Understanding Euler’s Concept of Column Buckling

Column buckling is a phenomenon where a column, under load, faces the risk of collapsing due to buckling. Euler’s theory, formulated by Leonhard Euler in 1757, provides a method to estimate the critical buckling load – the maximum load a column can endure before buckling.

Euler’s Theory in a Nutshell:

Euler’s theory asserts that the stress caused by direct loads on a column is minimal compared to the stress arising from buckling failure. The formula derived from this principle helps calculate the critical buckling load, emphasizing bending stress while neglecting direct stress due to loads.

Key Assumptions:

To apply Euler’s theory, certain assumptions are made:

1. The column starts perfectly straight.
2. The cross-section remains uniform throughout the column’s length.
3. The load is axial and passes through the centroid of the section.
4. Stresses stay within the elastic limit.
5. The material is homogeneous and isotropic.
6. Self-weight of the column is ignored.
7. Buckling is the only cause of failure.
8. Column length significantly exceeds cross-sectional dimensions.
9. Ends are frictionless.
10. Axial compression-induced shortening is negligible.

Limitations of Euler’s Theory:

Despite its utility, Euler’s theory has limitations:

• Ignores crookedness and the possibility of non-axial loads.
• The formula neglects axial stress, potentially overestimating the critical buckling load.

Understanding Critical Buckling Load:

Critical buckling load, denoted as Pcr, is the maximum axial load a column can bear before buckling. The formula is derived as follows:

Pcr=EI(π/KL)2Equation 1Pcr=Equation 1EI(π/KL)2​

Where:

• $Pcr$: Critical buckling load
• $E$: Modulus of elasticity
• $I$: Moment of inertia (cross-sectional area multiplied by radius of gyration)
• $L$: Length of the slender column
• $K$: Effective length factor based on support conditions

End Conditions:

The support conditions depicted in Fig. 1 influence the effective length factor ($K$).

Failure of Columns:

Buckling occurs when the column’s cross-section is relatively small compared to its height. The formula in terms of the radius of gyration is given by:

Pcr=Ear2(π/KL)2Equation 2Pcr=Equation 2Ear2(π/KL)2​

Or

Mean compressive stress on columnE=π2(KL/r)2EMean compressive stress on column​=(KL/r)2π2​

Equation 3, in terms of the slenderness ratio ($KL/r$), is convenient for presenting theoretical and experimental results.

Buckling Failure Scenarios:

• Pure axial compression leads to failure if the stress reaches the material’s yield stress.
• Buckling failure occurs before reaching the yield stress if the critical buckling stress is lower.

Sign Convention for Bending Moments:

• Positive bending moments present convexity towards the initial center line.
• Negative bending moments present concavity towards the initial center line.

Conclusion:

Euler’s theory offers a valuable framework for estimating critical buckling loads in columns, acknowledging its assumptions and limitations. Understanding the interplay of factors like length, support conditions, and material properties is essential for practical applications in engineering.