#7489. Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports

October 2026publication date
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Journal’s subject area:
Civil and Structural Engineering;
Mechanical Engineering;
Mechanics of Materials;
Condensed Matter Physics;
Materials Science (all);
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Abstract:
This paper presents exact closed-form solutions for free vibration of discretely supported Euler-Bernoulli (DEB) and Timoshenko beams (DTB) in the presence of an arbitrary number of intermediate elastic constraints. The exact eigenvalue equations and mode shapes of the DEB and DTB are derived using the generalized function method for the first time, which expands the general solution of mode shapes as a combination of the standard trigonometric/hyperbolic functions with integration constants extended to generalized functions. The second-order eigenvalue equation is formulated from the perspective of the entire domain of the beam without enforcement of any continuity conditions.
Keywords:
Courants maximum-minimum principle; Discretely supported beam; Eigenvalue equation; Exact closed-form solution; Generalized function method; Support optimization

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