Beams in Shear - Unified Theory of Concrete Structures - page 364

Figure 8.14 Analytical model used for calculating web-shear capacities of beams

In Section 8.3.3, we will derive a simple formula for shear strength V n that does not involve
the prestressing force and the failure crack angle.

8.3.3 UH Shear Strength Equation
8.3.3.1 Shear Model

The concept of shear resistance developed by Loov (2002) (Figure 8.14) was used to derive
the ultimate shear capacity. According to this model, the contribution of concrete to the shear
capacity of the beams stems from the shear stress in the concrete along a failure crack,
represented by the force S . Loov's concept is very different from the concept of existing shear
design methods (ACI, 2008; AASHTO, 2007) which assume that the concrete contribution to
the shear capacity of beams is derived from the tensile stress across the cracks.
Assuming the failure surface to be an inclined plane, and taking the force equilibrium of
the free body along the crack direction (Figure. 8.14), the shear capacity of the beam V n can
be calculated as:

F V

S − T sin α 1

V n = + (8.14)

cos α 1

where F V is the summation of vertical forces of the stirrups intersected by the failure crack
at the ultimate load. T is the tensile force in the prestressing tendons at the ultimate load of
the beams; and α 1 is the angle between the normal to the failure crack and the longitudinal

axis. The α 1 angles of beams B1-B3, which failed in web-shear, were observed in tests to be

approximately 45 ◦ .
The term ( S

α 1 in Equation (8.14) is the 'contribution of concrete in shear'
( V c ). In order to avoid the excessive complexity involved in the calculation of S , T and

− T sin α 1 ) / cos

α 1 ,itwas

decided to derive the V c term directly from tests. In Equation (8.14) F V is the 'contribution
of steel in shear' denoted as V s . Thus,

V n = V c + V s (8.15)