Civil Engineering Reference
In-Depth Information
The magnitude of the compression resultant
C
can be expressed in the following manner.
Let
x
be the distance from the neutral axis to a level where the strain is
ε
and the stress is
σ
(Figure 3.9b and c). From similar triangle of the strain diagram,
x
=
(
c
/ε
u
)
ε
. Differentiating
x
gives d
x
=
(
c
/ε
u
)d
ε
and the resultant
C
can be expressed by integrating the compression
stress block:
b
c
ε
u
c
ε
u
C
=
b
σ
d
x
=
σ
d
ε
(3.60)
o
o
k
1
f
c
bc
into Equation (3.60) gives the expression for
k
1
:
Substituting
C
=
ε
u
1
f
c
ε
u
k
1
=
σ
d
ε
(3.61)
o
Substituting the concrete stress
σ
from Equation (3.59) into Equation (3.61) and integrating
results in
1
k
1
=
ε
u
ε
o
1
3
ε
u
−
(3.62)
ε
o
0.75.
The location of the resultant
C
is defined by
k
2
c
measured from the top surface of the
cross-section. The coefficient
k
2
is determined by taking moment about the neutral axis:
Inserting the ACI value of
ε
u
=
0.003 and
ε
o
=
0.002 into Equation (3.62) gives
k
1
=
b
c
ε
u
2
c
ε
u
C
(
c
−
k
2
c
)
=
b
σ
x
d
x
=
σε
d
ε
(3.63)
o
o
Substituting
C
from Equation (3.60) into Equation (3.63) gives
o
σε
ε
u
d
ε
1
ε
u
k
2
=
−
1
(3.64)
o
σ
ε
u
d
ε
Again, substituting the concrete stress
σ
from Equation (3.59) into Equation (3.64) and
integrating results in
3
1
−
8
(
ε
u
/ε
o
)
2
3
k
2
=
1
−
(3.65)
1
3
(
1
−
ε
u
/ε
o
)
Inserting the ACI value of
ε
u
=
0.003 and
ε
o
=
0.002 into Equation (3.65) gives
k
2
=
0.4167.
The values of
k
1
=
0.4167 can be inserted into Equations (3.55)-(3.57), so
that these equilibrium equations can be used in connection with the compatibility equation
and the constitutive laws.
0.75 and
k
2
=
