A similarity transformation leading to an exact transfer matrix for the composite beam-column with refined zigzag kinematics: A benchmark example

摘要

The Refined Zigzag Theory (RZT), developed by Tessler et al. (2009) is among the most promising approaches for analysing composite structures today. Since its appearance many contributions have been published dealing with finite elements for laminated structures based on the efficient kinematic of RZT. The governing equations can be assembled in a first order differential system and could be solved by different numerical strategies, e.g., finite difference methods, Runge-Kutta methods or the transfer matrix method. In this work a new approach for the analysis of laminated composite beams will be presented which can be considered as exact. The first order differential system is solved here by following two parallel approaches in establishing the transfer matrix: a series solution and a similarity transformation. The proposed similarity transformation is based on a Jordan Decomposition using the eigensystem of the system matrix and leads to an explicit and analytical form of the transfer matrix for the static case. The solution procedure - as presented in Wimmer & Nachbagauer (2018) - is reflected here in short. As the main contribution of this work, the main advantages of the proposed similarity transformation are carved out and used to produce a text example leading to exact results. On the one hand, with the well-known relations between the transfer- and the stiffness matrix an exact version of the latter one can be obtained. On the other hand, as a further advantage, this procedure provides the exact values of the first derivatives of the essential kinematic variables. Therefore, the exact values can be used to calculate the strains and stresses in the laminate exactly in the framework of the RZT. The input data as well as the results for a representative sandwich beam example, which can serve as a benchmark test for other approximate solutions, is presented here in detail.