Civil Engineering Reference
In-Depth Information
For a column section with a single layer of steel there are three possible
conditions. By drawing simple sketches similar to those in
Figure 10.16
, the
reader can verify that for two layers of steel there will be 3+2 conditions, for
three layers there will be 3+2+2 and so on. Therefore, for a column section
with n layers of steel, there will be
3+2(
n
-1)=2
n
+1
possible conditions, within which some, or all, of the reinforcement may
reach their yield strengths. It follows that the values of the parameters
K
1
to
NN
, and hence the coefficients
b
and
c
, have at most 2
n
+1 possible
combinations. The condition of compatibility is satisfied, if the root of Eqn
10.23 is within the
x/h
interval, where
b
and
c
are calculated. If the root of
Eqn 10.23 so calculated satisfies Eqn 10.19, the condition of equilibrium
is achieved.
10.4.3.4
Preparation of
curvesÏAn interval technique
The procedure
for preparing the moment-deflection (
) curves may be summarised as
follows.
Step 1:
Determine
a
unity
and
a
max
(see (iv) and (v) of Section 10.4.3.2).
Step 2:
Select a convenient positive
a
value, say
a
1
, such that
a
1
<
a
max
,
where
a
max
is determined in Step 1.
Step 3:
Determine the concrete strain ratio
e
c
/
e
cu
at
x/h
=1, i.e. [
e
c
/
e
cu
]
x/h
=1
, for
the
a
chosen in Step 2 (see (ii) of Section 10.4.3.2).
Step 4:
Determine the initial portion of the
curve for the
uncracked
section
(i.e. curve 0'
AB
of
Figure 10.14
).
(a)
With reference to
Figure 10.9
a
, select a concrete strain ratio
e
c
/
e
cu
, such that 0<
e
c
/
e
cu
£
[
e
c
/
e
cu
]
x/h
=1
, where [
e
c
/
e
cu
]
x/h
=1
is
determined in Step 3.
(b)
With reference to Figure 10.16 calculate, for each layer of
steel, the ratios [
x/h
]
yti
and [
x/h
]
yci
from Eqns 10.26a and b.
Therefore, for a section with n layers of reinforcement, there
are 2n such
x/h
ratios.
(c)
From the
x/h
ratios calculated in Step 4b, select those with
values greater than or equal to 1. Then arrange the selected
x/h
ratios in ascending order. For example,
where
A, B, C
and so on are the selected
x/h
ratios.
