Electronic
Journal of Structural Engineering, Vol. 1, No.1 (2001) 214 
Professor
and Head, School of Civil and Environmental Engineering
The University of New South Wales, Sydney,
NSW, 2052
Email: i.gilbert@unsw.edu.au
Abstract
This
paper addresses the effects of shrinkage on the serviceability of concrete
structures. It outlines why
shrinkage is important, its major influence on the final extent of cracking and
the magnitude of deflection in structures, and what to do about it in design.
A model is presented for predicting the shrinkage strain in normal and
high strength concrete and the timedependent behaviour of plain concrete and
reinforced concrete, with and without external restraints, is explained. Analytical procedures are described for estimating the final
width and spacing of both flexural cracks and direct tension cracks and a
simplified procedure is presented for including the effects of shrinkage when
calculating longterm deflection. The
paper also contains an overview of the considerations currently being made by
the working group established by Standards Australia to revise the
serviceability provisions of AS36001994, particularly those clauses related to
shrinkage.
Keywords
Creep; Cracking; Deflection; Reinforced concrete; Serviceability; Shrinkage.
For a
concrete structure to be serviceable, cracking must be controlled and
deflections must not be excessive. It must also not vibrate excessively.
Concrete shrinkage plays a major role in each of these aspects of the service
load behaviour of concrete structures.
The
design for serviceability is possibility the most difficult and least well
understood aspect of the design of concrete structures. Service load behaviour
depends primarily on the properties of the concrete and these are often not
known reliably at the design stage. Moreover, concrete behaves in a nonlinear
and inelastic manner at service loads. The nonlinear behaviour that complicates
serviceability calculations is due to cracking, tension stiffening, creep, and
shrinkage. Of these, shrinkage is the most problematic. Restraint to shrinkage
causes timedependent cracking and gradually reduces the beneficial effects of
tension stiffening. It results in a gradual widening of existing cracks and, in
flexural members, a significant increase in deflections with time.
The
control of cracking in a reinforced or prestressed concrete structure is usually
achieved by limiting the stress increment in the bonded reinforcement to some
appropriately low value and ensuring that the bonded reinforcement is suitably
distributed. Many codes of practice specify maximum steel stress increments
after cracking and maximum spacing requirements for the bonded reinforcement.
However, few existing code procedures, if any, account adequately for the
gradual increase in existing crack widths with time, due primarily to shrinkage,
or the timedependent development of new cracks resulting from tensile stresses
caused by restraint to shrinkage.
For
deflection control, the structural designer should select maximum deflection limits that are appropriate to the structure and
its intended use. The calculated deflection (or camber) must not exceed these
limits. Codes of practice give general guidance for both the selection of the
maximum deflection limits and the calculation of deflection. However, the
simplified procedures for calculating deflection in most codes were developed
from tests on simplysupported reinforced concrete beams and often produce
grossly inaccurate predictions when applied to more complex structures. Again,
the existing code procedures do not provide real guidance on how to adequately
model the timedependent effects of creep and shrinkage in deflection
calculations.
Serviceability
failures of concrete structures involving excessive cracking and/or excessive
deflection are relatively common. Numerous cases have been reported, in
Australia and elsewhere, of structures that complied with code requirements but
still deflected or cracked excessively. In a large majority of these failures,
shrinkage of concrete is primarily responsible. Clearly, the serviceability provisions embodied in our codes
do not adequately model the inservice behaviour of structures and, in
particular, fail to account adequately for shrinkage.
The
quest for serviceable concrete structures must involve the development of more
reliable design procedures. It must also involve designers giving more attention
to the specification of an appropriate concrete mix, particularly with regard to
the creep and shrinkage characteristics of the mix, and sound engineering input
is required in the construction procedures. High performance concrete structures
require the specification of high performance concrete (not necessarily high
strength concrete, but concrete with relatively low shrinkage, not prone to
plastic shrinkage cracking) and a high standard of construction, involving
suitably long stripping times, adequate propping, effective curing procedures
and rigorous onsite supervision.
This
paper addresses some of these problems, particularly those related to designing
for the effects of shrinkage. It outlines how shrinkage affects the inservice
behaviour of structures and what to do about it in design. It also provides an
overview of the considerations currently being made by the working group
established by Standards Australia to revise the serviceability provisions of
AS36001994 [1], particularly those clauses related to
shrinkage.
When
designing for serviceability, the designer must ensure that the structure can
perform its intended function under the day to day service loads.
Deflection must not be excessive, cracks must be adequately controlled
and no portion of the structure should suffer excessive vibration. Shrinkage
causes timedependent cracking, thereby reducing the stiffness of a concrete
structure, and is therefore a detrimental factor in all aspects of the design
for serviceability.
Deflection
problems that may affect the serviceability of concrete structures can be
classified into three main types:
(a)
Where excessive
deflection causes either aesthetic or functional problems.
(b)
Where excessive
deflection results in damage to either structural or nonstructural element
attached to the member.
(c)
Where dynamics
effects due to insufficient stiffness cause discomfort to occupants.
Examples of deflection problems of type (a) include objectionable visual sagging (or hogging), and ponding of water on roofs. In fact, any deflection that prevents a member fulfilling its intended function causes a problem of this type. Type (a) problems are generally overcome by limiting the total deflection to some appropriately low value. The total deflection is the sum of the shortterm and timedependent deflection caused by the dead load (including selfweight), the prestress (if any), the expected inservice live load, and the loadindependent effects of shrinkage and temperature changes.
When the
total deflection exceeds about span/200 below the horizontal, it may become
visually unacceptable. The designer must decide on the maximum limiting value
for the total deflection and this limit must be appropriate for the particular
member and its intended function. A total deflection limit of span/200, for
example, may be appropriate for the floor of a carpark, but is inadequate for a
gymnasium floor which may be required to remain essentially plane under service
conditions.
Examples
of type (b) problems include deflections resulting in cracking of masonry walls
or other partitions, damage to ceiling or floor finishes, and improper
functioning of sliding windows and doors. To avoid these problems, a limit must
be placed on that part of the total deflection that occurs after the attachment
of such elements. This incremental
deflection is usually the sum of the longterm deflection due to all the
sustained loads and shrinkage, the shortterm deflection due to the transitory
live load, and any temperatureinduced deflection. AS 3600 (1994) [1]
limits the incremental deflection for members supporting masonry partitions to
between span/500 and span/1000, depending on the provisions made to minimise the
effect of movement.
Type (c)
deflection problems include the perceptible springy vertical motion of floor
systems and other vibrationrelated problems. Very little quantitative
information for controlling vibration is available in codes of practice. ACI
31899 [2] places a limit of span/360 on the shortterm
deflection of a floor due to live load. This limit provides a minimum
requirement on the stiffness of members that may, in some cases, be sufficient
to avoid problems of type (c).
Excessively
wide cracks can be unsightly and spoil the appearance of an exposed concrete
surface; they can allow the ingress of moisture accelerating corrosion of the
reinforcement and durability failure; and, in exceptional cases, they can reduce
