Civil Engineering Reference
In-Depth Information
3.10 Appendix - effective lengths of compression
members
3.10.1 Intermediate restraints
Astraightpin-endedcompressionmemberwithacentralelasticrestraintisshown
in Figure 3.16a. It is assumed that when the buckling load
N
cr
is applied to the
member,itbucklessymmetricallyasshowninFigure3.16awithacentraldeflec-
tion
δ
, and the restraint exerts a restoring force
αδ
. The equilibrium equation for
this buckled position is
EI
d
2
v
d
x
2
=−
N
cr
v
+
αδ
2
x
for 0
≤
x
≤
L
/
2.
The solution of this equation which satisfies the boundary conditions
(
v
)
0
=
(
d
v
/
d
x
)
L
/
2
=
0 is given by
x
L
−
v
=
αδ
L
2
N
cr
sin
π
x
/
k
cr
L
2
(π/
2
k
cr
)
cos
π/
2
k
cr
,
where
k
cr
=
L
cr
/
L
and
L
cr
is given by equation 3.23. Since
δ
=
(
v
)
L
/
2
,it
follows that
3
π
2
k
cr
cot
π
2
k
cr
=
α
L
3
16
EI
.
(3.65)
2
k
cr
cot
π
π
2
k
cr
−
1
The variation with the dimensionless restraint stiffness
α
L
3
/
16
EI
of the dimen-
sionless buckling load
2
π
2
EI
/
L
2
=
4
N
cr
π
2
k
cr
(3.66)
π
2
which satisfies equation 3.65 is shown in Figure 3.16c. It can be seen that the
buckling load for this symmetrical mode varies from
π
2
EI
/
L
2
when the restraint
is of zero stiffness to approximately 8
π
2
EI
/
L
2
when the restraint is rigid. When
the restraint stiffness exceeds
α
L
=
16
π
2
EI
/
L
3
,
(3.31)
the buckling load obtained from equations 3.65 and 3.66 exceeds the value of
4
π
2
EI
/
L
2
for which the member buckles in the antisymmetrical second mode
shown in Figure 3.16b. Since buckling always takes place at the lowest possible
load, it follows that the member buckles at 4
π
2
EI
/
L
2
in the second mode shown
in Figure 3.16b for all restraint stiffnesses
α
which exceed
α
L
.
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