Chapter 13: Buckling of Columns
Objectives
✓ Understand the behavior of columns and concept of critical load
and buckling
✓ Determine the axial load needed to buckle a so-called ‘ideal’
column
✓ Determine the ‘effective length’ of a column with various end-
conditions
Solid Mechanics (edited by Byoung-Ho Choi) Page 1
13.1 Critical Load
• Columns: is long slender members subjected to an axial compressive
force
• Buckling: is the lateral deflection that occurs
➢ Critical Load, Pcr
• The maximum axial load that a column can support when it is on the
verge of buckling is called the critical load.
Solid Mechanics (edited by Byoung-Ho Choi) Page 2
13.1 Critical Load
• From the free-body diagram: kL
2 P tan = F = k = k (L / 2) • Stable equilibrium : P
4
• For small θ, tan θ ≈ θ,
kL
2 P = k (L / 2 ) • Unstable equilibrium : P
4
P = kL / 4 (independent of )
Solid Mechanics (edited by Byoung-Ho Choi) Page 3
13.2 Ideal Column with Pin Supports
➢ Ideal Column
• It is perfectly straight before loading
• Both ends are pin-supported
• Loads are applied throughout the centroid of the cross section
Solid Mechanics (edited by Byoung-Ho Choi) Page 4
13.2 Ideal Column with Pin Supports
• Some slender pin-connected
members used in moving
machinery, such as this short
link, are subjected to
compressive loads and thus
act as columns.
• Typical interior steel pipe
columns used to support the
roof of a single story building.
Solid Mechanics (edited by Byoung-Ho Choi) Page 5
13.2 Ideal Column with Pin Supports
➢ Behavior
• When P < Pcr, the column remains straight.
• When P = Pcr,
d 2v
EI 2 = M = − Pv
dx
d 2v P
+ v = 0
dx 2 EI
P P
v = C1 sin x + C2 cos x
EI EI
• Since v=0 at x=0, then C2 = 0
P
• Since v=0 at x=L, then 1 C sin L = 0
EI
P
• Therefore, sin L = 0
EI
P
• Which is satisfied if sin L = n
EI
n 2 2 EI
• Or P = L2 where n = 1,2,3,...
Solid Mechanics (edited by Byoung-Ho Choi) Page 6
13.2 Ideal Column with Pin Supports
• A column will buckle about the principal axis of the cross section having
the least moment of inertia.
• A column having a rectangular cross section,
like a meter stick, as shown in Fig. 13-6,
will buckle about the a-a axis, not the b-b axis.
Solid Mechanics (edited by Byoung-Ho Choi) Page 7
13.2 Ideal Column with Pin Supports
2 EI
• Smallest value at P is when n=1, thus Pcr =
L2
2E
• Corresponding stress is cr =
(KL / r )2
• Where r = I / A is called ‘radius of gyration’
• (L/r) is called the ‘slenderness ratio.’
• The critical-stress curves are hyperbolic,
valid only for σcr is below yield stress
Solid Mechanics (edited by Byoung-Ho Choi) Page 8
13.3 Columns Having Various Types of
Supports
• The tubular columns used to support
this water tank have been braced at
three locations along their length to
prevent them from buckling.
Solid Mechanics (edited by Byoung-Ho Choi) Page 9
13.3 Columns Having Various Types of
Supports
➢ Columns having various end-conditions
• Consider the moment-deflection equation for the cantilevered column,
which is fixed at the base.
d 2v
EI 2 = P( − v )
dx
P
EIV + 2 v = 2 where 2 =
EI
• The solution is v = C1 sin (x ) + C2 cos(x ) +
• Since v = 0 at x = 0, so that C2 = −
• Also, v' = C1 cos(x ) − C2 sin (x )
• Since v’ = 0 at x = 0, so that C1 = 0
Solid Mechanics (edited by Byoung-Ho Choi) Page 10
13.3 Columns Having Various Types of
Supports
• Hence v = 1 − cos(x )x = 0
• Since v = δ at x = L, thus
cos(L ) = 0
n
cos(L ) = 0 or L =
2
• The smallest critical load occurs when n = 1, thus
2 EI 2 EI
Pcr = or Pcr = (with K = 2)
4L 2
(KL )2
• K is called the ‘effective-length factor’
Solid Mechanics (edited by Byoung-Ho Choi) Page 11
13.3 Columns Having Various Types of
Supports
➢ Specific values of K
Solid Mechanics (edited by Byoung-Ho Choi) Page 12
