Civil Engineering Reference
In-Depth Information
Examination of the four unknowns in the four equations, (3.142)-(3.145), suggests that the
best method of solution is the generic trial-and-error procedure:
Step 1:
Check
e
>
e
b
to confirm the tension failure of the column section.
Step 2:
Assume a value of the depth
a
, which should be less than the balanced
a
b
, and calculate
c
/β
1
.
Step 3:
Determine the compression steel strain
=
a
ε
s
from the compatibility equation (3.144), and
the compression steel stress
f
s
from the stress-strain equation (3.145).
Step 4:
Calculate the force
N
n
from the moment equilibrium equation (3.143).
Step 5:
Insert
f
s
and
N
n
into the force equilibrium equation (3.142), and solve for a new value
of the depth
a
.
Step 6:
If the new depth
a
is equal to the assumed depth
a
, a solution is obtained. If not,
assume another depth
a
and repeat the cycle. The convergence is usually quite rapid.
It would be instructive to mention here that the problem posed above is to give the eccentricity
e
and to find the force
N
n
. If the problem is reversed by giving
N
n
and finding
e
, the solution
procedure will be simplified, because Equation (3.142) will contain only two unknowns rather
than three. This simplified solution procedure is left as an exercise for the reader.
3.3.4 Compression Failure
The analysis of column sections failing in compression is more complicated than that of failing
in tension, because two types of analysis are required to deal with two very different situations.
The first situation occurs when the neutral axis lies within the concrete section (
c
h
)as
shown in Figure 3.21. The second situation occurs when the neutral axis lies outside the
concrete section (
c
≤
>
h
) (Figure 3.22). These two situations will be treated separately below.
When
c
h
(Figure 3.21)
When the neutral axis lies within the concrete cross-section, the ACI method of converting
an actual compression stress block into an equivalent rectangular one remains valid. That is
to say, the coefficient
≤
β
1
for the depth
c
and the coefficient 0.85 for the stress
f
c
are still
applicable. Consequently, the six equations, (3.133)-(3.138), remain available for the analysis
of columns failing in compression with
c
≤
h
.
Figure 3.21
Compression failure (c
≤
h) in column sections
