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56
than with conventional Grade 60 steel. Consequently, the Pa wL2
Ma = + (Eq. 15)
service load reinforcing strains are greater (i.e., s = fs/Es). This 2 8
larger strain affects deflection and crack widths at service
loads. In the following sections, discussion focuses on the Where:
behavior at loads corresponding to longitudinal reinforcing P = total applied load in four-point bending (sum of two
bar stresses of 36, 60, and 72 ksi, representing service load lev- point loads);
els (i.e., 0.6fy) for steel having fy = 60, 100, and 120 ksi, respec- w = self weight of beam;
tively. At these service load stresses, the use of Es = 29000 ksi L = length of simple span, 240 in. in all cases;
for all steel grades is acceptable (see Section 1.3.2.1) although a = length of shear span, 102 in. in all cases.
experimentally determined R-O curves have nevertheless In the formulations of effective moment of inertia (Equa-
been used in all cases to calculate stress from measured rein- tions 1 and 2), the moment to cause cracking is calculated as
forcing bar strains. 80% of the moment corresponding to modulus of rupture.
2.9.1 Deflections of Flexural Members 7.5 fcI g 6 fcI g
M cr = 0.80 = ( psi ) (Eq. 16)
)
y y
Table 28 summarizes the midspan deflections of all flexural
beam specimens (F1 through F6) corresponding to longitu-
Where:
dinal bar stresses of 36, 60, and 72 ksi. The experimentally
Ig = moment of inertia of gross concrete section, nominally
measured deflections include any support settlement but
4096 in.4;
do not include deflection due to self-weight. Also shown in
y = neutral axis distance from the tensile face for gross
Table 28 are the deflections calculated using both the Branson
concrete section, nominally 8 in.
(Equation 1) and Bischoff (Equation 2) formulations (see
Chapter 1) for effective moment of inertia (Ie). In calculating The use of the reduced value of Mcr accounts for cases where
the applied moment (Ma in Equations 1 and 2), the self- the applied moment (Ma) is only slightly less than the unre-
weight of the beam is accounted for; thus, the effective strained Mcr (based on 7.5 fc ) since factors such as shrinkage
moment of inertia is based on the appropriate cracked section and temperature may still cause a section to crack over time
for the load level considered. (Scanlon and Bischoff 2008).
Table 28. Comparison of experimental and calculated deflections at
service load levels.
Deflection
Beam and Bar Ma
= As/bd Experimental Branson Bischoff
Stress
(kip-in.) (in.) (in.) (in.)
F1 @ 36 ksi 0.012 898 0.582 0.372 0.365
F1 @ 60 ksi 0.012 1318 1.145 0.600 0.590
F1 @ 72 ksi 0.012 1553 1.400 0.723 0.713
F2 @ 36 ksi 0.016 1038 0.527 0.318 0.312
F2 @ 60 ksi 0.016 1726 1.145 0.567 0.561
F2 @ 72 ksi 0.016 2084 1.450 0.695 0.690
F3 @ 36 ksi 0.007 645 0.527 0.269 0.288
F3 @ 60 ksi 0.007 900 0.855 0.478 0.482
F3 @ 72 ksi 0.007 1099 1.182 0.633 0.629
F4 @ 36 ksi 0.016 895 0.625 0.286 0.280
F4 @ 60 ksi 0.016 1405 1.146 0.501 0.492
F4 @ 72 ksi 0.016 1650 1.354 0.601 0.592
F5 @ 36 ksi 0.023 1313 0.688 0.330 0.326
F5 @ 60 ksi 0.023 2096 1.271 0.551 0.547
F5 @ 72 ksi 0.023 2517 1.583 0.669 0.666
F6 @ 36 ksi 0.012 569 0.458 0.156 0.166
F6 @ 60 ksi 0.012 1012 0.938 0.429 0.424
F6 @ 72 ksi 0.012 1242 1.229 0.561 0.552
