Experimental path-following of equilibria using Newton’s method. Part I: Theory, modelling, experiments

Abstract

Modern numerical path-following techniques provide a comprehensive suite of computational tools to study the bifurcation behaviour of engineering structures. In contrast, experimental testing of load-bearing nonlinear structures is still performed using simple force control (dead loading) or displacement control (rigid loading). This means that established experimental methods cannot trace equilibrium manifolds in their entirety because structures snap to alternative equilibria at limit points in the forcing parameter and because branch switching to alternative equilibria cannot be controlled and performed reliably. To extend current testing methods, in Part I of this paper, we implement an experimental path-following method that uses tangent quantities (stiffness and residual forces) and Newton’s method to continue along stable and unstable equilibrium paths and traverse limit points. In addition to enforcing the displacement at primary load-introduction points, the overall shape of the structure is controlled via secondary actuators and sensors. Small perturbations of the structure using the secondary actuators allow an experimental tangent stiffness to be computed, which is then used in a control algorithm. As a pertinent test case, the experimental method is applied to a transversely loaded shallow circular arch. Due to the complexity of the test setup, the experiment is first designed using a virtual testing environment based on a surrogate finite element model. Experimental results demonstrate the robustness of the proposed experimental method and the usefulness of virtual testing as a surrogate, but also highlight that experimental efficiency and the effects of noise and sensor uncertainty is of particular concern. In Part II, we present perspectives on future research directions and novel testing capabilities that are enabled by extending the methodology to pinpointing of critical points, tracing of critical boundaries, and branch switching.