Analysis of Deep to Moderately Deep Beams Using Timoshenko Theory and 2D Plane-Stress Elasticity

This study examines rectangular cantilever beams, clamped at one end and subjected to a transverse tip load at the other, with aspect ratios \( L/h = 1\text{--}5 \)—a range in which shear deformation and stress field nonuniformities have a pronounced influence on structural behavior. A plane-stress finite element formulation is employed to capture the full two-dimensional elastic response, and the results are systematically compared with closed-form solutions derived from Timoshenko beam theory. The comparison highlights the limitations of classical beam assumptions within this aspect ratio range. Based on numerical evidence, approximate expressions for the maximum deflection under end loading are proposed for selected values of Poisson’s ratio, offering improved accuracy for moderately deep and deep beams. In addition, a MATLAB\textsuperscript{\textregistered} code is provided for estimating the maximum deflection for arbitrary values of Poisson’s ratio.