 # Buckling Load Numerical | Introduction | Important Formulas

### 1. Introduction

The maximum limiting load at which the column tends to have lateral displacement / tends to buckle is called Buckling Load. It is also known as Crippling Load.

The buckling takes place about the axis having a minimum radius of gyration or least moment of inertia.

 Pcr = ( π2 EI ) / L2

It is Euler’s formula for buckling load.

 End Condition Effective length Euler’s Buckling Both ends pinned / hinged Le = L Pcr = π2 EI / Le2,Pcr = π2 EI / L2 One end Fixed, other and free Le = 2 L Pcr = π2 EI / Le2,Pcr = π2 EI / 4L2 One end fixed, other end hinged / pinned Le = L / √2 Pcr = π2 EI / Le2,Pcr = 2π2 EI / L2 Both ends fixed Le = L / 2 Pcr = π2 EI /Le2,Pcr = 4π2 EI / L2

### 2. Important Formulas

 1. Pcr = ( π2 EI ) / Le2

 3. Moment of Inertia (I) = π ( D4 – d4 ) / 64

 5. E = FL / A Δl

Δl= Elongated length

 6. Slenderness ratio (λ) = l / Rmin

1. A solid round bar 60 mm in diameter and 2.5 m long is used as a strut, one end of the strut is fixed while its other end is hinged. Find the safe compressive load for this strut using Euler’s formula. Assume E= 200 GN/m2 and factor of safety 3.

Given,

d = 60 mm = 0.06 m

l = 2.5m

E= 200 GN/m2 = 200 x 109 N/m2

FOS = 3

Le = L / √2 = 2.5 / √2 = 1.77 m

we have,

Buckling Load (Pcr) = π2 EI / Le2

= π2 200 x 10π ( 0.064) / 64 x 1.772

= 400.8 KN

= 400.8 / 3

= 133.6 KN

2. A slender pin-connected aluminum column 1.8m length of the circular cross-section is to have an outside diameter of 50 mm. Calculate the necessary internal diameter to prevent failure by buckling if the actual load is applied is 13.6 KN and the critical load applied is twice the actual load. Take E for aluminum as 70 GN/m2.

Given,

L = 1.8 m

Outside diameter (D) = 50 mm

Internal diameter (d) = ?

E = 70 GN/m2

= 70 x 109 N/m2

Actual load applied (P) = 13.6 KN

= 2 x 13.6

= 27.2 KN

Moment of inertia (I) = π ( D4 – d4 ) / 64

= π ( 0.054 – d4 ) / 64

End condition: both ends pinned

Effective length (Le) = L = 1.8 m

Buckling Load (Pcr) = π2 EI / Le2

27.2 x 103 = π2 70 x 109 x π ( 0.054 – d4 ) / 64 x 1.82

= 564019.2 = 13565246.05 – 2.17 x 1012d4

d4 = 13001226.85/ 2.17 x 1012

d = 0.049m

3. An I section joist 400 mm x 200 mm and 6 m long is used as a strut with both ends fixed. What is Euler’s crippling load for columns? Take Young’s modulus of joist as 200 Gpa.

All dimensions are in mm.

Given, x̄ = 100 mm

ȳ = 200 mm

x₁ = 100 mm           x₂ = 100 mm

y₁ = 10 mm             y₂ = 200 mm

x₃ = 100 mm           y₃ = 390 mm

Buckling Load (Pcr) = π2 EI / Le2

Here, I is Minimum of Iₓₓ and Iᵧᵧ

Iₓₓ = ( Iₓₓ )₁ + ( Iₓₓ )₂ + ( Iₓₓ )₃

= [ 200x 203 / 12 + 4000 (10-200)2 ] + [ 200x 3603 / 12 + 1200 (200-200)2 ] + [ 200x 203 / 12 + 4000 (390-200)2 ]

= 366826666.7 mm4

Iᵧᵧ = ( Iᵧᵧ )₁ + ( Iᵧᵧ )₂ + ( Iᵧᵧ )₃

= [ 20x 2003 / 12 + 4000 (100-100)2 ] + [ 360 x 203 / 12 + 0 ] + [ 20x 2003 / 12 + 0 ]

= 26906666.67 mm4

Iᵧᵧ < Iₓₓ so, I = 26906666.67 mm4

Buckling Load (Pcr) = π2 EI / Le2

= π2 200 x 109 x 26906666.67  / 32

= 5901.289 KN

4. A hollow tube 4 m long with external & internal diameter 40 mm & 25 mm respectively was found to extend 4.8 mm under a tensile load of 60 KN. Find the buckling load for the tune with both ends pinned. Also, Find the safe load on the tube taking FOS as 5.

Given,

L= 4m

D= 40 mm

d= 25 mm

FOS = 5

Δl = 4.8 mm

Both ends are pinned so, Le= L =4m

Buckling Load (Pcr) = π2 EI / Le2      …………(i)

Moment of Inertia (I) = π ( D4 – d4 ) / 64

= π ( 0.044 – 0.0254 ) / 64

= 1.6 x 10-7 mm4

E = FL/ AΔl

= ( 60 x 10x 4 x 4 ) / (0.042 – 0.0252 ) x 4.8 x 10-3

= 6.53 x 1010 N/m2

From eqn (i),

Pcr = π2 6.53 x1010 x 1.6 x 10-7 / 4

=4270 N

= 4270/5

5. Determine the ratio of buckling strength of 2 columns of circular cross-section one hollow and other solid when both are made of the same material, have the same length, same cross-sectional area, and same end conditions. The internal diameter of the hollow column is half of its external diameter.

Given,

 Hollow Solid Eᴴ E lᴴ l Aᴴ A

dᴴ = Dᴴ / 2

Buckling Load (Pcr)ᴴ = π2 Eᴴ Iᴴ / Le2       ………….(i)

Buckling Load (Pcr) = π2 E I / Le2         …………(ii)

(Pcr)ᴴ / (Pcr) = Iᴴ / I

= ( D4 – d4 )  / D4

= 0.938 D4 / D…………..(iii)

Since the cross-sectional area is same of both columns,

A = A

π D2 / 4 =  π D2 / 4

D2 / D2 = 1.333

Dᴴ / D = 1.1547

from (iii), we get

(Pcr)ᴴ / (Pcr) = 0.938 (1.1547)4

(Pcr)ᴴ / (Pcr) = 1.66