Use of Variational Calculus to Evolve Third Order Functionals for Continuum Analysis

Abstract

This paper presents use of variational calculus to evolve third order functionals for continuum analysis. The governing equilibrium equation of forces of a line continuum was integrated in the open domain with respect to deflection to obtain three different valid forms of total energy functional for the continuum. They are second order (Ritz energy functional), fourth order (work error functional) and any functional hereinafter called the third order energy functional. Third order energy functional for a rectangular plate was also formulated. These third order energy functionals were subjected to direct variation (differentiating with respect to the coefficient of deflection) to obtain the weak form equilibrium of forces of continuums. Line continuum of four different boundary conditions and a plate with one edge clamped and the other three edges simply supported were used to test this new third order energy functional. In this numerical study, pure bending, buckling and free vibration analysis were performed. The results obtained indicated that the values obtained using this new method are exactly the same as the values obtained using either Ritz or work error energy functionals. Thus, one can comfortably and confidently use the third order energy functionals in continuum analysis