1.0 CRACK CRACK CONTROL CONTROL AND AND CALCULATION IN REINFORCED CONCRETE STRUCTURES 1.1 INTRODUC INTRODUCTION TION
The presence of the crack does not necessary mean that the structure will collapse, but it is a signal that the behaviour of the structure should should be considered more carefully. Cracking may detract from the appearance of the structure, permit ingress of moisture and lead to corrosion of steel. Excessive cracking and wide cracks will thus affect durability of the structure. An accurate prediction of the initiation of cracks and the subsequent monitoring of their behaviour during the structure response play a significant role in the
prediction of the general general safety safety of the structure. 1.2 SEQUENCY SEQUENCY OF OF CRACK CRACK FORMATION
!C element experience cracks in the tension face when the tensile strength of the concrete is exceeded. The tensile strength of concrete is generally less than "#$ of the compressive strength. %rimary cracks form first and as applied moment increases secondary cracks follow. The width of secondary crack width at appoint on the surface of a beam is affected by& 'urf 'urfac acee st strain rainss wh which ich ca can be be fou found nd by analy(ing the section assuming the section remains plane in accordance with )ernoulli*s beam theory " The di dista stance nce of of the the point oint from rom a point oint of (ero crack width. The points of (ero "
crack width are the neutral axis and the surface of longitudinal reinforcing bars. The larger this is, the larger the cracks width will be as shown in +igure . below.
Figure 1.1: Critical ocations of Cracking in a beam
- - At A equi equidis distan tantt betwe between en the neutra neutrall axis and the bar surface. -" -" At ) equ equid idis ista tant nt bet betwe ween en the the bar bar.. -/ -/ At C on on the the cor corne nerr of the the bea beam. m. 1.3 CODE PROVISION FOR CRACKS CRACKS
The following sections in )' 0#& %art "."./1 "."./.2.1 /.".." and )' 0#& %art "& 'ections /.0.1 'ection 0 deal with
/
crack phenomena in reinforced concrete structures.
1. SOURCES OF CRACKS IN REINFORCED CONCRETE
-a 3ncorrect detailing is the most common source of cracking in !.C. 'tructures. +or example, wrong or no placement of movement 4oint during detailing and construction. 5ovement 4oints are provided to reduce or eliminate cracking. They permit relative movement to occur without impairing structural integrity. 5ovement 4oints are covered in )' 0#& part ", 'ection 0. The various types of movement 4oints as shown in +igure ." are& i. Contraction 4oints ii. Expansion 4oints iii. 'liding 4oints 2
iv. 6inged 4oints v. 'ettlement 4oints1
Figure 1.2 7oints in !.C. 'tructures
-b Through errors in design calculation and detailing1 -c !einforcing bar 'pacing limits in tension (ones1 -d Amount of reinforcing bars1 -e %oor Construction methods such as& i. 3ncorrect placement of steel leading to insufficient cover ii. 3n adequate cover to reinforcement iii. %oor curing& loss of water can cause shrinkage cracking. 8uring 9
curing the concrete should be kept damp and covered1 -f External physical and mechanical factors such as& i. !estraint of elements against movement causes cracking. The movement might be due to elastic deformation, creep under constant load, shrinkage on drying and settlement of foundations1 Changes in temperature and moisture content. ii. :verloading the structure1 :verloading is brought about by change of function of the building or room i.e. classroom changed to library increases 3mposed oading which was not catered for in the earlier design. iii. 'tructure settlement namely& 8ifferential settlement of foundation can cause cracking,
;
'hrinkage of clays from ground dewatering, 8rying out in droughts, Tree roots causing disruptions,
)' 0#& %art 'ection "."./.2. specifies two methods of crack control. -a imiting maximum bar spacing in the tension (one of members under normal case. 5aximum spacing is specified in )' 0#& %art 'ection /.".." Table /."0 -b 'pecial case by use of the formula provided in )' 0#& %art " 'ection /.0 for assessing the design crack width. 6owever in addition the following should be complied with& • Allowance to be made for stiffening effects of concrete in tension (one, for creep and shrinkage1 • 5inimum areas of steel to control cracking in beams>)' 0#& %art 'ection /.".9./, Table /."91 • )ar spacing control rules to limit crack width to #./mm
0
i. )ars of diameter less than #.29 of the largest diameter should be ignored unless when considering beam side face bars. ii. !efer to +igure ./ below for bar spacing control to limit crack width.
