232 CHAPTER 6: CONTROL OF CRACK WIDTHS “Surface cracking is inevitable but, with proper structural design and detailing, the cracks are very narrow and barely perceptible.” -A.M. Neville This chapter describes a series of experiments designed to investigate the effects of bar diameter of skin reinforcing steel on crack widths in the webs of large flexural elements. This chapter also describes experiments to investigate the effects of skin reinforcement on the shear behaviour of flexural elements, and whether the shear strength is related to the vertical spacing between skin reinforcing bars. It is found that the diameter of skin reinforcing bars has a clear effect on crack widths, and that the ACI skin reinforcement provisions should specify a minimum bar diameter. It is also found that the shear strength of large members is not entirely related to the vertical spacing between skin reinforcement, and a modified method by which the SMCFT should calculate s x is recommended, based on the effective depth of the member. 6.1 General The low tensile strength of concrete relative to its compressive strength means that most non-prestressed concrete in service is cracked to some degree. In zones of tension, the steel reinforcement is engaged primarily when a crack occurs, and design of reinforced concrete structures is carried out based on the fact that significant portions of the structure are cracked. However, the widths of these cracks must be limited for appearance, durability and structural integrity. It is important to limit crack width so as to ensure adequate shear behaviour. As crack widths increase, their ability to transfer shear stresses by aggregate interlock decreases. Members in which there is insufficient reinforcement to control crack widths are at risk of developing wide cracks that may result in a premature shear failure. This is of particular concern for very thick members without stirrups, as cracks widths within the web can be considerably greater than those at the level of the steel. It is an aspect that has typically not been addressed by previous studies on crack widths in reinforced
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
232
CHAPTER 6: CONTROL OF CRACK WIDTHS
“Surface cracking is inevitable but, with proper structural design and detailing, the cracks are very narrow and barely perceptible.” -A.M. Neville This chapter describes a series of experiments designed to investigate the effects of bar diameter of skin reinforcing steel on crack widths in the webs of large flexural elements. This chapter also describes experiments to investigate the effects of skin reinforcement on the shear behaviour of flexural elements, and whether the shear strength is related to the vertical spacing between skin reinforcing bars. It is found that the diameter of skin reinforcing bars has a clear effect on crack widths, and that the ACI skin reinforcement provisions should specify a minimum bar diameter. It is also found that the shear strength of large members is not entirely related to the vertical spacing between skin reinforcement, and a modified method by which the SMCFT should calculate sx is recommended, based on the effective depth of the member.
6.1 General
The low tensile strength of concrete relative to its compressive strength means that most
non-prestressed concrete in service is cracked to some degree. In zones of tension, the
steel reinforcement is engaged primarily when a crack occurs, and design of reinforced
concrete structures is carried out based on the fact that significant portions of the
structure are cracked. However, the widths of these cracks must be limited for
appearance, durability and structural integrity.
It is important to limit crack width so as to ensure adequate shear behaviour. As crack
widths increase, their ability to transfer shear stresses by aggregate interlock decreases.
Members in which there is insufficient reinforcement to control crack widths are at risk
of developing wide cracks that may result in a premature shear failure. This is of
particular concern for very thick members without stirrups, as cracks widths within the
web can be considerably greater than those at the level of the steel. It is an aspect that
has typically not been addressed by previous studies on crack widths in reinforced
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
233
concrete, as the focus has generally been on the effects of crack widths on durability and
appearance.
The 1995 version of the ACI-318 code included provisions for crack control based on
crack width limits of 0.4mm (0.016in.) and 0.33mm (0.013in.) for interior and exterior
applications, respectively. However, ACI Committee 318 is now of the opinion that
crack width is not directly related to long-term durability, with cover depth and concrete
quality being of greater importance (ACI Committee 224, 1993). Furthermore, ACI
Committee 318 now believes that, given the inherent variability of crack widths in
concrete structures, it can be misleading to use a design method that purports to
effectively calculate crack widths. Hence, crack control requirements in the ACI code
have evolved over ten years, with the 2005 crack control requirements representing a
considerable departure from the 1995 requirements.
A particular aspect of the 2005 requirements that is worthy of further study is the skin
reinforcement requirements. Skin reinforcement is provided within the web of thick
members so as to control the width of flexural cracks as they extend above the tension
steel. See, for example, the three 20M bars provided on each face of the transfer girders
described in Figure 1-4. However, as discussed in the following sections, the 2005 ACI
318 code no longer requires a minimum bar diameter for skin reinforcement, based on
research suggesting that spacing of skin reinforcing bars is the primary variable affecting
flexural crack widths in the webs of thick members. It is thus possible to use, for
example, D4 wires in place of No. 5 skin reinforcing bars, at the same spacing, and still
meet the 2005 ACI 318 skin reinforcement requirements. This is despite the fact that the
area of steel has been reduced by 87%.
Based on the above discussion, the intention of this chapter is to investigate the skin
reinforcement requirements of the 2005 ACI-318 code. The effects of crack control
reinforcement on the shear behaviour of thick slabs will also be investigated, and the
ability of the 2004 CSA A23.3 code to account for these effects will be assessed.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
234
6.2 Crack Control in the ACI-318 Code
6.2.1 Crack Control at the Level of the Tensile Steel
In the 1995 ACI-318 code, crack control requirements at the level of the tensile
reinforcement were based on the well-known Gergely-Lutz expression (Gergely and Lutz,
(1968)), which was derived from regression analyses on data from several crack width
studies:
s3
bb RfAt0.076w = (6.1)
where wb = crack width on the bottom (soffit) of the member, tb = cover from bottom of member to centre of lowest level of steel= dc, R = (h-kd)/((1-k)d) = factor to account for strain gradient (ratio between strain at bottom of member and strain at level of reinforcement) fs = steel stress A = 2b’(h-d)/m = effective area of concrete in tension surrounding the reinforcement b’ = width of member at centroid of steel m = number of tensile reinforcing bars
In their analysis of flexural crack widths at the level of the reinforcement and on the
bottom face of the member, Gergely and Lutz found that:
“1. The steel stress is the most important variable. 2. The cover thickness is an important variable but not the only consideration. 3. The bar diameter is not a major variable. 4. The size of the side crack width is reduced by the proximity of the compression zone in flexural members. 5. The bottom crack width increases with the strain gradient. 6. The major variables are the effective area of concrete, Ac, the number of bars, m, the side or bottom cover, and the steel stress.”
