Novel method for analyzing the behavior of composite beams with non-smooth interfaces

  • Chein-Shan Liu
  • Botong LiEmail author
  • Chung-Lun Kuo


An almost exact solution is derived for the forced vibration of a composite beam with periodically varying non-smooth interface through a moderate weak-form formulation. The material property of a non-uniform beam is characterized by its flexural rigidity function R(x). In the novel method, R(x) is relaxed to be an integrable function rather than a \({\mathcal{C}}^2\) smooth function in the usual approach. The R(x)-orthogonal bases in the linear span of all boundary functions are derived such that the second-order derivatives of the bases elements are orthogonal with respect to the weight function R(x). When the deflection of the beam is expressed in terms of the bases, the expansion coefficients can be determined exactly in closed form owing to the R(x)-orthogonality of the bases. The solution obtained is almost exact, since its accuracy can be up to the order \(10^{-15}\). This powerful method is used to analyze the forced vibration behavior of composite beams with three different periodic interfaces.


Composite beams Moderate weak-form formulation Forced vibration Non-smooth interface 


  1. Abu-Hilal, M.: Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions. J. Sound Vib. 267, 191–207 (2003)CrossRefzbMATHGoogle Scholar
  2. Balduzzi, G., Aminbaghai, M., Auricchio, F., Fussl, J.: Planar Timoshenko-like model for multilayer non-prismatic beams. Int. J. Mech. Mater. Des. 14, 51–70 (2018)CrossRefGoogle Scholar
  3. Banerjee, J.R., Su, H., Jackson, D.R.: Free vibration of rotating tapered beams using the dynamic stiffness method. J. Sound Vib. 298, 1034–1054 (2006)CrossRefGoogle Scholar
  4. Bruant, I., Proslier, L.: Optimal location of piezoelectric actuators for active vibration control of thin axially functionally graded beams. Int. J. Mech. Mater. Des. 12, 173–192 (2016)CrossRefGoogle Scholar
  5. Datta, A.K., Sil, S.N.: An analytical of free undamped vibration of beams of varying cross-section. Comput. Struct. 59, 479–483 (1996)CrossRefzbMATHGoogle Scholar
  6. Dong, X.Y., Bai, Z.B., Zhang, S.Q.: Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Probl. 5, 1–15 (2017)MathSciNetzbMATHGoogle Scholar
  7. Huang, Y., Li, X.F.: A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J. Sound Vib. 329, 2291303 (2010)Google Scholar
  8. Li, F.S.: Global existence and uniqueness of weak solution for nonlinear viscoelastic full Marguerre–von Karman shallow shell equations. Acta Math. Sin. 25, 2133–2156 (2009)CrossRefzbMATHGoogle Scholar
  9. Li, H., Balachandran, B.: Buckling and free oscillations of composite microresonators. J. Micromech. Syst. 15, 42–51 (2006)CrossRefGoogle Scholar
  10. Li, B., Liu, C.-S.: A new method for the deflection analysis of composite beams with periodically varying interfaces. Z. Angew. Math. Mech. 98, 718–726 (2018)MathSciNetCrossRefGoogle Scholar
  11. Li, B., Dong, L., Zhu, L., Chen, X.: On the natural frequency and vibration mode of composite beam with non-uniform cross-section. J. Vibroeng. 17, 2491–2502 (2015)Google Scholar
  12. Li, B., Liu, C.-S., Zhu, L.: Vibration analysis of composite beams with sinusoidal periodically varying interfaces. Z. Nat. A 73, 57–67 (2018a)Google Scholar
  13. Li, B., Liu, C.-S., Zhu, L.: A general algorithm on the natural vibration analysis of composite beams with arbitrary complex interfaces. Z. Nat. A 73, 995–1004 (2018b)Google Scholar
  14. Liu, C.-S., Li, B.: A fast new algorithm for solving a nonlinear beam equation under nonlinear boundary conditions. Z. Nat. A 72, 397–400 (2017a)Google Scholar
  15. Liu, C.-S., Li, B.: An upper bound theory to approximate the natural frequencies and parameters identification of composite beams. Compos. Struct. 171, 131–144 (2017b)CrossRefGoogle Scholar
  16. Myint-U, T., Debnath, L.: Partial Differential Equations for Scientists and Engineers, 3rd edn. Prentice-Hall, Englewood Cliffs (2007)zbMATHGoogle Scholar
  17. Raja, S., Venkata, R.K., Gowda, M.T.: Improved finite element modeling of piezoelectric beam with edge debonded actuator for actuation authority and vibration behaviour. Int. J. Mech. Mater. Des. 13, 25–41 (2017)CrossRefGoogle Scholar
  18. Sahoo, S.R., Ray, M.C.: Analysis of smart damping of laminated composite beams using mesh free method. Int. J. Mech. Mater. Des. 14, 359–374 (2018)CrossRefGoogle Scholar
  19. Sears, A., Batra, R.: Macroscopic properties of carbon nanotubes from molecular-mechanics simulations. Phys. Rev. B 69, 235406 (2004)CrossRefGoogle Scholar
  20. Shaat, M., Abdelkefi, A.: Buckling characteristics of nanocrystalline nano-beams. Int. J. Mech. Mater. Des. 14, 71–89 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center of Excellence for Ocean EngineeringNational Taiwan Ocean UniversityKeelungTaiwan
  2. 2.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina

Personalised recommendations