nonlinear seismic analysis of a masonry arch bridge · nonlinear seismic analysis of a masonry arch...

Elisabeth Scheibmeir

NONLINEAR SEISMIC ANALYSIS OF A

MASONRY ARCH BRIDGE

Barcelona, 2012

Nonlinear seismic analysis of a masonry arch bridge

Erasmus Mundus Programme

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i

DECLARATION

Name: Elisabeth Scheibmeir

Email: [email protected]

Title of the

Msc Dissertation:


Supervisor(s): Luca Pelà

Year: 2012

I hereby declare that all information in this document has been obtained and presented in accordance

with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I

have fully cited and referenced all material and results that are not original to this work.

University: Universitat Politècnica de Catalunya

Date: 16.07.2012

Signature: ___________________________



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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii

ACKNOWLEDGEMENTS

The present work has been developed at the Universitat Politècnica de Catalunya. Foremost the help of

Luca Pelà and Pere Roca have made this study possible. Especially I would like to thank Mr. Luca Pelà for

guidance, support and supervision.

I would also like to thank all the lecturers who participated during the classes of the “Master of

Structural Analysis of Monuments and Historical Buildings” in Guimarães from Sep. 2011 to March 2012

for sharing their great knowledge.

This study has been carried out with the financial help of the Erasmus Mundus Program of the European

Union, for which the author expresses its gratitude.



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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v

ABSTRACT

Seismic safety evaluation of existing masonry arch bridges in Italy is a major concern brought

forward by recent earthquake events in 2009 and 2012. Standard computation methods for

seismic safety evaluation include static and dynamic procedures, ranging from linear static to

the more sophisticated non-linear dynamic analysis. The difficulty concerning historic masonry

lies in the determination of representative material properties and the definition of an

appropriate constitutive law, taking into account that masonry, formed as a composite of units

and mortar, has some intrinsic complex mechanical properties especially in the nonlinear

range.

The present work determines the influence of different constitutive material laws and non-

elastic parameters for masonry on the seismic performance level of an existing triple-arched

masonry arch bridge built after World War II close to the village of San Marcello Pistoiese in

Italy. Seismic evaluation has been carried out with the help of DIANA FE software applying the

method of static pushover analysis; additionally a nonlinear dynamic analysis has been

conducted.

The different constitutive models applied include a simple Drucker-Prager criterion and a

model combining Drucker-Prager in compression and smeared cracking in tension;

furthermore two additional smeared cracking models which combine compressive crushing

and tensile cracking incorporated in the FE code were analysed. The variation of the material

parameters has been conducted following references found in relevant literature.

It will be shown how different laws of energy dissipation and the variation of parameters in

order to describe this dissipation qualitatively change the bridges behaviour regarding seismic

impact. The study points the necessity of experimental data in the non-linear range for the

correct assessment of the safety status and hints how small variations on the applied

parameters can change the output of safety evaluation.

Keywords: Nonlinear seismic analysis, masonry arch bridge, constitutive material laws



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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii

RESUMEN

La evaluación de la seguridad sísmica de puentes de fábrica en Italia es una de las principales

preocupaciones puestas de manifiesto por eventos sísmicos recientes en los años 2009 y 2012.

Los métodos estándar de cálculo para la evaluación de la seguridad sísmica incluyen

procedimientos estáticos y dinámicos, desde la estática lineal, hasta el más sofisticado análisis

no lineal dinámico. La dificultad respecto a la albañilería histórica radica en la determinación

de propiedades representativas y la definición de una ley constitutiva adecuada, teniendo en

cuenta que la albañilería, formado como compuesto de unidades y mortero, tiene algunas

propiedades mecánicas intrínsecamente complejas, especialmente en el rango no lineal.

El presente trabajo determina la influencia de diferentes leyes constitutivos y parámetros

inelásticos de materiales de fábrica sobre el nivel de desempeño sísmico de un puente

existente de fábrica en tres arcos, construido después de la Segunda Guerra Mundial cerca del

pueblo San Marcello Pistoiese en Italia. La evaluación sísmica se llevó a cabo con la ayuda del

programa de elementos finitos DIANA aplicando el método de análisis estático no-lineal;

adicionalmente se realizó un análisis dinámico no-lineal.

