<p>Due to vibration and impact isolation performances, functionally graded porous (FGP) polygon built-up beam structures are often encountered in engineering cases. However, material nonhomogeneity and multi-directional coupling pose challenges for dynamic modeling of these structures. Therefore, this paper proposes an energy method for vibration analysis of arbitrary polygon built-up FGP beam structures resting on elastic foundations. The energy functional of individual beams is formulated using the first-order shear deformation theory (FSDT). The coordinate transformation at the joint points is derived, and a unified expression for the connection of beams at arbitrary angles is established. A modified variational method is developed to reinforce the interaction and continuity constraints in the built-up beam. Accurate solutions of the governing equations using arbitrary admissible functions are therefore ensured. Concave hexagonal FGP built-up beams are developed to show good convergence and accuracy of the method, compared with experimental and FE results. Combined influences of geometric and material properties on the natural frequencies of the built-up beam are discussed. The effects of one parameter are dependent on another ones, and nonlinear combined effects can be observed. The presented results can help to find the best combination of material and geometric coefficients, which is of significance for optimal design of these built-up structures.</p>