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)

200 Shear, hand and tarsian
where Vis the shear force acting on the beam section, d the effective depth
and b v the beam width. For a rectangular beam, b v is taken as the width b;
for a f1anged beam, it is taken as the web width b w . Section 6.4 will show
how design shear stresses are used in practice.
In the meantime, let us retum to Fig. 6.2-1(a) and study how the
rectangular beam fails eventually as the shear force V is increased. Many
tests [1, 2] have established that the failure mode is strongly dependent on
the shear-span/depth ratio avfd:
(a) avfd > 6: Beams with such a high avfd ratio usually fail in bending.
(b) 6> avfd > 2.5: Beams with avid lower than about 6 tend to fail in
shear. With reference to Fig. 6.2-1(a), as the force Vis increased,
the ftexural crack a-b nearest the support would propagate towards
the loading point, gradually becoming an inclined crack, which is
known as a f1exure-shear crack but which is of ten referred to simply
as a diagonal crack (Fig. 6.2-2: crack a-b-c). With further increase
in V, failure usually occurs in one of two modes. It the avf d ratio is
relatively high, the diagonal crack would rapidly spread to e, resulting
in collapse by splitting the beam into two pieces. This mode of failure
is of ten called diagonal-tension failure; for such a failure mode, the
ultimate load is sensibly the same as that at the formation of the
diagonal crack. It the avf d ratio is relatively low, the diagonal crack
tends to stop somewhere at j (Fig. 6.2-2); a number of random
cracks may develop in the concrete around the longitudinal tension
reinforcement. As Vis further increased, the diagonal crack widens
and propagates along the level of the tension reinforcement (Fig.
6.2-2: crack g-h). The increased shear force presses down the
longitudinal steel and causes the destruction of the bond between the
concrete and the steel, usually leading to the splitting of the concrete
along g-h. It the longitudinal reinforcement is not hooked at the
end, the destruction of bond and the concrete splitting will cause
immediate collapse. It hooks are provided, the beam behaves like a
two-hinge arch until the increasing force in the longitudinal reinforcement
destroys the concrete surrounding the hooks, whence
Fig. 6.2-2