S"rinkage, !racking and 8eflection of !oncrete St ructures
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
ig. / - First cracking in a restrained direct tension member.
%o determine t"e crack &idt" w and t"e concrete and steel stresses in Fig. , t"e distance s distance so over &"ic" t"e concrete and steel stresses var, needs to be kno&n and t"e restraining force N force N cr needs to be calculated. /n a##roximation for s for so mabe obtained using t"e follo&ing e+uation, &"ic" &as #ro#osed b Favre et al. -14: 63 637 for a member containing deformed bars or &elded &ire mes"9 so C d $ 10 r
-3
&"ere d $ is t"e bar diameter, and r is is t"e reinforcement ratio + ratio + s +c. Base and 'urra -14:2 used a similar ex#ression. Hilbert -1442 s"o&ed t"at t"e concrete and steel st resses immediatel after first cracking are
A
A and
-;
&"ere C 1 C 2 s 2 so - 7 ( 2 s 2 so. If n is t"e modular ratio, E ratio, E s 8 E c, t"e restraining force immediatel after first cracking is
-: >it" t"e stresses and deformations determined immediatel after first cracking, t"e subse+uent long(term be"aviour as s"rinkage continues must next be determined. /fter first cracking, t"e concrete is no longer full restrained since t"e crack &idt" can increase &it" time as s"rinkage continues. / state of #artial restraint t"erefore exists after first cracking. Subse+uent s"rinkage &ill cause furt"er gradual increases in t"e restraining force N force N -t -t and in t"e concrete stress a&a from t"e crack, and a second crack ma develo#. /dditional cracks ma occur as t"e s"rinkage strain continues to increase &it" time. o&ever, as eac" ne& crack forms, t"e member becomes less stiff and t"e amount of s"rinkage re+uired to #roduce eac" ne& crack increases. %"e #rocess continues until t"e crack #attern is establis"ed, usuall in t"e first fe& mont"s after t"e commencement of dring. %"e concrete stress "istor in an uncracked region is s"o&n diagrammaticall in Fig. 5. 5. %"e final average crack s#acing, s#acing, s s,, and t"e final average crack &idt", w, de#end on t"e +uantit and distribution of reinforcement, t"e +ualit of bond bet&een t"e concrete and steel, t"e amount of s"rinkage, and t "e concrete strengt". et t"e final s"rinkage(induced restraining force be N be N -. -.
S"rinkage, !racking and 8eflection of !oncrete St ructures
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
ig. 4 - !oncrete stress "istor in uncracked *egion 1 -Hilbert, 1442 6: 6 :7
/fter all s"rinkage "as taken #lace and t"e final crack #attern is establis"ed, t"e average concrete stress at a distance greater t"an s t"an so from t"e nearest crack is sK c1 and t"e steel stresses at a crack and at a distance greater t"an s t"an so from a crack are sK s2 and sKs1, res#ective res#ectivel l.. Hilbert Hilbert -1442 6: 6:7 develo#ed t"e follo&ing ex#ressions for t"e final restraining force N force N - - and t"e final average crack &idt" w9 =rovided t"e steel +uantit is sufficientl large, so t"at i elding does not occur at first cracking or subse+uentl, t"e final restraining force is given b
-4 is t"e t"e fina finall s"ri s"rink nkag agee stra strain inAA
is t"e t"e fina finall effe effect ctiv ivee modu modulu luss of t"e t"e conc concre rete te and and is give given n b
A
is t"e t"e fina finall cree cree# # coef coeffi fici cien entA tA n6 is t"e effective modular ratio
A C 2 C 2 so- s (
2 soA and s and sav is t"e average stress in t"e uncracked concrete -see Fig. 5 5 and ma be assumed to be - s - sc/ L f L f t 2 . %"e maximum maximum crack crack s#acing s#acing is
-10
and * and * is given b
-11
%"e final steel stress at eac" crack and t"e f inal concrete stress in *egions 1 -furt"er t"an s t"an so from a crack are, res#ectivel, sKs2 C N C N - - + + s
and
sKc1 C N C N --1 --1 L C 2 + +c G f t
-12
=rovided t"e steel at t"e crack "as not ielded, t"e final crack &idt" is gi ven b
-1 >"en t"e +uantit of steel is small, suc" t"at ielding occurs at first cracking, uncontrolled and unserviceable cracking &ill result and t"e final crack &idt" is &ide. In t"is case,
A
A
and
-15
and t"e final crack &idt" is
-1< &"ere 7 &"ere 7 is is t"e l engt" of t"e restrained member. Nu5erical E7a58le9
!onsider a < m long and 1<0 mm t"ick reinforced concrete slab, full restrained restrained at eac" end. %"e slab contains 12(mm diameter deformed deformed longitudinal bars at 00 mm centres in bot" t"e to# and bottom of t"e slab -/s C ;<0 mm2m. %"e concrete cover to t"e reinforcement is 0 mm. Estimate t"e s#acing, s, and s, and final average &idt", w, of w, of t"e restrained s"rinkage cracks. %ake
C 2.<,
C ( 300 x 10(3, f t 5 2.0 '=a, E '=a, E c C 2<000 '=a, E '=a, E s C 200000 '=a, n C : and f and f y C 500 '=a. %"e reinforcement ratio is r C 0.00< and from E+uation 3,
mm.