Figure 1.3: )ar spacing rules
a b should not be greater than values specified in Table /."0 of the code ?
' b should not be greater than "9# mm =#### β Clear spacing ≥ f ≤ /## @here β +or b B #. 5aximum a b B /## mm for f y B "9# Dmm" and 5aximum a b B ;# mm for f y B 2;# Dmm" iii. An alternative method of computing a b is& 2=### Clear spacing ≥ f ≤ /## @here f s is the service stress established from equation 0 Table /.# " f A f X / A β Equation 0 of )' 0#& %art 1 ??= iv. ac should not be greater than #.9 of a b v. 3f h exceeds =9# add bars as indicated in +igure ./ above. -'ection /."..".; of )' 0#& %art & ??= b
y
b =
Moment after distributi on
Moment before redistribu tion
s
s =
y
sreq
s prov .
b
#
vi. The si(e of the bar should not be less than that specified in )' 0#& %art 'ection /.".9.2 ab b f y
vii. The maximum clear spacing for slab is given in )' 0#& %art 'ection /."..=. :ther control measures are outlined in /."..0 for slabs 1." CRACK #IDT$ CALCULATION
1.6.1 General The equations for assessing crack width for flexure and direct tension is covered in )' 0#& %art " 'ection /.0. The computation of the crack width for flexure and direct tension shall be treated separately in this course. 1.6.2 Crack widt calculation for r.c. elements in fle!ure
The design crack width, , at any point on the surface of the tension (one of the element is calculated by equation " of the w
cr
code&
wcr =
/a cr
ε m
a cr − c min − !
+ "
.
@hereacr 8istance of the point considered to the surface of the nearest longitudinal bar, m Average strain at the level where the cracking is being considered, Cmin 5inimum cover to the tension steel, h :verall depth of the element, x 8epth of the neutral axis. This is provided the strains st in the tension reinforcement is not greater than #.0 f y " s
Calculatin# $m. An approximate method is given in )' 0# %art " 'ection /.0./ "
+or a rectangular tension (one use ( − )( − ) = − equation / of the code& ( − ) @here 'train at level being considered, calculated while ignoring the stiffening effect of the concrete in the tension (one. a* 8istance from the compression face to the point at which the crack with is being consideredDcalculated. bt @idth of the section at the centroid of the tension steel. Es 5odulus of elasticity of the reinforcement As Area of tension reinforcement ε m
ε
bt
! a
/ " s A s d
F
!
!
3f the whole section is in tension an effective value of -h G x can be estimated by interpolation between the following conditions&
/
-a @here the .A. is at the least compressed face -h>x B h i.e. x B# -b +or axial tension h>x B "h :TE . 3f m H # the section is not cracked. ". The modulus of elasticity of the concrete is taken as #.9 of the instantaneous value. -see Table =." %art " )' 0# /. +or high drying shrinkage ie I #.###;, m should be increased by adding 9#$ of the shrinkage strains. :therwise in normal cases shrinkage may be neglected. %mplications of equation 1& of '( )11*+ ,art 2 6is can be illustrated diagrammatically as shown in +igure .2 below&
2
.
Along the bottom edge in the region of maximum tension a * B h b ( − ! ) = − a constant 'o m becomes. / " A ( d − ! ) value. 'o in the crack width formula acr is the only variable and as the maximum value of acr is midway between bars, the maximum crack width occurs at this point ". 3mmediately below the reinforcing bar. acr is a minimum and equals to Cmin, so crack width B / Cmin m and is the minimum value. /. At the corner acr B ac which is greater than Cmin so the crack in wider. 2. 5oving up the side of the beam m decreases linearly.from a maximum at the corner of the beam to (ero at the . A. The value of a cr decreases to a minimum at the level of the reinforcement and then increases up to the level of .A. 3f no further longitudinal bars are encountered. 3t "
ε m
ε
t
s
9
s
will be found that the maximum value of the crack width occurs about D/ of the distance between the longitudinal steel and the .A. ote that these comments assume that the maximum tensile stress is in the span of the element. @here the maximum stress occurs in the top of the element, e.g. over support, the diagram will be inverted.
