A steel column has a slenderness ratio of 120. If the yield strength of steel is 250 MPa and Young’s modulus is 200 GPa, what is the critical stress according to Euler’s formula?
- A108 MPa
- B137 MPaCorrect
- C166 MPa
- D195 MPa
Solution
Concept
Euler critical buckling stress of steel column
Formula
σcr = (π2 × E) / λ2
Step-by-step solution
1. Euler's critical buckling stress is given by σcr = (π2 × E) / λ2. 2. Substitute E = 200 GPa = 200,000 MPa and slenderness ratio λ = 120 into the formula. Answer: B
Explanation
Euler's critical stress is calculated using the formula σcr = (π² × E) / λ², where λ is the slenderness ratio. Substituting E = 200 GPa and λ = 120 into the formula yields approximately 137 MPa. Thus, option B is the correct choice.
Why the other options are wrong
A) 108 MPa underestimates the critical buckling stress calculated by Euler's column formula. D) 195 MPa exceeds the theoretical critical buckling stress for the given slenderness ratio.
Shortcut
Euler critical stress = (π² × E) / λ² = (9.87 × 200000) / 120² ≈ 137 MPa.
Common mistake
Substituting yield strength instead of Young's modulus or failing to square the slenderness ratio in Euler's formula.
Correct answer: B) 137 MPa