Bulletin of Earthquake Engineering

, Volume 13, Issue 5, pp 1347–1376 | Cite as

Parametric model for capacity curves

  • Luis G. PujadesEmail author
  • Yeudy F. Vargas-Alzate
  • Alex H. Barbat
  • José R. González-Drigo
Original Research Paper


A parametric model for capacity curves and capacity spectra is proposed. The capacity curve is considered to be composed of a linear part and a nonlinear part. The normalized nonlinear part is modelled by means of a cumulative lognormal function. Instead, the cumulative Beta function can be used. Moreover, this new conceptualization of the capacity curves allows defining stiffness and energy functions relative to the total energy loss and stiffness degradation at the ultimate capacity point. Based on these functions, a new damage index is proposed and it is shown that this index, obtained from nonlinear static analysis, is compatible with the Park and Ang index obtained from dynamic analysis. This capacity based damage index allows setting up a fragility model. Specific reinforced concrete buildings are used to illustrate the adequacy of the capacity, damage and fragility models. The usefulness of the models here proposed is highlighted showing how the parametric model is representative for a family of capacity curves having the same normalized nonlinear part and how important variables can be tabulated as empirical functions of the two main parameters defining the capacity model. The availability of this new mathematical model may be a powerful tool for current earthquake engineering research, especially in seismic risk assessments at regional scale and in probabilistic approaches where massive computations are needed.


Capacity curves Parametric model Stiffness degradation Energy loss Fragility curves Damage assessment 



The thoughtful comments and suggestions of an anonymous reviewer helped to significantly improve the manuscript; his thorough review is highly appreciated and acknowledged. This work has been partially funded by the Spanish Government, by the European Commission and with FEDER funds, through the research projects: CGL2008-00869/BTE, CGL2011-23621, SEDUREC-CONSOLIDER-CSD2006-00060, INTERREG:POCTEFA 2007-2013/73/08, MOVE-FT7-ENV-2007-1-211590 and DESURBS-FP7-2011-261652. Y.F. Vargas-Alzate, has been the holder of a scholarship and a contract funded by the Geological Institute of Catalonia and the Polytechnic University of Catalonia by means of a bilateral agreement. Several quotes related to simplicity concept have been extracted from Wikipedia.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Luis G. Pujades
    • 1
    Email author
  • Yeudy F. Vargas-Alzate
    • 2
  • Alex H. Barbat
    • 1
  • José R. González-Drigo
    • 3
  1. 1.Polytechnic University of Catalonia, BarcelonaTechBarcelonaSpain
  2. 2.Universidad Nacional de Colombia, Sede ManizalesManizalesColombia
  3. 3.Polytechnic University of Catalonia, BarcelonaTechBarcelonaSpain

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