the contribution of the concrete to the shear strength of a member. Excessively
wide cracks in floor systems and walls may often be avoided by the inclusion of
strategically placed contraction joints, thereby removing some of the restraint
to shrinkage and reducing the internal tension. When cracking does occur, in
order to ensure that crack widths remain acceptably small, adequate quantities
of well distributed and wellanchored reinforcement must be included at every
location where significant tension will exist.
The
maximum crack width that may be considered to be acceptable in a given
situation, depends on the type of structure, the environment and the
consequences of excessive cracking. In corrosive and aggressive environments,
crack widths should not exceed 0.1  0.2 mm. For members with one or more
exposed surfaces, a maximum crack width of 0.3 mm should provide visual
acceptability. For the sheltered interior of most buildings where the concrete
is not exposed and aesthetic requirements are of secondary importance, larger
crack widths may be acceptable (say 0.5 mm or larger).
If concrete members were free to shrink, without restraint,
shrinkage of concrete would not be a major concern to structural engineers.
However, this is not the case. The contraction of a concrete member is often
restrained by its supports or by the adjacent structure. Bonded reinforcement
also restrains shrinkage. Each of these forms of restraint involve the
imposition of a gradually increasing tensile force on the concrete which may
lead to timedependent cracking (in previously uncracked regions), increases in
deflection and a widening of existing cracks. Restraint to shrinkage is probably
the most common cause of unsightly cracking in concrete structures. In many
cases, these problems arise because shrinkage has not been adequately considered
by the structural designer and the effects of shrinkage are not adequately
modelled in the design procedures specified in codes of practice for crack
control and deflection calculation.
The advent of shrinkage cracking depends on the degree of
restraint to shrinkage, the extensibility and strength of the concrete in
tension, tensile creep and the load induced tension existing in the member.
Cracking can only be avoided if the gradually increasing tensile stress induced
by shrinkage, and reduced by creep, is at all times less than the tensile
strength of the concrete. Although the tensile strength of concrete increases
with time, so too does the elastic modulus and, therefore, so too does the
tensile stress induced by shrinkage. Furthermore, the relief offered by creep
decreases with age. The existence of load induced tension in uncracked regions
accelerates the formation of timedependent cracking. In many cases, therefore,
shrinkage cracking is inevitable. The control of such cracking requires two
important steps. First, the shrinkageinduced tension and the regions where
shrinkage cracks are likely to develop must be recognised by the structural
designer. Second, an adequate quantity and distribution of anchored
reinforcement must be included in these regions to ensure that the cracks remain
fine and the structure remains serviceable.
Shrinkage
of concrete is the timedependent strain measured in an unloaded and
unrestrained specimen at constant temperature. It is important from the outset
to distinguish between plastic shrinkage, chemical shrinkage and drying
shrinkage. Some high strength concretes are prone to plastic shrinkage, which
occurs in the wet concrete, and may result in significant cracking during the
setting process. This cracking occurs due to capillary tension in the pore
water. Since the bond between the plastic concrete and the reinforcement has not
yet developed, the steel is ineffective in controlling such cracks. This problem
may be severe in the case of low water content, silica fume concrete and the use
of such concrete in elements such as slabs with large exposed surfaces is not
recommended.
Drying
shrinkage is the reduction in volume caused principally by the loss of water
during the drying process. Chemical (or endogenous) shrinkage results from
various chemical reactions within the cement paste and includes hydration
shrinkage, which is related to the degree of hydration of the binder in a sealed
specimen. Concrete shrinkage
strain, which is usually considered to be the sum of the drying and chemical
shrinkage components, continues to increase with time at a decreasing rate.
Shrinkage is assumed to approach a final value,
, as time approaches infinity and is dependent on all the factors which affect
the drying of concrete, including the relative humidity and temperature, the mix
characteristics (in particular, the type and quantity of the binder, the water
content and watertocement ratio, the ratio of fine to coarse aggregate, and
the type of aggregate), and the size and shape of the member.
Drying
shrinkage in high strength concrete is smaller than in normal strength concrete
due to the smaller quantities of free water after hydration. However, endogenous
shrinkage is significantly higher.
For
normal strength concrete (
MPa), AS3600 suggests that the design shrinkage (which includes both drying and
endogenous shrinkage) at any time after the commencement of drying may be
estimated from
(1)
where
is
a basic shrinkage strain which, in the absence of measurements, may be taken to
be 850 x 10^{6} (note that this value was increased from 700 x 10^{6}
in the recent Amendment 2 of the Standard); k_{1}
is obtained by interpolation from Figure 6.1.7.2 in the Standard and depends on
the time since the commencement of drying, the environment and the concrete
surface area to volume ratio. A hypothetical thickness, t_{h}_{ }= 2A/
u_{e}, is used to take this into account, where A
is the crosssectional area of the member and u_{e}
is that portion of the section perimeter exposed to the atmosphere plus half the
total perimeter of any voids contained within the section.
AS3600
states that the actual shrinkage strain may be within a range of plus or minus
40% of the value predicted (increased from ±
30% in Amendment 2 to AS36001994). In
the writer’s opinion, this range is still optimistically narrow, particularly
when one considers the size of the country and the wide variation in shrinkage
measured in concretes from the various geographical locations. Equation 1 does
not include any of the effects related to the composition and quality of the
concrete. The same value of e_{cs}_{
} is
predicted irrespective of the concrete strength, the watercement ratio, the
aggregate type and quantity, the type of admixtures, etc. In addition, the
factor k_{1} tends to
overestimate the effect of member size and significantly underestimate the rate
of shrinkage development at early ages.
The
method should be used only as a guide for concrete with a low watercement ratio
(<0.4) and with a well graded, good quality aggregate. Where a higher
watercement ratio is expected or when doubts exist concerning the type of
aggregate to be used, the value of e_{cs}_{
}predicted by
AS3600 should be increased by at least 50%. The method in the Standard for the
prediction of shrinkage strain is currently under revision and it is quite
likely that significant changes will be proposed with the inclusion of high
strength concretes.
A
proposal currently being considered by Standards Australia, and proposed by
Gilbert (1998) [9], involves the total shrinkage strain, e_{cs},
being divided into two components, endogenous shrinkage, e_{cse},
(which is assumed to develop relatively rapidly and increases with concrete
strength) and drying shrinkage, e_{csd}
(which develops more slowly, but decreases with concrete strength). At any time
t (in days) after pouring, the endogenous shrinkage is given by
e_{cse }= e^{*}_{cse}
(1.0  e^{0.1t})
(2)
where e^{*}_{cse}
is the final endogenous shrinkage and may be taken as
e^{*}_{cse}
, where
is
in MPa. The basic drying shrinkage
is given by
(3)
and at
any time t (in days) after the commencement of drying, the drying shrinkage may
be taken as
(4)
The
variable
is
given by
(5)
where
and
is
equal to 0.7 for an arid environment, 0.6 for a temperate environment and 0.5
for a tropical/coastal environment. For an interior environment, k_{5}
may be taken as 0.65. The value of k_{1} given by Equation 5 has
the same general shape as that given in Figure 6.1.7.2 in AS3600, except that
shrinkage develops more rapidly at early ages and the reduction in drying
shrinkage with increasing values of t_{h}
is not as great.
The
final shrinkage at any time is therefore the sum of the endogenous shrinkage
(Equation 2) and the drying shrinkage (Equation 4). For example, for specimens
in an interior environment with hypothetical thicknesses t_{h} =
100 mm and t_{h} = 400 mm, the shrinkage strains predicted by the
above model are given in Table 1.
Table 1 Design shrinkage strains predicted by proposed model for an interior environment.


