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As demonstrated in Example 11.4, imperfections may reduce the critical loads sub-
stantially. Therefore, a safe design involves the imposition of possible imperfections
that lead to the lowest critical loads. However, the shapes and the amplitudes cor-
responding to the imperfections are usually not known in advance and have to be
assumed. The buckling mode corresponding to the lowest eigenvalue of the system is
considered to be the worst shape, where the amplitudes can be taken from appropriate
design codes.
EXAMPLE 11.12 Simply Supported Beam with Shear Deformation
Find the buckling load of the beam of Fig. 11.40a. Assume displacements of the form
u
=
u
=
C
1
(
x
+
L
/
2
)
→
C
1
w
=
C
2
(
L
2
/
4
−
x
2
)
→
w
=−
2
C
2
x
(1)
→
θ
=
θ
=
C
3
x
C
3
which are sketched in Fig. 11.40b. These displacements satisfy all displacement boundary
conditions
. Substitution of these trial functions
into the principle of virtual work expression of Eq. (11.59) gives
(
u
(
−
L
/
2
)
=
0
,
w(
−
L
/
2
)
=
w(
L
/
2
)
=
0
)
=
EA
0
0
C
1
0
L
/
2
0
(
−
2
x
)(
−
P
)(
−
2
x
)
(
−
2
x
)
k
s
GAx
dx
C
2
0
(2)
+
(
−
2
x
)
k
s
GA
(
−
2
x
)
−
L
/
2
C
3
0
0
xk
s
GA
(
−
2
x
)
EI
+
xk
s
GAx
where
N
0
has been replaced by the compressive axial force
−
P
. Integration of this
FIGURE 11.40
A simply supported beam with shear deformation.
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