2E4: SOLIDS & STRUCTURES2E4: SOLIDS & STRUCTURES

Lecture 15Lecture 15

Dr. Bidisha Ghosh

Notes:

http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solid

s & Structures

Buckling of ColumnsBuckling of Columns

What is buckling?

Buckling is a large deformation produced under compressive load in a

direction or plane normal to the direction of application of the load.

Buckling is a form of instability, it occurs suddenly with large changes

in deformation but little change in loading. For this reason it is a

dangerous phenomenon that must be avoided in structural design.

Buckling is not failure through yielding. Due to the shape of a

structural element it can buckle under a load below the ultimate

strength.

Whether a column will buckle or not depends on the

length of the column.

Long ColumnsLong Columns

Long columns usually fail by elastic buckling.

The failure load is below ultimate strength of the

material.

The Euler formula is used to calculate failure strength in

long columns.

Short ColumnsShort Columns

Short columns generally dont fail by elastic buckling.

The failure stress is close to yield stress of the material.

The true short-columns do not have much practical

application.

How do we know which is a short/long column?How do we know which is a short/long column?

eLr

=

Ir

A=

Classification of ColumnsClassification of Columns

CD is Euler curve showing behaviour of long columns

Euler formula should not be used for slenderness ratio

End ConditionsEnd Conditions

What was the pinned-pinned condition mentioned in connection

with Slenderness ratio?

Euler derived all formulae related to column buckling for pinned-

pinned condition and later for other end support conditions, those

formula were altered by using a constant, C

Instead of the actual length of the columns a new length termed as the

effective length was used.

Effective length, 2

2e

LLC

=

Different EndDifferent End--ConditionsConditions

Check the load required to buckle!

Failure StressFailure Stress

2

2cre

EIPLpi

=

2

2cr

cr

P EA

pi

= =

The radius of gyration provides a measure of the resistance provided

by a cross-section to lateral buckling.

The radius of gyration is not a physical entity in itself. It is a

relationship derived to make prediction of column behaviour easy. The

radius of gyration is related to the size and shape of the cross-section.

Columns will buckle in the direction of least cross-sectional

stiffness (minimum value of I ).

A rectangular column will buckle in the direction of the smaller

dimension in cross-section. A square column cross-section will be

equally prone to buckling in both x and y directions. This is because the

cross section will offer equal resistance to buckling in the direction x

and y.

x

y

z

y

z

I y > I z

Cross-sectionP

P

Buckling Direction

A

b

h

CrossCross--sectional Shape & Bucklingsectional Shape & Buckling

Example Example

Determine the thickness of a round steel tubular strut,

375mm external diameter and 2.5m long, pin-jointed at

the ends, to withstand an axial load of 39000kN.

E=200GPa.

Moment curvature equationMoment curvature equation

A quick touch on bending before learning buckling!

1.Moment at any section

2.Moment-curvature equation:

3.Buckling is an effect of combined compression & bending

2

2d yM EIdx

=

Compression on A Slender ColumnCompression on A Slender Column

From knowledge of bending,

Solve this equation..

1.

2. So,

2

2

2

2 0

d yM EI Pydx

d yEI Pydx

= =

+ =

22 2

2 0; where d y Pydx EI

+ = =

sin cos ;From boundary conditions, B=0 & sin 0y A x B x

A L

= +

=

Compression on A Slender ColumnCompression on A Slender Column

For any buckling to happen the second condition has to be true and

that means,

2 2

2

0, ,2 ,3 .....

, ( 1,2,...)

L

Pand L n n

EIn EIP

L

pi pi pi

pi

pi

=

= = =

=

sin 0means, either 0 , sin 0A L

A or L

=

= =