Original PDF Flash format 224.2r-92-cracking-of-concrete-members-in-direct-tension  


224.2r 92 Cracking Of Concrete Members In Direct Tension

ACI 224.2R-92
(Reapproved 1997)
Cracking of Concrete Members in Direct Tension
Reported by ACI Committee 224
David Darwin*
Andrew Scanlon*
Grant T. Halvorsen
Chairman
Peter Gergely*
Secretary
Subcommittee
Co-Chairmen
Alfred G. Bishara
Fouad H. Fouad Milos Polvika
Howard L. Boggs
Tony C. Liu
Lewis H. Tuthill*
Merle E. Brander
LeRoy Lutz* Orville R. Werner
Roy W. Carlson
Edward G. Nawy Zenon A. Zielinski
William L. Clark, Jr.*
* Members of the subcommittee who prepared this report.
Committee members voting on this minor revision:
Grant T. Halvorsen
Randall W. Poston
Chairman
Secretary
Florian Barth
Peter Gergely
Keith A. Pashina
Alfred G. Bishara
Will Hansen
Andrew Scanlon
Howard L. Boggs
M. Nadim Hassoun
Ernest K. Schrader
Merle E. Brander
William Lee
Wimal Suaris
David Darwin
Tony C. Liu
Lewis H. Tuthill
Fouad H. Fouad
Edward G. Nawy
Zenon A. Zielinski
David W. Fowler
Harry M. Palmbaum
This report is concerned with cracking in reinforced concrete caused
CONTENTS
primarily by direct tension rather than bending. Causes of direct tension
cracking are reviewed, and equations for predicting crack spacing and
Chapter 1-Introduction, pg. 224.2-2
crack width are presented. As cracking progresses with increasing load,
axial stiffness decreases. Methods for estimating post-cracking axial stiffness
Chapter 2-Causes of cracking, pg. 224.2-2
are discussed. The report concludes with a review of methods for
controlling cracking caused by direct tension. 2.1-Introduction
2.2-Applied loads
2.3-Restraint
Keywords: cracking (fracturing); crack width and spacing; loads
(forces); reinforced concrete; restraints;
stiffness; strains; stresse .
Chapter 3-Crack behavior and prediction equations, pg.
tensile stress; tension; volume change.
224.2-3
3.1-Introduction
3.2-Tensile strength
ACI Committee Reports, Guides, Standard Practices, and
Commentaries are intended for guidance in designing, plan-
The 1992 revisions became effective Mar. 1, 1992. The revisions consisted of
ning, executing, or inspecting construction and in preparing
removing year designations of the recommended references of standards-pro-
ducing organizations so that they refer to current editions.
specifications. Reference to these documents shall not be
Copyright 1986, American Concrete Institute.
made in the Project Documents. If items found in these
All rights reserved including rights of reproduction and use in any form or by
documents are desired to be part of the Project Documents
any means, including the making of copies by any photo process, or by any elec-
they should be phrased in mandatory language and in-
tronic or mechanical device, printed, written, or oral, or recording for sound or
corporated into the Project Documents.
visual reproduction or for use in any knowledge or retrieval system or device,
unless permission in writing is obtained from the copyright proprietors.
224.2R-2 ACI COMMITTEE REPORT
3.3-Development of cracks
Concrete members and structures that transmit loads
3.4-Crack spacing
primarily by direct tension rather than bending include
3.5-Crack width
bins and silos, tanks, shells, ties of arches, roof and
bridge trusses, and braced frames and towers. Members
Chapter 4-Effect of cracking on axial stiffness, pg.
such as floor and roof slabs, walls, and tunnel linings may
224.2R-6
also be subjected to direct tension as a result of the
4.1-Axial stiffness of one-dimensional members
restraint of volume change. In many instances, cracking
4.2-Finite element applications
may be attributed to a combination of stresses due to
4.3-Summary
applied load and restraint of volume change. In the fol-
lowing sections, the effects of applied loads and restraint
Chapter 5-Control of cracking caused by direct tension,
of volume change are discussed in relation to the for-
pg. 224.2R-9
mation of direct tension cracks.
5.1-Introduction
5.2-Control of cracking caused by applied loads
2.2-Applied loads
5.3-Control of cracking caused by restraint of volume
Axial forces caused by applied loads can usually be
change
obtained by standard analysis procedures, particularly if
the structure is statically determinate. If the structure is
Notation, pg. 224.23-10
statically indeterminate, the member forces are affected
by changes in stiffness due to cracking. Methods for est-
Conversion factors-S1 equivalents, pg. 224.2R-11
imating the effect of cracking on axial stiffness are
presented in Chapter 4.
Chapter 6-References, pg. 224.2R-11
Cracking occurs when the concrete tensile stress in a
6.1-Recommended references
member reaches the tensile strength. The load carried by
6.2-Cited references
the concrete before cracking is transferred to the rein-
forcement crossing the crack. For a symmetrical member,
the force in the member at cracking is
CHAPTER l-INTRODUCTION
Because concrete is relatively weak and brittle in
tension, cracking is expected when significant tensile
in which
stress is induced in a member. Mild reinforcement and/or
prestressing steel can be used to provide the necessary
= gross area
tensile strength of a tension member. However, a number
= steel area
of factors must be considered in both design and con-
f '
= tensile strength of concrete
t
struction to insure proper control of cracking that may
n = the ratio of modulus of elasticity of the steel
occur.
to that of concrete
A separate report by ACI Committee 224 (ACI 224R)
p
= reinforcing ratio =
covers control of cracking in concrete members in gen-
eral, but contains only a brief reference to tension
After cracking, if the applied force remains un-
cracking. This report deals specifically with cracking in
changed, the steel stress at a crack is
members subjected to direct tension.
Chapter 2 reviews the primary causes of direct tension

