Civil Engineering Reference
In-Depth Information
a‡a
unity,
the entire curve corresponds to uncracked sections
can be found from Eqn 10.19 with
e
c
/
e
cu
=1 and
x/h
=1
.
v)
It is convenient, and sufficiently accurate, to consider that the
nth
layer of reinforcement yields in compression. The value of
a
max
can be found from Eqn 10.19 with
e
c
/
e
cu
=1 and
x/h
=[
x/h
]
ycn
, where
[
x/h
]
ycn
is the
x/h
ratio at which the
nth
layer of reinforcement yields
in compression (see Eqn 10.26b). When
a
>
a
max
, the section may be
considered to have crushed; hence equilibrium is not possible and
Eqn 10.23 is not solvable.
10.4.3.3
Solution of Equation 10.23
On a curve for any particular
value of
a
, if the concrete strain ratio
e
c
/
e
cu
is known at any point, then the
corresponding value of
x/h
ratio at that point can be found by solving Eqn
10.23. Suppose for the time being, two simplifying assumptions are made:
Assumption (i)
No reinforcement reaches its yield strength, i.e.
f
si
<
f
yi
at all
points on the
curve for the particular value of
a
.
Assumption (ii)
For any positive values assigned to
e
cu
, Eqn 10.23
is solvable for a real and positive root (i.e. for a real and
positive
x/h
).
a
and
e
c
/
c
become constant. Suppose the concrete strain ratio
e
c
/
e
cu
at a certain point on
a curve for a given value of
a
is known, then the corresponding
x/h
ratio
at the point can be determined from Eqn 10.23 as follows:
Case 1:
e
c
/
e
cu
£
[
e
c
/
e
cu
]
x/h
=1
Reference to Figure 10.14 makes it clear that the point lies to the
left of
B;
that is
x/h
‡
1 and the section is
uncracked
. Hence, in Eqn
10.24a, the limit of integration
e
Ó
c
is itself a function of
x/h
(see
Section 10.4.2.3). Therefore, in Eqn 10.23 the coefficient a is a
function of
x/h;
the solution of Eqn 10.23 requires an iterative
method, say the bisection method (Conte and Boor, 1980).
Case 2:
e
c
/
e
cu
>[
e
c
/
e
cu
]
x/h
=1
The point now lies to the right of
B
in Figure 10.14; that is
x/h
<1
and the section is
cracked
. Hence, in Eqn 10.24a,
e
Ó
c
=0 (see
Figure 10.11
b). Eqn 10.23 is therefore a quadratic equation, the
roots of which are
(10.25)
