Mechanics of Materials-Part 49 (Columns - Euler's buckling load) | BME301

Introduction: Welcome to this tutorial on columns, an essential structural element in engineering design. In this video, we will explore the difference between short and long columns, their failure modes, and specifically focus on how long columns fail due to buckling under compressive loads. We'll also dive into the derivation of Euler's formula for calculating the buckling load, which predicts the load at which a slender, long column will fail based on its end conditions. Today, we will particularly focus on the case where the column has both ends hinged. A column is a structural member designed primarily to withstand axial compressive loads. Columns can be classified into two categories based on their length: Short Columns are typically stocky and fail when the direct compressive stress exceeds the material's permissible stress. The failure in short columns is due to direct crushing and occurs at a load equal to or slightly greater than the material’s capacity to withstand compressive forces. Long, slender columns behave differently. Instead of failing due to material strength being exceeded, they tend to fail due to buckling. Buckling is a form of instability that occurs at loads much lower than what would typically cause a short column to fail. What is Buckling? Buckling is the sudden lateral deflection of a long column under axial load. This happens when a column is slender enough that lateral deformation becomes significant, leading to instability and failure even if the material strength is not fully utilized. This makes understanding and predicting buckling critical in the design of long columns. Euler's formula provides a theoretical calculation for the critical buckling load, at which a long, slender column will buckle. This critical load depends not just on the properties of the material but also on the boundary conditions of the column. Both ends pinned (hinged): The column can freely rotate but not translate. This is the most common case we will focus on in this tutorial. One end fixed, the other free: This setup creates a longer effective length, increasing the column's tendency to buckle. Both ends fixed: Here, the column's ends are fully restrained from rotation, reducing its effective length and increasing its resistance to buckling. Derivation of Euler’s Formula for a Hinged Column: In this video, we will derive Euler's formula for a long column with both ends hinged, meaning both ends can rotate but not translate. This is the most common case in practical engineering applications. We begin with the fundamental equation of bending for columns under compressive load. Apply boundary conditions specific to the hinged case, where both ends are free to rotate but restrained from any lateral movement. From this, we arrive at the expression that relates the critical load to the physical properties of the column and its length. Understanding buckling and Euler's formula is vital for the safe design of long columns in structures. In this video, we’ve explored how different end conditions affect a column’s buckling load and derived the formula for the most common case of a column with both ends hinged. This knowledge allows engineers to predict when a column will fail due to buckling and ensure structural integrity in their designs.