CONCRETE LIBRARY OF JSCE NO. 34, DECEMBER 1999
FRACTURE OF CONCRETE COVER - ITS EFFECT ON TENSION
STIFFENING AND MODELING (Reprinted from proceedings of JSCE, No.613/V-42, February 1999)
.-»
“
A
"
-- -<2-:"=:=:=$>
1;:-"1-I--=1:~>:-:-:1:-.
~.-:~:-.~.;.
..
“"
".“1:Z;.;.
151’
I-I
1;! ‘s
F?‘-.
I511 ‘$21:
:+:;:;-:-1;
> ' :I1‘~;'
- 113".‘ ;.
I ;:_:;:
?' :-.-i
I<I~-1' ‘T
;l'~‘~-1'
.- .
~.
r::_./_:;7;:;;-Q3->_v,_.__
' >,;;:~;-:-I-.1:2;;.':§.}’;'
t
.‘;;;' 6; ‘“?:I;I;1:-¢:;;;;; :-:.
153531“ .
t- '21"'--@'?“»::1§':"i73;3'
‘-1 "
.:‘:""1.'.;.;.;:§;1:-" \- -;:k~-;:
'
1 -'
'
§-:1:1;3i1-
;j_1;¢
;I:'
_;~':g~
,1
.:
31+:
!:I1?.1I:I‘I" -- I‘ '1.-=‘l;i
. .,.I:!:?.IL- -‘
- .
'~I3I- -.,=
;t‘;>:>:'::' 1.‘ - .
.-‘-'-~:-:~‘:3:-'.
..3:3:3:l:-‘:‘~::.' .
-' ‘ .1‘ :'-‘3.i1?:3:-‘:?:"¢'
;._:;:;;;:;;;:;1;:-.-.,
.
.;;;;t-;-;:;:;l;i§¢§_
.jI;f;l1I;.-Z{I;f;Z;I;I;J;§;l‘I;.>" _
'
.' :;.::;:;:;:;:§§r
';5I3:3I>-i ."7‘3I3t::3;1:7;':‘!5f3:5:3‘1:?1-1
;i;l.i.-.~:5:'r3;5:i:3:7:3:1:-:5
=:;:;:;:;:;.;.;:;._»:;:;$;;:§:§£$;!;§:;:¢ ; ~ ;.;:;1-"':-:;:;';.§;:;:;1g;:;*
in I;
'3:151iI§7E3;i:1:I:?:Ii7:3:7I5l?iI:-17'1I11 5;3:7:"'§1}5i?1Y'Ii -I31i5I515I"‘
£.31E:E¢E=';
=;£1E;E4E1ijI~jE;2r3;‘>5‘-1512*:-"E'==-1:E?£¢Er£"~.¢’E=E‘E?i?I.5’--='~?1?i?Y5ilE555EFEii
_;:;1;.;1;:;:;:-'
EIii?I:1.3;1:~;;.1:§;5Zf:§2$;§§%5_§z=9$'I - I I:-. .‘;'t1;§:§:§:1:‘;;§1§
E5E55E3E:{:E:E:E.jl%1E¢§1§I§=~-.-_-:9"ET}:\.;:}:;:i i:i1EiE¢EIE=E1Z=?IE'
‘E55355555E?Er§i
§s;&n;a~;;<4s$§§e??$§§%;
i5E5i5Ei:£1ErE1if
.;.;.;.;-;-;-:-;1+1‘.-.
.1 <.;.;.7.\.,¢.;.;.;~;¢;i;-;-:-:~.
.4,-,._-_-,,_v_',~_-_-..-.-. ..>. ,,_._._._._._.,_-_-,-r
'I;§:}:§:;:;:E:;:;?§3?3§§7~1‘1;§¢§;§;§1§:§;§;{§$:§:;:;:;:
' "~;,;.;:;:;:;:_ ‘;‘~..,.M.‘Z_L;I'Z;Z;I\‘l‘l;I;7:l_'_‘
§;§:§t§f§'§"'
'
‘;:‘;;§:f:_:'.5:§L;“{:‘~."
.~
V _
_
,
‘iv’-L-’~‘-I-'-I-"
'
\‘7;§5'<'=Z'I~<¢:1i3?1“"
-.;.-;._-;:-:;:;:;:;:; .if-V
-. .F"?
t‘-\'
flare
" '
-- .-.
-I
t
-1'1.-g: -,
Hamed SALEM
"F512;"
K¢§E55§?335£5€55§i§i;I;i~E".~‘55l?,E1*5I-:i 1'?l E-E53551“
'~Z~'*Z~Z>I'I"“
' ‘-‘-‘.<Ii"?-I‘I~‘~I~I\I~I-‘-‘
~~
i
x
_.-.\_._~ .-
~,-.-.‘.-.-.‘.v.~.\. \~.=.-.>.~~.».>..>
";*I~I-Z-I~Z~I'2 ' ..<-......S'. ’*'='~’-‘§~;-I-;~'
-;;:;:;;3;“;£,;§;;;;;;;;-:-..;;;;-it>-$,'~‘;:—
.
V
. g;:;_.--1;;-;; xr;¢;;;.~§:~';:-;-
-..;_ ' ;-. i
. .:-;-: ..
/
-~-.-.‘.-.*--.-
;?s1:€s‘s:='E1%2¢i: i éie?§i&i;Es?sisi§EsisisE2%;%§i1i;%;?si2‘21'?
-:€;'>.=E‘-Ei:I;I“f-‘EI:iEE‘E¢E1§I¢§~.-:i'
Y"‘
':<5$:3$:¥:?:¥$:7:i:Y:i:‘:;::~':i-I-If..
'.‘I:T§':l'I:-‘Fr?
