Buckling of Columns

Abstract

If a common hacksaw blade is held between the palms and a gradually increasing compressive load is applied, it quickly collapses into a bowed shape. This is an example of column buckling; the blade fails as a structural element under compressive loads that are tiny compared to the compressive loads necessary to cause yielding of the steel blade. The chapter begins with the derivation of the buckling load of a pin-ended prismatic bar subjected to compression, known as the Euler buckling load. In theory, the bar can buckle into a one-half cycle sine shape (the familiar bowed shape), a full cycle sine shape, a one-and-one-half cycle sine shape, etc., with increasing corresponding buckling loads. These different solutions are called buckling modes. (As intuition indicates, the higher modes are unstable and the bar actually buckles in the first mode.) Analyses are presented of the buckling of bars with other types of end conditions, which can greatly affect the resulting buckling load. A concept known as the effective length is introduced. Suppose that a given column buckles in a particular mode and the buckling load is P. The effective length is the length of a column whose Euler buckling load equals P. In some cases, the effective length of a buckled column can be determined or approximated by observation and used to determine the buckling load. The final topic is the behavior of a bar subjected to eccentric axial loads, that is, loads that are a specified distance from the centroid of the cross section. Such bars do not buckle but assume a bowed shape whose maximum deflection increases with the magnitude of the axial load. An expression for the maximum compressive stress in the bar, known as the secant formula, is derived.