .p lain concretes under
sustained stresses of several hundred pounds per square inch3 however-^
Over 95 percent of
probable t o ta l flow took place m tten 5 years^
In a series of te s ts on rein f dreed concrete columns under load for' %ofe
than 5 years^ I t m s found that* in terms of stresses in th e steel or corn
Crete, the movements due to drying: shrinkage o r to plastic- flow are of
p ractica l importance during the- f i r s t year,
-
fhe p la stic flow o f plain Concrete cylinders, which have been, under ■
load fo r 10 years, are shown in Figure I*-' One group of specimens was ■
loaded at the- age of
days, and the other at the age of 3 months; prior
to loading, the concrete was moist cured, and a f te r loading, i t was stored
in a ir at. 70°
fhe sustained compressive stresses were 3G0, 600, 900
and 1200 pounds per square Inch*
‘
In ta b le ,I are summarized the. re su lts of te s ts upon reinforced concrete
columns under load fo r about 51 years*' Thera are shown'the. stre sse s' in
the longitudinal ste e l and in the -concrete due to the combined effect of
. .
,
,
1
i
elastic, and p la stic strain s produced by the sustained load and of length
changes due to causes other than applied load*
As flow progressed, more
and more load was transferred from the concrete to the s te e l, .and the ra te
of flow decreased in greater proportion.than would be the- case fo r plain
concrete sustaining th e -same in i t i a l stre ss,
.
I t w ill be noted th at while under conditions', of wet storage«, the stress
in- the .steel has not greatly increased during the period of sustained load;
under dry conditions fo r the columns- with the smaller percentage -of rein­
forcement the s tre s s in th e ' -steel has increased more than five times and . ;;
fo r the columns with, the larg er percentage' of reinforcement, th e ' concrete
i s actually in tension,=
12
£ 1400
- - - L - -
f
*
4
--1—
i-
I —I -
* *c}
*
--- T r —
,!!I
t$Et
8988
tkU
BSS#
—
H-
Days 10
Xears
4—
20 30
— f H— ft~f
:;
•
i
A
200300! 5007001000 2pOO30005000
I
2
5
KJ
.Time Since Application of Stress to Specimen, log scale
Ku; I
'10
50 70 iOO
Flow U nder Umg-Sustaine<l Stress
15 20
30 40 5060 80 100
150 200 300400 500600
Time Since Application of Stress to Specimen, days, log scale
F ig . £
KSect of A ggregate C em ent R atio and W ater-C em ent R atio Upon Flow
■1 3 *
Sffgpt of AggeeRatG-Geaent Ratio and Water-Qgiasnt Ratio gpoa F lastig Flow*
Tests to determine the effect of t?e.tey»eenstxf ratio and aggregate^
cement ra tio upon p la stic flow indicated th a t the more th e cement paste^
the le ss the flow and th a t th is variable was very important* Within the
range of normal concretes te sted , i t w s observed th at with pastes with
the same water cement ra tio , the flow was p ractically proportional to
the paste content of the concrete*
For a more thorough understanding of th e effects of th e water-cement
and aggregate-cement ra tio s upon the p la stic flow, re fe r to Figure 2*
Flow in Axial Tension and Compression
Th te s ta to, compare the flow of concrete in axial tension and com­
pression, i t was observed.that, a t le a st during the early ages, the flow
under te n s ile stress was greater than th a t under compressive stress*
At
the la te r ages, the rate of flow was■leas under te n sile than under com-,
preseivo stress*
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 Figures 3 and 4,
.^trnlns Under Sustained F lem ral load.
Similar results as those obtained In determining the flow in aadLal
tension and compression, ware found in te s ts to determine the fiber strain s
In plain concrete beams under constant sustained bending moment*
Effect, of Fineness- and Oomnosition of Cement uuon P lastic Flow*
In te s ts to determine the effect- of fineness and composition of cement
upon p la stic flow, i t was found th a t a low-heat type * of portland cement
® A low-heat cement has s low percentage of- trlcslciu® almsiimte, and a
14
Hormal Poriland - Tension
Time Since Application of Stress to Specimen,days
Fir. 3
Flow in Cnmprrsninn and Tensi n
f i 200
r
0
]
20
40
Mass < .m l ( 'ni
'
Jgf1^ i rLoodedor Tdors fo W lb perse m
60
60
IOO
l?0
140
160
180
Time Since Application of Stress to Specimen, days
F ig . ^
Flow in Compression and Tension
Standard-C ured Concrete.
15»
exhibited markedly .greater p la stic propertied^ p artio u larly at the early
agesj than a normal portland cement»
'i'^srmal .Stregb .Btndiee./
In te s ta tede nnder aaes-curing conditions,, the stresses developed
due to thermal changes in -large concrete cylinders tinder complete axial
restraint* were meastirad*
It was indicated that tbs degree to which flow
takes place has m important influence upon the stresses developed.
re la tiv e ly ■Iotr percentage of irica lcim t s ilic a te which lib e ra te s medium .
amount of heat during hydration, but has & re la tiv e ly high percentage of
dicalcitia, s ilic a te which lib e ra tes a small amount of heat during hydrat;Lon>
I t i s slow in hardening and has' a satisfacto ry ultimate strength. I t s
m in purpose is to reduce.the amount of heat generated by the hydrating ■
dement thus decreasing the amount of expansion.and subsequent contraction
of the 'concrete*' - .
>
■!■^ _ .'--V,:,-.,
%
IGo ,
' '
Bamsaas ims&oRm mm m am m m ea a m mmTio mow
Slirinka^e
.
■.
As was mentioned before., the (hying out of concrete ’varies sdth. time
and with the esiposure of the member* A plain- concrete specimen w ill
bhrihlc as i t dries out but th e re fa ll, be- no stress in the member due to
shrinkage.