.
zi:4-2:-:-:~:-:-§;-:-:-:‘:+:~:->:-:-:~:-:-;T:-..»:-
._
‘ -~-'?~‘El§i§i=.i1 I.1:=E=?I‘ ;.-:.~.:1:*.~:51:1:1§Y:1:151:1§$r1r=:1:l:111'
‘.3;E;5;E;};%;i1£;E;E;E;E§£3312;EgE;E5i;$i;:;:;E5:3‘
'
-;:;:;:;:;:;:-:;:; .1;
-:I.-.-' 1'1‘?!-1-1-.-II-.-I~I'Z-Z-.»l:»‘.$:i:¢:':<:!:~:~:i:i:.~ ~ . " 3;: ; - ~ '.1.l:i:
i i *1"
:-1-:1
.».-:-:-:_»
:
.
;-.'.
".-:§}'1:1rlg11;?;i;I:i;l:!=>1II'1.1‘:"':- --‘Iii:-:1:1:‘-"
r'.{~_c;.->*___:;:;:¢:gg:»g:;:5-=-;‘;¢-. _ ;;_:-;;‘
._
",:;:»=:;:;:;t_:
$151‘:
-
‘:a:€?:1;i:11"-:I:- 3;‘:-.3;3;;;"-":1;
3'3=4:-:1.-.-:i:1.~:i.-:1
'~‘~'~'-*-~.-.~t-"~r~r\r¢»:»:»:->
.-:-:-;-:-:-.:-;-:-"- 1»; 1_1:;‘;1;I;:;:;r;-r,
"?.‘€1Z71¢"E<I'13I3I3If3' I ‘
- »
;.\:;:{-[:§:§:§:§§:§t}._
'.1;‘<‘.1'1:.:'
‘. H ' ' ..:~1:it1.111:i:?:-23:1‘? ..
1:;.;:-:_;.,.,.,g.
. ;__ , , , , ;‘».;i_I;t;i;2;2~I;I;
~'l'.-'63 .
.
1*-L---"~21
.;:;I;:;
;__t,;__t,t, .._.;.;:;:;:3"-;I,§:§.;;';
§$.;:;:;:;:~,.;.,;,;.;.;.;-_-.-.-.-‘
,.,;»;.;\_:_v_\_ {§;;‘_;1-_!i‘
'33‘‘5?3I3Iil"'“' “"<§7l3'~.~1;3i1§5‘n
":i.‘:1$"
2;‘.
;!
;:;.';:-‘:
2151 1 I\'.11?.5:’§3§?il§':3i3'f5;1" 21:"-‘:7:?:l:1:?:?1?:3:1:‘:':3§'t-211511;112,! j1j.r;_g.;I;§§;E;I§;:;£5E3."1-. I=§i;¢;i;*;‘*~.=5§;EgE;:;g3£;ig£;E z
;-:;1_V‘;1:;:;:;:;:;:;;;:;:;:;t;:;:<:;" '_1;§t;:;:;:;;;.1:;:;;;:;;~*¢;:;:;:;:_$5
:'5:3:1:?:5:""; ' : ;5:'.-;-.-.-..i-:1:l:1;?2YI§‘3‘5'§‘§:3f¢:57'EE1t3I7i}.-I513
Z1.533=31}rE¢E1?;:1i
=ErE;E1?Er5'55E"E155?
v _ 1. '=j;55€:€=1>- .~
.--:~::.:-:-‘.-.-.~
~ . A " ' .-.'--.».-~:<:»:»:'t'::~:"
I‘ ‘I:-'
.:;:;:;:;:;;;._.-:-:.;;:gg§:;:;:;:;:;:;;;:;:;:;:;'=- ;—;;:»
-- -
i‘-1:1:I:':._
ei zieai;z2aa=::s~.-.,.-.;¢,v"T:1:!~‘
==z;a;iz=”¥s&z%z¢aEa=I'
v'~'~>Z-1&2 l' “
'i=i1§13ZYi? ~ ' -I:-1'-E3313 ‘ 1 5512151355
I . .f-- IrI.= t i
.
~.--.:.r.~.-.-__.;';...
<.‘.- '- A .‘-:~.v:"~:-:~;-:»;»
' .-:2~‘-1-F:-:-:i:»:»:
4 1:<:<. '»;-1-1..-:>6$::»:’<+;»:-::-;-;-
Bernhard HAUKE
";§"\v\':>\.
has ' ¢-‘Q
.
.
‘--'-3
__ .;.;:_;§ :-r; --:;:;:;:;:;:;:;:;t;:;:;:;:;.,.,,
.. -.4-1-:1:I:1:1~1$:?_;’3.. ":§¢:~..-.
--:~:?I=:i:3!7' I-:-.\;;~'
'?'1:¥:?:1:1:"?‘1:l'~v,-.~.'.-.~.-.-.-ma’-.~.-.>.'-Sit
,- *'.1";-.~.
5-ct:-:1:-:-*-6'*>'~**l~';:::?$
‘*"‘~’;:,*;5;:;:;:--" "';:;:;!;- 14:» ~, _ v-5‘-:
-?1~1-¢-:-'<.
~»'~\ ~~\/.-.->. ».-.~.
=1~.~.~.\.~.-..-;
-Q3‘. <.- 453
"‘»3y§%."'$.'§~;<1‘1._
‘$152213!’
:?:;~ ‘
'2
'3 “ ‘Q
.=1:L;~1 2:2-zsr-=a>
=»=:r- I ‘ ii YT -.~'»<=-§<,~$\»"‘-§.»;.5.71,»3>
' 4..
?c~’
~‘=':-:.<§§§<a<<i-* .
;\,§<§"{tQ.§.§f-¢',\'%t~°».
> *~ .‘
.- it
\ .
“rr-=3».
-‘
l
<11
'
.
_\__¢;
-‘-7-.-.*'
"“'" 51*’
.-:1‘.-:~.
n
,->1
‘:‘ '
6
Q
‘§§;’Z>$:.
