Seismic Assessment of a Medieval Masonry Tower in Northern Italy by Limit, Nonlinear Static, and...

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Seismic Assessment of a MedievalMasonry Tower in Northern Italy byLimit, Nonlinear Static, and Full DynamicAnalysesGabriele Milani a , Siro Casolo a , Andrea Naliato b & Antonio Tralli ba Dipartimento di Ingegneria Strutturale (DIS), Politecnico di Milano,Milan, Italyb Dipartimento di Ingegneria, Università di Ferrara, Ferrara, Italy

Available online: 26 Oct 2011

To cite this article: Gabriele Milani, Siro Casolo, Andrea Naliato & Antonio Tralli (2012): SeismicAssessment of a Medieval Masonry Tower in Northern Italy by Limit, Nonlinear Static, and Full DynamicAnalyses, International Journal of Architectural Heritage: Conservation, Analysis, and Restoration,6:5, 489-524

To link to this article: http://dx.doi.org/10.1080/15583058.2011.588987

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SEISMIC ASSESSMENT OF A MEDIEVAL MASONRYTOWER IN NORTHERN ITALY BY LIMIT, NONLINEARSTATIC, AND FULL DYNAMIC ANALYSES

Gabriele Milani,1 Siro Casolo,1 Andrea Naliato,2

and Antonio Tralli21Dipartimento di Ingegneria Strutturale (DIS), Politecnico di Milano, Milan, Italy2Dipartimento di Ingegneria, Università di Ferrara, Ferrara, Italy

A comparative numerical study on a 12th-century masonry tower located in northern Italyis described. To assess the safety of the tower under seismic loads, different numerical analy-ses have been performed: nonlinear static, limit, and nonlinear full dynamic analyses. In thefirst two cases, a full three-dimensional (3D) detailed finite element model (FEM) is adopted,changing the seismic load direction and assuming different hypotheses for the intercon-nection between the core and the external walls. When dealing with the FEM incrementalanalysis, a commercial code is utilized assuming for masonry a smeared crack isotropicmodel. For limit analysis, a noncommercial full 3D code developed by the authors is uti-lized. It provides good estimates of limit loads and failure mechanisms to compare withstandard FEM results. The dynamical analyses have been performed by a specific two-dimensional (2D) rigid body and spring model (RBSM), accounting for the asymmetriesalong the thickness and the irregularities of thickness of both the external and internal wallsin an approximate but realistic way. Four different accelerograms are utilized—passing fromlow to high seismicity zones—to evaluate the performance of the tower under dynamic loads.From numerical results, the role played by the actual geometry of the tower is envisaged, aswell as a detailed comparison of failure mechanisms provided by the incremental FEM andlimit analysis is provided. In all cases, the numerical analysis has given a valuable pictureof possible damage mechanisms providing useful hints for the introduction of structuralmonitoring.

KEY WORDS: Masonry tower, pushover, limit analysis, dynamic analysis, three-dimensional (3D) finite elements, two-dimensional (2D) rigid body and spring model (RBSM)

1. INTRODUCTION

A detailed comparative numerical study on the seismic vulnerability of a medievalmasonry tower is presented. The example under consideration represent a good benchmarkto show that the most complete set of numerical analysis that today can be performed forthe safety assessment of monumental buildings comprises pushover, limit and dynamicnonlinear analyses.

Received April 12, 2011; accepted May 10, 2011.Address correspondence to Gabriele Milani, Dipartimento di Ingegneria Strutturale (DIS), Politecnico di

Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Phone: +39 02 2399 4290, Fax: +39 022399 4220.E-mail: milani@stru.polimi.it

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490 G. MILANI ET AL.

The Italian norms for constructions (Nuove normative tecniche per le costruzioni[NTC], 2008; Ordinanza del Presidente del Consiglio dei Ministri [OPCM], 2003) haverecently classified as seismic zone a relevant part of the Italian territory previouslyunclassified, for example, most of the Emilia-Romagna region, where the structure islocated. As a consequence, the seismic vulnerability analysis of both strategic buildingsand monuments has to be evaluated for the first time.

The tower (Figure 1 and Figure 2) rises in Bondeno, a small but strategically impor-tant town—in the Medieval Age—located near Ferrara, the ancient capital of the Estense

Figure 1. The “Matildea” bell tower analyzed. Pictures of the overall tower, details and drawings describing theinclination (color figure available online).

Figure 2. Sections, planar views and plants at increasing levels of the Matildea bell tower.

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 6(5): 489–524

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MEDIEVAL MASONRY TOWER IN ITALY 491

Dukedom, near Panaro river’s banks, not far from its confluence into the Po river (thelongest river in Italy).

A moderate inclination on the vertical is present, due to foundation differential dis-placements, namely 1.6◦ west–east and 3◦ north–south (Figure 1). The inclination resultsin an out-of-center placement of the top section centroid of approximately 160 cm and85 cm along the X and Y direction, respectively. It is expected that this inclination slightlyreduces the resistance of the tower for horizontal loads.

Approximately 10 years ago (in 2000), the municipality arranged a restoration of themonument; thus, the present state of conservation appears to be good. The tower has a slen-derness—defined as the ratio between the overall height of the structure and the smallestdimension of the base cross section— approximately equal to 4. Therefore it is expectedthat the tower suffers little of the typical stability problems connected to second ordereffects of the vertical loads, due to possible tilting. Moreover, an expedited vulnerabilityanalysis, performed according to Italian Guide Lines (Ministero per i Beni e le AttivitàCulturali, 2006), shows a near sufficient vulnerability index for the seismic zone underconsideration. Therefore, more sophisticated analyses are needed to assess the stability ofthe structure under seismic excitation. For this reason, the tower was chosen as a paradig-matic case study to perform a wide range of the most sophisticated numerical analysesproposed in the technical literature. Due to some asymmetries in the structure, 3D-FEManalyses have been performed where possible.

In addition to a standard modal analysis, which may be conducted by any practitionerwith common software, also performed have been both incremental nonlinear simulationsconducted with the commercial code DIANA (TNO DIANA DV, FEMGV, Delft, TheNetherlands, 2008), assuming an elasto-plastic damaging constitutive model for masonryand a full 3D-limit analysis (Milani et al. 2006, 2008; Milani 2009).

As a matter of fact, the first approach represents a refined pushover analysis, requiredby the new Italian norms (NTC, 2008). The response of several submodels is investigatedin detail. Specifically, evaluated are the structural performance along two orthogonal direc-tions, in presence and absence of the actual inclination, changing material properties, andconsidering or neglecting the interlocking of the internal stairs.

It is shown how both approaches provide quite similar results in terms of maxi-mum loads and collapse mechanisms, which essentially interest the lower part of thetower, where an accurate survey has shown a structural discontinuity. The response ofboth numerical models is similar also considering the moderate inclination of the tower.Finally, 2D-nonlinear dynamic analyses, based on a rigid body and spring masonry model(RBSM), have been performed (Casolo 2004; 2006; 2009; Casolo and Uva, 2011). As amatter of fact, nonlinear dynamic analysis methods for masonry structures require furtherresearch efforts before they can be confidently used in standard design, but certainly repre-sent the most reliable method to validate the pushover analysis for such type of structuresif experimental results are not available.

Dynamic results confirm that earthquakes of intensity slightly greater than thatexpected in the seismic zones under consideration would result in severe damages anda probable collapse of the structure. However, damages and failure mechanisms obtainedwith the dynamic analyses appear, as expected, visibly different to those obtained using astatic procedure. While pushover simulations envisage a collapse of the tower due to theformation of a combined flexural and shear hinge near the base of the structure, nonlineardynamic analyses exhibit severe and diffused damage of the tower along the height of thestructure. In presence of ground accelerations representative of Italian seismic zone 2, the

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tower collapses for the formation of a shear-flexural hinge at the first floor level, whereasin seismic zone 1 the premature and complete destruction of the bell cell occurs, due to theimportance of the higher modes of vibration.

