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Chapter 12
Elastic Stability of Columns
• Axial compressive loads can cause a
sudden lateral deflection (Buckling)
• For columns made of elastic-perfectly
plastic materials, Pcr
– Depends primarily on E and I
– Independent of syield and sult
• For columns made of elastic strain-
hardening material, Pcr
– Will also depend on the inelastic stress-strain
behavior
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• Ideal column
– Perfectly straight
– Load lies exactly along central longitudinal
axis
– Weightless
– Free of residual stresses
– Not subject to
• a bending moment or
• a lateral force
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1. Introduce some basic concepts of column
buckling
2. Physical description of the elastic buckling
of columns
a. For a range of lateral deflections
b. For both ideal and imperfect slender columns
3. Derive Euler formula for a pin-pin column
4. Examine the effect of constraints
5. Investigate Local Buckling of thin-wall
flanges of elastic columns with open cross
sections
• When an initially straight, slender column with pinned ends is
subject to a compressive load P, failure occurs by elastic buckling
when P = Pcr
𝑃𝑐𝑟 =𝜋2𝐸𝐼
𝐿2 (12.1)
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12.1 Introduction to the Concept of Column Buckling
• When an ideal column has P < Pcr,
– Column remains straight
– A lateral force will cause the beam to move laterally,
but beam will return to straight position upon
removal of the force
– Stable Equilibrium
• When an ideal column has P = Pcr,
– Column can be freely moved laterally and remain
displaced after removal of the lateral load
– Neutral Equilibrium
• When an ideal column has P > Pcr
Unstable
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• Magnitude of the buckling load is a function of the boundary conditions
• Buckling is governed by the SMALLEST area moment of inertia
• Real materials experience:
− plastic collapse or fracture (unrestrained lateral displacements)
− jamming in assembly (restrained lateral displacements)
12.2 Deflection Response of Columns to Compressive Loads
12.2.1 Elastic Buckling of an Ideal Slender Column
• Consider a straight slender pinned-end column made of a
homogeneous material
• Load the column to Pcr
• Lateral deflection is represented by Curve 0AB in Fig. 12.3a
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Fig. 12.3 Relation between load and lateral deflection for columns
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where
• r is the radius of gyration (r2 = I/A)
• L/r is the slenderness ratio
• For elastic behavior, scr < syield
Fig. 12.3 Relation between load and lateral deflection for columns
𝑃𝑐𝑟 =𝜋2𝐸𝐼
𝐿2 , 𝜎𝑐𝑟 =
𝑃𝑐𝑟
𝐴=
𝜋2𝐸
𝐿
𝑟
2 (12.2)
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Large Deflections
• Southwell (1941) showed that a very slender column can sustain a load
greater than Pcr in a bent position
– Provided the average s < syield
• The load-deflection response is similar to curves BCD
• For a real column, the syield is exceeded at some value C due to axial and
bending stresses
Fig. 12.3 Relation between load and lateral deflection for columns
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By elementary beam theory: 𝑀 𝑥 =𝐸𝐼
𝑅(𝑥) (12.4)
From calculus: 1
𝑅= ±
𝑑2𝑦
𝑑𝑥2
1+𝑑𝑦
𝑑𝑥
2 3/2 ≈ ±𝑑2𝑦
𝑑𝑥2 (12.5)
From Eqs. 12.4 and 12.5:
𝑀 𝑥 = ±𝐸𝐼𝑑2𝑦
𝑑𝑥2 (12.6)
By Eqs. 12.3 (𝑀 𝑥 = −𝑃𝑦) and 12.6 after dividing
by EI:
𝑑2𝑦
𝑑𝑥2+ 𝑘2𝑦 = 0 (12.7)
where
𝑘2 =𝑃
𝐸𝐼 (12.8)
Fig. 12.4 Column with
pinned ends
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12.3 The Euler Formula for Columns with Pinned Ends
Five methods:
1. Equilibrium
2. Imperfection
3. Energy
4. Snap through (more significant in buckling of
shells than of beams)
5. Vibration (beyond scope of course)
12.3.1 The Equilibrium Method
By equilibrium of moments about Point A:
𝑀𝐴 = 0 = 𝑀 𝑥 + 𝑃𝑦
𝑀 𝑥 = −𝑃𝑦 (12.3)
Eq. 12.3 represents a state of neutral
equilibrium
Fig. 12.4/5 Column with pinned
ends and FBD of lower portion
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Fig. 12.6 Sign convention for internal moment.
(a) Positive moment taken CW.
(b) Positive moment taken CCW.
The b.c.’s associated with Eq. 12.7 are:
y = 0, for x = 0, L (12.9)
For arbitrary values of k, Eq’s. 12.7 and 12.9 admit only the trivial
solution y = 0. However, nontrivial solutions exist for specific
values (eigenvalues) of k.