cracking, applied loads, and restraint of volume change.
- 1
(2.2)
Chapter 3 discusses crack mechanisms in tension mem-
bers and presents methods for predicting crack spacing
For
= 10,
= 500 psi (3.45 MPa). Table 2.1 gives
and width. The effect of cracking on axial stiffness is
the steel stress after cracking for a range of steel ratios
discussed in Chapter 4. As cracks develop, a progressive
assuming that the yield strength of the
has not
reduction in axial stiffness takes place. Methods for
been exceeded.
estimating the reduced stiffness in the post-cracking
range are presented for both one-dimensional members
Table 2.1-Steel stress after cracking for various steel
and more complex systems. Chapter 5 reviews measures
ratios
that should be taken in both design and construction to
control cracking in direct tension members.
1- - l + n
ksi (MPa)
CHAPTER 2-CAUSES OF CRACKING
2.1-Introduction *Assumes <
CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-3
Table 3.1-Variability of concrete tensile strength: Typical results5
Mean
Standard deviation Coefficient
strength,
within batches,
o f
psi psi
variation,
Type of test (MPa) (MPa)
percent
Splitting test 405 (2.8) 2 0 (0.14)
5
Direct tensile test 275 (1.9) 19 (0.13)
Modulus of rupture 605 (4.2) 36 (0.25)
Compression cube test 5980 (42) 207 (1.45)
31/2
Table 3.2-Relation between compressive strength and tensile strengths of concrete6
Strength ratio
Compressive
Direct tensile
Direct tensile
strength
Modulus of rupture*
strength to
strength to
of cylinders,
to compressive
compressive
modulus of
psi (MPa)
strength
strength
rupture*
1000 (6.9)
0.23
0.11
0.48
2000 (13.8)
0.19
0.10
0.53
3000 (20.7)
0.16
0.09
0.57
4000 (27.6)
0.15
0.09
0.59
5000 (34.5)
0.14
0.08
0.59
6000 (41.4)
0.13
0.08
0.60
7000 (48.2)
0.12
0.07
0.61
8000 (55.1)
0.12
0.07
0.62
9000 (62.0)
0.11
0.07
0.63
*Determined under third-point loading.
For low steel ratios, depending on the grade of steel,
restraint. This point is demonstrated in Tam and Scan-
yielding occurs immediately after cracking if the force in
lon’s numerical analysis of time-dependent restraint force
the member remains the same. The force in the cracked
due to drying shrinkage.3
member at steel yield is
2.3-Restraint
CHAPTER 3-CRACK BEHAVIOR AND
When volume change due to drying shrinkage, thermal
PREDICTION EQUATIONS
contraction, or another cause is restrained, tensile
stresses develop and often lead to cracking. The restraint
3.1-Introduction
may be provided by stiff supports or reinforcing bars.
This chapter reviews the basic behavior of reinforced
Restraint may also be provided by other parts of the
concrete elements subjected to direct tension. Methods
member when volume change takes place at different
for determining tensile strength of plain concrete are
rates within a member. For example, tensile stresses
discussed and the effect of reinforcement on devel-
occur when drying takes place more rapidly at the ex-
opment of cracks and crack geometry is examined.
terior than in the interior of a member. A detailed
discussion of cracking related to drying shrinkage and
3.2-Tensile strength
temperature effects is given in ACI 224R for concrete
Methods to determine tensile strength of plain con-
structures in general.
crete can be classified into one of the following cate-
Axial forces due to restraint may occur not only in
gories: 1) direct tension, 2) flexural tension, and 3) in-
tension members but also in flexural members such as
direct tension
. .4. Because of difficulties associated with
floor and roof slabs. Unanticipated cracking due to axial
applying a pure tensile force to a plain concrete spec-
restraint may lead to undesirable structural behavior such
imen, there are no standard tests for direct tension.
as excessive deflection of floor slabs1 and reduction in
Following ASTM C 292 and C 78 the modulus of rup-
buckling capacity of shell structures.2 Both are direct
ture, a measure of tensile strength, can be obtained by
results of the reduced flexural stiffness caused by
testing a plain concrete beam in flexure. An indirect
restraint cracking. In addition, the formation of cracks
measure of direct tensile strength is obtained from the
due to restraint can lead to leaking and unsightly con-
splitting test (described in ASTM C 496). As indicated in
ditions when water can penetrate the cracks, as in
Reference 4, tensile strength measured from the flexure
parking structures.
test is usually 40 to 80 percent higher than that measured
Cracking due to restraint causes a reduction in axial
from the splitting test.
stiffness, which in turn leads to a reduction (or relax-
Representative values of tensile strength obtained
ation) of the restraint force in the member. Therefore,
from tests and measures of variability are shown in
the high level stresses indicated in Table 2.1 for small
Tables 3.1 and 3.2.
steel ratios may not develop if the cracking is due to
ACI 209R suggests the following expressions to esti-
224.2R-4 ACI COMMITTEE REPORT
mate tensile strength as a function of compressive
same stress level. Clark and Spiers14 estimated that the
strength
first major crack forms at about 90 percent of the aver-
age concrete tensile strength and the last major crack at
modulus of rupture: =