Point 3 appears to be counterintuitive, but the effect of bar diameter is, in fact, taken into
account by other parameters. Reducing the bar diameter by using a large number of
small diameter bars (at a reduced spacing) would be expected to produce smaller crack
widths than would the use of a small number of large diameter bars at the same steel area
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
235
and steel stress. This effect is accounted for by the variable m. The bar diameter also
affects crack widths when the total steel area is reduced by using smaller diameter bars,
and this effect is accounted for by use of the steel stress term fs, which will increase.
Equation (6.1) was developed to calculate the most probable crack width on the bottom
of the flexural member. A second expression was also derived to calculate the most
probable crack width on the side face of the member at the level of the reinforcement.
The ACI implementation of the Gergely-Lutz expression used R=1.2 and required the
calculation of a “z-factor” as outlined below, in which z was limited to 175kips/in for
interior exposure and 145kips/in for exterior exposure. These limits correspond to crack
widths of 0.016 and 0.013in (0.4 and 0.33mm). The CSA-A23.3 code also uses the z-
factor for controlling crack widths at the tensile reinforcement, with z being limited to
30,000N/mm and 25,000N/mm for interior and exposure conditions respectively.
3cs Adfz = (6.2)
A challenge posed by the z-factor is that it promotes the use of smaller covers below the
level of the reinforcement so as to reduce dc. Yet, it is generally understood that larger
covers are very effective at improving long-term durability, possibly even more effective
than limiting crack widths (ACI Committee 224 (1993)). Furthermore, while a range of
dc values from 0.75-3.31in. (19 – 84mm) was used by Gergely and Lutz to derive their
expression for bottom-face cracking, there were only three data points with a cover
greater than 2.5in (64mm). As such, it can be difficult to meet the requirements of Eq.
(6.2) at covers exceeding 2in (50mm). The commentary to Clause 10.6.1 in the CSA
code suggests that in situations with large covers, it is not necessary to use a value for
clear cover greater than 50mm when calculating dc and A. In these situations, it is better
to allow thicker covers at the expense of wider surface crack widths. In these cases,
crack widths at the level of the steel will remain small, with the wider surface crack
widths therefore becoming essentially an aesthetic issue. This simple solution was not
implemented in the ACI code.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
236
Frosch (1999) noted the difficulty in meeting the requirements of the z-factor for larger
covers, and developed a new approach to crack control at the level of tension
reinforcement. Reviewing the work of Broms (1965), Frosch noted that the spacing of
cracks depends on the concrete cover, and calculates the crack spacing as follows:
*sc dΨS = (6.3)
where Sc = crack spacing, d* = controlling cover distance (Figure 6-3) Ψs = crack spacing factor =1.0 for minimum crack spacing =1.5 for average crack spacing =2.0 for maximum crack spacing Noting that the crack width at the level of the reinforcement wc=εsSc, Frosch derived an
equation for the maximum crack width on the bottom of the beam as follows:
22c
s
sc (s/2)dβ
Ef
2w += (6.4)
where Es = Young’s Modulus of steel, β = equivalent to Gergely-Lutz R-value = 1.0 + 0.08dc as a simplification.
Frosch rearranged this equation to solve for the permissible bar spacing, s, as a function
of the permissible maximum crack width, wc:
2c
2
s
sc dβ2f
Ew2s −⎟⎟
⎠
⎞⎜⎜⎝
⎛= (6.5)
A permissible crack width of between 0.016 and 0.021in (0.4-0.53mm) was chosen by
Frosch, a service load steel stress of 0.6fy was assumed, and simplified design curves
generated as shown in Figure 6-2 (Frosch Design Curves).
Figure 6-1: Controlling Cover Distance
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
In implementing the design recommendation of Frosch (1999), ACI Committee 318
chose more conservative design curves as shown in Figure 6-2 (ACI Design Curves), and
the expression for calculating minimum bar spacing is shown below (in ksi, inch units).
The ACI expression is formulated in terms of the clear cover, cc, and entered use in the
1999 ACI-318 design code.
( )scs
f36122.5cf
540s ≤−= (6.6)
where s = centre-to-centre spacing of tension reinforcement fs = calculated stress in longitudinal reinforcement at service loads (in ksi). In lieu of direct calculation, it is permitted to take fs=60% of the specified yield strength cc = clear cover from surface to tensile steel.
ACI Committee 318 now believes that, given the inherent variability of crack widths in
concrete structures, it can be misleading to use a design method that purports to
example, notes that crack spacing (and, hence, crack widths) can vary by a factor of 2.
The 1978 CEB-FIP code (CEB 1978) suggests that the 95th percentile of crack widths is
equal to 1.7 times the average crack width. A distinction is no longer made between
interior and exterior exposure conditions, as the committee has accepted that crack widths
are not directly related to durability.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
238
Due to changes in φ-factors and load combinations in the 2002 version of the ACI code,
Equation (6-6) was reformulated for the 2005 version of the code to take into account the
higher service load stresses in flexural steel:
( )scs
f40122.5cf
15(40)s ≤−= (6.7)
where fs = calculated stress in longitudinal reinforcement at service loads (in ksi). In lieu of direct calculation, it is permitted to take fs=2/3 of the specified yield strength
Despite an increase in service load stresses of 10%, the required spacing of the tensile
reinforcement was not changed. For the case of a 2in. (50mm) clear cover, the required
spacing in both versions of the expression is 10 in. (254mm). The maximum spacing for
tensile reinforcement is 12 in. (300mm).
6.2.2 Skin Reinforcement
It has long been recognized that flexural crack widths can increase in width as the cracks
extend into the web of a deep member (Figure 6-3), and it is argued in Chapter 5 that this
is the primary cause of the size effect in shear. The 1977 version of the ACI-318 code,
for example, required that an area of steel equal to 10% of the tensile reinforcement be
distributed along the side faces of deep members to control crack widths in the web.
