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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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