A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar

An approximate Hamilton principle is established for the transverse vibration of a reinforced concrete pillar by considering the dissipation energy, and a generalized Boussinesq equation is obtained. The exp-function method is adopted to solve the equation, and its solution properties are discussed and elucidated, including solitary solution, blowup solution, and discontinuous solution. In order to study the effect of a porous structure on the vibration property, fractal calculus is used to derive the fractal Boussinesq equation, and a fractal variational principle is also established. The fractal model confers many attractive properties, which can not be revealed by the traditional protocol. The effect of the nanofiber-reinforced concrete structure on its wave morphology is discussed and illustrated. A blowup solution can be converted into a flat solution by adjusting the value of the fractal derivative order. The paper sheds new light on the design of reinforced concrete pillars to avoid vibration damage.