Figure 6-3: Side-Face Cracking in Large Beams (adapted from Frantz and Breen (1980))
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
239
The logic of this requirement in the 1977 ACI code is weak, however, as identified by
Frantz and Breen (1976, 1980a,b). For a given factored moment, increasing the effective
depth of a member will result in a lower area of tensile steel and a lower area of skin
reinforcement, where in fact a deeper section would logically require at least the same, or
probably additional, skin reinforcement. Thus, Frantz and Breen carried out an extensive
series of tests in which crack widths in deep flexural members with various skin
reinforcement configurations were measured and analyzed, and recommended a design
procedure for skin reinforcement. They found a very clear relationship between the skin
reinforcement ratio, ρsk, and the maximum crack width in the web. They recommend that
the expression described in Figure 6-4 be used when determining the required amount of
skin reinforcement.
For 30 < d < 100inches, ρsk = 0.00024 (d-30) d > 100inches, ρsk = 0.011 + 0.000058d
Figure 6-4: Frantz and Breen (1980a) Skin Reinforcement Requirements
ACI 318-02
The results of Frantz and Breen’s study formed the basis of the ACI skin reinforcement
requirements up until the 2002 ACI-318 design code. As shown in Figure 6-5, it is
reflected in the requirement that that the spacing, ssk, between skin reinforcing bars not
exceed 1000Ab/(d-30), where Ab is the area of an individual bar. In previous versions of
the code this requirement was formulated such that the area of skin reinforcement per
foot height of web per side exceed 0.012(d-30) in2/ft. Both expressions, however, are
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
240
mathematically equivalent. No distinction was made between interior and exterior
exposure conditions, unlike the z-factor for the flexural steel.
10.6.7 — If the effective depth d of a beam or joist exceeds 36in., longitudinal skin reinforcement shall be uniformly distributed along both side faces of the member for a distance d/2 nearest the flexural tension reinforcement. The spacing ssk between longitudinal bars or wires of the skin reinforcement shall not exceed the least of d/6, 12in. and 1000Ab/(d-30). It shall be permitted to include such reinforcement in strength computations if a strain compatability analysis is made to determine stress in the longitudinal bars or wires. The total area of longitudinal skin reinforcement in both faces need not exceed one-half of the required flexural tensile reinforcement.
The ACI 318-02 maximum spacings, ssk, are shown in Figure 6-6a) as a function of the
beam depth, d, for various bar diameters. The resulting skin reinforcement ratios, ρsk, are
shown in Figure 6-6b) for a value of cc+0.5db=2in, where ρsk is calculated as Abar/(ssk x
(2cc+db)), as opposed to the method by which Frantz and Breen calculate ρsk. This is the
method used to calculate ρsk in the CSA code, and ρsk calculated using this method can be
converted to Frantz and Breen’s ρsk by multiplying it by the ratio (No. of bars per side
/(No. of bars per side + 1). The Frantz and Breen expressions are also shown in Figure
6-6b), and have been modified for the different method of calculating ρsk.
Inspection of Figure 6-6a) will show that, for all bar sizes, the maximum spacing initially
increases as a function of the depth by virtue of the d/6 spacing limit, until the point at
which the 1000Ab(d-30) limit governs. At this point, the required spacing decreases.
The d/6 limit thus serves to prevent designs with both large bar spacings and large bar
diameters at effective depths close to 36in. An efficient use of steel would result by
using No. 3 bars as skin reinforcement for depths from 36-48in., No. 4 bars for depths
from 48-60in., and No. 5 bars for depths exceeding 60in. (Figure 6-6b)).
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
241
0
2
4
6
8
10
12
0 12 24 36 48 60 72 84 96 108 120
Effective Depth, d
Req
uire
d Sp
acin
g of
Ski
n R
einf
orce
men
t, s
0
50
100
150
200
250
300
0 1000 2000 3000
D4D10
No. 3
No.4
No. 5
d/6 Limit
Skin Rft. not req'd in either code
(in.)
(mm)
(in.) (mm)1000Ab
(d-30)Limit
d=55in.
ACI-318-05
c c =3in.
c c =2in.
c c <1.2in.
ACI-318-02
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 12 24 36 48 60 72 84 96 108 120
Effective Depth, d
Skin
Rei
nfor
cem
ent R
atio
, ρsk
(%)
0 1000 2000 3000
(in.)
(mm)
D4
D10No. 3
No. 5CSA Exterior
CSA Interior
Range of ρsk
for d=55in.
d=55in.
No. 4
Frantz&Breen -D4
Frantz&Breen -No.5
ACI-318-02
Figure 6-6: ACI 318-02 and ACI-318-05 Skin Reinforcement Requirements
ACI 318-05
In an attempt to unify the flexural reinforcement spacing requirements of clause 10.6.4
and the skin reinforcement spacing requirements of clause 10.6.7, Frosch (2002)
undertook a review of Frantz and Breen’s study. In the review, Frosch suggested that the
spacing of skin reinforcing bars has a far greater effect on the crack widths in the web of
flexural elements than does the diameter of the bars. In deriving a new design method for
skin reinforcement, Frosch (2002) predicts that “the bar size does not significantly affect
crack width” and that “any size bar can be used successfully.” Frosch also quotes
Gergely and Lutz, who found that the bar diameter was not major variable. The clear
effect of ρsk found by Frantz and Breen was thus suggested by Frosch to be a result
primarily of changes in the bar spacing rather than the bar diameter.
a) b)
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
242
Based on further analytical work, Frosch found that the crack widths on the side face of
deep flexural elements are related to their distances from the bars. As shown below,
Frosch developed an equation for the side face crack width, ws, at a distance x below the
neutral axis (Figure 6-7):
*xxss dεΨw = (6.8)
where ws = crack width located a distance x below the neutral axis, εx = longitudinal strain at a distance x below the neutral axis dx* = controlling cover distance at a distance x below the neutral axis = 2
s2 d+)'x(
Ψs =1.0 for minimum crack spacing, =1.5 for average spacing, =2.0 for max spacing x` =vertical distance from point x to nearest reinforcing bar
The depth of the neutral axis, c, is calculated based on an elastic analysis of the
transformed section.