~"=2-zr§;=¢.==
'__.,;, 4
,..5;,;-s"-. _ ._-,, -#1;
T'\\,‘r*-:1i‘§.Q;w_~.,~;;Y-<1.f_
I;
Koichi MAEKAWA
The aim of this study is to investigate the effect of insufficient concrete cover on the tension stiffness of
reinforced concrete. Splitting cracks are predicted by simultaneously solving for equilibrium among radial bond
stresses, softening tensile stresses of splitting concrete planes, and transverse stress on the reinforcement. The
behavior of bond after splitting cracks occur is the point of the study. An analytical model is derived from the
micro-bond characteristics. An experimental program was carried out to verify the proposed post-crack bond
model, and the analysis is in fairly good agreement with reality
Keywords: Bond-slip-strain, tension stzflenirtg, crack spacing, confining pressure, splitting crack, cover
Hamed M. M. Salem is an assistant professor at the Structural Engineering Department, Cairo University, Giza,
Egypt. He obtained his Ph.D. from the University of Tokyo in 1998. After obtaining his Ph.D., he worked as a
JSPS post-doctoral fellow in the Concrete Laboratory, The University of Tokyo. His research interests relate to
nonlinear mechanics and the constitutive laws of reinforced concrete as well as the seismic analysis of reinforced
concrete. He is a member of the JSCE.
Bernhard HAUKE is an engineer at HOCHTIEF AG in Frankfurt, Germany. He obtained his Ph.D. from the
University of Tokyo in 1998. His research interests relate to the nonlinear mechanics of reinforced concrete and
analysis of steel composite structures. He is a member of the JSCE.
Koichi Maekawa is a professor at The University of Tokyo. He obtained his D.Eng. from the University of
Tokyo in 1985. His research interests relate to nonlinear mechanics and the constitutive laws of reinforced
concrete as well as seismic analysis of structures and concrete thermodynamics. He is a member of the JSCE.
-17]-
1
. INTRODUCTION
When there is insufficient
concrete, longitudinal
"splitting"
cracks, form parallel to the reinforcing
bars. The
occurrence of these cracks is a result of three-dimensional
bond transfer mechanisms. The lugs of deformed bars
induce bearing stresses in the surrounding concrete, resulting in conical compressive struts as shown in Fig. 1.
The conical bond actions between bar and concrete can be resolved into radial and tangential
components.
Usually, the tangential component per unit area of the reinforcing bar's surface is called the bond stress, whereas
the radial one is called the confining stress. The radial stresses may be seen as analogous to hydraulic pressure
acting on a thick-walled concrete ring. When the tensile ring stress illustrated
in Fig. 1 exceeds the cracking
strength, a-splitting crack is formed as shown in Fig. 2.
The bond behavior of concrete with such cracks was studied by Gambarova et al.5). Gambarova tested specimens
with artificially
induced splitting
cracks. By relating the splitting crack width to the confining pressure on the
bars, an empirical formula was proposed for bond stresses after longitudinal
splitting
of the concrete cover.
Abrishami and Mitchell105 studied the effect of splitting cracks on the tension stiffening
of concrete. Specimens
with shallow depth were targeted. Here, the concrete cover was insufficient
on both sides of the reinforcing bars
concerned. Most members of civil structures are deep in general and the small cover of either side of the
structural reinforcement does matter. Therefore, splitting cracks would have less effect. Salem and Maekawan)i
12) derived macroscopic tension stiffening from local bond stress development by assuming thick cover, which
would lead to the full performance of bond stress formulated by Shima et a!4).
' Tensile Ring
Stress Released Zone
3!«
1-Elastic
Un-Cracked
Surrounding
Fig. 1 Tensile
2-Interior
Splitting
Crack
3-Completely
Cracked
Concrete
ring force caused by diagonal
bond struts
Fig. 2 Radial tensile
stresses
and splitting
cracks
The aim of this study is to derive a smeared model for reinforced concrete in tension from microscopic behavior,
taking into account the possible reduction in bond stresses due to insufficient
cover accompanying longitudinal
splitting
cracks. This study is an extension of the authors' formulation of macro tension-stiffness1!)>
12) covering
the case where pre-matured splitting cracks are induced.
2
. SPLITTING
BOND STRESS
(1) Members without transverse
reinforcement
The principal direction of the bond force transferred between deformed reinforcing bars and the surrounding
concrete is at an angle with the bar axis. The bond forces can be resolved into radial and tangential components.
Usually, the tangential component is called the bond stress, whereas the radial one is called the confining stress
or pressure. The angle of inclination
denoted by a (see Fig. 1) ranges from 45 to 80 degrees as reported by
Goto2). The radial stresses due to bond action act like hydraulic pressure acting on a thick-walled concrete ring.
772
An elastic solution
for the stresses
in a thick-walled
Timoshenko0 and also by Avalle et al.9) as,
1 -
C7r=pR
cylinder
subjected
to internal
R '
pressure
by
R '
1 +
a
cr
-R'
R '
(1)
,=pR'
where, ar, <jt: radial and tangential
stresses, at radial.distance
r from the centre of the bar; p: radial
radius of cracked concrete zone; Rmax:cover of concrete + <3>/2; and <£>: bar diameter.
These equations are valid for uncracked
develops as illustrated
in Fig. 3.
concrete.
However, in cracked
Fractured
Plastic
Elas tic
C
is given
concrete,
a tension
pressure;
fracturing
Rcr:
zone
ft^
Un-Cracked
(Elastic Solution)
racked
(Tension Fracturing)
Crack Opening
Fig.
3 Tangential
concrete
stress
developing
in cracked
Fig. 4 Elasto-plastic
and uncracked
and fracturing concrete in tension
According to Avalle et al.9), the bond pressure which causes a splitting
crack of radius Rcr can be computed by
equilibrating
the bond pressure p with the hoop tensile stresses in both the cracked and uncracked concrete as,
fR
n
t.
