Civil Engineering Reference
In-Depth Information
When equations 6.81 and 6.82 are both satisfied at all points along the beam,
then the deflected and twisted position is one of equilibrium. Such a position is
defined by the buckled shape
π
2
EI
z
/
L
2
φ
=
δ
sin
π
x
M
cr
v
=
L
,
(6.1)
inwhichthemaximumdeflection
δ
isindeterminate.Thisbuckledshapesatisfies
the boundary conditions at the supports of lateral deflection prevented,
(
v
)
0
=
(
v
)
L
=
0,
(6.83)
twist rotation prevented,
(φ)
0
=
(φ)
L
=
0,
(6.84)
and ends free to warp (see Section 10.8.3),
d
2
φ
d
x
2
d
2
φ
d
x
2
=
=
0.
(6.85)
0
L
Equation6.1alsosatisfiesthedifferentialequilibriumequations(equations6.81
and 6.82) when
M
cr
=
M
zx
, where
π
2
EI
z
L
2
GI
t
+
π
2
EI
w
L
2
M
zx
=
(6.3)
which defines the moment at elastic lateral buckling.
6.12.1.2 Beams with unequal end moments
The major axis bending moment
M
y
and shear
V
z
in the beam with unequal end
moments
M
and
β
m
M
shown in Figure 6.4a are given by
M
y
=
M
−
(
1
+
β
m
)
Mx
/
L
,
and
V
z
=−
(
1
+
β
m
)
M
/
L
.
When the beam buckles, the minor axis bending equation is
EI
z
d
2
v
d
x
2
=−
M
y
φ
,
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