Skip to main content
Log in

A nonlinear SDOF model for the simplified evaluation of the displacement demand of low-rise URM buildings

  • Original Research Paper
  • Published:
Bulletin of Earthquake Engineering Aims and scope Submit manuscript


The determination of displacement demands for masonry buildings subjected to seismic action is a key issue in the performance-based assessment and design of such structures. A technique for the definition of single-degree-of-freedom (SDOF) nonlinear systems that approximates the global behaviour of multi-degree-of-freedom (MDOF) 3D structural models has been developed in order to provide useful information on the dependency of displacement demand on different seismic intensity measures. The definition of SDOF system properties is based on the dynamic equivalence of the elastic properties (vibration period and viscous damping) and on the comparability with nonlinear hysteretic behaviour obtained by cyclic pushover analysis on MDOF models. The MDOF systems are based on a nonlinear macroelement model that is able to reproduce the in-plane shear and flexural cyclic behaviour of pier and spandrel elements. For the complete MDOF models an equivalent frame modelling technique was used. The equivalent SDOF system was modelled using a suitable nonlinear spring comprised of two macroelements in parallel. This allows for a simple calibration of the hysteretic response of the SDOF by suitably proportioning the contributions of flexure-dominated and shear-dominated responses. The comparison of results in terms of maximum displacements obtained for the SDOF and MDOF systems demonstrates the feasibility and reliability of the proposed approach. The comparisons between MDOF and equivalent SDOF systems, carried out for several building prototypes, were based on the results of time-history analyses performed with a large database of natural records covering a wide range of magnitude, distance and local soil conditions. The use of unscaled natural accelerograms allowed the displacement demand to be expressed as a function of different ground motion parameters allowing for the study of their relative influence on the displacement demand for masonry structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30

Similar content being viewed by others


  • Azarbakht A, Dolšek M (2011) Progressive incremental dynamic analysis for first-mode dominated structures. J Struct Eng 137(3):445–455

    Article  Google Scholar 

  • Chopra AK, Goel RK (2001) Direct displacement-based design: use of inelastic vs. elastic design spectra. Earthq Spectra 17(1):47–64

  • Galasco A, Lagomarsino S, Penna A (2006) On the use of pushover analysis for existing masonry buildings. In: Proceedings of the 1st European conference on earthquake engineering and seismology. Geneva, Switzerland

  • Graziotti F (2013) Contributions towards a displacement-bases seismic assessment of masonry structures, Ph.D. thesis: IUSS, Pavia, Italy

  • Hall JF (2006) Problems encountered from the use (or misuse) of Rayleigh damping. Earthq Eng Struct Dyn 35:525–545

    Article  Google Scholar 

  • Housner GW (1959) Behavior of structures during earthquakes. J Eng Div (ASCE) 85(EM14):109–129

    Google Scholar 

  • Lagomarsino S, Penna A, Galasco A, Cattari S (2013) TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng Struct 56:1787–1799

    Article  Google Scholar 

  • Luco N, Bazzurro P, Cornell A (2004) Dynamic versus static computation of the residual capacity of a mainshock-damaged building to withstand an aftershock. In: Proceedings of the 13th WCEE. Vancouver, Canada

  • Magenes G, Calvi G, Kingsley G (1995) Seismic testing of a full-scale, two-story masonry building: test procedure and measured experimental response. Report of University of Pavia, Italy

  • Miranda E (2001) Estimation of inelastic deformation demands of SDOF systems. J Struct Eng 127:1005–1012

    Article  Google Scholar 

  • MIT (2008) Italian building code. DM 14.01. 2008: Norme Tecniche per le Costruzioni. Italian Ministry of Infrastructures and Transportation, Rome

    Google Scholar 

  • MIT (2009) Italian Building Code, Commentary. Circ. 617 02.02. 2009: Istruzione per l’applicazione delle nuove Norme Tecniche per le Costruzioni. Italian Ministry of Infrastructures and Transportation, Rome

    Google Scholar 

  • Moehle JP (1992) Displacement-based design of RC structures subjected to earthquakes. Earthq Spectra 8(3):403–428

    Article  Google Scholar 

  • Mouyiannou A, Penna A, Rota M, Graziotti F, Magenes G (2014) Implications of cumulated seismic damage on the seismic performance of unreinforced masonry buildings. Bull NZ Soc Earthq Eng 47(2):157–170

    Google Scholar 

  • Penna A, Lagomarsino S, Galasco A (2014) A nonlinear macroelement model for the seismic analysis of masonry buildings. Earthq Eng Struct Dyn 43(2):159–179

    Article  Google Scholar 

  • Priestley MJ (1996) Displacement-based seismic assessment of existing reinforced concrete buildings. Bull NZ Soc Earthq Eng 29(4):256–272

    Google Scholar 

  • Shome N, Cornell CA, Bazzurro P, Carballo J (1998) Earthquakes, records, and nonlinear responses. Earthq Spectra 14(3):469–500

  • Smerzini C, Galasso C, Iervolino I, Paolucci R (2014) Ground motion record selection based on broadband spectral compatibility. Earthq Spectra 30(4):1427–1448

    Article  Google Scholar 

  • Vamvatsikos D, Cornell CA (2005) Direct estimation of seismic demand and capacity of multidegree-of-freedom systems through incremental dynamic analysis of single degree of freedom approximation. J Struct Eng 131(4):589–599

    Article  Google Scholar 

Download references


This work was carried out with the partial financial support of the EUCENTRE Executive Project 2012–2014 e3 “Seismic vulnerability of masonry buildings”, funded by the Italian Department of Civil Protection, and the bilateral cooperation project between Italy and Slovenia, funded by Italian Ministry of Foreign Affairs “Protection of cultural heritage buildings against earthquakes” (2011). Special thanks go to Prof. M. Dolšek, Dr. A. Galasco, Dr. M. J. Fox, Dr. A. Mouyiannou and Dr. I. Senaldi for the theoretical and practical support.

Author information

Authors and Affiliations


Corresponding author

Correspondence to F. Graziotti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Graziotti, F., Penna, A. & Magenes, G. A nonlinear SDOF model for the simplified evaluation of the displacement demand of low-rise URM buildings. Bull Earthquake Eng 14, 1589–1612 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: