Bulletin of Earthquake Engineering

, Volume 16, Issue 7, pp 3031–3056 | Cite as

Fragility analysis of the nave macro-element of the Cathedral of Santiago, Chile

  • Wilson Torres
  • José Luis Almazán
  • Cristián Sandoval
  • Fernando Peña
Original Research Paper


This paper presents the fragility analysis of a typical nave macro-element of the Metropolitan Cathedral of Santiago, Chile. The analysis is carried out by using the rigid body spring model approach, in which rigid elements are connected to each other by means of axial and shear springs. The 2D model generated is initially verified by comparing modes with a 3D finite element model previously calibrated in DIANA software. The methodology used in this study is based on a set of eleven real seismic records corresponding to four major earthquakes that have affected Santiago city. Nonlinear incremental dynamic analyses together with a damage index based on stiffness degradation, which considers the relation between shear at the base and deformation of the macro-element, are used to generate the fragility curves. As a result of this study, the probability of exceedance for different damage states has been obtained based on a possible peak ground acceleration of the site. In particular, the results of the study demonstrate that the proposed damage index satisfactorily describes the damage suffered by some of the nave transverse sections of the Cathedral after the 2010 Maule earthquake (PGA 2.11 m/s2—Santiago Centro station).


Heritage buildings Fragility analysis Nonlinear analysis Masonry building Rigid body spring model (RBSM) 

List of symbols


Effective acceleration for the site


Distance between the axial and shear springs in a vertical interface


Distance between the axial and shear springs in an horizontal interface


Random variable representing the limit state of the structure


Yielding displacement


Ultimate displacement


Young’s modulus of brick masonry


Young’s modulus of reinforced brick masonry


Young’s modulus of stone masonry


Elastic Young’s modulus of axial stress in the interface


Non elastic Young’s modulus for loading and unloading of axial stress in the interface


Elastic shear modulus in the interface


Non elastic shear modulus for loading and unloading in the interface


Stiffness degradation index


Stiffness degradation index based on change of frequency of modes


Spring stiffness values for compression loading


Spring stiffness values compression unloading


Stiffness of shear springs in horizontal direction


Stiffness of shear springs in vertical direction


Stiffness of axial spring in X direction


Stiffness of axial spring in Y direction


Modal mass participation of the ith mode


Value for generation of Chilean spectrum


Probability of event C that has full compliance given a PGA value of x


Variable related to the level of seismic intensity expressed in terms of PGA


Periods for generation of Chilean spectrum. Where, i could be a, b, c, or d


Weighting factor based on natural frequency error


Weighting factor for each mode


Weighting factor based on modal shape error


PGA for which the cumulative probability is calculated


Factor of seismic zonification


Factors for generation of Chilean spectrum. Where, J could be A, V, or D


PGA logarithmic standard deviation for compliance with the limit state C


Strain at peak compression strength


Strain at the start of the residual stage in tension


Strain at peak tension strength


Maximum strain reached by the shear spring


Strain at peak shear strength


Normal cumulative distribution


Change in the base shear for each cycle


Displacement of control point for each cycle


PGA for which the structure reaches 50% of the cumulative probability


Peak compression strength


Strength at the start of the residual stage in tension


Peak tension strength


Frequency of the ith mode before the earthquake


Frequency of the ith mode after the earthquake


Damage measure


Engineering demand parameter


Modal assurance criterion


Rigid body spring model



The first author acknowledges the support of the Secretary of Higher Education, Science, Technology and Innovation of Ecuador (SENESCYT), through contract number 20120011. Additionally, the first author also wants to thank the financial support given by the Vicerrectoría de Investigación (VRI) of the Pontificia Universidad Católica de Chile for his research stage at the Engineering Institute, UNAM, of Mexico.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  • Wilson Torres
    • 1
  • José Luis Almazán
    • 2
  • Cristián Sandoval
    • 2
    • 3
  • Fernando Peña
    • 4
  1. 1.Faculty of EngineeringPontificia Universidad Católica del EcuadorQuitoEcuador
  2. 2.Department of Structural and Geotechnical EngineeringPontificia Universidad Católica de ChileSantiagoChile
  3. 3.School of ArchitecturePontificia Universidad Católica de ChileSantiagoChile
  4. 4.Instituto de IngenieríaUniversidad Nacional Autónoma de MéxicoMexico CityMexico

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