Civil Engineering Reference
In-Depth Information
8.5
Conclusion
In the preceding sections coinciding lower and upper bound solutions have
been presented for deep beams subjected to point loading. Basically, the
analysis shows that the ultimate load is determined by the flexural capacity,
expressed in terms of the clearance
a
by Eqn (8.11), and in terms of the span
l
by Eqn (8.16). Note, however, that when the compression zone reaches
mid-depth (
y
o
=
h
/2) the beam becomes over-reinforced. Thus for
/2 the
ultimate load is governed by the strength of the inclined compression strut,
which is found by putting
y
=
h
/2 irrespective of the yield force of the
reinforcement.
On the other hand, the attainment of the flexural capacity requires a
certain relationship between the length
s
of the support platen and the level
c
of the reinforcement centroid
s
f
>
n
s
l
where
s
l
is given by Eqn (8.14) in terms of
a,
and by Eqn (8.19) in terms of l. For
s
<
s
l
the reinforcement does not yield,
and the ultimate load is determined in terms of
a
by Eqn (8.15), and in terms
of l by Eqn (8.20) (By solving for
x
=
V
/
bvf
c
).
The latter case
s
<
s
l
corresponds to the generally observed shear failure,
and it typically arises when the reinforcement is placed close to the bottom
face of the beam. The stress distribution is shown in
Figure 8.7
,
except that
the length of the support platen will normally be designed according to the
load (i.e.
s
=
x
) The capacity is determined by the inclined concrete strut, and
as the stresses are concentrated at the extremities the collapse mode may
also be classified as bearing failure.
The result is a significant loss of load-carrying capacity, unless the
support platen is very large. As shown by the example in Section 8.4.1 it is
beneficial to increase the cover to the reinforcement, the small loss in
flexural capacity being offset by a large gain in shear strength.
Shear failure of deep beams is, however, also observed in cases where the
load is governed by the flexural capacity. This is due to the fact that the
effectiveness factor for the concrete is smaller for the sliding failure of the
shear mechanism
(
Figure 8.10
) than for the crushing failure of the flexural
mechanism
(
Figure 8.11
). For larger shear span ratios this effect is drowned
by the influence of the neglected tensile concrete strength, wherefore slender
beams are likely to fail in flexure, (
cf.
the discussion in Section 8.4.3).
The lower effectiveness factor for shear failure means that the introduction
of shear reinforcement is also beneficial for deep beams which nominally
attain their flexural capacity. The strength may be estimated by Eqn (8.27), but
this upper bound is not backed by a lower bound solution for
‡
/2.
The well known observation that horizontal web reinforcement has little
or no effect on the shear strength is readily explained by the fact that the
relative displacement rate at failure is close to the vertical.
It may be concluded that the theory of plasticity for structural concrete
gives an insight into the behaviour of deep beams at failure, in addition to
providing reasonable predictions of the ultimate loads.
f
<
n