Los diferentes modelos constitutivos aplicados incluyen un simple criterio de Drucker-Prager y

un modelo que combina Drucker-Prager en compresión con craqueo manchado en tensión;

por otro lado fueron analizados otros dos modelos de craqueo manchado que se componen de

un comportamiento de trituración en compresión y craqueo en tensión, incorporados en el

código de FE utilizado. La variación de los parámetros de los materiales se llevó a cabo

siguiendo referencias de literatura pertinente.

El estudio demuestra cómo las diferentes leyes de disipación de energía y la variación de los

parámetros aplicados para simular esta disipación de forma cuantitativa cambian el

comportamiento del puente ante las acciones sísmicas. El estudio señala la necesidad de

obtener datos experimentales en el rango no-lineal para el análisis estructural correcto y cómo

pequeñas variaciones en los parámetros aplicados cambian el resultado de la evaluación de

seguridad.

Palabras clave: Análisis no-lineal sísmica, puente de fábrica en arco, leyes constitutivas de

albañilería



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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS ix

TABLE OF CONTENTS

ABSTRACT ...................................................................................................................................... V

1. INTRODUCTION ..................................................................................................................... 1

2. MASONRY ARCH BRIDGES AND SEISMIC ASSESSMENT ........................................................ 3

3. CASE STUDY: S. MARCELLO PISTOIESE BRIDGE ................................................................... 11

3.1.BRIDGE GEOMETRY ................................................................................................... 11

3.2.FINITE ELEMENT MODELLING ................................................................................... 12

3.3.MATERIAL PROPERTIES ............................................................................................. 14

3.4.NATURAL FREQUENCIES AND MODE SHAPES ........................................................... 16

4. CONSTITUTIVE MATERIAL MODELS .................................................................................... 19

4.1.DRUCKER-PRAGER ..................................................................................................... 21

4.2.DRUCKER-PRAGER WITH TENSION SOFTENING ........................................................ 23

4.3.TOTAL STRAIN CRACK FIXED/ROTATING ACC. TO DIANA (DIANA ©, 2010) ............. 27

4.4.SOIL-BRIDGE INTERFACE ELEMENTS ......................................................................... 30

4.5.CYCLIC BEHAVIOUR ................................................................................................... 30

5. SEISMIC ASSESSMENT ......................................................................................................... 35

5.1.NONLINEAR STATIC ANALYSIS ................................................................................... 35

5.2.NSA RESULTS AND DISCUSSION ................................................................................ 39

5.3.SEISMIC PERFORMANCE RESULTS AND DISCUSSION ................................................ 58

5.4.NONLINEAR DYNAMIC ANALYSIS .............................................................................. 63

5.5.NDA RESULTS AND DISCUSSION................................................................................ 65

6. CONCLUSIONS ..................................................................................................................... 67

REFERENCES ................................................................................................................................ 72

A. ANNEX I .............................................................................................................................. A-1



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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1

1. INTRODUCTION

The masonry arch has been part of architectural heritage throughout history. Evolving from

simple post/lintel and corbeled arch structures, built as a sequence of self-stable cantilevers,

masonry arches have been widely used as a structural feature which upon applying self-load

only develops compressive stresses. The mechanical properties of masonry (low tensile

strength and high compressive strength) make it an ideal component for the construction of

arches and can be found in Roman semi-circular arches, Gothic pointed arches or Moorish

arches just to name a few of them. If the voussoir consists of natural stone like granite,

limestone or sandstone, persistency of the structure is additionally enhanced by the durability

of the material.

Cutting, centering, bedding, placing and pointing of cut-stone arches requires extraordinary

craftsmanship skills, which were replaced with more time and resource saving methods in

modern building industry. The masonry arch has changed its use from a functional solution of

spanning a structure to a more symbolic expression of decorative elements (Boothby &

Anderson Jr., 1995). However, the masonry arch, apart from spanning door or window

openings, was been widely used in still existing bridges all over the world. The number of

masonry arch bridges in Europe is estimated to approximate 200.000 individual structures

serving the railway network; 300.000 including bridges for the national road system (Brencich

& Morbiducci, 2007). Most of these structures have spans with less than 70m and consist of

natural cut-stone with mortar joints, generally lime or cement-based. Most of these bridges

were constructed in a short time frame. According to Resemini (Rota, 2004) modern Italian

masonry arch bridges were constructed between 1830 and 1930 (example in Figure 1).

