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)

364 Prestressed concrete simple beams
v e = the design shear stress taken from Table 6.4-1, in which
As is now interpreted as the sum of the area Aps and that
of any ordinary longitudinal reinforcement bars that may
be present;
bv = the width of the beam as defined for eqn (9.6-1);
d = the effective depth to the centroid of the tendons.
M0 = the moment necessary to produce zero stress in the
concrete at the extreme tension fibre. For the purpose of
this equation, M 0 is to be calculated as 0.8[fp 1 1/y] where
fpt is the concrete compressive stress at the extreme
tension fibre due to the effective prestressing force, I the
second moment of area of the beam section; andy is the
distance of the extreme tension fibre from the centroid of
the beam section (see Step 3 of Example 9.6-1);
V and M = the shear force and bending moment respectively at the
Comments
section considered, due to ultimate loads (ignoring the
vertical component of the tendon force if any).
(a) The derivation of eqn (9.6-2) is given on pp. 42-47 of Reference 2,
which also explains the pre-cracking and post-cracking behaviour of
prestressed concrete beams. Until recently, it was thought adequate
to use elastic theory for shear design; that is, to calculate the principal
tensile stresses under service condition and limit them to a specified
value. On p. 20 of Reference 2, four major reasons are given to
explain why the elastic theory is not adequate.
(b) BS 8110 states that, in using eqn (9.6-1), f 1 is to be taken as
0.24~ feu. The tensile strength of concrete is usually between
o.3Heu and o.4~feu
Therefore, for design purposes it is reasonable to take f 1 as
0.3~(/eufYm) = 0.24~feu
when 1.5 is substituted for the partial safety factor Ym· Note also that
in eqn (9.6-1), BS 8110 applies a factor of 0.8 to fer.· This is because
the value 0.24~feu (for ft) includes a factor of 1/vfl.S =i= 0.8.
(c) Equation (9.6-1) might at first sight appear to be applicable to
rectangular sections only, since its derivation is based on the equation
3VcO
Vc0 = 2b h
v
For a general section, the shear stress is of course given by the wellknown
formula
VeoA.Y
VcO = bT
v
where the product A.Y is the statical moment (taken about the
centroidal axis of the entire cross-section) of the area above the level
at which vc0 occurs, bv is the beam width at that level, and I is the
second moment of area of the entire cross section taken about the