Two-variables shear theory for bio-inspired helicoidal composite and carbon nanotube-reinforced composite beams resting on a general viscoelastic Winkler-Pasternak foundation

This paper presents a Hermite polynomial-based two-variable shear theory and a Ritz method for analysing the buckling, bending, free vibration, and damped vibration of carbon nanotube-reinforced composite (CNTRC) and bio-inspired helicoidal composite (BIHC) beams. These beams rest on a general viscoelastic Winkler-Pasternak foundation characterised by four parameters, incorporating the stiffness and damping effects of both the Winkler and Pasternak layers. The foundation model can be simplified into traditional forms such as Winkler, Pasternak, or their viscoelastic variants. Numerical examples are provided to validate the proposed Ritz method and theory. Additionally, the influences of lamination stacking sequence, slenderness ratio, material anisotropy, boundary conditions, and the four foundation parameters on the structural behaviour of CNTRC and BIHC beams are examined in detail. The results demonstrate that foundation effects play a crucial role in the mechanical behaviour of BIHC and CNTRC beams, with the Pasternak foundation model exerting a more significant influence than the Winkler model. This study also presents the first-ever analysis of damped vibration behaviour for these structures, providing novel insights into how foundation properties and damping interact to affect structural vibration. These findings advance the understanding necessary for applying BIHC and CNTRC beams in engineering designs where vibration and damping effects are critical.