A rheological model for concrete

Torroja Miret, Eduardo
A rehological model for concrete /
Eduardo Torroja Miret. - [S.l. :
sn.], [19-?]
19 h., [6] h. de gráf. ; 30 cm
Manuscrito mecanografiado
perteneciente al Archivo Torroja
depositado en CEHOPU
Sig. TC-33
10. RHEOLOG rCAL MODEL FOR CONCRETE
by
Prof. Dr. h. c. E. Torroja
"
Re i nforcod
cre t e
technol o ~J 9
concrete ~
a n d efOlpecially pre:Tcressed con-
r e quires every day a more de-cailed knowledg e
o í s"c r ain phe nomena? or more gener a lly ~ oí the rheolog ic8.1 beha"
viour o í -Che material.
'1'he non el o,s t ic deiormation oí concrote i s so impor ,
"c a,n'!;?
"chat in many calcul e.t ions and projects it i s essentia l to
t ake ac co1.1n'l; of i t. Thi s re qui r es the ava ilabili t y oE mathemo;ci- ,
cal expressions capab le oí describing with s ufficient a ccura cy
a nd g enerality -Che l aws rclating the non e l a st ic defor mat ions and
thoir c aus es.
Uníortuna tely the ultima te and d otailed c a uses oí
thi s -eype o f phenome na s uch as t heir Ilhysica l ? or physico-- chemic a l origi n are nor yet known. Bosides 9 the technoloG'ica l l aws ob
t a incd by va rious resoarch vvorke:rs a r e t oo var ied "to Derve as
b as i s o f a general tho ory.
Both shrinkage and non el as tic deformations · ·and even
e l as"ci c c.le f ormacion i"Gsol:í:' ·· depond on so many var iablo s tha"i; ~ for
t ho
pro8o nt~
i"i;
is pr a otio a lly impossi 1)1 0 t o detormino the
iIú1u~
c o oí e o.o11. of them. Each vari abl o aífects the 6thorD , and tho rosulting comploxity makos i t v ory c1ifficult , ií' not altogother im ··
p ossible to obtain go nol'al rosults by mo a n s o±' stra i g htforward
technolog ioa l rosoaroh
01'1
con crot e tos t spo cimons ? howevor numo···
r ou s -cho tostD .
Tllis is empha sizod i f it is
l~o a lised
that t he r0 8ul"03
aT'O ii:l:::' luonced a l l oast by tho f ollovri ng f a ctors : tho ohomioa l
~',¡:!d phy sica l composition of tho o or.lent~ tho ,aa-cer/ c emol1t rat io ~
tho rn'opol' -cion ing of aggrog ato ¡ th o shapo of tho pobblc s , "choir
PO 'i; l' ographio composi tion? tho type oí curing a nd consor vation,
')
,1
i 1uc0 c o oi &11 tho co v ari a b l es so
CO Q~ lic 2.toc
t ho pho nomonon'lliat
"CO oo vor 2.11 the vory co mp lica"cod phonoL1ona, and 'chon to disei:1"Cé:'11
gl o this
[ ru 1 t i ~ licity
o f oXDeriDontal data, i nto 2nd or d ere d
p ~~
t UT'O .
l E t }:ce lJas ic cause 01' 'cher::o phe,lOlilona io to be :('01.líld
in tl, o ce ü el1';/v2.ter P <él wc e ? it will be logi c a l 'co disc ovel' "iha t
ho,',lOgene ity anci. sloymess c oxw ed by
-ee r3t pie ce
) OlJ,n Clf::
OJ:
C':Lél
di :f~'ur:Jion
phollO':1ené.1. wi 'chin "che
:Je avoiclecl by experimel1til1g on the sma ll o u'c 1J08 '·
t ho s o ':; C8E1Gl1t. hy mdk ing iul l u se)
o :~·
the poworfun
Only a:éJcer 'Cüo :.c,'ho olog ic C'- l 18.:\,8 01' tilO pa sto a r o
and
ü ili 'cy o E ülvostiga.ting sucODsfully tho 1110:CO complox c a so oi'
}1.o:r' E1a.. l
s i zod l-:l ortél. I'
2.11.cl
cOllcre ta ' t U~J ·t f:3 poci1l1CJI1S
o
A g roa-e doa1 has boon a chiovcd in t hi s ini ti a l
-eho p a th of l'u rthor progro s s so tor-euous and int ricato that tha
:;'" 0
ü :; o.lv;r ay s dc:mgor o ] llover
02 P UI'SOVO:L'2.l1c o ,.
l"c
lS
c~chiov i ng
i 'inD,l s uc o ss? du o 'co 1 '::,01':
:9 08siblo, ¿cI so? tha t duo
'co 'ch e
o :{Q up G~O,
- 3-
nally r ap id growth oí' tochnology ? this ultimatosuooss mny only
bo ob-ca ined whon constructiona l mothods aro ' orienta God towards
noY!
ma tori als ~
thus causing thi s typo oí' r osoarch to loso int o·-
ros t.
