A Nonlinear Parametric Resonance Analysis of Planar Beam Structures Using Vector Form Intrinsic Finite Element Method

Abstract The parametric resonance of a frame structure may lead to a disastrous consequence. Structural engineers need a straightforward and practical method to calculate the large nonlinear parametric resonance responses of the frame structure for assessing structural safety. This paper reports on a new Vector Form Intrinsic Finite Element (VFIFE) method to analyse the nonlinear parametric resonance of planar beam structures. By taking into account the geometric stiffness effect of the internal axial force for the frame element, a relationship between internal nodal forces and deformation components is firstly established based on the VFIFE principle and the parametric resonance mechanism of frame structures. The VFIFE scheme is then developed to analyse the nonlinear parametric resonance of frame structures. To demonstrate the efficacy of this approach, a simply-supported beam is used as a numerical example, and the resulting VFIFE calculations are compared to the solutions obtained by current analytical methods. A parametric resonance test of the cantilever beam is conducted to verify the applicability of the VFIFE method. The numerical results show that this approach can overcome the limitations of existing analytical methods, well simulate the nonlinear phenomena of parametric resonances, and produce predictions that are consistent with experimental observations. The numerical stability boundary of parametric resonance agrees well with the experimental boundary. Possible causes of deviations between the numerical and experimental results are discussed. The proposed VFIFE method is a simple and effective means of analyzing the stability and nonlinear responses of parametric resonance of planar beam structures.