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

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Abstract

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).

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Abbreviations

Ao:

Effective acceleration for the site

bx :

Distance between the axial and shear springs in a vertical interface

by :

Distance between the axial and shear springs in an horizontal interface

C:

Random variable representing the limit state of the structure

dY :

Yielding displacement

dU :

Ultimate displacement

Ebm:

Young’s modulus of brick masonry

Erm:

Young’s modulus of reinforced brick masonry

Esm:

Young’s modulus of stone masonry

E0 :

Elastic Young’s modulus of axial stress in the interface

E*:

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

G0 :

Elastic shear modulus in the interface

G*:

Non elastic shear modulus for loading and unloading in the interface

ik :

Stiffness degradation index

ik2 :

Stiffness degradation index based on change of frequency of modes

kA :

Spring stiffness values for compression loading

kA* :

Spring stiffness values compression unloading

k Qh :

Stiffness of shear springs in horizontal direction

k Qv :

Stiffness of shear springs in vertical direction

k Ax :

Stiffness of axial spring in X direction

k Ay :

Stiffness of axial spring in Y direction

mpi :

Modal mass participation of the ith mode

p:

Value for generation of Chilean spectrum

Pc:

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

Q:

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

Ti:

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

wf :

Weighting factor based on natural frequency error

wi :

Weighting factor for each mode

wϕ :

Weighting factor based on modal shape error

x:

PGA for which the cumulative probability is calculated

Z:

Factor of seismic zonification

αJJ :

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

β:

PGA logarithmic standard deviation for compliance with the limit state C

εAc :

Strain at peak compression strength

εAr :

Strain at the start of the residual stage in tension

εAt :

Strain at peak tension strength

εQ* :

Maximum strain reached by the shear spring

εQc :

Strain at peak shear strength

φ[]:

Normal cumulative distribution

ΔVb :

Change in the base shear for each cycle

Δδ:

Displacement of control point for each cycle

μ:

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

σAc :

Peak compression strength

σAr :

Strength at the start of the residual stage in tension

σAt :

Peak tension strength

ωio :

Frequency of the ith mode before the earthquake

ωif :

Frequency of the ith mode after the earthquake

DM:

Damage measure

EDP:

Engineering demand parameter

MAC:

Modal assurance criterion

RBSM:

Rigid body spring model

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Acknowledgements

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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Authors and Affiliations

  1. Faculty of Engineering, Pontificia Universidad Católica del Ecuador, Apartado Postal 17 01 21 84, Quito, Ecuador

    Wilson Torres

  2. Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

    José Luis Almazán & Cristián Sandoval

  3. School of Architecture, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

    Cristián Sandoval

  4. Instituto de Ingeniería, Universidad Nacional Autónoma de México, Edificio 2–401, Circuito Escolar, Ciudad Universitaria, 04510, Mexico City, Mexico

    Fernando Peña

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  1. Wilson Torres
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Correspondence to Wilson Torres.

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Torres, W., Almazán, J.L., Sandoval, C. et al. Fragility analysis of the nave macro-element of the Cathedral of Santiago, Chile. Bull Earthquake Eng 16, 3031–3056 (2018). https://doi.org/10.1007/s10518-017-0292-6

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