Civil Engineering Reference
In-Depth Information
Figure 3.22
Compression failure (c > h) in column sections
An obvious observation is available, however, to simplify the analysis of a column section
failing in compression with
c
≤
h
. The compression steel strain is expected to lie within the
ε
s
ε
s
is close to the
plastic range,
≥
ε
y
, because the neutral axis is low enough such that
ultimate concrete strain
ε
u
of 0.003. This is true for all columns of practical size, except those
very small ones with height
h
close to the lowest allowable limit of 305 mm (12 in.), and
depth
a
close to the balanced
a
b
. Bernoulli's compatibility condition, which relates
ε
s
to
ε
u
,
becomes irrelevant to the solution of the stress-type variables. As a result, the compression
steel is expected to yield
f
s
=
f
y
and the stress-strain curve of compression steel is not
required in the solution. Because of this simplification, analysis of column sections failing in
compression involves only twelve variables,
b
,
d
,
A
s
,
A
s
,
N
n
,
e
,
f
s
,
ε
u
and
c
(or
a
) (Figure 3.21). The number of available equations is reduced to four, in which two are from
the equilibrium condition, one is from the compatibility condition of tension steel and one is
from the stress-strain relationship of tension steel. Therefore, eight variables must be given
before the four remaining unknown variables can be solved by the four available equations.
The problem posed for the analysis of column sections failing in compression with
c
ε
s
,
f
y
,
f
c
,
≤
h
is:
Given:
b
,
d
,
A
s
,
A
s
,
e
,
f
y
,
f
c
,
ε
u
=
0.003
ε
s
, and
a
(or
c
=
/β
1
)
Find:
N
n
,
f
s
,
a
The four available equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
A
s
f
y
Equilibrium of forces
N
n
=
0
.
−
A
s
f
s
+
N
n
f
s
a
(3
.
146)
0
85
f
c
ba
d
2
a
.
−
n
e
=
Equilibrium of moment about
T
N
n
a
(3
.
147)
A
s
f
y
(
d
d
)
+
−
ε
s
ε
u
=
d
−
c
Compatibility of tension steel
(
c
=
a
/β
1
)
ε
s
a
(3
.
148)
c
Constitutive law of tension steel
f
s
=
E
s
ε
s
for
ε
s
≤
ε
y
f
s
ε
s
(3
.
149)
