Civil Engineering Reference
In-Depth Information
1
)/k
2
L
2
)
is a moment magnification factor accounting for the
effects of the axial compressive force,
P
. The secant function can be expanded in a
power series as (Beyer, 1984)
where
(
8
(
sec
kL/
2
−
kL
2
2
kL
2
4
kL
2
6
sec
kL
1
2
5
24
61
720
2
=
+
+
+
+···
1
.
(8.42)
=
√
P/EI
,
2
EI/L
2
(Euler buckling load) and
k
Since
P
e
= π
P
P
e
kL
2
=
2
.
(8.43)
Substitution of Equation 8.43 into Equations 8.41 and 8.42 provides
1
(8.44a)
1.028
P
P
e
1.031
P
P
e
2
1.032
P
P
e
3
wL
2
8
M
z
=
L
/
2
=
+
+
+
+···
1
1
.
1.028
P
P
e
1.003
P
P
e
1.004
P
P
e
2
wL
2
8
=
+
+
+
+···
(8.44b)
Equation 8.44b may be approximated as
1
1
1.028
P
P
e
P
P
e
P
P
e
2
P
P
e
3
wL
2
8
M
z
=
L/
2
≈
+
+
+
+
+···
(8.45)
1
+
1.028
P
P
e
wL
2
8
1
≈
(8.46)
1
−
(
P/P
e
)
,
wL
2
8
1
≈
(8.47)
1
−
(P/P
e
)
where 1
/(
1
(P/P
e
))
is an approximate moment magnification factor appropriate
for use in design.
−
8.4.2.1.2 Axial Compression and Bending from a Concentrated
Transverse Load
Qz(L
−
a)
M
p
(z)
=
M
Q(z)
=
for 0
≤
z
≤
a
,
(8.48a)
L
Qa(L
−
z)
M
p
(z)
=
M
Q(z)
=
for
a
≤
z
≤
L
.
(8.48b)
L
