Civil Engineering Reference
In-Depth Information
The lowest buckling mode corresponds to a value of
n =
1, which is the
case of single curvature shown. The other (higher) modes can also be
found with the other values of
n
.
cr
=
p
2
EI
L
P
2
The critical bucking load is sometimes written as a critical buckling stress
as follows:
P
A
p
EI
AL
2
p
2
EIr
IL
2
p
2
E
c
== =
F
=
(
)
cr
2
2
L
2
r
2
This is known as the Euler buckling stress and was derived by Leonhard
Euler in 1757.
Example 3.12
Column buckling with difference operator
Calculate the critical buckling load,
P
cr
, for a 25 foot long fixed end col-
umn using central difference operator of order of error
h
2
at 1/6th points.
The column has
E
= 29,000 ksi and
I
= 1000 in
4
.
This problem has the same model as Example 3.11 except an axial
load is applied instead of a uniform lateral load and is shown in Figure 3.9.
The central difference expressions with error of order
h
2
will be used
to solve for the values. Since the load is known, we will use the fourth
derivative relationship between load and deflection. This can be found
Z
P
X
A
B
25′-0″
Y
3
Y
2
Y
1
Y
0
Y
1
Y
2
Y
3
Y
2
Y
1
Y
0
Y
1
Y
2
Y
3
Figure 3.9.
Example 3.12 Column buckling with difference operator.
