Equivalence of refined theory and decomposed theorem of an elastic plate

A connection between Cheng's refined theory and Gregory's decomposed theorem is analyzed. The equivalence of the refined theory and the decomposed theorem is given. Using operator matrix determinant of partial differential equation, Cheng gained one equation, and he substituted the sum of the general integrals of three differential equations for the solution of the equation. But he did not prove the rationality of substitute. There, a whole proof for the refined theory from Papkovich-Neuber solution was given. At first expressions were obtained for all the displacements and stress components in term of the midplane displacement and its derivatives. Using Lur'e method and the theorem of appendix, the refined theory was given. At last, using basic mathematic method, the equivalence between Cheng's refined theory and Gregory's decomposed theorem was proved,i. e., Cheng's bi-harmonic equation, shear equation and transcendental equation are equivalent to Gregory's interior state, shear state and Papkovich-Fadle state, respectively.