analysis and interpretation of the structural … · 2014. 7. 28. · t.trombetti 1, s. silvestri...

SAHC2014 – 9th International Conference on Structural Analysis of Historical Constructions

F. Peña & M. Chávez (eds.) Mexico City, Mexico, 14–17 October 2014

ANALYSIS AND INTERPRETATION OF THE STRUCTURAL BEHAVIOR OF THE ROSE WINDOW OF THE CATHEDRAL OF

MODENA (ITALY)

T.Trombetti1, S. Silvestri1, G. Gasparini1, M. Palermo1 and S. Baraccani1

1 University of Bologna Viale Risorgimento 2, 40136 Bologna, Italy

e-mail: [email protected]

[email protected]

[email protected]

[email protected]

[email protected]

Keywords: Modena Cathedral, Rose window, in-plan analysis, out-of-plan analysis.

Abstract. The rose window is a peculiar character of the main facade of Romanesque cathe-drals. A beautiful example of rose window can be found in the Cathedral of Modena, Italy. The faced of the Cathedral, built at the end of the XI century, and declared as UNESCO her-itage sites in the 1997, was originally built without the great rose window which was added during the XIII century, together with some lateral doors, to increase the brightness inside the building. It is made of various stones typologies which are arranged in order to suggest the shape of a rose. Due to the quite slender shapes of the stones forming the spokes, the rose could be subjected to high stress and susceptible to instability. In this paper a preliminary assessment of the structural safety of the rose window is carried out. The structural behavior is studied considering both the in-plane and out-of plane actions. The in-plane analyses are developed through limit schematizations. The out-of-plane analyses are carried out through finite element models which are compared with simple hand calcula-tions. The analyses results indicate that the rose window is in a safety state.

T. Trombetti, S. Silvestri, G. Gasparini, M. Palermo and S. Baraccani

2

1 INTRODUCTION

The Cathedral of Modena, designed by architects Lanfranco and Wiligelmus, is one of the most important example of Italian Romanesque cathedral. In 1997 the Cathedral was declared as “UNESCO World Heritage” site and therefore preservation became a fundament issue. The central part of the façade is occupied by a great rose window, an architectural elements of high artistic value, Fig. 1. The rose window was not present from the beginning of the history of the Cathedral and it was inserted between 1230-1244 in substitution of pre-existing win-dows to increase the brightness inside the building [1]. Its insertion required the construction of a discharging arch of 6 meters diameter. At the mid of XIX century, due to a number of factors such as material degradations and environmental pollutants, the stones of the rose window were in a warning state of degradations. In 1893, to prevent the failure of some de-cayed stones of the lower part of the external circle, the first restoration works of the rose window were undertaken by replacing the decayed stone blocks with new ones carved to imi-tate the original ones [2]. After the first works, further restoration interventions were neces-sary in 1983 to fill some deep cracks by injections of a mixture of acrylic (Paraloid) and silicone resins; moreover to protect the stones from rainwater penetration, a waterproof treat-ment based on acrylic resins was laid on [3,4]. Again, in 2007 the lower part of the decorated external circle exhibited strong degradation, consisting in diffused cracks and erosion. The assessment of possible causes leading to the observed strong degradation of the carved round frame of the rose window was the objective of the work by Sandrolini, et al. 2011 [2]. In their work, the authors identified the freezing-thawing cycles as the main cause of the cracks and material loss. Nonetheless, they also performed some structural analyses in order to quantify the stress levels in the stones. In this paper, the structural behavior of the rose window is analyzed in order to assess its structural safety. Both in-plane and out-of-plane analyses have been carried out.

Figure 1: The façade of the Cathedral of Modena and the rose window.

Interpretation and analysis on the structural behavior of the rose window of the Cathedral of Modena (Italy)

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2 GEOMETRICAL AND MATERIALS CHARACTERIZATION OF THE ROSE WINDOW

The rose window, with an external diameter of approximately 8 meters, is located in the central part of the facade of the Cathedral at approximately 16 m high from the ground. It has a complex geometry, composed of stone elements of different shapes and materials. It is com-posed of an external circle frame made of carved stones. A tracery of spokes radiates from the central roundel, with inside a cross. The spokes has decorated capitals, which are connected to the external circle frame by a system of arches. Figure 2 highlights in blue colours the differ-ent components (or structural elements):

- outer circle (Fig.2b); - petals: the system of arches (Fig.2c); - spokes or columns (Fig.2d); - central disk (Fig.2e); - steel linchpin (Fig.2f);

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: (a) the rose window, (b) outer circle, (c) petals, (d) spokes, (e) central disk, (f) steel linchpin.

The different types of stones are shown in the color map of Figure 3, while their main physical and mechanical properties (density γ, compressive strength fc, Young's modulus E,) are listed in Table 1 [5].