T>
-i
^
\
n
Kcr
\
r ii n_
1+
I
/___/__\\
(wini
I
Idr
I
K
If,I
$
R ,
+R,
*/2
/ ,
(2)
where,/ is the tensile strength, w(r) is the splitting
crack width at radius r, and ac(w(r)) is the residual tensile
stress corresponding to a crack width equal to w(r). The tension softening model adopted here is given by
Uchida
et al.7) as,
crc(w(r))=
/j l + 0.5f-^lw(r)
I
lGfJ
;
(3)
where Gf is the fracture energy ranging from 0.1 to 0.15 N/mrafor plain normal concrete 7).
In Eq. 2, Avalle assumed two propagating
splitting
cracks. This assumption agrees with the experimental
observations of Morita and Kaku3) who reported that two or three splitting
cracks propagate to the surface of a
773
concrete cylinder in pull-out tests. Moreover, in structural members, this is usually the case where splitting
cracks propagate towards the side with less cover. Avalle also assumed tangential strain compatibility
by
equating the circumferential
elongation atthe surface of the reinforcing bar and at the crack propagation front
with the concrete elasticity denoted by Ec as,
YoJ*
I Z.l-
/,_
Using Eq. 4, the splitting
crack width at the reinforcing
X(Wmax)
(4)
bar's surface, denoted by wmax,is computed.
Acrack width distribution
has to be assumed in order to integrate the second part in the right hand term of Eq. 2.
The authors assume the splitting
crack width distribution
to be linear, ranging from wmax
at the surface of the
reinforcement to zero at the crack front, as follows.
<£
=wr
(5)
! -à"
$
w(r)
-
R
When the splitting
ultimate pressure,
cracks reach the concrete surface, Rcr becomes equal to Rmax.Thus, from Eq. 2 we have the
pult, as,
à"^
txmax
(6)
Pui, = -r J<Jc(w(r))dr
By substituting
Eq. 5 into Eq. 3 we have,
A
1
c(w(r))
+ 0.5/f
wmax[Rmax
-r]
(7)
= /r
$
a
-3
^f|Kmax
Substitution
-'
of Eq. 7 into Eq. 6 yields,
V3
= 2 $｣
P
0 .5 / r w m ax [R m ax - r ]
2
ult
(8)
we have,
1
Finally
dr
<｣>
G , R .
P ult
=
2b
(9)
1 -1 4-HP
A
à" UAXmax
-174-
_b3>\
-T- I
I
where,
2/,
=
0 -5/,
b
3>
w,
=
(10)
&
a
f
Rmax
^
z
G
The foregoing equations assume that concrete is an elastic-damaging
material in tension. But in reality, the rapid
relaxation of tensile stress at higher levels is observed in concrete as a time dependency, and a plastic-flow
deformation may take place close to the cracking stress (Bujadham et al.I4)). Therefore, a simple form of
concrete plasticity
is introduced with respect to a yielding plateau equal to double the cracking strain, as
proposed by Okamura and Maekawa6). Figure 4 illustrates
the idealized
concrete plasticity
in computing
confining pressure.
The solution is derived by considering an exact elastic solution and determining the position with a tangential
stress equal to f, relative to the position with tangential
stress equal to 2 ft. At location r = Rcr,the fictitious
tangential stress equals to 2ft and, for r = Rp, the substantial
tangential stress must be the same asf,. Then, by
substituting
these boundary conditions into Eq. 1, we have,
R.
/
.-|
IT
_j_ ~-max
0.
"i"
(12)
R ,
R ,
the radial pressure p is computed as,
2_-D
2/,
iv max
(R.-R.J+
<&
R,
R,
2^
crc (w(r))
ix p
+R/
I
<J>/
2
dr
(13)
/,
/
Consequently,
(ll)
R ,
R ,
=R,
R
'
R
1
R ,
The splitting pressure when concrete plasticity
is considered is the same as that when plasticity
is neglected. This
is due to the fact that the contribution made by the uncracked concrete is zero when the crack finally reaches the
outer surface of the concrete, as shown in Fig. 5.
ComputedConfining Pressure p (MPa)
25
10
20
30
40
50
Cracked Radius Rcr (mm)
Fig. 5 Effect of concrete plasticity
prior to cracking on splitting
175~
crack radius
Rcr
I
Tangential
Stirrup
Strain Compatability
[Eq.3-4]
S
=S
+0.047(fu
-fy)(es
-Esh)
A
Crack Width at Tie Position
S
=ES(2+3500ss)
Virtual Crack
Width Distribution
_*<?'
^\^~
&
&'
J»
cjx _cv>.cy
§>" ,x&*~
*
l/^'
V
"&
<ey(!2+3500ey)
a=a(e)
e
^
Sh
Strain,
e,
ps=2.As.a/d
Fig. 6 Scheme for computing confining
rei n forcement
(2)
.^
'\f
E=e(slip)
Tie Slip
c'S-'
c5*
-XJ"
/^^^#
Tie = Ws/2.0
Rcr
/
/
,
^ ..^"^
.~ .A-v.&«'
<&à"'
^5
<à"=A."
(Ws)
'c&-
Slip.of
'.A
^
Max.Crack Width (Wmax)
Fig. 7 Slip-strain
pressure due to hoop
relationship
(Okamura and Maekawa6))
Members with transverse reinforcement
If transverse reinforcement is present, the resistance to splitting
cracks increases and the confining pressure on
bars is accompanied. To consider the effect of hoops, the same analysis as adopted in the previous section is
used, and the confining stress produced by the hoops is added (Fig. 6). The splitting
crack distribution
in this
case will not be linear due to the existence of hoop reinforcement, as shown in Fig. 6. The crack width vanishes
at the hoop Iocation8). However, the real slip of the hoop reinforcement, which equals to the deformation of the
concrete adjacent to the hoop, can be obtained directly from the linear distribution.
