Civil Engineering Reference
In-Depth Information
From the stress diagram in Figure 3.9(c), we recognize that the external moment
M
n
is
resisted by the internal moment, consisting of a pair of equal, opposite and parallel forces,
T
and
C
, located at a distance
jd
from each other. This idea of bending resistance is frequently
referred to as the
internal couple concept
. From the force equilibrium
T
=
C
we have
k
1
f
c
bc
A
s
f
s
=
(3.55)
From the moment equilibrium, the two most convenient forms are:
M
n
=
T
(
jd
)or
M
n
=
C
(
jd
), resulting in:
M
n
=
A
s
f
s
(
jd
)
=
A
s
f
s
(
d
−
k
2
c
)
(3.56)
k
1
f
c
bc
(
jd
)
k
1
f
c
bc
(
d
M
n
=
=
−
k
2
c
)
(3.57)
Although we have written three equilibrium equations (3.55)-(3.57), only two of these
equations are independent, meaning that they could only be used to solve two unknown
variables.
Using two of the three equilibrium equations, the compatibility equation (3.54), and the
two stress-strain curves of concrete and steel, we can solve the variable
c
and determine the
location of the neutral axis.
Finally, from the linear strain distribution given in Figure 3.9(b) the ultimate curvature
φ
u
is expressed as
φ
u
=
ε
u
c
(3.58)
Equations (3.56) and (3.58) establish the moment-curvature relationship (
M
n
and
φ
u
)at
ultimate strength.
The nonlinear constitutive relationships of concrete will be elaborated in Section 3.2.1.2.
The detailed solution procedures using Equations (3.54)-(3.57) and the nonlinear constitutive
laws of concrete and steel will be explained in Sections 3.2.2, 3.2.3 and 3.2.4 for singly
reinforced, doubly reinforced and flanged sections, respectively.
3.2.1.2 Nonlinear Constitutive Relationship of Concrete
Theoretical Derivation
The distribution of stresses in the concrete compression stress block (Figure 3.9c), is assumed
to have the same shape as the stress-strain curve of concrete (Figure 3.9e), obtained from the
tests of standard 6
12 in. concrete cylinders. The stress-strain curve of concrete is frequently
expressed by a parabolic equation:
×
2
ε
ε
o
2
ε
ε
o
f
c
σ
=
−
(3.59)
where
σ
=
compressive stress of concrete;
ε
=
compressive strain of concrete;
f
c
=
maximum compressive stress of concrete obtained from standard cylinders;
strain at the maximum stress
f
c
, usually taken as 0.002.
ε
o
=
