Civil Engineering Reference
In-Depth Information
Figure 3.11
Under-Reinforced sections at ultimate
The strain and stress diagrams for a ductile beam are shown in Figure 3.11(b) and (c),
respectively. The analysis and design of ductile beams are considerably simplified for two
reasons. First, the tensile steel stress is known to yield,
f
s
=
f
y
, and the stress-strain curve
of steel is not required in the solution. Second, the tensile steel strain is expected to lie within
the plastic range,
ε
s
≥
ε
y
. Bernoulli's compatibility condition, which relates the steel strain
ε
s
to the maximum concrete strain
ε
u
at the top surface, becomes irrelevant to the solution
of other stress-type variables. Because of these two simplifications, the analysis and design
of under-reinforced beams involve only eight variables,
b
,
d
,
A
s
,
M
u
,
f
y
,
f
c
,
ε
u
and
a
(or
c
).
The available equations are now down to two, the only two from the equilibrium condition.
Therefore, six variables must be given before the two remaining unknown variables can be
solved by the two equilibrium equations.
The analysis problem to find the moment is posed as follows:
Given:
b
,
d
,
A
s
,
f
y
,
f
c
and
ε
u
Find:
M
u
and
a
As mentioned previously, three forms of equilibrium equations are frequently used in the
parallel, coplanar force system of bending action, but only two are independent. However,
the three equations are given below for the purpose of finding the most convenient solution.
Remember that only two of these three equations will be selected.
Type of equations
Equations
Unknowns
85
f
c
ba
Equilibrium of forces
A
s
f
y
=
0
.
a
(3
.
84)
A
s
f
y
d
a
2
Equilibrium of moment about
C
=
ϕ
−
M
u
a
(3
.
85)
u
85
f
c
ba
d
a
2
Equilibrium of moment about
C
=
ϕ
0
.
−
M
u
a
(3
.
86)
u
