A High-Frequency Model of a Rectilinear Beam with a T-Shaped Cross Section

Abstract

This paper derives an analytical model of a straight beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 35 kHz. The web, the right part of the flange, and the left part of the flange of the T-beam are modeled independently with two-dimensional elasticity equations for the in-plane motion and Mindlin flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions. These algebraic equations are then solved to yield the wave propagation coefficients and thus produce a solution to the displacement field in all three directions. An example problem is formulated and compared with solutions from fully elastic finite element modeling, a previously derived analytical model, and Timoshenko beam theory. It is shown that the accurate frequency range of this new model is significantly higher than that of the analytical model and the Timoshenko beam model, and, in the frequency range up to 35 kHz, the results compare very favorably to those from finite element analysis.