Symplectic Transfer-Matrix Method for Bending of Nonuniform Timoshenko Beams on Elastic Foundations

This paper proposes a new symplectic transfer-matrix method (STMM) to solve the bending problem of nonuniform beams. The STMM is a hybrid method that combines the symplectic dual solution system and the traditional transfer-matrix method (TMM). It is first necessary to obtain the benchmark solution of uniform beams in a symplectic system. On this basis, this paper derives the transfer matrices under various nonuniform conditions and discusses the mathematical structure of these transfer matrices. After obtaining the global transfer equation, the equation is solved by changing its rows and columns. Three numerical examples are given to verify the correctness and applicability of the STMM. This paper proves that the transfer matrix in the symplectic system is a symplectic matrix in mathematics, whether it is a field transfer matrix, a point transfer matrix, or a global transfer matrix. The STMM reveals the mathematical property of the transfer matrix and provides a theoretical basis for the standardized application of the "transfer type" method in structural analysis.