Figure 6-7: Effect of Skin Reinforcement According to Frosch (2002)
The effect of side face steel on crack widths is modelled by the reduction of the
controlling cover distance, dx*. Cracks are thus predicted to be locally wider at s/2 from
a skin reinforcing bar, and locally narrower directly beside a bar (where x’=0). Noting
that side face cracks exhibit considerable variability in widths, it was suggested that the
largest crack width can be calculated using Ψs=2, and the narrowest crack width
calculated using Ψs=1. Showing generally good agreement between predicted and
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
243
experimental crack widths from the Frantz and Breen study, Frosch recommended that
Equation (6-7) be applied to the design of skin reinforcement in addition to flexural
reinforcement. Clause 10.6.7 was thus rewritten for the 2005 version of the ACI code
based on the work of Frosch (2002). See Figure 6-8.
Code Commentary 10.6.7 — Where h of a beam or joist exceeds 36in., longitudinal skin reinforcement shall be uniformly distributed along both side faces of the member. Skin reinforcement shall extend for a distance h/2 from the tension face. The spacing s shall be as provided in 10.6.4, where cc is the least distance from the surface of the skin reinforcement or prestressing steel to the side face. It shall be permitted to include such reinforcement in strength computations if a strain compatibility analysis is made to determine stress in the individual bars or wires.
R10.6.7 — For relatively deep flexural members, some reinforcement should be placed near the vertical faces of the tension zone to control cracking in the web.10.20,
10.21 (See Fig. R10.6.7.) Without such auxiliary steel, the width of the cracks in the web may exceed the crack widths at the level of the flexural tension reinforcement. This section was modified in the 2005 edition to make the skin reinforcement spacing consistent with that of the flexural reinforcement. The size of the skin reinforcement is not specified; research has indicated that the spacing rather than bar size is of primary importance.10.21 Bar sizes No. 3 to No. 5 (or welded wire reinforcement with a minimum area of 0.1 in.2 per foot of depth) are typically provided. (emphasis added)
Crack Widths at Mid-Height (mm) Max. Crack Width (mm)
Zurich Visual
6.3.4 Crack Widths as a Function of Steel Stress
Maximum Crack Widths
The maximum single crack widths measured at any point in the 48in. wide midspan
regions of specimens L-10N1 + L-10N2 (no skin reinforcement), L-50N1 (D4@10in.), L-
50N2 (15M@10in.) and L-20D ([email protected].) are plotted in Figure 6-12a) versus the
steel stress at midspan. The maximum single crack widths measured at any point in the
48in. wide midspan regions of specimens L-10H (D4@10in.) and L-10HS (15M@10in.)
are plotted in Figure 6-12b). The crack widths in Figure 6-12 were measured visually
using a crack comparator gauge at load stages. The steel stresses were calculated based
on an elastic analysis of the section at midspan. Since all specimens in Figure 6-12 other
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
252
than L-20D and L-10HS failed prior to the steel stresses reaching the maximum service
load steel stress of 67% of fy, linear extrapolations have been added to the data. Also
shown is the ACI 318-99 maximum crack width of 0.016in. (0.4mm) for interior
exposure. ACI 318-99 required that crack widths not exceed this width at service loads.
Figure 6-12 Maximum Crack Widths in 48in. Wide Midspan Region, Measured Visually
It can be clearly seen in Figure 6-12 that unacceptably wide cracks occurred in L-10N1
and L-10N2 under service loads. For example, at the maximum service load steel stress
(0.67fy) the widest crack in L-10N1 would be expected to be about 0.6mm (0.024in.).
Indeed, the maximum crack widths in L-10N1 and L-10N2 reached 0.4mm (0.016in.) at a
steel stress of about 50% of fy. Clearly, members of this depth are in need of additional
reinforcement to control crack widths at service loads.
However, the provision of D4 bars spaced at 10in. did not provide adequate control of
crack widths. Based on the linear extrapolations in Figure 6-12, it is expected that the
maximum crack width in both L-50N1 and L-10H would reach 0.45mm at a steel stress
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
253
of 67% of the yield stress. Unacceptably wide cracks thus occurred in the midspan
regions of L-50N1 and L-10H, even though these regions contained skin reinforcement
designed to conform to ACI 318-05.
Using 15M bars spaced at 10in. successfully controlled crack widths in specimens L-
50N2 and L-10HS. Compare the crack widths measured in L-50N2 to those measured in
L-50N1, and the crack widths measured in L-10HS to those measured in L-10H. The
maximum measured crack width in L-10HS was 0.25mm at a steel stress of 31.8ksi
(357MPa), representing 86% of the yield stress. Using 10M bars at 8.9in. also
successfully controlled crack widths in L-20D. Any differences in the crack widths
between L-20D and L-50N2 fall within the precision of the crack comparator gauge.
Crack Widths at Midheight
The averages of the crack widths at the midheight (h/2) of the 48in. wide midspan regions
are plotted in Figure 6-13 and Figure 6-14 versus the midspan steel strains. The averages
of the crack widths measured using the comparator gauge are plotted in Figure 6-13,
while the average based on the zurich target data are plotted in Figure 6-14.
The maximum crack widths in the middle 48in. of the specimens did not occur at mid-
height, but, rather, were found to occur at between 450mm and 670mm (18-26.5in.) from
the bottom face. Nevertheless, the average crack widths at the midheight of the beams
without skin reinforcement were unacceptably wide. Provision of D4 skin reinforcement
reduced the average crack widths at mid-height, though the 15M skin reinforcement was
far more effective.
The analysis of crack widths plotted in Figure 6-12, Figure 6-13 and Figure 6-14 has
shown that that the D4 skin reinforcement had only a small impact on flexural crack
widths. Crack widths were still unacceptably wide in the specimens with D4 skin
reinforcement, even though this skin reinforcement met the requirements of ACI 318-05.
