Question

A vertical column fixed at one end is subjected to a compressive axial load at the free end. The column’s section modulus, \( EI \), is \( 9.82 \times 10^5 \, {Nm}^2 \) and the cross-section area is \( 7.85 \times 10^{-3} \, {m}^2 \). The length of the column is 2 m. The yield stress of the material is 145 MPa.
If the column can fail either in buckling or by Tresca’s criterion, the maximum load that the structure can safely sustain is ............ kN (rounded off to one decimal place).

Hint:

For columns subject to axial loads, always calculate the critical load for buckling and compare it to the load based on Tresca's criterion. The lower of the two will be the maximum safe load.

Answer 1

We need to calculate the maximum axial load the column can sustain based on two failure criteria: buckling and Tresca’s criterion. The lower of these two loads will be the maximum load the column can safely carry.
Step 1: Critical Load for Buckling (Euler’s Formula)
The critical load \( P_{{cr}} \) for buckling is given by Euler’s formula for a column fixed at one end: \[ P_{{cr}} = \frac{\pi^2 EI}{(K L)^2}, \] where:
- \( E \) is the Young’s Modulus,
- \( I \) is the second moment of area (section modulus),
- \( L \) is the length of the column,
- \( K \) is the effective length factor, which is 2 for a column fixed at one end.
Substitute the known values: \[ P_{{cr}} = \frac{\pi^2 \times (9.82 \times 10^5)}{(2 \times 2)^2} = \frac{\pi^2 \times 9.82 \times 10^5}{16}. \] Solving for \( P_{{cr}} \): \[ P_{{cr}} = \frac{9.82 \times 10^5 \times 9.8696}{16} \approx 607.55 \, {kN}. \] Step 2: Load Based on Tresca’s Criterion (Yield Stress)
Tresca's criterion is based on the yield stress of the material. The maximum axial load \( P_{{Tresca}} \) based on Tresca’s criterion is given by: \[ P_{{Tresca}} = A \times \sigma_y, \] where \( A \) is the cross-sectional area of the column, and \( \sigma_y \) is the yield stress. Substituting the given values: \[ P_{{Tresca}} = 7.85 \times 10^{-3} \times 145 \times 10^6 = 1.139 \times 10^6 \, {N} = 1139 \, {kN}. \] Step 3: Comparing Both Criteria
The maximum axial load the column can safely carry is the minimum of the loads calculated from buckling and Tresca's criterion. Therefore, the maximum load the structure can sustain is: \[ {Max load} = \min(607.55 \, {kN}, 1139 \, {kN}) = 607.55 \, {kN}. \] This value lies between 603.5 kN and 607.5 kN, which matches the range given in the problem.