Civil Engineering Reference
In-Depth Information
Figure 3.20
Tension failure in column sections
Second, the tensile steel strain is expected to lie within the plastic range,
ε
s
≥
ε
y
. Bernoulli's
compatibility condition, which relates the tensile steel strain
ε
s
to the maximum strain of
concrete
ε
u
becomes irrelevant to the solution of the stress-type variables. Because of these two
simplifications, analysis of column sections failing in tension involves only twelve variables,
b
,
d
,
A
s
,
A
s
,
N
n
,
e
,
f
s
,
ε
s
,
f
y
,
f
c
,
ε
u
and
a
(or
c
), Figure 3.20. The available equations are now
down to four, in which two are from the equilibrium condition, one is from the compatibility
condition of compression steel and one is from the stress-strain relationship of compression
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 columns failing in tension is:
Given:
b
,
d
,
A
s
,
A
s
,
e
,
f
y
,
f
c
and
ε
u
=
0.003
Find:
N
n
,
f
s
,
ε
s
and
a
(
c
=
a
/β
1
)
The four available equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
N
n
=
0
.
−
A
s
f
y
N
n
f
s
Equilibrium of forces
a
(3
.
142)
A
s
f
s
+
0
85
f
c
ba
d
2
a
.
−
N
n
e
=
N
n
f
s
Equilibrium of moment about
T
a
(3
.
143)
A
s
f
s
(
d
d
)
+
−
ε
s
ε
u
=
d
c
−
ε
s
a
Compatibility of compression steel
(
c
=
a
/β
1
)
(3
.
144)
c
f
s
E
s
ε
s
ε
s
≤
ε
y
f
s
ε
s
Constitutive law of compression steel
=
for
(3
.
145a)
f
s
ε
s
>ε
y
f
s
ε
s
=
f
y
for
(3
.
145b)