4

Figure 3: The identification of the main stones types used for the rose window

Table 1: The main physical and mechanical properties of the main stones

Stone γ [Kg/m3] fc [Mpa] E [Mpa] Granitello 2500 150 60000 Ammonitico 2700 130 40000 Sandstone(Pantano) 2700 156 30000 Scabiazza 2700 156 30000

3 IN-PLAN ANALYSIS OF THE ROSE WINDOW

In-plan analysis, considering the vertical loads transmitted by the upper part of the wall fa-cade and the self-weight, has been carried out based on limit schematizations dealing with the loads (i.e. how the loadings flow from the outer circle to the internal core). In both cases it is assumed that only a triangular portion of the wall facade weight is applied to the rose window (see the schemes of figures 4).

3.1 Vertical loads acting on the rose window

The total weight of the triangular portion of masonry (with a height equal to one half of its length and assuming a masonry self-weight, mγ of 1200 kg/m2) is equal to:

2

, 192004

mtr tot

DQ kg

γ ⋅= = (1)

In addition to the loads transmitted by the upper masonry the element self-weight has been considered (Table 2).


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Table 2: Vertical loads applied on the rose window

Element: Weight (P) Roofing 700 kg/m2 Turret 4600 kg/elem Wall 1200 kg/m2 Self-weight of structural elements of rose window: outer circle 56406 kg Petals 61 kg/elem Spokes 35 Kg/elem central disk 3000 kg

3.2 Assumptions and limit schematizations

Masonry has been assumed as a homogeneous, isotropic material. Two limit schematiza-tions have been assumed to account for the uncertainties related to the effective mechanisms of load transfer (from the external circle to the internal disk) and for the effective degree of connections between the different elements:

1. load-bearing arch mechanism: the outer circle behaves like an arc able to carry the vertical loads (Fig.4a); the other elements carries only their self-weight.

2. load-bearing spokes mechanism: outer circle is not able to carry the self-weight and vertical loads. The vertical loads are carried by the system composed of arches, col-umns and internal disk (Fig.4b).

(a) (b)

Figure 4: (a) Load path: load- bearing arch; (b) Load path: load- bearing spokes

3.3 Main results

The stress states of the various elements of the rose window (outer circle, petals, spokes cen-tral disk, steel linchpin) have been calculated for both limiting schematizations. Results are summarized in Table 3 (limit scheme 1) and Table 4 (limit scheme 2). The central disk has been instead analyzed considering the Theory of Elasticity [6]. In the limit scheme 1, the disk has been considered as a full disk submitted to uniform pressure p (Fig.5a) The total load P acting on the upper half of the central disk is estimated equal to 3800kg. The uniform radial stress whose resultant is equal to the total load P is equal to


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0.37kg/cm2 (by assuming a thickness of 20 cm). Instead, in the limit scheme 2 the disk has been considered subjected to two equal and opposite forces P acting along the vertical diame-ter (Fig.5b). The stress at any point is obtained by superposing a uniform tension in the plane of the disk of the magnitude 2P/πd on the above two simple radial stress distributions (from [3]):

34 cos 2

yP P

r d

θσπ π

= − + (2)

The maximum compressive stress along the horizontal diameter is at the center of the disk:

, max6

yP

dσ

π= (3)

Considering the concentrated load P composed of the external load and the weight of disk, the maximum radial stresses acting on the central disk are equal to:

, max

2,

21000

612,2

est disk

yest dik disk

P Q Q Kg

P Kg

d s cmσ

π

== +

= ≅⋅ ⋅

(4)

Where de,disk is the disk diameter, sdisk is the disk thickness.

(a) (b)

Figure 5: (a) full disk submitted to uniform pressure; (b) Load path: load- bearing spokes; (c)Trend of stresses

Table 3: Stress levels for the different elements, limit case 1.

Element: normal stress

[Kg/cm2] bending stress

[Kg/cm2] Shear stress [Kg/cm2]

σc [Kg/cm2]

outer circle

1.3

1560

petals 160 1300 spokes 110 5 0.23 1500 steel linchpin 2000 83 3000


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Table 4: Stress levels for the different elements, limit case 2.

Element: Compressive

strength [Kg/cm2]

Flexural strength [Kg/cm2]

Shear strength [Kg/cm2]

σc [Kg/cm2]

outer circle

1.22 1560

petals ,max

,max

161

32c

t

σσ

==

1300

spokes 111 4.4 0.23 1500 steel linchpin 2000 83 3000

The stress levels in the various elements are well below material strengths.

4 OUT-OF-PLAN ANALYSIS OF THE ROSE WINDOW

Out-of-plan analysis of the rose window, considering the horizontal loads (in particular wind load), has been performed on Finite Element (FE) models and compared with hand cal-culations.

4.1 Wind load

The horizontal equivalent static loads due to the effect of wind have been evaluated ac-cording to the Italian NTC08 [7]. In particular, the following expression of the equivalent stat-ic pressure due to the effect of wind is used:

2

100b e p dkg

p q c c cm

= ⋅ ⋅ ⋅ = (5)

Where qb is the reference pressure, ce is the exposure coefficient, cp is the shape coefficient, cd is the dynamic coefficient (assumed equal to 1.0).