Skin reinforcement consisting of 15M bars at the same spacing and bar cover, however,
was extremely effective at controlling crack widths.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
254
Figure 6-13 Average Crack Width at Midheight of Midspan Region, Measured Visually
Figure 6-14 Average Crack Widths at Midheight Midspan Region, -Zurich Target Data
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
255
6.3.5 Crack Widths at Maximum Service Load Steel Stress
While crack widths below the maximum service load are useful to investigate, the crack
widths at the ACI maximum service load steel stress of 0.67fy are the most useful. Since
a number of the specimens shown in the previous figures failed at loads below those
corresponding to a steel stress of 0.67fy, it is necessary to estimate the crack widths at
fs=0.67fy had the beams not failed. This was accomplished using the extrapolations
shown in Figure 6-12, and the resulting estimated maximum crack widths at fs=0.67fy are
shown in Figure 6-15. Extrapolated crack widths for specimens L-20N1, L-20N2, L-
40N1 and L-40N2 are also shown. Using an extrapolation is an appropriate method for
estimating the maximum crack width at fs=0.67fy since a stable crack pattern formed in
the specimens, in which existing cracks widened with increasing load, with few new
cracks forming.
Figure 6-15 clearly shows that at a constant clear cover and a constant spacing of 10in.,
decreasing the skin reinforcement ratio by decreasing the bar diameter can result in
unacceptably wide cracks at service loads. At a constant spacing and cover, the bar
diameter thus has a very clear effect on flexural crack widths.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 0.4 0.8 1.2 1.6 2Skin Reinforcement ratio, ρsk
0
0.004
0.008
0.012
0.016
0.02
0.024
0.028L-10N SeriesL-20N SeriesL-40N Series
Max
imum
Exp
ecte
d C
rack
Wid
th a
t fs=
0.67
f y (mm) (in.)
Normal f'cSeries
High f'cSeries
Range in Crack Widths at ρsk=0 c=2in (50mm)
Figure 6-15: Expected Crack Widths at fs=0.67fy
(L-50N Series, L-20D) (L-10H, L-10HS)
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
256
The concrete strengths in the specimens without skin reinforcement ranged from 28.1-
41MPa (4075-5945psi), resulting in neutral axis depths of 393-426mm (15.5-16.8in.).
Thus, some of the scatter in crack widths at ρsk=0 might be expected to have occurred due
to differences in the concrete strength. An increase in concrete strength is associated
with a decrease in neutral axis depth, resulting in slightly larger longitudinal strains
below the neutral axis and consequent crack widths at otherwise constant crack spacings
and steel strains. While the differences in concrete strengths between the specimens
would result in only very small differences in longitudinal web strains, this effect is
evident in the data at ρsk=0 and ρsk=0.095%, in that there is a general trend of increasing
crack widths with increasing concrete strengths. An opposite effect is noted, however, at
ρsk=0.67%. Other researchers, however (for example, Hognestad (1962)) have not found
that concrete strength affects crack widths beyond the initial cracking load.
In specimens of similar concrete strength, the maximum expected crack width decreases
by about 20% in specimen L-50N1 (reinforced with D4 wires) relative to specimens L-
10N1 and L-10N2 (with no skin reinforcement). Nevertheless, it is instructive to note
that the crack widths in the specimens reinforced with D4 wire fall within the scatter of
maximum crack width data from specimens without skin reinforcement. It thus
interesting to note that provision of D4 skin reinforcement at a spacing of 10in. was about
as effective at reducing crack widths as reducing the concrete strength by about 30%.
6.3.6 Predictions of Web Crack Width
Crack Width Profiles
The widths of the single widest flexural crack in the six specimens shown in Figure 6-11
are plotted in Figure 6-16. Using the cracking model developed by Frosch (2002) and
described by Equation 6-8 and in Figure 6-7, it is possible to predict the measured crack
width profiles shown in Figure 6-16.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
257
The predicted widest (Ψ=2) and narrowest (Ψ=1) cracks in specimens L-10N1 and L-
10N2 are shown in Figure 6-16a) and b). It can be seen that Frosch’s model is generally
good at calculating the overall width profile of the widest cracks in these specimens.
While the maximum crack width was observed to occur lower in the section than what is
predicted by Frosch’s model, this maximum width was calculated reasonably accurately
using Ψ=2 in Equation 6-8.
The predicted crack width profiles for the specimens with D4 reinforcement are shown in
Figure 6-16c) and d). The predicted crack width profiles for the specimens with 15M
reinforcement are shown in Figure 6-16e) and f). Two sets of predictions have been
generated. The first set, shown in solid lines, is for the case where the controlling cover
distance, dx* in Equation 6-8, is calculated based on skin reinforcement provided at
10inch spacing. Since all four specimens in Figures 6-18c)-f) have skin reinforcement
present, these are the predictions that apply to these beams. The second set, shown in the
dashed lines, was generated neglecting the skin reinforcement, and resemble the crack
width predictions for L-10N1 and L-10N2. The predicted effect of reducing the
controlling cover distance, dx*, by providing skin reinforcement can be clearly seen by
comparing these two sets of predictions. It is predicted (Figure 6-7) that cracks will
locally widen between skin reinforcing bars, but the overall width is predicted to be
considerably reduced.
It can be seen in Figure 6-16e) and f) that the measured crack widths in the specimens
with 15M skin reinforcement are considerably smaller than those measured in L-10N1
and L-10N2 (Figure 6-16a) and b)). For these two beams, calculating dx* based on the
presence of the skin reinforcement resulted in accurate predictions of the maximum crack
width from Frosch’s model. While the precision of the crack comparator gauge did not
allow for the detailed mapping of the variation in crack widths over the height of
specimens L-10HS and L-50N2, the maximum crack width was accurately calculated
using Ψs=2.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
258
Figure 6-16: Widest Crack in Midspan Region and Crack Width Predictions by Eq. 6-8
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
259
Figure 6-16c) and d) indicate that calculating dx* based on the presence of skin
reinforcement did not accurately predict the crack widths and overall crack width profiles
in the two specimens with D4 skin reinforcement. For L-10H and L-50N1, the measured
crack widths fell well outside the band of expected crack widths defined by 1<Ψs<2. For
example, the maximum measured width in these specimens was 0.3mm, and in both this
maximum width was observed to occur directly beside a skin reinforcing bar, where
Frosch’s model predicts a local narrowing of the crack. By observation, in fact, it can be
seen that the predicted crack width profiles generated neglecting the presence of the D4
bars (the dotted lines) more accurately predict the measured crack width profiles.
