<p>This paper presents a Ritz method utilising Gegenbauer polynomial to analyse functionally graded triply periodic minimal surface (FG-TPMS) beams supported on a viscoelastic foundation. The generalised foundation model incorporates four parameters, including two parameters related to stiffness and two parameters associated with damping coefficients. A higher-order shear deformation theory, based on Chebyshev polynomial, is employed. The governing equations are solved via the Gegenbauer polynomial-based Ritz method. The study meticulously investigates the impacts of various factors on the frequency, critical buckling load, deflection, and damped vibration responses of FG-TPMS beams. Results demonstrate excellent convergence and superior computational efficiency for the present method. The research reveals that foundation stiffness parameters significantly impact the dynamic and static characteristics of FG-TPMS beams, with the Pasternak stiffness coefficient exerting a more pronounced effect than the Winkler coefficient. Furthermore, damping coefficients play a key role, with the Pasternak damping coefficient dominating for lower slenderness ratios (<i>L/h</i> = <i>2</i>), while the Winkler coefficient becomes dominant for larger slenderness ratios (<i>L/h</i> = <i>10</i>). These findings provide valuable insights for the design and optimisation of FG-TPMS structures in engineering applications.</p>