Structural Concrete - Hassoun

214 Chapter 5 Shear and Diagonal Tension
where V and M are the external shear and moment, respectively, and α is the slope angle of one
face of the beam relative to the other face. The plus sign is used when the beam depth decreases
as the moment increases, whereas the minus sign is used when the depth increases as the moment
increases. This formula is used for small slopes, where the angle α ≤ 30 ∘ .
A simple form of Eq. 5.26 can be formed by eliminating the j value:
v = V bd ± M (tanα) (5.27)
2
bd
For the strength design method, the following equation may be used:
v u = V u
φbd ± M u
(tanα) (5.28)
2
φbd
For the shearing force,
φV n = V u ± M u
(tanα) (5.29)
d
Figure 5.18 shows a cantilever beam with a concentrated load P at the free end. The moment and
the depth d increase toward the support. In this case a negative sign is used in Eqs. 5.27, 5.28, and
5.29. Similarly, a negative sign is used for section t in the simply supported beam shown, and a
positive sign is used for section Z, where the moment increases as the depth decreases.
Figure 5.18
Beams with variable depth: (a) moment diagrams and (b) typical forms.