The general solutions to Eq. 12.7:
𝑦 = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥 (12.10)
where A and B are constants determined from the boundary
conditions in Eq. 12.9. Thus, from Eq. 12.10:
𝐴 sin 𝑘𝐿 = 0, 𝐵 = 0 (12.11)
For a nontrivial solution (𝐴 ≠ 0), Eq. 12.11 requires that 𝑘𝐿 = 0,
or: 𝑘 =𝑃
𝐸𝐼
2=𝑛𝜋
𝐿, 𝑛 = 1, 2, 3, …
For each value of 𝑛, by Eq. 12.10, there exists a nontrivial solution
(eigenfunction):
𝑦 = 𝐴𝑛 sin𝑛𝜋𝑥
𝐿 (12.13)
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From Eq. 12.12, the corresponding Euler loads are:
𝑃 =𝑛2𝜋2𝐸𝐼
𝐿2, 𝑛 = 1, 2, 3, … (12.14)
The minimum 𝑃 occurs for 𝑛 = 1. This load is the smallest load
for which a nontrivial solution is possible—the critical load for the
column. By 12.12/12.14, with 𝑛 = 1:
𝑃 =𝜋2𝐸𝐼
𝐿2= 𝑃𝑐𝑟 (12.15)
Euler formula for buckling of a column
with pinned ends.
The buckled shape of the column is determined from Eq. 12.13
with 𝑛 = 1 :
𝑦 = 𝐴1 sin𝜋𝑥
𝐿 (12.16)
But, 𝐴1is indeterminate. The maximum amplitude of the buckled
column cannot be determined by this approach. 𝐴1 must be
determined by the theory of elasticity. 13
12.3.2 Higher Buckling Loads; n >1
Higher buckling loads than Pcr are possible
if the lower modes are constrained
By Eq. 12.14 for n=2
𝑃 = 4𝜋2𝐸𝐼
𝐿2= 4𝑃𝑐𝑟, 𝜎𝑐𝑟(2) =
𝑃
𝐴= 4
𝜋2𝐸
𝐿
𝑟
2
(12.17)
By Eq. 12.14 for n=2
𝑃 = 9𝜋2𝐸𝐼
𝐿2= 9𝑃𝑐𝑟, 𝜎𝑐𝑟(3) =
𝑃
𝐴= 9
𝜋2𝐸
𝐿
𝑟
2
(12.18)
In general:
𝑃 = 𝑛2𝜋2𝐸𝐼
𝐿2= 𝑛2𝑃𝑐𝑟, 𝜎𝑐𝑟(𝑛0) =
𝑃
𝐴= 𝑛2
𝜋2𝐸
𝐿
𝑟
2
(12.19)
In practice, n=1 is the most significant. 14
Fig. 12.7 Buckling modes:
n=1, 2, 3
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• Real columns nearly always possess deviations from ideal conditions
• Unless a column is extremely slender, it will fail by yielding or fracture
before failing by large lateral deflections
• An imperfect column may be considered as a perfect column with an
eccentricity, e
• For small e, 0’B’FG represents the Load-d curve (max load close to Pcr)
• For large e, 0”B”IJ represents the Load-d curve
(max load can be much lower than Pcr)
12.2.2 Imperfect Slender Columns
Fig. 12.3 Relation between load and lateral deflection for columns
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• The load-d relations for columns of intermediate slender ratios are
represented by the curves in Fig. 12.3c
• For such columns, a condition of instability is associated with Points B,
F and I
• At these points, inelastic strain occurs and is followed, after only a
small increase in load, by instability collapse at relatively small lateral
deflections
Failure of Columns of Intermediate Slenderness Ratio
Fig. 12.3 Relation between load and lateral deflection for columns
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• Two potential types of failures
1. Failure by excessive deflection before plastic collapse or fracture
2. Failure by plastic collapse or fracture
• Pure analytical approach is difficult
Empirical methods are usually used in conjunction with
analysis to develop workable design criteria
Which Type of Failure Occurs?
Fig. 12.3 Relation between load and lateral deflection for columns
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• Acknowledge that a real column is usually loaded eccentrically, e
• Hence, the Imperfection Method is a generalization of the Equilibrium
Method
By equilibrium of moments about Point A
𝑀𝐴 = 0 = 𝑀 𝑥 + 𝑃𝑒𝑥
𝐿+ 𝑃𝑦 𝑀 𝑥 = −𝑃𝑒
𝑥
𝐿− 𝑃𝑦
12.3.3 The Imperfection Method
Fig. 12.8 Eccentrically loaded pinned-end columns
𝑀 𝑥 = −𝑃𝑒𝑥
𝐿− 𝑃𝑦 (12.20) Recall, 12.6: 𝑀 𝑥 = ±𝐸𝐼
𝑑2𝑦
𝑑𝑥2
Thus,
𝑀 𝑥 = +𝐸𝐼𝑑2𝑦
𝑑𝑥2= −𝑃𝑒
𝑥
𝐿− 𝑃𝑦 (12.20)
Dividing by 𝐸𝐼 and recalling: 𝑘2 =𝑃
𝐸𝐼 gives:
𝑑2𝑦
𝑑𝑥2+ 𝑘2𝑦 = −
𝑘2𝑒𝑥
𝐿 (12.21)
The b.c.’s are: 𝑦 = 0 for 𝑥 = 0, 𝐿 (12.22)
Giving the general solution of Eq. 12.21:
𝑦 = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥 −𝑒𝑥
𝐿 (12.23)
where 𝐴 and 𝐵 are constants determined by the boundary
conditions. Hence, from Eq.’s 12.22 and 12.23:
𝑦 = 𝑒sin(𝑘𝑥)
sin(𝑘𝑙)−𝑥
𝐿 (12.24)
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Rewriting 12.24: 𝑦 = 𝑒sin(𝑘𝑥)
sin(𝑘𝑙)−𝑥
𝐿 (12.24)
• As the load P increases, the deflection of the column increases
• When sin(kL) = 0 for kL=np, n=1,2,3, …, y
• The Imperfection Method gives the same result as the
Equilibrium Method.