Steely on.the other hand3 does not shrink as th e concrete
dries out^ so i t s re s tra in t tends to reduce the volume change of rein-*
forced concrete. After a. certain tim ? the concrete w ill have te n sile ‘'
stresses and the s t e e l 'i d l l W i n compression^'
'
fo compute the effect of shrinkage upon the stresses in a reinforced
concrete Biember5- consider Figure
which' skews a. member of u n it lengthy
of a certain cross-sectional area As , .and symmetrically reinforced with ■
ste e l in amount pA, I f th e member were without reinforcements the short­
enings due to shrinkage would fee s but the action- of ste e l reduces the
actual change -as. shorn in .th e figure by a certain amount*
From Figure 5> i t Can be easily seen th a t the f in a l length' of the
s te e l and concrete must be the Ssrne5 so equating the strains? we gets
M&i a S'.IS ■
Se
W
S 9 IS. / |S
Es; -Ec
wheres
fe - te n sile stre ss in-'concrete
f s - compressive'stress in s te e l , ' '
fhe stresses a t a giyen section must be in equilibrium? so equating ■
the compressive force on the ste e l to the tension in. the concrete? we haves
fepA * foAo * fo
A *
0)
SolvMig Equation 3 ■for f b - . ■
•
ts
,
” Zd(I-B)A a: fo(l-p) "
PA '
p
■ • .. .
Sabs^ituilng t h is value o f f s in Equation I
2S& el
• _xSe
Solving th is fo r fe
(4 )
S im il a r ^
*** I
(S)
S e u se 'p f 'these relatio n s i s d iffic u lt because o f the im possibility
of %B9R&ag exactly the value of the Shrinkage poeffiolebt &* g&aeg the -
length of -the member w,s assumei to be unity* s i s given in inches per
S
inch5 a f a ir value is approximately Q6OOOA for protected members in bull*
dings*
#La@t&@f lo w ' .
■■ .
J f a Constant lead i s maintained on a concrete member, there is a
sloi'7 increase in th e -in itia l, deformation due to flow which re su lts in a
progressive transfer' of stress from concrete to Steel5 a process which.
la s ts 'f o r about 6 years ©r more before equilibrium -is reached*
1 The 'effect of p la stic flow on the concrete -stress may be- approximately
Ag
* in columns# p » A and % * p&, but Ae
As| so Aa ^ A - pA ■* O p M
18
analysed as follows,
fhe modulus of e la s tic ity of 'concrete increases with
the age of the concrete.
constant»'
.
In the derivation below, i t i s assumed to be
.
Considering figure 6, which shows a block of. concrete with a' central
reinforcing rod, a load B i s applied ax ially which makes the concrete
block shorten by an.amount
' Owing to the p la stic action, which in­
creases with time, the column s t i l l shortens as time goes on* Using the
following notation?
Cp- The p la stic flow of a unit length of plain concrete under unit
stress from the time of loading to any given time under considera­
tio n .
'
do -• The p la stic flow o f ' a unit length of plain, concrete fo r a given
infinitesim al time interval dt due to a' unit stress in ten sity ,
f c ,“ The in ten sity of stre ss in the concrete, a t the time in te rv a l dh
due both to the applied load and the effects of p la stic flow up
■to th a t tim e, and assumed constant through the. small interval d t,
I f the block were of plain concrete,- i t would shorten p la stic a lly an
amount fed© in addition to the original e la s tic deformation» Assuming
perfect bond between the ste e l and concrete, the shortening of the. concrete
becomes less, due to the resistance offered, by' the ste e l.
l e t dfe: be the change o f's tre ss in concrete and tifs be the change o f
.
stress in ste e l,
The fin a l lengths of both the ste e l and concrete w ill be
the same. Then,
f# o /
*
.
(6 )
^ Pins sign denotes 'compression and minus sign tension.
19
£
IA r l
Umt
't Leng th __
Kio. f
Fi' C
20
E q u atin g th e t o t a l change i n lo a d c a r r ie d by th e s t e e l t o th e change
c a r r ie d by th e c o n c r e te ,
A sdfe = -A cd fc
(7 )
S o lv in g e q u a tio n s 6 and 7 by e lim in a tin g d f s , we o b ta in
_ __=dc___
fc _ A c _ / l _
AeEe
Ec
I n te g r a tin g
/
1Hf0' =-AS_C/ i AbEs
k
Ec
when c = o , f c becomes f c 0 , th a t i s , th e o r i g in a l c o n c r e te s t r e s s due t o
e l a s t i c a c tio n o n ly , and k » I n f c 0 ,
Hence,
ln fc^ J E T n i
AsEs
or
^
fCo
* lnfcO
Ec
= . - — zo
Ac
AsEs
J_
Ec
and
fc .=
_____ f c o
e —
^c
AsEe
S u b s t it u t in g Es _
Ec ~
to
n?
/ L.
Ec
As = pAj and Ac * A ( l - p ) , th e above e q u a tio n red u ces
' " e -
Bscp
(8)
p (n -l) / I
L et f s, eq u a l a c t u a l u n it s t e e l s t r e s s com bining e l a s t i c and p l a s t i c
e f f e c t s } f s o e q u a l o r i g in a l s t e e l s t r e s s from e l a s t i c a c t io n o n ly ; ^ f c
e q u a l t o t a l change in c o n c r e te s t r e s s due t o p l a s t i c f lo w , and ^ f e eq u a l
21
t o t a l change in s t e e l s t r e s s due t o p l a s t i c flo w .
fs , * fso /
ZXfs
- Z l f c a f c o - f c ; but f c , =
e - Z lfc = f c o -
e -
Ec
f1 - e ArEs
d fs = - 4 fc
and s in c e f c o =
^ L.
