Test: Struts & Columns - 3 - Mechanical Engineering MCQ

Euler's formula to calculate crippling load in the column

\({P_{cr}} = \frac{{{\pi ^2}EI_{mini}}}{{L_e^2}}\)

Where P = Crippling load, E = young's modulus, I_mini = Minimum area moment of inertia of column, L_e = Effective length of a column

The assumption made while developing Euler Formula for the crippling load of a column does not meet in the real-life practice due to which therefore Euler's formula gives 5 to 10 % error compared to experimental results or true value.

The following assumptions are made in Euler's column theory : 

• The column is initially perfectly straight and the load is applied axially. 
• The cross-section Of the column is uniform throughout its length. 
• The column material is perfectly elastic, homogeneous, and isotropic and obeys Hooke's law. 
• The length of the column is very large as compared to its lateral dimensions. 
• The direct stress is very small as compared to the bending stress.
• The self-weight of the column is negligible.
• The column will fail by buckling alone. 
• Pin joints are free from friction.

(i) Both ends hinged 
(ii) One end fixed and other end free 
(iii) Both ends fixed 
(iv) One end fixed and other end hinged 

For Euler’s formula to be applicable the critical stress must not exceed the proportional limit.
Now crushing stress in mild steel
= 3300 kg/cm²
= 330 N/mm²
But stress at proportional limit in mild steel
= 250 M/mm²
Euler’s buckling stress


λ = 88.55 ≈ 89
Thus slenderness ratio should be more than or equal to 89 ideally. Option (d) is the most close one.

Slenderness ratio, 
where r is the least radius of gyration and L_eff is effective length

∴ 

For same material, same end condition and same length



 

Original load = 
when one end of hinged column is fixed and other free. New L_θ = 2L
∴ New load

Direct stress