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Can solve the problem using the Rayleigh method by reducing
the problem to a single DOF, e.g. y(x) = A sin(p x / L)
A more general form is to use a Fourier series:
𝑦 𝑥 = 𝑎𝑛 sin𝑛𝜋𝑥
𝐿 (12.27)
Eq. 12.27 satisfies the BCs y=0 @ x=0 and x=L 26
The Energy Method is based on the first law of thermodynamics
The work that external forces perform on a system plus the heat energy
that flows into the system equals the increase in internal energy of the
system plus the increase in the kinetic energy of the system:
𝛿𝑊 + 𝛿𝐻 = 𝛿𝑈 + 𝛿𝐾 (12.25)
12.3.4 The Energy Method
For column buckling, assuming an adiabatic system 𝛿𝐻 = 0
If beam is disturbed laterally, then it may vibrate, but 𝛿𝐾 << 𝛿𝑊
Implies 𝛿𝑊 = 𝛿𝑈 (12.26)
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(12.33)
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12.12. Determine the Euler load for the column shown in
Figure 12.8c. See the discussion on the imperfection method
in Section 12.3.
Fig. 12.8c
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12.4 Euler Buckling of Columns with Linear Elastic End
Constraints
• Consider a straight elastic column with
linear elastic end constraints
• Apply an axial force P
• The potential energy of the column-spring
system is
Fig. 12.10
Elastic column with linear elastic
end constraints
(12.34)
• The displaced equilibrium position of the
column is given by the principle of
stationary potential energy
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• By Eq. 12.34, set dV=0
• Eq. 12.37 is the Euler equation for the column
• Eq. 12.38 are the BCs (Includes both the natural (e.g. y”=0 implies
no moment at a pin) and forced (specified, e.g. y=0 at ends) BCs)
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(12.40)
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• If any of the end displacements (y1,y2) and the end slopes (y’1, y’2)
of the column are forced (given),
– then they are not arbitrary
– and the associated variations must vanish
• These specified conditions are called forced BCs
(also called geometric, kinematic, or essential BCs)
• e.g., for pinned ends
– y1=0 @ x=0 and y2=0 @ x=L
– Therefore, dy1=dy2=0
– Then the last two of Eqs. 12.38 are identically satisfied
• The first two of Eqs. 12.38 yield the natural (unforced) BCs for the
pinned ends
– Because y’1 and y’2 and hence dy1’ and dy2’ are arbitrary (i.e. nonzero)
– Also for the pinned ends K1=K2=0
– Therefore, Eqs. 12.38 give the natural BCs (because EI>0)
y”1 = y”2 = 0 (12.42)
• Eqs. 12.39, 12.41 and 12.42 yield B = C = D = 0 and A sin KL = 0,
i.e. the result Pcr=p2EI/L2
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• For specific values of K1, K2, k1 and k2 that are neither zero nor
infinity
– The buckling load is obtained by setting the determinate D of the
coefficients A, B, C and D in Eq. 12.40
– Usually must be solved numerically
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12.5 Local Buckling of Columns
• Consider a column that is formed with several thin-wall parts
e.g., a channel, an angle or a wide-flange I-beam
• Depending on the relative cross-sectional dimensions of a flange or
web
– Such a column may fail by local buckling of the flange or web, before it
fails as an Euler column
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• Consider the example
– If the ratio t/b is relatively large, the column buckles as an Euler column
(global buckling)
– If t/b is relatively small, the column fails by buckling or wrinkling, or more
generally, Local Buckling
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• Local buckling of a compressed thin-wall column may not cause
immediate collapse of the column. However,
– It alters the stress distribution in the system
– Reduces the compressive stiffness of the column
– Generally leads to collapse at loads lower than the Euler Pcr
• In the design of columns in building structures using hot-rolled steel,
local buckling is controlled by selecting cross sections with t/b ratios s.t.
the critical stress for local buckling will exceed the syield of the material
– Therefore, local buckling will not occur before the material yields
• Local buckling is controlled in cold-formed steel members by the use of
effective widths of the various compression elements
(i.e., leg of an angle or flange of a channel) which will account for the
relatively small t/b ratio.
– These effective widths are then used to compute effective (reduced) cross-
section properties, A, I and so forth.
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Fig. 12.11
Buckling loads for local buckling and Euler buckling for columns made
of 245 TR aluminum (E=74.5 GPa)
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