Lecture 12
8th February 2021
Buckling of Columns
Intended Learning Outcomes
At the end of the Lecture, students are expected to be able to:
Discuss the behavior of columns
Discuss the method used to design concentrically loaded column made of
common engineering materials
Discuss the method used to design eccentrically loaded column made of
common engineering materials
Solve related problems
1-3
Examples of Columns in Real Life Applications
Some slender pin-connected members used in moving machinery,
such as this short link, are subjected to compressive loads and
thus act as columns.
1-4
Examples of Columns in Real Life Applications
Typical interior steel pipe columns used to support the roof of a
single story building.
1-5
Examples of Columns in Real Life Applications
The tubular columns used to
support this water tank have been
braced at three locations along
their length to prevent them
from buckling.
1-6
Examples of Columns in Real Life Applications
1-7 Laboratory test showing a buckled column.
Examples of Columns in Real Life Applications
1-8
Examples of Columns in Real Life Applications
The leg in (a) acts as a column
and can be modeled (b) by the two
pin-connected members that are
attached to a torsional spring
having a stiffness k.
1-9
Introduction
Whenever a member is designed, it is necessary that it satisfy specific
strength, deflection, and stability requirements.
In the preceding Lectures, we have discussed some of the methods used
to determine a member’s strength and deflection, while assuming that the
member was always in stable equilibrium.
Some members, however, may be subjected to compressive loadings, and
if these members are long and slender, the loading may be large enough
to cause the member to deflect laterally or sidesway.
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Introduction
To be specific, long slender members subjected to an axial compressive
force are called columns, and the lateral deflection that occurs is called
buckling.
Quite often the buckling of a column can lead to a sudden and dramatic
failure of a structure or mechanism, and as a result, special attention must
be given to the design of columns so that they can safely support their
intended loadings without buckling.
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Introduction
The maximum axial load that a column can support when it is on the
verge of buckling is called the critical load, Fig. 13–1a.
Any additional loading will cause the column to buckle and therefore
deflect laterally as shown in Fig. 13–1b.
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Introduction
In order to better understand the nature of this instability, consider a two-
bar mechanism consisting of weightless bars that are rigid and pin
connected as shown in Fig. 13–2a.
When the bars are in the vertical position, the spring, having a stiffness k,
is unstretched, and a small vertical force P is applied at the top of one of
the bars.
1 - 13
Introduction
1 - 14
Introduction
1 - 15
Introduction
1 - 16
Introduction
1 - 17
Introduction
even greater load than Pcr. Unfortunately, however, this loading may require
the column to undergo a large deflection, which is generally not tolerated in
engineering structures or machines. For example, it may take only a few
newtons of force to buckle a meterstick, but the additional load it may
support can be applied only after the stick undergoes a relatively large lateral
deflection.
1 - 18
Introduction
1 - 19
Ideal Column With Pin Supports
The column to be considered is an ideal column, meaning one that is
perfectly straight before loading, is made of homogeneous material, and
upon which the load is applied through the centroid of the cross section.
It is further assumed that the material behaves in a linear-elastic manner
and that the column buckles or bends in a single plane.
In reality, the conditions of column straightness and load application are
never accomplished; however, the analysis to be performed on an “ideal
column” is similar to that used to analyze initially crooked columns or
those having an eccentric load application.
1 - 20
Ideal Column With Pin Supports
Since an ideal column is straight, theoretically the axial load P could be
increased until failure occurs by either fracture or yielding of the material.
However, when the critical load Pcr is reached, the column will be on the
verge of becoming unstable, so that a small lateral force F, Fig. 13–4b,
will cause the column to remain in the deflected position when F is
removed, Fig. 13–4c.
Any slight reduction in the axial load P from Pcr will allow the column to
straighten out, and any slight increase in P, beyond Pcr, will cause further
increases in lateral deflection.
1 - 21
Ideal Column With Pin Supports
1 - 22
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Euler’s Formula for Pin-Ended Beams
Extension of Euler’s Formula
Effective Length
As stated previously, the Euler formula, was developed for the case of a
column having ends that are pinned or free to rotate.
In other words, L in the equation represents the unsupported distance
between the points of zero moment.
This formula can be used to determine the critical load on columns
having other types of support provided “L” represents the distance
between the zeromoment points.
Extension of Euler’s Formula
Effective Length
This distance is called the column’s effective length, Le. Obviously, for a
pin-ended column Le = L.
For the fixed- and free-ended column, the deflection curve, Eq. 13–8, was
found to be onehalf that of a column that is pin connected and has a
length of 2L.
Thus, the effective length between the points of zero moment is Le = 2L.
Examples for two other columns with different end supports are also
shown in the next slide with their corresponding effective lengths.
