Which of the following column formulas is applicable to cast iron columns?
a) Euler’s formula
b) J.B.Johnson’s formula
c) Secant formula
d) Straight line formula
The Correct Answer and Explanation is :
Correct Answer:
b) J.B. Johnson’s formula
Explanation:
Cast iron columns are typically short and stocky rather than slender, meaning they fail primarily due to crushing rather than buckling. This behavior makes Euler’s formula unsuitable because Euler’s formula is used for long, slender columns that fail due to elastic buckling. Cast iron, being brittle, does not exhibit significant elastic deformation before failure. Instead, it fails suddenly under compressive stresses.
To account for the inelastic buckling and crushing characteristics of cast iron columns, J.B. Johnson’s formula is used. This formula is an empirical modification of Euler’s formula that considers both elastic and inelastic buckling effects, making it suitable for columns with intermediate slenderness ratios. The formula is expressed as:
[
P_{cr} = \frac{\sigma_c}{1 + \frac{\sigma_c}{\pi^2E} \left(\frac{L}{K}\right)^2}
]
Where:
- ( P_{cr} ) = Critical load
- ( \sigma_c ) = Compressive stress at failure (yield stress for ductile materials or crushing stress for brittle materials)
- ( E ) = Modulus of elasticity
- ( L ) = Effective length of the column
- ( K ) = Radius of gyration
This formula transitions between Euler’s buckling theory for long columns and Rankine’s crushing theory for very short columns, making it more applicable to cast iron columns, which are generally short or intermediate in length.
Why Not Other Formulas?
- Euler’s Formula is only valid for long, slender columns where buckling is elastic, which does not apply to brittle cast iron.
- Secant Formula is used for eccentrically loaded columns, which is not the primary concern for cast iron columns.
- Straight Line Formula is a simplified empirical approach but does not accurately describe the transition between buckling and crushing, which J.B. Johnson’s formula does.
Thus, J.B. Johnson’s formula is the most applicable for cast iron columns.