On tho other hand in H s presen-c stage of c1evolop -
,t
mont cons -cru ctional "cochnique ? a lthough setting itsolf conti
'/
nuo s l y a higher standard of accura cy? is no"c so advanced in
thi s l'OSp oct t hat i t can aflord to i gnore
any attempJIi to osta ··
bli s1:. a pproxi mat o goner cll l aws ? hov:evor rough tbis:l:'irst approxima--c ion ? may be.
Cons oqu entlY 9 f ortif iod by -cho courage of my mm
ignor an co 1 malee bold to prosont this r hoologica l theory? It
",lUst be t ako n a s a pa stimo ? wi "chout othor va luo than to sorvo
t o ompha si zo -Che marit of othar papers.
In a ccordanco "vith experiment al rosults? it may bo
a ssumod tha t in a short- timo compre ssion test to dostruc-cion
6 increa sos moro r apic11y than the stress T. It may
a l so be assumod tha -c tho divergence from tho hookoan lawbecomes
gro o.tor o.s strossos approa ch tho í'ailing s tross? until fin ally?
j Uf.¡ -G bef ore f ailuro s-cr a in incr02.s os indefini tely wi thout an
incromont of s t ros s. I f "liheso exporiment a l f a cts aro a dmittod :it
i 8 oa sy to es t ablish a gonernl law for -cho phenomonon o It suff i cos to moasuro off a long tho ordina tos tho r olativo s-crossose
G = T/R (i. o . 9 the r a tio oí -Che a ctual stros s T to the maximun
t 1.10 s t rain
{
,
f' e.iling st re s s TI of "cho ma t aricl) ? a nd a long tho abs cissa e the
r ol c'c ive st r a in
€ =
&/f3 (Lo o? the ratio of "che a C'(¡ual stra in
-co 'cho s tra in cOrr eS1Jonc1ing to tho maximun stress R? montionad
prov iou i.:>ly ). In tho rosul ting diagr nm tho values obt a inod by
var ious rosoarch work ors can bo alignod ( with no ver"J groat
sco.tt or ing ) nlong a par abola? whose degroo is betv,re on 1.8 a nd
2. 5 ( l i g . 1).
- 4 -
For "[¡ho momont 9 to simplify iha
oxprossione~
let it
be nssumod thnt this pnr abola is of dogroo 2 1 so tha t tho l aw
may bo vITitton thus:
1 ,)
€
"- es r2"
(1
( 1)
In thi s expre s ion rupturo corre sponds to "::;ho valuos
es
=
1 und
€
= 1. Honco
tJ;lO corrosponcing strossjstra in :rel é1tio~
a ccording to Hooke's l aw, will be given by
es
2
and tho non Hookonn st rain will bo
[ 1
1
2
:'; '1'ou t Ilo scanty oxporimon"Gal data "availablo i t appoc.r s that this
typo o E strain romains constant 'lihon "liho dura:!iion of thotost i n
roduced ("li his reduces tho breaking strain as well)? so it can be
ox"::; r o..p olat od until tho tost dura;[¡ion t onds to zoro, providod
inortia phonomona aro noglectod, which is moroly a thoorotica l
cons i dorcJ:c ion.
Besides this, when the load is removed a residual
l'
strain, ,or permanont set, remains, "which increases steeply f or
increas ing applie d relative stresses. All this leads one to
suppos e "Ghat this non Hookoan type of strain may be expressed
as a function of frictional f orcos, which vary with the s tress.
Use vd ll now b e mado of a physical modol, but only
with the purposo oi' making it oaoior to work out the mathomati
cal troatmont. Lot us t hink of a basic unH (Fig. 2) consisting
of a cylinder ? insido whi ch thoro are a numbor of discs " Each
of thoso discs has a diffo r 0xct fr:l.0t ion coeff iciont with the
l~
-
../
.-
cyIinder waII. '1'ho d iscs aro connoctod ono to tho othor by moans
of oClu a l r::prings o If a f orco
i s appliod to tho ii1echanisf:1? a s
Ú
shoy;n on }::'ig. 2? thc firs-c spring wiII bo s ub joct -CO a tonsila
20:cco
vrhOI'O
r 1 is -e ho frictional force acting on the first disco Behlnd
the nth disc -Che tensile f orce wilI b e
I n order to o.clopt the cOídjinui ty of cha nge required
by diff erentiaI ca IcuIus ? it may by supposed "ella"e -Che r rictional
forcees opera:l;ing on each disc?
