Building Design and Construction Handbook - Merritt - Ventech!

5.58 SECTION FIVE
be taken into account; but unless the curvature is sharp, its effect on deformations
may be neglected. So only Eq. (5.86) and the first term in Eq. (5.85) need be used.
(S. Timoshenko and D. H. Young, ‘‘Theory of Structures,’’ McGraw-Hill Publishing
Company, New York.) See also Arts. 5.14.1 to 5.14.3.
5.7 BUCKLING OF COLUMNS
Columns are compression members whose cross-sectional dimensions are relatively
small compared with their length in the direction of the compressive force. Failure
of such members occurs because of instability when a certain axial load P c (called
critical or Euler load) is equated or exceeded. The member may bend, or buckle,
suddenly and collapse.
Hence the strength P of a column is not determined by the unit stress in Eq.
(5.21) (P � Aƒ) but by the maximum load it can carry without becoming unstable.
The condition of instability is characterized by disproportionately large increases
in lateral deformation with slight increase in axial load. Instability may occur in
slender columns before the unit stress reaches the elastic limit.
FIGURE 5.43 Buckling of a pin-ended long
column.
5.7.1 Stable Equilibrium
Consider, for example, an axially loaded
column with ends unrestrained against
rotation, shown in Fig. 5.43. If the member
is initially perfectly straight, it will
remain straight as long as the load P is
less than the critical load P c. If a small
transverse force is applied, the column
will deflect, but it will return to the
straight position when this force is removed.
Thus, when P is less than P c,
internal and external forces are in stable
equilibrium.
5.7.2 Unstable Equilibrium
If P � P c and a small transverse force
is applied, the column again will deflect,
but this time, when the force is removed,
the column will remain in the
bent position (dashed line in Fig. 5.43).
The equation of this elastic curve can be obtained from Eq. (5.62):
2 dy
2
EI ��Py c
(5.87)
dx
in which E � modulus of elasticity
I � least moment of inertia
y � deflection of the bent member from the straight position at a distance
x from one end