Figure 1: Prarolo arch bridge (1850 – 1853) on the railway line Genoa-Turin: mixed masonry

(stone blocks and bricks), 40 m single span and two truncated-cone abutments (Lagomarsino &

Resemini, 2007)




Safety and collapse prevention of these essential elements of our modern infrastructure

system are a major concern. Recent events in Italy have shown the importance of assessing the

seismic vulnerability of existing structures. The latest Italian building code from 2008, NTC08

(CS.LL.PP., 2008), defined for the first time specific criteria for existing masonry structures

regarding the degree of uncertainty resulting from a global assessment of the structure

(geometry, constructive details, material properties) and has been a significant improvement

concerning the criteria of safety/performance assessment for masonry structures, taking into

account local mechanisms and the non-feasibility of including masonry to the conceptual

framework “ductile mechanism/brittle mechanism” applied for materials like reinforced

concrete or steel (Marcari, 2012).

On April 6, 2009, shortly after the NTC08 came into force, a major earthquake (Mw 6,3)

occurred in the centre of Italy. It was the third strongest earthquake recorded in Italy since

1972, after the 1976 Friuli (Mw 6.4), and 1980 Irpinia (Mw 6.9) and should become tragically

famous for a high number of victims and destroyed buildings especially in the city of L´Aquila.

Although the area has been known for high seismic hazard, the experimental data obtained

during the earthquake and aftershocks showed, that the acceleration spectra evaluated in the

short period range are higher than those considered by NTC08 for the collapse prevention

performance target (Sabetta, 2011). The severe damages experienced in L´Aquila together

with the even more recent earthquakes in Emilia-Romagna on May 20 and May 29, 2012 (the

strongest with Mw 5,9), will probably lead to further debates concerning the current codes and

especially seismic spectra in the near future.

The present work aims to determine the influence of different constitutive material laws for

masonry on the seismic performance level of an existing masonry arch bridge. The lack of

detailed information concerning the material properties in the nonlinear range makes the

application of different scenarios an interesting tool for safety evaluation. However it should

be reminded at this point that safety evaluation of historic buildings cannot be based solely on

the results of calculations. The ICOMOS / ISCARSAH guidelines (ISCARSAH, s.d.) clearly indicate

to combine a historical, a qualitative, an analytical and an experimental approach for safety

evaluation of historic buildings. The present study only addresses the analytical approach

through structural analysis, since the other approaches were already addressed in previous

research works Pelà et al (Pelà, et al., 2009), (Pelà, et al., 2012) and Leprotti et al (Leprotti, et

al., 2010).




2. MASONRY ARCH BRIDGES AND SEISMIC ASSESSMENT

A typical masonry bridge (Figure 2) consists of one or more arches (also denominated vaults),

depending on the width of the valley/river to be crossed, which is the key structural element

of the bridge. The vaults are in some cases covered with cornices from the outside and rest on

abutments which themselves are supported either on the ground or on piers. In order to

provide a horizontal decking, the bridge is covered with some kind of filling material, which

usually consists of granular material from the excavation zone or from the river underneath. In

domes and vaults of buildings sometimes more sophisticated filling materials like hollow

concrete spheres were applied in order to increase the stiffness without adding too much load.

However such materials are not to be found in bridges. The filling material is embedded

between the spandrel walls, usually only on the external side of the vault but sometimes

additional spandrel walls can be found internally, and topped by parapet walls for practical

reasons.

Figure 2: Axonometric section plane of a typical masonry bridge (Galasco, et al., 2004)

The influence of the apparently non-structural elements like filling and spandrel wall on the

load-bearing capacity has long been underestimated. Traditional analysis methods like static or

kinematic limit analysis only take into account vault, abutment and piers, leading to

conservative results. When it comes to vertical loading, the filling material always increases

the failure load, distributing concentrated loads from the deck of the bridge and increasing

stability by introducing initial compression to the arch (Ford, et al., 2003). However seismic

actions might have a different effect on the influence of filling material and spandrel walls.

