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57
In the calculation of deflection, the self-weight is neglected width measured at the height of the extreme tension steel
since this component of the deflection is also not included in from all cracks in the constant moment region. Figure 37b
the experimentally determined deflections, against which com- provides the maximum crack width measured in this region.
parisons are made. For the beams considered, the deflection The ratio of maximum to average measured crack widths for
associated with beam self-weight is approximately 19%, 11%, all specimens at all stress levels is 1.8, consistent with avail-
and 9% of the deflections corresponding to applied load at bar able guidance for this ratio, which tends to range between
stress levels of 36, 60, and 72 ksi, respectively. The midspan 1.5 and 2.0 (CEB-FIP 1993). In all cases, the ratio of maximum
deflections associated with the applied four-point bending are to average crack width falls with increasing bar stress. At
calculated as follows: approximately 36 ksi, this ratio is 1.7, falling to 1.6 at 60 ksi
and 1.5 at 72 ksi.
PL3 a a
3
The data shown in Figure 37 clearly show that at all consid-
= - L
3 4 (Eq. 17
7)
48 Ec I e L ered service load levels ( fs < 72 ksi), average crack widths are
all below the present AASHTO de facto limits for Class 1 and
The Branson and Bischoff formulations yield very similar Class 2 exposure (0.017 in. and 0.01275 in., respectively; see
results for the specimens tested. The correlation between the Section 1.3.7.2). Indeed, with the exception of beam F2,
formulations is not as good for the lower reinforcing ratio maximum crack widths also fall below the Class 1 threshold
of 0.007 (F3). This difference is consistent with the observa- through bar stresses of 72 ksi. Crack width is largely unaf-
tion that Branson's Equation underestimates short-term fected by the reinforcing ratio within the range given. It is
deflection for concrete members when the reinforcing ratio noted that all 12-in. wide beams had four bars (#5 or #6) in
is less than approximately 1% (Bischoff 2005). Although the lowermost layer; thus, crack control reinforcing would be
both equations are suitable for calculating deflections, the considered excellent for these beams. Considering the mea-
Bischoff approach is based on fundamental mechanics and sured crack widths in this experimental study, it appears that
may therefore be applied for any type of elastic reinforcing the inherent conservativeness in existing equations allows
material. The Branson formulation is empirical and cali- present specifications to be extended to the anticipated higher
brated for mild steel. service level stresses associated with the use of high-strength
reinforcing steel.
2.9.2 Crack Widths Using Equation 6 (as discussed in Chapter 1, this equation
was derived from the present AASHTO LRFD provisions for
Extensive crack width data were collected in the flexural crack control given in Equation 5), the expected crack width (w)
test series (F1 to F6). To assess the effects of using higher for a given reinforcing bar strain (s) is calculated. Figures 38a
strength steel, the crack widths corresponding to a variety and 38b show the calculated crack width for both Class 1 and
of stresses in the reinforcing steel were determined and are 2 exposure conditions, respectively, compared with measured
plotted in Figure 37. Figure 37a provides the average crack average crack width from specimens F1 to F6. The generally
(a) Average Crack Widths (b) Maximum Crack Widths
Figure 37. Measured crack widths with longitudinal bar reinforcing bar stress for flexural beams.