Ec
c
-AS- / 1_
AbEs
- ZXfc = f c o
AsiiS
Ec
As
, we have
Zlfs = fso -AS (I nAs
e -
L / i_
( 10 )
AsEs r Ec
The f i n a l s t r e s s e s may be ap p ro x im a tely found by g iv in g n in th e or­
d in a ry form ulae an in c r e a se d v a lu e , a f a c t in d ic a te d by exam ining eq u a tio n
8#
I f t h i s i s r e w r itte n and th e denom inator expanded in t o a s e r i e s ,
(ex
ZTr ZIr S
------- ) , and th e f i r s t two term s
ta k en (b ecau se th e o th e r term s are r a th e r n e g l i g i b l e ) , th e r e s u l t i s :
Es
fc »
fc o
m fc o (Ac /
As)
„
P
Ac / T E / cEs)As
“ Ac / nr As
I /
-AL. / I -
Astis r Sc
' Vtic
(U )
88,
BereiS the numerator represents the o rig in al e la stic concrete stress
by the transformed area with a *
th is equals the load on the -column«,
The denominator represents the transformed, areas "using a modified n / ■
called nr " This ■combines e la s tic and p la stic effects# with. nr ~ ™, / cls»
SG
According-to Gla^fvllles to take care of th e e ffe c t of p la stic flow#
the same method as fo r purely e la stic deformation may fee used with an in-*
ff
ff
creased. Value of n of about 40 without an appreciable error#
' 20,
ms
This subject is mainly based on the hypothesis th at the stress-stra in
diagram for concrete under short time loading may be idealised to consist
of two lin e ar parts, one 'representing e la s tic behavior and the other representing plastics behavior»
The f i r s t part is measured by what we know as the cmodulus ratio", "
which is th e ra tio of the modulus of e l a s tic ity ' of s te e l to th at of the
concrete, ^
The second part of the diagram, on the other hand, i s measured by the
F p lastieity ra tio " , which i s defined to be- the ra tio of the- p la stic strain
to the to ta l a t rupture of the concrete.
An empirical equation has been derived; by professor V*'P* donees,of
the University of. I llin o is to express the relationship between, the p la stic
c ity ra tio and the compressive strength of concrete,
The. idealised s tre s s-s tra in curves-for concrete1-of Professor: % P , :
Jensen are shown in Figure Y fo r reference.
* By modulus of elasticity ..o f concrete, i t s o riginal modulus..Of e la stic
city is. m e a n t a s i t varies with time:,
i
- mgsoir o? mAario mow m # im&tm
ow m m m .
In a routine reinforced concrete deeIgns the designer bases Ms work
on. the accepted methods and standard specifications:«, With the .march of .
time 5 more, and More knowledge has been acquired on the properties and be-1'
h atior of concrete and consequently the accepted methods and specifications
must be altered to comply.with the new findings in th is field*
With reference to f* 1« Bhankt s paper^ 0* f* 'Morris^ in his paper on
the rtEffectg of p la stic flow and volume changes- on design*^ gives the
following useful formulae and information^
'In general^ the p la stic flow of concrete may be expressed •by an equa­
tion of th is form,
a,
, y & C /^ x .
(is)
in whicht
y 3 the p la stic flow in m illionths of - an inch' per inch for a u n it:
stress of one pound per square Inqhs
x - Time after loading in days*.- ,
" ■ ■'
a, and g,~ Gonstants varying with the material* '
The shape of the curve as expressed by the above equation i s f a ir ly
accurate under controlled conditions fo r time up to one year after loading;
fo r longer period®, y should be progressively reduced u n til, a fte r a period
of fiv e years; no fu rth er p la stic flow need be considered*. The rate of
p la stic flow in air- is quite sensitive to .-changes In humidity*. .
- The p la stic flow of concrete leaded -at 28 day®,-for periods- of five
years or more M il not exceed the flow at. one year by more than 30 percent,
and. for concretes loaded a t 3 months, the -excess over the flow a t one year
may reach 40 percent*
.
P laetle flew la ta so- far available may be m pect#! te be as aacb as
50 percent in erro r when considering external conditions and-any specific
concrete mixture„
The plastic- flow for ordinary portland cement 'concretes made from or­
dinary aggregate materials# natural sands# natural gravels and lim estonesP
loaded in a i r a t 2S days may be (Impressed w ith sufficient accuracy byt
y a q iM tlk e
(1 3 )
For the same concretes in,water# equation (12) may be w ritten
y * 0 ,0 6 9 W "
(1 4 )
To take into account# the effect of p la stic flow# a reduced- modulus of
e la s tic ity should be Used;# obtained by the following equation.
l e x 's . 33*.;.
I / Icy
in which*
(1^ )
;
Bcx w The effective modulus of e la s tic ity ,
Be' 55' Instantaneous modulus of elasticity *
y # Plastic, flow deformation fo r the, time under consideration ob*»
tained from equations (IS )s (13)# or ( Ii) 0
26,
PLASTIC TBBQBZ OP BEIBFOBGBD GOROBETB DBBION
Sinee concrete is not a purely e la s tic materials, the theory of elas­
t i c i t y which assumes a straig h t-lin e d istrib u tio n of stresses, is too in­
flex ib le and inaccurate to be en tirely satisfacto ry ,
As i t m s mentioned before, i t is rath er impossible to determine the
o
p la stic flow and shrinkage effects with any accuracy fo r any given struc­
ture , so i t becomes a hopeless task to attempt to determine th e stresses
set up under a given load t-dth. any degree of accuracy.
