Skip to main content
SpringerLink
Log in

Dynamic analysis of partial-interaction Kant composite beams by weak-form quadrature element method

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

In this paper, the Kant higher-order beam theory is applied to model each segment of the partial-interaction composite beams, aiming to capture as possible fidelity as the plane stress model. On this basis, the weak-form equation is obtained through the principle of virtual work. Besides, the weak-form quadrature element (WQE), as a counterpart of the conventional finite element (CFE), is derived and implemented to more efficiently solve problems, including free vibration eigenvalue analysis and dynamic responses prediction to moving loads. After the verification of all the programs developed, a series of numerical examples are given to investigate the WQE’s superiority on convergence rate and numerical smoothness over the CFE. At the end of the paper, the influences of structural damping and loads’ moving speed on impact factor of two-span continuous beams are analyzed. Numerical results show that the proposed WQE, due to the variable-order interpolation of the element, possesses overwhelmingly higher computational efficiency than the CFE, and the numerical smoothness problem in the internal force analysis is significantly alleviated by WQE method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Newmark, N.M., Siess, C.P., Viest, I.M.: Tests and analysis of composite beams with incomplete interaction. Proc. Soc. Exp. Stress Anal. 9(1), 75–92 (1951)

    Google Scholar 

  2. Xu, R., Wang, G.: Variational principle of partial-interaction composite beams using Timoshenko’s beam theory. Int. J. Mech. Sci. 60(1), 72–83 (2012)

    Article  Google Scholar 

  3. Xu, R., Wang, G.: Bending solutions of the Timoshenko partial-interaction composite beams using Euler–Bernoulli solutions. J. Eng. Mech., ASCE 139(12), 1881–1885 (2013)

    Article  Google Scholar 

  4. Ecsedi, I., Baksa, A.: Analytical solution for layered composite beams with partial shear interaction based on Timoshenko beam theory. Eng. Struct. 115, 107–117 (2016)

    Article  Google Scholar 

  5. Schnabl, S., Saje, M., Turk, G., Planinc, I.: Analytical solution of two-layer beam taking into account interlayer slip and shear deformation. J. Struct. Eng., ASCE 133(6), 886–894 (2007)

    Article  Google Scholar 

  6. Ranzi, G., Bradford, M.A.: Direct stiffness analysis of a composite beam-column element with partial interaction. Comput. Struct. 85(15–16), 1206–1214 (2007)

    Article  Google Scholar 

  7. Nguyen, Q.-H., Martinelli, E., Hjiaj, M.: Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction. Eng. Struct. 33(2), 298–307 (2011)

    Article  Google Scholar 

  8. Hou, H., He, G.: Static and dynamic analysis of two-layer Timoshenko composite beams by weak-form quadrature element method. Appl. Math. Model. 55, 466–483 (2018)

    Article  MathSciNet  Google Scholar 

  9. Chakrabarti, A., Sheikh, A.H., Griffith, M., Oehlers, D.J.: Dynamic response of composite beams with partial shear interaction using a higher order beam theory. J. Struct. Eng., ASCE 139(1), 47–56 (2013)

    Article  Google Scholar 

  10. Chakrabarti, A., Sheikh, A.H., Griffith, M., Oehlers, D.J.: Analysis of composite beams with partial shear interactions using a higher order beam theory. Eng. Struct. 36, 283–291 (2012)

    Article  Google Scholar 

  11. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51(4), 745–752 (1984)

    Article  Google Scholar 

  12. He, G., Yang, X.: Dynamic analysis of two-layer composite beams with partial interaction using a higher order beam theory. Int. J. Mech. Sci. 90, 102–112 (2015)

    Article  Google Scholar 

  13. He, G., Yang, X.: Analysis of higher order composite beams by exact and finite element methods. Struct. Eng. Mech. 53(4), 625–644 (2015)

    Article  Google Scholar 

  14. He, G., Yang, X.: Finite element analysis for buckling of two-layer composite beams using Reddy’s higher order beam theory. Finite Elem. Anal. Des. 83, 49–57 (2014)

    Article  Google Scholar 

  15. Manjunatha, B.S., Kant, T.: New theories for symmetric/unsymmetric composite and sandwich beams with C\(^{0}\) finite elements. Compos. Struct. 23(1), 61–73 (1993)

    Article  Google Scholar 

  16. Kant, T., Owen, D.R.J., Zienkiew, O.C.: A refined higher-order C\(^{0}\) plate bending element. Compos. Struct. 15(2), 177–183 (1982)

    Article  MathSciNet  Google Scholar 

  17. Kant, T., Gupta, A.: A finite element model for a higher-order shear-deformable beam theory. J. Sound Vib. 125(2), 193–202 (1988)

    Article  Google Scholar 

  18. Kroker, A.M.: Becker W Closed-form analysis of a higher-order composite box beam theory. Proc. Appl. Math. Mech. 9(1), 213–214 (2009)

    Article  MathSciNet  Google Scholar 

  19. Kroker, A.M.: Becker W A higher-order composite beam theory for closed-form analysis of beams with box and I cross-section. Proc. Appl. Math. Mech. 10(1), 179–180 (2010)

    Article  MathSciNet  Google Scholar 

  20. Carrera, E., Pagani, A.: Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models. Int. J. Mech. Sci. 75, 278–287 (2013)

    Article  Google Scholar 

  21. Carrera, E., Cinefra, M., Petrolo, M., Zappino, E.: Finite Element Analysis of Structures Through Unified Formulation. Wiley, New Delhi (2014)

