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32
be directly calculated by Mohr's circle of stress, as shown in
f pc
Vcw = ft 1 + bw d + Vp Figures B-4 and F-2 in the appendixes; see Equation 18.
ft (Eq. 15)
Av f y d
Vs = cot
where d is suggested to be not less than 0.8h. s (Eq. 17)
The tensile strength of the concrete, ft, can be taken as
somewhere between 2 fc and 4 fc where f c is in psi units. f pc
where: cot = 1 + (Eq. 18)
In the proposed provisions, stress is expressed in ksi but it is ft
considered more useful to present the proposal with the stress
given in psi units. A tensile cracking strength close When fpc = 0, the axial stress is zero, or if flexure-shear
to 4 fc is believed to provide a more accurate estimate of cracking governs, then cot = 1 ( = 45 degrees).
the diagonal cracking strength in the design of the end
regions of a fully prestressed member in which there is no
effect of flexure while a value of 2 fc is a better estimate of 3.1.2 Proposed Simplified Provisions
the diagonal cracking load in a reinforced concrete member
The proposed simplified provisions are given here in both
or a prestressed member with a low level of prestressing. A
ksi and psi units. In order not to imply a greater level of pre-
transition between those two levels is a function of the level
cision in the procedure than can be justified, the coefficients
of the prestress and the axial load.
for the expressions in ksi units, as currently used in the LRFD
specifications, are rounded off.
Flexure-Shear Cracking Strength,Vci
For flexure-shear cracking of prestressed beams, the Web-Shear Cracking Strength
expression used in the AASHTO Standard Specification is
Vcw = (0.06 fc + 0.30 f pc )bv d v + Vp
Vi Mcr (where stress is in ksi units) (Eq. 19)
Vci = 0.6 fcbw d + Vd + 1.7 fcbw d
M max (Eq. 16)
which is equivalent to
where the sum of the second and third terms is an estimate Vcw = (1.9 fc + 0.30 f pc ) bv d v + Vp
of the shear force at the time of flexural cracking while
the first term is the increase in shear that has been observed (where stress is in psi units) (Eq. 20)
in experiments for a flexural crack to propagate into a diag-
onal crack. Flexure-Shear Cracking Strength
Although the concrete contribution, Vc, is taken as an esti-
mate of the diagonal cracking load, it must also be a lower Vi Mcr
Vci = 0.02 fcbv d v + Vd + 0.06 fcbv d v
bound estimate of the concrete contribution to shear resis- M max
tance at the ultimate limit state. At the ultimate limit state,
the concrete contribution is the sum of the shear carried in the (where stress is in ksi units) (Eq. 21)
compression zone, the shear carried across diagonal crack which is equivalent to
due to shear-friction (aggregate interlock), direct tension
across diagonal cracks, dowel action, and arch action. Many Vi Mcr
Vci = 0.632 fcbv d v + Vd + 1.9 fcbv d v
factors influence the contributions of each of these mecha- M max
nisms and attempts to reasonably account for them lead to
(where stress is in psi units) (Eq. 22)
complicated expressions for Vc. Thus, the approach taken by
the research team in developing this simplified approach has
The 0.06 coefficient in Equation 19 establishes a uniform
been to use a lower bound estimate of the diagonal cracking
minimum Vc contribution over the length of the member
load that, when added to the calculated stirrup contribution
independent of whether a web-shear or flexure-shear region
to shear resistance, is shown to provide a conservative esti-
is being designed. The coefficient of 0.06 (ksi units) is also
mate of the capacity of test beams presented and discussed in
Section 3.3. very close to the traditional coefficient of 1.7 (psi units) when
it is considered that dv = 0.9d.
Contribution of Shear Reinforcement, Vs Vci ,min = 1.7 fcbv d v (psi units)
1.7 fc / 1000 / 0.9 0.0597 fcbv d v ( ksi units)
The contribution of the shear reinforcement to shear resis-
tance is given in Equation 17. The angle of shear cracking can (Eq. 23)