(x
10^{6}) 
(x
10^{6}) 
Strain
at 28 days (x
10^{6}) 
Strain
at 10000 days (x
10^{6}) 








100 
25 
25 
900 
23 
449 
472 
25 
885 
910 
50 
100 
700 
94 
349 
443 
100 
690 
790 

75 
175 
500 
164 
249 
413 
175 
493 
668 

100 
250 
300 
235 
150 
385 
250 
296 
546 

400 
25 
25 
900 
23 
114 
137 
25 
543 
568 
50 
100 
700 
94 
88 
182 
100 
422 
522 

75 
175 
500 
164 
63 
227 
175 
303 
478 

100 
250 
300 
235 
38 
273 
250 
182 
432 
Drying shrinkage is greatest at the surfaces exposed to
drying and decreases towards the interior of a concrete member. In Fig.1a,
the shrinkage strains through the thickness of a plain concrete slab, drying on
both the top and bottom surfaces, are shown. The slab is unloaded and
unrestrained.
The mean shrinkage strain, e_{cs} in Fig. 1, is the average contraction. The nonlinear strain labelled De_{cs} is that portion of the shrinkage strain that causes internal stresses to develop. These selfequilibrating stresses (called eigenstresses) produce the elastic and creep strains required to restore compatibility (ie. to ensure that plane sections remain plane). These stresses occur in all concrete structures and are tensile near the drying surfaces and compressive in the interior of the member. Because the shrinkageinduced stresses develop gradually with time, they are relieved by creep. Nevertheless, the tensile stresses near the drying surfaces often overcome the tensile strength of the immature concrete and result in surface cracking, soon after the commencement of drying. Moist curing delays the commencement of drying and may provide the concrete time to develop sufficient tensile strength to avoid unsightly surface cracking.
Fig. 1  Strain
components caused by shrinkage in a plain concrete slab. 
The elastic plus creep strains caused by the eigenstresses
are equal and opposite to De_{cs}
and are shown in Fig. 1b. The total strain distribution,
obtained by summing the elastic, creep and shrinkage strain components, is
linear (Fig. 1c) thus satisfying compatibility. If the
drying conditions are the same at both the top and bottom surfaces, the total
strain is uniform over the depth of the slab and equal to the mean shrinkage
strain, e_{cs
}. It is this quantity that is usually of significance in the analysis
of concrete structures. If drying occurs at a different rate from the top and
bottom surfaces, the total strain distribution becomes inclined and a warping of
the member results.
In concrete structures, unrestrained contraction and
unrestrained warping are unusual. Reinforcement embedded in the concrete
provides restraint to shrinkage. As the concrete shrinks, the reinforcement is
compressed and imposes an equal and opposite tensile force on the concrete at
the level of the reinforcement. If the reinforcement is not symmetrically placed
on a section, a shrinkageinduced curvature develops with time. Shrinkage in an
unsymmetrically reinforced concrete beam or slab can produce deflections of
significant magnitude, even if the beam is unloaded.
Consider the unrestrained, singly reinforced, simplysupported concrete beam shown in Figure 2a and the small beam segment of length Dx. The shrinkage induced stresses and strains on an uncracked and on a cracked crosssection are shown in Figures 2b and 2c, respectively.
Fig. 2  Shrinkage
warping in a singly reinforced beam. 
As the concrete shrinks, the bonded reinforcement imposes a tensile restraining force, DT, on the concrete at the level of the steel. This gradually increasing tensile force, acting at some eccentricity to the centroid of the concrete crosssection, produces curvature (elastic plus creep) and a gradual warping of the beam. It also may cause cracking on an uncracked section or an increase in the width of existing cracks in a cracked member. For a particular shrinkage strain, the magnitude of DT depends on the quantity of reinforcement and on whether or not the crosssection has cracked.
Shrinkage strain is independent of stress, but shrinkage
warping is not independent of the load and is significantly greater in a cracked
beam than in an uncracked beam, as indicated in Fig. 2.
The ability of the concrete section to carry tensile stress depends on whether
or not the section has cracked, ie. on the magnitude of the applied moment,
among other things. DT
is much larger on the uncracked section of Fig. 2b than on
the cracked section of Fig. 2c. Existing design procedures
for the calculation of longterm deflection fail to adequately model the
additional cracking that occurs with time due to DT
and the gradual breakdown of tension stiffening with time (also due to DT),
and consequently often greatly underestimate final deformations.
Compressive reinforcement reduces shrinkage curvature. By
providing restraint at the top of the section, in addition to the restraint at
the bottom, the eccentricity of the resultant tension in the concrete is reduced
and, consequently, so is the shrinkage curvature. An uncracked, symmetrically
reinforced section will suffer no shrinkage curvature. Shrinkage will however
induce a uniform tensile stress which when added to the tension caused by
external loading may cause timedependent cracking.
Structural interest in shrinkage goes beyond its tendency
to increase deflections due to shrinkage warping. External restraint to
shrinkage is often provided by the supports of a structural member and by the
adjacent structure. When flexural members are also restrained at the supports,
shrinkage causes a buildup of axial tension in the member, in addition to the
bending caused by the external loads. Shrinkage is usually accommodated in
flexural members by an increase in the widths of the numerous flexural cracks.
However, for members not subjected to significant bending and where restraint is
provided to the longitudinal movements caused by shrinkage and temperature
changes, cracks tend to propagate over the full depth of the crosssection.
Excessively wide cracks are not uncommon. Such cracks are often called direct
tension cracks, since they are caused by direct tension rather than by
flexural tension. In fully restrained direct tension members, relatively large
amounts of reinforcement are required to control the load independent cracking.
Consider the fullyrestrained member shown in Fig.
3a. As the concrete shrinks, the restraining force N(t)
gradually increases until the first crack occurs when N(t)
= A_{c }f_{t}, usually
within two weeks from the commencement of drying, where A_{c} is the crosssectional area of the member and f_{t}
is the tensile strength of the concrete. Immediately after first cracking, the
restraining force reduces to N_{cr},
and the concrete stress away from the crack is less than the tensile strength of
the concrete. The concrete on either side of the crack shortens elastically and
the crack opens to a width w, as shown
in Fig. 3b. At the crack, the steel carries the entire
force N_{cr} and the stress in
the concrete is obviously zero. In the region immediately adjacent to the crack,
the concrete and steel stresses vary considerably and there exists a region of
partial bond breakdown. At some distance s_{o}
on each side of the crack, the concrete and steel stresses are no longer
influenced directly by the presence of the crack, as shown in Figs
3c and 3d.
In Region 1, where the distance from the crack is greater than or equal to s_{o}, the concrete and steel stresses are s_{c1} and s_{s1}, respectively. Since the steel stress (and hence strain) at the crack is tensile and the overall elongation of the steel is zero (full restraint), s_{s1} must be compressive. Equilibrium requires that the sum of the forces carried by the concrete and the steel on any crosssection is equal to the restraining force. Therefore, with the force in the steel in Region 1 being compressive, the force carried by the concrete (A_{c }s_{c1}) must be tensile and somewhat greater than the restraining force (N_{cr}). In Region 2, where the distance from the crack is less than s_{o}, the concrete stress varies from zero at the crack to s_{c1} at s_{o} from the crack. The steel stress varies from s_{s2} (tensile) at the crack to s_{s1} (compressive) at s_{o} from the crack, as shown.