    Book  Google Scholar 

  22. Carrera, E., Giunta, G., Petrolo, M.: Beam Structures Classical and Advanced Theories. Wiley, London (2011)

    Book  Google Scholar 

  23. Szabo, B.A., Mehta, A.K.: \(p\)-convergent finite element approximations in fracture mechanics. Int. J. Numer. Methods Eng. 12(3), 551–560 (1978)

    Article  Google Scholar 

  24. Pozrikidis, C.: Introduction to Finite and Spectral Element Methods Using Matlab. CRC Press, New York (2014)

    MATH  Google Scholar 

  25. Wang, X.: Differential Quadrature and Differential Quadrature Based Element Methods Theory and Applications. Butterworth-Heinemann, Oxford (2015)

    MATH  Google Scholar 

  26. Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 15(15), 791–798 (1992)

    Article  Google Scholar 

  27. Zhong, H., Wang, Y.: Weak form quadrature element analysis of Bickford Beams. Eur. J. Mech. A-Solid 29(5), 851–858 (2010)

    Article  Google Scholar 

  28. Zhang, R., Zhong, H.: Weak form quadrature element analysis of planar slender beams based on geometrically exact beam theory. Arch. Appl. Mech. 83(9), 1309–1325 (2013)

    Article  Google Scholar 

  29. Zhong, H., Zhang, R., Xiao, N.: A quaternion-based weak form quadrature element formulation for spatial geometrically exact beams. Arch. Appl. Mech. 84(12), 1825–1840 (2014)

    Article  Google Scholar 

  30. Zhang, R., Zhong, H.: Weak form quadrature element analysis of spatial geometrically exact shear-rigid beams. Finite Elem. Anal. Des. 87, 22–31 (2014)

    Article  MathSciNet  Google Scholar 

  31. Jin, C., Wang, X.: Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method. Compos. Struct. 125, 41–50 (2015)

    Article  Google Scholar 

  32. Wang, Y., Wang, X.: Free vibration analysis of soft-core sandwich beams by the novel weak form quadrature element method. J. Sandw. Struct. Mater. 18(3), 294–320 (2016)

    Article  Google Scholar 

  33. Wang, X., Yuan, Z.: A novel weak form three-dimensional quadrature element solution for vibrations of elastic solids with different boundary conditions. Finite Elem. Anal. Des. 141, 70–83 (2018)

    Article  MathSciNet  Google Scholar 

  34. Shen, Z., Zhong, H.: Static and vibrational analysis of partially composite beams using the weak-form quadrature element method. Math. Probl. Eng. 2012, 1–23 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Zhong, H., Yue, Z.: Analysis of thin plates by the weak form quadrature element method. Sci. China Ser. G. 55(5), 861–871 (2012)

    Article  Google Scholar 

  36. Liao, M., Zhong, H.: Application of a weak form quadrature element method to nonlinear free vibrations of thin rectangular plates. Int. J. Struct. Stab. Dyn. 16(1), 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  37. Zhang, R., Zhong, H.: Weak form quadrature element analysis of geometrically exact shells. Int. J. Nonlinear Mech. 71, 63–71 (2015)

    Article  Google Scholar 

  38. Yuan, S., Zhong, H.: Consolidation analysis of non-homogeneous soil by the weak form quadrature element method. Comput. Geotech. 62, 1–10 (2014)

    Article  Google Scholar 

  39. Yuan, S., Zhong, H.: A weak form quadrature element formulation for coupled analysis of unsaturated soils. Comput. Geotech. 76, 1–11 (2016)

    Article  Google Scholar 

  40. Liu, B., Ferreira, A.J.M., Xing, Y.F., Neves, A.M.A.: Analysis of functionally graded sandwich and laminated shells using a layerwise theory and a differential quadrature finite element method. Compos. Struct. 136, 546–553 (2016)

    Article  Google Scholar 

  41. Liu, B., Ferreira, A.J.M., Xing, Y.F., Neves, A.M.A.: Analysis of composite plates using a layerwise theory and a differential quadrature finite element method. Compos. Struct. 156, 393–398 (2016)

    Article  Google Scholar 

  42. Wang, X., Yuan, Z., Jin, C.: Weak form quadrature element method and its applications in science and engineering: A state-of-the-art review. ASME Appl. Mech. Rev. 69(3), 030801 (2017)

    Article  Google Scholar 

  43. Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)

    Article  MathSciNet  Google Scholar 

  44. Bellman, R.E., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34(2), 235–238 (1971)

    Article  MathSciNet  Google Scholar 

  45. Jin, C., Wang, X., Ge, L.: Novel weak form quadrature element method with expanded Chebyshev nodes. Appl. Math. Lett. 34, 51–59 (2014)

    Article  MathSciNet  Google Scholar 

  46. Bathe, K.J.: Finite Element Procedures. Prentice-Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

  47. Huang, C.W., Su, Y.H.: Dynamic characteristics of partial composite beams. Int. J. Struct. Stab. Dyn. 8(4), 665–685 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Shanghai University, Shangda Road 99, Shanghai, 200444, China

    Chao Fu & Xiao Yang

Authors
  1. Chao Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chao Fu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, C., Yang, X. Dynamic analysis of partial-interaction Kant composite beams by weak-form quadrature element method. Arch Appl Mech 88, 2179–2198 (2018). https://doi.org/10.1007/s00419-018-1443-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1443-1

Keywords

Search

Navigation