Inelastic Base Shear Reconstruction from Sparse Acceleration Measurements of Buildings

  • Boya Yin
  • Henri GavinEmail author
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


This paper presents a novel method for recovering base shear forces of building structures with unknown nonlinearities from sparse seismic-response measurements of floor accelerations. The method requires only direct matrix calculations (factorizations and multiplications); no iterative trial-and-error methods are required. The method requires a mass matrix, or at least an estimate of the floor masses. A stiffness matrix may be used, but is not necessary. Essentially, the method operates on a matrix of incomplete measurements of floor accelerations. In the special case of complete floor measurements of systems with linear dynamics and real modes, the principal components of this matrix are the modal responses. In the more general case of partial measurements and nonlinear dynamics, the method extracts a number of linearly-dependent components from Hankel matrices of measured horizontal response accelerations, assembles these components row-wise and extracts principal components from the singular value decomposition of this large matrix of linearly-dependent components. These principal components are then interpolated between floors in a way that minimizes the curvature energy of the interpolation. This interpolation step can make use of a reduced-order stiffness matrix, a backward difference matrix or a central difference matrix. The measured and interpolated floor acceleration components at all floors are then assembled and multiplied by a mass matrix. A sum (or weighted sum) of the resulting vector of inertial forces gives the base shear. The proposed algorithm is suitable for linear and nonlinear hysteretic structural systems.


Base shear recovery Nonlinear hysteretic behavior Seismic isolation Sparse acceleration measurement Singular spectrum analysis 



This material is based in part upon work supported by the National Science Foundation under Grant Number CMMI-1258466. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


