MECHANICS OF

MATERIALS

Dr. Eddy El Tabach

Assistant professor

NDU University

Barsa campus

CHAPTER

11 Buckling of

Columns

BAU

MECHANICS OF MATERIALS

11 - 2

Stability of Structures

• In the design of columns, cross-sectional area is

selected such that

- allowable stress is not exceeded

allA

P

- deformation falls within specifications

specAE

PL

• After these design calculations, may discover

that the column is unstable under loading and

that it suddenly becomes sharply curved or

buckles (lateral deflection).

MECHANICS OF MATERIALS

11 - 3

Stability of Structures

The maximum axial load that a column can support when it is on

the verge of buckling is called the critical load, Pcr.

Any additional loading will cause the column to buckle and

therefore deflect laterally.

MECHANICS OF MATERIALS

11 - 4

Euler’s Formula for Pin-Ended Beams

• Consider an axially loaded beam.

After a small perturbation, the system

reaches an equilibrium configuration

such that

02

2

2

2

yEI

P

dx

yd

yEI

P

EI

M

dx

yd

• Solution with assumed configuration

can only be obtained if

2

2

2

22

2

2

rL

E

AL

ArE

A

P

L

EIPP

cr

cr

r is the radius of gyration

MECHANICS OF MATERIALS

11 - 5

Euler’s Formula for Pin-Ended Beams

s ratioslendernesr

L

tresscritical srL

E

AL

ArE

A

P

A

P

L

EIPP

cr

crcr

cr

2

2

2

22

2

2

• The value of stress corresponding to

the critical load,

• Preceding analysis is limited to

centric loadings.

MECHANICS OF MATERIALS

11 - 6

Extension of Euler’s Formula

• A column with one fixed and one free

end, will behave as the upper-half of a

pin-connected column.

• The critical loading is calculated from

Euler’s formula,

length equivalent 2

2

2

2

2

LL

rL

E

L

EIP

e

e

cr

ecr

MECHANICS OF MATERIALS

11 - 7

Extension of Euler’s Formula

MECHANICS OF MATERIALS

11- 8

Sample Problem 10.1

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

efficient design against buckling.

b) Design the most efficient cross-section for

the column. L = 20 in.

E = 10.1 x 106 psi

P = 5 kips

FS = 2.5

MECHANICS OF MATERIALS

11 - 9

Sample Problem 10.1

• Buckling in xy Plane:

12

7.0

1212

,

23121

2

a

L

r

L

ar

a

ab

ba

A

Ir

z

ze

zz

z

• Buckling in xz Plane:

12/

2

1212

,

23121

2

b

L

r

L

br

b

ab

ab

A

Ir

y

ye

yy

y

• Most efficient design:

2

7.0

12/

2

12

7.0

,,

b

a

b

L

a

L

r

L

r

L

y

ye

z

ze

35.0b

a

SOLUTION:

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.

MECHANICS OF MATERIALS

11- 10

Sample Problem 10.1

L = 20 in.

E = 10.1 x 106 psi

P = 5 kips

FS = 2.5

a/b = 0.35

• Design:

2

62

2

62

2

2

cr

cr

,

6.138

psi101.10

0.35

lbs 12500

6.138

psi101.10

0.35

lbs 12500

kips 5.12kips 55.2

6.138

12

in 202

12

2

bbb

brL

E

bbA

P

PFSP

bbb

L

r

L

e

cr

cr

y

ye

in. 567.035.0

in. 620.1

ba

b