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11
would occur; thus, confining reinforcement is critical in devel- an equation for elastic deflection. Both AASHTO (2007) and
oping higher strength bars. ACI 318 (2008) prescribe Branson's Equation (Branson 1963)
Seliem et al. (2009) assessed the present empirical develop- to determine an equivalent moment of inertia (Ie) of a cracked
ment length equations prescribed by ACI 318 (2008) and ACI concrete section as follows:
408 (2003) when applied to developing A1035 bars. ACI 318
was found to underestimate development length requirements M
m
M m
I e = cr I g + 1 - cr I cr I g (Eq. 1)
when no confining reinforcement was present and was only Ma Ma
marginally improved when confining reinforcement was used.
The ACI 408 recommendations were found to be adequate Where:
whether confinement was present or not. Present AASHTO Ig = moment of inertia of gross concrete section;
requirements were not assessed although these can be shown Icr = moment of inertia of fully cracked concrete section;
to result in comparable development lengths to the require- Mcr = moment to cause cracking;
ments of ACI 408 in cases where confinement is present. Peter- Ma = applied moment at which Ie is calculated; and
freund (2003), in a study of A1035 reinforcement for bridge m = factor as defined below.
decks (#4 and #5 bars only), concluded to the contrary, that is,
the ACI 318 requirements for development length were ade- Setting m = 4 accounts for tension stiffening effects at the
quate to develop A1035 bars with no confining reinforcement critical section along a span, while calculations are conven-
present. However, in his study, Peterfreund used the simplified tionally made setting m = 3 to reflect the "average" stiffness
ACI equation which results in development lengths almost across the entire span. Equation 1 is found to be generally sat-
twice as long as the more rigorous approach used by Seliem isfactory for beams having typical amounts of non-prestressed
et al. and others. Seliem et al. recommended that confining reinforcement; indeed this equation was originally calibrated
based on beams having a reinforcement ratio of = 0.0165
reinforcement always be used when developing A1035 or other
(Branson 1963). The value of Ie calculated using Equation 1 is
high-strength reinforcing steel.
only slightly smaller than Ig in cases where Ma is only margin-
ally larger than Mcr. This case generally happens in members
1.3.6.1 Development of Standard Hooks having a low reinforcement ratio, typically < 0.006. For such
members, the calculated value of Ie is very sensitive to changes
Ciancone et al. (2008) evaluated the behavior of standard
of Mcr (Gilbert 1999). Thus, Equation 1 may overestimate the
hooks made using #5 and #7 A1035 steel. No confinement
effective moment of inertia for lightly reinforced flexural
reinforcement was provided in the specimens. While the #5
members having an Ig/Icr ratio greater than 3 (Scanlon et al.
hooks were able to develop their yield capacity of 100 ksi, the
2001, Bischoff 2005, and Gilbert 2006). As decreases, Ig/Icr
#7 hooks were not. This result suggests an effect of bar size and increases exponentially and Ma/Mcr decreases. The result is
supports the need for confining reinforcement when develop- that the effective moment of inertia, Ie, is overestimated on the
ing A1035 bars. order of 200% when = 0.007 but by only about 10% at =
0.025 (Nawy and Neuwerth 1977). Bischoff (2005) reports
1.3.7 Serviceability Considerations that Branson's Equation underestimates short-term deflection
for concrete members when the reinforcing ratio is less than
A fundamental issue in using A1035 or any other high- approximately 1% and the Ig/Icr ratio is greater than 3.
strength reinforcing steel is that the stress at service load (fs; Several attempts have been made by different investigators
assumed to be on the order of 0.6fy) may be greater than to modify Branson's Equation, aiming to improve the accu-
with conventional Grade 60 steel. Consequently, the service racy of the predicted deflection (Grossman 1981, Rangan
load reinforcing bar strains are greater (i.e., s = fs /Es). This 1982, Al-Zaid et al. 1991, Al-Shaikh and Al-Zaid 1993, Fikry
larger strain impacts deflection calculations and crack con- and Thomas 1998). With the exception of Rangan (1982),
trol parameters. Regardless of this discussion, as discussed none of these modifications has been adopted into building
previously, most studies of members reinforced with A1035 codes; Branson's Equation remains the standard calculation
steel exhibit serviceability performance, as measured by both for computing effective moment of inertia.