Geometrically,
the virtual
splitting
crack width at the hoop position, as computed from the linear distribution,
is equal to twice the pull-out
of hoops from the concrete. To compute the hope strain at the crack location, the slip-strain relation, obtained by
integrating
the steel strain based on the local bond-slip-strain
model, was used (Fig. 7).
Hence, the confinement provided by the hoops can be coupled with concrete fracturing. As can be seen from Fig.
8, before the tip of a splitting crack reaches the concrete surface, the confining pressure rises wherever the hoop
becomes closer to the bar. This is because, the closer the hoop to the bar, the higher the load required to develop
the same crack width. The authors believe that, after the crack reaches the concrete surface, a different crack
width distribution
may exist with a wider crack opening at the concrete surface. Hence, the closer the hoop to the
bar, the lower the confining stress.
Confining Pressure p (MPa)
30
Splitting
With 6 mmstirrup having
a cover of40 mm
25
20
15
10
5
'0
20
30
40
50
60
Cracked Radius Rcr (mm)
Fig. 8 Effect of hoop confinement on splitting crack radius
-176-
70
3
. SIZE EFFECT SIMULATION
Many experimental works have exhibited a size effect in which the nominal splitting
stress decreases with the
increase in the bar diameter15). The present model successfully simulates this size effect of splitting
pressure.
Since the splitting
crack width is proportional
to the bar diameter, as shown in Fig. 9, tension softening and
hence the splitting
pressure is reduced in large-scale specimens. Figure 10 shows the computed size effect on
splitting for geometrically similar specimens.
4
. BOND BEHAVIOR AFTER SPLITTING CRACK QCCURENCE
After splitting cracks occur, the bond stress tends to be sensitively influenced by the reinforcement confinement.
This confining action may be provided by the residual stresses transmitted between the faces of the split concrete
and by hoop transverse reinforcement distributed
along the main bar, as illustrated
in Fig. ll.
If the confining agent is not adequately apportioned around the bars, splitting cracks develop abruptly along the
re-bars and a sudden splitting failure may occur. On the other hand, if adequate confining action is passively
induced by the fracture of the concrete cover, the bond stress may increase until pull-out failure or rupture of rebar as illustrated
in Fig. 12. However, pull-out failure is not possible in practice. This kind of failure is possible
only when the bonded length is very short, as in some experimental simulations of bond mechanisms in which
only one or a few lugs are bonded.
M
ax. Crack Width at Splitting
0
.100
0
.080
Splitting
(mm)
Pressure p (MPa)
20
15
0.060
10
0.040
5
0.020
0
.000
0
10
20
30
40
Bar Diameter
Fig. 9 Size effect on splitting
50
60
C/<3>= Constant
10
20
30
Fig. 10 Size effect on splitting
crack width
W
Reinforcement
Fiig. ll Confining
40
50
60
Bar Diameter (mm)
(mm)
Concrete
pressure acting on reinforcement
Splitting
Failure
Fig. 12 Possible
-777-
pressure
<2r
Pull-out
kinds of failure
W
Failure
of reinforcing
<2r
Rupture Failure
bars in tension
Gambarova et al.5) developed an empirical formula for bond stress after formation of splitting
cracks. Their
model represents the bond stress as a function of splitting
crack width, wmax,
and bar confinement denoted by p
as,
T
=/c'(0.042-
0.288(wmax
0 .258
/cD))-
-1.018
(14)
((w^/30+O.ll)
wherefc is the compressive strength
of concrete.
1
The bond-slip-strain
model of Shima et al.4), which is used in the analysis, does not take into account the effect
of splitting
cracks. In fact, in the experiments on which the model is based, splitting
failure was avoided by
adopting a thick concrete cover. Thus, the model of Shima is simply modified in this study by changing the
intrinsic slip function as follows.
T(e,s)
T
0(s) = /c' k [ln(l + 5S)]C
(15)
= T0(s)
before
splitting
(orig
inal form)
(16)
T
0(S)
after splitting
=T1
(proposed
where, S is the non-dimensional slip=1000 s/<3>, s is slip, <I> is bar diameter,
and TI =T0(S=Si) where S} is the non-dimensional slip at splitting.
model)
k is constant
=0.73, c is constant = 3,
The slip function after splitting might be higher or lower than the assumed one depending on the splitting crack
width. The authors decided to tentatively
assume plastic behavior of the slip function after splitting.
This
simplified
model is then discussed by comparing the analytical
and experimental macroscopic behavior of
reinforced concrete later.
When the bond stress computed using Gambarova's model exceeds that of the original Shima model, in the case
of a very small crack width, the original Shima model is used. This is because the bond stresses after splitting
will not be higher than the bond stresses before spitting.
T
o(s)
O
riginal
Model ( No Splitting
)
TI
Proposed Modified Model
(After Splitting)
slip(s)
Fig. 13 Extension
-775-
of Shima model4
5
. ANALYSIS ALONG AXIS OF REINFORCEMENT
Based on the microscopic bond behavior, Salem and Maekawan)'12) computationally
derived the macroscopic
behavior of reinforced concrete in tension as illustrated
in Fig. 14. In this analysis, local stresses of both concrete
and reinforcement are evaluated. Hence, the average strains and stresses are computed over the whole of the
analysis domain.
Whenthere is insufficient
concrete cover, the cover may split and a possible reduction in bond stresses also has
to be checked. Here, both the Gambarova model and the modified Shima model are used with coupling as
explained in the previous section. As illustrated
in Fig. 14, five governing equations are simultaneously
solved
after the formation of splitting cracks. One more unknown (splitting
crack width, w) accompanies the additional
equation
(Eq.