2. LITERATURE REVIEW AND TOWER DESCRIPTION

At present, a number of studies are available in the technical literature dealing withthe numerical/experimental analysis of masonry towers. A robust tradition is available inItaly and in Europe in general, with a variety of analyses ranging from, for example:

� utilization of nonlinear FE codes (Abruzzese et al. 2009; Carpinteri et al. 2006; Rivaet al. 1998; Bernardeschi et al. 2004; Pena et al. 2010;);

� implementation of very specific fiber-element models for the 3D nonlinear dynamics ofslender towers (Casolo 1998; 2001);

� combined eigenvalues and experimental identification studies (Ivorra and Pallares 2006;Russo et al. 2010);

� 2D limit analyses performed by means of a no-tension material approach (Heyman1992);

� experimental and in-situ tests (Anzani et al. 2010; Binda et al. 2005);� repairing and rehabilitation proposals (Lourenço 2005; Modena et al. 2002)

Researchers have identified that one of the most important parameters that determines thevulnerability index of such kind of structures is the slenderness. As known, the Asinellitower (Bologna), the Mangia’s Tower in Siena, San Marco in Venice, the Torrazzo inCremona (the highest masonry tower in the world) and the Qtub Minar in India are fiveof the most investigated examples of slender towers, being the Asinelli tower the mostslender (slenderness equal to 11).

To perform a full literature review of all the contributions regarding the experimental-numerical analysis of masonry towers is almost impossible. However, in a partial review ofthe literature, it should be mentioned that the great part of big historical structures has beenmodeled with a macro-modeling strategy, defined as the heterogeneous masonry substi-tuted at a structural level through a fictitious material with average mechanical properties(either orthotropic or isotropic) representing globally the response of brickwork underincreasing loads. In some cases, sophisticated material models have been used, rangingfrom elasto-plastic with softening and damaging models, which are the only one suitable tohave an insight into the pushover behavior of a masonry structure making use of generallypurpose commercial software.

For example, Buti’s bell tower (Italy) studied by Bernardeschi et al. (2004) wasanalyzed through two different analyses, first the bell tower subjected solely to its ownweight, and, second, subjected to self-weight and horizontal loads mimicking seismicaction. An 8th-century masonry tower (Sineo in Alba, Italy) was numerically analyzed andmonitored by Carpinteri et al. (2006), because of dangerous emerging damage patterns.Several dynamic structural characterizations are also available in the specialized litera-ture, for example, the study conducted by Ivorra and Pallares (2006) on the bell tower of‘Nuestra Señora de la Misericordia Church (Valencia, Spain). Subsequent to the geometri-cal analysis of the bell tower structure, different numerical models were calibrated basedon dynamic tests to determine the bending and torsion frequencies of the tower.

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A presentation is given of the fundamental design choices and of the selection ofthe most appropriate materials and techniques that have been made for strengthening theMonza cathedral bell tower by Modena et al. (2002). The investigation carried out onsite and in the laboratory on the materials and structure of Monza bell tower allows thedetection of the details of damage, which is evolving toward the failure.

Seismic analysis of the Asinelli Tower in Bologna (Italy) has been studied by Rivaet al. (1998). In this study, an assessment of the tower’s stability with respect to compatibleseismic events with the region seismicity is carried out by means of a nonlinear dynamicanalysis on a simplified model, essentially constituted by beam elements with nonlinearstress-strain behavior.

The application of georadar to the detection of three main structural problems for thebell tower (Torrazzo) of Cremona is studied by Binda et al. (2005). Their study demon-strates the need and the potential of a multidisciplinary collaboration for the solution ofmorphological and diagnostic problems by use of non-destructive investigations. Also,a case study is fully detailed, including the aspects of historical, damage and geometricinvestigations, of advanced numerical analysis, of justification of remedial measures andof detailing the adopted strengthening of Outeiro Church in Portugal by Lourenço (2005).In addition, many useful studies have investigated old historical structures (e.g., Bayraktaret al. 2010; Abruzzese et al. 2010).

Compared with such critical case studies, the Matildea tower under consideration,with its height of 30 m and a base edge of around 7.20 m, has a slenderness equal to 4, threetimes smaller with respect to the aforementioned examples. Therefore the tower suffersless of the typical stability problems associated with the second order effects of the verticalloads, due to possible tilting of the tower. The tower (Figure 1) may be considered as anisolated construction, with very little connection at the base with the contiguous church, inagreement with the medieval building practice in Italy where the tradition of isolating thechurch survived until the Renaissance age. Sections and cross-sections at different heightsare sketched in Figure 2. The connection is now secured by a small modern corridor builtwith thick masonry.

The internal structure is typical for this region and may be found in several othertowers around that region (e.g., San Mercuriale in Forlì, San Marco in Venice), and on intotwo coaxial vertical box structures. Stairs are built between the internal and external wallsby means of small masonry barrel vaults, contemporarily conceived with a sustaining andceiling role.

The internal space is subdivided along the height into four superimposed floors,which can be accessed by means of the stair system. Two floors (the first and the sec-ond) are constituted by masonry barrel vaults, whereas the remaining floors (the third andfourth) are two light timber structures. The internal floors probably had the original roleto host the guardhouse. The construction of the tower started around the year 1100 andwas concluded in the early 1300s. The original function of the tower was twofold: defenseand sighting. Then, two additional stories were added, to raise the height of 30 m and thestructure was converted to bell tower.

The first and second level (until a level of around 14 m) are from the 1100s, whenthe structure was used as sighting tower. The base is approximately square, with angularreinforcing pillars. In correspondence of the second floor, the tower was built in multileafwith smaller bricks and a different technology, suggesting that raising was needed duringthese years. The thickness of the walls diminishes from 1.50 m to 0.90 m.

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The bell cell was built later and architectonically is influenced by the last gothicperiod. A large gothic arch on all faces is present there. Under the arch, a mullioned win-dow with three lights is placed, sustained by slender masonry pillars with cubic shapecapitals. Small circular holes between the arch and the window are also present, in order toboth lighten the structure and allowing for a better acoustic diffusion of the bell sound. Thelast level indeed was conceived and added with the precise idea to convert the structure,immediately near the new church.

A 12-m spire on the top was initially planned but never built, as demonstrated by thepresence of three small superimposed arches on the internal corners at the top of the tower,immediately under the roof. Instead, a very light wooden roof was realized to cover thebell cell. From a structural point of view, the tower is conceived as two coaxial cantileverbeams, with approximately square section and interconnected one each other by the stairs,masonry vaults and timber beams. The internal walls are sustained at the base throughsquat circular arches, an issue that increases considerably the seismic vulnerability of thestructure. The barrel vaults are approximately 20 cm thick, whereas the external walls aremulti-leaf walls realized with small “Bolognese” clay bricks of approximate dimensions28 x 5 x 10 (length x height x thickness). Timber floors and masonry barrel vaults help fora 3D behavior of the structure, being disposed orthogonally one each other, to strengthenthe structures against possible horizontal loads.

3. BELL TOWER FE DISCRETIZATION AND MATERIAL PROPERTIES

Figure 3 shows some perspective views of the 3D FE discretization adopted to per-form both the pushover and the limit analysis on the bell tower. As a matter of fact, masonryis a material which exhibits distinct directional properties due to the mortar joints, actingas planes of weakness. Depending on the level of accuracy and the simplicity desired, it ispossible to use the following modeling strategies:

1. Micro-modeling: Units and mortar in the joints are represented by continuum elementswhereas the unit mortar interface is represented by discontinuous elements. To limit thecomputation effort, in simplified micro-modeling, units are expanded and modeled by

Figure 3. FE discretization of the bell tower by means of four noded tetrahedron elements (24285 brick elementsand 7161 nodes) (color figure available online).

