Civil Engineering Reference
In-Depth Information
Solution for lowest
critical load
tan
mL
mL
mL
p
3
p
2
2
F
IGURE
21.8
Solution of a
transcendental equation
Also
v
=
0at
x
=
0. Then
0
=
tan
µ
L
−
µ
L
or
µ
L
=
tan
µ
L
(21.20)
Equation (21.20) is a transcendental equation which may be solved graphically as
shown in Fig. 21.8. The smallest non-zero value satisfying Eq. (21.20) is approxi-
mately 4.49.
This gives
20
.
2
EI
L
2
P
CR
=
which may be written approximately as
2
.
05
π
2
EI
L
2
P
CR
=
(21.21)
It can be seen from Eqs (21.5), (21.10), (21.15) and (21.21) that the buckling load in
all cases has the form
K
2
π
2
EI
L
2
P
CR
=
(21.22)
in which
K
is some constant. Equation (21.22) may be written in the form
π
2
EI
L
e
P
CR
=
(21.23)
in which
L
e
(
L
/
K
)isthe
equivalent length
of the column, i.e. (by comparison of
Eqs (21.23) and (21.5)) the length of a pin-ended column that has the same buckling
load as the actual column. Clearly the buckling load of any column may be expressed
in this form so long as its equivalent length is known. By inspection of Eqs (21.5),
=