Whereas the arched structure can be modelled in 2D for vertical loading, the hypothesis of 2D

behaviour is not applicable anymore for seismic actions leading to load-redistributions in a 3D




way (Galasco, et al., 2004). It has been shown (Rota, 2004) that a typical failure mechanism for

masonry arch bridges under seismic action consists of the overturning of the spandrel walls,

which are pushed out-of-plane by the mass of the filling.

The comparison of experimental and numerical results concerning the number of vaults

(Brencich & Sabia, 2008) and the influence of the height of the arches rise (Brencich, et al.,

s.d.), indicate their importance on the load-bearing capacity. It has been shown that the

number of vaults significantly influences the mode shapes and the damping values (Brencich &

Sabia, 2008); hence models of multi-span bridges should not be simplified by modelling only a

self-repeating part of the bridge. Regarding the height of the rise it has been shown, that the

higher the arch, the less influence will the number of vaults have on the load-bearing capacity

(Brencich, et al., s.d.). However this is true for vertical loads, the results are not valid for

seismic action.

As documented in Roca and Orduña (Roca & Orduña, 2012), the strength of arches and vaults

depends fundamentally on the geometry of the structure and its support conditions whereas

the material properties only have a minor influence. This is reflected in the fact that ancient

empirical criteria before the scientific revolution were based on mere geometric approaches,

which are still valid nowadays disregarding that they had no scientific or rational base. Later on

the first rational approaches led to graphic statics inventing the concept of thrust lines (Figure

3), which are still applied in modern limit analysis within the concept of lower bound theorem.

(a) Load condition (b) Inverted catenary model

(c) Thrust line and corresponding hinges (d) Kinematics of resulting mechanism

Figure 3: Concept of Thrust lines (Roca & Orduña, 2012)




Although geometric approaches are still valid for structural analysis of masonry arches,

modern expression of limit analysis, like the one formulated by Heyman (Roca & Orduña,

2012), (Gilbert, 2007), includes also the material strength. In case of masonry Heymans

hypotheses suggests null tensile strength, infinite compression strength and no sliding

between the stone blocks, which can be seen as the base for further development of modern

constitutive laws.

As summarized by Roca et al (Roca, et al., 2010) and Lourenço (Lourenço, 2001), an enormous

effort has been made in the last decade within the formulation of numerical methods and

programs, making an accurate analysis of complex structures feasible by applying Discrete

Element method (DEM), Discontinuous Micro-Modelling (FEM), Continuous Macro-Modelling

(FEM) or Macro blocks just to name the most popular analysis methods used particularly for

seismic assessment. However, material science has suffered a slower evolution (Lourenço,

2001) and detailed material parameters in literature, especially for historic materials, are

scarce. As explained by Roca et al (Roca, et al., 2010), recent progresses for understanding the

complex behaviour of historic masonry have been achieved by many researchers working in

this field.

Historic masonry exhibits complex mechanical phenomena characterized by its composite

nature combining the properties of a unit (brick, block or stone) with a mortar, brittle

behaviour in tension, high compressive strength and governed by friction in shear.

Furthermore masonry is an anisotropic material, requiring a high number of parameters for

the detailed description especially in the post-yielding range, which can hardly been obtained

in practice.

According to Lourenço (Lourenço, 1996), masonry can be modelled by different strategies

depending on the level of accuracy desired in the results. Figure 4 depicts the representation

of these strategies: in the detailed micro-modelling units and mortar are represented by

(separate) continuum elements whereas the unit-mortar interface is represented by

discontinuous elements; the simplified micro-models – sometimes also referred to as meso-

level - expand the units and combine the mortar and the interfaces in a discontinuous

element. Finally the macro-model approach does not distinguish between units, mortar and

interface and all the components are modelled together as a continuum. Whereas micro-

modelling helps to understand local behaviour and structural details, macro-model approaches

are more practical when it comes to the analysis of large structures.




Figure 4: Modelling strategies for masonry structures according to Lourenço (Lourenço, 1996):

(a) masonry sample; (b) detailed micro-modelling; (c) simplified micro-modelling; (d) macro-

modelling

In order to obtain parameters for macro-modelling analysis, either the parameters from the

individual components can be homogenized or a sufficiently large masonry specimen can be

described experimentally. A standard homogenization approach for the compressive strength

of modern masonry is given in Eurocode 6 (UNI ENV 1996-1 (Eurocode6), November 2005):

(1)

where fk is the characteristic strength of masonry, fb the brick strength, fm the mortar strength

and K, α and β are constants depending on the brick and mortar material and configuration.

