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

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COLuMNS
Another way of expressing the idea of an effective column length is to consider points
of zero internal bending moment. The pin-ended column, by definition, has zero internal
bending moments at each end. The length L in the Euler buckling equation, therefore, is the
distance between successive points of zero internal bending moment. All that is needed to
adapt the Euler buckling equation for use with other end conditions is to replace L with L e ,
where L e is defined as the effective length of the column—that is, the distance between two
successive points of zero internal bending moment. A point of zero internal bending
moment is termed an inflection point.
The effective lengths of four common columns are shown in Figure 16.7. The
pin-ended column is shown in Figure 16.7a, and by definition, the effective length L e
of this column is equal to its actual length L. The fixed–pinned column is shown in
Figure 16.7b, and as was concluded in the preceding discussion, its effective length is
L e = 0.7L.
The ends of the column in Figure 16.7c are fixed. Since the deflection curve is symmetrical
for this column, inflection points occur at distances of L/4 from each fixed end.
The effective length is therefore represented by the middle half of the column length. Thus,
the effective length L e of a fixed–fixed column for use in the Euler buckling equation is
equal to one-half of the actual length of the column, or L e = 0.5L.
The column in Figure 16.7d is fixed at one end and free at the other end; consequently,
the column has a zero internal bending moment only at the free end. If a mirror image of
this column is visualized below the fixed end, however, the effective length between points
of zero moment is seen to be twice the actual length of the column (L e = 2L).
Effective-Length Factor
To simplify critical load calculations, many design codes employ a dimensionless coefficient
K called the effective-length factor, which is defined as
Le = KL
(16.15)
P
P
P
P
B
B
B
L
B
Le=
L
Le=
0.7L
L e= 0.5L
A
Le=
2L
A
A
A
(a) Pinned–pinned
column: K = 1
(b) Fixed–pinned
column: K = 0.7
(c) Fixed–fixed
column: K = 0.5
(d) Fixed–free
column: K = 2
FIGURE 16.7 Effective lengths L e and effective-length factors K for ideal columns with various
end conditions.