Simplified Closed-Form Method to Large Deflection Stability Analysis of Thin Rectangular Plate

Abstract

At the moment, the analyses of postbuckling behavior of thin rectangular plates of constant thickness are generally based on von Karman’s large deflection equations which make use of stress functions and trigonometric variables. These equations are coupled, nonlinear partial differential equations of fourth order each. Getting a closed-form solution of these equations is near impossible and tedious. The present work presents a simplified closed-form general equation using a variational approach for large deflection analysis. The approach adopted here is devoid of Airy’s stress functions. A new strain-displacement equation is formulated, and the total potential energy equation is minimized. The resulting compatibility equation was solved to obtain the general governing stability equation under large deflection. This governing equation was applied to a plate simply supported all-around using polynomial displacement function and numerical results were obtained. To validate the numerical results obtained, they were compared with the values obtained by Levy whose results are generally acclaimed as exact and with two others. It was observed that the minimum and maximum percentage differences were 0% and 41.56% at stress parameters 3.66 and 21.45 respectively. Also, for deflection to thickness ratio (w/t), the present results showed a close agreement with those of Levy with a minimum and maximum percentage difference of 0.07% and 41.56% at w/t of 0 and 3.376 respectively. Importantly, the present result lies between two other research results. We, therefore, conclude that the present work is adequate and a new simple closed-form approach to understanding and predicting the postbuckling strength of thin rectangular plates.