Civil Engineering Reference
In-Depth Information
where the pair [
, ¦] are the coordinates at a point on the
curve
for
a
.
With reference to
Figure 10.17
and based on the above discussions, it can
be concluded that the equilibrium of a slender column can be related to the
value of
D
, calculated from Eqn 10.33 as follows.
Condition 1:
D
>0
This corresponds to stable equilibrium, as shown at the point
b
in Figure 10.17a.
Condition 2:
D
<0
This corresponds to the condition that equilibrium is
impossible, as shown in Figure 10.17b.
Condition 3:
D
=0
This corresponds to unstable equilibrium, as shown at the
point
c
in Figure 10.17c.
10.4.4.2
Analytical expressions for instability failures
As explained in
Section 10.4.4.1, the slope at a point on a
curve for
a
is defined by the
derivative
where ¦ is given by Eqn 10.20. The derivative can be
rewritten as
(10.34)
Considering the derivatives
d
¦/
d(x/h)
and
d(x/h)/d
(
) separately, it can be
shown that (Wong, 1987a)
a
c
are as defined before.
If the value of the concrete strain
and
k
2
,
K
3
, and
e
c
and that of the neutral axis depth
x
(i.e. the values of the pair [
e
c
/
e
cu
,
x/h
] at a certain point on the
curve
for
a
are known, Eqn 10.35 can be used to calculate the slope of the
curve at that point.
Suppose the slope at a point on the
curve for a particular value of
a
is equal to
a
(Eqn 10.27), then substituting Eqn 10.19 (for
a
) and Eqn 10.35
(for)
into Eqn 10.27b, and rearranging