Fig. 3  First cracking in a restrained direct tension member. 
To determine the crack width w and the concrete and steel stresses in Fig. 3, the distance s_{o} over which the concrete and steel stresses vary, needs to be known and the restraining force N_{cr} needs to be calculated. An approximation for s_{o} maybe obtained using the following equation, which was proposed by Favre et al. (1983) [6] for a member containing deformed bars or welded wire mesh:
s_{o} = d_{b} / 10 r
(6)
where d_{b}
is the bar diameter, and r
is the reinforcement ratio A_{s} /
A_{c}. Base and Murray (1982)
used a similar expression.
Gilbert
(1992) showed that the concrete and steel stresses immediately after first
cracking are
;
; and
(7)
where C_{1}
= 2 s_{o} /(3L  2 s_{o}). If n is the
modular ratio, E_{s} / E_{c},
the restraining force immediately after first cracking is
(8)
With the
stresses and deformations determined immediately after first cracking, the
subsequent longterm behaviour as shrinkage continues must next be determined.
After first cracking, the concrete is no longer fully restrained since the crack
width can increase with time as shrinkage continues. A state of partial
restraint therefore exists after first cracking. Subsequent shrinkage will cause
further gradual increases in the restraining force N(t) and in the concrete stress away from the crack, and a second
crack may develop. Additional cracks may occur as the shrinkage strain continues
to increase with time. However, as each new crack forms, the member becomes less
stiff and the amount of shrinkage required to produce each new crack increases.
The process continues until the crack pattern is established, usually in the
first few months after the commencement of drying. The concrete stress history
in an uncracked region is shown diagrammatically in Fig. 4.
The final average crack spacing, s, and the final average crack width, w, depend on the quantity and distribution of reinforcement, the
quality of bond between the concrete and steel, the amount of shrinkage, and the
concrete strength. Let the final shrinkageinduced restraining force be N(¥).
Fig. 4  Concrete
stress history in uncracked Region 1 (Gilbert, 1992) [8] 
After
all shrinkage has taken place and the final crack pattern is established, the
average concrete stress at a distance greater than s_{o} from the nearest crack is s*_{c1}
and the steel stresses at a crack and at a distance greater than s_{o} from a
crack are s*_{s2}
and s*_{s1},
respectively. Gilbert (1992) [8]
developed the following expressions for the final restraining force N(¥)
and the final average crack width w:
Provided the steel quantity is sufficiently large, so that
yielding does not occur at first cracking or subsequently, the final restraining
force is given by
(9)
is the final shrinkage strain;
is the final effective modulus of
the concrete and is given by
;
is the final creep coefficient; n*
is the effective modular ratio
; C_{2
}= 2s_{o}/(3s  2s_{o});
and s_{av}
is the average stress in the uncracked concrete (see Fig. 4)
and may be assumed to be (s_{c1}
+ f_{t })/2 . The
maximum crack spacing is
(10)
and x is given
by
(11)
The final steel stress at each crack and the final concrete stress in Regions 1 (further than s_{o} from a crack) are, respectively,
s*_{s2}
= N(¥)/A_{s}
and
s*_{c1}
= N(¥)(1
+ C_{2 }) /A_{c}
< f_{t}
(12)
Provided
the steel at the crack has not yielded, the final crack width is given by
(13)
When the
quantity of steel is small, such that yielding occurs at first cracking,
uncontrolled and unserviceable cracking will result and the final crack width is
wide. In this case,
;
;
and
(14)
and the
final crack width is
(15)
where L
is the length of the restrained member.
Numerical Example:
Consider
a 5 m long and 150 mm thick
reinforced concrete slab, fully restrained at each end. The slab contains 12mm
diameter deformed longitudinal bars at 300 mm centres in both the top and bottom
of the slab (A_{s} = 750 mm^{2}/m). The concrete cover to the
reinforcement is 30 mm. Estimate the spacing, s, and final average width, w,
of the restrained shrinkage cracks.
Take
= 2.5,
=  600 x 10^{6}, f_{t} = 2.0 MPa, E_{c}
= 25000 MPa, E_{s}_{ }=
200000 MPa, n = 8 and f_{y}
= 400 MPa. The reinforcement ratio is r
= 0.005 and from Equation 6,
mm.
The
final effective modulus is
MPa and the corresponding effective modular ratio is
. The constant C_{1} = 2
s_{o} /(3L  2
s_{o}) = 2 x 240/(3 x 5000  2 x 240) = 0.0331 and from
Equation 8, the restraining force immediately after first cracking is
N/m
The
steel stress at the crack s_{s2}
= 161300/750 = 215 MPa and the concrete stress
is obtained from Equation 7:
MPa.
The
average concrete stress may be approximated by s_{av} = (1.11 +
2.0)/2 = 1.56 MPa and from Equation 11:
The
maximum crack spacing is determined using Equation 10:
mm
The
constant C_{2
}is obtained from C_{2 }= (2 x 240)/(3 x 839  2 x240) = 0.236 and the final restraining force is calculated using Equation 9:
N/m
From
Equation 12, s*_{s2}
= 323 MPa, s*_{c1}
= 1.99 MPa and, consequently, s*_{s1}
= 76.4 MPa. The
final crack width is determined using Equation 13:
mm.
Tables 2
and 3 contain results of a limited parametric study showing the effect of
varying steel area, bar size, shrinkage strain and concrete tensile strength on
the final restraining force, crack width, crack spacing and steel stress in a
150 mm thick slab, fullyrestrained over a length of 5 m.
Table 2
Effect
of steel area and shrinkage strain on direct tension cracking (f*=
2.5, f_{t} = 2.0 MPa and d_{b}
= 12 mm)
A_{s} mm^{2} 
r 
=  0.0006 
=  0.00075 
=  0.0009 

N(¥) kN 
s_{s2}* MPa 
s mm 
w mm 
N(¥) kN 
s_{s2}* MPa 
s mm 
w mm 
N(¥) kN 
s_{s2}* MPa 
s mm 
w mm 

375 
.0025 
150 
400 
 
1.37 
150 
400 
 
2.03 
150 
400 
 
2.68 
450 
.003 
180 
400 
 
1.35 
180 
400 
 
2.01 
180 
400 
 
2.66 
600 
.004 
240 
400 
 
1.22 
234 
390 
913 
0.49 
216 
360 
717 
0.50 
750 
.005 
243 
324 
837 
0.31 
220 
294 
601 
0.33 
197 
264 
469 
0.34 
900 
.006 
233 
259 
601 
0.23 
206 
229 
427 
0.24 
179 
199 
332 
0.24 
1050 
.007 
224 
214 
453 
0.18 
193 
184 
320 
0.18 
161 
154 
247 
0.19 
1200 
.008 
215 
170 
354 
0.14 
179 
149 
248 
0.15 
143 
119 
191 
0.15 
Table 3
Effect
of bar diameter and concrete tensile strength on direct tension cracking.
(f*=
2.5, e_{cs}*=
0.0006, A_{s} = 900 mm^{2}
and r
= 0.006)
d_{b} (mm) 
f_{t} = 2.0 MPa 
f_{t} = 2.5 MPa 