Extension of Euler’s Formula
Example 1
An aluminum column of length L and
rectangular cross section has a fixed end at B
and supports a centric load at A. Two smooth
and rounded fixed plates restrain end A from
moving in one of the vertical planes of
symmetry but allow it to move in the other
plane.
a) Determine the ratio a/b of the two sides of
the cross section corresponding to the most
L = 0.5 m efficient design against buckling.
E = 70 GPa b) Design the most efficient cross section for
the column.
P = 20 kN
FS = 2.5
SOLUTION TO EXAMPLE 1
The most efficient design occurs when the
resistance to buckling is equal in both planes of
symmetry. This occurs when the slenderness
ratios are equal.
• Buckling in xy plane:
1 ba3 2
2 I a a
rz 12
z rz
A ab 12 12
Le, z 0.7 L
• Most efficient design:
rz a 12
Le, z Le, y
• Buckling in xz plane: rz ry
1 ab3
Iy b2 b 0.7 L 2L
ry2 12 ry
A ab 12 12 a 12 b / 12
Le, y 2L a 0.7 a
0.35
ry b / 12 b 2 b
• Design:
L = 0.5 m
E = 70 GPa
P = 20 kN
FS = 2.5
a/b = 0.35
Eccentric Loading: The Secant Formula
The Euler formula was derived assuming the load P is always applied
through the centroid of the column’s cross-sectional area and that the
column is perfectly straight.
This is actually quite unrealistic, since manufactured columns are never
perfectly straight, nor is the application of the load known with great
accuracy.
In reality, then, columns never suddenly buckle; instead they begin to
bend, although ever so slightly, immediately upon application of the load.
As a result, the actual criterion for load application should be limited
either to a specified deflection of the column or by not allowing the
maximum stress in the column to exceed an allowable stress.
To study this effect, we will apply the load P to the column at a short
eccentric distance e from its centroid, refer to Fig. on next slide.
Eccentric Loading: The Secant Formula
Denoting by e the eccentricity of the load, i.e., the distance between the
line of action P and the axis of the column (Fig. 10.18a ), we replace the
given eccentric load by a centric force P and a couple MA of moment
MA = Pe (Fig. 10.18b).
It is clear that, no matter how small the load P and the eccentricity e, the
couple MA will cause some bending of the column (Fig. 10.19).
As the eccentric load is increased, both the couple MA and the axial force
P increase, and both cause the column to bend further.
Viewed in this way, the problem of buckling is not a question of
determining how long the column can remain straight and stable under an
increasing load, but rather how much the column can be permitted to
bend under the increasing load, if the allowable stress is not to be
exceeded and if the deflection ymax is not to become excessive.
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
Eccentric Loading: The Secant Formula
EXAMPLE 2
The uniform column consists of a 2.4-m section
of structural tubing having the cross section
shown.
a) Using Euler’s formula and a factor of safety
of two, determine the allowable centric load
for the column and the corresponding
normal stress.
b) Assuming that the allowable load, found in
E = 200 GPa part a, is applied at a point 18 mm from the
geometric axis of the column, determine the
horizontal deflection of the top of the
column and the maximum normal stress in
the column.
SOLUTION TO EXAMPLE 2
• Maximum allowable centric load:
Effective length,
Critical load,
Allowable load,
• Eccentric load:
End deflection,
- Maximum normal stress,
Design of Columns Under Centric Load
• Previous analyses assumed
stresses below the proportional
limit and initially straight,
homogeneous columns.
• Experimental data demonstrate
for large Le/r, scr follows
Euler’s formula and depends
upon E but not sY.
for small Le/r, scr is
determined by the yield
strength sY and not E.
for intermediate Le/r, scr
depends on both sY and E.
Design of Columns Under Centric Load
Structural Steel —Allowable Stress Design • For Le/r < Cc
American Inst. of Steel Construction 2E s cr
s cr s all
Le / r 2 FS
FS 1.67
• For Le/r > Cc
Le / r 2 s cr
s cr s Y 1 2
s all
2Cc FS
3
5 3 L / r 1 L / r
FS e e
3 8 Cc 8 Cc
• At Le/r = Cc
2 2 E
s cr 1s
2 Y
Cc2
sY
Design of Columns Under Centric Load
Aluminum
Aluminum Association, Inc.
• Alloy 6061-T6
• Alloy 2014-T6
Design of Columns Under Centric Load
Aluminum
Aluminum Association, Inc.
Symmetrical and Asymmetrical Loading
(Already covered in previous Lectures of which overview is
presented in today’s class)
Wish You Good Luck in Your Exams
References
1. Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F.
Mazurek, Mechanics of Materials, McGraw-Hill.
2. R.C. Hibbeler, Mechanics of Materials, 8th edition, ISBN 10: 0-13-
602230-8 0134319656, Pearson.
THANKS
(Questions if any)
