Civil Engineering Reference
In-Depth Information
Substituting Equation (5.24) into Equation (5.26) and rearranging,
2
a
d
a
d
(
)
−− −
20.208
QQ
+
2.77
Q
=
0
(5.27)
1
1
2
f
f
where
f
f
d
d
y
Q
1
=ρ
(5.28)
s
cf
f
f
d
d
s
(5.29)
Q
1
=ρ
s
cf
d
d
QQQ
=−Ψ−
and
Q
fromEquation(5.16)
(5.30)
2
2
1
f
2
f
a
d
= −−
(5.31)
Assuming
B
≈
2
1
12.77
Q
2
f
Q
2
in Equation (5.30) may be directly applied to find
d
f
if compression
steel has yielded (which is typically assumed and then checked using the strain
compatibility, Equation [5.22]). If this yielding happens, the solution is com-
plete. Otherwise, Equation (5.22) is substituted into Equations (5.29) and (5.30),
resulting in a cubic expression,
3
2
a
d
a
d
a
d
β
d
d
(
)
1
−
2
+
2.77
Q
+
2.77
QQ
−
=
0
(5.32)
2
2
2
f
f
f
f
where
87
d
d
d
d
QQ
=−ρ
Ψ −
when U.S. customary units are used
(5.33a)
2
s
f
f
cf
f
600
d
d
d
d
QQ
=−ρ
Ψ −
when S.I. units are used
(5.33b)
2
s
f
f
c
f
f
Similar equations were first derived by Rasheed and Pervaiz (2003) for doubly
reinforced sections with
Ψ
f
=
1.0.
Example 5.6: Analysis
Redo Example 5.3 assuming a doubly reinforced concrete section and considering
f
y
=
83.5 ksi as well.
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