N(¥) kN 
s_{s2}* MPa 
s mm 
w mm 
N(¥) kN 
s_{s2}* MPa 
s mm 
w mm 

6 
237 
263 
317 
0.12 
323 
359 
482 
0.14 
10 
234 
260 
508 
0.19 
320 
356 
758 
0.23 
12 
233 
259 
601 
0.23 
319 
354 
889 
0.27 
16 
232 
257 
781 
0.30 
317 
352 
1141 
0.35 
20 
230 
256 
956 
0.37 
315 
350 
1385 
0.42 
The
control of deflections may be achieved by limiting the calculated deflection to
an acceptably small value. Two
alternative general approaches for deflection calculation are specified in
AS3600 (1), namely ‘deflection by
refined calculation’ (Clause 9.5.2 for beams and Clause 9.3.2 for slabs)
and ‘deflection by simplified calculation’ (Clause 9.5.3 for beams
and Clause 9.3.3 for slabs). The
former is not specified in detail but allowance should be made for cracking and
tension stiffening, the shrinkage and creep properties of the concrete, the
expected load history and, for slabs, the twoway action of the slab.
The
longterm or timedependent behaviour of a beam or slab under sustained service
loads can be determined using a variety of analytical procedures (Gilbert, 1988)
[7], including the AgeAdjusted Effective Modulus Method (AEMM),
described in detail by Gilbert and Mickleborough (1997) [12].
The use of the AEMM to determine the instantaneous and timedependent
deformation of the critical crosssections in a beam or slab and then
integrating the curvatures to obtain deflection, is a refined calculation method and is recommended.
Using
the AEMM, the strain and curvature on individual crosssections at any time can
be calculated, as can the stress in the concrete and bonded reinforcement or
tendons. The routine use of the AEMM in the design of concrete structures for
the serviceability limit states is strongly encouraged.
However,
in most design situations, the latter approach (deflection by simplified
calculation) is generally used and its limitations are discussed in detail
below.
The
instantaneous or shortterm deflection of a beam may be calculated using the
mean value of the elastic modulus of concrete at the time of first loading, E_{cj},
together with the effective second moment of area of the member, I_{ef}. The effective second moment of area involves an
empirical adjustment of the second moment of area of a cracked member to account
for tension stiffening (the stiffening effect of the intact tensile concrete
between the cracks). For a given
crosssection, I_{ef} is
calculated using Branson’s formula (Branson, 1963):
I_{ef} = I_{cr} + (I  I_{cr})(M_{cr}/M_{s})^{3}
I_{e,max}
(16)
where I_{cr}
is the second moment of area of the fullycracked section (calculated using
modular ratio theory); I is the second
moment of area of the gross concrete section about its centroidal axis; M_{s} is the maximum bending moment at the section, based on
the shortterm serviceability design load or the construction load; and M_{cr}
is the cracking moment given by
M_{cr}
= Z(f'_{cf}
 f_{cs} + P/A_{g})
+ Pe) ³
0.0
(17)
Z is the section modulus of the uncracked section,
referred to the extreme fibre at which cracking occurs; f'_{cf}
is the characteristic flexural tensile strength of
concrete; f_{cs} is the
maximum shrinkage induced tensile stress on the uncracked section at
the extreme fibre at which cracking occurs and may be taken as
f_{cs} =
(18)
where
p is the reinforcement ratio (A_{st}/bd)
and e_{cs} is the final design shrinkage strain.
The maximum value of I_{ef} at any crosssection, I_{e,max} in Equation 16, is I when
p = A_{st}/bd ³ 0.005 and 0.6I when p < 0.005.
Alternatively,
as a further simplification but only for reinforced concrete members, I_{ef}
at each nominated crosssection for rectangular sections may be taken as equal
to
(0.02 + 2.5p)bd^{3
} when
p ³ 0.005
and (0.1
– 13.5p)bd^{3}
when p
< 0.005.
The value of
I_{ef} for the member is determined from the value of I_{ef} at midspan for a simplesupported beam. For interior spans of
continuous beams, I_{ef}
is half the midspan value plus one quarter of the value at each support,
and for end spans of continuous strips, I_{ef} is half the midspan value plus half the value at the
continuous support. For a cantilever, I_{ef}
is the value at the support.
The
term f_{cs}
in Equation 17 was introduced in Amendment
2 to AS3600 to allow for the tension that inevitably develops due to the
restraint to shrinkage provided by the bonded tensile reinforcement. Equation 18
is based on the expression proposed in Gilbert (1998) [9],
by assuming conservative values for the elastic modulus and the creep
coefficient of concrete and assuming about 40% of the final shrinkage has
occurred at the time of cracking. In the calculation of f_{cs}
using Equation 18, the final or longterm value of e_{cs}
in the concrete should be used.
This
allowance for shrinkage induced tension is particularly important in the case of
lightly reinforced members (including slabs) where the tension induced by the
full service moment alone might not be enough to cause cracking. In such cases,
failure to account for shrinkage may lead to deflection calculations based on
the uncracked section properties. This usually grossly underestimates the actual
deflection. For heavily reinforced sections, the problem is not so significant,
as the service loads are usually well in excess of the cracking load and the
ratio of cracked to uncracked stiffness is larger.
For the
calculation of longterm deflection, one of two approaches may be used.
For reinforced or prestressed beams, the creep and shrinkage deflections
can be calculated separately (using the material data specified in the Standard
and the principles of mechanics). Alternatively, for reinforced concrete beam,
longterm deflection can be crudely approximated by multiplying the immediate
deflection caused by the sustained load by a multiplier k_{cs}
given by
k_{cs} = [2  1.2(A_{sc}/A_{st})]
³
0.8
(19)
where the
ratio A_{sc}/A_{st} is
taken at midspan for a simple or continuous span and at the support for a
cantilever.
The current simplified approach for the calculation of final deflection fails to adequately predict the longterm or timedependent deflection (by far the largest portion of the total deflection in most reinforced and prestressed concrete members). Shrinkage induced curvature and the resulting deflection is not adequately accounted for when using k_{cs} and no account is taken of the actual creep and shrinkage properties of the concrete. The introduction of f_{cs} in the estimation of the cracking moment is a positive step in improving the procedure, by recognising that early shrinkage can induce tension that significantly reduces the cracking moment and significantly reduces the instantaneous stiffness with time. However, the gradual reduction in I_{ef} with time due to shrinkage and cyclic loading is still not fully accounted for. To better model the breakdown of tension stiffening with time, Equation 18 should be replaced by Equation 20, which was originally proposed by Gilbert (1999a) [10] but was modified by Standards Australia (for political, rather than technical, reasons).
f_{cs} =
(20)
A further criticism of the simplified approach is the use of the second moment of area of the gross concrete section I in Equation 16. It is unnecessarily conservative to ignore the stiffening effect of the bonded reinforcement in the calculation of the properties of the uncracked crosssection.
The use of the deflection multiplier k_{cs} to calculate timedependent deflections is simple and convenient and, provided the section is initially cracked under short term loads, it sometimes provides a ‘ballpark’ estimate of final deflection. However, to calculate the shrinkage induced deflection by multiplying the load induced shortterm deflection by a longterm deflection multiplier is fundamentally wrong. Shrinkage can cause significant deflection even in unloaded members (where the shortterm deflection is zero). The approach ignores the creep and shrinkage characteristics of the concrete, the environment, the age at first loading and so on. At best, it provides a very approximate estimate. At worst, it is not worth the time involved in making the calculation.
It is, however, not too much more complicated to calculate longterm creep and shrinkage deflection separately. As mentioned previously, well established and reliable methods are available for calculating the timedependent behaviour of reinforced and prestressed concrete crosssections (Gilbert, 1988) [7]. A simple method suitable for routine use in design is outlined below.
The load
