Principle Component Analysis Based Approach to Model Deformation of Web Core Beam

Abstract

The purpose of the present study was to develop a systematic way to determine the modes of deformation found in architected materials. Specifically, the mode of deformation for the web within a web core beam structure is studied. Localized deflection information was gathered and PCA was used to determine the common mode of deformation. This was used to develop a energy based model for a unit cell to determine the values of kinematic variables. The model was homogenized so a structure with many unit cells could be easily analyzed by being discretized into elements much larger than the unit cell size. Using Castigliano’s first theorem the values of the kinematic variables can be found. To minimize the objective function, MATLABs optimization tool box was utilized. For a web core beam it was found that the webs deform as a cubic function and 3 PCs were adequate to approximate the deformation. This information was used to develop the beam model and the accuracy of the center line deflection for a fixed-fixed beam was analyzed to validate the model. While the work done does not provide a robust answer to what shape function to use in all unidirectional sandwich beam structures, it does provide a systematic method to determine the modes of deformation before the development of a analytical model is developed. This method can also be applied to other architected structures, and assist in future work related to determining deformation, failure, and instabilities within the micro-structure of architected materials. The work done provides a systematic method to determine the shape function of unidirectional sandwich beam structures with any cross section. This method can also be applied to other types of architected structures. It can assist in future work related to determining deformation, failure, and instabilities within the micro-structure of architected structures. Furthermore, it shows that the implementation of an energy based method can be used rather than the typical system of equations. This allows for a more simple implementation when changing the structure’s geometry, external loads, or boundary conditions.