14).
Figure 15 illustrates
a sample of computations, in which the splitting crack width and the bond stress are plotted
along a bar of 13 mmdiameter, 750 mmlength, and with a concrete cover of 5 mm on both sides. The hoop
tension crack width increases close to the transverse cracks, where the bond stresses are significantly
decreased.
C
:£|I
Local Bond Stress (MPa)
.L
10
Ribr
I I I ITh-r -~J ^>i<lTI
^
(a+Aa)As^
Original
'
_^T
(without
any splitting
Front of Splitting
Crack
BC:S=0.0,T=0.0
Proposed
/
crAs
crack}
200
X (mm)
Hoop Tension Crack Width (mm)
S=Z(e
Ax)
16
£8?
'
1.200
to"=». Sa
0.800
0.400
T=T(w)
0
.000
0
4) Slip Compatability
5) Bond StressSplitting
Crack WidthConfining Pressure Model
1.600
'
-
s
o $
1) Constitutive
Equation
<j=cr(£)
2) Equilibrium
A0.As=t.7Td.Ax
3) Bond-Slip-Strain
Model
T=To(S).g(e)
4v
200
X (mm)
OL
*
>//>..
N "«*
,a%
Fig.14
300
Splitting/
400
Crack
D13mm
<fn
Scheme for solving bond governing equations with finite
dis creti zation
Fig.15 Analytical example for splitting
of bond stresses
crack width and reduction
Figure 16 illustrates
the analysis of three members with the same reinforcement ratio but with different
geometry and cover thickness. Cross-sections of 100 x 100 mm,40 x 250 mm,and 30 * 330 mmare used. The
corresponding concrete covers are 40, 10, and 5 mm, respectively.
No hoop reinforcement is assumed in the
analysis. For the specimens with 40 mmcover, the model predicts no splitting crack and three transverse cracks
over the total length of 1,000 mm.For the 10 mm case, analysis predicts splitting
cracks with only one
transverse crack, while for the 5 mmcase, no transverse crack is expected and longitudinal
cracks are predicted.
Tension stiffening is affected by the longitudinal
cracks and the corresponding reduction of bond stresses. The
smaller the cover concrete, the larger the reduction in tension stiffening of the concrete, as shown in Fig. 17.
-179-
(Average Stress / ft)
1 .0
<t> =19mm
Cover =40 mm
No Splitting Cracks
<S> =19 mm
Cover =40 mm
H 1!
0 .1
0.2
0.3
0.4
0 .5
Average Strain ( % )
*
=19mm
'
nun Cover=5mm
Fig.16
6
Fig.17
Case study for cover effect on tension stiffness
. EXPERIMENTAL
(1) Abrishami
Case study for cover effect on tension stiffness:
results
analytical
VERIFICATION
and Mitchell's
Experiments10*
For experimental verification,
specimens tested by Abrishami and MitchellIO) were analyzed. Abrishami tested
five tension specimens. All had a length of 1,500 mm.A single reinforcing bar, with a minimum concrete cover
on two faces of each specimen of 40 mm,wasprovided. A reinforcement ratio of 1.23 percent was used in all
specimens. Reinforcing bars of diameters 1 1.3, 16, 19.5, 25.2, and 29.9 mmwereused, respectively.
Figures 18
through 22 compare the authors' analysis with the experiments. These figures show both member behavior and
the tension stiffening
behavior of the concrete. It should be mentioned that the experimental tension stiffening
could be computed only before yielding of reinforcing bars, since the steel is in the elastic range and the concrete
contribution
to load carrying can be computed by subtracting the reinforcement contribution as follows.
a
c =at-p(sEs)
(17)
a
t =T/Ag&
e=Al/1
where, T is the tensile load, Ag is the cross sectional area, Al is the elongation, and 1 is the total length
specimen. It has also to be mentioned that initial drying shrinkage of the specimens is not considered
author's analysis. The analytical results show fair agreement with the experimental results.
(2) Authors'
of the
in the
Experiments
For a further experimental verification,
two specimens of two-meters length were tested. The specimens details
are shown in Fig. 23. Each specimen was reinforced with two 10 mmdeformed bars. A 20 mm-thick steel plate
was punched as shown in Fig. 23 and Fig. 24, and the main reinforcement was attached to the plate by means of
nuts. A PC tendon was attached to the center of the plate by a nut as also shown in Fig. 23 and Fig. 24. The test
machine load was applied directly
to this tendon. Specimen elongation
was measured by a box-type
displacement transducer fixed to the steel plate, as illustrated
in Fig. 23.
180
-
The specimens cross sections, reinforcement, and concrete cover were identical. However, one of them had no
hoops, while the other was transversely reinforced with 6 mmhoops (Fig. 23 and Fig. 25) The ratio of cover to
bar diameter in both specimens was 1.0, which would give no tension stiffening
and no transverse cracks
according to Abrishami and Mitchell10). The authors deemed that Abrishami's model might be valid primarily for
the cases of shallow depth and insufficient
cover on both sides. The specimens tested in this study represent the
morecommoncase of large-sized reinforced concrete members. The behavior was expected to be deviant from
Abrishami's model, since different confining and bond properties can be expected.
Figures 26 through 29 illustrate
the analytical
and experimental results. Figures 26 and 28 show the loadelongation relationship,
while Figs. 27 and 29 show the tension stiffening
of concrete in comparison with
Abrishami's model. From these, figures,, it can be concluded,.that.the
author predictions
are in a fairly good
agreement with the experiment and that Abrishami's model underestimates the tension stiffening.
The analysis indicates a splitting load of 49 kN in specimen (1) and no splitting in specimen (2). The observed
splitting
load of specimen (1) was 45 kN with a deviation of 8% from the analytical value, while no splitting
crack was observed in specimen (2) reinforced with transverse reinforcement as shown in Fig. 30. Also, the
predicted crack spacing was close to the experiment with deviation of 12% and 19%, respectively.