induced curvature, k(t),
(instantaneous plus creep) at any time t due to sustained service actions may be
expressed as
k(t)
= k_{i}(t)(1
+ f/a)
(21)
where k_{i}(t)
is the instantaneous curvature due to the sustained service moment M_{s}
(k_{i}(t)
= M_{s}/E_{c}I_{ef}
); for an uncracked crosssection I_{ef}
should be taken as the second moment of area of the uncracked transformed
section, while for a cracked section, I_{ef} should be calculated from Equation 16 (with f_{cs} calculated using Equation 20 when estimating the
final longterm curvature); f
is the creep coefficient at time t; and α is a term that accounts for the
effects of cracking and the ‘braking’ action of the reinforcement on creep
and may be estimated from Equations 22a, 22b or 22c.
For a
cracked reinforced concrete section in pure bending, a
= a_{1},
where
a_{1}
= [0.48 p^{  0.5}] [1 + (125
p + 0.1)(A_{sc}/A_{st})^{1.2}]
(22a)
For an
uncracked reinforced or prestressed concrete section, a
= a_{2},
where
a_{2}
= [1.0  15.0 p] [1 + (140 p  0.1)(A_{sc}/A_{st})^{1.2}]
(22b)
where p
= A_{st}/b
d_{o} and A_{st} is the equivalent area of bonded reinforcement in the
tensile zone at depth d_{o}
(the depth from the extreme compressive fibre to the centroid of the outermost
layer of tensile reinforcement). The area of any bonded reinforcement in the
tensile zone (including bonded tendons) not contained in the outermost layer of
tensile reinforcement (ie. located at a depth d_{1} less than d_{o})
should be included in the calculation of A_{st}
by multiplying that area by d_{1}/d_{o}).
For the purpose of the calculation of A_{st},
the tensile zone is that zone that would be in tension due to the applied moment
acting in isolation. A_{sc}
is the area of the bonded reinforcement in the compressive zone.
For a
cracked, partially prestressed section or for a cracked reinforced concrete
section subjected to bending and axial compression, a
may be taken as
a
= a_{2}
+ (a_{1}
 a_{2})(d_{n1}/d_{n})^{2.4}
(22c)
where d_{n}
is the depth of the intact compressive concrete on the cracked section and d_{n1}
is the depth of the intact compressive concrete on the cracked section ignoring
the axial compression and/or the prestressing force (ie. the value of d_{n}
for an equivalent cracked reinforced concrete section containing the same
quantity of bonded reinforcement).
The
shrinkage induced curvature on a reinforced or prestressed concrete section can
be approximated by
(23)
where D
is the overall depth of the section, A_{st}
and A_{sc} are as defined under Equation 22b above, and the
factor k_{r }depends on the quantity and location of the bonded
reinforcement and may be estimated from Equations 24a, 24b, 24c or 24d.
For an
uncracked crosssection, k_{r}
= k_{r1}, where
k_{r1}
= (100 p
 2500 p^{2})
when p
= A_{st}/b
d_{o} £
0.01 (24a)
k_{r1}
= (40 p
+ 0.35)
when
p = A_{st}/b
d_{o} > 0.01
(24b)
For a
cracked reinforced concrete section in pure bending, k_{r} = k_{r2},
where
k_{r2}
= 1.2
(24c)
For a
cracked, partially prestressed section or for a cracked reinforced concrete
section subjected to bending and axial compression, k_{r} may be taken as
k_{r} = k_{r1} + (k_{r2}
 k_{r1})(d_{n1}/d_{n})
(24d)
where d_{n}
is the depth of the intact compressive concrete on the cracked section and d_{n1}
is the depth of the intact compressive concrete on the cracked section ignoring
the axial compression and/or the prestressing force (ie. the value of d_{n}
after cracking for an equivalent cracked reinforced concrete section containing
the same quantity of bonded reinforcement).
Equations
22, 23 and 24 have been developed from parametric studies of a wide range of
crosssections analysed using the AgeAdjusted Effective Modulus Method of
analysis (with typical results of such analyses presented and illustrated by
Gilbert ,2000).
When the
load induced and shrinkage induced curvatures are calculated at selected
sections along a beam or slab, the deflection may be obtained by double
integration. For a reinforced or prestressed concrete continuous span with the
degree of cracking varying along the member, the curvature at the left and right
supports,
and
and the curvature at midspan
may be calculated at any time after
loading and the deflection at midspan Δ may be approximated by assuming a
parabolic curvature diagram along the span,
:
(25)
The
above equation will give a reasonable estimate of deflection even when the
curvature diagram is not parabolic and is a useful expression for use in
deflection calculations.
Example 1
A reinforced concrete beam of rectangular section (800 mm deep and 400 mm wide) is simplysupported over a 12 m span and is subjected to a uniformly distributed sustained service load of 22.22 kN/m. The longitudinal reinforcement is uniform over the entire span and consists of 4 Y32 bars located in the bottom at an effective depth of 750 mm (A_{st} = 3200 mm^{2}) and 2 Y32 bars in the top at a depth of 50 mm below the top surface (A_{sc} = 1600 mm^{2}). Calculate the instantaneous and longterm deflection at midspan, assuming the following material properties:
f'_{c} = 32 MPa; f'_{cf}
= 3.39 MPa; E_{c} = 28,570
MPa; E_{s} = 2 x 10^{5}
MPa; f = 2.5; and
e_{cs} =
0.0006.
For each
crosssection, p = A_{st}/bd = 0.0107.
The
section at midspan:
The
sustained bending moment is M_{s} = 400 kNm. The second moments of area
of the uncracked transformed crosssection, I,
and the fullcracked transformed section, I_{cr},
are I = 20,560 x 10^{6} mm^{4}
and I_{cr} = 7,990 x 10^{6}
mm^{4}. The bottom fibre section modulus of the uncracked section is Z
= I/y_{b} = 52.7 x 10^{6}
mm^{3}. From Equation 20,
f_{cs} = = 2.09 MPa
and the timedependent cracking moment is obtained from Equation 17:
M_{cr} = 52.7 x 10^{6}
(3.39  2.09) = 68.5 kNm.
From
Equation 16, the effective second moment of area is
I_{ef}
= [7990 + (20560  7990)(68.5/400)^{3}] x 10^{6} = 8050 x
10^{6} mm^{4}
The instantaneous curvature due to the sustained service moment is therefore
k_{i}(t)
=
=
= 1.74 x 10^{6} mm^{1}.
From
Equation 22a:
a_{1}
= [0.48 x 0.0107^{0.5}][1 + (125 x 0.0107 + 0.1)(1600/3200)^{1.2}]
= 7.55
and the
load induced curvature (instantaneous plus creep) is obtained from Equation 21:
k(t)
= 1.74 x 10^{6} (1 + 2.5/7.55) = 2.32 x 10^{6} mm^{1}.
From
Equation 24c:
k_{r} = k_{r2}
=
= 0.96
and the
shrinkage induced curvature is obtained from Equation 23:
mm^{1}
The
instantaneous and final timedependent curvatures at midspan are therefore
k_{i}
= 1.74 x 10^{6} mm^{1}
and k
= k(t)
+ k_{cs}
= 3.04 x 10^{6} mm^{1}.
The
section at each support:
The
sustained bending moment is zero and the section remains uncracked. The
loaddependent curvature is therefore zero. However, shrinkage curvature
develops with time. From Equation 24b:
k_{r} = k_{r1} =
= 0.276
and the
shrinkage induced curvature is estimated from Equation 23:
mm^{1}
Deflections:
The
instantaneous and final longterm deflections at midspan, D_{i}
and D_{LT},
respectively, are obtained from Equation 25:
mm
mm
(= span/260)
It is of
interest to note that using the current approach in AS3600, with k_{cs}
= 1.4 (from Equation 19), the calculated final deflection is 60.9 mm.
Example 2
A posttensioned concrete beam of rectangular section (800 mm deep and 400 mm wide) is simplysupported over a 12 m span and is subjected to a uniformly distributed sustained service load of 38.89 kN/m. The beam is prestressed with a single parabolic cable consisting of 15/12.7mm diameter strands (A_{p} = 1500 mm^{2}) with d_{p} = 650 mm at midspan and d_{p} = 400 mm at each support. The duct containing the tendons is filled with grouted soon after transfer. The longitudinal reinforcement is uniform over the entire span and consists of 4 Y32 bars located in the bottom at an effective depth of 750 mm (A_{s} = 3200 mm^{2}) and 2 Y32 bars in the top at a depth of 50 mm below the top surface (A_{sc} = 1600 mm^{2}). For the purpose of this exercise, the initial prestressing force in the tendon is assumed to be 2025 kN throughout the member and the relaxation loss is 50 kN. Calculate the instantaneous and longterm deflection at midspan, assuming the following material properties:
f'_{c} = 32 MPa; f'_{cf}
= 3.39 MPa; E_{c} = 28,570
MPa; E_{s} = 2 x 10^{5}
MPa; f = 2.5; and
e_{cs} =
0.0006.
The
section at midspan:
The
sustained bending moment is M_{s} = 700 kNm. The centroidal axis of the
uncracked transformed crosssection is located at a depth of 415.7 mm below the
top fibre and the second moment of area is I
= 21,070 x 10^{6} mm^{4}. The top and bottom fibre concrete