(3.1)
about 110 percent of the average tensile strength.
yaji and Shah15 used a bilinear stress-strain diagram for
direct tensile strength: =

(3.2)
concrete in tension to model the formation of cracks
along the member at increasing load levels. They as-
where
sumed that the tensile strength beyond first cracking was
= unit weight of concrete (lb/ft3)
a function of the strain gradient in the concrete along the
= compressive strength of concrete (psi)
length of the bar.
= 0.60 to 1.00 (0.012 to 0.021 for
in
Induced tensile stresses caused by restrained concrete
kg/m3
in MPa)
shrinkage affect the amount of cracking that is visible at
= 0.33 (0.0069)
a given tensile force. This has been made apparent by
tensile tests conducted to compare the performance of
Both the flexure and splitting tests result in a sudden
Type I cement and Type K shrinkage-compensating)
failure of the test specimen, indicating the brittle nature
16
cement in concrete specimens.
Specimens placed under
of plain concrete in tension. However, if the deformation
the same conditions of environment and loading had
of the specimen is controlled in a test, a significant
markedly different cracking behavior.
descending branch of the tensile stress-strain diagram can
When specimens made with Type I cement had fully
be developed beyond the strain corresponding to maxi-
developed external cracks, the specimens made with Type
mum tensile stress. Evans and Marathe7 illustrated this
K cement exhibited fewer and narrower external cracks.
behavior on specimens loaded in direct tension in a test-
The Type K specimens exhibited first cracking at a higher
ing machine modified to control deformation. Fig. 3.1
load than the Type I specimens, and in some tests no
shows tensile stress-strain curves that include unloading
visible cracking was evident in Type K specimens.
beyond the maximum tensile stress. More recent work by
The compressive stress induced in the concrete by the
Petersson8 shows that the descending branch of the curve
restrained expansion of the Type K cement was appar-
is controlled primarily by localized deformatiou across
ently responsible for increasing the loads both at first
individual cracks, indicating that there are large dif-
cracking and at which cracking was fully developed. Thus,
ferences between the average strain (Fig. 3.1) and local
efforts to compensate for concrete shrinkage also appear
strains.
to help reduce cracking.
3.3-Development of cracks
3.4-Crack spacing
When a reinforced concrete member is subjected to
As a result of the formation of cracks in a tension
tension, two types of cracks eventually form (Fig. 3.2).
member, a new stress pattern develops between the
One type is the visible crack that shows at the surface of
cracks. The formation of additional primary cracks con-
the concrete, while the other type does not progress to
tinues as the stress increases until the crack spacing is
the concrete surface. Broms9 called cracks of the first
approximately twice the cover thickness as measured to
type primary cracks and those of the second type se-
the center of the reinforcing bar.”
condary cracks.
Each of the two types of cracks has a different geo-
metry. The primary or external cracks are widest at the
500
surface of the concrete and narrowest at the surface of
mix w/c age
reinforcing bars.10-12 The difference in crack width be-
(4)_ 1:1:2 0.45 65 days
(5)_1::4 0.60 270 "
tween the concrete surface and the reinforcing bar is
(6)_1:3:6 0.90 70 "
small at low tension levels (just after crack formation),
and increases as the tension level increases; therefore,
the crack width at the reinforcing bar increases more
slowly than the width at the concrete surface with an
increase in load. The deformations on the reinforcing
bars tend to control the crack width by limiting the slip
between the concrete and the steel.
The secondary, or internal, cracks increase in width
with distance away from the reinforcement before nar-
800
1200 1600 2000 2400
longitudinal tensile strain x 106
rowing and closing prior to reaching the concrete surface.
More detail on internal crack formation is presented in
Reference 13.
Because of the variability in tensile strength along the
Fig. 3.1-Tensile stress-strain diagrams for concrete7
length of a tension member, cracks do not all form at the
(includes unloading portion)
CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-5
cover) at high steel stress by the average strain in the
reinforcement. When tensile members with more than
one reinforcing bar are considered,” the actual concrete
cover is not the most appropriate variable. Instead, an
effective concrete cover is used. is defined as a
function of the reinforcement spacing, as well as the
concrete cover measured to the center of the reinfor-
cement. 11 The greater the reinforcement spacing, the
greater will be the crack width. This is reflected as an
increased effective cover. Based on the work of Broms
and Lutz,11 the effective concrete cover is
2
(3.3)
in which
= distance from center of bar to extreme
tension fiber, in., and s = bar spacing, in.
The variable is similar to the variable
used in
the Gergely-Lutz crack width expression for flexural
members,17 in which A = area of concrete symmetric
with reinforcing steel divided by number of bars (in.2).
Using it is possible to express the maximum crack
width in a form similar to the Gergely-Lutz expression.
Due to the larger variability in crack width in tension
members, the maximum crack width in direct tension is
expected to be larger than the maximum crack width in
flexure at the same steel stress.
The larger crack width in tensile members may be due
Fig. 3.2-Primary and secondary cracks in a reinforced
to the lack of crack restraint provided by the compression
concrete tension member
zone in flexural members. The stress gradient in a flex-
ural member causes cracks to initiate at the most highly
There is, of course, a considerable variation in the
stressed location and to develop more gradually than in
spacing of external cracks. The variability in the tensile
a tensile member that is uniformly stressed.
strength of the concrete, the bond integrity of the bar,
The expression for the maximum tensile crack width
and the proximity of previous primary cracks, which tend
developed by Broms and Lutz” is
to decrease the local tensile stress in the concrete, are
the main cause of this variation in crack spacing. For the
Wmax = 4 =
x 10-3
(3.4)
normal range of concrete covers, 1.25 to 3 in. (30 to 75
mm), the average crack spacing will not reach the limit-
Using the definition of given in Eq. (3.3), Wmax may be
ing value of twice the cover until the reinforcement stress
expressed as
reaches 20 to 30 ksi (138 to 200 MPa).11
The expected value of the maximum crack spacing is
about twice that of the average crack spacing.11 That is,
W
= 0.138
+
x 1O-3
(3.5)
the maximum crack spacing is equal to about four times
c
the concrete cover thickness. This range of crack spacing
is more than 20 percent greater than observed for flex-

