Civil Engineering Reference
In-Depth Information
derivative {
d(A)/d(x/h)
} (and hence the coefficients
a
5
,
a
4
,
a
3
,
a
2
) depend on
the value of
x
(i.e. of
x/h
). It follows that, for a given value of
e
cu
, Eqn
10.40 is a non-linear equation in
x/h,
the solution of which requires an
iterative procedure such as the bisection method (Conte and Boor, 1984).
For a
cracked section
(i.e.
x/h
<1), the area A (see Eqn 10.5) under the
stress-strain curve in
Figure 10.9
a, between
e
c
/
e
=
e
Ó
c
=0 and
e
=
e
c
, is completely
e
cu
,
A
is constant and
hence the derivative
d(A)/d(x/h)
is equal to zero. It follows that the
coefficient
a
5
becomes zero and the coefficients
a
4
,
a
3
,
a
2
become constant.
Therefore, for a cracked section, Eqn 10.40 becomes a quartic equation (i.e.
an algebraical equation of the fourth degree). That is,
defined by the concrete strain
e
c
. For a given value of
e
c
/
(10.42)
where
(10.43)
and
a
3
,
a
2
,
a
1
and
a
0
are as defined by Eqns 10.41c, d, e and f respectively.
Following the argument in Section 10.4.3.3, it should be noted that there
are at most
2n
+1 possible combination of values for the coefficients [
a
1
,
a
0
]
(as they depend on
K
2
,
K
3
,
K
4
,
MM
) irrespective of whether the section is
uncracked or cracked, where n is the number of layers of reinforcement.
It is now clear that Eqns 10.40 and 10.42 define the relationship between
the concrete strain
e
cu
and
x/h
) at a
point on a moment-deflecting curve, where the slope is equal to the
values of
a
for constructing the curve. However, at that point on the
curve it is not known whether the section is uncracked (i.e.
x/h
‡
1) or
cracked (i.e.
x/h
<1). Therefore, for a given concrete strain ratio
e
c
/
e
cu
, a trial
and error procedure similar to that described in Section 10.4.3.3 is required
to determine the corresponding neutral axis depth ratio
x/h
at that point on
the curve. Further details of solving Eqns 10.40 and 10.42 are given
elsewhere (Wong, 1987a).
e
c
and the neutral axis depth
x
(i.e.
e
c
/
10.4.4.3
Procedure for determining column buckling loads
The procedure for determining column buckling loads can be outlined as
follows:
Step 1:
Select a convenient value for the concrete strain ratio
e
c
/
e
cu
between
the interval [0,1].
Step 2:
Solve Eqn 10.27 (i.e. Eqn 10.40 or 10.42) for the correct value of
x/
h,
as explained in Section 10.4.4.2.
Step 3:
For a concrete strain ratio
e
c
/
e
cu
selected in Step 1 and the neutral
axis depth ratio
x/h
determined in Step 2, the values of
a
, ¦, and
are calculated from Eqn 10.19, 10.20 and 10.16, respectively. The