Average Stress (MPa)
.0 r
2
8
Elongation
0
10
2
( mm )
4
Comparison between authors' analysis
1 .0
0 .5
0 .5
o
0
( mm )
0.020.040.060.08
and experimental
work of Abrishami
A
Analysis
.oJi
0
Average Strain %
b)Tension
Stiffening
120i
0.02
UC -1 0
0.04
0.06
0.08
Average Strain
of Concrete
and Mitchell10':
( Specimen
UC-10)
Average Stress (MPa)
verage Stress (MPa)
2 .0
nalysis
.0r
1 .0
04-0.02
Load (kN)
A
verage Stress (MPa)
2
UC-1(P|
1 .5
a) Member Response
Fig.18
xperiment
A
1 .5
6
Elongation
E
2
.0 r
A
UC-15
nalysis
UC-15
100
I 1-"3
/ 1500
2
4
Elongation
6
8
10
"
0
( mm )
2
Only
Transverse
Cracks
4
Elongation
6
8
10
( mm )
Comparison between authors' analysis
1 .5
1 .0
1 .0
0 .5
0 .5
O
.C
o
"-0.05
0
0.05
0.1
0.15
Average Strain
a) Member Response
Fig.19
1 .5
0.2
%
b)Tension
and experimental
-181~
work of Abrishami
0.25
.o!
""0
0.05
0.1
0.15
Average Strain
Stiffening
and Mitchell10':
of Concrete
( Specimen
UC-15)
0.2
%
0.25
Load (kN)
Average Stress (MPa)
A
verage Stress (MPa)
140
120
100
60
40
20
0
0
2
4
6
Elongation
8
10
0
2
( mm )
4
6
Elongation
8
10
0
0.05
0.1
a) Member Response
Fig.20
0.15
0.2
0
0.05
and experimental
work of Abrishami
Stiffening
0.2
0.25
%
( Specimen
UC-20)
Average Stress (MPa)
2
Experiment
0.15
of Concrete
and Mitchell101:
Average Stress (MPa)
2 .0f
0.1
Average Strain
b)Tension
Comparison between authors' analysis
0.25
Average Strain %
( mm )
UC-25
.0r
1 .5
1 .0
0 .5
8
Elongation
Fig.21
2
10
4
( mm )
Elongation
Comparison
between authors' analysis
6
8
10 Oftos
( mm )
E
xperiment
0.05
A
and experimental
Load (kN)
400
0
UC-30
0.1
0.15
0.2
verage Strain
work of Abrishami
0.25°'°0
0.05
0.15
0.2
0.25
Average Strain
%
and Mitchell10':
( Specimen
UC-25)
A
verage Stress (MPa)
Average Stress (MPa)
1
.6i
1.6i
.
Analysis
UC -30
300
200
100
-2
0
2
4
6
Fig.22
8
10
12
-2
0
2
4
( nun )
Elongation
Comparison
between authors' analysis
Elongation
6
8
10
12
( mm)
-0.05
0
0.05
0.1
0.15
Average Strain
and experimental
work ofAbrishami
0.2
0.25
0.3
%
and Mitchell101:
0
0.05
0.1
0.15
0.2
Average Strain
( Specimen
0.25
0.3
%
UC-30)
782
Displacement Transducer
0.3 mmStainless
Steel Wire
220 mm
20 mmSteel Plate
Fig. 23 Details
of test specimens
Fig. 24 PC tendon connection to the specimen
Fig. 25 Reinforcement details
Load (kN)
Average Stress (MPa)
60
2 .5
50
2 .0
40
of test specimens
à"!t
1.5
30
1.0
A
nalytical
Crack Spacing = 250 mm
Observed Crack Spacing = 220 mm
Analytical Splitting
Load = 49 kN
Observed Splitting
Load = 45 kN
20
10
0.5
0.0
0
10
15
Elongation
Fig. 26 Results
20
25
0 .02
30
0.06
0.08
Average Strain
(mm)
of specimen (1): load-elongation
0.04
relationship
Fig.
-183-
27 Results of specimen
(before yielding)
(1):
tension
0.1
0.12
0.14
(%)
stiffness
of concrete
Load (kN)
Average Stress (MPa)
60
2 .0
Experiment
Authors' Analysis
Abrishami's
Model
50
E xperim en t
B are B ar
A uthors' A nalysis
40
30
1 .5
i
A n alytical C rack Spacin g = 218 m m I
O bserved C rack Spacing = 175 m m I
N o Splitting in A n alysis
N o Splitting in E xperim ent
20
10
0
»
1.0
I*
0.5
0.0
10
15
20
Elongation
Fig. 28 Results
25
30
0 .02
0.04
0.06
0.08
0.1
0 .12
0.14
Average Strain ( %)
(mm)
of specimen (2): load-elongation
Fig.
relationship
C
29 Results of specimen
(before yielding)
(2):
tension
stiffness
of concrete
onfining Pressure on Bar (MPa)
40
Cracked
Fig. 30 Observed crack pattern of test specimens
60
80
100
Radius Rcr (mm)
Fig. 31 Confining of bars in specimen (1)
Bond Stress (MPa)
Bond Stress (MPa)
JOOmin
zu
CL
100 m m C f-
15
15
jg cr
p= T tan (a ) (=n5.
