Bulletin of Earthquake Engineering

, Volume 17, Issue 2, pp 985–1007 | Cite as

A probabilistic simplified seismic model of masonry buildings based on ambient vibrations

  • D. SpinaEmail author
  • G. Acunzo
  • N. Fiorini
  • F. Mori
  • M. Dolce
Original Research


The paper presents a new simplified mathematical model for predicting the structural seismic response of buildings. The model, denominated Seismic Model from Ambient Vibrations (SMAV), is based on the experimental modal parameters identified from ambient vibration and only a few information about the geometry and the structural typology of the building. After a short review of the Multi Rigid Polygons model, recently illustrated and validated in an other paper by some of the authors, that allows to estimate the seismic participation factors of the experimental modes also for buildings characterized by complex shaped plan and structural irregularity along the height, the attention is focused on a new stochastic approach for modeling the seismic response of masonry buildings. In particular this new approach aims to take into account the non linearity occurred during a seismic event so that the nonlinear behaviour of the building is considered by reducing its modal frequencies according to the response amplitude. The reduction of the natural frequencies, extracted by Operational Modal Analysis from ambient vibrations, is computed according to specific probabilistic curves: the Frequency Shift Curves (FSCs). This curves provide the percentage reduction of the natural frequencies as a function of the maximum roof drift reached during the strong motion and they are obtained, for some specific masonry typologies, through a Monte Carlo analysis carried out using a simple mechanical model of a masonry panel with geometric and mechanical parameters that vary according to their probabilistic distributions. The seismic response of the building is then computed through a linear equivalent analysis in which an iterative algorithm updates the resonant frequencies according to the specific FSC curve. The concept of structural serviceability index (IOPS), expressing the probability of the building to remain operational, is also introduced. Finally, a comparison between the seismic response computed by the model and the experimental seismic response of the Pizzoli town hall, a masonry existing building endowed with a permanent accelerometer monitoring system, is illustrated.


Seismic vulnerability Masonry building Frequency shift Operational modal analysis 

List of symbols


Vector of natural frequencies


Matrix of mass scaled mode shapes


Vector of modal damping ratios


Subscrit denoting that the quantity refers to the k-th mode


Mass matrix of the Multi Rigid Polygon (MRP) model


Matrix of the linear transformation from phisical to MRP degree of freedom

\(\bar{\varPsi }\)

Matrix of the mode shapes of the MRP model


Vector of modal participation factors



\({\mathbf{\ddot{u}}}_{g} \left( t \right)\)

Acceleration time history on the ground

\(\varvec{u}\left( t \right)\)

Vector of the dinamic displacements of the structure


Structural period

\(S_{\varvec{a}} \left( T \right)\)

Pseudo-acceleration response spectrum of the seismic input at \(T\)

\(S_{\varvec{d}} \left( T \right)\)

Displacement response spectrum at \(T\)


Roof drift


Yeld roof drift


Maximum roof drift


Maximum Interstory Drift Ratio


Vector of geometrical and mechanical parameters which characterize the wall panel

\(\rho_{f} (\delta ;\, \varvec{q} )\)

Frequency Shift Curve (FSC)

\(F(\delta )\)

Force-drift relationship associated to this assumed mechanical model


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Civil ProtectionRomeItaly
  2. 2.Department of Mathematics and Physics of Roma Tre (Previously, CNR-IGAG)RomeItaly
  3. 3.Department of Civil Protection External consultant (Previously, CNR-IGAG)RomeItaly
  4. 4.National Research Centre CNR-IGAGMontelibretti (Rome)Italy

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