When using Frosch’s model to predict the flexural crack widths in the two beams with
D4 skin reinforcement, it appears that it is more accurate to neglect the skin
reinforcement than it is to consider it in the model.
Experimentally Determined Ψs Values
At each load stage, it is possible to calculate an experimentally-determined Ψs value. It
can be calculated by dividing the maximum crack width measured at any point on the
crack by the largest width on the predicted crack width profile generated using Ψs = 1.
Separate experimental Ψs values can be determined by both considering and neglecting
the skin reinforcement.
To further examine how the crack widths measured in L-10H and L-50N1 lie outside the
range of crack widths predicted by Frosch’s model, Figure 6-17 was created. Each data
point represents the average of experimentally-determined Ψs values for load stages
where the steel stress exceeded one-third of the yield stress. This limit on the steel stress
was chosen to ensure a stable crack pattern had formed. This figure shows that the
maximum measured crack widths in specimens with ρsk=0.4% and 0.67% lie within the
scatter band defined by Frosch, with experimental Ψs values ranging from 1.3 to 1.6.
Likewise, the maximum measured crack widths for ρsk=0% lie within the scatter band.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
260
The maximum measured crack widths in the specimens reinforced with D4 wire lie well
outside the expected range of Ψs values, with Ψs values of 2.4 and 2.5. However, the key
to this figure is to note that the maximum measured crack widths in these specimens
actually fall within the range of Ψs values generated assuming no skin reinforcement. In
this case, the experimental values of Ψs are both 1.23. Thus, when using Frosch’s model
to predict the crack widths in the specimens with D4 skin reinforcement, assuming that
no skin reinforcement is present produces more accurate predictions of the maximum
crack width than assuming skin reinforcement to be present.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 0.2 0.4 0.6 0.8 1.0
Skin Reinforcement Ratio, ρsk (%)
Expe
rimen
tal C
rack
Spa
cing
Fac
tor, Ψ
s
Range of Ψs in Frosch Model
Assuming no Skin Reinforcement Provided
Assuming Skin Reinforcement Provided
Figure 6-17: Experimentally Determined Values of Ψs
2.5 2.4
1.23
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
261
6.3.7 Another Look at Frantz and Breen
Average and Maximum Crack Widths Frantz and Breen tested a total of 44 inverted-T specimens, studying numerous
combinations of web depths, flange depths, reinforcement ratios ρw and ρsk, web widths,
bar diameters and clear covers. The majority of them were “reduced segment” specimens,
which were 72in. (1830mm) long, and which were loaded using hydraulic rams to
directly apply tension to the main reinforcing bars and compression above the neutral
axis. The location of the compression force was determined by an analysis of the cracked
transformed section. The specimens were thus loaded in pure moment, and were tested
upside-down to facilitate observations of the cracks.
Despite the impressively large number of specimens in the test program, only one series
of five specimens was tested in which all variables were kept constant, with only db being
varied. Results from this series of tests are summarized in Figure 6-18, in which average
and maximum web crack widths are plotted as a function of steel stress. In these
specimens, the centre-to-centre bar spacing, clear cover and section dimensions were kept
constant, while the skin reinforcement was varied from Swedish Grade 77 deformed
6mm bars (Specimen A-7) to No. 3 bars and No. 4 bars (Specimens A-8 and A-9). These
bars resulted in ρsk values of 0.34%, 0.81% and 1.34%. Two specimens, A-1 and A-2,
were tested without skin reinforcement.
In reviewing the data summarized in Figure 6-18, Frosch (2002) notes that there was
essentially no difference between the measured crack widths in the specimens reinforced
with No. 3 bars and No. 4 bars, and thus concludes that any bar size can be successfully
used to control crack widths. The considerably wider cracks in the specimen reinforced
with 6mm bars were attributed by Frosch to the fact that the bars were Swedish, with
deformations that were less pronounced than the No. 3 and 4 bars, and which, according
to Frosch, did not conform to the ASTM A615 standard on deformations. Thus, the
increase in crack widths was attributed by Frosch to differing bond properties between
the Swedish and US bars. The fact that smaller deformations can be used on smaller bars
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
262
because of the larger ratio of perimeter to cross-sectional area (and thus inherently better
bond properties) is not addressed. Also note that the accuracy of the gauge used to
measure the crack widths in Frantz and Breen’s experimental program was 0.001in.
Hence, the maximum crack widths in A-9 (No. 4 bars) were larger than those in A-8 (No.
3 bars) by only one increment on the gauge.
0.00
0.10
0.20
0.30
0.40
0.50
25 30 35 40 45Bar Stress
Cra
ck W
idth
0
4
8
12
16
20170 210 250 290
#3 Bars ρsk=0.81%
(mm)
(x10
-3in
.)
(ksi)
(MPa)
#4 Bars ρsk=1.41%
6mm Bars ρsk=0.34%
No Skin Rft. ρsk=0%
a) Average Web Crack Width
0.00
0.10
0.20
0.30
0.40
0.50
25 30 35 40 45Bar Stress
Cra
ck W
idth
0
4
8
12
16
20170 210 250 290
#3 Bars
(mm)
(x10
-3in
.)
(ksi)
(MPa)
#4 Bars
6mm Bars
No Skin Rft.
