Civil Engineering Reference
In-Depth Information
N
n
is located at an eccentricity
e
measured from the plastic centroid or at an eccentric-
ity of
e
measured from the centroid of the tensile steel. The latter is frequently more
convenient for the analysis. The difference between
e
and
e
is the distance
d
p
given by
Eqution (3.132).
The mild steel has an elastic-perfectly plastic stress-strain curve, as shown in Figure 3.19(e).
In a mild steel reinforced column, therefore, the tensile steel may or may not yield at ultimate.
If the tension steel is in the plastic range (
ε
s
≥
ε
y
) when the concrete crushes at ultimate load
(
ε
u
=
0.003), the column is said to fail in tension. If the tension steel is in the elastic range
(
ε
s
<ε
y
) when the concrete crushes, the column is said to fail in compression. In order to
divide these two types of columns, we define the 'balanced condition' when the steel reaches
the yield point (
ε
s
=
ε
y
) simultaneously with the crushing of concrete (
ε
u
=
0.003).
The eccentric loading of an unsymmetrically reinforced rectangular column involves thirteen
variables
b
,
d
,
A
s
,
A
s
,
N
n
,
e
,
f
s
,
f
s
,
f
c
,
ε
s
,
ε
s
,
ε
u
and
a
(or
c
), as shown in Figure 3.19(a)-(c).
The coefficient
β
1
is not considered a variable because it has been determined independently
from tests. A total of six equations are available, two from equilibrium (Figure 3.19c), two
from Bernoulli compatibility (Figure 3.19b), and two from the constitutive law of mild steel
(Figure 3.19e). Therefore, seven variables must be given before the remaining six unknown
variables can be solved by the six equations.
3.3.2 Balanced Condition
The problem posed for the balanced condition is:
Given:
b
,
d
,
A
s
,
A
s
,
f
c
,
ε
s
=
ε
y
,
ε
u
=
0.003
Find:
N
n
,
e
,
f
s
,
f
s
,
ε
s
,
a
(or
c
=
a
/β
1
)
The six available equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
N
n
=
0
.
−
A
s
f
s
f
s
f
s
Equilibrium of forces
N
n
a
(3
.
133)
A
s
f
s
+
0
85
f
c
ba
d
2
a
.
−
Equilibrium of moment about
TN
n
e
=
N
n
e
f
s
a
(3
.
134)
A
s
f
s
(
d
d
)
+
−
a
β
1
d
=
ε
u
ε
u
+
ε
y
Compatibility of tension steel
a
(3
.
135)
Compatibility of compression
steel
ε
s
ε
u
=
d
c
−
ε
s
(
c
−
a
/β
1
)
a
(3
.
136)
c
Consttutive law of tension steel
f
s
=
E
s
ε
s
for
ε
s
≤
ε
y
f
s
(3
.
137a)
f
s
=
f
y
for
ε
s
>ε
y
f
s
(3
.
137b)
Constitutive law of compression
steel
f
s
E
s
ε
s
ε
s
≤
ε
y
,
f
s
,ε
s
=
for
(3
.
138a)
f
s
ε
s
>ε
y
f
s
,ε
s
=
f
y
for
(3
.
138b)
