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

•The Hamilton principle is used to derive the governing equation for transverse vibration of a reinforced concrete pillar.•A Boussinesq-like equation is obtained and its resonance-like and solitary solutions are discussed.•A fractal Boussinesq equation is discussed for vibration design by taking int...

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Bibliographic Details
Published in:Applied Mathematical Modelling 2020-06, Vol.82, p.437-448
Main Authors: Ji, Fei-Yu, He, Chun-Hui, Zhang, Jing-Jing, He, Ji-Huan
Format: Article
Language:English
Subjects:
Blasting demolition
Boussinesq equations
Concrete structures
Energy dissipation
Fractal concrete
Fractal models
Fractals
Hamilton's principle
Morphology
Nanofibers
Railway bridge monotoring
Reinforced concrete
Solitary wave, two-scale transform
Transverse oscillation
Vibration control
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Summary:•The Hamilton principle is used to derive the governing equation for transverse vibration of a reinforced concrete pillar.•A Boussinesq-like equation is obtained and its resonance-like and solitary solutions are discussed.•A fractal Boussinesq equation is discussed for vibration design by taking into account the porous structure of concrete.•The morphology of a travelling wave can be effectively controlled by the fractal dimensions of fractal concrete. 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.
ISSN:0307-904X
1088-8691
0307-904X
DOI:10.1016/j.apm.2020.01.027