Uniformly Loaded Logarithmic Beam Mode with Spatially Varying Flexural Rigidity

This analysis explores natural leading modes represented by logarithmic functions, achieved by imposing four boundary constraints at the ends of an elastic inhomogeneous beam. The beam possessing constant material inertia, is assumed to be uniformly loaded, and is composed of material with variable stiffness. It is sought analytical expressions for beam deflections in terms of logarithmic functions. Our findings demonstrate that such formulae can be derived for a beam under axially uniform load and with spatially distributed flexural rigidity. Subsequently, the beam shapes and material properties for four specific scenarios are identified: free-free logarithmic beam, cantilevered logarithmic beam, simply-supported logarithmic beam, and simply-supported sliding logarithmic beam. Explicit logarithmic beam responses, governed by a limited number of shape parameters, are illustrated graphically using normalized deflections with respect to the maximum deflection. Highly deflected elastic logarithmic modes emerge as a consequence of high flexural rigidity influenced by the uniformly applied transverse load. These elucidated logarithmic beam modes offer potential practical applications in the structural design of functionally graded materials. They also serve as valuable testing platforms for numerical techniques employed in the analysis of more complex beam problems.