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continuum elements, whereas the behavior of the mortar joints and unit-mortar inter-face is lumped in discontinuous elements (e.g., Lourenço and Rots, 1997; Orduna andLourenço 2005). For the problem at hand, micro-modeling is inapplicable, due to theneed of limiting degrees of freedom in the nonlinear analyses.

2. Homogenization (e.g., Luciano and Sacco 1997; Milani et al. 2006): It replaces thecomplex geometry of the basic cell using at a structural level a fictitious homogeneousorthotropic material, with mechanical properties deduced from a suitable boundaryvalue problem solved on a suitable unit cell, which generates by repetition the entirestructure. For the problem at hand, where multi-leaf and multi-head walls are present,the identification of a unit cell is however questionable and the utilization of orthotropicmodels would complicate the structural analysis without a sufficiently based hypoth-esis on the actual texture present. Furthermore, it is expected that orthotropy is acrucial issue for masonry walls loaded within their plane or façade eminently loadedout-of-plane. In this case, the behavior is more near to a squat cantilever beam.It is therefore preferable the utilization of isotropic materials, also in agreement withexisting literature.

3. Macro-modeling (e.g., Lourenço et al., 1997): Units, mortar and unit-mortar interfaceare directly smeared into a continuum (either isotropic or orthotropic), with mechanicalproperties deduced at the micro-scale by means of experimental data available. Thislatter approach is probably the most suitable in this case.

The structural analysis conducted by means of the commercial code DIANA 9.3.2. (TNODIANA DV, FEMGV, Delft, The Netherlands, 2008) is performed assuming a smearedcrack total strain material model for masonry, which allows for a nonlinear behavior inves-tigation of the bell tower. Although the model is specifically suited for a fragile isotropicmaterial (as is the case of concrete), its basic constitutive laws can be adapted for reproduc-ing masonry properties in the inelastic range. The “concrete model” basic characteristics,indeed, well reproduce uniaxial masonry behavior near collapse, as, for example:

1. tensile failure due to cracking and consequent softening branch (Figure 4);2. compression crushing failure (with compressive strength higher with respect to tensile

strength, as is the case of masonry);3. strain softening during compression crushing until an ultimate strain value is reached,

at which the material totally fails.

Since typical brickwork anisotropy at failure is not reproducible with the model at hand,averaged values between horizontal and vertical strengths are adopted with reference tosome specific data available in the technical literature. The assumption of an isotropicmodel is justified by the following considerations:

1. 3D elements are used. A full characterization of the compressive-tensile behaviorof masonry along geometrical x-y-z axes is not available and would require costlyexperimental campaigns.

2. Texture of the walls is a multi-leaf one and does not maintain constant all over the tower.Therefore, different macroscopically equivalent materials should be used in severaldifferent zones of the tower.

3. Little differences between the present model and sophisticated softening orthotropicmodels are expected in this case. Indeed, the nonlinear behavior of the tower is strongly

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influenced by masonry self-weight and differences in the tensile resistance do not affectsignificantly the global behavior of the structure under increasing static loads, as notedin the following discussion.

Mechanical properties adopted in the smeared crack model are summarized in Table 1.Two values of tensile peak strength are considered, namely 0.30 and 0.10 MPa. The firsthypothesis probably overestimates the actual experimental behavior of the masonry underconsideration, whereas the second is believed to be more realistic, also in light of theexperience accumulated by the authors on similar structures in the same geographic area.In any case, a comparison between results provided by the two assumptions may represent auseful benchmark on the influence of masonry peak tensile strength on the overall behaviorof the structure. As it will be shown hereafter, the difference in terms of pushover curvesis minimal, confirming the intuition that self-weight plays is predominant.

Masonry elastic modulus is deduced in agreement with existing literature in thisfield (e.g., Lourenço and Rots, 1997), and Italian code of practice specifics (OPCM, 2005,chapter 11.D). The nonlinear behavior of the material in tension is ruled by the fractureenergy in mode I GI and scaled by the dimension of the element h, Menin et al. (2009), tosecure mesh objectivity in the softening branch.

Linear softening model

-cr el

Figure 4. DIANA non-linear behavior in uniaxial tension to analyze the bell tower.

Table 1. Mechanical properties adopted for the pushover analyses (smeared crack model)

Property Unit of measure Masonry Filler Timber

Young modulus [MPa] 2800 930 4000Poisson ratio [ – ] 0.15 0.15 0.15Specific weight [Kg/mc] 1800 1800 600Tensile strength (ft) [MPa] 0.30/0.10 1 5Compressive strength (fc) [MPa] 3.0/1.5 10 15β (shear retention factor) 0.33 0.33 0.33εcr

u [‰] 3.5 3.5 10

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Three materials are used in the model, namely ordinary masonry (external and inter-nal walls and vaults), filler (disposed over the vaults as commonly done in building practiceto both increase vaults strength and to realize planar floors) and timber beams (for twoof the floors and some small section lintels of the windows). A linear softening modelis assumed in tension, where the stress-strain relationship in the crack is defined by thefollowing expression in Equation 1:

σ cr (εcr) ={

(1 − εcr

)→ 0 < εcr < εult

0 → εult ≤ εcr < ∞(1)

Where εult = 2GI

ftand εcr = ε − εel, being ε and εel the total and the elastic strain

respectively.In the smeared crack models, it is adopted a reduction parameter β for the transverse

stiffness of the material known as “shear retention factor”. The shear stiffness is henceexpressed by Equation 2:

Dsec = β

1 − βG (2)

where G is the shear elastic modulus. Following the recommendations found inthe literature (NTC, 2008; Menin et al. 2009), in the numerical examples analyzedβ = 0.33.

In compression, the smeared crack model is combined with a plasticity model, whichdescribes the crushing of the material. In particular an associated elasto-plastic modelwith a Mohr-Coulomb failure criterion is used, featuring under uniaxial loads hardeningfollowed by softening. Since for the case under consideration compression crushing isscarcely active, details of the model are not reported and the reader is referred to DIANAUser’s Guide (DIANA, 2008).

3.1. Limit Analysis Model

A full 3D homogeneous approach is used within limit analysis to model the belltower, which is discretized by means of four-noded tetrahedrons rigid infinitely resistant,by means of the same mesh used in the commercial code simulations. Here, differentfrom standard FEM, possible discontinuities of the velocity field along the edges of adja-cent elements may occur, where all plastic dissipation takes place. As shown with severalexamples by a number of authors (e.g., Milani et al. 2006; Krabbenhoft et al. 2005; Sloanand Kleeman, 1995) the introduction of jumps of velocities on discontinuities leads toan improvement of the numerical collapse multipliers obtained when cohesive-frictionalmaterials are treated, which is the case of masonry.

Kinematic variables for each four-noded tetrahedron element E are represented bythree centroid velocities (uE

x , uEy , uE

z ) and three rotations around centroid G (�Ex , �E

y , �Ez ),

as shown in Figure 5. Denoting with the symbol �e3 the edge surface of the element E

connecting P1, P2 and P4 nodes, then �e3 is triangular and jump of velocities on it is linear.

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P (xP,yP,zP)

Masonry element: M Masonry element: N

Interface I

Stress action on a point P of interface I

Masonry tetrahedron: E Interface I ( edge)3E

P (xP,yP,zP)

M-N interface I

Figure 5. 4-noded tetrahedron element (left) used to model masonry and three-noded interface (right) betweencontiguous elements where plastic dissipation takes place (global and local frame of reference).