1113
we II as the Ienght of the
spr ings? tenel. to zero ? s o tha t t Le f orce on the
-canc e x 1 f rom the top wiII be
Ci m
Ci -
f
spl~ing7
a t a dis
r (x) dx
If for -Che sake of goneralitY9 i t i s is supposed tha-e
.
,
E;,
nU:"jb or o f dises at "che t op ol" the oylinder h a.ve no fric"i;ion
then? the previous oxpr oss ion v!ilJ. become
¡X I
(ú m) ><1
=
ú
~J'
r (x) dx
a
v¡here 1'(><) d~( ? i s t he elomont a ry fj:, ictiona l forco op e r a ting on
the diff'e rential disc at any depth x? so that a
If
'NO
< x < x1
0
suppos e? t o oxpress the f rictional f or c o func-
tion ? "c ho s i mpl est l aw:
r
CJ
G-
Thon tho formor expross ion becomes
and its value vanishes f'or
.f
1 0
o 0 9 under tho action of the f orc o
u p to a depth
Xi ?
los s than
X
ú
Ú
only -Ghe springs situatod
Ol~
g experionce a ny strain
movo·-
;nent lrom thoir orig ina l p osi tion. In the se circumstances the
total strai n 01 tho system i se
1:: ==
_~_ aú +
E
\ Xú [O
Ja
E
11 the sum oí all t ho diffo rontial frictional farces
is equ a t o d ...t o -tiho max i mum fa iling stress ú
J>!
dx = 1
Lot tho further condi t ion bo impo so d tha t
==
1 0 Thenz
T'~
b
-
1::
1 for ú
==
1; this
g ivo s on simplification
y + 2
1::
1 _ (1 _
r:Ch i s agrooa vvHh expr ossion (1) i f n
va l uo8 a r o
ú) ~;~'+-T~
1/2 9 so that -tiho folloV'.r:i.ng
obt a inod~
E ==
b
==
a
2
"
))
=
3
Tho Hookoan stra in v
i s p ictu r od by t h o
of springs a nd discs without f riction 9
SO ·(;
whi l st tho non Hookoa n doforma t ion, givon by
1
Sn
S -
=
SE
=
1 _ (1 _ ú
)-2 ~
-g-
i s produced by tho sy s·Gom of s pring s a nd disc submi tt e d to f rio-tion.
Thi s cor rospondonc o vlould havo no
paJ.~ tioular
into·-
r es t ? i f i t woro no t that t ho propo so d modol makos i t a lso pos -sib l o to des cr ibo t ho bohaviour of concr e to undor succossivo
J. oadi ng a nd unloa ding cyc l os .
Por in f a c t? i f a f tor aJd a ining a load ú this is
appli e d~
pJ70gTos s i voly l'omove d ? until f ina lly no load i s
h av ing p2,ssod through int ormodia t o Ioadings
aftor
ú' ? tho springs
JGond to roturn to th o ir ini ti a l po s itions. How tho f rictional
f or c os act i n t h c oppo s it o sonso , and th o l ong th xú ' (Fig. 3)
over which ? tho rocovory motion of the d iscs is producod 9 ovor-comi ng t ho s o fr ict ional roact ions is g iven by tho oClu a tions
Xú '
ú --
1
a
r,/d><
"
=
Thi s g ivos a now v a lu e f or x
1
=
[ 4 _ 2 (ú _ ú') j 2" - -12
(6)
Honc o t ho f ina l s tra i n? incIuiding t ho hooke n
ú'
s t r a ins
2
will bo
Tho first torm 1s tho hookoan s train p tho socond
torm is tho strain botwoon
~
and
><0-" and tho third torm tho
strain botwoon ><O .. and Xc" e
Aft.cr intograt.ing and simplifying
-~hi s
1
€ .. =
Ú
ú ..
