A condensed spectral superelement enabling hybrid assembly in vibration and buckling analysis of axially functionally graded nonprismatic beams and frames

To address the high computational cost of conventional finite-element analysis for axially functionally graded nonprismatic Timoshenko beams and the difficulty of extending many analytical and semi-analytical approaches to complex beam-frame assemblies, this study presents a condensed spectral superelement (CSSE) method for vibration and linear buckling analyses of axially functionally graded nonprismatic beams and frames, including Winkler-Pasternak foundation effects. High-order C⁰ hierarchical basis functions constructed from Lobatto polynomials are employed to separate boundary and internal modes, and an independent-order interpolation strategy is introduced to alleviate shear locking. Through dynamic condensation, the internal spectral degrees of freedom are reduced to a standard 6 × 6 nodal dynamic-stiffness matrix, enabling direct hybrid assembly with conventional beam elements while retaining high spectral accuracy for members with continuously varying material and sectional properties. Combined with the Wittrick–Williams algorithm, the formulation provides a unified procedure for extracting natural frequencies and critical linear-buckling loads. Numerical examples demonstrate good agreement with reference solutions, rapid spectral-order convergence, and substantial reduction in global system size, confirming the effectiveness of the proposed method for nonuniform axially functionally graded beam and frame structures.