In particular, velocity field of a generic point P with global coordinates (xP, yP, zP), on �e3

is expressed in the global frame of reference as shown in Equation 3:

U(P) =[ ux

⎡⎣ uEx

⎤⎦ +⎡⎣ 0 −�E

y �Ez

�Ey 0 −�E

−�Ez �E

⎤⎦[ xP − xG

yP − yG

zP − zG

E + RE (P − G)

where U (P) is the point P velocity, UGE is the element E centroid velocity and RE is element

E rotation matrix.From Equation (EQ3), jump of velocities [U(P)] at a point P on the interfaces

I between two contiguous elements N and M can be evaluated as difference betweenvelocities of P regarded belonging respectively to N and M as shown in Equation 4:

[U (P)] = UGM − UG

N + RM (P − GM)− RN (P − GN) (4)

Also, introduced for each interface �ei (i = 1, . . . , 4) between contiguous elements is the

vector field tI , defined as tIT = [ τ I1 τ I

2 σ I ] and representing the stress acting along localaxis rI

1(τ I1), rI

2(τ I2) and sI(σ I), as indicated in Figure 5.

Power dissipated at the interface can be evaluated analytically as shown inEquation 5:

PI =∫�e

r1�r1 + σ Ir2�r2 + σ I�s

)d�e

where �r1, �r2 and �s are velocities jumps (two tangential and mutually orthogonaland one perpendicular to the interface [Figure 5]) in the local coordinate system rI

1-rI2-

sI . Velocities jumps in the local system may be easily evaluated from Equation 4 once thatthe rotation matrix RI for rI

1-rI2-sI is at disposal, as shown in Equation 6:

�U (P) =[�r1

]= RI [U (P)] (6)

where �U (P) is the jump of velocities vector in the local system.

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The rotation matrix RI , which permits to pass from the global to the local coordinatesystem is evaluated once that unitary vectors rI

1-rI2-sI are expressed in terms of the e1-e2-e3

global frame of reference. We assume that rI1 and rI

2 are two axes laying on the interfaceplane and mutually orthogonal, whereas sI is the third axis perpendicular to the interface.In this way, unitary vectors rI

1-rI2-sI can be expressed in the global coordinate system as

shown in Equation 7: ⎡⎢⎢⎢⎢⎢⎢⎣rI

1 = P4 − P1

‖P4 − P1‖rI

2 = P2 − P1

‖P2 − P1‖s̃I = (̃

rI2 × rI

) �⇒ sI = s̃I/∥∥̃sI

∥∥rI

2 = rI1 × sI

⎤⎥⎥⎥⎥⎥⎥⎦ (7)

RI is finally obtained rearranging numerically Equation 7, interface by interface.For each interface �e

3 of area 3 connecting nodes 1-2-4 (Figure 5), is assumed tohave at disposal a linearized strength domain for the interface in the local coordinate systemrI

1-rI2-sI and constituted by mI planes.

In particular, a generic linearization plane qI is assumed to have the equation AqI

r1τI1 +

s σIs + AqI

r2τI2 = CqI

I 1 ≤ qI ≤ mI . Introducing plastic multipliers fields at the interface(one for each linearization plane) from Equation 5, power dissipated at the interface can bere-written as shown in Equation 8:

PI =∫�e

λ̇qI

I (r1, r2)(

r1τI1 + AqI

s σI + AqI

r2τI2

)d3 (8)

Obviously, field λ̇qI

I (r1, r2) assumes the same analytical expression found for the velocity

field, i.e., is linear in r1-r2, as shown in Equation 6. Therefore, λ̇qI

I (r1, r2) field is fullydetermined introducing only three plastic multipliers for each internal interface and foreach linearization plane, corresponding, for example, to nodes 1, 4, 2.

In another regard, the numerical evaluation of integral (Equation 8) is trivial andplastic dissipation at a generic interface can be obtained as shown in Equation 9:

PI = 3

mI∑qI

I,1qI + λ̇

I,4qI + λ̇

where all the symbols are self-explaining or have been already introduced.To be kinematically admissible, and thus provide an upper bound of the collapse

load, the velocity field must satisfy the set of constraints imposed by an associated flowrule. Since 3D elements are rigid infinitely resistant, constrains must be imposed onlyon triangular interfaces. Plastic flow constraints in discontinuities can be written analo-gously to the continuum case when a linearization of the failure surface is provided for theinterfaces, as in Equation 10:

�U (P) =[�r1

]= λ̇ (r1, r2)∇FI (10)

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where λ̇ (r1, r2) is the plastic multiplier rate field on the interface I (a linear function inr1 and r2) and FI is the eventually nonlinear failure surface of the interface (expressed as afunction of stress components τ I

1, τ I2 and σ I).

For masonry interfaces, a piecewise linear approximation with npl= 25 planes of aMohr-Coulomb failure criterion with tension cutoff and cap in compression in adopted(Figure 6) is adopted (cohesion c = 0.10/0.30 MPa, tensile strength f t = 0.10/0.30 MPa,compressive strength f c = 3.0/1.5 MPa, friction angle � = 30◦, shape of the linearizedcompressive cap �2 = 60◦), as shown in Equation 11:

FM1 = [

σ Is τ I

1 τ I2

]:⎧⎪⎪⎪⎨⎪⎪⎪⎩

√(τ I

12 + τ I

∓ (c + σ I tan�

) = 0 ≈ Ai−Ms σ I + Ai−M

r1 τ I1 + Ai−M

r2 τ I2

= Ci−M i = 1, . . . , npl

∃i|Ai−Ms σ I + Ai−M

r1 τ I1 + Ai−M

r2 τ I2 = bi−B and

Aj−Ms σ I + Aj−M

r1 τ I1 + Aj−M

r2 τ I2 ≤ Cj−M ∀j �= i

⎫⎪⎪⎪⎬⎪⎪⎪⎭FM

2 = [σ I τ I

1 τ I2

]:{σ I ≤ ft

}FM = FM

1 ∪ FM2

: compression linearized cap

c : cohesion

fc : compression strengthft : tensile strength

: friction angle

Tetrahedrons interfacesLinearized Lourenço and Rots

(1997) failure criterion edge

Triangular interfaces

Figure 6. Linearized strength domains adopted for masonry interfaces

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where:� - Ai−M

s , Ai−Mr1 and Ai−M

r2 are the coefficients of the i − th linearization plane;� - Ci−M is the right hand side of the i − th linearization plane.

Full sensitivity analysis will be performed varying tensile strength, as it will be dis-cussed later, in order to evaluate the role played by the finite resistance of masonry intension.

External power dissipated can be written as Pex = (PT

0 + λPT1

)w, where P0 is the

vector of permanent loads, λ is the load multiplier for the structure examined, PT1 is

the vector of variable loads and w is the vector of assembled centroid elements veloc-ities. As the amplitude of the failure mechanism is arbitrary, a further normalizationcondition PT

1 w = 1 is usually introduced. Hence, the external power becomes linearin w and λ.

A linear programming problem is obtained, after some elementary assemblage oper-ations, where the objective function is the total internal power dissipated minus the powerdissipated by external loads not dependent on the load multiplier, as shown in Equation 12:⎧⎪⎨⎪⎩

Pin,assI λ̇

I,assT − PT0 w

}such that

{AeqU = beq

λ̇I,ass ≥ 0

where:� U is the vector of global unknowns and collects the vector of elements centroids veloc-

ities (w) and rotations (�) of masonry elements and the vector of assembled interface

plastic multiplier rates (λ̇I,ass

). λ̇I,ass

collects plastic multiplier rates of masonry-masonryinterfaces.

� Aeq is the overall constraints matrix and collects normalization conditions, velocityboundary conditions and constraints for plastic flow in velocity discontinuities.

� Pin,assI is a row vector that collects contributions to the internal dissipation of masonry-

masonry interfaces (Equation 9).

