1) The document describes numerical modeling of the static and seismic behavior of the Church of San Francesco in Lucca using the NOSA-ITACA finite element code. 2) Reinforcement operations including steel bars were performed on the masonry and roof of the church. 3) Numerical models of the existing and reinforced church were created to evaluate the effectiveness of the reinforcement by comparing modal analyses, stress fields, and collapse loads between the two models.

1. Numerical modelling of the static and seismic behaviour of
historical buildings: the church of San Francesco in Lucca
1
The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
Maria Girardi
Cristina Padovani
Giuseppe Pasquinelli
Laboratory of Mechanics
of Materials and Structures
ISTI-CNR
Pisa

2. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
The NOSA-ITACA version for static analyses will be freely
downloadable by http://www.nosaitaca.it/it/software/
•The NOSA-ITACA 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
• Dynamic analyses
• Thermo-mechanical analyses
• Modal analyses
• Stress fields
• Collapse loads
• Elastic, fracture and crushing strain fields
• Displacement fields
• Time- histories
• Eigenvalues and eigenvectors
• NOSA-ITACA library: beam, truss, shell, 2D, 3D elements (35 elements)

3. 33
the infinitesimal strain tensor,
the Cauchy stress tensor,
the elastic part of the strain,
the fracture strain,
the crushing strain,
the modulus of elasticity and the Poisson’s ratio,
the masonry maximum compressive stress.
T
c
E
E
f
E
e
E
E,
e
s0
s
0 0 
DE
ˆ ˆT = T(E), T(E)
Given E, find Ef, Ec, T such that

e f c
f c
E = E + E + E ,
E E = 0,
E
1+ 1-2
e e
T E tr( E )I ,

 
 
   
 f c
0T E T I E 0,    
0-T I 0, , c
T E 0,
f
E 0
The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013

4. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
The Church of San Francesco

5. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
The Church of San Francesco: reinforcement operations
On masonry
On the roof
NPL 150X18 NPL 150X18
NPL 150X18 NPL 150X18 NPL 150X18
diagonali 20 diagonali 20
NPL 150X18 NPL 150X18
NPL 150X18 NPL 150X18
diagonali 20 diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20
NPL 150X18
diagonali 20diagonali 20
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
150X150x8
On the cloister

6. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
Evaluation of the seismic vulnerability and
the effectiveness of the reinforcement operations
Step 1
•In situ tests: geometric survey, flat jacks tests, georadar scans
•Laboratory tests: mechanical characterization of materials and soil
Step 2
Numerical model of the existing structure: modal analyses, definition of the
horizontal loads, static nonlinear analyses, definition of the collapse loads
Step 3
Numerical model of the reinforced structure: modal analyses, definition of
the horizontal loads, static nonlinear analyses, definition of the collapse loads
Step 4
Existing and reinforced structure: comparison of numerical results

7. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
g1
g2
The Church of San Francesco: The finite element mesh
/2
11 11
/2
d ,
h
h
N T 

 
/2
22 22
/2
d
h
h
N T 

 
/2
11 11
/2
d ,
h
h
M T  

 
/2
22 22
/2
d .
h
h
M T  

 
11 1 1,T  g Tg 22 2 2 ,T  g Tg
{ ' | , , ( )},' ζ     p p p n p n n p
[ / 2, / 2].h h  
Stresses
Normal forces
Bending moments

8. 0.5 1 1.5 2 2.5 3
T sec
1
2
3
4
a m sec 2
T1r=1,04 sT1=1,26 s
Without reinforcing With reinforcing
The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
The Church of San Francesco: modal analyses
Elastic Response Spectrum
for the Lucca site
z
x

9. The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
The Church of San Francesco: permanent loads
Stresses T22Displacements ux

10. Without reinforcing
With reinforcing
Stresses T22
SLU
The Church of San Francesco: evaluation of the seismic vulnerability
HU=6,5% Weigth
HU=4,1% Weigth

11. The Church of San Francesco: evolution of the maximum values of normal stresses
-2,5
-2
-1,5
-1
-0,5
0
-6,E+06 -5,E+06 -4,E+06 -3,E+06 -2,E+06 -1,E+06 0,E+00
T22[Mpa]
Total horizontal force H [N]
without reinforcement with reinforcement

12. Displacements ux
SLU
Without reinforcing
With reinforcing
The Church of San Francesco: evaluation of the seismic vulnerability
HU=6,5% Weigth
Uxmax /h= 0,0078
Uxmax /h= 0,005
HU=4,1% Weigth

13. Eccentricities e22
SLU
Without reinforcing
With reinforcing
The Church of San Francesco: evaluation of the seismic vulnerability
e22 = M22/N22

14. Stresses T11
SLU
Without reinforcing
With reinforcing

15. Fracture strains
Without reinforcing
With reinforcing
11
f
E

16. Without reinforcing
Stresses T22
With reinforcing

17. Normal force in the windbracing at the ultimate limit state
The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013

18. The Church of San Francesco: evaluation of the seismic vulnerability
0
2
4
6
8
10
12
14
16
18
20
-0,50 -0,40 -0,30 -0,20 -0,10 0,00
load
increments
horizontal displacements [m]
without reinforcing
with reinforcing

19. 0.5 1 1.5 2 2.5 3
T s
0.5
1
1.5
2
Sd
m
s2
TR = 475 (reference)
TR = 140 (reinforced)
TR = 50 (not reinforced)
50
0.1
475
SI  
475
0.75
0.4
1.86
SLU
a
a
f
a
  
140
0.3
475
SI  
475
1.16
0.6
1.86
SLU
a
a
f
a
  
0.041SLUH
W

The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
0.065SLUH
W

Without reinforcing With reinforcing

20. 20
The Fourteenth International Conference on Civil, Structural and Environmental
Engineering Computing, Cagliari 3-6 September 2013
Conclusions
•The NOSA - ITACA code is a finite element code for static and dynamic
nonlinear analyses of masonry structures. The version for static analyses
will be freely downloadable by the end of the year.
•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
church of San Francesco in Lucca is assessed by means of a nonlinear static
analysis conducted via the NOSA–ITACA code.
The financial support of the Region of Tuscany (project “Tools for modelling
and assessing the structural behaviour of ancient constructions: the NOSA-
ITACA code”, PAR FAS 2007-2013) is gratefully acknowledged.