Short Answer:

Euler’s theory for buckling of columns is based on several simplifying assumptions to make the analysis easy and accurate for long, slender columns. It assumes that the column is perfectly straight, homogeneous, elastic, and loaded axially through its centroid. The load is applied gradually and remains within the elastic limit of the material.

In simple terms, Euler’s theory assumes ideal conditions — no initial bends, no eccentric loading, and no material defects. These assumptions ensure that the column buckles only due to instability and not because of material failure or imperfections.

Detailed Explanation :

Assumptions in Euler’s Theory

Euler’s theory of buckling, developed by Leonhard Euler, provides a mathematical formula to determine the critical buckling load of a long, slender column under axial compression. However, the theory is based on a few ideal assumptions that simplify the real behavior of columns. These assumptions help in deriving the Euler’s formula accurately for theoretical analysis, though in practice, real columns may deviate from these ideal conditions.

Below are the major assumptions made in Euler’s theory:

1. The Column is Initially Perfectly Straight

One of the most important assumptions is that the column is perfectly straight before any load is applied. This means there are no initial bends, imperfections, or curvature along its length.

In reality, manufacturing and assembly processes often introduce small bends or irregularities, but Euler’s theory assumes that the column is ideal. This simplifies the analysis because the buckling behavior depends only on the applied load and not on initial imperfections.

2. The Load is Axial and Acts Through the Centroid

The theory assumes that the compressive load is applied axially, i.e., along the central longitudinal axis of the column, and passes through the centroid of the cross-section. This ensures that the load produces only pure compression without any bending moments at the start.

If the load is applied eccentrically (away from the axis), bending stresses appear even before buckling, and the Euler’s formula no longer holds true. Hence, the theory considers perfectly axial loading for simplicity.

3. The Column Material is Homogeneous and Isotropic

Euler’s theory assumes that the column is made of a homogeneous material, meaning its composition and structure are uniform throughout. It is also assumed to be isotropic, meaning the material has the same properties in all directions.

This assumption ensures that the material reacts uniformly to the applied stress, making the deformation behavior predictable and uniform along the entire column length.

4. The Material Obeys Hooke’s Law

The material of the column is assumed to behave elastically, following Hooke’s Law, which states that stress is directly proportional to strain within the elastic limit.

Here,  is the modulus of elasticity.
This assumption implies that when the load is removed, the column returns to its original shape, and no permanent deformation occurs. Hence, buckling occurs only within the elastic range and not due to plastic deformation.

5. The Deflection is Small Compared to the Column Length

Euler’s theory assumes that the deflection or lateral displacement of the column during buckling is very small compared to its overall length. This allows engineers to use small-angle approximations, such as:

This simplification makes the differential equations of equilibrium linear and easier to solve. However, in practice, larger deflections can occur, making the theory less accurate for very slender columns near collapse.

6. The Column Cross-Section Remains Plane and Constant

The cross-section of the column is assumed to remain plane, uniform, and unchanged throughout loading. This means there is no warping, twisting, or local deformation of the cross-section.

This assumption allows the bending moment and stress distribution to be analyzed easily and ensures that the buckling shape can be described mathematically using simple functions.

7. The Column is Long and Slender

Euler’s theory applies only to long and slender columns where buckling occurs before crushing. For short columns, failure happens due to material crushing, and Euler’s formula cannot predict their behavior accurately.

A column is considered slender if its slenderness ratio (L/r) is high.

where  is the effective length and  is the radius of gyration.
Only when the column is long and slender, buckling becomes the dominant failure mode, satisfying Euler’s assumptions.

8. The Column is Perfectly Elastic and Free from Residual Stresses

It is assumed that the column is completely elastic and free from residual stresses or internal imperfections caused by previous loading or manufacturing defects. These residual stresses can alter the load-carrying capacity and affect the actual buckling load.

Hence, Euler’s theory considers the column as perfectly stress-free before applying any load.

9. The Ends of the Column are Idealized

Euler’s theory considers ideal end conditions, such as:

• Both ends pinned (hinged)
• Both ends fixed
• One end fixed and other free
• One end fixed and other hinged

These boundary conditions help define the effective length of the column and determine the critical buckling load. In reality, supports may not be perfectly fixed or hinged, which causes deviations from theoretical results.

10. The Load is Static and Applied Gradually

The theory assumes that the compressive load is static, meaning it is applied slowly and gradually, not suddenly or with impact. This ensures that the column reaches equilibrium at every stage and fails only by static buckling rather than due to dynamic effects.

Significance of These Assumptions

These assumptions simplify the mathematical derivation of Euler’s buckling formula:

They also make it possible to predict the exact critical load for a perfect column. However, in real-world applications, some assumptions do not hold true because of imperfections, eccentricities, and material variations. Therefore, engineers often apply safety factors or use modified formulas like Rankine’s formula for more accurate design.

Limitations of the Assumptions

• Not suitable for short or intermediate columns.
• Does not consider imperfections or eccentric loading.
• Neglects material non-linearity and plastic deformation.
• Assumes ideal support conditions which rarely exist in practice.

Despite these limitations, Euler’s theory forms the foundation of column design and is essential for understanding stability in compression members.

Conclusion

Euler’s theory of buckling is based on several simplifying assumptions to analyze long, slender columns under axial loads. It assumes that the column is perfectly straight, elastic, homogeneous, and centrally loaded with small deflections and ideal supports. These assumptions make the theory mathematically simple and provide a clear understanding of buckling behavior. Although not entirely applicable to real-world conditions, Euler’s assumptions remain fundamental in mechanical and structural engineering for analyzing and designing compression members.