Civil Engineering Reference
In-Depth Information
Figure 3.12
Doubly reinforced rectangular sections at ultimate
two additional equations are available, one for the compatibility of compression steel and the
other for the stress-strain relationship of the compression steel.
The analysis and design of doubly reinforced rectangular sections will be limited to ductile
beams in this section, using the stress-strain relationships of concrete and steel shown in Figure
3.12(f) and (g). In this type of problems the tensile steel will be in the yield range,
f
s
=
f
y
, and
the tensile steel strain,
ε
s
, is irrelevant to the solution of stress-type variables. Correspondingly,
the compatibility equation for the tensile steel and the stress-strain relationship of the tensile
steel are not required. As a result, we now have eleven variables,
b
,
d
,
A
s
,
A
s
,
M
u
,
f
y
,
f
s
,
f
c
,
ε
s
,
ε
u
and
c
(or
a
), and four available equations. Two of these four equations come from the
equilibrium condition and the other two from the compatibility and stress-strain relationship
of compression steel. If seven variables are given, the remaining four unknown variables can
be solved by the four available equations.
The analysis and design of doubly reinforced beams will now be treated separately.
3.2.3.1 Analysis of Ductile Sections
Analysis problems to find moment are posed as follows:
Given:
b
,
d
,
A
s
,
A
s
,
f
y
,
f
c
,
ε
u
Find:
M
u
,
f
s
,
ε
s
and
a
(or
c
=
a
/β
1
)
