A Modified Transfer Matrix Method for Static and Dynamic Analysis of Beams that Eliminates the Need to Compute the Inverse of the Zero Matrix

The classical transfer matrix method has been widely used to study the static and dynamic analysis of uniform and non-uniform beams in various structural engineering applications; however, this method implies that to calculate the transfer matrix, it is necessary to compute the inverse of the zero matrix of each element, which drastically increases the computational cost when the size of the zero matrix and the number of elements increases. This paper proposes a modified transfer matrix method that directly calculates the transfer matrix without the need to compute the inverse of the zero matrix. The equilibrium equations, constitutive laws, and boundary conditions of the beam are derived with a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. The proposed modification consists of calculating the transfer matrix with a Laplacian approach, where initially a Laplace transform is applied to the system of equilibrium equations, and then an inverse Laplace transform is applied to express the displacements and internal forces in terms of the displacements and internal forces of the zero point of the origin of coordinates. This method allows for obtaining the transfer matrix directly without the necessity of calculating the inverse of the zero matrix, thereby minimizing the computational cost of the method. The results of the numerical applications demonstrate the validity and superiority of the proposed modification to the classical transfer matrix method.