2 (1 - --_:_-~-)
2
-2""
reads
1
~. 1 ~ (1 -
ú)
-2
( 8)
When the load is totally removed? there remains a:r:e
sidual strain or permanent set:
1
1
€~ = 2 (1 _ _ú.._) ~2_ 1 _ (1 _ ú)-2
2
1 This corresponds to a va lue of X gi ven by
=
><0-"= o =
(4 - 2úf -2---}. If after being unloaded, the modelis again loa-
ded, for increasing v a lues of ú ll
<ú
the position of the firs"ti
disc whi ch is not displaced will be at a depth
Xo:i'
de f ined by
9
the con dition
( 10)
From thi s )(,1" is g i ven expl i ci ty by
><0-"
=
[4 - 2 (cr
u
_
1
·)J-- -2"
Ú
1
2
(11 )
In order to obtain the total strain €" produced by
this reload process , it is useful to suppose the model divided
in three rangesc In the first, corresponding to depths limited
by
Xr<
v
and >'cí' tho stress at a dopth
(5m = (5
YIhilst if ><cí' )' x
(5 m =
~In X
-i=lna' ....(5"
:. X
ú'
=
><es'
ls ~
-J arxdx
;>
+
> a
Úm
where ><o- '> x>
x~
'x
><CS n
;
f
tho stross bocomes
x r x dx
a
(5" - J:rxdX
Tho total strain 8" will be tho suro of dof orma tions
duo to JGhoso throo rangos of stross 9 togo·- thor 'VIll.1;h tho hookoan
strain produood by tho load (5" f inaIly appliedn Honco aocording
to (3) e
( 12)
Yiork ing out J¡;ho intograls and simplifying? this bocomos
1
8"
=
1
-1
-~
'2
[4 ~· 2 ((5 - (5)] 2_[4 _ 2 ((511_(5')] - (1-(5)
1 +
(1 3)
If the lo ad (5'= o
i s mado zara (i ooo? tho dovioo is entiroly
unloadod beforo roloa ding)? 13 0 bocomos
1
8"(5'
=
'o = 1 + (4 - 2(5)-2_
1
(4~2(5;1)2_
-1
(1_0)2
(14)
If during tho second loading up oycle ? the load sur·paS8es the value reaohed in the f irst oyole, the stress-otrain
diagTam, i n this reloading oyole? is the same as if the load were
applied for the firs t time when? 0"
I f the diagrams (5
= f
>
00
(8) resulting f roID these expre~
s ions are expressed graphically ? fig. 4 is obtainedo This shows
narrow hysteresi s cy oles in the unloading and reloading prooessa3.
'l'he mean slopes of these repeated diagrams inorease slighJ¡¡ly as
-;;ho total s tress inoreasos. iJ110 n strossos' are low thi o slopo ooin
- 10 -
cides with tho initial elasticity modulus (E = 1/2). AIl this
agre os closely with experimental data.
This samo imaginary device may servo to explaincreop
phenomonae 9 als0 9 to which concrete under long periods of comprES
sive stress is subject.
In fact 9 so far data is still vory incomplete as to
how this plas tic flow vary in torms of the appliod stross. Bu"ti
.'
ava'ilablo information indicates that for same intervals of time
and equal increments of rolative stress 7 tho groator -cho relatiVB
stross, "tiho groater the incroment of creep. In the absencc of
;r oatol~
dotail of informa:l;ion 9 i t appoars advisable to assume
that croop is "tiho samo as that for short timostrains, whon -Che
duration oí loading tends to zer(). If "chis were not
S07
i t would
only be nocossary to change the law oí variation oí friction, as
adoptod proviously . Though actually lacking experimental justifi
cation 9 it sooms bot-cor to accopt the earlier law of variation.
Evidontly it i8 now necossary to tako accoúnt of viE,.
cosity. Consoquontly tho uni t previously adopted will bo
complo·~
tod by adding to it a piston without íriction. Its movomont will
forco the liquid out of tho cylindor 'through ono orifico. ').'his
will dofino a singlo viscosity constant for the unit. This unit
will be comlOctod in sories wi th tho oarlor crooploss sys-Gom so
that the nov¡ doformations that will now occur in tho courss oí
timo will bo addod to those alroady rocordod oarlior.
Lot
bo tho part oí the load Ú that actuates on
tho viscous liquid (again horo inertia force s will be negloct ed).
Thol1 the behaviour of an inolastic crooploss unit, such as that
0v
alroady considorod 9 with variablo fri ction and subject to a
doublo doformation coofficion"i; (1.0. 9 tho spring is assumed, for
the sake oi convonionco in l a tor oporations to bo twico as flexi
blo) impli os the f ollowing rosult (obtained fr om (2) or (4) "
._- 11 -
Hon co
( 16)
vrhoro
(J v
=
d8
(2r¡)
dt
Horo t is tho variablo roforr ing to tho loading timo and
is a visoosi ty constant
(2? )
-)~).
Hence -tihe following difforential oquation is obtainod~
(J
Its solution is
~81 (1 + ~t=(j) -
I{fl (1
+
1[T'='éJ)
+
(J
(J
whon apply ing tho limi-tiing condition that
8J. = o when t
= 0 0
This oxprossion can also bo W:l' iJ¡¡ton in the form
t
---~
e 2 fj
I~)~~.~
1;
S
1
r~_.::' ___ :_ .J'.~l
+ 1
'1 + ~
(
- ~ . ___ ~
- ~:~ ___
.~ __.___
- (1_ _ ~ ~ _0)
~- - 1
_ _'_
.