The tensile and shear strength are derived in a similar way. However there exist more

sophisticated methods for homogenization of a self-repeating masonry unit cell, see Ref.

(Lourenço, 1996) and (Zucchini & Lourenço, 2006) for instance.

A standard experimental description of the compressive strength of masonry parallel to the

bed joints is given by RILEM, consisting of a masonry specimen with a length of minimum 2

units and a height of minimum 5 units (Pech & Kolbitsch, 2005). Still, a comprehensive

standardization of historic masonry description is a challenging task as the properties of

masonry depend a large number of factors, such as material properties of the units and

mortar, arrangement of bed and head joints, anisotropy of units, dimension of units, joint

width, quality of workmanship, degree of curing, environment and age (Lourenço, 1996).




Independently of the type of strategy (micro-, meso- or macro-level) applied, the accuracy of

the structural analysis will rely on the availability of experimental data of the materials.

Although 3D FEM approaches nowadays can incorporate 3D anisotropic non-linear constitutive

models for masonry, the reliability is suspicious based on the lack of experimental data. For

this reason practical studies often apply simplified isotropic constitutive laws defining only the

elastic modulus, compressive (and tensile) strength and Poisson´s ratio. Post-peak behaviour in

the plastic regime is often neglected. However, these simplifications can be crucial when it

comes to the safety evaluation of an historic masonry structure with intrinsic non-linear

properties. Especially life safety or collapse prevention evaluation within a seismic analysis

cannot be accomplished without taking into account the behaviour in the post-peak regime.

Regarding seismic safety evaluation, current Seismic codes like the Eurocode 8 or the Italian

NTC08 (chapter 7) suggest a variety of different analysis methods and incorporate rules for

existing and new structures together with specific regulation for different building materials,

including masonry. As explained in chapter 1, NTC08 defines specific criteria for masonry

entailing specific confidence factors which take into account the level of knowledge concerning

the structure by means of geometric parameters, morphology and material properties.

Depending on the structural characteristics of the building and/or on the trend of uncertainty

within the results admitted, seismic codes recommend to use either lateral force methods,

modal response spectrum, nonlinear static (or pushover) or nonlinear dynamic analysis. The

Italian code furthermore specifies other methods like linear and nonlinear kinematic analysis

for existing masonry structures (Marcari, 2012).

Static lateral force methods only are applicable if the building is approximately regular in plan

and in elevations regarding its mass and stiffness distribution. Although this method gives a

prior idea about the seismic behaviour of a building, it may result in very conservative and

inaccurate results. Generally speaking it is not appropriate for historic masonry structures

(Máca & Oliveira, 2012).

Nonlinear Static Analysis (NSA) consists of applying a predefined lateral load-pattern along the

height of the structure. The load is then monotonically increased by multiplication with an

incrementally increasing load-factor until the target displacement (e.g. de displacement

expected in an earthquake) is reached or until collapse of the structure. NSA is widely

recommended in most seismic codes - EC8 (UNI ENV 1998-1 (Eurocode8), December 2004),

FEMA 440 (FEMA 440, 2005) - and is the most popular method for performance-based design




applied at different design levels (Kalkan & Kunnath, 2007), (Magliulo, et al., 2007), (Mwafy &

Elnashai, 2001), (Krawinkler & Seneviratna, 1998), (Chopra & Goel, 2002), (Chopra, et al., s.d.).

As explained more detailed in chapter 5.1, the accuracy of this approach depends essentially

on the choice of the load-pattern and higher mode effects are not taken into account.

Improvements have been achieved by adaptive pushover analysis taking into account the

contribution of higher modes.

In Modal Analysis or Response-Spectrum Analysis the seismic action is represented by a

response spectrum (see Figure 5) which serves to excite the structure to be analysed. The

advantage is that this method requires very little input data and low computational effort.

However it is more suitable for modal calibration than performance-based design and fails to

capture the strong non-linearity behaviour of masonry structures (Máca & Oliveira, 2012).

