Civil Engineering Reference
In-Depth Information
Eqn (3.5) (Timoshenko, 1956) is an alternative presentation to Eqn (3.3) and
is an interaction type of equation for failure criterion. Just prior to failure,
classical elastic stress analysis does not hold good, because of redistribution
of stresses at higher loads in concrete, resulting from the post-cracking
behaviour of concrete. As a result, evaluation of
s
3
will be a difficult
problem in as much as these principal stresses are dependent on
s
1
and
s
y
and
t
ry
which, in turn, could not be precisely measured just before failure. Herein
lies the difficulty in using the Eqn (3.5). Therefore, for calculation of the
sliding strength of reinforced concrete deep beams, Eqn (3.3) is invariably
preferred.
From
Figure 3.9
the characteristic constants
c
and
s
x
,
and other
relationships are evaluated, using geometry only, and Eqn (3.3) for the
Mohr-Coulomb failure criterion assumes the form
f
(3.6)
(3.7)
Eqn (3.6) represents the failure criterion which will be utilised in developing
the ultimate strength of deep beams.
3.10
Ultimate shear strength
A typical solid deep beam with main and web reinforcement and a plane of
rupture is shown in
Figure 3.10a.
A part of the beam separated by the potential
diagonal crack is shown as the free body diagram in Figure 3.10b. The
penetration of the crack is considered to extend to the full depth although
usually this crack stops at one-fifteenth to one-tenth of the depth of the beam
from the top and acts in a manner similar to the compression zone of a tied arch.
Considering the plane of rupture:
N
(=Normal force)=
bD
cosec ß×
s
(3.8)
¯
T
(=Tangential force)=
bD
cosec ß×
t
(3.9)
¯
From Eqns (3.3, 3.8 and 3.9), it may be stated that:
T
=(
cbD
/sin
ß
)+
N
tan
f
or
T
=
T
c
+N
tan
f
(3.10)
T
c
is the cohesive force of concrete along the inclined plane
=
cbD
/sinß
(3.11)
Since
N
is taken as a tensile normal force, Eqn (3.10) may be written as
