Structural Concrete - Hassoun

768 Chapter 19 Introduction to Prestressed Concrete
When the factored shear force, V u ,islessthan 1 2 φV c, minimum shear reinforcement is required.
19.8.2 Shear Strength Provided by Concrete (Prestressed)
The ACI Code, Section 22.5.8.2, presents a simple empirical expression to estimate the nominal
ultimate shear capacity of a prestressed concrete member in which the tendons have an effective
prestress, f se , of at least 40% of the specified tensile strength, f pu :
(
V c = 0.6λ √ f c ′ + 700 V )
ud p
b
M w d (19.50)
u
where
V u , M u = factored shear and moment at section under consideration
b w = width of web
d p (in term V u d p /M u ) = distance from compression fibers to centroid of prestressing steel
d p (in V ci or V cw equations) = larger of above d or 0.8h (ACI Code, Section 22.5.2.1)
The use of Eq. 19.50 is limited to the following conditions:
1. The quantity V u d p /M u ≤ 1.0 (to account for small values of V u and M u ).
2. V c ≥ (2λ √ f c)b ′ w d p (minimum V c ).
3. V c ≤ (5λ √ f c)b ′ w d p (maximum V c ).
The variation of the concrete shear capacity for a simply supported prestressed concrete beam
subjected to a uniform load is shown in Fig. 19.8. Note that the maximum shear reinforcement may
be required near the supports and near one-fourth of the span where φV s reaches maximum values.
In contrast, similar reinforced concrete beams require maximum shear reinforcement (or minimum
spacing) only near the support where maximum φV s develops.
The values of V c calculated by Eq. 19.50 may be conservative sometimes; therefore, the
ACI Code, Section 22.5.8.3, gives an alternative approach for calculating V c that takes into consideration
the additional strength of concrete in the section. In this approach, V c is taken as the
smaller of two calculated values of the concrete shear strength V ci and V cw (Fig. 19.8). Both are
explained next.
The shear strength, V ci , is based on the assumption that flexural-shear cracking occurs near
the interior extremity of a flexural crack at an approximate distance of d/2 from the load point in the
direction of decreasing moment. The ACI Code, Section 22.5.8.3.1, specifies that V ci be computed
as follows:
V ci =(0.6λ √ f ′ c)b w d p + V d + V iM cr
M max
(19.51)
but it is not less than (1.7λ √ f c)b ′ w d,where
V d = shear force at section due to unfactored dead load
V i = factored shear force at section due to externally applied loads occurring simultaneously
with M max
M max = maximum factored moment at section due to externally applied loads
M cre = moment causing flexural cracking at section due to externally applied load