it is very difficult to determine the exact buckling load in complex structures using the Euler formula, due to the difficulty in determining the constant K. Therefore, maximum buckling load is often approximated using energy conservation and referred to as an energy method in structural analysis. The first step in this method is to assume a displacement mode and a function that represents that displacement. This function must satisfy the most important boundary conditions, such as displacement and rotation. The more accurate the displacement function, the more accurate the result. The method assumes that the system (the column) is a conservative system in which energy is not dissipated as heat, hence the energy added to the column by the applied external forces is stored in the column in the form of strain energy. In this method, there are two equations used (for small deformations) to approximate the "strain" energy (the potential energy stored as elastic deformation of the structure) and "applied" energy (the work done on the system by external forces). where is the displacement function and the subscripts and refer to the first and second derivatives of the displacement. Single-degree-of-freedom models Using the concept of total potential energy, , it is possible to identify four fundamental forms of buckling found in structural models with one degree of freedom. We start by expressing where is the strain energy stored in the structure, is the applied conservative load and is the distance moved by in its direction. Using the axioms of elastic instability theory, namely that equilibrium is any point where is stationary with respect to the coordinate measuring the degree(s) of freedom and that these points are only stable if is a local minimum and unstable if otherwise (e.g. maximum or a point of inflection). These four forms of elastic buckling are the saddle-node bifurcation or limit point; the supercritical or stable-symmetric bifurcation; the subcritical or unstable-symmetric bifurcation; and the transcritical or asymmetric bifurcation. All but the first of these examples is a form of pitchfork bifurcation. Simple models for each of these types of buckling behaviour are shown in the figures below, along with the associated bifurcation diagrams. Engineering examples Bicycle wheels A conventional bicycle wheel consists of a thin rim kept under high compressive stress by the (roughly normal) inward pull of a large number of spokes. It can be considered as a loaded column that has been