The reader is referred to Sloan and Kleeman (1995) for a critical discussion of efficient(classic) linear programming tools suited for solving (Equation 12).

In contrast, it is worth noting that recent trends in limit analysis have demonstratedthat the linearization of the strength domain can be circumvented using conic/semi-definiteprogramming (e.g., Makrodimopoulos and Martin, 2006; Krabbenhoft et al. 2007). Thistool is more powerful in terms of processing time with respect to classic linear program-ming (LP) and could lead to a numerical efficiency improvement for the structural analyses.Both free (e.g., SeDuMi, available at: http://sedumi.mcmaster.ca/) and commercial (e.g.,Mosek, available at: www.mosek.com) standalone tools are available today. However, sincethe aim of this study is mainly concentrated on the structural aspects related to the limitanalysis of very complex reinforced masonry elements, the classic interior point LP routineavailable in Matlab is used for the sake of simplicity.

3.2. Rigid Body and Spring Model (RBSM)

The present RBSM (Casolo, 2004; Casolo, 2009) is based on a mechanistic approachin which the heterogeneous solid material is imaged as an assemblage of rigid masses and

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nonlinear damaging springs, and can be classified in the vast family of discrete elementsmodels (Lemos, 2007). In particular, for RBSM the discretization is obtained by meansof quadrilateral rigid elements where the mass is concentrated, interconnected by nonlin-ear shear and normal springs, as schematically illustrated in Figure 9 and Figure 10. Thisapproach is specifically designed for macro-scale applications, and thus the nonlinear anddamaging material response is defined in order to perform efficient and reliable analyseseven when dealing with a rather crude discretization (Casolo and Peña, 2007). Additionally,anisotropic materials (such as masonry) with possible micro-structure (Casolo, 2006)may be handled, and in this sense a main feature is the possibility of assigning a fullorthotropic nonlinear shear response at the macro-scale level. In practice, this meansthat we can define different cohesions and internal friction coefficients when consider-ing shear deformations that activate sliding along vertical or horizontal plane of fractures(Casolo, 2009). While some dynamical RBSM applications are available in the liter-ature (Casolo and Peña, 2007), this one is particularly relevant because it deals witha masonry tower with a very complex geometry, where meaningful thickness changes,internal double curvature elements (vaults) and a bell tower with wide openings aremodeled.

From authors’ experience in this field, it can be deduced that the overall geometricaland mechanical properties of the structure under consideration will result into complexdamage patters under dynamic actions, such as the combined failure of the central part forshear action (being the slenderness low and the external walls relatively thin), localizedcracks consequent to the redistribution of internal actions due to the vaults presence andthe complex geometry of the bell cell at the top.

The geometry and the kinematics of the RBSM are described by considering aglobal Cartesian coordinate frame {O, x, y} that is placed with the x-axis parallel to thebed-joints. The domain is partitioned into quadrilateral elements such that no vertex ofone quadrilateral lies on the edge of another quadrilateral. As shown in Figure 10a, alocal reference frame

{ξ i, ηi

}is fixed in the element centroid Oi, with the ξ i-axis ini-

tially parallel with the global x-axis. The deformed configuration of the discrete modelis described by the variations of position of these local reference frames with respectto the global one. The element Lagrangian coordinates are assembled into the 3n vec-tor UT = [ u1 v1 ψ1 u2 v2 ψ2 . . . un vn ψn ], where n is the total numberof elements present in the mesh and ui, vi and ψi are the centroid horizontal and verticaldisplacement and the element in-plane rotation respectively.

Three connection points named P, Q and R are defined for each side in commonbetween two adjoining rigid elements (Figure 9) and the corresponding measures of axialand shear strain are assigned to a volume of pertinence opportunely associated to each ofthese points.

The present formulation is implemented in the dynamical field, thus in orderto consider the inertial forces, the mass of each element mi and the rotational iner-tia about the centroid ii, are assembled in a diagonal generalized mass matrix: M =diag [ m1 m1 i1 m2 m2 i2 . . . mn mn in ].

Mechanical behavior of the nonlinear springs (axial and shear) are depicted inFigure 10b. Numerical values adopted in the present dynamic simulations are summarizedin Table 2 and correspond globally to material properties adopted in the nonlinear staticanalyses (e.g., same Young modulus, similar low tensile strength, isotropic material). Thenonlinear behavior in tension and shear are characterized by a marked softening branch,

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Table 2. Mechanical properties adopted for dynamic analyses

Point E Y U S

CompressionE11 = E22 [%] Sii/Eii 0.1 0.2 0.4S11 = S22 [MPa] 1 2 1.75 1

TensionE11 = E22 [%] Sii/Eii 0.1 —S11 = S22 [MPa] 0.2 0.1 —

Shear springs 12 (friction angle � = 15◦)E12 [%] S12/2E12 0.0070 0.0107 0.0900S12 [MPa] 0.054 0.088 0.097 0.020

Shear springs 21 (friction angle � = 3◦)E21 [%] S21/2E21 0.0050 0.0187 0.1650S21 [MPa] 0.054 0.088 0.165 0.020

and some micro-structure effects that are typical of the most part of masonry textures, havebeen accounted by specifying different values for the vertical and horizontal shear springs.The phenomenological constitutive laws of individual springs can be obtained from theexperimental evidence of three main types of post-elastic and hysteretic behavior (Casoloand Peña, 2007; Casolo, 2006):

1. Tension: in this case the strength is very low, the response is brittle, and earlymechanical degradation is observed, without significant energy dissipation under cyclicloading.

2. Compression: in this case the material exhibits its maximum bearing capacity withlimited ductility and a progressive mechanical degradation once the peak valueof strength is reached. The energy dissipation is essentially related to progressivecrushing.

3. Shear: this case is governed by the response of mortar joints; once the maximumstrength is reached, the residual shear capacity depends on the vertical axial load,obeying a Coulomb-type relation. Energy dissipation for repeated cycles of loadingmay be significant. The shear spring behavior is inspired by the Takeda hystereticmodel with degrading strength and stiffness. The skeleton curve is tri-linear, and thehysteretic rules are schematically described by following the loading path shown inFigure 10.

These observations were condensed into the formulation of two different classes of con-stitutive behavior for the axial and shear connecting springs. For example, in Figure 10b,the assumed hysteretic rules are illustrated by following the paths indicated by the pointslabeled as 1-2-. . .-9.

Since the structure under consideration is not a standard typology, the applicationof a RBSM analysis requires particular attention in the discretization definition and in theequivalent thickness to assign to the elements. A proposal that partially takes into accountall this elements is depicted in Figure 9.

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4. THE NUMERICAL MODELS ANALYZED

Loads applied to both limit analysis and commercial FE models when dealingwith the pushover analysis are: masonry self-weight, bells weight, wooden and masonryfloors weight (dead and live loads) and seismic action. Two sets of pushover analy-ses are performed, assuming a seismic action acting along X or Y direction (Figure 3shows axes direction). Horizontal loads distribution is assumed with a shape accord-ing to the first mode of the structure, assuming the maximum value at the top of thetower. Due to the impossibility of automatically applying diffused forces on the cen-troid of the single elements in the commercial code, concentrated forced are appliedin correspondence of two corners of the structure, following a first mode scheme(Figure 7).

Several different models are critically investigated and compared:

� Model A is the 3D model of the bell tower, with the presence of the stairs and theinternal vaults interconnecting the core walls and the skin walls. The tower is supposedcompletely isolated from the context and pinned at the base.

� Model A1 is similar to model A, except that all the elements connected with the corridorinterconnecting the tower to the contiguous church are pinned.

Vault backfill

Timber beams

Figure 7. Application of the horizontal loads in the commercial FE code (color figure available online).

