Modal decomposition of coupled instabilities: The method of the equivalent nodal forces

A solution to the ‘modal decomposition problem’ encountered within the context of the stability analysis of thin-walled structural members is presented. The proposed method achieves decomposition of a randomly deformed shape into a number of constituent modes, which have the physical meaning of the classical local, distortional and global buckling modes, augmented with two additional classes of shear and transverse extension modes. The basis vectors of these five classes are created by defining sets of nodal forces which, when applied to the member in a first order linear elastic problem, generate shapes commensurate with specific mechanical criteria defining the local, distortional, global, shear and transverse extension modes. In a second step the basis vectors of a given class are used to define a constrained stability problem, where the solution is restricted to a linear combination of these basis vectors, in order to obtain the buckled shapes under a given loading. The full set of buckling modes spanning the five classes forms an orthonormal basis of the complete deformation space. Consequently, decomposition can be achieved by projecting the shape which is to be decomposed onto the basis vectors. Two examples are provided to illustrate the method.