There is a maximum height of a uniform vertical column made of any material that can support itself without buckling, and it is independent of the cross-sectional area (why?). Calculate this height for ordinary steel (density 7.8 x 10³ kg/m³).

Verified step by step guidance

1. Understand the problem: The maximum height of a vertical column that can support itself without buckling is determined by the material's density and the acceleration due to gravity. This height is independent of the cross-sectional area because the weight of the column increases proportionally with the cross-sectional area, but so does the supporting force (stress).
2. Start with the condition for equilibrium: The column can support itself if the stress at the base due to its own weight does not exceed the material's compressive strength. The stress is given by \( \sigma = \rho g h \), where \( \rho \) is the density of the material, \( g \) is the acceleration due to gravity, and \( h \) is the height of the column.
3. Set the stress equal to the compressive strength of the material: \( \rho g h = \sigma_{\text{max}} \), where \( \sigma_{\text{max}} \) is the maximum compressive strength of steel. Rearrange this equation to solve for \( h \): \( h = \frac{\sigma_{\text{max}}}{\rho g} \).
4. Substitute the known values: Use the density of steel \( \rho = 7.8 \times 10^3 \ \text{kg/m}^3 \), the acceleration due to gravity \( g = 9.8 \ \text{m/s}^2 \), and the compressive strength of steel (a typical value is around \( \sigma_{\text{max}} = 2 \times 10^8 \ \text{Pa} \)).
5. Perform the calculation: Plug the values into the formula \( h = \frac{\sigma_{\text{max}}}{\rho g} \) to determine the maximum height. Ensure the units are consistent (e.g., \( \text{Pa} = \text{N/m}^2 \), \( \text{kg/m}^3 \), and \( \text{m/s}^2 \)).

Key Concepts

Buckling

Buckling is a failure mode that occurs when a structural member is subjected to compressive stress, leading to a sudden change in shape. For columns, this phenomenon is critical as it limits the height a column can achieve before it becomes unstable. The maximum height is determined by the material's properties and the load it bears, rather than its cross-sectional area.

Euler's Buckling Formula

Euler's Buckling Formula provides a theoretical framework to calculate the critical load at which a slender column will buckle. The formula shows that the critical load is inversely proportional to the square of the column's effective length and directly proportional to the material's modulus of elasticity. This relationship indicates that the maximum height of a column is influenced by its material properties and geometry, rather than its cross-sectional area.

Material Properties

Material properties, such as density and modulus of elasticity, play a crucial role in determining a column's ability to support itself without buckling. For ordinary steel, with a density of 7.8 x 10³ kg/m³, these properties dictate how much load the material can withstand before failing. Understanding these properties allows for the calculation of the maximum height a steel column can achieve while remaining stable under its own weight.