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Hint:
Use identities of the sort
1
L
L
0if
i
sin
i
π
x
sin
k
π
x
=
k
dx
=
1
/
2 f
i
=
k
L
L
0
If
β
=
0, the cross-section is uniform, and for
m
=
1 you should obtain
P
cr
=
2
EI
0
/
L
2
.If
π
β
=
0 and
m
=
1, i.e., only one term is retained,
L
2
EI
0
1
2
=
π
8
β
P
cr
+
3
π
1.
11.7 Find the critical axial load
P
in the stepped column of Fig. P11.7.
This is less than 1.5% too high for
β
=
L
2
Answer:
P
cr
≈
4
.
3
EI
/
FIGURE P11.7
FIGURE P11.8
11.8 Find the buckling load of the column of Fig. P11.8, with a spring support at one end.
L
2
Answer:
P
cr
=
ε
0
EI
/
where
ε
0
is the lowest non-zero solution of tan
ε
=
0
−
ε
k
EI
L
2
ε
(
1
)
0
11.9 Find the critical load for the spring supported column with constant
EI
shown in
Fig. P11.9.
11.10 For a fixed-fixed column of length
L
, calculate the buckling load using a four-element
model.
=
.
/
L
2
Answer:
P
cr
39
47
EI
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