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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

2

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

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12.1 Introduction to the Concept of Column Buckling

When an initially straight, slender column with pinned

ends is subject to a compressive load P, failure occurs

by elastic buckling when P=Pcr

(12.1)

When anideal column has P < Pcr,

Column remains straight

A lateral force will cause the beam to move laterally, but

beam will return to straight 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

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12.2 Deflection Response of Columns to CompressiveLoads

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

Elastic Buckling of an Ideal Slender Column

12.212.2

12.2.1

Fig. 12.3 Relation between load and lateral deflection for columns

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(12.2)

where

ris the radius of gyration (r2 = I/A)

L/r is the slenderness ratio

For elastic behavior, scr< syield


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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


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12.3 The Euler Formula for Columns with Pinned Ends

Five methods

Equilibrium

Imperfection

Energy

Snap through (more significant in

buckling of shells than of beams)

Vibration (beyond scope of course)

12.3.1 The Equilibrium Method

By equilibrium of moments about Point A

(12.3)

Eq. 12.3 represents a state of neutral

equilibrium Fig. 12.4 Column with pinned ends

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By elementary beam theory

(12.5)

From calculus

(12.4)

Eqs. 12.4 and 12.5 give

(12.6)

By Eqs. 12.3 and 12.6 after dividing by EI

(12.7)

Fig. 12.4 Column with pinned endswhere (12.8)

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Fig. 12.6 Sign convention for internal moment.(a) Positive moment taken CW.

(b) Positive moment taken CCW.

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Higher buckling loads than Pcrare

possible if the lower modes are

constrained

By Eq. 12.14 for n=2

12.3.2 Higher Buckling Loads; n >1

Fig. 12.7 Buckling modes: n=1, 2, 3

By Eq. 12.14 for n=3

In general

In practice, n=1 is the most significant

(12.19)

(12.17)

(12.18)

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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, 0BFG represents the Load-d curve (max load close to Pcr)

For large e, 0BIJ represents the Load-d curve

(max load can be much lower than Pcr)

12.2.2 Imperfect Slender 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


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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?


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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

SMA =M(x) +Pex/L+Py = 0

12.3.3 The Imperfection Method

Fig. 12.8 Eccentrically loaded pinned-end columns

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As the load P increases, the deflection of the column increases

Whensin(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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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 pf the

system

12.3.4 The Energy Method

(12.25)

For column buckling, assuming an adiabatic system dH=0

If beam is disturbed laterally, then it may vibrate, but dK

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(12.33)

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12.4 Euler Buckling of Columns with Linear Elastic EndConstraints

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 and forced BCs)

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(12.40)

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If any of the end displacements (y1,y2) and the end slopes (y1, y2)

of the column are forced (given),

then they are not arbitrary

and the associated variations must vanish

These specified conditions are called forcedBCs

(also calledgeometric, kinematic, or essentialBCs)

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 y1 and y2 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)

y1 = y2 = 0 (12.42)

Eqs. 12.39, 12.41 and 12.42 yieldB = C = D = 0 andA sin KL = 0,

i.e. the resultPcr=p2EI/L2

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For specific values ofK1,K2, k1 and k2 that are neither zero norinfinity

The buckling load is obtained by setting the determinate D of the

coefficientsA,B, CandD 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 P cr

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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22311--chapter12-v2010-v1

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