  1. 1.
    Abe, M., Fujino, Y., Yoshida, J.: Dynamic behaviour and seismic performance of base-isolated bridges in observed seismic records. In: Proceedings of 12th World Conference on Earthquake Engineering (2000)Google Scholar
  2. 2.
    Ahmadi, G., Fan, F., Noori, M.: A thermodynamically consistent model for hysteretic materials. Iran. J. Sci. Technol. 21(3), 257–278 (1997)Google Scholar
  3. 3.
    Alhan, C., Gavin, H.: A parametric study of linear and non-linear passively damped seismic isolation systems for buildings. Eng. Struct. 26(4), 485–497 (2004)CrossRefGoogle Scholar
  4. 4.
    Cadzow, J.A.: Signal enhancement-a composite property mapping algorithm. IEEE Trans. Acoust. Speech Signal Process. 36(1), 49–62 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Celebi, M.: Successful performance of a base-isolated hospital building during the 17 January 1994 Northridge earthquake. Struct. Des. Tall Build. 5, 95–109 (1996)CrossRefGoogle Scholar
  6. 6.
    Chaudhary, M., Abe, M., Fujino, Y.: Identification of soil–structure interaction effect in base-isolated bridges from earthquake records. Soil Dyn. Earthq. Eng. 21(8), 713–725 (2001)CrossRefGoogle Scholar
  7. 7.
    Chaudhary, M.T.A., Abe, M., Fujino, Y., Yoshida, J.: System identification of two base-isolated bridges using seismic records. J. Struct. Eng. 126(10), 1187–1195 (2000)CrossRefGoogle Scholar
  8. 8.
    Ding, Y., Law, S., Wu, B., Xu, G., Lin, Q., Jiang, H., Miao, Q.: Average acceleration discrete algorithm for force identification in state space. Eng. Struct. 56, 1880–1892 (2013)CrossRefGoogle Scholar
  9. 9.
    Erlicher, S., Point, N.: Thermodynamic admissibility of Bouc-Wen type hysteresis models. Comptes Rendus Méc. 332(1), 51–57 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Furukawa, T., Ito, M., Izawa, K., Noori, M.N.: System identification of base-isolated building using seismic response data. J. Eng. Mech. 131(3), 268–275 (2005)CrossRefGoogle Scholar
  11. 11.
    Gavin, H.P., Scruggs, J.T.: Constrained optimization using lagrange multipliers. CEE 201L. Duke University (2012)Google Scholar
  12. 12.
    Golyandina, N., Nekrutkin, V., Zhigljavsky, A.A.: Analysis of Time Series Structure: SSA and Related Techniques. CRC, New York (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Huang, M.-C., Wang, Y.-P., Lin, T.-K., Chen, Y.-H.: Development of physical-parameter identification procedure for in-situ buildings with sliding-type isolation system. J. Sound Vib. 332(13), 3315–3328 (2013)CrossRefGoogle Scholar
  14. 14.
    Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.-C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 454, pp. 903–995. The Royal Society (1998)Google Scholar
  15. 15.
    Hyvärinen, A.: Complexity pursuit: separating interesting components from time series. Neural Comput. 13(4), 883–898 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis bouc-wen model, a survey. Arch. Comput. Meth. Eng. 16(2), 161–188 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Juang, J.-N., Pappa, R.S.: An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control. Dyn. 8(5), 620–627 (1985)CrossRefzbMATHGoogle Scholar
  18. 18.
    Juang, J.-N., Phan, M., Horta, L.G., Longman, R.W.: Identification of observer/kalman filter Markov parameters-theory and experiments. J. Guid. Control. Dyn. 16(2), 320–329 (1993)Google Scholar
  19. 19.
    Kampas, G., Makris, N.: Time and frequency domain identification of seismically isolated structures: advantages and limitations. Earthq. Struct. 3(3–4), 249–270 (2012)CrossRefGoogle Scholar
  20. 20.
    Kijewski, T., Kareem, A.: Wavelet transforms for system identification in civil engineering. Comput.-Aided Civ. Infrastruct. Eng. 18(5), 339–355 (2003)CrossRefGoogle Scholar
  21. 21.
    Ljung, L.: Prediction error estimation methods. Circuits Syst. Signal Process. 21(1), 11–21 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Loh, C.-H., Weng, J.-H., Chen, C.-H., Lu, K.-C.: System identification of mid-story isolation building using both ambient and earthquake response data. Struct. Control. Health Monit. 20(2), 139–155 (2013)CrossRefGoogle Scholar
  23. 23.
    Mahmoudvand, R., Zokaei, M.: On the singular values of the Hankel matrix with application in singular spectrum analysis. Chilean J. Stat. 3(1), 43–56 (2012)MathSciNetGoogle Scholar
  24. 24.
    Markovsky, I.: Structured low-rank approximation and its applications. Automatica 44(4), 891–909 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Markovsky, I.: Low Rank Approximation: Algorithms, Implementation, Applications. Springer Science & Business Media, New York (2011)zbMATHGoogle Scholar
  26. 26.
    Nagarajaiah, S., Xiaohong, S.: Response of base-isolated USC hospital building in Northridge earthquake. J. Struct. Eng. 126(10), 1177–1186 (2000)CrossRefGoogle Scholar
  27. 27.
    Oliveto, N.D., Scalia, G., Oliveto, G.: Time domain identification of hybrid base isolation systems using free vibration tests. Earthq. Eng. Struct. Dyn. 39(9), 1015–1038 (2010)Google Scholar
  28. 28.
    Peeters, B., De Roeck, G.: Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Signal Process. 13(6), 855–878 (1999)Google Scholar
  29. 29.
    Siringoringo, D.M., Fujino, Y.: Seismic response analyses of an asymmetric base-isolated building during the 2011 great east Japan (Tohoku) earthquake. Struct. Control. Health Monit. 22(1), 71–90 (2015)CrossRefGoogle Scholar
  30. 30.
    Staszewski, W.: Identification of damping in MDOF systems using time-scale decomposition. J. Sound Vib. 203(2), 283–305 (1997)CrossRefGoogle Scholar
  31. 31.
    Stewart, J.P., Conte, J.P., Aiken, I.D.: Observed behavior of seismically isolated buildings. J. Struct. Eng. 125(9), 955–964 (1999)CrossRefGoogle Scholar
  32. 32.
    Takewaki, I., Nakamura, M.: Stiffness-damping simultaneous identification under limited observation. J. Eng. Mech. 131(10), 1027–1035 (2005)CrossRefGoogle Scholar
  33. 33.
    Takewaki, I., Nakamura, M.: Temporal variation in modal properties of a base-isolated building during an earthquake. J. Zhejiang Univ. Sci. A 11(1), 1–8 (2010)CrossRefzbMATHGoogle Scholar
  34. 34.
    Trefethen, L.N., Bau III, D.: Numerical Linear Algebra, vol. 50. SIAM (1997)Google Scholar
  35. 35.
    Xu, C., Chase, J.G., Rodgers, G.W.: Physical parameter identification of nonlinear base-isolated buildings using seismic response data. Comput. Struct. 145, 47–57 (2014)CrossRefGoogle Scholar
  36. 36.
    Yang, J.N., Lei, Y., Pan, S., et al.: System identification of linear structures based on Hilbert-Huang spectral analysis. part 1: normal modes. Earthq. Eng. Struct. Dyn. 32(9), 1443–1468 (2003)Google Scholar
  37. 37.
    Yang, Y., Nagarajaiah, S.: Output-only modal identification with limited sensors using sparse component analysis. J. Sound Vib. 332(19), 4741–4765 (2013)CrossRefGoogle Scholar
  38. 38.
    Yoshimoto, R.: Damage detection of base-isolated buildings using multi-inputs multi-outputs subspace identification. Ph.D Thesis, Department of System Design Engineering, Keio University (1776)Google Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2016

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringDuke UniversityDurhamUSA

Personalised recommendations