I t i s , however, more practicable to adopt the ^plastic theory# in­
stead. of the #stra ig h t-lin e theory'# fo r design purposes, because i t takes
into account the p la s tic ity of the m aterial,
The equations derived from
th is theory are much simpler than the standard formulae and agree b etter
with te s t re su lts,
They are based on the cylinder strength of the concrete
a, %Uf
and the yield strength of the ste e l without the use of modulus ra tio fij3 which
has been shown to be meaningless.
The classic conception of the stre ss-stra in curve is th a t of a para­
bola with i t s lower part approximately straig h t and ending at the top tan­
gent to the horizontal. The lower th ird of the curve being nearly stra ig h t,
i t formed the basis of the straig h t-lin e formula in which working stresses
are used.
The en tire curve was used. in. the parabolic formula based on
ultimate strength.
Both of the formulae involve Ioung1s modulus of elas­
t i c i t y of the concrete, which i s inaccurate because the stre ss-stra in re­
la tio n i s an indeterminate variable depending on the concrete, the manner
and in ten sity of loading- and on tim e,
Change in stra in with time is p ri­
marily due to p la stic flow and shrinkage.
On account of the v a ria b ility
of th is s tre s s -s tra in relatio n ^ i t i s ro t possible to determine the imit
stresses a t any p articu lar point in the concrete member by measuring
strains^, except to a very lim ited degree „
M almost universal misconception of the behavior of concrete in flex ­
ure is the assumption that at maximum Ioad5 the highest u n it stress i s at
the outer surface.
The descending branch (see Figure 8) of the compression
stre s s-s tra in curve has been generally disregarded because i t is d iffic u lt
to measure and has been considered of no p articu lar interest*
BectanKular Beams
The stress d istrib u tio n in, a reinforced concrete a t fa ilu re is assumed
to have the shape of the cylinder stre s s-s tra in curve as shorn in Figure 9»
The to ta l compression O5 i s the area bounded by the curve.
The line of
action of G lie s through the center of .gravity of th is area. For the sake
of sim plicity in deriving a workable expression5 the actu al stress curve
fo r the concrete may' be replaced by an. equivalent rectangular area of
ttidth equal to 0*8$fc^ (fe^ being the strength of a te a t cylinder) and a
depth equal to a (as shown in Figure 9)* The location of the center of
gravity of th is rectangle corresponds closely with th at of the actual area.
The actual irreg u lar stress block i s ' replaced only fo r sim plicity with
7
a rectangular one of equal else having an average stress in te n sity of O0Shfe'!
without meaning th a t th is method i s based, on an assumption of rectangular
stress distribution*
The use of a rectangle i s merely a mathematical de­
vice to approximate the effect of true distribution and to simplify the
calculations,.
I t should be noted th a t the depth'a ''does not correspond with the depth
20
0.0016
0 .0 0 2 4
Unif Stro n, i t
W /:
00032
0xX)08
C 0016
0 .0 0 2 4
U nit S t f-Q ini. ( c
R g. 7 ~ id e o liz e d stress-strain curves ( c ro n c re t^ of different strengths
Fio. # 4*» f
89
t o th e n e u tr a l a x i s , kd, and i t b ea rs no d e f i n i t e r e l a t i o n t o i t .
The
v a lu e , kd, i s determ ined by th e s t r a in s a t th e to p and bottom o f th e
beam, w hereas a ''is d eterm ined by th e s tr e n g th o f th e m a te r ia l and i s l e s s
th an th e d is t a n c e kd.
C o n sid erin g F ig u re 1 0 , s in c e T e q u a ls C
-
A sfs
( 16 )
" o.esbfc'
S in c e C = 0 .8 $ b fc ^
(a r e a bounded by cu rve)
M * Ce • 0 .8 5 f c bac
(1 7 )
in which
As = Area o f t e n s i l e s t e e l ,
f s = Y ie ld p o in t s t r e s s o f s t e e l ,
b • Width o f beam.
f C7 * Com pressive str e n g th o f c o n c r e te c y lin d e r .
A stu d y o f t e s t r e s u l t s ^
shows t h a t we may ta k e a /d = 0.537# and
c /d = 0 .7 3 2 , w hich r e s u l t s in
M - 0 .3 3 fc 'b d 2
(1 8 )
I f th e beam i s r e in fo r c e d in com p ression a s w e ll a s in t e n s io n . F ig u re
1 1 , th e com pression d is t a n c e b e in g p la ced a t a d is ta n c e d^ from th a t in
t e n s io n , th e moment o f r e s is t a n c e w i l l be in c r e a s e d , becom ing
M = 0 .3 3 f c Zbd2 / f S A sV z
(1 9 )
where f s i s th e e l a s t i c l i m i t s t r e s s as b e fo r e and As^ i s th e area o f th e
com p ression s t e e l .
M = Tc = fsA s0 .7 3 2 d
i s - _ _ S -------.
0 .7 3 2 f sd
0 .7 3 2 fs d
T O.A56— -bd
fs
(20)
30
4000
------fStress Strain Curve
-------- t o Concrete Cyfinder
12000
0.001
0 002
0.003
0.004
Unit Strain in Concrete,# , Inches per Inch
0.006
S U
■
Sinee in the p la stic theory the designs are done on th e basis at
ultim ate strength., i t i s necessary to apply a definite factor of safety
suitab le to the conditions=
I t Ie based on the yield point of ste el and
'
the cylinder strength of concrete*.
