Understanding Euler's Formula and Column Buckling

Euler's column formula is a fundamental concept in structural mechanics used to predict the critical buckling load for long columns. Buckling is a sudden lateral instability that occurs when a column is subjected to an axial compressive load.

The formula derived by Euler is given by:

$$P_{cr} = \frac{\pi^2 E I}{(L_{eff})^2}$$

Where:

• \(P_{cr}\) is the critical buckling load (the load at which buckling is expected to occur).
• \(E\) is the modulus of elasticity of the column material.
• \(I\) is the minimum moment of inertia of the column's cross-section.
• \(L_{eff}\) is the effective length of the column, which depends on the boundary conditions.

This formula can also be expressed in terms of critical stress (\(\sigma_{cr}\)) by dividing the critical load by the cross-sectional area \(A\):

$$\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E I}{A (L_{eff})^2}$$

We know that the radius of gyration \(r = \sqrt{I/A}\), so \(I/A = r^2\). Also, the slenderness ratio (\(\lambda\)) is defined as \(\lambda = L_{eff}/r\). Substituting these into the critical stress equation:

$$\sigma_{cr} = \frac{\pi^2 E r^2}{(L_{eff})^2} = \frac{\pi^2 E}{(L_{eff}/r)^2} = \frac{\pi^2 E}{\lambda^2}$$

So, the critical buckling stress according to Euler's formula is inversely proportional to the square of the slenderness ratio.

Limitations of Euler's Formula for Mild Steel Columns

Euler's formula is based on several key assumptions:

• The column material is perfectly elastic and obeys Hooke's Law.
• The column is perfectly straight initially.
• The load is applied perfectly axially.
• The failure occurs solely due to elastic buckling.

For long columns, these assumptions are generally reasonable. The critical stress predicted by Euler's formula is relatively low, and the column buckles elastically before the material reaches its yield strength.

However, for shorter columns or columns with lower slenderness ratios, the critical stress predicted by Euler's formula becomes higher. If this predicted critical stress exceeds the material's yield strength (\(\sigma_y\)), the material will yield or crush before the column can buckle elastically according to Euler's prediction. In such cases, the column fails due to a combination of yielding (or crushing) and buckling, and Euler's formula is no longer valid.

Slenderness Ratio Threshold for Mild Steel

For mild steel, there is a slenderness ratio threshold that differentiates between long columns (where Euler's formula is applicable) and intermediate or short columns (where it is not). This threshold is typically found by setting the critical stress from Euler's formula equal to the material's yield strength:

$$\sigma_y = \frac{\pi^2 E}{\lambda_{threshold}^2}$$

$$\lambda_{threshold} = \sqrt{\frac{\pi^2 E}{\sigma_y}}$$

Using typical values for mild steel (E ≈ 200 GPa, \(\sigma_y\) ≈ 250 MPa), the calculated threshold slenderness ratio is approximately:

$$\lambda_{threshold} \approx \sqrt{\frac{\pi^2 \times 200 \times 10^3 \text{ MPa}}{250 \text{ MPa}}} \approx \sqrt{\frac{197392}{250}} \approx \sqrt{789.5} \approx 28.1$$

Wait, this calculation gives a very low value. This threshold calculation using yield strength sets the boundary between elastic buckling and inelastic behavior. However, the *practical* boundary for applying Euler's formula is often higher because of imperfections and the transition from purely elastic to inelastic buckling. For mild steel, experimental results and design codes typically indicate that Euler's formula is generally applicable only for slenderness ratios greater than a certain value, often cited in the range of 80 to 100 or more.

If the slenderness ratio is less than this value (typically < 80 or < 100), the column is considered intermediate or short, and failure occurs predominantly due to yielding or crushing combined with buckling. In this range, formulas like Rankine-Gordon, Johnson's parabolic formula, or code-specific design curves are used instead of Euler's formula.

Conclusion on Euler's Formula Validity

Euler's formula is based on elastic buckling and is valid for long columns where the critical buckling stress is below the material's yield strength. For mild steel, this condition is typically met when the slenderness ratio is high, usually considered to be above 80 to 100.

Therefore, Euler's formula is not valid for mild steel columns when the slenderness ratio is below this threshold, i.e., when the slenderness ratio is relatively low. Looking at the options:

• Less than 80
• More than 80
• More than 120
• More than 30

Euler's formula is valid for slenderness ratios more than the threshold (e.g., > 80). It is not valid for slenderness ratios less than the threshold (e.g., < 80).

Thus, Euler's formula is not valid for mild steel column when slenderness ratio is Less than 80.

Revision Table: Euler's Formula and Column Types

Additional Information on Column Buckling Formulas

While Euler's formula provides the theoretical elastic buckling load for ideal long columns, real-world columns have imperfections and material non-linearity at higher stresses. Other formulas are used to predict the strength of intermediate and short columns:

• Rankine-Gordon Formula: This is an empirical formula that attempts to bridge the gap between Euler's formula (for long columns) and the crushing strength (for short columns). It considers both buckling and crushing failure modes.
• Johnson's Parabolic Formula: This is another empirical formula, typically used for intermediate columns, which provides a better fit to experimental data in the range where inelastic buckling occurs.
• Design Codes (e.g., AISC, Eurocode, IS Code): Structural design codes provide specific formulas and curves for column design based on extensive testing and analysis, taking into account material properties, imperfections, and residual stresses. These are generally used in practical engineering design.

Understanding the range of applicability of Euler's formula based on the slenderness ratio and material properties like yield strength and modulus of elasticity is crucial for correctly analyzing and designing columns.