S"rinkage, !racking and 8eflection of !oncrete St ructures
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
%"e final effective modulus is '=a and t"e corres#onding effective modular ratio is E+uation :, t"e restraining force immediatel after first cracking is
. %"e constant C 1 C 2 s 2 so - 7 ( 2 s 2 so C 2 x 250- x <000 ( 2 x 250 C 0.01 and from
m %"e steel stress at t"e crack s crack ss2 C 13100;<0 C 21< '=a and t"e concrete stress
is obtained from E+uation ;9
'=a.
%"e average concrete stress ma be a##roximated b s b sav 5 -1.11 5 -1.11 L 2.02 C 1.<3 '=a and from E+uation 119
%"e maximum crack s#acing is determined using E+uation 109
mm %"e constant C 2 is obtained from C 2 C -2 x 250- x :4 ( 2 x250 C 0.23 and t"e final restraining force is calculated using E+uation 49
m From E+uation 12, sK s2 C 2 '=a, sK c1 C 1.44 '=a and, conse+uentl, sKs1 C (;3.5 '=a. %"e final crack crack &idt" is determined determined using E+uation 19
mm. %ables %ables 2 and contain results of a limited #arametric stud s"o&ing t"e effect of varing steel area, bar si?e, s"rinkage strain and concrete tensile strengt" on t"e final restraining force, crack &idt", crack s#acing and steel stress in a 1<0 mm t"ick slab, full(restrained over a lengt" of < m. *a#le 2 Effect of steel area and s"rinkage strain on direct tension cracking -fKC 2.<, f 2.<, f t 5 2.0 '=a and d $ C 12 mm
+ s
r N - -
mm2
k ;< 5<0 300 ;<0 400 10<0 1200
.002<
.00 .005 .00< .003 .00; .00:
1<0 1:0 250 25 2 225 21<
C ( 0.0003 s
ss2K 9Pa 500 500 500 25 2<4 215 1;0
w
N - -
mm
mm
k
( ( ( :; 301 5< <5
1.; 1.< 1.22 0.1 0.2 0.1: 0.15
1<0 1:0 25 220 203 14 1;4
C ( 0.000;< s
w
ss2K 9Pa 500 500 40 245 224 1:5 154
mm
mm
k
( ( 41 301 52; 20 25:
2.0 2.01 0.54 0. 0.25 0.1: 0.1<
1<0 1:0 213 14; 1;4 131 15
*a#le / Effect of bar diameter and concrete tensile strengt" on direct tension cracking.
-fKC 2.<, ecsKC (0.0003, + (0.0003, + s C 400 mm2 and r C 0.003 d $
f t 5 2.0 '=a
N - -
f t 5 2.< '=a
C ( 0.0004 s
ss2K 9Pa 500 500 30 235 144 1<5 114
w
mm
mm
( ( ;1; 534 2 25; 141
2.3: 2.33 0.<0 0.5 0.25 0.14 0.1<
S"rinkage, !racking and 8eflection of !oncrete St ructures
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
%"e above e+uation &ill give a reasonable estimate of deflection even &"en t"e curvature diagram is not #arabolic and is a useful ex#ression for use in deflection calculations. 4./ Deflection alculations - orke! orke! E7a58les9 E7a58les9 E7a58le 1
/ reinforced concrete beam of of rectangular section -:00 mm dee# and 500 mm &ide &ide is sim#l(su##orted over a 12 m s#an and is sub$ected to a uniforml distributed sustained service load of 22.22 km. %"e longitudinal reinforcement is uniform over t"e entire s#an and consists of 5 V2 bars located in t"e bottom at an effective de#t" of ;<0 mm + -+ st C 200 mm2 and 2 V2 bars in t"e to# at a de#t" of <0 mm belo& t"e to# surface - + sc C 1300 mm2. !alculate t"e instantaneous and long(term deflection at mids#an, mids#an, assuming t"e follo&ing material material #ro#erties9 < '=aA f C f@ c C 2 '=aA f@ '=aA f@ cf C .4 '=aA E '=aA E c C 2:,<;0 '=aA E '=aA E C 2.
