For the symmetrically supported Euler buckling column with both ends hinged the classical stability theory yields simple trigonometric functions as buckling modes, i.e. w(x) = A sin αx. The eigenvalues α are just multiples of π. In comparison, the analysis of the asymmetrically supported Euler buckling column with one end clamped and the other end hinged is more complicated: The buckling modes are a combination of trigonometric functions in form of w(x) = A (sin αx − αx cos (αL)). The eigenvalues follow from a transcendental equation.
Applying a geometrically exact theory to the aforementioned Euler buckling problems, a similar relation in the complexity of the analyses will naturally arise. Using, e.g., the elastica model the buckling behavior of the symmetrically supported column is represented by elliptic integrals. However, the determination of the buckling behavior of the asymmetrically supported column turns out to be much more complex and elaborate. This article presents a direct comparison of the symmetrically and asymmetrically supported buckling columns regarding their analyses by means of classical stability theory and by the geometrically exact theory of the elastica. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)