Figure 1.: 'ignificant factors in calculating crack width. ;
1.6.& Crack widt calculation for r.c. elements in direct tension 8irect tension as a predominant force is unlikely to occur in normal building structures, but direct tension forces and bending moments may occur. 3n this case neutral axis will be calculated taking into account the tension1 this is more complicated that as carried out in the previous example. 6aving doe this, the calculations for flexural crack width will be as before. The limit of flexure being predominant is where the neutral axis is at the top face of the member shown in +igure .2, i.e. when xB#. The equation for the crack width now becomes wcr =
/a cr
ε m
a cr − c min
+ "
And the tension stiffening effect =
=
bt " / " s A s d
@hen the whole section is in tension it is suggested that the modification suggested in /.0./-b for axial tension is used. The equation for the crack width now becomes "b = − w = /a and / " A @here As is the total area of steel in the section, equally divided between the two faces. cr
ε m
cr ε m
ε
t
s
s
The most obvious case of axial tension is the hoop tension in a circular tank containing water. 3n the Code for liquid> retaining structure the maximum crack width is #."mm. 8esigners of these structures are interested primarily in serviceability limit state of cracking. A calculation is required at ultimate limit state, but if the walls are cracked and water is leaking out, the tank is unserviceable and a factor of safety at ultimate is irrelevant.
0
The have been prepared whereby a designer can select an arrangement of bars to suit a particular wall thickness, tensile force, cover and crack width requirement. To illustrate the calculation procedure the following example uses values from these tables. 1.% E&LES IN CRACK #IDT$ CALCULATION
1.-.1Crack widt calculation for a rectan#ular beam Calculate design crack widths at critical positions for the internal support section of the following two>span continuous beam for which 9$ redistribution has been allowed at the ultimate limit state.
?
Figure 1.!:
+rom the diagram it can be seen that the distance between the bars is /#mm, which is less than the value given in Table /./# of the Code. The formula below the table gives /0.;mm. The distance to the corner of he beam is ;0.9mm, which is also within the limit. - Calculate internal support moment due to service load. M s =
- "" + 9 X # " 0
= 2;/ k.m
-" Calculate properties of equivalent " = " transformed section with " c
s
"#
" c
=
"0 k.
" e
=
2 k.
mm
"
mm
"
from Table =." in %art " so =
α e
d
so
" c
=
"## 2
α e ρ =
so
!
" s
= 2
2 X "9# = #.= /## X ;?#
= α e ρ + [α e ρ ( " + α e ρ ) ] = #.2/=
! = /#" mm
and / = 90? mm
-/ Calculate average surface strain at top of beam f s =
M s A s /
=
2;/ X # ; "9# X 90?
= // .
mm
Figure 1.":
"
"
(< #.0 f = /;0 . mm ) y
"
ε s
ε
=
f s " s
=
=
− ! d − !
// "## X # /
ε s
ε m = ε −
=
220 /00
= #.##9;9
X #.##9;9 = #.##0
bt - − ! " / " s A s - d − !
= #.##0 −
/##- 220 " / X "## X # / X "9# X /00
= #.##0 − #.### = #.##=
-2 Calculate position
crack
widths
at
critical
0a op of beam 5aximum crack width occurs midway between bars, where am =
( ;#
"
c min = 2# mm
+ 09 " ) − "# = 02 mm
wcr =
-top face
/a cr ε m
a cr − c min − !
+ "
=
/ X 02 X .= X # −/
02 − 2# 220 = #./; mm - > #./ mm + "
0b At corner of beam ""
ac = ;0.9 mm
wcr =
/a cr ε m
a cr − c min − !
