This chapter presents a new technique called Variable Order Secant Matrix (VOSM) for the kinematically exact postbuckling analysis of structural components such as beams and frames. Using Reissner’s strain–displacement relations, the VOSM approach uses higher-order secant matrices to obtain accurate results in the postbuckling behavior of plane beams, frames, and arches. The concept and the formulation of the VOSM are introduced using an example of an inextensible elastica. Kinematically exact Reissner’s beam element is formulated and applied to standard benchmark problems to validate its capabilities. The numerical results illustrate the VOSM technique’s effectiveness in representing postbuckling responses of plane beams, frames, and arches. This chapter also presents and validates many benchmark problems to help researchers and professionals.