deflections and crack widths, similar to that of members re- The following two approaches have been proposed to
inforced with A615 bars. modify Branson's Equation to address its efficacy when used
with lower reinforcing ratios:
1.3.7.1 Deflection Calculations
· Introduce a coefficient, , into the first term of Equation 1
Deflection of reinforced-concrete flexural members is most to modify Ig (Gao et al. 1998). is less than unity and
typically determined using an equivalent moment of inertia in is calculated based on reinforcing bar modulus (for

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softer reinforcing materials such as FRP) (Theriault and having very large concrete cover (ACI 224 2001). Addition-
Benmokrane 1998, Masmoudi et al. 1998) or reinforcing ally, Beeby (1983) showed no conclusive evidence linking
ratio relative to the balanced ratio (i.e., /b) (Yost et al. reinforcement corrosion with crack width while Poursaee
2003). The latter approach is necessary when considering et al. (2010) show that a crack as small as 0.004 in. acts as a free
high-strength steel reinforcement. surface with respect to water ingress. Despite the latter asser-
· Adjust the exponent m (Dolan 1989) as a function of the tion, the simplified versions of the Frosch approach adopted by
reinforcing ratio (Toutanji and Saafi 2000; Al-Zaid et al. AASHTO and ACI implicitly assume a maximum crack width
1991) or simply increase the value of m (Brown and of 0.017 in. which was also the value assumed for exterior expo-
Bartholomew [1996] propose m = 5). sure conditions when applying the Gergely-Lutz approach
prior to 1999.
Other methods involving finding an effective modulus of The ACI 318 version of the Frosch equation for determin-
the beam have been proposed by Murashev (1940), Rao ing the maximum spacing of flexural reinforcement to affect
(1966), and CEB-FIP (1993). Finally, approaches involving adequate crack control is as follows:
integrating curvature along a beam have been proposed by
40, 000 40, 000
Ghali (1993), Toutanji and Saafi (2000), Rasheed et al. (2004), s 15 - 2.5cc 12
and Razaqpur et al. (2000). fs fs
Bischoff (2005), in addition to providing a thorough review ( fr in psi; cc in inches ) (Eq. 3)
of all deflection investigations briefly summarized above, pro-
poses a method of calculating the effective moment of inertia Where:
at a section that better captures the effects of tension stiffening cc = minimum concrete cover measured to center of rein-
particularly for "soft" sections having low reinforcing ratios. forcing bar closest to the extreme tension face and
This method is summarized in Equation 2. fs = service load stress in reinforcing bar closest to the
I cr extreme tension face.
Ie = 2
(Eq. 2)
M Equation 3 may be rewritten in terms of reinforcing bar
1 - cr strain (s), assuming the material obeys Hooke's Law, and cal-
Ma
ibrated for any desired crack width (w) (Ospina and Bakis
Where: 2007), as follows:
= 1 - I cr I g
w w
k3 2
s 1.15 - 2.5cc 0.92
s s
(w in inch
hes ) (Eq. 4)
I cr = cr + n (1 - kcr ) bd 3
3
Thus, the relationship between crack width, reinforcing
kcr = (n)2 + 2n - n bar strain, and longitudinal bar spacing required to control
n = modular ratio Es/Ec and cracking is demonstrated in a relatively simple format consis-
= reinforcing ratio. tent with present design practice. The relationship is material
Moment-curvature relationships may then be predicted independent, only assuming a linear behavior is present.
using M = EcIe. The derivation of Equation 2 is presented in Available data comparing the cracking behavior of steel and
Bischoff (2005) and is further shown to be essentially equiv- FRP-reinforced members confirm the implications of this
alent to the Murashev (1940) equation, of which the Branson approach (e.g., Creazza and Russo 2001, Bischoff and Paixao
Equation is a simplification. 2004). Ospina and Bakis conclude that the use of Equation 3
is valid, if not conservative, for beams having large elastic
reinforcing bar strains.
1.3.7.2 Crack Control AASHTO Equation 5.7.3.4-1 (AASHTO 2007) takes the
The traditional "z-factor" or Gergely-Lutz (1968) same form as the ACI equation, as follows:
approach of directly assessing cracking behavior of concrete
700 d
beams was dropped by ACI 318 in 1999 and by AASHTO s - 2dc ( f s in ksi; dc in inches ) (Eq. 5)
s fs
in 2005 in favor of a simplified version of the alternative
approach proposed by Frosch (1999 and 2001) that prescribed
Where:
spacing limits for longitudinal reinforcing steel thereby indi-
rectly controlling crack width. The empirically tuned Gergely- dc = minimum concrete cover measured to center of rein-
Lutz approach was considered inadequate to address cases forcing bar;