Frem the standpoint of economy*, the design of p la stic theory method
seems to be more economical than th e straig h t-lin e method because the
■sections of members thus found require le ss area of concrete and, more area
of s te e l, a facto r indicating th a t the designs based on th e p la stic theory
method are cheaper since ste el is le s s -expensiye than concrete in the
United S tates9 Other important facto rs th a t affect the economy of a struc­
tu re designed, by th is method considerably a re ; the headroom saving,, the
o v erall.building height, which i s reduced due to the smaller sections and
Slabs used* the dead load reduction*etc,.■
As stated 1# the foreword;, the soopo of th is work wss to find the
aotnal stress variation in the concrete oh, the compression side o f.b ea m s, ■
f o r t h i s p u rp o s e , two s h o r t te am s, each fo u r f e e t lo n g , wore a s s ig n e d
to carry s load of 13,000 pounds,
■
;
•
•
"fho strength-of the eosorete, t«*sa taken n s.-SOOQ ^duna© per Sqnare Indh
with a w a t e r ofaent r a tio of 8 gallons per sack of cement and a omeht^asnd
atone proportion of I S ^ - S ^ hy volumet
•,
-
Both o f the heams (as show in, .the deaign BoI qw) were designed with
the pig's t ie theory method except for the stirrups 9 which for one beam were
designed with the standard str a ig h t-lin e method and for the other one with
the plastic, theory method,
®u.a to improper forms, one of the beams m& poured to he slig h tly
■
wider a t the middle p o rt;.a factor that the w riter thinks w ill not cause
appreciable error In the fin a l te s t of the beam*
®» find the actu al strength o f the concrete„ 5 standard -sins cylinder®
were made of the some wai or- eament ra tio and same cemeht-fiad aggregate9 ■
eQanse*-3ggr®gcito proportion*
D e riv a tio n of B b e a riu s and pond, ?ormi,il-ao '
,
$he variation of shear Intenoity in a,- rectangular reinforced concrete
beam may- be ascertained by considering a. small portion of such a beam be­
tween any two sections a small d is ta n c e 4 s -, apart, ns. shown in iiguro 15
the breadth of boas being taken -,as b inches*
- . - r n r - j n . . i.
f - . . -
■
■
.- r .
i;< Deed weight 1.® neglected.
t:,rT
i'he forces acting on this
33
T able I.—U nit Stresses in R eineorced-C oncff.te Columns.
Nominal
Strength of
Concrete at
28 Days,
lb. per sq. in.
Vnit Stresses, Ib per sq. in.
Total Load
Steel
to
Ratio,^ I Applied
Column,
lb.
^alUm'o/l^d1i 1Tr Und"
Steel
in
Concrete j
Aia o r 50 per
Steel
cent
Concrete
Steel
R ela tiv e H umidity
at
Concrete
Steel
Concret.
70 F.
19
'
22 300
14 20 0
21 8 0 0
875
MO
I
97 5
2 6 900
34 80 0
3 7 sou
-2 0 °
60
395
j
27 40 0
35 700
40 400
■ -5 0 »
I
45
340
: 28 0 0 0
37 100
70 0
j 41
I
X
w
2 0 0 0 .............
2 0 0 0 .............
4 0 0 0 ...............
#10
I
C olumns St o ie d
Sf yr. Under
Load
3Vr Und" l-'»d
C olumns S tored U nder W ater AT 7 0 F .
2 0 0 0 ............... . 5 0
2 0 0 0 ..........
I 9
4 0 0 0 ...............
I 9
I
I
19 20 0
13 650
20 600
I
7200
5400
7320
81 0
605
925
i
10 0 5 0
7 981)
10 590
j
665
555
865
1
10 890
9 OAO
11 6 7 0
620
535
840
11
400
9 480
12 120
590
52 5
83 5
• Minus sign indicates tension
N o t e .—F lo w tests series No. 8 S|>ecimens, reinforce<l-concrete columns S in in diameter by 48 in long. Aggregat«
O to 4-in. local gravel. Fineness modulus, 4 47 Cement, normal portland. .Xggregate cement ratio, by weight (a) f«>
2000-lb« cosKTete, 8.40; (S) for 4000 lb. concrete, 4.80. Water-cement ratio, by weight (<ij for 2000-lb. concrete, 0.8.1; (f>
for 4000-lb. concrete, 0.53. Consistency, 6-in. slump. Curing, 21 days moist at 70 F. (21 C.), then stored as indicated
Age at loading, 21 days. Column loads determined in accordance with provisions of 1924 Report of Joint < mmittee «»:
Spedbcationa for Concrete and Reinforced Concrete.
Ac/
Fm /3
V
.34,
M t of beam consist of the normal CS ,and ? and the shear,?; i t is assumed
th at the moment a t BBf i s larger than th a t a t M* s and th at the sections
'
are so close together th at the two shears may be considered equal.
Ihese
forces arc In eqnilibrium, end I f the condition ZM = o is applied,, there
re s u lts ;
■
.
. 4 ? % jd a Yds
The te n d e n c y o f th e sm a ll .p o rtio n o f th e beam., cdBA, to be p u lle d to
th e r i g h t i s r e s i s t e d by the, h o r iz o n ta l s h e a r on 'th e crl p la n e , t h i c k may
be e x p re sse d as th e i n t e n s i t y o f s h e a r , v , assumed t o be u n ifo rm , m u lti­
p l i e d by th e a re a M s .
Then
hr »- b » d s 3 A t
Combining th e s e two e q u a tio n s g iv e n t
v * b » 4s s
antX '
T «' y
b jd
Consideration of S1Igmro:13 shows,that the total bond, the grlp'Of the
concrete on tho tension reinforcement, which prevents slippage, equals the
difference in to ta l ste e l stress a t the two sections., 4 and B$A T 5 v . b . ds
Assuming uniform in ten sity , u, of bond stre s s, %a have also. AT * u *Xo » ds4
■where
o equals the sum. of the M r perimeters, th is resu lts in*
Vb
*
v
or
* = 3%ro
T he■tra n s f o r m a tio n o f th o above fo rm u lae so t h a t th e y Can, be a p p lie d
i n th e d e sig n by th o p l a s t i c th e o ry method i s sh o rn i n th e p a r t under th e
su b -h e a d in g o f "Design o f Beams” when ch o ck in g f o r s h e a r and bond.