for a single layer of reinforcement
ural members.
see Fig. 3.3), which is approximately equal
The number of visible cracks can be reduced at a
to

for S/d, between 1 and 2. Thus,
given tensile force by simply increasing the concrete
for tensile cracking
cover. With large cover, a larger percentage of the cracks
will remain as internal cracks at a given level of tensile
Wmax = 0.10
x 10-3
(3.6)
force. However, as will be discussed in Section 3.5, in-
creased cover does result in wider visible cracks.
Eq. (3.6) can be used to predict the probable maximum
crack width in fully cracked tensile members. As with
3.5-Crack width
flexural members, there is a large variability in the
The maximum crack width may be estimated by mul-
maximum crack width. One should fully expect the maxi-
tiplying the maximum crack spacing (4 times concrete
mum crack width to be 30 percent larger or smaller than
224.2R-6 ACI COMMITTEE REPORT
tributed between the reinforcing steel and concrete in
proportion to their respective stiffnesses. Total load at
strain is given by
2d x 2s =
2
P =
+ =
+

=
(1 - p + n
=
(4.1)
in which
= modulus of elasticity of concrete. Loads
carried by the concrete and reinforcing steel are, re-
spectively
Fig.
parameter in terms of bar spacing
the value obtained from Eq. (3.6).

P

1 +
The maximum flexural crack width expression17
and
W
=

10-3
(3.7)

p
(4.3)
in which = ratio of distance between neutral axis and

1 +
tension face to distance between neutral axis and cen-
troid of reinforcing steel
1.20 in beams may be used
Cracking occurs when the strain corresponds to the ten-
tocompare the crack widths obtained in flexure and ten-
sile strength of concrete. If the ascending branch of the
sion.
tensile stress-strain curve is assumed to be linear = Et
Using a value of = 1.20, the coefficient
in
=
in which
is the tensile stress causing the first
Eq. (3.7) becomes 0.091 compared to a coefficient of 0.10
crack. The total load at cracking
is carried across the
in Eq. (3.6) for tensile cracks. This indicates that for the
crack entirely by the reinforcement. If the applied force
same section and steel stress the maximum tensile
remains unchanged, steel stress after the crack
crack will be about 10 percent wider than the maximum
is given by
flexural crack.
The flexural crack width expression in Eq. (3.7), with
= 1.2, is used in the following form in ACI 318
=
=

- 1 +
(4.4)
The load carried across the crack by the reinforcement
Z
(3.8)
is gradually transferred by bond to the concrete on each
A maximum value of z = 175 kips/in. (30.6 MN/m) is
side of the crack. As the applied load increases, addi-
permitted for interior exposure, corresponding
tional cracks form at discrete intervals along the member
to a
limiting crack width of 0.016 in. (0.41 mm). ACI 318
as discussed in Chapter 3. The contribution of concrete
limits the value of z to 145 kips/in. (25.4 MN/m) for
between cracks to the net stiffness of a member is known
exterior exposure, corresponding to a crack width of
as tension stiffening. The gradual reduction in stiffness
0.013 in. (0.33 mm). To obtain similar crack widths for
Axial Load
tensile members, the z-factors of 145 and 175 for flexural
members should be multiplied by the ratio of coefficients
in Eq. (3.7) and (3.6) (= 0.91). Using the same definition
of z for both tensile and flexural members, this produces
z-values of 132 and 160, respectively, for tensile members.
Rizkalla and Hwang18
reported recently on tests in
steel plus concrete
direct tension and presented an alternative procedure for
computing crack widths and crack spacing based on ex-
pressions given by Beeby19 and Leonhardt.20
shaded area represents
CHAPTER 4-EFFECT OF CRACKING
contribution of concrete
to overall stiffness
ON AXIAL STIFFNESS
4.1-Axial stiffness of one-dimensional members
Axial Strain
When a symmetrical uncracked reinforced concrete
member is loaded in tension. the tensile force is dis-