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� Model B differs from model A in that the stairs and the interconnecting small vaults arenot modeled here. The core and the skin are therefore interconnected by the barrel vaultpresent under the first floor and by the bell cell at the top.

� Model B1: For this model, similar to model A1, the tower is supposed pinned both at thebase and in correspondence to the linking corridor between the tower and the contiguouschurch.

Here it is worth noting that it is expected that the interconnection of the stairs with theinternal and external walls does not change drastically the structural behavior of models A

Model A Model B

Model A1 Model B1

Figure 8. Synopsis of the different models investigated (color figure available online).

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Figure 9. RBSM discretization of the bell tower by means of quadrangular rigid elements (438 elements) (colorfigure available online).

Figure 10. -a: couple of rigid elements with the adopted notation for the displacements and notation adopted forthe external forces and the springs. –b: Scheme of the hysteretic rules of the axial (top) and of the shear spring(bottom) (color figure available online).

and B under horizontal loads, in terms of the overall stiffness, strength, and ductility. Littledifferences are therefore expected between the models. The same considerations may berepeated for the effect induced by the inclination of the tower, which is not critical in thiscase, thanks to the low slenderness of the tower.

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Depending on the load case and the model considered, the processing times neededto perform the simulations on a PC equipped with an Intel Core 2 Duo CPU @ 2 GHzand 4 GB RAM were approximately: 30 minutes for each limit analysis simulation underMatlab, 70 minutes for the pushover analyses under DIANA and 120 minutes for the fulldynamic non linear analyses under an Intel Fortran 77 code.

5. NUMERICAL RESULTS

In this section, results obtained by means of all the models proposed (3D modalanalysis, pushover analysis, limit analysis, dynamic nonlinear analysis) are shown and dis-cussed in detail, starting from the preliminary (standard) modal analysis and concludingwith the nonlinear dynamic output.

5.1. Preliminary Modal Analysis

Preliminarily to the nonlinear analysis of the tower, a 3D modal analysis is performedin order to determine the tower eigenvalues and eigen-modes. To compare results witha simple at hand model, firstly the mass of the bells (0.0215 kg/cm2) and of the roof(0.0200 kg/cm2) are neglected. The periods of the first 3 modes of a natural frequenciesanalysis are shown in Table 3 for all the four models investigated. Modal deformed shapesare shown in Figure 11 for model A only, being almost identical for all models. Verysimilar results are obtained in presence of vertical inclination, due to the small out-of-vertical exhibited by the tower, as well as in presence of the mass of the bells. The first andsecond periods are obviously very similar, since the first period refers to a direction +45◦in the X-Y plane, the second to a −45◦ direction.

The eigenvalue analysis is extremely useful in light of the pushover simulations toperform later. Indeed, to reduce the structure to a 1DOF system, the Italian code requiresthe knowledge of the fundamental mode normalized displacement vector. Data used arethose referred to the presence of the bells and the roof.

As expected, model A has slightly shorter periods with respect to model B (absenceof interconnection between core and skin walls), as well as model A1 and model B1 haveperiods that are shorter with respect to model A and B, respectively, obviously a con-sequence of the constraints added in a portion of the lateral edge near the base of thetower.

A comparison of all the results with a cantilever beam approach is also provided—except for the torsional third mode—and may be useful for practical purposes, to confirm

Table 3. First three periods of the structure from a natural frequencies analysis, compared with the formulaprovided by a cantilever beam hypothesis

Period T (sec) First mode Second mode Third mode

Model A 0.4130 0.4101 0.1329Model A1 0.4122 0.3783 0.1287Model B 0.4250 0.4243 0.1351Model B1 0.4250 0.3939 0.1322— X direction first mode Y direction first mode —Cantilever beam prediction 0.4137 0.4137 —

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Period T = 0.414 sec Period T = 0.411 sec Period T = 0.133 sec

X excited mass 45.00% X excited mass 11.85% X excited mass 0.18%

Y excited mass 12.01% Y excited mass 45.06% Y excited mass 0.07%

X excited mass 0.015% X excited mass 20.23% X excited mass 0.06%

Y excited mass 20.66% Y excited mass 0.03% Y excited mass 0.16%

Figure 11. Model A. Eigenvalues analysis (color figure available online).

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that the natural frequencies of the tower may be evaluated with acceptable accuracy evenwithout the utilization of complex 3D FE analyses. In particular the frequency of the tower,assuming a cantilever beam hypothesis is given by the following simple formula shown inEquation 13:

fi = αi

2π L2

μA(13)

where μ(=1529 kg/m3) is the average density of the structure (accounting for internalholes and changes in thickness along the height), E (=2800 N/mm2) is the Young modulus,A (=25.158 m2) and I (=164.107 m4) are the base cross section area and inertia moment,L the tower height (around 28.30 m) and αi is a coefficient associated to the i − th modeinspected. For example, for i = 1 (first mode frequency) αi = 3.5156, for i = 2 αi = 22.035,for i = 3 αi = 61.698.

The inertia moments along X and Y direction are almost equal, thanks to the quasi-square cross-section of the tower. Therefore, in the cantilever beam approach they are keptequal. Authors experienced that the mass of the bells (approximately 4500 kg account-ing also for the steel frame sustaining the bells), in practice, does not influence the periodof the tower. While the bells weight may be easily taken into account in the FE models,within the cantilever beam assumption, it is obviously disregarded. In another regard, itis worth noting here that the bell cell is highly perforated and both the weight and theinertia of the upper part of the tower can’t be modeled with sufficient accuracy within thecontext of Equation 13. As a matter of fact, both the inertia and the cross-area—as wellas the weight—of the bell cell result overestimated in the cantilever beam approach. Suchoverestimation turns out to compensate totally the effect induced by the bells. In addi-tion, in Equation 13 some other important structural aspects typical of the tower underconsideration (e.g., internal vaults, openings, stairs) are disregarded.

Eigenvalues analyses are conducted also with the RBSM model. The first three vibra-tion modes of the tower are shown in Figure 12, with the corresponding natural periods. It isworth noting the almost-perfect agreement between the first period and modal shape foundwith the 3D and the RBSM model, meaning that the calibration of the 2D discretizationis adequate to perform full nonlinear dynamic analyses. The shape and frequency of thesecond natural mode are also in excellent agreement with the corresponding 3D analysis.

5.2. 3D Nonlinear and Limit Analysis

In the past few years, simplified procedures have been proposed for the nonlinearstatic analysis of masonry structures. These methods, generally known as pushover anal-yses, have recently assumed a large relevance, especially for the assessment of existingbuildings, also in light of code of practice requirements. Basically, a computational modelof the structure is loaded up with a proper distribution of horizontal static loads, whichare gradually increased with the aim of “pushing” the structure into the nonlinear field.The resulting response conveniently represents the envelope of all the possible structuralresponses, and can so be used to replace a full nonlinear dynamic analysis.

Here full 3D pushover analyses are conducted on all models, investigating also therole played by the actual inclination of the tower along X and Y axes. Resultant capacity

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Figure 12. RBSM model. Eigenvalues analysis (color figure available online).

curves of the structure are depicted in Figure 13 and Figure 14 for all models (A-A1 andB-B1 with and without inclination) for seismic action along X and Y directions, respec-tively. A comparison with limit analysis results is also reported. Two different hypothesesare made for the tensile peak strength of the masonry material, in order to evaluate theinfluence of masonry peak tensile strength on the pushover curve of the structure. In thefirst model, ft is assumed equal to 0.30 MPa, whereas in the second it is assumed equal to0.10 MPa. The first hypothesis is more realistic for the tower under consideration, where itis expected a rather high resistance of joints, realized with good mortar and well-disposedbricks (e.g., presence of transversal blocks, continuous horizontal joints), whereas the sec-ond approximates realistically an almost no tension material behavior. Results in terms ofpushover curves are summarized in Figure 15 for models A and B (seismic direction X andY), with limit analysis results. As it is possible to notice, the difference in terms of bothpeak strength and ductility is negligible, meaning that masonry self-weight is predominantin this case study.