.
05,
D esig n o f Beams
U s O0SSfc1Me
10000 z B x 18%. 130000 ia(&~BotwA8
AssiMiia^ d ~ 7 incites and substituting a
0«53?&, c ™0,7584 and
tQ* - 0*48 x 8000 Cfaeter of safety) the above equation becomes#
180000 s 0.88 x 0,45 X 8000 x 7
X
0*5574 x 0,7324
180000 = 81004s
4 ^
v
SlOO
» 9. BS inches
We 11, inches
------ -—
'S = '®e s W sO
180000
'=•
SOOOOAS
x
0 ,7 5 8 x 11
' As * ——— S 1,12 sq u are in c h e s ,
161000
Ose 6-1* inch 0 rods With an area of 1,80 square inches,
Oheek for shear
(Standard method)
v # w t* * —
^ - 118 pounds p e r sq u a re in c h ,
7 % 6 ,8 6 8 z 11 .
W in g deform ed bars allowable Io
r
40 < 118
O0OSfc S' 0*0.2 x 8000 - 40 pounds p e r s q u a re ' in c h ,
wob re in fo rc e m e n t needed,
118 - 40 = 78 pounds p e r ' in c h
2
c a r r i e d by s t i r r u p s o r 78 x 8 x 7 x 18 3 '12100 pounds ( h a l f s p a n ) .
f in c h 0 W s ta r r u p a , each S ti r r u p c a n c a r r y
2 X 0 ,0 9 x 20000 Z 2000 p o u n d s,'
Use 6
- ; |
lum ber o f s t i r r u p s needed = ~~~rT 3 0 ,0 5
2000
in c h 0 Wstirrups f o r h a l f o f sp an ,
■ 20000.x 0 ,0 5 x'S S’ 78 x 78
'
SOOO
8 5 — * s 0 ,9 7 ' in c h e s -fo r sp a c in g o f s t i r r u p s ,
• 504
.
■
to be
W in g
36»
Ohoek fo r bondt
' ' '
•
u s I W = K S e i i r u T s C T ^ ' ?8-5 *°®de SW itoll$
80.5 C 112
O .& .
Check fo r shear (P la s tic theory method}'
v e '
or ^
, (See fig u re 10} .
but c s O*70Sd» th e re fo re ;
Y
7500
7 " 0.792ba " 0.702 z''7'% 11 = ISO
9
per inch'd
133 * 40 = 93 pounds per inch® to be c a rrie d by s tirru p s
or
90 x 2 a ? % 18 « 15640 pounds,
tfeing again the same W -Stirrups3 each s tir r u p c a r r ie s ,
,80000 % 8 % 0,05 s 8000 pounds,
Kumber o f s tir r u p s needed?
» 7,88 s tirru p s ■
Wse 8*4 iucb0 s tir r u p s f o r h a lf Span6
2000 % 0.05 % 8 s 93 % 78
S ”
' POU S 3,08 inches f o r spacing.
Oheok for hon5g
.
* 2JL =
u
aro ■0»752dIO
s'—------— = 98,7 pounds per Inch®
0.732 x 11 z 9,48
98,7 <C 112 pounds per inch® o.k.
A se c tio n o f the beam is shown in Figure 12
37
Fm. //
JL-+~U
I
T
. I
T
I
------------- —
---
/
—A\
I
hI
I»
\ \
-O
T
-t— —
El
PSa n
I
F /* .
/a
Table II - 8 Stirrup Beam
Load in
Pounds
S tr a in in
S te e l
in /in
S tr a in in C oncrete in i n . / i n .
1 M from to p
3 ” from to p
5" from top
0
0
0
0
4000
0.00004
0 .0 0 0 1
0 .0 0 0 1
0 .0 0 0 1
6000
0 .0 0 0 0 ?
0 .0 0 0 2
0 .0 0 0 2
0 .0 0 0 1
8000
0 .0 0 0 1 2
0 .0 0 0 3
0 .0 0 0 4
0 .0003
10000
0 .0 0 0 1 8
0 .0 0 0 4
0 .0 0 0 5
0 .0003
12000
0 .0 0 0 2 5
0 .0 0 0 5
0.0006
0 .0 0 0 4
14000
0.00Q31
0 .0 0 0 7
0 .0 0 0 6
0.0005
16000
0 .0 0 0 3 8
0 .0 0 0 9
0 .0 0 0 7
0 .0 0 0 6
18000
C .00045
0 .0 0 1 0
0 .0 0 0 9
0 .0 0 0 8
20000
0.00053
0 .0 0 1 2
0 .0 0 1 0
0.0009
38.
0
Table III - 6 Stirrup Beam
Load in
Pounds
S t r a in in
S teel
in ./in .
S tr a in in C oncrete in i n . / i n .
I" from to p
3 a from tqp
5 " from top
0
0
0
0
0
ZtOOO
0.00003
0.0001
0.0001
0.0001
6000
0 .0 0 0 0 8
0.0002
0 .0 0 0 3
0.0 0 0 2
8000
0 .0 0 0 1 6
0.0003
0 .0 0 0 4
0.0003
12750
0 .00 0 3 3
0 .G005
0.0005
0.0005
16000
0 .0 0 0 4 4
0 .0 0 0 8
0 .0 0 0 8
0 .0 0 0 ?