For eac" cross(section, & cross(section, & C C + + st $d C C 0.010;. *he section at 5i!s8an9
%"e sustained bending moment is ' s C 500 km. %"e second moments of area of t"e uncracked transformed cross(section, ( cross(section, ( , and t"e full(cracked transformed section, ( section, ( cr , are ( are ( C C 20,<30 x 103 mm5 and ( and ( cr C ;,440 x 103 mm5. %"e bottom fibre section modulus of t"e uncracked section is ? is ? C (8y C (8y$ C <2.; x 103 mm. From From E+uat E+uation ion 20, 20,
f cs C
C 2.04 '=a
and t"e time(de#endent cracking moment is obtained from E+uation 1;9 9 cr C <2.; x 103 -.4 ( 2.04 C 3:.< km. From E+uation 13, t"e effective second moment of area is ( ef C 6;440 L -20<30 ( ;440-3:.<5007 x 103 C :0<0 x 103 mm5 %"e instantaneous curvature due to t"e sustained service moment is t "erefore
k i-t C From E+uation 22 22a9
C
C 1.;5 x 10(3 mm(1.
a1 C 60.5: x 0.010;(0.<761 L -12< x 0.010; L 0.1-13002001.27 C ;.<<
and t"e load induced curvature -instantaneous #lus cree# is obtained from E+uation 219 k-t C 1.;5 x 10(3 -1 L 2.<;.<< C 2.2 x 10(3 mm(1.
From E+uation 25c9
' r C ' r2 C
C 0.43
and t"e s"rinkage induced curvature is obtained from E+uation 29
mm(1
%"e instantaneous and final time(de#endent curvatures at mids#an are t"erefore
S"rinkage, !racking and 8eflection of !oncrete St ructures
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
and t"e load induced curvature -instantaneous #lus cree# is obtained from E+uation 219 k-t C 0.;< x 10(3 -1 L 2.<1.22 C 1.15 x 10(3 mm(1.
From E+uation 25b9
' r C ' r/ C
C 0.<3
and t"e s"rinkage induced curvature is obtained from E+uation 29
mm(1
%"e instantaneous and final time(de#endent curvatures at mids#an are t"erefore k i C 0.;< x 10(3 mm(1
and
k C k-t L kcs C 1.<5 x 10(3 mm(1.
*he section at each su88ort9
%"e sustained bending moment is ?ero and t"e section remains uncracked. uncracked. %"e centroidal axis of t"e uncracked transformed cross(section cross(section -&it" + -&it" + & located at a de#t" of 500 mm is located at a de#t" of 504.5 mm belo& t"e to# fibre and t"e s econd moment of area is ( is ( C C 20,<30 x 103 mm5. %"e #restressing steel is located 4.5 mm above t"e centroidal axis of t"e transformed section, so t"at t"e #restressing force induces a small instantaneous #ositive curvature. S"rinkage -and cree# curvature develo#s &it" time. %"e instantaneous curvature is
k i-t C
C 0.02 x 10(3 mm(1.
C
From E+uation 22a, &it" + &it" + sc C 1300 mm2 , + st C + s/ L + & d & 8d o C 200 L 1<00x500;<0 C 5000 mm2 and, t"erefore & t"erefore & C + st $ d o C 5000-500x;<0 C 0.019 9
a2 C 61.0 ( 1<.0 x 0.01761 L -150 x 0.01 ( 0.1-130050001.27 C 1.2;
and t"e load induced curvature -instantaneous #lus cree# is obtained from E+uation 219 k-t C 0.02 x 10(3 -1 L 2.<1.2; C 0.04 x 10(3 mm(1. From E+uation 25b9
' r C ' r/ C
C 0.4:
and t"e s"rinkage induced curvature is estimated from E+uation 29
mm(1
%"e instantaneous and final time(de#endent curvatures at t"e su##orts are t"erefore k i C 0.02 x 10(3 mm(1
and
k C k-t L kcs C 0.4 x 10(3 mm(1.
Deflections9
%"e instantaneous and final long(term deflections at mids#an, 8i and 8%, res#ectivel, are obtained from E+uation 2<9
S"rinkage, !racking and 8eflection of !oncrete St ructures
Sen! Discussions
"tt#9&&&.e$se.org/rc"ivesFulltext2001010220010102."tm
Electronic Journal of Structural Engineering, Vol. 1, No.1 (2001) 2-14 © EJSE International 2001
