Civil Engineering Reference
In-Depth Information
A supporting plane to the yield surface is a plane in stress space with the
equation:
(
Q
i
) is a linear yield function.
A rigid, perfectly plastic body can now be defined as a body with the
following properties:
p
(
Q
i
)=0 where
p
i) There exists a
convex yield surface F
(
Q
i
)=0 such that non- zero strain
rates qi are only possible for stress states
Q
i
o
for which
F
(
Q
i
o
)=0
ii) The strain rates
q
i
o
are governed by the
associated flow rule,
which
may be expressed:
q
i
o
=
ldp
/
d
Q
i
, where
l
is a non-negative constant
and
(
Q
i
)=0 is a supporting plane to the yield surface through the
point
Q
i
=
Q
i
o
.
p
The associated flow rule is also called the
normality condition,
because if
the strain rates
q
i
are represented as a vector in generalised stress space,
q
i
o
is an outwards directed normal to the yield surface at the corresponding
stress point
Q
i
=
Q
i
o
if the point is regular. If the yield surface is not
differentiable at
Q
i
=
Q
i
o
the direction of
q
i
o
is confined by the normals to the
adjoining parts of the yield surface.
It appears from the above that the limit analysis theorems reflect sound
engineering concepts of structural response, but that the formal proof is based
upon the assumption of plastic material behaviour, in particular the conditions
of convexity and normality. For a more comprehensive review of the theory of
plasticity reference is made to standard textbooks, e.g. Martin (1975).
8.3
Structural concrete plane elements
8.3.1
Concrete modelling
In many reinforced concrete structures, including deep beams, the concrete can
reasonably be assumed to be in a state of plane stress. This means that the
principal stresses
s
2
) may be taken as generalised stresses, the
corresponding generalised strain rates being the principal strain rates
s
i
=(
s
1
,
e
2
).
The uniaxial strength of concrete in compression is termed and, assuming
that the strength in biaxial compression is independent of the lateral stress,
two yield functions have been identified:
f
1
=
-f
c
*
-
e
i
=(
e
1
,
s
2
. The tensile
strength of concrete is small and unreliable, and is prudently neglected in
plastic analysis of plane elements. Thus we have the additional yield
functions:
f
3
=
s
1
and
f
2
=
-f
c
*
-
s
2
.
The four yield functions
f
k
(
s
1
and
f
4
=
0 constitute a yield condition for concrete
in plane stress, and the corresponding yield locus in the principal stress plane is
shown in
Figure 8.1
, which also indicates the associated flow rule.
The well-known square yield locus of Figure 8.1 corresponds to a
more comprehensive material model for concrete, know as the Coulomb
failure criterion, modified by a zero tension cut-off. The modified
Coulomb criterion (also with a non-zero tension cut-off) was introduced
s
1
,
s
2
)
£
