RANKINE’S HYPOTHESIS FOR STRUTS/ COLUMNS

W

H

Y

?

Before we start Rankine’s hypothesis for struts/columns……..

So why we require Rankine’s hypothesis?

So there should be some limitations of Euler’s theory. So let us study first the limitations of Euler’s theory.

L

I

M

I

TAT

I

O

N

S

O

F

E

U

L

E

R

’

S

T

H

E

O

R

Y

In practice the ideal condition are never reached (i.e. the strut is initially straight and the end load being applied axially through centroid).

And Euler’s theory is applicable for ideal condition only.

And Euler’s theory is applicable for the Slenderness ration greater than 80.

H

I

S

T

O

RY

So to overcome the limitation of Euler’s theory , Tredgold suggested an empirical formula , which was adopted by Gordan which is also known as semi-empirical formula , and the final form of the formula was given by Rankine . Hence it is often known as Rankine’s formula.

N

o

t

a

t

i

o

n

s

1. σc :maximum possible compressive stress.

2. A :cross sectional area.

3. Pc :crippling load.

4. E :young’s modulus.

5. I :least moment of inertia, where I=A��.

6. PRankine :is the actual crippling load for a strut.

7. le :Equivalent length.

8. K : Radius of gyration.

9. PEuler :Euler’s load.

10. a :constant.

D

e

r

i

v

a

t

i

o

n

For a very short strut, collapse will result from direct crushing, and

Crippling load is :

• Pc = σc . A …………………(1)

For a long strut the Euler’s formula applies,

• PEuler = ����

��� =

������

��� = ����

�

��

�……………(2)

According To The Rankine hypothsis is,

•�

� =

�

�������� =

�

�� +

�

������……………(3)

D

e

r

i

v

a

t

i

o

n

Peuler Peuler SHORT

OR LONG

SHORT OR

LONG STRUT STRUT

CONDITIONS

If the strut is very (short)

1

������= 0

P=Pc

If the strut is very (long)

1

������= ∞

P=Peuler

D

e

r

i

v

a

t

i

o

n

Then it may be assumed that if the Rankine hypothsis is true for both very long and very short strut,it will also be true for struts of other dimensions.

Substituting, we have

1

�=

1

σ��+

1

�� ���

�

��

�

�=

1

σ�+

1

�� ���

�

�

D

e

r

i

v

a

t

i

o

n

�

�=

�

�

���

���

�����

.

�

�=

σ�

1 +σ�

������

� .

Prankine = ���

������

� .

This is Rankine formula for the mean breaking stress of a strut/ column, where a=

��

���.

D

e

r

i

v

a

t

i

o

n

Material ��(MN/m2) a=��

���

Mild

steel

320 1

7500

Cast iron 550 1

1600

Wrought

iron

250 1

9000

Strong

timber

50 1

750

• The following values shows the different values of

σc and a for different materials.

D

e

r

i

v

a

t

i

o

n

Prankine = ���

������

� .

Prankine = �������� ����

������

�

The facetor 1+a��

�

� has thus been introduced

to take into account the buckling effect.

C

O

N

D

I

T

I

O

N

Sometimes it is required to find out the length of a column which shall give the same value of buckling load by Euler and Rankine formulae . This is obtained as follows:

Equating the two formulae of Rankine and Euler, we get:

Peuler = Prankine

����

��� =

���

������

�

D

E

R

I

VAT

I

O

N

� �EI × 1 + ���

�

� = σ�� ��

�

σ�� ��� −

����� ���

�� = � �EI

��� σ�� −

���� � ��

�� = � �EA ��.

��� =

� �E ��.�� � � ���

�� =� �E ��.

�� � � ���

��⁄

.

D

E

R

I

VAT

I

O

N

Peuler = Prankine

��

� =

���

��� ����

�

,

At this slenderness ratio , Peuler = Prankine.

It may be noted that the value of ‘a’ in this equation should be substituted for hinged ends only and the length so obtained will be for hinged ends only. If the problem pertains to end conditions other than the hinged ends, and ‘a’ substituted is for the hinged ends, the value of �� so obtained will be the equivalent length for the given case , and it can be converted into the actual length of the column.

A

P

P

L

I

C

AT

I

O

N

Rankine suggested that for strut of a slenderness ratio 80 to 120 fails at smaller loads than predicted values by Euler’s theory , an empirical formula can be used(based on experimental data).

Rankine’s formula can be used for all Slenderness ratio.

Rankine’s formula can be used for practical purpose too.

Rankine’s formula can be used for long, short, intermediate columns (strut) also.

L

I

V

E

E

X

A

M

P

L

E

Short columns

Slenderness ratio < 32.

L

I

V

E

E

X

A

M

P

L

E

Intermediate columns

32 < Slenderness ratio < 120..

L

I

V

E

E

X

A

M

P

L

E

Long columns

Slenderness ratio > 120.

Source:

Strength of material-by r.k.rajput.

Solid mechanics-by s.m.a kazimi.

Internet.