F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Stresses in service: elastic theory 345
Shear in prestressed concrete beams
Figure 9.2-S(a) shows a beam with a curved tendon profile; such a profile
is commonly used in post-tensioned beams. At a typical section, the
vertical component of the tendon force Pe produces a shear force acting on
the beam:
Vp = -Pesin{3 = -Pe[~] (9.2-22)
where the negative sign is consistent with the sign convention in Fig.
9.2-S(b). Therefore the net shear force acting on the concrete section is
Vc = V + Vp (9.2-23)
where V is the shear force due to the imposed load and the dead load.
Specifically.
(9.2-24)
Vcmin = V;min + Vd + VP (9.2-25)
where Vcmax is the maximum net shear force acting on the concrete at that
section, V;max is the maximum shear force due to the imposed load, and so
on. Ideally, the tendon profile should be such as to result in net shear
forces of the least magnitude; this occurs if Vcmax = - Vcmin• i.e. as V;min
changes to V;max• the shear force Vc is exactly reversed. Putting Vcmax =
- Vcmin in eqn (9.2-24) gives
VP = -![Vimax + V;min + 2Vd]
Using eqn (9.2-22),
(9.2-26)
~ = 2~ [V;max + Vimin + 2Vd] (9.2-27)
e
l.:hus, the ideal tendon profile for shear is one having a slope at any point
given by eqn (9.2-27). For simply supported beams, the load distribution
producing V;max (V;m;n) usually also produces the moments Mimax (M;m; 0 ),
and a further simplification of eqn (9.2-27) is possible, by integrating with
respect to x:
J d = _l_J[dMimax + dMimin + 2dMd]dx
es 2P dx dx dx
X ·I
c
Fig. 9.2-5
(a)