Civil Engineering Reference
In-Depth Information
The generic trial-and-error procedure can be used to solve these four equations:
Step 1:
Check
e
<
e
b
to confirm the compression failure of the column section.
Step 2:
Assume a value of the depth
a
, which should be greater than the balanced
a
b
, and
calculate
c
/β
1
.
Step 3:
Determine the tensile steel strain,
=
a
ε
s
, from the compatibility equation (3.148) and the
tensile steel stress
f
s
from the stress-strain equation (3.149).
Step 4:
Calculate the force
N
n
from the moment equilibrium equation (3.147).
Step 5:
Insert
f
s
and
N
n
into the force equilibrium equation (3.146) 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.
Step 7:
Check the assumption for the yielding of the compression steel,
ε
s
≥
ε
y
, by Equation
(3.136). In all practical cases, this step is not necessary.
Incidentally, the problem posed here 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 (3.146) will contain only two unknowns rather than three. The reader should
now be able to work out this simplified solution procedure if required.
When
c
h
(Figure 3.22)
When the neutral axis lies outside the concrete cross-section, the ACI rule to convert the actual
compression stress block to an equivalent rectangular one is no longer valid. In other words,
the coefficient
>
β
1
derived from pure bending, is not applicable. Therefore, a column section
failing in compression with
c
h
requires special analysis.
When the neutral axis leaves the bottom surface, failure begins to change from a compression
failure at the extreme fiber (say
>
ε
u
=
0.003) to a concentric compression failure of the whole
cross-section (
0.002). In this range of transition no simple method is available to determine
the magnitude and the location of the concrete compression stress resultant
C
(Figure 3.22c).
Fortunately, tests have shown that the moment of
C
about the bottom steel is approximately
equal to the moment of a rectangular stress block shown in Figure 3.22(d) about the same
bottom steel. The rectangular stress block has a stress of 0
ε
o
=
.
85
f
c
throughout the whole cross-
section. Hence,
85
f
c
bh
d
h
2
Cg
=
0
.
−
(3.150)
where
g
is the distance from the compression resultant
C
to the centroid of the bottom steel.
Taking all the moments about the bottom steel gives:
85
f
c
bh
d
h
2
N
n
e
=
A
s
f
y
(
d
d
)
0
.
−
+
−
(3.151)
Equation (3.151) shows that
N
n
e
is no longer a function of the depth
c
(or
a
). That is to
say, the position of the neutral axis can not be determined.
