For which type of columns is Euler's equation for buckling most applicable?

Question

Question

This question was previously asked in

BHEL Supervisor Trainee Mechanical 21 Jan 2024 Official Paper

Answer 1

Explanation:

Euler's Equation for Buckling

Euler's equation for buckling describes the critical load at which a slender, straight column under axial compression will buckle.
This equation is applicable to idealized columns that are long and slender, with ends that are pinned (hinged) or fixed, and it assumes that the column is perfectly straight and homogeneous.
Buckling is a failure mode characterized by a sudden lateral deflection of a column under axial load. Euler's formula predicts the critical load (P_cr) at which a column will buckle. The critical load is given by:

Formula:

\(P_{cr} = \frac{{{n^2}{\pi ^2}EI}}{{{L^2}}}\)

Where:

P_cr = Critical load at which buckling occurs
E = Modulus of elasticity of the material
I = Moment of inertia of the cross-section about the axis of buckling
K = Column effective length factor (depends on end conditions, e.g., 1 for pinned-pinned, 0.5 for fixed-fixed)
L = Actual length of the column

Applicability:

Euler's equation is most applicable to long, slender columns where buckling occurs before the material yields. These columns are characterized by having a high slenderness ratio (length to radius of gyration). The formula is derived based on the assumption that the column will fail due to buckling rather than crushing or yielding of the material.

Advantages:

Provides a simple and reliable method to predict the buckling load for slender columns.
Helps in the design of structures by ensuring that columns are prevented from buckling under expected loads.

Disadvantages:

Not applicable for short, stocky columns where yielding may occur before buckling.
Assumes ideal conditions (perfectly straight, homogeneous material), which may not be true in real-world applications.

Applications:

Euler's buckling theory is widely used in the design of structural elements in buildings, bridges, towers, and other structures where long, slender columns are used. It helps in determining the safe load limits to avoid buckling failure.