An Exact Finite Element Formulation For Static Bending Analysis of Timoshenko Beams Based on Analytical Homogeneous Solutions

Mohammed Ali Hjaji; Hasan M.  Nagiar

doi:10.66411/jer.v41i2.457

Authors

Mohammed Ali Hjaji Applied Mechanics Division, Department of Mechanical and Industrial Engineering, University of Tripoli, Tripoli, Libya Author
Hasan M. Nagiar Applied Mechanics Division, Department of Mechanical and Industrial Engineering, University of Tripoli, Tripoli, Libya Author

DOI:

https://doi.org/10.66411/jer.v41i2.457

Keywords:

Timoshenko beam, Exact finite element, Analytical homogeneous solution, Exact Shape Functions, Static bending, Shear deformation

Abstract

An exact finite element formulation is developed for the static bending analysis of shear-deformable Timoshenko beams under distributed transverse loads and bending moments. The coupled field equations are obtained by applying the principle of minimum potential energy to the Timoshenko beam formulation., and their analytical homogeneous solutions are employed to construct interpolation functions that satisfy the governing equations exactly throughout the element domain. This eliminates interpolation errors within each element and provides improved accuracy and convergence compared with conventional polynomial-based beam elements. An exact two-node beam element incorporating four degrees of freedom is established, and explicit closed-form expressions for the corresponding element stiffness matrix and consistent load vector are derived directly from the exact interpolation functions. The proposed formulation is validated through benchmark problems involving cantilever, simply supported, clamped–pinned, clamped–roller, and clamped–clamped beams. The numerical results show excellent agreement with analytical solutions, Euler–Bernoulli beam predictions, and highly refined Abaqus finite element models, requiring a substantially smaller number of degrees of freedom. The proposed element provides an accurate and computationally efficient framework for the static analysis of thick and moderately thick Timoshenko beams and establishes a foundation for future extensions to dynamic, stability, and thermoelastic problems.

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