A novel discrete mechanical model for static, stability, and free vibration analysis of tapered 2D-FGM portal frames

This paper presents a discrete mechanical formulation for the static, buckling, and free-vibration analyses of a tapered two-dimensional functionally graded material portal frame. The structure is composed of two columns and a top beam, whose cross-sectional dimensions may vary along the member axis and whose material properties are graded both through the thickness and along the axial direction. The thickness-wise gradation is described by a power-law distribution, whereas the axial variation is modeled by an exponential law. The frame is discretized by a reduced number of generalized coordinates, leading to a total of 3N+1 degrees of freedom. The structural flexibility is modeled by discrete rotational springs, while the inertia is represented by lumped masses located at the discrete stations of the frame. This reduced-order representation is a major advantage of the proposed approach, since it preserves the main geometric and mechanical features of complex frame structures while keeping the formulation simple and computationally inexpensive. The model is therefore particularly attractive for large parametric investigations. The governing equations are derived from the total strain energy, kinetic energy, and geometric energy associated with axial compression in the columns. The linear stiffness matrix, mass matrix, and geometric stiffness matrix are obtained in explicit form and assembled into a unified discrete system. The formulation enables the prediction of static response, critical buckling loads, and natural frequencies within the same framework. Due to its simplicity, efficiency, and ability to capture the effects of geometric non-uniformity and bidirectional material gradation, the proposed discrete model provides an effective tool for the analysis of advanced tapered portal frames.