Civil Engineering Reference
In-Depth Information
It can be seen that the solution of the second type of analysis problems is the same as the first
type in steps 1 and 2, i.e. in solving the position of the neutral axis. The only small difference
is in step 3 in the calculation of different unknowns by equilibrium equations.
3.1.2 Transformed Area for Reinforcing Bars
The key problem in the linear bending analysis (Section 3.1.2) is to find the location of the
neutral axis, represented by
k
, using Bernoulli's compatibility condition, the force equilibrium
condition and Hooke's law. A short-cut method to find
k
will now be introduced. The method
utilizes the concept of the transformed area for the rebars.
Based on the assumption that perfect bond exists between the rebar and the concrete, the
rebar strain
ε
s
should be equal to the concrete strain at the rebar level
ε
cs
(Figure 3.1b):
ε
s
=
ε
cs
(3.19)
Applying Hooke's law,
f
s
=
E
s
ε
s
and
f
cs
=
E
c
ε
cs
, to Eq. (3.19) gives:
E
s
E
c
f
s
=
f
cs
=
nf
cs
(3.20)
where
f
cs
is the concrete stress at the rebar level.
Using Equation (3.20) the tensile force
T
can be written as
T
=
A
s
f
s
=
(
nA
s
)
f
cs
(3.21)
Equation (3.21) states that the tensile force
T
can be thought of as being supplied by a
concrete area of
nA
s
in connection with a concrete stress of
f
cs
. Physically, this means that
the rebar area
A
s
can be transformed into a concrete area
nA
s
, as long as the rebar stress
f
s
is simultaneously converted to the concrete stress
f
cs
. A cross-section with the transformed
rebar area is shown in Figure 3.1(d).
The transformation of the rebar area
A
s
to a concrete area
nA
s
has a profound significance.
A flexural beam, which is made up of two materials, steel and concrete, can now be thought
of as a homogeneous elastic material made of concrete only. For such a homogeneous elastic
beam, the neutral axis coincides with the centroidal axis. So, instead of solving the neutral axis
from the five equations of equilibrium, compatibility and constitutive laws, as illustrated in
Section 3.1.1.4, we can now locate the neutral axis using the simple and well-known method
of finding the centroidal axis of a homogeneous beam. In other words, the static moments of
the stressed areas, shaded in Figure 3.1(d), about the centroidal axis must be equal to zero.
Hence,
1
2
b
(
kd
)
2
−
nA
s
(
d
−
kd
)
=
0
(3.22)
The variable
k
in Equation (3.22) can be solved by two methods. The first method is the
trial-and-error procedure. A value of
k
is assumed and is inserted into Equation (3.22). If the
equation is satisfied, the assumed
k
is the solution. If the equation is not satisfied, another
k
value is assumed and the process repeated. This trial-and-error method is very efficient for an
experienced engineer, who can closely estimate the
k
value in the first trial.