~~ ~~_
( 19)
r
. ·
1 ~ ~
(*) Following ge nora l practico (2?) is takon, instoad of
~
lJormally tho di mo nsions of viocosity aro (voloci ty) ><
( s tross x long th) - 1 = (stress x timo) · ·1 0 But sinco in our
caso stross is r ol a tivo stross, and rz is the invorso oí the
viscosity, t he dimon s ion oí
is timo .
?
- 12 -
ancl so?
10::'
t =
-choro corrosponds tho value
0 0
8
1
=
o.
If 0 i a assumod -co be constant ) thero i3 a valuo of
81' and honce~ aC':co rding to (16)" also a v a lue of (0
ú v ), wh:1ch
corrospond to oach valuo of t. Fig. 6 illustrato s graphically
)
t h i G rol a tionship.
ThOl~O
aro iour distinct cas es that may be considcred
vrllOn ocho ini tially a:pp liod forco 01 varios. Lot tI be the dura-tion oi action of tho f ir st load 019 and let O
2 bo tho load
subooquontly app llod.
Ac c crding to -cho exprossions alroady est a blished,
tho doforDation E1 obt a inod after this first perlod tI will
aatisfy tho condi"Cion.
wi th \;"1
:1.:11 this caso the cloforwation law will follow the
aboyo go nol"al
lavl ~
wi-c h no othor moclification 'chan that conso ·
quont upon tho chiJ.nge of boundary conditions o For
of €:l = o
whon t
=
vlh:)11.
-C =
0,
now ~
inG"eoad
tho opora tivo condi tion is that El = E1
tI o :3u1-:Jst i-Guting tho nOVí v a lue of tho intograting cons ·
t ant., th o fo ll owing o::prossion
r es ults ~
-
1} -
(21)
1 11
¡...~ 2
-'-l'
V llS
It is evident that (19) is a particular case of ( 21 1
f Ol" whi ch 81
=
o
¡¡
t1
=
o
The nev¡ loading
po riod i s le ss t han
=
2l{f1
=
tinue t o
8 -¡
Ó
Ú
2 applied a f ter the first loading
1? but greater than the lo ading
Ú
'1 - úV1 =
supported by the spr ings of this uni t o Strains con·..
i l1crease~
though at smaller rate ? a s if tending more
r ap i dl y to a pos ition of eQuilibriul11 o Equation (21) canalso be
applied without moc1ificat ion in this case.
I f aftor the firs t loading proce s s, with 10adingó1
anc1 duration t 1? the load i s reduced to the value
,,
(22)
and this i s kept c omd;ant ? the system will suff er no modifica-tiono The spri ngs and disc s vri ll be kept i n eQuilibrium with the
new loading s o the vi scous damping mechanism will exert no
forc~
and "che whol e syst em will r ema i n s t at ica lly balanced , without
e~
periencing any f urt hel" creep . In this manner the system atta ins
i t s limiting s tabili ty 9 and 81 remains constant throughout time.
.- 14 -
In cOl1.-crast wi th previou s cases, the new
defol~ma tions
i n this case are docreas ing. Tha new applied loading a2 is less
than the l oaeling a"1 .~
2~
sí
=
S1 acting on tha first spring
at the end oí th o lirst per iod tI. Henc a tha piston 9 and the
be:~'ins
I7hole c ro ep system
.>
av ., =
to moyo back -eowards a new position of
oCJ.uilibrium. frho provi ous l y established l Ol 'mula are 110t longer
operativo sinca the fric tion f orc as of t he various eliscs now act
in tho opp o s ita sonse. Tha 1m'\' of eloformation dofinod by (1 5)
\lhich do f inos tha movor:1Gnt o l - th'o eliscs is not applicablo. It
iJUst bo s ulJst ituto d by oxprossion (8) e This detorminos tho elas-tic dof"ormat ion
a/2
and a fac tor of 2 i s appliad to tha whole ox--
pross ion 9 in tho sama faannor as was clonD to obt a in oxprossion
( 15) :e-rom (4).
Honco tho l aw ol deforma tion for the spring s and
discs in this case in g i van by
sI
=
(23)
41 ~ ~1
.... [(01 ". a v 1)
=
~.