M
10
10
Deep
No Stirrup
5
<I>= 10mm
71
a
ft = 2.0 MPa
20
40
Cracked
Fig. 32 Concrete plasticity
in specimen (1)
60
80
1 00
120
140
Radius Rcr (mm)
0
10
Cracked
effect on confinement and bond of bars
-184-
I
v ^
D eep
MCom
axiput
m um
ed
I 6 m m S tirru ps
IJ ^ = 1Q m m
Bond Stress
U t = 2 .O M P a
20
30
Radius Rcr (mm)
Fig. 33 Bond Stresses of bars in specimen (2)
In the analysis of the two specimens, the bond stresses were not affected by splitting
cracks. In fact, the original
Shima bond model was used, since Gambarova's model gave bond stresses higher than Shima's model due to the
huge confinement of bars even after the occurrence of the splitting
cracks. Figure 31 illustrates
the confining
pressure on the bars of specimen (1). If we consider the bar cross section as dividing the concrete into two zones,
where one is on the shallow side while the other is on the deep side, it can be seen that the confinement of the
deep side of concrete is predominant. In Abrishami's experiment, this confining action developing inside the
specimen does not exist, leading to the great reduction in bond stresses.
Figure 32 illustrates
the computed bond stress on bars in specimen (1) as a function of the cracked radius, both
with and without taking the concrete plasticity
in tension into consideration.
As can be seen from this figure, the
effect of concrete plasticity
cannot be neglected. If. it is neglected, the computed splitting
bond stress would be
6.3 MPa instead of 8.4 MPa and the computed splitting load would be 32 kN instead of 49 kN, which is far from
the experimental observations.
Figure 33 illustrates
the computed bond stress on bars in specimen (2) in comparison with that in the case of
specimen (1). Due to the confining action of the steel ties in specimen (2), a 30% increase in bond stress required
to cause splitting of the cover is computed. The computed maximumlocal bond stress in this specimen did not
reach the splitting
bond stress, as is clear in Fig. 33. Based on this, the splitting
load increased and no
longitudinal
cracks were expected. This agrees well with the experimental observations.
7. CONCLUSIONS
Based on the tension softening of plain concrete, the radial
memberswith insufficient
cover has been predicted.
bond stresses,
and hence the splitting
The effect of splitting
cracks on the bond properties and tension stiffening
reduction in tension stiffening due to longitudinal
cracks has been predicted
was discussed,
load of tension
and the possible
The effect of splitting cracks on the bond properties and tension stiffening is very large for structural members
with shallow thickness, such as thin shells, where the concrete cover is insufficient
on all sides. However, this
effect is negligible
for deep structural members, like beams, even if the concrete cover is not sufficient. This is
due to the effect of the confining action of concrete on the deep side.
References
[1] Timoshenko, S.: Strength of Materials. Part II: Advanced Theory and Problems, Princeton, N. J., D. Van
Nostrand Company Inc., pp. 205-210, 1956.
[2] Goto, Y.: Cracks formed in concrete around deformed tension bars, ACI Journal, Proceedings Vol. 68, No. 4,
pp.
244-251,
[3]
Morita,
Vol.
[4]
The
[5]
and
76,
1971.
S. and Kaku, T.: Splitting
pp. 93-110,
bond failure
of large deformed reinforcing
bars, ACI Journal,
Proceedings
1979.
Shima, H., Chou, L., and Okamura, H.: Micro and macro models for bond in reinforced
concrete, Journal of
Faculty of Engineering,
The University
of Tokyo (B), Vol. 39, No 2, 1987.
Gambarova, P., Rosati, G., and Zasso, B.: Steel-to-concrete
bond after concrete splitting:
constitutive
laws
interface deterioration,
Materials and Structures, Vol. 22, pp. 347-356,
1989.
[6] Okamura, H. and Maekawa, K.: Nonlinear Analysis
and Constitutive
Gihodo, Tokyo, 1991.
[7] Uchida,
Y., Rokugo, K. and Koyanagi, W.: Determination
of tension
means of bending tests, Proc. ofJSCE, Vol. 14, No. 426, pp. 203-212,
1991.
[8] Maekawa, K. and Qureshi,
J.: Stress transfer
across interfaces
interlock
and dowel action, Journal of Materials,
Concrete Structures
pp 159-172,
1997.
Models
softening
of Reinforced
diagrams
Concrete,
of concrete
by
in reinforced
concrete due to aggregate
and Pavements, JSCE, Vol.34, No. 557,
785
[9] Avalle, M., lori, I. and Vallini, P.: Analisi numerica del comportamento di element! in conglomerate armato
semplicemente tesi, Studi E Ricerche, Vol. 14, pp. 29-54, 1993 ( In Italian ).
[10]
Abrishami,
H. and Mitchell,
D.: Influence
of splitting
cracks on tension stiffening,
ACI Structural
Journal,
Vol.93,
[1 1]
Salem,
characteristics,
No.6,
pp. 703-710,
1996.
H. and Maekawa, K.: Tension stiffness
for cracked
Proc. ofJCI, Vol.19, No.2, pp. 549-554,
1997.
reinforced
concrete
derived
from micro-bond
[12]
Salem, H. and Maekawa, K.: Computational
model for tension stiffness
of cracked reinforced
concrete
derived
from micro-bond characteristics,
Proc. Of the Sixth East Asia-Pacific
Conference on Structural
Engineering & Construction, Taipei, Vol.3, pp. 1929-1934,
1998.
[13]
Salem, H., Hauke, B. and Maekawa, K.: Concrete cover effect on tension stiffness
of cracked reinforced
concrete,
[14]
tensile
Proc. ofJCI,
Vol.
20, No.3,
pp. 97-102,
1998.
Bujadham,
B., Harada, S. and Maekawa, K.: Temperature and tensile
strength of concrete, Proc. ofJCI, Vol.10, No.3, pp. 721-726,
1988.
[15]
Saitoh,
S., Yoshii, Y. and lizima, M.: Anchorage capacity
Proc. ofJSCEAnnual
Convention, Vol.50, pp. 800-801,
1995.
-186-
of ribbed
stress
steel
path
pipes
dependent
embedded
models
in concrete,
for