Civil Engineering Reference
In-Depth Information
give rise, for purpose of equilibrium, to tensile stresses at right angles to the
direction of the trajectories with the tensile stress resultant acting in the region
of the largest cross-section as indicated in
Figure 2.16.
However, the inclined
crack, which starts in this region when the local strength of the material is
exceeded, is insufficient to cause collapse of the beam. The inclined crack
starts at a load level often several times lower than the collapse load (Kong
and Evans, 1987). With increasing load the inclined crack extends
simultaneously towards the loading and support points and such an extension
should inevitably result in a continuous stress redistribution in the region of
the crack tips. The stage is reached, however, when such redistributions
cannot maintain the stress levels below critical values and therefore crack
extension continues in an unstable fashion simultaneously towards the top and
bottom faces and leads to collapse.
The cracking process of structural concrete under increasing load has
been investigated analytically by means of nonlinear finite element analysis
(Kotsovos and Newman, 1981b; Kotsovos, 1981; Bedard and Kotsovos,
1985; 1986). The results obtained from such investigations for the case of
deep beams under two-point loading are shown in
Figure 2.19
which shows
the crack pattern of a typical beam at various load levels up to ultimate.
Cracking not only always starts in regions subjected to a critical
combination of compressive and tensile stresses, but also propagates into
regions subjected to similar states of stress. The Figure also indicates that
the region of the loading point, which is subjected to a wholly compressive
state of stress, reduces in size as the applied load increases above the level
which causes crack initiation. This is due to stress redistribution which
transforms the state of stress at the periphery of this region from a wholly
compressive state of stress to a state of stress with at least one of the
principal stress components being tensile. When the strength of concrete
under this latter state of stress is exceeded cracking occurs and the size of
the compressive region further reduces. In all cases investigated collapse
occurs before the strength of concrete in the compressive region is
exceeded. Such behaviour is in compliance with the conclusions of the
experimental information discussed in Section 2.3.
The mode of failure shown in
Figure 2.18b
is also characterised by a
number of flexural and inclined cracks. However, although the inclined
cracks extend towards the compressive zone within the middle span as do
the inclined cracks that characterise type III behaviour, they differ from the
latter in that their depth is similar to that of the flexural cracks. For type IV
behaviour, the presence of such cracks may lead to an alternative mode of
failure which is characterised by failure of the compressive zone of the
middle span of the beam. The causes underlying this mode of failure should
be similar to those described earlier in the section for type III behaviour:
volume dilation of the concrete in the region of the section that coincides
with the tip of the deepest flexural or inclined crack will induce tensile
stresses in adjacent regions. It is failure of these regions under the combined