(a -1 ay)] '- 2
frhat is to say 9 tho loacling ú -1
~ Ú
v
~'I
-
(a1 -CíV1 )
~ 2(a1 -
av)
acting on the springs and
disc s at a ny timo, i s givon by
2
••
~
21 ~
1
(24)
(a 1 - Ú VI )' -
2~V2 ~ 2 -~:I
~-a1-~-a~) -
- (0'1 -.av ¡r-E1-.'-
SI
The 10ac1ing act i ng on the p i ston which oporates on
vis cous liCJ.u i cl i s
Tho no gat i v o s i g n do s c1'i bes t ho chango in tho sense oí movemo nt.
~3 inco
doformation dec l'oaso D tho L.1cromonts dS l 9-ro nogativo for
pos itivo valuo s of
av
-- 15 -
v i 8 s u bs t itut od in oxpross ion
ocho di ffcrontL:ü oquation do:;.'ining tho movomont 01' the
I f t h i s v a luo of
( 24 )
9
croo p
sys "~Gm
Ú
i 8 ob t a i nod? "chus.
Tho so lution oí thi s oquat ion
is~
( 26 )
Th0 i ntograJ~i 6n c ons tant
whcn t
==
t1 7 t:
1
s ion 26 b0coffiese
t
~
e
i s dotor mi n ad b~/ tho con ditior'l tha t
t: 1 a ncl onc e this v a l ue is i n tro du c e d ? expr e s -
t -¡
o
the l a"i! is obta ine d
.-,'hi ch de f i n e s t h e do l a:;red str a i n, after removing t he loading
I t i s s u ppo sed 9
fo:i.~
Ú
1"
thi s 1 m! to ope:Cü:co 9 t ha t t ho loading ú 1 h a d
beon a pplie d over a perio d t 1 ? a nd that él"C the end oE thi 8 periai
the maxi mum de:forme-tion was t: _¡ , A part of thhJ deformation i s
,
•
,
.
r ec overed 9 i n the cou rse 01' t i mo o '1'hi 8 prov ides -Che well knovm
"cype oE 1lhono,l1e no n call eO. do l ayed ol a8t i c i t y " The curve obtai ned
by c orre l at i ng the cha.ngo
0 ';0
delo rmat ion? ( aft er removing t h e
loa (ci ng ) ago.ins·c t i me ? t 0nds a oymp to ti cal ly t owar dB a p o s i tion
of stability o The equdtion of "ch i s ilsymp"C o to i 3
( 28 )
16 -
~.
7
rig e
showo the diagramo.tic representation of this rlelayecl do-
?orl!lat ion phenomonon o 10i g . 8 s hows curves corrosponding to various
periocls of t ime cluring vrhich th e loading ha s been removecL
11.11 theso l cws corresponcl to act u a l ooserved phono"
mena a ffecting concrete subjectod to s imple compression. But if
the result s shall havo l' oal valuos? it vvil l bo n eco ssary to assumo not ono croep systom? 'ou t sev ural of thom ? act ing under a t
l oQ,s -c tvro distinct vi sco s ities o
F or i f an i mo,gin2.ry basic modol bo s upp osed to consist of -c h o ho okoan uni t
0 1'
spring vvhoso deform2. tion is sE 9 con·,·
nectod? in sorios viÍth éln inel as tic croopless unit, who se cleforilli:1t ion i s
? Q,nd this aga i n connoc'cod in serios ,'ii th 4.2 cree p
n
systems with viscosi ty 1, = 7 minut o s? which in turn El,l'O s orially
8
I
connoctod wi th 9'.6 systoms, y/hoso vi s co sity is ~ = 2 years (t in
yeal's ) ? then the curvos obtcüned exporimentally "by Glanvillo (*)
a re ropr o ducod ,Ti th su:;:':;:'iciont a ccuracy.
It is possiblo that moro numorou s o.nd homogonous
t osts may mako i:e a dvis élblo to chango the valuos of tho numorfual
co e fic i ents ju s t rnent ionodo But
l S
8o ems? novertheless, tha t the
schomu tic arr angements pro po sed ab oye can des cribo fairly woll
this t ypo of phonomona, élS
TIO
know thom a t presont, and as dos-
cr i bed by tho a vorago l avvs o 0:[' courso, 2011 thoso numorical fa cr
•
,
.
tO:L' S nlUst dopond on many othor v o.:ciablos a nd should doscribe
l a;Hs t ha t ",:co ovon moro compl ox and,diff icul t to hcmdlo.
'.['hero ar o
t e re ~j ting
-0"1'10
further p o ss i b ilitios which i t is in·
to i nv es tigating i n rol at ion to tho basic modol pro-
viOfioly pro posod as a menns oí i n t orproting rh oolog ical phonome-
nCl. .
e
X- )
o H o Glcmvill o o Dept, oí S ci ontific al1d Indu s tri a l Rosearch,
Tochnical Pa pers, TIº 12 ancl 21 ~ London.
