<p>This research utilizes a combination of the shear deformation theory and the nonlocal theory to investigate the bending and natural vibration behaviors of a nano-beam with a changing thickness. The beam is supported by a viscoelastic foundation, and its stiffness parameter changes along its length. The beam is composed of a substance that has porosities arranged in two distinct and separate patterns. This study employs two types of shear deformation functions in conjunction with a two-node beam element. Each node is characterized by two degrees of freedom for displacement and two degrees of freedom for the derivatives of these displacements. The equation is constructed using the finite simulation method and subsequently solved to determine the bending and natural vibration responses. The beam’s bending displacement has both real and imaginary components as a result of the foundation’s viscous damping characteristic. This study also demonstrates that the reduction in bending displacement is influenced by several parameters, with the thickness variation parameter having a substantial impact on this reduction. Furthermore, this work demonstrates the impact of various geometric parameters, rules governing porosity distribution, the form of the shear strain function, boundary conditions on the bending response, and the natural vibration of nano-beams with changing thickness. The findings of this work provide a helpful scientific foundation for engineers to practically design and produce nano-beams with varying thicknesses.</p>