Civil Engineering Reference
In-Depth Information
N
cr
N
cr
N
cr
N
cr
N
cr
L
L
cr
L
cr
L
L
cr
L
L
L
L
cr
L
cr
L
cr
2
2
2
2
2
2
2
2
2
2
N
cr
π
=
L
cr
=
L
(a)
EI
/
L
N
cr
=
L
cr
=
L
/2
(b)
4
π
EI
/
L
N
cr
≈
2
π
EI
/
L
N
cr
=
L
cr
=
L
/2
(d)
4
π
EI
/
L
N
cr
π
=
L
cr
= 2
L
(e)
EI
/
L
L
cr
0.7
L
(c)
≈
Figure 3.15
Effective lengths of columns.
as shown in Figure 3.15b, then its buckled shape
v
is given by
v
=
δ
sin2
π
x
/
L
,
(3.29)
and its elastic buckling load
N
cr
is given by
N
cr
=
4
π
2
EI
L
2
.
(3.30)
The end supports of a compression member may also differ from the simple sup-
ports shown in Figure 3.15a which allow the member ends to rotate but prevent
themfromdeflectinglaterally.Forexample,oneorbothendsmayberigidlybuilt-
insoastopreventendrotation(Figure3.15candd),oroneendmaybecompletely
free (Figure 3.15e). In each case the elastic buckling load of the member may be
obtained by finding the solution of the differential equilibrium equation which
satisfies the boundary conditions.
All of these buckling loads can be expressed by the generalisation of
equation3.23whichreplacesthememberlength
L
ofequation3.2bytheeffective
length
L
cr
. Expressions for
L
cr
are shown in Figure 3.15, and in each case it can
be seen that the effective length of the member is equal to the distance between
the inflexion points of its buckled shape.
Theeffectsofthevariationsinthesupportandrestraintconditionsonthecom-
pression member buckling resistance
N
b
,
Rd
may be accounted for by replacing
the actual length
L
used in equation 3.2 by the effective length
L
cr
in the calcula-
tion of the elastic buckling load
N
cr
and hence the generalised slenderness
λ
, and
using this modified generalised slenderness throughout the resistance equations.
It should be noted that it is often necessary to consider the member behaviour in
each principal plane, since the effective lengths
L
cr
,
y
and
L
cr
,
z
may also differ, as
well as the radii of gyration
i
y
and
i
z
.
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