+ "
=
/ X ;0.9 X .= X # −/
;0.9 − 2# 220 = #./ mm -> #./ mm + "
0c 3n side of beam Critical position approximately -d>xD/ from reinforcement, which is "9?mm from neutral axis a F = /#" + "9? = 9; mm
ε m
"9? a F − ! = ε m = X #.##= = #.###?0 − ! 220
(;9 " + -d − a F " ) − "# =
a cr =
c min = 29 mm
wcr =
- side face
/a cr ε m
a cr − c min − !
+ "
=
/ X "2.9 X ?.0 X # −/
"2.9 − 29 + " 220
= #."= mm
-< #./ mm
"/
-;9 " + "? " − "# = "2.9 mm
+rom these calculations it can be seen that although the spacing of the bars comply with the bar spacing rules. The calculated crack widths in 2-a and -b exceed #./mm. This is one of the cases where the calculation does improve on the rules, and in deep beams this does to be the case. A better condition would be achieved by using /D/" bars
1.-.2Crack widt calculation for a 4beam The section and reinforcement at mid>span of a simply supported T>beam are shown in +igure .=. The total moment at the section due to service loads is .; km. The materials are
"2
midway between the neutral axis and the surface of the tension reinforcement. The properties of the transformed section are computed first. The values for the module of elasticity are as follows& " "## k. !einforcement mm " = X "; = / k. Concrete mm " " "## = = = 9.2 5odular ratio " / The transformed section is shown in +igure .=-b. The neutral axis is located first1 s
=
"
"
c
α c
(
C
="9 ! " + ;?- ! − 29 = "";;?-/## − !
'olve to give x B 0#.?mm
"9
Figure 1.%: -a 'ection& -b transformed section& -c crack locations and dimensions -d stress diagram The moment of inertia about the neutral axis is % !!
0#.? / " = 29# X + ;? X /9.? / 0 2 = /."" X # mm
+
"";;? X "? ."
The stress in the tension steel is& f t =
.; X # ; X "?. X 9.2
/."" X # = "02.0 . mm "
";
0
The strain in the tension steel is& ε s
=
=
"02 .0 "## X # /
.22 X #
/
−
eglect the stiffening effect of the concrete in tension in the flange of the T> beam. 0a Crack widt at A The strain in the concrete at A is& .2"2 X # / X -/9# − 0#.? /## −0#.? / = .=2? X # −
ε =
−
The strain reduction due to the stiffening effect of the concrete in the tension (one, where = = is& aF
/9# mm
"9# -/9# − 0#.? " /
/ X "## X # X 2=" -/## − !
=
?./9 X #
9
−
The average strain at the crack location is therefore& ε m = -.=2? − #.#?2 X # −/ = .;/9 X # −/
The design surface crack width at A where = /=.9 mm a = 90." mm and c is cr
min
"=
wcr =
/a cr ε m
a cr − c min − !
+ "
=
/ X 90." X .;99 X #
/
−
90." −/=.9 /9# −0#.?
+ "
= #."9 mm
0b Crack widt at ' 05i#ure 1.0c The dimension strain ε =.;99 X #
and the average
acr = 9# mm
−/
m
Therefore crack width B #."/mm 0c Crack widt at C C is midway between the neutral axis and the surface of the reinforcement 05i#ure 1.-0c. The location of C is found by successive trail. 3f C is #0.=mm from the neutral axis, it is also #0.=mm from the surface of the bar and a * B 0?.;mm. The strain in the concrete at C is& .2"2 X # / X #0 .= ε = /## − 0# .? / = #.=#; X # −
−
"0
The strain reduction due to the stiffening effect of the concrete is& "9# -/9# − 0#.? " -0? .; − 0#.? /
/ X "## X # X 2=" -/## − 0# .?
=
/.=0 X #
9
−
The average strain at the crack location is& ε m
= -#.=#; − #.#/0 X # −/ = #.;;0 X # −/
The design surface crack width at C where acr B #0.=mm is& wcr =
/a cr ε m
a cr − c min − !
+ "
=
/ X #0.= X #.;;0 X # −/
#0.= −/=.9 + " /9# −0#.?
= #.2 mm
All crack width are less than #./mm and are thus satisfactory 1.-.&Crack widt calculation for an element in direct tension The hoop tension force in the wall of a tank is 9=#kDm. The wall is "## thick "?