In Figure 16, deformed shapes at peak (or collapse) provided by the commercial codeand limit analysis for model A (seismic direction X and Y) are compared. As it is possibleto notice, the tower fails in both cases for the formation of a flexural-shear hinge near thebase of the structure, Figure 17. The slip surface provided by limit analysis is rather clearand may be isolated manually identifying the interfaces with maximum plastic dissipation(Figure 18). From simulations results, it appears that the cracked surface at the base of thetower is not horizontal and slightly asymmetric, a consequence of the limited shear strengthof masonry and the asymmetric distribution of the stairs inside the tower. A small but ratherevident contribution of the limited shear resistance may be clearly identified in the limitanalysis model. When dealing with the commercial FE code, the inelastic deformation εnn

perpendicular to the crack direction may be plotted (Figure 17), with the correspondingstress level σnn. While it is not strictly possible to identify a slip surface in a continuumFE approach, the cracked zones patch is anyway able to give an approximate idea of the

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0 2 4 6 8 10 12

Roof displacement [cm]

X direction

Model AModel A1Model BModel B1Model A InclinedModel A1 InclinedModel B InclinedModel B1 InclinedLimit analysis model ALimit analysis model A inclinedLimit analysis model BLimit analysis model B inclined

Figure 13. Seismic direction X. Synopsis of all models with and without inclination (ft = 0.30 MPa) (colorfigure available online).

failure mechanism, which seems to approximate well the limit analysis results (apart aconcentration of cracks near the bell cell).

From a synoptic analysis of all results collected, the following conclusions may bedrawn:

1. Tensile resistance of masonry affects slightly the maximum base shear of the tower.This effect was largely expected, since the overall strength of the base section is largelydependent on masonry self-weight. Simple equilibrium models assuming the rocking ofthe tower around one base edge may lead to rather accurate results.

2. The inclination of the tower obviously turns out to reduce slightly the maximum hori-zontal load that the tower can withstand prone to collapse. Nonetheless, the reductionis negligible from an engineering point of view.

3. Limit analysis results are in very good agreement with those provided by the nonlinearincremental code.

4. A quasi perfectly plastic behavior is obtained with standard FE, even in presence of amarked brittle behavior in tension for masonry. This is not surprising, because the roleplayed by self-weight in the overall resistance of the structure is predominant.

5. Crack pattern provided by the commercial code compared with limit analysis failuresurfaces (Figure 17 and Figure 18) envisages a full 3D analysis is needed for a reliableevaluation of the actual behavior of masonry towers, which rarely exhibit full symmetryin plan.

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0 2 4 6 8 10 12

Roof displacement [cm]

Y direction

Model AModel A1Model BModel B1Model A InclinedModel A1 InclinedModel B InclinedModel B1 InclinedLimit analysis model ALimit analysis model A inclinedLimit analysis model BLimit analysis model B inclined

Figure 14. Seismic direction Y. Synopsis of all models with and without inclination (ft = 0.30 MPa) (color figureavailable online).

6. The overall resistance of the tower along X direction is smaller with respect to thatprovided by the models along Y direction. Again, this 3D effect cannot be kept bysimplified 2D models and is justified exclusively by the asymmetry of the cross sectionnear the foot of the tower.

7. Model A and B results are comparable. This means that the interlocking between coreand skin walls is secured by the barred vaults of the first and second floor instead ofthe system of small vaults sustaining stairs. In any case, the differences between thetwo models, both in terms of peak base shear, ductility at failure and initial stiffness arenegligible from an engineering point of view.

5.3. RBSM Dynamic Analyses

A step-by-step nonlinear dynamic analysis is definitely the most appropriate non-standard numerical procedure to evaluate the actual behavior of masonry structures duringearthquakes. However, at present, adequate commercial numerical codes are not easilyavailable and the analysis itself requires experienced users, especially with regard to theconstitutive aspects (the post-elastic, hysteretic behavior of the structure, and the conse-quent energy dissipation, should be consistently reproduced). The choice of significantinput accelerograms is still a controversial question. When speaking of historical andmonumental buildings, the question is even more critical: nearly all the structural analysis

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Different tensile strength pushover curves

3000X direction Model A

ft = 0.30 MPa commerical code

ft = 0.30 MPa Limit analysis

Roof displacement [mm]

0 2 4 6 8 10 120

3000Y direction Model A

ft = 0.30 Mpa Limit analysis

0 2 4 6 8 10 12

3000X direction Model B

0 2 4 6 8 10 120

0 2 4 6 8 10 12

Y direction Model B

ft = 0.30 Mpa Limit analysis

Figure 15. Model A and B. comparison of pushover curves assuming two different tensile strengths for masonry(color figure available online).

methods suffer from a lot of uncertainties and limitations, mainly derived from the incom-pleteness of the knowledge level, from the presence of widespread inhomogeneities andnonlinearities, and a complex constructive history. Despite the general drawbacks andlimitations aforementioned regarding the dynamic analyses, the non-commercial RBSMcode used by the authors allows to perform reliable numerical analyses on real scalemasonry structures, with a computation effort relatively small even using relatively refineddiscretization.

The full dynamic analyses were performed by using artificial accelerograms to repre-sent the ground motions. The reference design spectra used are those for the Ultimate LimitState of Life Safety for the zones 1, 2, 3 and 4 in which the Italian territory is classified,assuming an “A”–type soil and no topographic amplification effect (this corresponds to aprobability of exceedance PVR = 10 in 50 years, i.e., to a return period of the seismic actionTR = 475 years). For each zone, four artificial accelerograms were generated according tothe procedure proposed by Sabetta and Pugliese (1996). The resulting accelerograms havebeen further processed in order to improve the matching with the spectrum.

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Figure 16. Model A without inclination. Comparison between deformed shapes at peak and at collapse providedby the FE commercial code DIANA (top) and limit analysis (bottom) for X (left) and Y (right) horizontal action(color figure available online).

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X direction

Y direction

Figure 17. Model A. Map of the cracked elements from DIANA (deformation perpendicular to the crackedplane, stress perpendicular to the cracked plane) for X and Y seismic action (color figure available online).

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X direction

Y direction

Figure 18. Model A. Limit analysis and commercial code deduced failure surface near the base of the structure(color figure available online).

Here it is worth noting that in the model no vertical accelerogram has been appliedfor the dynamic analyses. While a vertical ground motion could be potentially rather impor-tant, especially for high buildings and in presence of bedrock foundation, it has been shownin Casolo (1998) that for low slenderness towers its influence tends to be negligible.

In Figures 19–22, the input accelerograms used respectively for Zone 1, 2, 3, and4 are shown. In the same figures, some significant outputs of the analyses are also repre-sented: the displacement time-history at the roof and bell levels (Figure 9), the plot of thebase shear against the roof level displacement, and the time history of the dissipated energy.For each zone, the deformed shape at the end of the dynamic analyses have been plotted inFigure 23. Additionally, in Figure 24a color patch of shear deformations E12 and E21 at theend of simulations is represented. The graphical representation of E12 and E21 total strainsthrough color patches helps in the determination of both failure mechanism (if any) reachedby the structure and the zones where severe damage concentrates. Finally in Figure 25, thedistribution of vertical stress (S22) at the end of the dynamic simulations is shown.

