DOI QR코드

DOI QR Code

An efficient seismic analysis of regular skeletal structures via graph product rules and canonical forms

  • Kaveh, A. ;
  • Zakian, P.
  • Received : 2015.06.02
  • Accepted : 2015.10.28
  • Published : 2016.01.25

Abstract

In this study, graph product rules are applied to the dynamic analysis of regular skeletal structures. Graph product rules have recently been utilized in structural mechanics as a powerful tool for eigensolution of symmetric and regular skeletal structures. A structure is called regular if its model is a graph product. In the first part of this paper, the formulation of time history dynamic analysis of regular structures under seismic excitation is derived using graph product rules. This formulation can generally be utilized for efficient linear elastic dynamic analysis using vibration modes. The second part comprises of random vibration analysis of regular skeletal structures via canonical forms and closed-form eigensolution of matrices containing special patterns for symmetric structures. In this part, the formulations are developed for dynamic analysis of structures subjected to random seismic excitation in frequency domain. In all the proposed methods, eigensolution of the problems is achieved with less computational effort due to incorporating graph product rules and canonical forms for symmetric and cyclically symmetric structures.

Keywords

seismic analysis;regular skeletal structures;graph product rules;canonical forms

References

  1. Brewer, J. (1978), "Kronecker products and matrix calculus in system theory", IEEE Trans. Circuits Syst., 25(9), 772-781. https://doi.org/10.1109/TCS.1978.1084534
  2. Chopra, A.K. (2012), Dynamics of Structures, 4th Edition, Pearson Education, Upper Saddle River.
  3. Clough, R.W. and Penzien, J. (2003), Dynamics of Structures, 3rd Edition, Computers and Structures, Berekely.
  4. El-Raheb, M. (2011), "Modal properties of a cyclic symmetric hexagon lattice", Comput. Struct., 89(23), 2249-2260. https://doi.org/10.1016/j.compstruc.2011.08.011
  5. He, Y., Yang, H., Xu, M. and Deeks, A.J. (2013), "A scaled boundary finite element method for cyclically symmetric two-dimensional elastic analysis", Comput. Struct., 120, 1-8. https://doi.org/10.1016/j.compstruc.2013.01.006
  6. Kangwai, R.D., Guest, S.D. and Pellegrino, S. (1999), "An introduction to the analysis of symmetric structures", Comput. Struct., 71(6), 671-688. https://doi.org/10.1016/S0045-7949(98)00234-X
  7. Kaveh, A. (2013), Optimal Analysis of Structures by Concepts of Symmetry and Regularity, Springer, Wien-New York.
  8. Kaveh, A. and Nemati, F. (2010), "Eigensolution of rotationally repetitive space structures using a canonical form", Int. J. Numer. Meth. Eng., 26(12), 1781-1796.
  9. Kaveh, A. and Rahami, H. (2010), "An efficient analysis of repetitive structures generated by graph products", Int. J. Numer. Meth. Eng., 84(1), 108-126. https://doi.org/10.1002/nme.2893
  10. Koohestani, K. and Kaveh, A. (2010), "Efficient buckling and free vibration analysis of cyclically repeated space truss structures", Finite Elem. Anal. Des., 46(10), 943-948. https://doi.org/10.1016/j.finel.2010.06.009
  11. Koohestani, K. (2011), "An orthogonal self-stress matrix for efficient analysis of cyclically symmetric space truss structures via force method", Int. J. Solid. Struct., 48(2), 227-233. https://doi.org/10.1016/j.ijsolstr.2010.09.023
  12. Kouachi, S. (2006), "Eigenvalues and eigenvectors of tridiagonal matrices", Electron. J. Linear Algeb., 15, 115-133.
  13. McDaniel, T.J. and Chang, K.J. (1980), "Dynamics of rotationally periodic large space structures", J. Sound Vib., 68(3), 351-368. https://doi.org/10.1016/0022-460X(80)90392-2
  14. Newland, D.E. (2012), An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd Edition, Dover Publications, Mineola, New York.
  15. Olson, B.J., Shaw, S.W., Shi, C., Pierre, C. and Parker, R.G. (2014), "Circulant matrices and their application to vibration analysis", Appl. Mech. Rev., 66(4), 040803-1-040803-41. https://doi.org/10.1115/1.4027722
  16. Rahami, H., Kaveh, A. and Shojaei, I. (2015), "Swift analysis for size and geometry optimization of structures", Adv. Struct. Eng., 18(3), 365-380. https://doi.org/10.1260/1369-4332.18.3.365
  17. Shi, C. and Parker, R.G. (2014), "Vibration modes and natural frequency veering in three-dimensional, cyclically symmetric centrifugal pendulum vibration absorber systems", J. Vib. Acoust., 136(1), 011014.
  18. Shi, C. and Parker, R.G. (2015), "Vibration mode structure and simplified modelling of cyclically symmetric or rotationally periodic systems", Proc. Roy. Soc. A, doi: 10.1098/rspa.2014.0672. https://doi.org/10.1098/rspa.2014.0672
  19. Thomas, D.L. (1979), "Dynamics of rotationally periodic structures", Int. J. Numer. Meth. Eng., 14(1), 81-102. https://doi.org/10.1002/nme.1620140107
  20. Williams, F.W. (1986), "Exact eigenvalue calculations for structures with rotationally periodic substructures", Int. J. Numer. Meth. Eng., 23(4), 695-706. https://doi.org/10.1002/nme.1620230411
  21. Yueh, W.-C. (2005), "Eigenvalues of several tridiagonal matrices", Appl. Math. E-Notes, 5, 210-230.
  22. Zingoni, A. (2012), "Symmetry recognition in group-theoretic computational schemes for complex structural systems", Comput. Struct., 94, 34-44.
  23. Zingoni, A. (2014), "Group-theoretic insights on the vibration of symmetric structures in engineering", Phil. Trans. R. Soc. A., 372, 20120037.