This document discusses the numerical modeling of masonry structures using the NOSA finite element code. It describes modeling masonry as a nonlinear isotropic material with no tensile strength. The document provides examples of static and dynamic analyses conducted on historic masonry towers and churches in Italy. Results shown include stress fields, displacement, and strain from dynamic analysis of the Rognosa tower in San Gimignano subjected to earthquake loading.

1. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
Numerical modelling of the dynamic behaviour of
masonry constructions
Laboratorio di Meccanica
dei Materiali e delle Strutture
Cristina Padovani
Giuseppe Pasquinelli
Maria Girardi
Istituto di Scienza e Tecnologie
dell’Informazione “A. Faedo”
ISTI-CNR Pisa
1

2. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
•The NOSA code is a finite element solver for nonlinear analyses.
•Masonry is modelled by a nonlinear isotropic elastic material with zero
tensile strength and limited compressive strength (masonry-like or no-
tension material). [G. Del Piero, Meccanica 1989; S. Di Pasquale, Meccanica 1992; M.
Lucchesi, C. Padovani et al., Masonry Constructions and Numerical Applications, Springer 2008].
• Static analyses • Stress fields
• Collapse loads
• Dynamic analyses
• Elastic, fracture and crushing strain fields
• Thermo-mechanical analyses
• Displacement fields
• Temperature fields
• Time- histories
• NOSA library: beam, shell, 2D, 3D elements (17 elements)
The NOSA version for static analyses is freely downloadable by
www.isti.cnr.it/research/unit.php?unit=MMS&section=software 22
Numerical modelling of the dynamic behaviour of masonry constructions

3. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The masonry-like constitutive equation
E the infinitesimal strain tensor,
s
T the Cauchy stress tensor,
Ee the elastic part of the strain,
Ef the fracture strain, e
Ec the crushing strain,
E,  the modulus of elasticity and the Poisson’s ratio,
 0  0 the masonry maximum compressive stress.
s0
Given E, find Ef, Ec, T such that
E = Ee+ E f + Ec ,
E f  E c = 0,
E  e  
T   E  tr( E e )I  , ˆ ˆ
T  T(E), DET(E)
1+  1-2 
T  E f  T   0 I   E c  0,
T, E c  0, T- 0 I  0, E  0
f
33
Numerical modelling of the dynamic behaviour of masonry constructions

4. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
Some example applications
• 1995 Battistero del Duomo, Volterra
• 1996 Arsenale Mediceo, Pisa
Static Analyses
• 1998 Teatro Goldoni, Livorno
• 1998 Chiesa Madre di S. Nicolò, Noto
• 2004 Chiesa di Santa Maria Maddalena, Morano Calabro
• 2005 Chiesa di San Ponziano, Lucca
Dynamic Analyses
• 2008 Chiesa Abbaziale di Santa Maria della Roccella, Roccella Ionica
• 2008 Torre “Rognosa” , San Gimignano
• 2010 Torre “delle Ore”, Lucca
Numerical modelling of the dynamic behaviour of masonry constructions

5. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano (St@rt project)
SEZ. A-A
+45,34 (Top)
+43,34
t
SEZ. A-A
y t
A
x
t Side overlooking
the Cathedral Square
z
+19,04 (Surrounding roofs level)
y t
A
x
Side overlooking
+8,60 (Masonry vaults level)
the Cathedral Square
z A A
+0,00 (Square level)
x
Numerical modelling of the dynamic behaviour of masonry constructions

6. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: digital acquisition of the geometry
(VC Lab – ISTI CNR)
Merging resolution=1 cm

7. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano:
- Static analysis The Tower is subjected to its own weight and to the
weight of the surrounding buildings
- Dynamic analysis The Tower subjected to the Nocera Umbra
earthquake in x - direction
Numerical modelling of the dynamic behaviour of masonry constructions

8. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
+43,34 (Top)
The “Rognosa” tower in San Gimignano: dynamic analysis
Tower SEZ. A-A
base section
t
+19,04 (Surrounding roofs level)
y t
A
x
Side overlooking
+8,60 (Masonry vaults level)
the Cathedral Square
z A A
+0,00 (Square level)
x
Numerical modelling of the dynamic behaviour of masonry constructions

9. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
+45,34 (Top)
+43,34
+19,04 (Surrounding roofs level)
+8,60 (Masonry vaults level)
z A A
+0,00 (Square level)
x
Numerical modelling of the dynamic behaviour of masonry constructions

10. CST 2010, The Tenth International Conference on Computational Structures Technology
14-17 September 2010, Valentia
Tower vertical section The bell chamber
+45,34 (Top)
+43,34
+19,04 (Surrounding roofs level)
+8,60 (Masonry vaults level)
z A A
+0,00 (Square level)
x
The “Rognosa” Tower in San Gimignano: digital acquisition and structural analysis

11. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
Compressive stresses Tzz
At time t=3,41 s: Minimum values reached
during the analysis :
Numerical modelling of the dynamic behaviour of masonry constructions

12. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
Crushing strain Eczz
At time t=3,41 s: Minimum values reached
during the analysis :
Numerical modelling of the dynamic behaviour of masonry constructions

13. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
Tangential fracture strain Eftt
At time t=3,41 s: Maximum values reached
during the analysis :
Numerical modelling of the dynamic behaviour of masonry constructions

14. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
Fracture strain Efzz
At time t=3,41 s: Maximum values reached
during the analysis :
Numerical modelling of the dynamic behaviour of masonry constructions

15. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The “Rognosa” tower in San Gimignano: dynamic analysis
Displacements ux
Maximum values reached during the analysis:
Numerical modelling of the dynamic behaviour of masonry constructions

16. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The NOSA-ITACA project
2011-2013
funded by the Region of Tuscany (PAR-FAS 2007-2013)
ISTI-CNR DCR
Pisa Firenze
Consulting
Basic Research NOSA -ITACA Case study
Service
CODE
Numerical modelling of the dynamic behaviour of masonry constructions

17. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
The NOSA-ITACA project
NOSA CODE: f.e.m. nonlinear solver
SALOME: pre-post processor
Case study: “Voltone”, Livorno
NOSA-ITACA code
CONSULTING SERVICE
Municipalities
Monuments and Fine Arts Offices 17
Professional offices

18. WONDERmasonry 2011, Facoltà di Ingegneria, Firenze
Conclusions
•The NOSA code is a finite element code for static and dynamic nonlinear
analyses of masonry structures. The version for static analyses is freely
downloadable.
•Masonry is modelled by means of a masonry-like constitutive equation with
zero tensile strength and finite or infinite compressive strength.
•A case study has been presented in which the seismic vulnerability of the
Rognosa Tower in San Gimignano is assessed by means of a dynamic
numerical analysis conducted via NOSA code.
• The NOSA-ITACA project aims to upgrade the NOSA code and disseminate
the use of numerical tools in the field of maintenance and restoration of the
architectural heritage.
Numerical modelling of the dynamic behaviour of masonry constructions