From an overall analysis of simulations results, it is possible to notice that the towerexhibits significant damage for earthquakes possible in zone 2 and zone 1. Some ele-ments undergo very large displacements in zone 1. While the code is based on a smalldisplacement assumption for the rigid elements, such a response indicates the activation ofa collapse mechanism. Comparing deformed shapes at the end of simulations, it appears

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Figure 19. RBSM model. Time histories: ground accelerations, displacements, force-displacement envelopes,and hysteretic energy dissipated. Zone 1.

rather clear that the tower could collapse in zone 2 for the formation of a shear-flexuralhinge in the lower region of the tower, immediately over the ground floor vault. Whilesome evident differences between present deformed shapes and nonlinear static ones maybe remarked, the dynamic behavior of the tower in zone 2 have some similarities with thatexhibited within the pushover analyses, such as the formation of a mixed shear and flexural

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hinge near the first floor level. Conversely, the behavior of the structure in zone 1 shows aclear collapse of the bell cell and again some meaningful cracks in correspondence of thefirst floor level. This finding is not surprising, since it is largely demonstrated that staticincremental analyses following a first mode deformation scheme may not reproduce par-tial collapses near the top, as it usually occurs for highly perforated bell cells. Finally, it is

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Figure 23. RBSM model. Deformed shapes with simple gravity load (at the left side, t = 0), and at the end ofthe dynamic analyses (Z4: zone 4, Z3: zone 3, Z2: zone 2, Z1: zone 1) (color figure available online).

Figure 24. RBSM model. Patch of E12 (-a) and E21 (-b) total deformations (Z4: zone 4, Z3: zone 3, Z2: zone 2,Z1: zone 1) (color figure available online).

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Figure 25. RBSM model. Patch of S22 vertical stress (Z4: zone 4, Z3: zone 3, Z2: zone 2, Z1: zone 1) (colorfigure available online).

worth remarking that in zone 3 the tower exhibits again some visible damage, with evidentvertical cracks running along the whole height of the structure, with a small damage of thebell cell. In zone 4, no visible damage occurs.

To summarize, from dynamic analysis performed on 4 artificial accelerograms cor-responding to 4 different seismic zones, it can be concluded that in zone 3 the tower is verynear its resistance limit (safety factor near one), whereas in zone 1 and 2 it could be highlyvulnerable.

6. CONCLUSIONS AND SAFETY ASSESSMENT CONSIDERATIONS

Before draw any conclusion through sophisticated numerical analyses on the exam-ple under consideration, it is interesting to evaluate the vulnerability index of the toweraccording to the Italian Guidelines on cultural heritage buildings (Direttiva del Presidentedel Consiglio dei Ministri [DPCM], 2007; Ministero per i Beni e le Attività Culturali,2006), which includes churches and bell towers.

The guidelines assume as safety index of the ratio shown in Equation 14:

Is = aSLU

γISag(14)

where aSLU is the soil acceleration which determines the collapse of the structure, γI theimportance factor of the structure (here, kept as 0.5 [Ministero per i Beni e le AttivitàCulturali, 2006]), S the soil stratigraphy coefficient and ag the site reference acceleration(usually evaluated using seismic risk maps). Is < 1 means that the vulnerability test is notsatisfied.

For towers with rectangular section, within the hypothesis that the normal pre-compression does not exceed 0.85fdAs, the ultimate bending moment at the base is shownin Equation 15:

Mu = σ0A

(b − σ0A

0.85afd

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where a is the transversal edge length of the base section, b the longitudinal length edge,A the section area, σ0 = W/A the average pre-compression (W: tower weight) and fd thedesign compressive strength.

External moment, within a cantilever beam hypothesis (subdivided into k elements),may be evaluated as shown in Equation 16:

Me = Feze

n∑k=1

with Fe = 0.85Sd (T1)W/g (Sd spectrum, T1 first period of the structure, g gravityacceleration).

With the equations shown in Equation 15 and Equation 16, it is possible to evaluateSd,SLU as shown in Equation 17:

Sd,SLU = Mug

0.85Wze(17)

and the corresponding aSLU as shown in Equation 18:

aSLU ={

0.4Sd,SLU T1 ≤ TC

0.4Sd,SLUT1

TcT1 > TC

where TC is a characteristic period depending exclusively on the spectrum used, see OPCM3431 (2005).

On the bases of the method proposed, a value equal to 1.047 for Is is obtained forthe seismic zone under consideration (zone 3). If the tower was located in zone 2 or 1,the corresponding Is would be 0.63 and 0.45 respectively (i.e., insufficient), in very goodagreement with full nonlinear dynamic analyses.

On the basis of the 3D nonlinear analyses, the seismic vulnerability is evaluated bymeans of a comparison between the displacement capacity dc and the displacement demanddd obtained by means of the pushover analysis, both referring to the same control point.Control points, in this case, are obviously placed in the middle of one of the edges of thetop section. The displacement demand is evaluated with respect to an equivalent singledegree-of-freedom system characterized by a bilinear behavior in a shear-displacementdiagram.

As a first step, the pushover curve is scaled by means of the so called participationfactor � =

∑mi�i∑mi�

2i, where �i is the i-th component of the eigenvector � and mi is the

mass of the node i. The fundamental eigenvector � is deduced from the modal analysis.Assuming as Fb and dc the actual base shear and corresponding displacement of the struc-ture respectively, the scaled values are F∗

b = Fb/� and d∗c = dc/�. Assuming as Fbu the

peak base shear, it follows that F∗bu = Fbu/�.

Once found the pushover curve of the equivalent system (d∗c -F∗

b), it is reduced toa bilinear elastic perfectly plastic diagram, where the elastic stiffness k∗ is calculated by

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0200400600800

10001200140016001800

F∗ [

d∗ [cm]

Model A1-X direction

Equivalent system1 DOF linear

1 DOF constant

0 0,5 1 1,5 2 2,5 3 0 0,5 1 1,5 2 2,5

d* [cm]

Model A1-Y direction

Equivalent system

1 DOF linear

1 DOF constant

d∗max = 1.18 cm < d∗

u = 3.01 cm d∗max = 1.23 cm < d∗

u = 2.61 cm

q* = 0.58 < 3 q* = 0.61 < 3

Figure 26. Evaluation of the seismic safety of the structure according to the pushover analysis (color figureavailable online).

drawing a line from the origin to the point of the equivalent capacity curve with a shearequal to the 70% of the maximum value F∗

y . The bilinear diagram is completed assumingan area equivalence between the equivalent and the bi-linear system, where the equivalentcurve is stopped at a displacement d∗

u corresponding to a base shear equal to 75% of thepeak shear. As usually occurs in complex 3D nonlinear analyses, a softening of about 15%is hardly reproducible. The numerical analysis is stopped for lack of convergence at around12 cm of displacement, reasonably assumed as the collapse of the structure. The plasticbase shear of the 1 DOF equivalent system is called F∗

y . In absence of a clear degradationof the base shear, it is assumed F∗

y = F∗bu.

After the evaluation of the equivalent mass m∗ of the system as∑

mi�i, the periodT∗ is equal to 2π

√m∗/k∗. Once known T∗, the Italian code allows to estimate the dis-

placement demand d∗max using the elastic displacement spectrum SDe (T). The base shear

corresponding to d∗max on the elastic one DOF system is hereafter called F∗

e .Once known d∗

max, it has to be checked if d∗max <= d∗

u . For masonry structures, theItalian code requires also that the ratio q∗ between the base shear evaluated using the elasticspectrum F∗

e and of the equivalent 1 DOF system F∗y does not exceed 3.

For the sake of clearness, in Figure 26, the reduction of the FE pushover curve formodel A1 (model with inclination), seismic direction X and Y, is represented, along withthe calculated displacement demand. In agreement with preliminary analysis and nonlineardynamic simulations in zone 3, it can be argued that the requirements of the Italian codeare satisfied and therefore the safety assessment of the tower under horizontal loads isverified also for the pushover procedure. However, some questions raise about the actualextra-resistance of the structure, which may not be quantified easily with such a procedure.

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