A Combined Spectral Element Method and Hilber–Hughes–Taylor Framework for Investigating the Transient Response of Functionally Graded Timoshenko Beams on Biparametric Vlasov Foundations

Functionally graded (FG) beams have been used more and more in highly designed structures under dynamic loading due to their graded mechanical properties and excellent performance. Their transient response on complex elastic foundations is, however, not easily predicted due to the material heterogeneity, shear deformation, rotary inertia, and coupled effect of the foundation parameters. The purpose of this study is thus to propose an accurate and efficient computational model for the dynamic analysis of FG Timoshenko beams supported by biparametric Vlasov foundations under harmonic excitation. The formulation takes into account the space-varying material properties, Timoshenko shear deformation, rotary inertia, and coupled Winkler–shear interaction of the Vlasov foundation. The governing equations are numerically solved in space with the high-order spectral element method (SEM) and in time with the Hilber–Hughes–Taylor (HHT) scheme. The resulting framework is used to study the transient displacement and vibration response with respect to the excitation frequency, material gradation index, and stiffness and damping properties of the foundation. The numerical results prove that the results converge quickly in space and time and also indicate that the dynamic response is significantly affected by the interaction between the gradation of material and the parameters of the foundation. The displacement amplitude, resonance behavior, and vibration characteristics are significantly altered by any variations in the gradation index and foundation characteristics. The results obtained with the proposed formulation are in good agreement with those available from the benchmark solutions, thus validating the correctness and reliability of the formulation. The SEM–HHT methodology offers a reliable, precise, and low-computational-cost solution for transient analysis of FG Timoshenko beams on biparametric Vlasov foundations under harmonic excitation. The proposed framework offers a powerful predictive tool for vibration analysis, response control, and design of advanced FG beam systems that can be applied in aerospace, marine, smart infrastructure, and other high-performance engineering structures.