Fig. 4.1-Tensile load versus strain diagram
CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-7
due to progressive cracking is referred to as strain Other methods for determining
are reviewed by
softening.
Moosecker and Grasser.21
The stiffening effect of the concrete between cracks
An alternative approach is to write the effective stiff-
can be illustrated by considering the relationship between
ness (EA) in terms of the modulus of elasticity of the
c
the load and the average strain in both the uncracked
concrete and an effective (reduced) area of concrete, i.e.
and cracked states. A tensile load versus strain curve is
shown in Fig. 4.1. In the range P = 0 to P = Pcr, the
(4.12)
member is uncracked, and the response follows the line
OA. The load-strain relationship [Eq. (4.1)] is given by
This approach is analogous to the effective moment of
inertia concept for the evaluation of deflections de-
P =
(I - +
=
(4.5)
veloped by
and incorporated in
318.
Using the same form of the equation as used for the
If the contribution to stiffness provided by the con-
effective moment of inertia, the effective cross-sectional
crete is ignored, the response follows the line OB, and
area for a member can be written as
the load-strain relationship is given by
P =
= n
=
(4.6)
A
+

e
(4.13)
For loads greater than Pcr, the actual response is inter-
mediate between the uncracked and fully cracked limits,
and response follows the line AD. At Point C on AD,
where
= gross cross-sectional area and Acr =
where P is greater than
a relationship can be de-
The term Ag could be replaced by the transformed
veloped between the load P and average strain in the
area Ae to include the contribution of reinforcing steel to
member
the uncracked system [A =
+
=
+ (n - 1)
t
The load-strain relationships obtained using the CEB
expression and the effective cross-sectional area [Eq.
The term
can be referred to as the effective
(4.13)] compare quite favorably, as shown in Fig. 4.2.
axial cross-sectional stiffness of the member. This term
A third approach that has been used in finite element
can be written in terms of the actual area of steel As, and
analysis of concrete structures involves a progressive
an effective modulus of elasticity
of the steel bars
reduction of the effective modulus of elasticity of con-
crete with increased cracking.
=
(4.8)
or
4.2-Finite element applications
Extensive research has been done in recent years on
the application of finite elements to modeling the be-
E
(4.9)
havior of reinforced concrete and is summarized in a
m
report of the ASCE Task Committee on Finite Element
Several methods can be used to determine
For
Analysis of Reinforced Concrete.23 Two basic ap-
example, the CEB Model Code gives
proaches, the discrete crack approach and the smeared
crack approach, have been used to model cracking and
tension stiffening.
(4.10)
In the discrete crack approach, originally used by Ngo
and Scordelis,24
individual cracks are modeled by using
separate nodal points for concrete elements located at
in which
is given by Eq. (4.4), =
=
cracks, as shown for a flexural member in Fig. 4.3. This
and k = 1.0 for first loading and 0.5 for repeated or sus-
allows separation of elements at cracks. Effects of bond
tained loading.
degradation on tension stiffening can be modeled by
Combining Eq. (4.9) and (4.10)
linear24 or nonlinear 25 bond-slip linkage elements con-
necting concrete and steel elements.
The finite element method combined with nonlinear
fracture mechanics was used by Gerstle, Ingraffea, and
=
26 to study the tension stiffening effect in tension
sm

(4.11)
members. The sequence of formation of primary and se-
condary cracks was studied using discrete crack modeling.
The CEB expression is based on tests of direct tension
A comparison of analyses with test results is shown in
members conducted at the University of Stuttgart.20
Fig. 4.4.
224.2R-8 ACI COMMITTEE REPORT
240
CEB
- - - - - - Effective Area, A ,
-
-
-

Steel Alone
180
Lood P
(kips)
120
l200
Axial Strain (millionths)
Fig. 4.2-Tensile load versus strain diagrams based on CEB and effective cross-sectional area expressions
3 0 0
Elastic analysis;
secondary cracking
ignored
2 5 0
-Analysis including
STEEL ELEMENT
Fig. 4.3-Finite element modeling by the discrete crack
approach24
In the smeared crack approach, tension stiffening is
modeled either by retaining a decreasing concrete
modulus of elasticity and leaving the steel modulus
Elongation, (mm)
unchanged, or by first increasing and then gradually
Fig. 4.4-Steel stress versus elongation curves for tension
decreasing the steel modulus of elasticity and setting the
specimen based on nonlinear fracture mechanics ap-
concrete modulus to zero as cracking progresses. Scanlon
proach26
and Murray’ introduced the concept of degrading con-
crete stiffness to model tension stiffening in two-way
same general approach can be used for members in
slabs. Variations of this approach have been used in
direct tension.
finite element models by a number of researchers.28-31
Gilbert and Warner31 used the smeared crack concept
4.3-Summary
and a layered plate model to compare results using the
Several methods have been proposed in the literature
degrading concrete stiffness approach and the increased
to estimate the axial stiffness of cracked reinforced
steel stiffness approach. Various models compared by
concrete members. The CEB Model Code approach in-
Gilbert and Warner are shown in Fig. 4.5. Satisfactory
volves the modification of the effective steel modulus of
results were obtained using all of the models considered.
elasticity and appears to be well-established in Europe.
However, the approach using a modified steel stiffness
An alternative approach, suggested in this report,
was found to be numerically the most efficient. More
involves an expression for the effective cross-sectional
recent works32 has shown that the energy consumed in
area that is analogous to the well-known effective
fracture must be correctly modeled to obtain objective
moment of inertia concept. Both of these approaches
finite element results in general cases. While most of
appear to be acceptable for the analysis of one-
these models have been applied to flexural members, the
dimensional members.
CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-9
effects. This chapter is intended to provide guidance to
Layer Containing the Tensile Steel
assist in achieving that objective.
Layer Once Removed from the Steel
The recommendations contained in ACI 224R apply
1