Crack Width Gauge Resolution=0.001in.b) Maximum Web Crack Width
Figure 6-18: Effect of Bar Diameter on Crack Width in Web (Frantz and Breen, 1976)
Crack Magnification Ratio and Extension of Cracks into the Web The ratios of the average web crack width (wweb) to the average crack width at the level
of the steel (wsteel) for all of Frantz and Breen’s specimens are plotted in Figure 6-19a) for
a steel stress of 35ksi. Frantz and Breen refer to wweb/wsteel as the crack magnification
ratio. The crack magnification ratios for specimens A-1, A-2, A-7, A-8 and A-9 have
been identified with open symbols. The crack magnification ratios for specimens A-7, A-
8 and A-9 are plotted in Figure 6-19b) at steel stresses of 25, 30, 35 and 40ksi. Figure
6-20 plots the percentage of the cracks at the level of the steel that extended into the web
in specimens A-1, A-2, A-7, A-8 and A-9, and crack diagrams have been reproduced.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
263
It can be seen in these figures that, in fact, using #4 bars instead of #3 bars affected
cracking in the webs in two important ways. Firstly, the crack magnification ratio was
lower in A-9 than in A-8 for all steel stresses. Although crack widths in the webs of A-8
and A-9 were similar, cracks at the level of the steel in A-9 were, on average, about 15%
wider than they were in A-8. The use of #4 bars reduced their widths in the web to
widths that were similar to the widths measured in A-8, resulting in the reduced crack
magnification ratios. It is interesting to note that a curve of best fit drawn through the
cloud of data points in Figure 6-19a) might be drawn almost right on top of the curve
connecting the points for A-7, A-8, A-9 and the average of A-1 and A-2.
Secondly, and more importantly, a considerably greater number of cracks extended into
the web in specimen A-9 than in A-8. Frantz and Breen report that twenty-eight cracks
occurred at the level of the steel in specimens A-7, A-8 and A-9 (counted on both sides of
the specimen), while twenty-six were counted in A-2 and nineteen in A-1. Sixteen of
these cracks extended into the web in A-9, versus ten in A-8, nine in A-7 and six in
specimens A-1 and A-2. The crack patterns reproduced in Figure 6-20 clearly indicate
how closely spaced the cracks were in A-9 as compared to other specimens. This
behaviour corresponds with Frantz and Breen’s findings with regards to all of their
specimens, whereby the percentage of flexural cracks that extended into the web
increased in direct proportion to the total area of skin reinforcing bars provided.
The fact that A-9 had 60% more cracks in the web than A-8, yet had web cracks that
were similar in width, leads to the conclusion that the longitudinal strain in the web was
considerably greater than it was in A-8. Specimens A-1 through A-9 were all loaded in a
similar manner, with the location of the hydraulic ram modified based on an elastic
analysis of the cross-section. Thus any differences in the strain profiles would be due to
differences in the locations of the neutral axes due to differences in the concrete strengths.
The concrete strength of A-9 was 5231psi (36.1MPa) and was greater than the concrete
strength of A-8, which was 4580psi (31.6MPa). However, this difference resulted in a
neutral axis depth in A-9 that was 97% of the neutral axis depth in A-8, an
inconsequential difference.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
Notes:(1) day of test(2) Calc. at d from face of loading plate, incl. self-weight (vexp = Vexp/bwd)(3) Average of L-10H, 10N1, 10N2, 20N1, 20N2, 40N1, 40N2, 50N1, 50N2, 50N2-R
0
100
200
300
400
500
600
700
800
0 2 4 6 8 10 12Mid-Span Deflection, Δ (mm)
App
lied
Load
(kN
)
ACI
SMCFT
L/800
4-30Mρw=0.84%
2-10M @ 225 Failure LoadL-20DR
L-20D
0
10
20
30
40
50
60
0.0 1.0 2.0 3.0 4.0Mid-Span Displacement, Δ (mm)
App
lied
Load
(kN
)
L/480
S-20D2
S-20D1
ACI -D2ACI -D1
SMCFT -D2SMCFT -D1
2-10Mρw=0.82%
Figure 6-24: Load vs. Midspan Displacement Curves, L-20D, S-20D1, S-20D2
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
274
Figure 6-25: Failure Crack patterns in L-20D and L-20N1
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
275
It is interesting to note that the ratio of Δ/0.5L to the shear strain, γult, in L-20D was 21%,
while the average for the specimens without crack control steel was 31% (with the lowest
value being 23%). Less of the total mid-span displacement was caused by shear strains in
L-20D and it exhibited a higher shear stiffness than the specimens without crack control
reinforcement. See Figure 6-26, in which the measured shear stress vs. shear strain
curves at the quarterspans are plotted for specimens L-20DR and L-20N2.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1Shear Strain (mm/m)
Shea
r Str
ess,
v (M
Pa)
L-20DR
L-20N2
L-20D Failure on East End
Figure 6-26: Shear Stress vs. Shear Strain for Specimens L-20DR and L-20N2,
Measured at Quarterspan
The stiffer response of L-20D compared to specimens without crack control steel can also
be seen by comparing the midspan deflections. At an applied load of 500kN, the
measured midspan deflection for L-20D was 7.2mm, and this is 22% less than the
average measured midspan deflection for L-20N1 and L-20N2 at the same load. Yet the
transformed moment of inertia for L-20D was 94,300x106mm4, versus an average value
of 95,500x106mm4 for L-20N1 and L-20N2.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
276
6.4.4 Effect of Dowel Action and Aggregate Interlock
It is suggested that the considerable increase in shear strength exhibited in L-20D and L-
20DR was primarily a result of the formation of narrower cracks that were better able to
transfer shear due to aggregate interlock. A portion of the increase can also be attributed
to increased dowel resistance.
Bhide and Collins (1987) note that a large number of small size bars offers a larger dowel
resistance than a small number of large bars, for the same reinforcement ratio and dowel
displacement, where the dowel displacement is calculated as “…the component of the
crack width in the direction at right angles to the reinforcing bar under consideration.”
Put another way, a large number of small bars offers a stiffer response than does a small
number of large bars. Thus, beams in which some of the longitudinal steel area
concentrated in the bottom has instead been distributed over the height would be
expected to exhibit a stiffer shear response, and this behaviour was observed in L-20D
and L-20DR. It would also be expected that a greater proportion of the total shear at a
section in such a beam would be carried by dowel action, for three reasons:
1) the cracks in the web are at a flatter angle than at the level of the main reinforcement,
with a larger component of the crack width oriented in the vertical direction. Analysis
of the cracks at the level of the main reinforcement in all the large specimens tested
indicated a wide range of crack angles ranging from perfectly vertical to identical to
the angle of the crack in the web. In general, however, the cracks in the web were
observed to be flatter than the cracks at the level of the main reinforcement.
