Application of a Direct Variational Principle in Elastic Stability Analysis of Thin Rectangular Flat Plates

ABSTRACT

The purpose of this work is to use the energy approach in the form of direct variational principle (Rayleigh-Ritz method) for buckling analysis of thin rectangular plates with various boundary conditions using Taylor series shape functions. To do this, thin rectangular flat plate of various boundary conditions with three dimensions Ly, Lx and t was analyzed in this research. Ly and Lx are secondary and primary in-plane dimensions respectively and t is the plate thickness. The boundary conditions covered in this research included SSSS, SSSC, SSCC, SCCC, SCSC, CCCC, SSSF, CCCF and SCFS plates. Rayleigh-Ritz method of direct variational approach for the plate analysis was adopted. The total potential energy functional of the method was derived from first principle by using equations and principles of theory of elasticity. Taylor-MacLaurin’s series was used to formulate the approximate shape functions for the plate with various boundary conditions. The shape functions from Taylor- MacLaurin’s series were substituted into the total potential energy functional, which was subsequently minimized to get the stability equations. Derived Eigen-value solver was used to solve the stability equations for plates of various aspect ratios (from 0.1 to 1 at the increment of 0.1) to get the buckling loads of the plates. The buckling loads from this study were compared with those of earlier researches. The results showed that the average percentage differences recorded for SSSS, CCCC, CSCS, CSSS, and SSFS plates are 0.069%, 3.54%, 3.071%, 6.25% and 4.14% respectively. The convergence of the shape function showed that for CSCS plate, the difference between the buckling loads when the Taylor- MacLaurin’s series were truncated at m = n = 4 and m = n = 5 is 1.11%. This difference is 0.878% for CCSS plates. These differences showed that the shape functions formulated by using Taylor-McLauruin’s series has rapid convergence and very good approximation of the exact displacement functions of the deformed thin rectangular plate under in-plane loading.