Accurate solutions of a thin rectangular plate deflection under large uniform loading

Abstract

The large deflection of a thin rectangular plate under uniform loading is a classic problem in solid mechanics. However, the equations are challenging to solve due to their strong nonlinearity. To the author’s best knowledge, existing studies have only reported solutions for rectangular plates with immovable clamped edges within the range of loadings a4q/(Dh)<500. In this study, we employ the homotopy analysis method (HAM) to address this problem and obtain accurate solutions even when the loading a4q/(Dh) reaches as high as 105. Additionally, we discover that the central deflection w(0,0)/h follows a scaling law of [a4q/(Dh)]1/3 for loadings a4q/(Dh)>104, which agrees with the scaling law for extremely large deflections. Consequently, we can utilize this scaling law to extrapolate our homotopy series solution from a4q/(Dh)=104 to even larger loadings. Our investigation also reveals that the bending and membrane stresses increase with the loading. The maximum bending stress is located at the edge, while the location of the maximum membrane stress shifts from the plate center towards the edge as the loading increases. Moreover, we find that even for large deflections, the bending stress remains significant at the edge, challenging the assumption of negligible bending stress adopted by the membrane equations. Thus, the membrane equation may over-predict the plate deflection even under large loadings. These findings advance our understanding of the large deflection of a thin rectangular plate with clamped edges, which has important applications in aerospace, ocean, chemical, and microelectronic engineering. The results presented in this paper can also be used to verify other algorithms designed for highly nonlinear problems.