Figure 5: Type 2 (near-field) response spectra proposed in Eurocode 8 for different soil types

(A=rock - E=soft soil; damping=5%) (UNI ENV 1998-1 (Eurocode8), December 2004)

The most powerful tool regarding seismic assessment is Nonlinear Dynamic Analysis (NDA) also

called Time-History Analysis. The method consists in applying a set of selected ground motion

records by means of accelerograms. The difficulty lies foremost in the selection of adequate

ground motions, as it will fundamentally influence the level of reliability of the results. EC8

suggests using 7 accelerograms and computing the average of the response quantities, in case

of using less time-histories (minimum 3), the most unfavourable response should be used. Still

the careful choice of ground motion records, whose spectral representation should match the

response spectra of the concerning code, is a sensitive task. Help is provided by recently

developed computer algorithms like REXEL (Iervolino, et al., 2010) , which overcome this




problem by applying a determinant sequence of code-matching selection criteria. REXEL

searches in the database of the European Strong Motion Database, the Italian Accelerometric

Archive or in SIMBAD for a user-determined number of real ground motion records (1, 7 or 30)

fitting a certain spectra. The advantage lies in the fact that real accelerograms are provided,

however only the combination of 7 or 30 records will easily fit a spectra, whereas the search

for a single record can be rather difficult. If only one single NDA should be performed it might

be easier to use an artificial accelerogram. Other computer freeware distributed over the

internet can help to compute such artificial time histories which will fit whatever spectrum

perfectly. The Italian SIMQE (Vanmarcke, et al., s.d.) or the Mexican “Acelerogramas Sintéticos

y Espectras de Respuesta” (Anon., s.d.) are examples for this kind of software. Still it has to be

taken into account that results from artificial time histories generally lead to non-conservative

estimation of the seismic response (Pelà, et al., 2012).

Although it has been shown that NDA provide the most reliable and accurate results (Máca &

Oliveira, 2012), (Magliulo, et al., 2007), (Mwafy & Elnashai, 2001), there are some reservations

concerning this method which are mainly related to the complexity and suitability for practical

applications due to the high computational effort.




3. CASE STUDY: S. MARCELLO PISTOIESE BRIDGE

The San Marcello Pistoiese bridge has been described in detail by Pelà et al (Pelà, et al., 2009),

(Pelà, et al., 2012) and Leprotti et al (Leprotti, et al., 2010). It concerns a triple-arched stone

bridge in the Tuscany region in a village called San Marcello Pistoiese, built after the Second

World War. Within the Italian Seismic hazard map for a return period of 475 years it is located

in zone VII in terms of a macroseismic intensity on the MCS (Mercalli-Cancani-Sieberg) scale or

classified as PGA=0,15-0,2 in terms of peak ground acceleration (see chapter 0).

3.1. BRIDGE GEOMETRY

The bridge has a total length of 72,50m, is 5,80m wide and has a height upon mean water level

of 23,25m. The main arch has a span of 21,50m and the lateral aches of 8,00m (Figure 6).

Bricks had been used for the construction of the vaults, whereas the rest of the structural

elements consist of sandstone blocks. Mortars used are lime mortar for the deeper parts of

the bridge and concrete mortar for the visible surfaces and the masonry vault courses. The

foundations are reinforced concrete footings. The fill above the vaults and in between the

spandrel walls consists of excavation material.

Figure 6: View and Geometry of the S. Marcello Pistoiese Bridge (Leprotti, et al., 2010)




3.2. FINITE ELEMENT MODELLING

An existing 3D finite element model of the bridge from STRAUS7 software (Pelà, et al., 2009)

was incorporated into DIANA software, a finite element program designed for civil engineering

purposes with an especially wide range of features for concrete, soil and masonry (TNO DIANA

BV ©, 2010).

The main advantage of using DIANA is given by a wide range of incorporated material models.

As explained before introducing advanced algorithms for constitutive models is often omitted

in standard FE programs as the materials behaviour is generally simplified for practical

applications. DIANA provides a well-appointed material library not only for structural

engineering, including elasticity and viscoelasticity, plasticity and cracking, creep and shrinkage

or liquefaction amongst many others (TNO DIANA BV ©, 2010).