Layer Twice Removed from the Steel
where applicable. This chapter deals more specifically
with members loaded in direct tension.
5.2-Control of cracking caused by applied loads
The main concern of crack control is to minimize
b)
maximum crack widths. In the past, tolerable crack
Gradually Unloading Response
After Cracking
widths have been related to exposure conditions (ACI
224R). However, at least in terms of protecting rein-
forcement from corrosion, the effect of surface crack
width appears to be relatively less important than be-
lieved previously (ACI 224.1R). For severe exposures, it
is preferable to provide a greater thickness of concrete
cover even though this will lead to wider surface cracks.
Tolerable crack widths may also be related to aesthetic
or functional requirements. Based on experience using
the z-factor for flexural cracking (ACI 318), a crack width
Alternative Stress-Strain Diagram for Concrete in Tension
of 0.016 in. (0.4 mm) may be acceptable for appearance
in most cases. Functional requirements such as liquid
storage (ACI 350R) may require narrower crack widths
such as 0.008 in. (0.2 mm) for liquid-retaining structures.
Eq. (3.6) and (3.8) for members in direct tension may
be used to select and arrange reinforcement to limit
crack widths.
Since crack width is related to tensile stress in rein-
forcement, cracks attributed to live loads applied for
short periods may not be as serious as cracks due to
sustained load, since the cracks due to live load may be
a ) I f
b) I f
expected to close or at least decrease in width upon
removal of the load. If acceptable crack control cannot
Material Modelling Law
be achieved by the use of mild reinforcement alone, pre-





stressing can be used to reduce tensile stresses in a





structure. Shrinkage-compensating concrete placed in






accordance with ACI 223 can also be effective.
While measures can be taken at the design stage to
control cracking, it is equally important to apply proper
Fig. 4.5-Tension stiffening models proposed for smeared
construction procedures to insure the intended perfor-
crack finite element approach31
= tensile strength of
mance of the structure. This requires avoiding over-
concrete,
= yield strength of reinforcment, = stress,
loading the structure during construction. Careful
and = strain)
placement of reinforcement is also essential, including
provision for properly designed lap splices (ACI 318).
For more complex systems, finite element analysis pro-
cedures have been used successfully to model the be-
5.3-Control of cracking caused by restraint of volume
havior of cracked reinforced concrete, using a variety of
change
stiffness models.
Cracking due to the restraint of volume change during
the early life of a structure can be minimized by pro-
tecting new concrete for as long as practical from drying
CHAPTER 5-CONTROL OF CRACKING
or temperature drop of the surface of the member that
CAUSED BY DIRECT TENSION
would cause tensile stress in the member greater than its
early age strength.
5.1-Introduction
Drying shrinkage at early ages can be controlled by
The three previous chapters emphasized predicting be-
using proper curing procedures. Shrinkage-compensating
havior when cracking due to direct tension occurs in re-
concrete can be very effective in limiting cracks due to
inforced concrete members. However, a major objective
drying shrinkage (ACI 223).
of design and construction of concrete structures should
Temperature changes related to the heat of hydration
be to minimize and/or control cracking and its adverse
can be minimized by placing concrete at lower than nor-
224.2R-10
ACI COMMITTEE REPORT
ma1 temperatures (precooling). For example, placing con-
that of concrete
crete at approximately 50 F (10 C) has significantly
P
= axial load
reduced cracking in concrete tunnel linings.33 It should
= axial load carried by concrete
be noted that concrete placed at 50 F (10 C) tends to
= axial load at which cracking occurs
develop higher strength at later ages than concrete
= axial load carried by reinforcement
placed at higher temperatures.
= bar spacing, in.
Circumferential cracks in tunnel linings (as well as
= effective concrete cover, in.
cast-in-place conduits and pipelines) can be greatly
= unit weight of concrete, lb/ft3
reduced in number and width if the tunnel is kept bulk-
w = most probable maximum crack width, in.
headed against air movement and shallow ponds of water
= factor limiting distribution of reinforcement
are kept in the invert from the time concrete is placed
= ratio of distance between neutral axis and
until the tunnel goes into service (see Fig. 35 of
tension face to distance between neutral axis
Reference 34).
and centroid of reinforcing steel 1.20 in.
To minimize crack widths caused by restraint stresses,
beams
bonded “temperature” reinforcement should be provided.
E
= average strain in member (unit elongation)
m
As a general rule, reinforcement controls the width and
= tensile strain in reinforcing bar assuming no
spacing of cracks most effectively when bar diameters are
tension in concrete
as small as possible, with correspondingly closer spacing
= reinforcing ratio =
for a given total area of steel. Fiber reinforced concrete
may also have application in minimizing the width of
cracks induced by restraint stresses (ACI 544.1R).
CONVERSION FACTORS-SI EQUIVALENTS
If tensile forces in a restrained concrete member will
result in unacceptably wide cracks, the degree of restraint
1 in. = 25.4 mm
can be reduced by using joints where feasible or leaving
1 lb (mass) = 0.4536 kg
empty pour strips that are subsequently filled with con-
1 lb (force) = 4.488 N
crete after the adjacent members have gained strength
1 lb/in.2 = 6.895 kPa
and been allowed to dry. Flatwork will be restrained by
1 kip = 444.8 N
the anchorage of the slab reinforcement to perimeter
1 kip/in.2 = 6.895 MPa
slabs or footings. When each slab is free to shrink from
all sides towards its center, cracking is minimized. For
Eq. (3.5)
slabs on ground, contraction joints and perimeter
supports should be designed accordingly (ACI 302.1R-
2
80). Frequent contraction joints or deep grooves must be
Wmax = 0.02
x 10-3
provided if it is desired to prevent or hide restraint
cracking in walls, slabs, and tunnel linings [ACI 224R-80
(Revised 1984), ACI 302.1R-80].
Eq. (3.6)
W
=