2) cracks in the web are wider than they are at the level of the main steel. This would
only be the case if the longitudinal crack spacing in the web was not reduced to a
degree such that crack widths reduced from a maximum at the steel to zero at the tips
of the cracks.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
277
3) reduced longitudinal stiffness in the main reinforcement results in wider cracks at the
level of the reinforcement, with the components of the widths in the vertical direction
increasing as a consequence.
At a load of 650kN, a total of eight 10M crack control bars crossed the critical shear
cracks in L-20D and L-20DR. Combined with the four 30M bars in the bottom of the
specimen, this resulted in a total cross-sectional area of 3600mm2 of steel crossing the
crack that was available to resist shear by dowel action. This cross-sectional area is 3%
greater than the area of steel crossing the crack in the specimens without crack control
steel, but more importantly there was a total of twelve bars crossing the crack as opposed
to five.
Bhide and Collins (1987) derived a series of expressions to estimate the dowel force in a
bar crossing a crack of width w. The dowel force, Fd, versus dowel displacement, δ,
relationships for a 10M and a 30M bar are shown in Figure 6-27, in which the cracking
pattern in L-20D at P=650kN has been reproduced. The measured crack widths at a
single crack are shown as well. This crack was inclined at about 60o, and measured
0.25mm wide over most of its height. At the level of the main reinforcement, it was
about 0.15mm wide. Based on comparisons with crack patterns in the other L-series of
specimens, had this beam not contained crack control steel, it likely would have failed at
this crack at a load that was considerably smaller than 650kN.
Figure 6-27: Analysis of Dowel Action in Specimen L-20D
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
278
The component of the crack widths in the vertical direction were 0.125mm and 0.075mm
in the web and at the main reinforcement, respectively. These dowel displacements
resulted in dowel forces of 2.5kN and 5.9kN in the 10M bars and 30M bars, respectively,
and a total of about 8 x 2.5 + 4 x 5.9 = 44kN of shear being transferred by dowel action at
the section. This represents 13% of the total shear at the section. Had five 30M bars
been placed in the bottom layer without crack control steel, and the same crack pattern
formed with identical inclinations and crack widths, the shear transferred by dowel action
would have been about 5 x 5.9 = 30kN, representing 9% of the total shear at the section.
Indeed, a slightly narrower crack might be expected to have formed at the level of the
steel due to the presence of an additional 30M bar, reducing the dowel forces slightly.
We thus see that increased dowel action may have accounted for about 44kN-
30kN=14kN of the total additional shear of (0.5x650 – 0.5x500) = 75kN. This represents
19% of the total increase in shear force. The geometry of the cracking patterns in L-20D
and L-20DR generally resembled the geometry of the cracking patterns in L-10N2, in
which about 24% of the total shear force was transferred in the uncracked compression
zone, and it is unlikely that crack control steel significantly alters this proportion.
We are therefore left with the conclusion that the majority of the additional shear strength
exhibited by L-20D and L-20DR, about (100%-24%-19%)=57%, was due to enhanced
aggregate interlock capacity along the narrower cracks.
6.4.5 Code Estimates of the Shear Strength
It can be seen in Figure 6-24 that both the ACI and SMCFT generally provided accurate
estimates of the failure loads of the small beams, but both methods were somewhat
unconservative when estimating the failure loads of the two large specimens. For these
large beams, the average ratio of vexp/vACI was 0.86, while the average ratio of vexp/vSMCFT
was 0.91.
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
279
Providing skin reinforcement spaced closer than that required by ACI 318-05 thus did not
completely eliminate the size effect in shear. Had bars of a diameter smaller than that of
a 10M rebar been used as skin reinforcement, such as #3 US rebar or small deformed
wires, the shear strength would have been reduced. Application of the ACI skin
reinforcement provisions, which do not require a minimum bar diameter, does not,
therefore, ensure that the size effect will be adequately accounted for when designing
slender beams and slabs without shear reinforcement.
The failure shear stresses of L-20D, L-20DR, S-20D1 and S-20D2 are plotted in Figure
6-28 versus the effective crack spacing, and have been normalized by (f’c)0.5 and the CSA
strain effect term. The CSA size effect term, 1300/(1000+sze), has been plotted as well,
along with the experimental points from the other S- and L- series specimens. This figure
shows that the size effect has not been completely eliminated through the use of crack
control steel with ρd=0.3%. Interestingly, the difference between the normalized shear
strengths of S-20D1/2 and L-20D/R resembles the difference between the normalized
shear strengths of the small and large specimens tested with stirrups (see Chapter 7).
Based on these observations, it seems that, while the CSA shear provisions perhaps need
to be improved for members with distributed reinforcement, it is unclear whether
increasing ρd will adequately address the situation.
Table 6-3 compares experimental results from fifteen tests of continuous and simply-
supported beams of various depths reported by Collins and Kuchma, all of which
contained crack control steel with ρd values well in excess of 0.3%, with the four tests in
the D series of this thesis. Examination of the vexp/vsmcft values indicates that a number of
the specimens failed at shear strengths that were less than those predicted by the SMCFT.
BHD100, BND100 and SE100B-45, for example, failed at shears that were 78%, 83%
and 77%, respectively, of the SMCFT predicted failure shears. With the exception of
SE50B-45, generally the deeper specimens exhibited reduced shear strengths. These
specimens had ρd values considerably higher than that used in L-20D as a result of both
smaller vertical spacings between the layers of crack control steel and larger bar
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
280
diameters, yet failed at lower ratios of vexp/vsmcft. Furthermore, these specimens were
shallower than L-20D. Thus, requiring a higher ρd value before sz can be set equal to the
vertical spacing between layers of crack control steel does not appear to be the correct
modification.
Figure 6-28: Size-Effect Factors for Members with Distributed Longitudinal Steel
Shear Behaviour of Large, Lightly-Reinforced Control of Crack Widths Concrete Beams and One-Way Slabs
281
Table 6-3: Summary of Experiments of Beams with Crack Control Reinforcement