4.2 The FE models

The rose window has been modeled through different FE models, differing for the element types (beam or bricks) and internal connections between the different structural elements. Figure 6 shows the two main FE models which have been developed:

• Model A: all elements are modelled as beam elements.

• Model B: all elements are modelled as brick elements (four elements along the thickness of the external circle).


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(a)

(b)

(c)

Figure 6: (a) Simplified geometric model (b) Model A: Beam Model (c) Brick Model

The connections between the different structural elements have been schematized through two limit cases (1 and 2) and an intermediate case (3):

1. the connections between the petals (arches) and the spokes, as well as the connec-tions between the spokes and the central disk, are modelled by an internal hinge (material continuity in the internal connection).

2. the connections between the petals (arches) and the spokes, as well as the connec-tions between the spokes and the central disk, are modelled by assuming material continuity.

3. the connections between the petals (arches) and the spokes, as well as the connec-tions between the spokes and the central disk, are modelled with rotational springs.

4.3 Main results

A total of six models have been developed and for each elements maximum stresses are reported in tables. The maximum stresses as obtained from Model A (Beam Model) are close to those showed by Model B (Brick Model) thus indicating that classical beam theory is accurate enough to model the geometry of the rose window. The results obtained by the FEM models have been also compared with simple hand calculations useful for qualitative evaluations. Highest stresses have been found in the most slender elements such as spokes and petals. Nevertheless, all stresses values are far from material capacity. As an illustra-tive example, table 5 collects the values of the maximum stresses for each element typolo-gy as obtained from Model A for each of the three schematizations of the connections used in the analyses.


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Table 5: Maximum stresses according to Model A.

Element: Limit case 1

stresses[kg/cm2] Limit case 2

stresses[kg/cm2] Intermediate case stresses[kg/cm2]

fc [Kg/cm2]

outer circle ,max

c,max

0.39

0.39tσ

σ== −

max

min

0.54

0.54

σσ

== −

max

min

0.47

0.47

σσ

== −

1560

petals max

min

4.8

4.8

σσ

== −

max

min

9.4

9.4

σσ

== −

max

min

7.3

7.3

σσ

== −

1300

spokes max

min

38.5

38.5

σσ

== −

max

min

38.1

38.1

σσ

== −

max

min

7.7

7.7

σσ

== −

1500

central disk max

min

9.9

0

σσ

==

max

min

0.24

0

σσ

==

max

min

4.3

0

σσ

==

1500

In addition to the evaluation of the maximum stresses, the strength capacity of the steel linchpins has been evaluated according to the formulation suggested by the Italian building code NTC08:

,2

0.6 346tk

v RdM

f AF kg

γ⋅= = (6)

With the material safety factor 2Mγ = 1.25, steel ultimate strength tkf =36·106 kg/m2 and area of steel linchpins A= 2·10-5 m2. All the connections are able to well sustain the applied loads provided that the maximum forces acting are equal to 68kg.

5 CONCLUSION

In this paper an assessment of the structural safety of the rose window of the Cathedral of Modena has been carried out by mean of in-plan (vertical loads) and out-of-plan (wind load) analyses. Due to the uncertainties related to the effective load path and the quality of connections between the different elements, several models have been developed, from simple limit schematizations to more complex FE models. Results indicate that stress lev-els are well below material strengths.

REFERENCES

[1] R. Salvini, Il Duomo di Modena e il romanico emiliano. Cassa di risparmio di Modena, Modena 1966.

[2] F. Sandrolini, E. Franzoni, I. Babuska, E. Sassoni, P.P. Diotallevi, The contribution of urban-scale environmental monitoring to materials diagnostics: A study on the Cathe-dral of Modena (Italy),Journal of Cultural Heritage ,December 441-450, 2011.

[3] C. Acidini Luchinat, L. Serchia, S. Piconi, I restauri del Duomo di Modena, Panini, Modena,1984.


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[4] G. Polidori, Il Duomo di Modena “Capolavoro del genio creatore umano: restauro del paramento lapideo”, P. Monari, A. Sardi (ed), Restauri in Emilia Romagna: attività de-gli istituti MiBAC nel 2008. Atti del Convegno organizzato dalla direzione Regionale per i Beni Culturali e Paesaggistici dell’Emilia Romagna nell’ambito del XVI Salone del Restauro della Conservazione dei Beni Culturali e Ambientali, Bologna, 2009.

[5] S. Lugli, G. Rossetti, Il volto della cattedrale: un mosaico di pietra il paramento lapi-deo nella facciata del duomo di Modena, tesi di laurea, Modena 2007

[6] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, Third Edition, McGraw-Hill Book Company, Singapore, 1970

[7] Norme Tecniche per le Costruzioni NTC08, Italian Ministerial Decree of January. Publi-shed in the Gazzetta Ufficiale n. 29 of 04 Feb 2008, Suppl. Ordinario n.30, Italia, 2008.

analysis and interpretation of the structural … · 2014. 7. 28. · t.trombetti 1, s. silvestri...

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