x 10-3
max
NOTATION
Eq. (3.7)
W
A
max =

x 10-3
area of concrete symmetric with reinforcing
steel divided by number of bars, in.’
W
effective cross-sectional area of concrete, in.’
in mm, in MPa, in mm, A in mm2, and in
gross area of section, in.2
mm.
A
area of nonprestressed tension reinforcement,
s
in.2
distance from center of bar to extreme tension
CHAPTER 6-REFERENCES
fiber, in.
modulus of elasticity of concrete, ksi
6.1-Recommended references
modulus of elasticity of reinforcement, ksi
The documents of the various standards-producing or-
E
effective modulus of elasticity of steel to that
ganizations referred to in this report are listed below with
of concrete
their serial designation.
modulus of rupture of concrete, psi
fs
stress in reinforcement, ksi
American Concrete Institute
f
steel stress after crack occurs, ksi
compressive strength of concrete, psi
209R
Prediction of Creep, Shrinkage, and Temper-
tensile strength of concrete, psi
ature Effects in Concrete Structures
n
the ratio of modulus of elasticity of steel to
223
Standard Practice for the Use of Shrinkage-
CRACKING OF CONCRETE MEMBERS IN DIRECT TENSION 224.2R-11
Compensating Concrete
Engineering Report No. 122, Department of Civil En-
224R
Control of Cracking in Concrete Structures
gineering, University of Alberta, Edmonton, Dec. 1984,
224.1R
Causes, Evaluation, and Repair of Cracks in
126 pp.
Concrete Structures
4. Neville, Adam M., Hardened Concrete: Physical and
302.1R
Guide for Concrete Floor and Slab Construc-
Mechanical Aspects, ACI Monograph No. 6, American
tion
Concrete Institute/Iowa State University Press, Detroit,
318
Building Code Requirements for Reinforced
1971, 260 pp.
Concrete
5. Wright, P.J.F., “Comments on Indirect Tensile Test
350R
Environmental Engineering Concrete Struc-
on Concrete Cylinders,” Magazine of Concrete Research
tures
(London), V. 7, No. 20, July 1955, pp. 87-96.
544.1R
State-of-the-Art Report on Fiber Reinforced
6. Price, Walter H., “Factors Influencing Concrete
Concrete
Strength,” ACI JOURNAL, Proceedings V. 47, No. 6, Feb.
1951, pp. 417-432.
7. Evans, R.H., and Marathe, M.S., “Microcracking
ASTM
and Stress-Strain Curves for Concrete in Tension,”
Materials and Structures, Research and Testing (RILEM,
C 78
Standard Test Method for Flexural Strength of
Paris), V. 1, No. 1, Jan.-Feb. 1968, pp. 61-64.
Concrete (Using Simple Beam with Third-
8. Petersson, Per-Erik, “Crack Growth and De-
Point Loading)
velopment of Fracture Zones in Plain Concrete and
C 293
Standard Test Method for Flexural Strength of
Similar Materials,” Report No. TVBM-1006, Division of
Concrete (Using Simple Beam with Center-
Building Materials, Lund Institute of Technology, 1981,
Point Loading)
174 pp.
C 496
Standard Test Method for Splitting Tensile
9. Broms, Bengt B., “Crack Width and Crack Spacing
Strength of Cylindrical Concrete Specimens
in Reinforced Concrete Members,” ACI JOURNAL, Pro-
ceedings V. 62, No. 10, Oct. 1965, pp. 1237-1256.
10. Broms, Bengt B., “Stress Distribution in Rein-
Euro-International du
Inter-
forced Concrete Members With Tension Cracks,” ACI
nationale de Ia
JOURNAL, Proceedings V. 62, No. 9, Sept. 1965, pp. 1095-
1108.
CEB-FIP Model Code for Concrete Structures
11. Broms, Bengt B., and Lutz, Leroy A., “Effects of
Arrangement of Reinforcement on Crack Width and
Spacing of Reinforced Concrete Members,”
ACI
These publications may be obtained from:
JOURNAL , Proceedings V. 62, No. 11, Nov. 1965, pp.
1395-1410.
American Concrete Institute
12. Goto, Yukimasa, “Cracks Formed in Concrete
PO Box 19150
Around Deformed Tension Bars,” ACI JOURNAL,
Detroit, MI 48219
Proceedings V. 68, No. 4, Apr. 1971, pp. 244-251.
13. Goto, Y., and Otsuka, K., “Experimental Studies
ASTM
on Cracks Formed in Concrete Around Deformed Ten-
1916 Race St.
sion Bars,” Technology Reports of the Tohoku University,
Philadelphia, PA 19103
V. 44, No. 1, June 1979, pp. 49-83.
14. Clark, L.A., and Spiers, D.M., “Tension Stiffening
Euro-International du
in Reinforced Concrete Beams and Slabs Under Short-
EPFL Case Postale 88
Term Load,” Technical Report No. 42.521, Cement and
CH 1015 Lausanne, Switzerland
Concrete Association, Wexham Springs, 1978, 19 pp.
15. Somayaji, S., and Shah, S.P., “Bond Stress Versus
Slip Relationship and Cracking Response of Tension
6.2-Cited references
