Civil Engineering Reference
In-Depth Information
(d)
Choose an interval from Step 4c, say interval No.1: [1, A].
Calculate the values of
K1, K
2
and
MM
from Eqns 10.21a, b
and e, and the coefficients
b
and
c
of Eqn 10.23 from Eqns
10.24b and c. Then solve Eqn 10.23 as a non-linear
equation, using an iterative method. If the root of Eqn 10.23
is positive and real, proceed to Step 4e; otherwise repeat
Step 4d for other intervals of
x/h
. If Eqn 10.23 is not
solvable in any one of the intervals of
x/h
determined in Step
4c, the selected concrete strain ratio
e
c
/
e
cu
is less than the
minimum possible value [
(see (iii)
of Section 10.4.3.2). Then return to Step 4a using a larger
value of
e
c
/
e
cu
]
min
for that chosen
a
e
c
/
e
cu
.
(e)
If the root obtained in Step 4d is within the interval of
x/h
chosen in Step 4d, the condition of compatibility is satisfied;
then proceed to Step 4f. Otherwise, return to Step 4d for other
intervals of
x/h.
(f)
Calculate the force parameter, say
a
Ó
1
by substituting the pair
[
e
cu
is chosen in Step
4a and the
x/h
is the root of Eqn 10.23 as obtained in Step 4d
and checked in Step 4e. The condition of equilibrium is
considered satisfied if |
e
c
/
e
cu
,
x/h
] into Eqn 10.19 where the
e
c
/
a
1
|<TOL, where a
1
is the axial
force ratio chosen in Step 2 and TOL is a small number, say
1.0ċ10
-4
. If the equilibrium is not satisfied, return to Step 4d
for other intervals of
x/h.
a
Ó
1
,-
(g)
For the
e
cu
chosen in Step 4a and the
x/h
determined in Step
4d and checked in Steps 4e and 4f, calculate
K
3
,
K
4
and
NN
from Eqns 10.21c, d and f. Then calculate the corresponding
values of ¦ and
e
c
/
from Eqns 10.20 and 10.16.
(h)
Repeat Steps 4a to 4g for other concrete strain ratios
e
c
/
e
cu
until sufficient pairs of [¦, ] are obtained for plotting the
initial portion of the moment-deflection curve, for instance,
curve 0'
AB
of
Figure 10.14
.
Step 5:
If the
a
value selected in Step 2 is less than
a
unity
(see (iv) of Section
10.4.3.2), proceed to Step 6. Otherwise, the moment-deflection
curve determined in Step 4 represents the entire moment-deflection
curve for the selected
a
value. Then return to Step 2 for other
a
values, if required.
Step 6:
Determine the portion of the
curve for the
cracked section
(i.e. curve
BCD
of Figure 10.14).
(a)
With reference to
Figure 10.9
a
, select a concrete strain ratio
e
c
/
e
cu
, such that [
e
c
/
e
cu
]
x/h
=1
<
e
c
/
e
cu
£
1. Then calculate the
coefficient a from Eqn 10.24a by putting
e
Ó
c
=0.
(b)
With reference to
Figure 10.16
calculate the 2n values of the
x/h
ratios from Eqns 10.26a and b as in Step 4b.
