Guided identification of nonlinear reduced-order models via the incorporation of von Kármán beam theory

Abstract

A reduction in the number of Degrees Of Freedom (DOFs) of a nonlinear structural model can significantly decrease the computational time of dynamic analyses, especially in problems where long analyses are required. A Reduced-Order Model (ROM) of a structural system is commonly generated using some combination of dimensionality reduction, such as substitution of a modal basis or static condensation of membrane DOFs. Perhaps the most common and widespread framework for developing low-order models of continuous systems is the Ritz method in conjunction with a smooth modal basis. The Ritz method requires access to the closed form governing differential equations, and it is difficult to implement for problems with complex boundary conditions. In recent decades, an alternative framework that instead constructs a low-order model from an existing Finite Element Model (FEM) has been developed. Such models are typically known as NonLinear Reduced-Order Models (NLROMs). Such NLROMs can be used to handle complex boundary conditions so long as the FEM can do the same, and they allow the analyst to start from an existing finite element package as opposed to directly working with the governing differential equations. The form of the NLROM restoring force function is assumed by the analyst prior to identifying the model, and the parameters of the function are identified to achieve a high-quality fit to the FEM load–displacement data. This research aims to examine the connection between classically derived models using the Ritz method and NLROMs in order to predict the coefficients of NLROMs and to develop simplifying assumptions about their structure. Specifically, the Condensed Von Kármán (CVK) beam equation is used as a basis for the study of nonlinear beam structures. The Ritz method is applied to the CVK beam equation, and the matrix form of the equilibrium equation is determined. The equivalence between the CVK coefficients and the NLROM coefficients is shown, and a simplified, tangent stiffness-based identification procedure is proposed to determine the coefficients of the CVK-based model. An arc-length continuation algorithm is used to predict the static load–displacement response of the models studied and to assess the accuracy of the ROMs. In the case of initially straight, post-buckled beams, the proposed identification procedure is shown to generate a model that generally outperforms a corresponding NLROM identified using a more traditional displacement-based procedure.