A Novel Two-Parameter Elastic Foundation Model for the Bending Analysis of Functionally Graded Beams

Aizhan Nurgoziyeva; Sungat Akhazhanov; Meiirrassul Bekov

doi:10.48084/etasr.18071

Authors

Aizhan Nurgoziyeva Laboratory of Applied Mechanics and Robotics, Buketov Karaganda National Research University, Karaganda, Kazakhstan
Sungat Akhazhanov Laboratory of Applied Mechanics and Robotics, Buketov Karaganda National Research University, Karaganda, Kazakhstan
Meiirrassul Bekov Laboratory of Applied Mechanics and Robotics, Buketov Karaganda National Research University, Karaganda, Kazakhstan

Volume: 16 | Issue: 3 | Pages: 35586-35594 | June 2026 | https://doi.org/10.48084/etasr.18071

Received: 9 February 2026 | Revised: 22 March 2026 | Accepted: 1 April 2026 | Online: 22 April 2026

Corresponding author: Sungat Akhazhanov

Abstract

This study presents an analytical bending analysis of Functionally Graded (FG) beams resting on a newly developed two-parameter elastic foundation. Unlike classical foundation models, such as the Winkler, Pasternak, and Vlasov formulations, the proposed foundation model incorporates two independent parameters, enabling a more flexible and physically realistic description of the coupled normal and shear interactions between the beam and the supporting medium. The material properties of the FG beam are assumed to vary continuously through the thickness according to a power-law distribution. The governing differential equations were derived within the framework of the beam theory and solved analytically. Closed-form solutions for transverse deflections, bending moments, and stress resultants are obtained, enabling a detailed assessment of the bending response of FG beams. A parametric study is performed to investigate the effects of material gradation, beam slenderness, and foundation parameters on displacement and stress distributions. The results demonstrate that the proposed foundation parameters significantly influence the bending behavior of FG beams, particularly for thick beams and higher material gradation indices, resulting in noticeable deviations from predictions based on classical elastic foundation models. Comparisons with available solutions from the literature for limiting cases, including the homogeneous metallic beam, show excellent agreement and confirm the accuracy and reliability of the present formulation. The smooth convergence to classical solutions verifies the mathematical consistency and physical validity of the proposed approach, which provides an effective framework for the advanced analysis of FG beams on elastic foundations.

Keywords:

functionally graded beam, two-parameter elastic foundation, analytical bending analysis, axial displacement, transverse displacement, transverse shear stress, axial stress

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