Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

The equation of the buckled column can be obtained by substituting C 2 = −C 1 kL [Equation (f)]
and B y /P = C 1 k [from Equation (e)] into Equation (16.11), yielding
683
THE EFFECT OF ENd
CONdITIONS ON
COLuMN buCkLINg
v = C sinkx − CkLcos kx + C k( L − x)
= C [sin kx − kLcos kx + k( L − x)]
= C
1 1 1
1
1
⎧ ⎛ x
x
x
sin 4.4934 ⎞ ⎡ ⎛
⎜ 4.4934 1 cos 4.4934 ⎞⎤⎫
⎨
⎟ + − − ⎜ ⎟
⎝ L ⎠
⎢
⎣ L ⎝ L ⎠
⎥⎬
⎩
⎦⎭
(16.14)
The expression inside the braces is the shape of the first buckling mode for a fixed–pinned
column. The constant C 1 cannot be evaluated; therefore, the amplitude of the curve is
undefined, although deflections are assumed to be small.
Effective-Length concept
From Equation (16.5), the Euler buckling load for a pinned–pinned column is
P
cr
2
π EI
=
2
L
and from Equation (16.13), the critical buckling load for a fixed–pinned column is
P
cr
=
2
2.0457π
EI
2
L
A comparison of these two equations shows that the form of the critical load equation for a
fixed–pinned column is nearly identical to the form of the Euler buckling load equation.
Indeed, the two equations differ by only a constant. This similarity suggests that it is possible
to relate the buckling loads of columns with various end conditions to the Euler buckling load.
The critical buckling load for a fixed–pinned column of length L is given by Equation
(16.13), and this critical load is greater than the Euler buckling load for a pin-ended column
of the same length L (assuming that EI is the same for both cases). What would the length
of an equivalent pin-ended column have to be in order for the equivalent pinned–pinned
column to buckle at the same critical load as the actual fixed–pinned column? Let L denote
the length of the fixed–pinned column, and let L e denote the length of the equivalent pinended
column that buckles at the same critical load. Equating the two critical loads gives
or
2
2.0457π
EI
2
L
L
e =
2
π EI
=
2
L
0.7L
Therefore, if the column length used in the Euler buckling load equation were modified to
an effective length L e = 0.7L, then the critical load calculated from Equation (16.5) would be
identical to the critical load calculated from the actual column length in Equation (16.13).
This idea of relating the critical buckling loads of columns with various end conditions to
the Euler buckling loads is known as the effective-length concept.
The effective length L e for any column is defined as the length of the equivalent
pin-ended column. But what is meant by “equivalent” in this context? An equivalent pin-ended
column has the same critical buckling load and the same deflected shape as all or part of
the actual column.
e