Civil Engineering Reference
In-Depth Information
P
4
P
k
EI
L
EI
L
(a)
(b)
F
IGURE
P.21.1
Subsequently it is required to carry an increased load and for that it is provided with a
lateral spring at the free end (Fig. P.21.1 (b)). Determine the necessary spring stiffness,
k
, so that the buckling load is 4
P
.
tan
µ
L
) where
µ
2
Ans. k
=
4
P
µ
/(
µ
L
−
=
P
/
EI
.
P.21.2
A pin-ended column of length
L
and flexural rigidity
EI
is reinforced to give a
flexural rigidity 4
EI
over its central half. Determine its lowest buckling load.
Ans.
24
.
2
EI
/
L
2
.
P.21.3
A uniform pin-ended column of length
L
and flexural rigidity
EI
has an initial
curvature such that the lateral displacement at any point between the column and the
straight line joining its ends is given by
a
4
x
v
0
=
L
2
(
L
−
x
)
where
a
is the initial displacement at the mid-length of the column and the origin for
x
is at one end.
Show that the maximum bending moment due to a compressive axial load,
P
,is
given by
sec
µ
L
1
P
EI
P.21.4
A compressionmember is made of circular section tube having a diameter
d
and
thickness
t
and is curved initially so that its initial deflected shape may be represented
by the expression
8
aP
(
µ
L
)
2
where
µ
2
M
max
=−
2
−
=
sin
π
x
L
v
0
= δ
in which
δ
is the displacement at its mid-length and the origin for
x
is at one end.
Show that if the ends are pinned, a compressive load,
P
, induces a maximum direct
stress,
σ
max
, given by
1
P
π
dt
1
4
d
σ
max
=
+
1
−
α
