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13. Theories of Column

Theory at a Glance (for IES, GATE, PSU)1. Introduction

• Strut: A member of structure which carries an axial compressive load.

• Column: If the strut is vertical it is known as column.

• A long, slender column becomes unstable when its axial compressive load reaches a value

called the critical buckling load.

• If a beam element is under a compressive load and its length is an order of magnitude larger

than either of its other dimensions such a beam is called a columns.

• Due to its size its axial displacement is going to be very small compared to its lateral

deflection called buckling .

• Buckling does not vary linearly with load it occurs suddenly and is therefore dangerous

• Slenderness Ratio: The ratio between the length and least radius of gyration.

• Elastic Buckling: Buckling with no permanent deformation.

• Euler buckling is only valid for long, slender objects in the elastic region.

• For short columns, a different set of equations must be used.

2. Which is the critical load?

• At this value the structure is in equilibrium regardless of the magnitude of the angle

(provided it stays small)

• Critical load is the only load for which the structure will be in equilibrium in the disturbed

position

• At this value, restoring effect of the moment in the spring matches the buckling effect of the

axial load represents the boundary between the stable and unstable conditions.

• If the axial load is less than Pcr the effect of the moment in the spring dominates and the

structure returns to the vertical position after a small disturbance – stable condition.

• If the axial load is larger than Pcr the effect of the axial force predominates and the structure

buckles – unstable condition.

• Because of the large deflection caused by buckling, the least moment of inertia I can be

expressed as, I = Ak2

• Where: A is the cross sectional area and r is the radius of gyration of the cross sectional area,

i.e. kmin =

minI

A

Page 364 of 429



Chapter-13 Theories of Column S K Mondal’s• Note that the smallest radius of gyration of the column, i.e. the least moment of inertia I

should be taken in order to find the critical stress. l/ k is called the slenderness ratio, it is a

measure of the column's flexibility.

3. Euler’s Critical Load for Long Column

Assumptions:

(i) The column is perfectly straight and of uniform cross-section

(ii) The material is homogenous and isotropic

(iii) The material behaves elastically

(iv) The load is perfectly axial and passes through the centroid of the column section.

(v) The weight of the column is neglected.

Euler’s critical load,

2

2

π

cr e

EI P

l=

Where e=Equivalent length of column (1st mode of bending)

4. Remember the following table

Case Diagram Pcr Equivalent

length(le)

Both ends hinged/pinned

Both ends fixed

One end fixed & other end free

2

2

π EI

2

2

4π EI

2

2

2

π EI

4

2

Page 365 of 429



/

hapter-13

ne end fixe

inged

. Slende

Sl∴

cr P

. Rankin

ankine the

• Shor

• Long

• Slen

• Crip

• P

whe

d & other e

ness Ra2

2

2

2

min

π

whe

π

nderness

⎛ ⎞⎜ ⎟⎝ ⎠

e

e

EI

L

EA

k

e’s Cripp

ry is applie

t strut /colu

Column (V

erness rati

π

=e

k

ling Load ,

cσ

1 '⎛

+ ⎜⎝

e

A

K k

e k' = Ran

d pinned

io of Co

2

mi

e

min

re I=A k

atio =

k

ling Loa

d to both

mn (valid u

alid upto S

o

2

e

E

P

2

⎞⎟ ⎠

ine consta

Theori

umn

min k lea=

to SR-40)

120)

(σ criti=e

c

2

σ

t = dπ E

s of Colu

t radius of

cal stress)

pends on

n

gyration

cr P

A

aterial & e

nd conditio

S K Mo

ns

2

ndal’s

66 of 429



Chapter-13 Theories of Column S K Mondal’s

cσ stress= crushing

• For steel columns

K’ =1

25000for both ends fixed

= 112500

for one end fixed & other hinged 20 100≤ ≤

e

k

7. Other formulas for crippling load (P)

• Gordon’s formula,

2

σ

b = a constant, d = least diameter or breadth of bar

1

=⎛ ⎞

+ ⎜ ⎟⎝ ⎠

c

e

AP

bd

• Johnson Straight line formula,

σ 1 c = a constant depending on material.⎡ ⎤⎛ ⎞

= − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

ecP A c

k

• Johnson parabolic formulae :

where the value of index ‘b' depends on the end conditions.

• Fiddler’s formula,

( ) ( )2

cσ σ σ σ 2 σ σ

⎡ ⎤= + − + −⎢ ⎥

⎣ ⎦c e e c e

AP c

C

2

e 2

π

where, σ =⎛ ⎞⎜ ⎟⎝ ⎠

e

E

k

8. Eccentrically Loaded Columns

• Secant formula

max 2σ 1 sec

2

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

c e

eyP P

A k k E A

Where maxσ =maximum compressive stress

P = load

u

PP

PP

M M

Page 367 of 429




c

A = Area of c/s

y = Distance of the outermost fiber in compression from the NA

e = Eccentricity of the load

el = Equivalent length

I

k = Radius of gyration = A

Modulus of elasticity of the material= E

e. .2k

Where M = Moment introduced.

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

P M P e Sec

EA

• Prof. Perry’s Formula

max 12

σ σ

1 1σ σ

⎛ ⎞⎛ ⎞− − =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠d c

d e

e y

k

maxWhere σ maximum compressive stress=

d

2

2

Loadσ

c/s area

Euler's loadσ

/ area

π

'

= =

= =

= =

ee

e

e

P

A

P

A c s

EI p Euler s load

1

' Versine at mid-length of column due to initial curvature

e = Eccentricity of the load

e ' 1.2

distance of outer most fiber in compression form the NA

k = Radius of gyration

=

= +

=c

e

e e

y

If maxσ is allowed to go up to f σ (permssible stress)

Then,

1

2η =

ce y

k

2

f

σ σ (1 ) σ σ (1 )σ σ σ

2 2

f e f e

d e

η η + + + +⎧ ⎫= − −⎨ ⎬

⎩ ⎭

• Perry-Robertson Formula

0.003

σ σ 1 0.003 σ σ (1 0.003

σ σ σ

2 2

η ⎛ ⎞

= ⎜ ⎟⎝ ⎠

⎛ ⎞ ⎧ ⎫+ + + +⎜ ⎟⎪ ⎪⎝ ⎠= − −⎨ ⎬⎪ ⎪⎩ ⎭

e

e e f e

f e

d e f

k

k k

Page 368 of 429




9. ISI’s Formula for Columns and Struts

• For

e

k =0 to 160

'

σ

1 0.2 sec4

=

⎛ ⎞×+ ⎜ ⎟⎝ ⎠

y

c

e c

fosP

fos p

k E

Where, Pc = Permissible axial compressive stress

Pc’ = A value obtained from above Secant formula

yσ = Guaranteed minimum yield stress = 2600 kg/cm2 for mild steel

fos = factor of safety = 1.68

el

k= Slenderness ratio

E = Modulus of elasticity =6 2

2.045 10 /kg cm× for mild steel

• For 160el

k>

Page 369 of 429

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