;,
- 17
Tho íiret of thoso poes i bilit i es is = Failuro corros ,
pondo in tho ine lastio oroopl08s unit (En) to tho total sliding
o f tha 8 0voablo part oí tho mechanism i . o., becauao a ll írict io n a l force
of
d
has beon ovorcomo \ivhen ú
1
ti
En
==
t But in tho case
o1'oop SyotOL1¡ ultimato failuro occure undor a loading which
is smallor ·;;ho longor the poriocl during vihich a load close to tho
:ca iling load is kopt a lJpli od. Thus 9 íor oxamplo? ií a load ie
kopt appliod during throo hours? failuro will occur aftor that
lJor iod ií -cho magni tu.do oí' tho load ia .88 oí tho load that will
caus o íailUl'o in a short chu'ation tos ';;. Ií oc ho load appliod is
.86 oí the short duration failing load ? than tho timo that will
ollapso boforo failuro will bo almos·;; indaf inito . All this coin-cidos v-vith tho oxperimontal rosu lts of Dhanko Thoso figuros may
be mocli l ioo. by v ary ing tho numbor 01 discs oí th8
Cl~OOp
systOGls.
f)G condly ? shrinkago may be closcri boo. by SUPIJosing trae
in the croop syst oms tho viscous liquid ol' oach unit (fig o 9) mo ·'
VGS
in"co a rosorvoir D? V/hos o li quicl l oyol cOl'responds to the hy
gromotric conditions
0:['
tho amb ion-c . \.hon -[;ho 113vol goes dovm in
tho rosorvoir a suction f or ce would act on the piston? and c ons&·
qu ently a shol~tOl'Üng talces placo " f1'his ie a movement tha-c prog~~.
si vely decl'aasoo with time a nc1 is only partly recoverod.
Fur-chormoro ; i f -Che actu a l test piece is supposad to
conois-c of a givon number oí' basic models? such as the one previouo
,, .
ly described? coup ledinparallel (rig. 10) ? so that the liquid has
to pass í'rom ono systom to another to re ach the rose1'voir which
describes th e h1.lmidity (*) ? it íollows that the shortenil1g in a
gi ven tiLle is smaller "c he greator -Che number ol baoic models in
p~
lel. 'rhat io to say? this shortening will be less the groa-ce:i' -Che
~~~.~~~.s..~.e.!::~~~._ dimonoi ons
of the "cost piece? as in fact i-l:; is tha Cé1Se o
( 7(- ) So that -ch8 ver ·c ical movoment of -Che Di:::vcon will not OX01't an
additional hydrostatic suction or com~ression (due to tho dil
forence in -Che lovel betwoen tho liquid and the piston, pro. ~­
ducod by -the vertical movOinent oí' the l a tter) 9 it vrill be sU.E,
posad that t ho group oí mcchanisms io placad on the same hori
?;ont a l plano? and that tho re so rvoir s a ro sufficiently amp16to r ender negli g iblo th o variation oí' level due to the displa
cod liquido
-
-
18 "-
Thi s a na logy will even expl a in the strange phenomen:n
obsorved by Duke anO. Davis , accor cling :to vrhich when evaporation
anO.
compressive lo ading oper a te s imultaneously con-cra ction i s
gro ater than when t hese t wo i nflu ences ac t in su ccession.
i~ O l'
in tho 18.;;;;"0e1' case the c1.elay in shrinkage due to
t he viscosity i s gr e a ter in t he intor na lly situaded units than
i n "che out or on(')s (i. e., in thoso closo "C O the reservoir, vrhich
ropr oc on"c -Che surrounding conditions ). Thus compressive
s t:i:' GSS
i B groa to r in tho centro than on tho oxternal layors. AnO. as iho
r ol a ti vo stross/8train di agr am is a curve (wi th s"hrains bocoming
l argor than t h080 proportiona l to tho strpps ? whon 'tl1e l a ttor
iE.
cre a s e s ) it follows tha t t ho avo r ago strain i p also groa ter than
tho s um of tho strains duo to oét ch cause (shrinka go Ecnd loa ding)
sopara-coly. This vfill coincido wi th tho ac count of th<;1 phonomonon
a dvancod by J?icküt-L
Tho rheolo gica l bohav iour of harc1enod concrot e is dG-pondon-c upon tho rhoologiccll bohaviour of the cOli1ont/wator pasto.
