A new perspective on shear deformation in a simply supported isotropic plates: static bending and critical load determination

This study advances the analysis of rectangular plates under transverse loading by employing a third-order shear deformation theory and a direct energy method. To address limitations of prior models, exact displacement fields were derived from a general variation of total potential energy, yielding closed-form expressions for deflection and transverse shear in the x and y directions via variational calculus. Subsequently, new formulas for critical lateral loads – applicable before excessive deflection or elastic yield – were developed. Numerical evaluations were conducted for aspect ratios 1 ≤ b a ≤ 2, spans of L = 1000, 3000, and 5000 mm, and thicknesses t = 5, 10, 12.5, and 15 mm. The results show that increasing thickness markedly enhances load capacity and safety margins, while larger aspect ratios and longer spans reduce stability. For example, a 15 mm plate with a 1000 mm span attains elastic-limit loads q i w > 74.7 N/mm at aspect ratio 1, decreasing to ~29.2 N/mm at aspect ratio 2, with yield loads q i p > 77.8 N/mm. In contrast, a 5 mm plate under the same spans has q i w values lowering from ~2.77 N/mm to ~1.08 N/mm as aspect ratio increases, while q i p remains in the range 8.39-12.53 N/mm. These trends indicate that thicker plates withstand higher elastic and plastic limits, expanding safe operating margins, whereas slender (high aspect ratio) configurations exhibit reduced capacity and greater susceptibility to instability. Under large-span/large-deflection conditions, negative or near-zero capacity values reveal potential instability modes such as buckling in slender plates. In sum, the proposed model enhances design effectiveness by integrating geometric, material, and reinforcement considerations to ensure safety and serviceability without relying on post hoc deflection or shear assessments. The framework provides reliable, accurate predictions of flexural and shear responses, supporting efficient, safe, and cost-effective structural design.