Civil Engineering Reference
In-Depth Information
We shall now assume a deflected shape having the equation
∞
A
n
sin
n
π
x
L
v
=
(21.67)
n
=
1
This satisfies the boundary conditions of
d
2
v
d
x
2
d
2
v
d
x
2
(
v
)
x
=
0
=
(
v
)
x
=
L
=
0
0
=
L
=
0
x
=
x
=
and is capable, within the limits for which it is valid and if suitable values for the constant
coefficients,
A
n
, are chosen, of representing any continuous curve. We are therefore
in a position to find
P
CR
exactly. Substituting Eq. (21.67) into Eq. (21.66) gives
L
4
∞
2
L
π
EI
2
n
2
A
n
sin
n
π
x
L
U
+
V
=
d
x
0
n
=
1
L
2
∞
2
L
π
P
CR
2
nA
n
cos
n
π
x
L
−
d
x
(21.68)
0
n
=
1
The product terms in both integrals of Eq. (21.68) disappear on integration leaving
only integrated values of the squared terms. Thus
∞
∞
π
4
EI
4
L
3
π
2
P
CR
4
L
n
4
A
n
−
n
2
A
n
U
+
V
=
(21.69)
n
=
1
n
=
1
Assigning a stationary value to the total potential energy of Eq. (21.69) with respect
to each coefficient,
A
n
, in turn, then taking
A
n
as being typical, we have
π
4
EIn
4
A
n
2
L
3
π
2
P
CR
n
2
A
n
2
L
∂
(
U
+
V
)
=
−
=
0
∂
A
n
from which
π
2
EIn
2
L
2
P
CR
=
as before.
We see that each term in Eq. (21.67) represents a particular deflected shape with a
corresponding critical load. Hence the first term represents the deflection of the col-
umn shown in Fig. 21.16 with
P
CR
=
π
2
EI
/
L
2
. The second and third terms correspond
to the shapes shown in Fig. 21.4(b) and (c) having critical loads of 4
π
2
EI
/
L
2
and
9
π
2
EI
/
L
2
and so on. Clearly the column must be constrained to buckle into these
more complex forms. In other words, the column is being forced into an unnatural
shape, is consequently stiffer and offers greater resistance to buckling, as we observe
from the higher values of critical load.
If the deflected shape of the column is known, it is immaterial which of Eqs. (21.65) or
(21.66) is used for the total potential energy. However, when only an approximate solu-
tion is possible, Eq. (21.65) is preferable since the integral involving bending moment