Th i s bohaviour i s r ol a ted in a complex mannor wi th olas-cic dofor-mation, romai ning stra ins or porma nont sot? cr oop, s hrinkago,
s wel ling , otco Howover it appears tha t it can be oxprossod by genor a l cnalytica:l "laws, which in turn may bo made to corrospond to
tho bohQviour ol a combina tion of i"l10chani sms? connocted in s ories
a nO. in para ll o1 o Thoso mochanisms, a s proposoCl by Dnginoor A. páoz
"
cons i s t of di ffe1'ont i a 1 01a 8t i o ol emonts, couplod to a viscous
pi st on, s ubj oc t to variablo f rict iona l for c os, g ivon by
sion
OXpl~OS· -
(3).
Tho moa sure of approx ima tion obtained b otwoon thos o
thooroti col r osults Qnd oxporimentol fac ts i s oncoura ging. \ilio n
oxporimont a l rosults b ocome more ac cura t 0 7 tho th oorotica l pott ern will havo to bo r oadjusto d (using tho samo bccsic pattern) by
-~
cOl~r o c-c i ng
¡
9 ..~
tho f l' C'O pa r ame tor s, or c orroct i n g t Ilo appli od l aws o
Por th o t i mo be i n g, h owe v e r , i t sooms jus t ifiabl e to ·
a cc cpt tho b a s ic mo do l cons i st i ng of ono h ook oan u n it ? to
bo e l as -ci c dc Í ormat io n ( ~ ) ~
co s ity, bu t wi t h
va~ i ab l o
(J.
desc~ ~
sG c ond mocha n i :JI'.1, a l s o wi t hout vis
f ri c ti onal f or ce s, to descr i bo i ne l as-
t i c oroop l os s bo haviour ( St ) ~ a nd fi n a lly t wo sy s toms , with s i mi
l a r l aws f or fr i c tion ¡ but vri t h 8opa ra t o v i s co s iti os .
)
To r e pr osent t h o bohav iou r oí o t h e r b o d i o s t h e dosc ri p t i v a patt orn will b e di f fo r ,';Ylt
tha t o c cupy t h o a t-c on ;; i on of
but t he maj or ity of bohaviOU'S
l~h o olo g i s t s
will c ons i s t oí' -cho
to gr a t ion or combination of st a t i o tica l l avíS
of
él.
l~ opr oso nt o d b~T
in~
moans
pattor n ma do up ol aot i c , p l a s t i c and vi s c ous un it s . Tho vvay
0 1' o s t a bli s h ing l aviS é,-lld nu mo ri c a l v a luos f o:c t h 0 freo
p ar amete~
wh i ch 1J1.rill n a:LT OV! t h e corres p on de nce be t ween th e theore tica l p aJ
t ern a nd "che exper i ment a l re s ul t 3, wi11 becoE1e more di:2f i cul t i n
pro p or tion to the cmlp1ex ity of behaviou r of the body . But may
b e cas e i nve s tig at ed :l'or concr e t e may h e l p oth e r s ongaged i n a
s i mi l ar pu rpos8 o
I t doe s n ot s e er,1 1 i ke l y that "chese pat t ern s a nd 1 avIS,
wi11 b e oí
gl~ e a t
u se t o t h e phys i c i s t who looles f or t h e intr ins i c
ca u s e s oí' ·che s e d e f orma t ions i n t he mi cr o mechani cal compo s i t ion
of the pa ste o Bu t they may pJ7 0Ve useful
p l~ O
tem to t he e ng ineer
who i s only concerne d with having i:1.v 2,i1 able approximat e 1aws vvith
v1h i ch to manage th e pr ac ti ca1 resul-c o of e xperi me n t
,1
9
anO. who i s
anx i ouo to p r ede termine roughly the mag nitude oí' deformations
tha t will o ccur in r e i nforced con cre-c e str uctur os .
0,9
0,8
0,7
0.6
(J
0.5
0.4
0,3
- 02
0.1
o·
-0.1
02
0.3
0,4
0,5
Q6
0, 7
08
0,9
E
FIG . l
(j
-- -L-
·1
(j
F/G . 2
I
I1
1
f
t~
lb
(/
RESULTA N T FR ICTlON LA WS
gcb l oaded (f :: db
g c ed un lo a dedO'= O
gcef r elo ade d ó "= df
1
FIG . 3
ti
~
J
1
o
E.
FIG 4
~
FIG 5
0,7
,_
QS
E
0,4
i""--
Ó-=O,85
I
,
~
I
~----r-
'j
I 6"=0. 80
0,3
u:: 0 ,75
I
u=O..70
-
0.2
--~---r""
-
0. 1
o
2'1
4(J
(
- +- ~--Qt5-e- -
oO.
.
,
,----t --
t
F I G. 7
f
FIG 8
D
FIG 9
.'
k"?:P
/ 7 ' ,-
1
}
,
~
I
f
+
:r:~
FIG 10