Free Vibration Analysis of Elastically Restrained Laminated Planar Frames

A wave-based model that incorporates the effects of shear deformation, rotary inertia and elastic coupling due to structural anisotropy, is developed to analyze the free vibrations of elastically restrained laminated planar frames. In this work, a generalized frame structure is represented as an assemblage of laminated beam segments that act as one-dimensional waveguides. The segments are assumed to undergo only in-plane motion, which upon applying Hamilton's principle, is described by a sixth order coupled differential equation. Dispersion analysis is conducted and the nature of the wavefields associated with the propagation matrix is discussed. Generally restrained boundaries and internal joints are considered, and the associated reflection and transmission matrices are derived. Using the principle of wave-train closure, a closed-form characteristic equation is obtained by systematically assembling the propagation, reflection and transmission matrices. The wave-based model is inherently deterministic, and solving the characteristic equation offers the advantage of determining the exact natural frequencies using conventional root finding algorithms. Application of the proposed model is demonstrated by analyzing an elastically restrained inclined laminated portal frame. Extensive computational analysis is conducted to illustrate the influence of stacking sequence, frame angle, relative frame length, orthotropicity ratios and spring stiffness on the exact natural frequencies (and in certain cases the mode shapes) of the frame. Independent finite element simulations conducted in ANSYS® APDL are consistently used to verify the validity of the analytical results.