Civil Engineering Reference
In-Depth Information
Inserting
a
b
from Equation (3.76),
A
sb
from Equation (3.77),
f
s
=
f
y
and the reduction
factor
ϕ
into the moment equilibrium equation (3.73), results in
1
f
c
bd
2
600
600
M
ub
=
ϕ
0
.
85
β
1
−
0
.
5
β
1
(3.79)
600
+
f
y
(MPa)
600
+
f
y
(MPa)
R
b
bd
2
, where
R
b
is the expression within
the bracket. The coefficient
R
b
is a function of the material properties
f
c
Equation (3.79) can be written simply as
M
ub
=
and
f
y
(tabulated in
some textbooks).
The balanced condition expressed by
a
b
(Equation 3.76),
ρ
b
(Equation 3.78) or
M
ub
(Equa-
tion 3.79), divides under-reinforced beams from over-reinforced beams. The balanced percent-
age of steel
ρ
b
is useful for the analysis type of problems when the cross-sections of concrete
and steel are given. The balanced factored moment
M
ub
is convenient for the design type of
problems when the moment is given. The balanced depth of rectangular stress block
a
b
could
be used either in analysis or design, but mostly in design in lieu of
M
ub
.
The application of the balanced condition discussed above can be summarized in the
following table:
Types of problem
Under-reinforced beams
Over-reinforced beams
Analysis
ρ<ρ
b
ρ>ρ
b
Design
M
u
<
M
ub
or
a
<
a
b
M
u
>
M
ub
or
a
>
a
b
3.2.2.3 Over-reinforced Beams
If the percentage of steel of a beam is greater than the balanced percentage (
ρ>ρ
b
)orthe
factored moment is greater than the balanced factored moment (
M
u
>
M
ub
), then the beam is
over-reinforced. The problem posed for the analysis of over-reinforced beams is:
Given:
b
,
d
,
A
s
,
f
c
,
ε
u
=
0.003
Find:
M
u
,
f
s
,
ε
s
<ε
y
, and
a
(or
c
)
The four available equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
Equilibrium of forces
A
s
f
s
=
0
.
f
s
a
(3
.
80)
A
s
f
s
d
a
2
Equilibrium of moments
M
u
=
ϕ
−
M
u
f
s
a
(3
.
81)
a
β
1
d
=
ε
u
ε
u
+
ε
y
Bernoulli compatibility
ε
s
a
(3
.
82)
Constitutive law for steel
f
s
=
E
s
ε
s
for
ε
s
≤
ε
y
f
s
ε
s
(3
.
83)
Based on the unknowns in each equations, the best strategy to solve the above four equa-
tions is as follows: (1) Substitute
ε
s
from the stress-strain relationship of Equation (3.83) into