18000
0 .0 0 0 5 2
0 .0 0 1 0
0 .0 0 0 9
0 .0 0 0 8
20000
0 .0 0 0 5 9
0.0011
0.0011
0.0009
Table IV - 8 Stirrup Beam
Load in
Pounds
Computed Theo r e tic a l
S tress o f
S t e e l in
lb s /in 2
Computed The o r e tic a l
S tress of
C oncrete in
lb s /in 2
Computed Ac­
tu a l S tr e ss
o f S t e e l in
lb s /in 2
0
Computed A ctual S t r e s s o f C oncrete
in l b s / i n 2
I" from to p
3" from top
5" from top
0
0
0
0
0
0
4000
4150
U2
1200
140
140
HO
6000
6220
212
2100
280
280
HO
8000
8300
283
3600
420
560
420
IOOCO
10360
354
5400
560
700
420
12000
12420
424
7500
700
840
560
UOOO
14500
495
9300
980
840
700
16000
16600
565
11400
1260
980
840
18000
18670
636
13500
HOO
1260
1120
20000
20700
705
15900
1680
UOO
980
The actual stresses in th e s t e e l and concrete were computed by multiplying their moduli
of elasticity, which were taken as 30 x IO^ psi and I.Ii x IO^ psi respectively, by the
measured strains.
In computing the actual concrete stresses, the shearing deformation was not subtracted
from the measured strains.
Table V - 6 Stirrup Beam
Ea = 30 x IO6
Load in
Pounds
Computed The­
o r e tic a l
S tr e ss o f
S t e e l in
I b s /in ^
Computed The o r e tic a l
S tr e ss o f
C oncrete in
I b s /in ^
Ec = 1.4 x IO6
Computed Ac­
tu a l S tr e ss
o f S t e e l in
l b s /i n ^
Computed A ctu al S t r e s s o f Concrete
in lb s /i n ^
I" from to p
3 ” from to p
5" from top
0
0
0
0
0
0
0
4000
4150
142
900
140
140
140
6000
6220
212
2400
280
280
280
8000
6300
283
4800
420
560
420
12750
13200
450
9900
700
700
700
16000
16600
565
13200
1120
1120
980
18000
18670
636
15600
1400
1260
1120
2X 00
20700
705
17700
1540
1540
1260
The a c t u a l s t r e s s e s in th e s t e e l and co n cre te were computed by m u ltip ly in g t h e ir moduli
of elasticity, which were taken as 30 x l e f t psi and 1.4 x 10& psi respectively, by the
measured strains.
In computing the actual concrete stresses, the shearing deformation was not subtracted
from the measured strains.
mscosBio# A&D coNG&ysiom
■Fi*ohi: the data and.' re su lts obtained from■the te s ts of the tvo beams,
the following things were observed.
With reference to tab les I? and f 5 . i t i s seen th a t a t low- stresses ,
there w s a- great difference between the computed stresses and the actual ■
stresses in the ste e l,
!b is i s prim arily due to the fact th a t a t low
stresses a considerable amount of the tension was taken by the concrete^
As the load increased* th is difference started decreasing because almost
a l l of the tension was carried by ste el and very l i t t l e or none' by the
concrete* which, started cracking and consequently could not carry any more'
tension.
For the concrete* a t low stresses both the computed and the actual
stresses were, close to each other*, whereas at .higher loads* the actual
stresses were much greater than the computed'.ones, th is was probably due,
to the fact th a t at high stresses* the concrete strain s started increasing
much more rapidly than the stresses,
A glance a t Figure 1 4 would show
that a t high stresses*, the stre ss-stra in curve of concrete s ta rts getting ■■
f la tt e r resulting in a greater strain-stress, ra tio .
From. Figures 15 and Ib 5 i t can be seen.that the neutral axis of the
cross-section o f t h e beams shifted upwards, as more load was applied^
!h is
oeeured. because a t higher stresses*, a l l of the tension was taken by the
s te e l and consequently* the strain s were much greater than those obtained
at low stresses'where a considerable amount of -tension was. taken by the
concrete,
!he actual distance from the top of the beam to the neutral-axis was
43.
45
46.
s m a lle r th a n th e one computed by th e " S tr a ig h t - l i n e m ethod".
I h is d i f ­
fe r e n c e in th e p o s it io n o f th e n e u tr a l a x is i s due t o th e e f f e c t o f p la s ­
t i c flo w , which r e s u l t s in a r e d is t r ib u t io n o f th e c o n c r e te s t r e s s e s pro­
d u cin g a s h o r te r moment arm and c o n se q u e n tly , r a is in g th e s t e e l s t r e s s e s
f o r a g iv e n a p p lie d moment.
The com p ressive f o r c e s i n th e beams were g r e a te r th an th e t e n s i l e f o r ­
ces.
T his d if f e r e n c e in t e n s io n and com pression s t r e s s e s may be due t o :
a ) rea d in g g r e a te r s t r a in s o f c o n c r e te th a n th e a c t u a l ones due t o
th e d i f f i c u l t i e s en cou n tered in m easuring th o se s c r a in s by th e
p rovided means.
b) rea d in g sm a lle r s t r a in s o f s t e e l b ecau se o f some p o s s ib le s lip p a g e
betw een th e ex ten so m eter and th e s t e e l b ar.
c ) U sing a g r e a te r v a lu e fo r th e modulus o f e l a s t i c i t y o f c o n c r e te ,
a f a c t o r t h a t i s v e r y hard t o d eterm in e w ith any a ccu ra cy due t o
each in c o n s is t e n c y , which would r e s u l t in g r e a t e r s t r e s s e s in th e
c o n c r e te .
In com puting th e s t r e s s e s o f c o n c r e te , t h e sh ea rin g d eform ation was
ta k en in t o c o n s id e r a t io n .