stresses immediately after first loading (due the applied moment and prestress)
are 10.11 MPa and 1.55 MPa, respectively (both compressive). The section
remains uncracked throughout.
The instantaneous curvature due to the sustained service moment is
k_{i}(t)
=
=
= 0.375 x 10^{6} mm^{1}.
From
Equation 22a, with A_{sc} =
1600 mm^{2 }, A_{st} =
A_{s} + A_{p} d_{p}/d_{o}_{ }= 3200 +
1500x650/750 = 4500 mm^{2} and, therefore p =
A_{st}/b d_{o}
= 4500/(400x750) = 0.015:
:
a_{2}
= [1.0  15.0 x 0.015][1 + (140 x 0.015  0.1)(1600/4500)^{1.2}] = 1.22
and the
load induced curvature (instantaneous plus creep) is obtained from Equation 21:
k(t)
= 0.375 x 10^{6} (1 + 2.5/1.22) = 1.14 x 10^{6} mm^{1}.
From
Equation 24b:
k_{r} = k_{r1}
=
= 0.536
and the
shrinkage induced curvature is obtained from Equation 23:
mm^{1}
The
instantaneous and final timedependent curvatures at midspan are therefore
k_{i}
= 0.375 x 10^{6} mm^{1}
and k
= k(t)
+ k_{cs}
= 1.54 x 10^{6} mm^{1}.
The
section at each support:
The
sustained bending moment is zero and the section remains uncracked.
The centroidal axis of the uncracked transformed crosssection (with A_{p}
located at a depth of 400 mm) is located at a depth of 409.4 mm below the top fibre and the second
moment of area is I = 20,560 x 10^{6}
mm^{4}. The prestressing steel is located 9.4 mm above the centroidal
axis of the transformed section, so that the prestressing force induces a small
instantaneous positive curvature. Shrinkage (and creep) curvature develops with
time.
The instantaneous curvature is
k_{i}(t)
=
=
= 0.032 x 10^{6} mm^{1}.
From
Equation 22a, with A_{sc} =
1600 mm^{2 }, A_{st} =
A_{s1} + A_{p} d_{p}/d_{o}_{ }= 3200 +
1500x400/750 = 4000 mm^{2} and, therefore p =
A_{st}/b d_{o}
= 4000/(400x750) = 0.0133:
:
a_{2}
= [1.0  15.0 x 0.0133][1 + (140 x 0.0133  0.1)(1600/4000)^{1.2}] =
1.27
and the
load induced curvature (instantaneous plus creep) is obtained from Equation 21:
k(t)
= 0.032 x 10^{6} (1 + 2.5/1.27) = 0.09 x 10^{6} mm^{1}.
From
Equation 24b:
k_{r} = k_{r1} =
= 0.398
and the
shrinkage induced curvature is estimated from Equation 23:
mm^{1 }
The
instantaneous and final timedependent curvatures at the supports are therefore
k_{i}
= 0.032 x 10^{6} mm^{1}
and k
= k(t)
+ k_{cs}
= 0.39 x 10^{6} mm^{1}.
Deflections:
The
instantaneous and final longterm deflections at midspan, D_{i}
and D_{LT},
respectively, are obtained from Equation 25:
mm
mm
In this
example, the ratio of final to instantaneous deflection is 4.3.
In AS36001994, the control of flexural cracking is deemed
to be satisfactory, providing the designer satisfies certain detailing
requirements. These involve maximum limits on the centretocentre spacing of
bars and on the distance from the side or soffit of the beam to the nearest
longitudinal bar. These limits do not depend on the stress in the tensile steel
under service loads and have been found to be unreliable when the steel stress
exceeds about 240 MPa. The provisions of AS36001994 oversimplify the problem
and do not always ensure adequate control of cracking.
With the current move to higher strength reinforcing steels
(characteristic strengths of 500 MPa and above), there is an urgent need to
review the crackcontrol design rules in AS3600 for reinforced concrete beams
and slabs. The existing design rules for reinforced concrete flexural elements
are intended for use in the design of elements containing 400 MPa bars and are
sometimes unconservative. They are unlikely to be satisfactory for members in
which higher strength steels are used, where steel stresses at service loads are
likely to be higher due to the reduced steel area required for strength.
Standards
Australia has established a Working Group to investigate and revise the crack
control provisions of the current Australian Standard to incorporate recent
developments and to accommodate the use of high of high strength reinforcing
steels. A theoretical and experimental investigation of the critical factors
that affect the control of cracking due to restrained deformation and external
loading is currently underway at the University of New South Wales. The main
objectives of the investigation are to gain a better understanding of the
factors that affect the spacing and width of cracks in reinforced concrete
elements and to develop rational and reliable designoriented procedures for the
control of cracking and the calculation of crack widths.
As an
interim measure, to allow the immediate introduction of 500 MPa steel
reinforcement, the deemed to comply crack control provisions of Eurocode 2 (with
minor modifications) have been included in the recent Amendment 2 of the
Standard. In Gilbert (1999b) [11]
and Gilbert et al. (1999) [13], the current crack control
provisions of AS 3600 were presented and compared with the corresponding
provisions in several of the major international concrete codes, including BS
8110, ACI 318 and Eurocode 2. A parametric evaluation of the various code
approaches was also undertaken to determine the relative importance in each
model of such factors as steel area, steel stress, bar diameter, bar spacing,
concrete cover and concrete strength on the final crack spacing and crack width.
The applicability of each model was assessed by comparison with some local crack
width measurements and problems were identified with each of the code models.
Gilbert et al (1999) conclude that the provisions of Eurocode 2 appear to
provide a more reliable means for ensuring adequate crack control than either BS
8110 or ACI 318, but that all approaches fail to adequately account the increase
in crack widths that occurs with time.
In
Amendment 2, Clause 8.6.1 Crack control
for flexure in reinforced beams has
been replaced with the following:
8.6.1
Crack control for flexure and tension in reinforced beams Cracking in reinforced beams subjected to flexure
or tension shall be deemed to be controlled if the appropriate requirements in
(a) and (b), and either (c) or (d) are satisfied. For the purpose of this
Clause, the resultant action is considered to be flexure
when the tensile stress distribution within the section prior to cracking is
triangular with some part of the section in compression, or tension
when the whole of the section is in tension.
(a)
The minimum area of reinforcement required in the tensile zone (A_{st.min})
in regions where cracking shall be taken as
A_{gt} = 3 k_{s} A_{ct} / f_{s }
where
k_{s}
=
a coefficient which takes into account the shape of the stress
distribution within the section immediately prior to cracking, and equals 0.6
for flexure and 0.8 for tension.
A_{ct
}=
the area of concrete in the tensile zone, being that part of the section
in tension assuming the section is uncracked; and
f_{s}
=
the maximum tensile stress permitted in the reinforcement after formation
of a crack, which shall be the lesser of the yield strength of the reinforcement
(f_{sy}) and the maximum steel
stress in Table 8.6.1(A) for the largest nominal bar diameter (d_{b})
of the bars in the section.
(b)
The distance from the side or soffit of a beam to the centre of the
nearest longitudinal bar shall not be greater than 100mm. Bars with a diameter
less than half the diameter of the largest bar in the crosssection shall be
ignored. The centretocentre spacing of bars near a tension face of the beam
shall not exceed 300 mm.
(c)
For beams subjected to tension, the steel stress (f_{scr}),
calculated for the load combination for the shortterm serviceability limit
states assuming the section is cracked, does not exceed the maximum steel stress
given in Table 8.6.1(A) for the largest nominal diameter (d_{b})
of the bars in the section.
(d)
For beams subjected to flexure, the
steel stress (f_{scr}),
calculated for the load combination for the shortterm serviceability limit
states assuming the section is cracked, does not exceed the maximum steel stress
given in Table 8.6.1(A) for the largest nominal diameter (d_{b})
of the bars in the tensile zone under the action of the design bending moment.
Alternatively, the steel stress does not exceed the maximum stress determined
from Table 8.6.1(B) for the largest centretocentre spacing of adjacent
parallel bars in the tensile zone. Bars with a diameter less than half the
diameter of the largest bar in the crosssection shall be ignored when
determining spacing.
TABLE 8.6.1(A)
TABLE 8.6.1(B)
MAXIMUM
STEEL STRESS
MAXIMUM
STEEL STRESS
FOR
TENSION OR FLEXURE IN BEAMS
FOR
FLEXURE IN BEAMS
Maximum
steel stress
(MPa) 
Nominal
bar diameter, d_{b}, (mm) 