Members,” ACI JOURNAL , Proceedings V. 78, No. 3, May-
1. Scanlon, Andrew, and Murray, David W., “Practical
June 1981, pp. 217-225.
Calculation of Two-Way Slab Deflections,” Concrete In-
16. Cusick, R.W., and Kesler, C.E., “Interim Report-
ternational: Design & Construction, V. 4, No. 11, Nov.
Phase 3: Behavior of Shrinkage-Compensating Concretes
1982, pp. 43-50.
Suitable for Use in Bridge Decks,” T. & A.M. Report No.
2. Cole, Peter P.; Abel, John F.; and Billington, David
409, Department of Theoretical and Applied Mechanics,
P., “Buckling of Cooling-Tower Shells: State-of-the-Art,”
University of Illinois, Urbana, July 1976, 41 pp.
Proceedings, ASCE, V. 101, ST6, June 1975, pp. 1185-
17. Gergely, Peter, and Lutz, Leroy A., “Maximum
1203.
Crack Width in Reinforced Concrete Flexural Members,”
3. Tam, K.S.S., and Scanlon, A., “The Effects of
Causes, Mechanism, and Control of Cracking in Concrete,
Restrained Shrinkage on Concrete Slabs,” Structural
SP-20, American Concrete Institute, Detroit, 1968, pp.
224.2R-1 2
ACI COMMlTTE E
REPORT
87-117.
ference on Bond in Concrete (Paisely, June 1982), Ap-
18. Rizkalla, S.H., and Hwang, L.S., “Crack Prediction
plied Science Publishers, London, 1982, pp. 97-106.
for Members in Uniaxial Tension,” ACI JOURNAL, Pro-
27. Scanlon, A., and Murray, D.W., “An Analysis to
ceedings V. 81, No. 6, Nov.-Dec. 1984, pp. 572-579.
Determine the Effects of Cracking in Reinforced Con-
19. Beeby, A.W., “The Prediction of Crack Widths in
crete Slabs,” Proceedings, Specialty Conference on the
Hardened Concrete,” Structural Engineer (London), V.
Finite Element Method in Civil Engineering, EIC/McGill
57A, Jan. 1979, pp. 9-17.
University, Montreal, 1972, pp. 841-867.
20. Leonhardt, Fritz, “Crack Control in Concrete
28. Lin, Cheng-Shung, and Scordelis, Alexander C.,
Structures,” IABSE Surveys No. S-4/77, International
“Nonlinear Analysis of RC Shells of General Form,”
Association for Bridge and Structural Engineering,
Proceedings, ASCE, V. 101, ST3, Mar. 1975, pp. 523-538.
Zurich, Aug. 1977, 26 pp.
29. Chitnuyanondh, L.; Rizkalla, S.; Murray, D.W.;
21. Moosecker, W., and Grasser, E., “Evaluation of
and MacGregor, J.G., “An Effective Uniaxial Tensile
Tension Stiffening in Reinforced Concrete Linear Mem-
Stress-Strain Relationship for Prestressed Concrete,”
bers,” Final Report, IABSE Colloquium on Advanced
Structural Engineering Report No. 74, University of
Mechanics of Reinforced Concrete (Delft, 1981), Inter-
Alberta, Edmonton, Feb. 1979, 91 pp.
national Association for Bridge and Structural
30. Argyris, J.H.; Faust, G.; Szimmat, J.; Warnke, P.,
Engineering, Zurich.
and William, K.J., “Recent Developments in the Finite
2 2 . Branson, D . E . , “Instantaneous and Time-
Element Analysis of Prestressed Concrete Reactor
Dependent Deflections of Simple and Continuous
Vessels,” Preprints, 2nd International Conference on
Reinforced Concrete Beams,” Research Report No. 7,
Structural Mechanics in Reactor Technology (Berlin,
Alabama Highway Department, Montgomery, Aug. 1963,
Sept. 1973), Commission of the European Communities,
94 pp.
Luxembourg, V. 3, Paper H l/l, 20 pp. Also, Nuclear
23. Finite Element Analysis of Reinforced Concrete,
Engineering and Design (Amsterdam), V. 28, 1974.
American Society of Civil Engineers, New York, 1982,
31. Gilbert, R. Ian, and Warner, Robert F., ‘‘Tension
545 pp.
Stiffening in Reinforced Concrete Slabs,” Proceedings,
24. Ngo, D., and Scordelis, A.C., “Finite Element
ASCE, V. 104, ST12, Dec. 1978, pp. 1885-1900.
Analysis of Reinforced Concrete Beams,” ACI JOURNAL.,
32. Bazant, Zdenek, and Cedolin, Luigi, “Blunt Crack
Proceedings V. 64, No. 3, Mar. 1967, pp. 152-163.
Band Propagation in Finite Element Analysis,”
25. Nilson, Arthur H., “Nonlinear Analysis of
Proceedings, ASCE, V. 105, EM2, Apr. 1979, pp. 297-315.
Reinforced Concrete by the Finite Element Method,”
33. Tuthill, Lewis H., “Tunnel Lining With Pumped
ACI JOURNAL , Proceedings V. 65, No. 9, Sept. 1968, pp.
Concrete,” ACI JOURNAL., Proceedings V. 68, No. 4, Apr.
757-766.
1971, pp. 252-262.
26. Gerstle, Walter; Ingraffea, Anthony R.; and
34. Concrete Manual, 8th Edition, U.S. Bureau of
Gergely, Peter, ‘‘Tension Stiffening: A Fracture
Reclamation, Denver, 1975, 627 pp.
Mechanics Approach,” Proceedings, International Con-

Document Outline