Wave propagation and Bragg gaps formation in periodically supported coupled bending-torsional thin-walled beams

Elastic wave propagation in periodically supported beams is an important topic in mechanics, with many engineering applications. This paper focuses on wave propagation in coupled bending-torsional thin-walled beams with monosymmetric cross section, resting on periodic elastic supports acting in the horizontal and vertical planes. Using a transfer matrix approach combined with Bloch's theorem, extensive parametric analyses are conducted to investigate the influence of geometrical parameters of the beam cross section and stiffness/location of the elastic supports on the dispersion diagram and, in particular, on size and attenuation properties of band gaps resulting from Bragg scattering. The parametric analyses unveil relevant aspects unaddressed in existing literature, markedly different from those observed in beams with doubly symmetric cross section. Moreover, for an exhaustive insight into the wave propagation properties, a number of complementary analyses are performed. Focusing on the relevant case of beams resting on rigid supports, free vibration analyses of the unit cell show that opening/closing frequencies of the Bragg gaps match a subset of natural frequencies of the unit cell, either simply supported or clamped at the ends. Notably, these conclusions serve to identify Bragg gaps, generalizing and validating concepts already established for wave propagation in beams with doubly symmetric and asymmetric cross section. Furthermore, considering the most general case of beams resting on elastic supports of arbitrary stiffness, the transmittance of the finite beam is calculated, confirming that wave propagation properties obtained for the infinite beam hold for the finite beam as well. To handle the dynamics of the finite beam, a computational framework is introduced, based on the theory of generalized functions to model shear-force and torque discontinuities attributed to the elastic supports.