Maximum
steel stress (MPa) 
Centretocentre
spacing (mm) 
160 
32 

160 
300 
200 
25 

200 
250 
240 
20 

240 
200 
280 
16 

280 
150 
320 
12 

320 
100 
360 
10 

360 
50 
400 
8 

Note: Linear interpolation may be
used. 

450 
6 

The
amendment is similar to the crack control provisions in Eurocode 2. In essence,
the amendment requires the quantity of steel in the tensile region to exceed a
minimum area, A_{st.min},
and places a maximum limit on the steel stress depending on either the bar
diameter or the centretocentre spacing of bars. As in the existing clause, a
maximum limit of 100 mm is also placed on the distance from the side or soffit
of a beam to the nearest longitudinal bar.
An
alternative approach to flexural crack control is to calculate the design
crack width and to limit this to an acceptably small value.
The writer has proposed an approach for calculating the design crack
width (Gilbert 1999b). This
approach is similar to that proposed in Eurocode 2, but modified to
include shrinkage shortening of the intact concrete between the cracks in the
tensile zone and to more realistically represent the increase in crack width if
the cover is increased.
The design crack width,
, may be calculated from
w = b_{m} s_{rm} ( e_{sm} + e_{cs.t} ) (26)
where s_{rm} is the average final crack spacing; b_{m} is a coefficient relating the average crack width to the design value and may be taken as b_{m} = 1.0 + 0.025c ³ 1.7; c is the distance from the concrete surface to the nearest longitudinal reinforcing bar; and e_{sm} is the mean strain allowing for the effects of tension stiffening and may be taken as
e_{sm }= (s_{s}/E_{s})[ 1  b_{1}b_{2} (s_{sr}/s_{s})^{2 }] (27)
where s_{s}_{ }is the stress in the tension steel calculated on the basis of a cracked section; s_{sr}_{ } is the stress in the tension steel calculated on the basis of a cracked section under the loading conditions causing first cracking; b_{1} depends on the bond properties of the bars and equals 1.0 for high bond bars and 0.5 for plain bars; and b_{2} accounts for the duration of loading and equals 1.0 for a single, shortterm loading and 0.5 for a sustained load or for many cycles of loading.
The average final spacing of flexural cracks, s_{rm} (in mm), can be calculated from
s_{rm} = 50 + 0.25 k_{1} k_{2} d_{b} / r (28)
where d_{b}
is the bar size (or average bar size in the section) in mm; k_{1}
accounts for the bond properties of the bar and, for flexural cracking, k_{1} = 0.8 for high bond bars and k_{1 }=1.6 for plain bars; k_{2} depends on the strain distribution and equals 0.5 for
bending; and r_{r}
is the effective reinforcement ratio, A_{s}/A_{c.eff}
where A_{s} is the area of
reinforcement contained within the effective tension area, A_{c.eff}. The
effective tension area is the area of concrete surrounding the tension steel of
depth equal to 2.5 times the distance from the tension face of the section to
the centroid of the reinforcement, but not greater than
of the depth of the tensile zone of
the cracked section, in the case of slabs.
e_{cs.t} is the shrinkage induced shortening of the intact concrete at the tensile steel level between the cracks. For shortterm crack width calculations, e_{cs.t} is zero. Using the ageadjusted effective modulus method and a shrinkage analysis of a singly reinforced concrete section, see Gilbert (1988), it can be shown that
e_{cs.t} = e_{cs} / ( 1 +3 p ) (13)
where p is the
tensile reinforcement ratio for the section (A_{st}/bd);
is the ageadjusted modular
ratio (E_{s}/E_{ef});
E_{ef} is the
ageadjusted effective modulus for concrete (E_{ef} =E_{c }/(1+0.8f));
and e_{cs}
and f
are final longterm values of shrinkage strain and creep coefficient,
respectively.
The above procedure overcomes the major deficiencies in
current code procedures and more accurately agrees with laboratory and field
measurements of crack widths. In
Tables 4 and 5, crack widths calculated
using the proposed procedure are presented for rectangular slab and beam
sections. In each case, e_{cs
}= 0.0006 and f_{
}= 3.0. In general, the calculated crack widths are larger than those
predicted by either ACI or EC2, but unlike these codes, the proposed model will
signal serviceability problems to the structural designer in most situations
where excessive crack widths are likely.
It should be pointed out that the steel stress under sustained service loads is usually less than 200 MPa for beams and slabs designed using 400 MPa steel. The range of steel stresses in Tables 4 and 5 are more typical of situations in which 500 MPa steel is used.
Table 4 Calculated final flexural crack widths in a 200 mm thick slab
Effective depth, d (mm) 
Bar diam d_{b} (mm) 
Area of tensile steel, A_{st} (mm^{2}/m) 
Bar spacing, s (mm) 
Crack width (mm) 

Steel stress, s_{s} (MPa) 

200 
250 
300 

174 
12 
1044 
108 
0.226 
0.279 
0.330 
172 
16 
1032 
195 
0.267 
0.331 
0.392 
170 
20 
1020 
308 
0.309 
0.384 
0.455 
168 
24 
1008 
449 
0.352 
0.438 
0.519 
166 
28 
996 
618 
0.394 
0.492 
0.585 
Table
5 Calculated final flexural crack widths for beam (b
= 400 mm and d = 400 mm)
Bar diam d_{b} (mm) 
No. of bars 
A_{st}
(mm^{2}) 
p
= A_{st}/bd 
Crack width (mm) 

Cover = 25 mm 
Cover = 50 mm 

Steel stress, s_{s} (MPa) 
Steel stress, s_{s} (MPa) 

200 
250 
300 
200 
250 
300 

20 
2 
620 
.0039 
.309 
.397 
.479 
.488 
.646 
.791 
20 
3 
930 
.0058 
.267 
.326 
.384 
.414 
.513 
.607 
20 
4 
1240 
.0078 
.231 
.280 
.327 
.349 
.425 
.498 
24 
2 
900 
.0056 
.314 
.386 
.455 
.480 
.596 
.707 
24 
3 
1350 
.0084 
.251 
.304 
.355 
.369 
.449 
.526 
24 
4 
1800 
.0113 
.214 
.258 
.301 
.304 
.367 
.430 
28 
2 
1240 
.0078 
.299 
.362 
.424 
.434 
.529 
.621 
28 
3 
1860 
.0116 
.234 
.281 
.329 
.325 
.393 
.459 
32 
2 
1600 
.0100 
.285 
.344 
.402 
.394 
.477 
.558 
The
effects of shrinkage on the behaviour of reinforced and prestressed concrete
members under sustained service loads has been discussed. In particular,
the mechanisms of
shrinkage warping in unsymmetrically reinforced elements and shrinkage cracking
in restrained direct tension members has been described. Recent amendments to
the serviceability provisions of AS3600 have been outlined and techniques for
the control of deflection and cracking are presented. Reliable procedures for
the prediction of longterm deflections and final crack widths in flexural
members have also been proposed and illustrated by examples.
This paper stems from a continuing study of the
serviceability of concrete structures at the University of New South Wales.
The work is currently funded by the Australian Research Council through
two ARC Large Grants, one on deflection control of reinforced concrete slabs and
one on crack control in concrete structures. The support of the ARC and UNSW is
gratefully acknowledged.
1. AS36001994, Australian Standard for Concrete Structures, Standards Australia, Sydney, (1994).
5. DD ENV199211 Eurocode 2, Design of Concrete Structures, British Standards Institute, 1992.
R.I. Gilbert,
BE Hon 1, PhD UNSW,
FIEAust, CPEng Ian Gilbert is Professor of Civil Engineering and Head of the School of Civil and
Environmental Engineering at the University of New South Wales.
His main research interests have been in the area of serviceability
and the timedependent behaviour of concrete structures.
His publications include three books and over one hundred refereed
papers in the area of reinforced and prestressed concrete structures.
He has served on Standards Australia’s Concrete Structures Code
Committee BD/2 since 1981 and was actively involved in the development of
AS3600. He is currently chairing two of the Working Groups (WG2 –
Anchorage and
WG7 – Serviceability) established to review AS3600.
Professor Gilbert was awarded the Chapman Medal by the IEAust in
2000 and is the 2001 Eminent Speaker for the Structural College, IEAust. 
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Journal of Structural Engineering, Vol. 1, No.1 (2001) 214 