At h ig h lo a d s , due t o th e g ro a t sh e a r in g s t r e s s e s
d e v e lo p e d , ev er y elem en t o f th e beam i s d i s t o r t e d , as shown i n F igu re 1 7 ,
which r e s u l t s in a d d it io n a l s t r a in s in th e c o n c r e te .
t
*—K'h"
-— z
i
/ t
J
I
F ig . 17
47,
!Phe- ,strains were eoii^uted. by the formula
-
.g-
-
-
where
£s * unit shear deformation
"
t - average shearing in te n sity computed from t » -E ts and •
hjd .
G- ~ shearing modulus of e la s t ic it y fo r concrete«,
In fig u res 15 and
16#
th e so lid lin e s represent th e s tr e s s distribu­
tion on the. .compression sid e o f the reinforced concrete beams when the
shearing- deformations were neglected), and th e dotted lin e s show - the stress
variation when the e ff e c t of the shearing deformation was taken in to account.
fhe applied moments were ; greater than the .resistin g ones,-. In th is case?
th e r e sis tin g moment' was -computed as "Tension times 'lever ar$k because the
■
w riter f e l t that the s te e l strains were more reliab le* -fo r the most favor­
able condition*, the error was'about 15 -percent* because of reasons mentioned
above*- In computing the th eo retica l moments* the- span o f,th e beams was
taken as 5 f e e t and. 4 inches* ''which was the clear ,-distance between -the -supr ■ports,
;■
■
Another factor that might a ffe c t the measured s tr a in s•s lig h tly i s
shrinkage*- because as the beams dried out there were, some te n s ile stresses
set up in the Concrete and compressive str e sse s in the s te e l before the load
was applied,
Conclusions
.
.,
■ ,.
,•,
'
...
,
‘
.
"
The r e su lts obtained from the two beams* i f not very sa tisfa c to ry , were .
quite helpful in determining th e actual str e ss variation oh the compression
side of reinforced 'concrete beams and in- proving, that the "Straight-Lino '
Theory" i s not much ju stified , ;
48
Designing wrbh the p la s tic theory met hoi seems to be more accurate
because i t i s based, on th e ultimate strength of reinforced concrete beams>
a value l i t t l e affected by shrinkage and flow since there i s a large red is­
trib u tion of str ess with the large strain s previous to failure. 'On the .
other hand, the straight lin o theory i s rather approximate because i t i s
almost impossible t o preduct with any accuracy the e ffe c ts of shrinkage and '
p la s tic flow and thus determine with any exactitude the str esse s set up
under any given load.
A bettor way of .getting more -accurate r esu lts concerning the d istrib u tio n of compressive stresses in reinforced concrete beams, would fee to attach
two or three extensomete r s at d ifferen t depths of a beast, in such a way thatmoving o f the measuring device for each reading w ill be avoided*, In the
author*s opinion, i f such a thing can be done, the p o s s ib ility of error in
reading the d ia l every time that i t i s set on' the 'beam between the two
p la te s, as shown in Figure IS, w ill be eliminated..
49.
r /6 . /0
50 =
LIiaaASMBG 01T&D AND OOBSWLTED
I 6 Bavisa E= B =3 Davis2 H= B*, Broxma SXvmod2 ttP la stie Flow and. FoXvbq
Ohanges of Ooncretolt2 Proeeodings2 A=Ooi6Maa Part XX2 FoX= 3?a
1937, P+ 318,
Zo
Xensen2 T= P62 ltXhe P la s tic ity Batio o f Concrete and. i t s E ffect on
, the Ultimate Strength of Beamsft2 XoumaX Amrican Concrete Insti­
tu te, June, 1943, P= 565=
3= Morisa Clyde T62 tsE ffoct o f P la stic Flow and Volume Changes on Design11a
Journal, American Concrete X nstitute9 November-Decembor2 1936, P= 123»
4= Peabody2 Dean2 llReinforeed Concrete Structures11, Xolm I-Jiley and Sons,
lev/ York, 1946=
So
Sutherland, Hale, Iieesea Raymond O2 tfBeinforced Concrete Design-*1, John
Wiley and Sons2 New York,,1943=
6o
Whitney, Charles S=, ffP la stic Theory o f Beinforeed Concrete Design*’,
Proceedings, A=S8T=M=, Vol= 66, 1940, p* 1749=
7» S aliger3 Budolfa ftBruchaustand und Sieherheit im Eisenbeton-balkentf,
Beton und Eisena October, 1936, p« 317=
8 . Hajnal-Kanyi2 K= and. discussion by oth ers, tlThe Modular Batio - A New
Method of Design Omitting mf,3 Concrete and .Constructional Engineer­
in g, January,
9= Coppee2 E=, "Considerations our Xe Calcul Qt la Securite des Pieces
Floehies Moments de Bupturelf2 Publications, International Associa­
tio n for Bridge and Structural Engineering, Vol= 3 , 1933, p. 19=
IO0 S la te r, W= A=, Lysea Xnge2 "Compressive Strength of Concrete 'in Flexure",
Journal, American Concrete I n s titu te , June, 1930, p= 831*
11» Empergera F r its VVa llDer Beiwertsa", Beton und Bison, Vol6 35, No= 19,
October 5, 1936, pp= 324*432,
12b Cox, Kenneth C6, "Tests of Beinforeed Concrete Beams with Becommenda­
tio n s fo r Attaining Balanced Design", ACI Journal, September, 1941,
pp« 65-80„ Duscussiona AOI Journal, June2 1942, pp6 (SO-I) to
(SO -Il)2 Proceeding V= 37=
13= Shank, J= R=, "The Mechanics of P la stic Plow of Concrete", Journal,
«
American Concrete I n s titu te , Kovember-Deeembor, 1935, Proceedings,
Vol* 32, p= 149,
MONTANA STATE UNIVERSITY LIBRARIES
762 1001 4690 9
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cop. 2
Lambropulos, Dimitri
The influence of
p la stic flo w . . .
gl'MCTTT
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