Civil Engineering Reference
In-Depth Information
Numerical solutions of these equations for the dimensionless buckling load
QL
2
/
√
(
EI
z
GI
t
)
are available [3, 13, 16, 42], and some of these are shown in
Figure 6.6, in which the dimensionless height
ε
of the point of application of the
load is generally given by
EI
z
GI
t
ε
=
z
Q
L
,
(6.8)
or for the particular case of equal flanged I-beams, by
2
z
Q
d
f
.
ε
=
K
π
6.12.1.4 Cantilevers with concentrated end loads
The elastic buckling of a cantilever with a concentrated end load
Q
applied at a
distance
z
Q
belowthecentroidcanbepredictedfromthesolutionsofthedifferential
equations of minor axis bending
EI
z
d
2
v
d
x
2
=−
M
y
φ
,
and of torsion
d
x
−
EI
w
d
3
φ
GI
t
d
φ
d
x
3
=
Q
(
v
−
z
Q
φ)
L
+
M
y
d
v
d
x
−
V
z
v
,
in which
M
y
=−
Q
(
L
−
x
)
and
V
z
=
Q
.
These solutions must satisfy the fixed end (
x
=
0) boundary conditions of
(
v
)
0
=
(φ)
0
=
(
d
v
/
d
x
)
0
=
(
d
φ/
d
x
)
0
=
0
and the condition that the end
x
=
L
is free to warp, whence
(
d
2
φ/
d
x
2
)
L
=
0.
Numericalsolutionsoftheseequationsareavailable[16,37,42,43],andsome
of these are shown in Figure 6.20.
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