The 8-node quadratic hexahedral elements in the STRAUS7 model were replaced by the similar

8-node isoparametric solid brick element HX24L in DIANA, 6-node wedge elements were

replaced by 6-node isoparametric solid wedge elements TL18L. The whole structure was

modelled with the solid brick elements with exception of the vault of the middle arch, which

was modelled with wedge elements. For the transition between rounded and straight

structure edges wedged elements were applied (see Figure 7).

Figure 7: Finite Element model (Model02)

L6TRU

TL18L

HX24L




For the determination of the natural frequencies and mode shapes, fixed abutments (all 6 DOF

restraint) and pillars were assumed for representing the boundary conditions (Model01),

whereas the nonlinear static and dynamic analysis were carried out by adding beam elements

to the abutments and pillars which allow better simulation of the soil-bridge interaction

(Model02 – see Figure 7). These beams transmit stresses and strains only in the vertical

direction, do not have any density and decrease the stiffness of the bridge. L6TRU elements

were applied for the simulation of the beams, which are two-node directly integrated truss

elements with a total of 6 DOF: three translations at each end of the truss. The different

elements applied are shown in Figure 8.

HX24L element TP18L element L6TRU element

Figure 8: Finite Elements applied in the model (Model01 and Model02). All the nodes have 3

DOF, one for a translation in each direction of the local coordinate system; the integration

scheme is explained in the DIANA User´s Manual (TNO DIANA BV ©, 2010)

The boundary conditions for Model02 consist of restraint horizontal displacements at the base

of the bridge (at the same location where in Model01 all 6 DOF were restraint = top of the

beam elements) and all 6 DOF restraint at the bottom of the beam elements. Although all

displacements with exception of translation in vertical direction are restraint at the base of the

bridge, the choice of the L6TRU element is based on the fact that displacements in the other

directions are possible from the element point of view; hence computational errors are

avoided in three-dimensional dynamic and geometrically nonlinear analysis.

For the sake of curiosity of the author incited by the DIANA User´s Manual (TNO DIANA BV ©,

2010), an additional model with 20-node and 15-node solid elements respectively was built

(Model03), in order to verify if the intrinsic shortcomings of elements with linear integration,

like parasitic shear and volumetric locking, would affect the results of the pushover analysis.

Therefor CHX60 and CTP45 elements were applied which are based on quadratic interpolation

(Figure 9).




HX24L element TP18L element L6TRU element

Figure 9: Finite Elements applied in Model03. All the nodes have 3 DOF, one for a translation in

each direction of the local coordinate system; the integration scheme is explained in the

DIANA User´s Manual [4]

3.3. MATERIAL PROPERTIES

The materials mechanical properties had been determined by compression and splitting tests,

leading to the elastic parameters fc, ft and Young´s Modulus as described by Ref. (Pelà, et al.,

2009), (Pelà, et al., 2012) and (Leprotti, et al., 2010). The mortar had been assessed by a

penetrometric mechanical in-situ test. Additionally suggestions from the Italian Guidelines had

been adopted by these authors for the backfill and the bricks. The material properties applied

in this study agree with the articles cited and are summarized in Table 1. For other than

hydrostatic pressure dependent material models (like the two-parameter Drucker-Prager

model), the friction angle φ and the cohesion c need to be converted into yield values fc and ft,

which is achieved by rearranging the yield function of the Drucker-Prager model according to

Chen (Chen & Han, 1988):

( )

( )

Regarding the tensile strength, two additional analyses have been carried out assigning ft as

1/2, respectively 1/4 of the tensile strength involved in the Drucker-Prager criterion.

For the simulation of inelastic material behaviour, several constitutive models have been

applied which are described in chapter 4. Additional parameters are needed for those models

which incorporate quasi-brittle hardening/softening behaviour, in order to describe the

stiffness change of the material after exceeding of the yield value. Owing to the lack of




experimental data, several scenarios have been analysed changing parameters for inelastic

behaviour in compression, tension and shear.

The parameters featuring the hardening/softening function considered in this study are: the

fracture energy Gc and a hardening/softening function for compression, the fracture energy Gf

and a softening function for tension and the shear retention factor ß. The fracture energy is

defined to be the integral of the stress-displacement diagram for uniaxial stress and can also

be explained to be the energy necessary to create a unit area of a fully developed crack

(Lourenço, 1996). The shear retention factor accounts for the residual strength (or friction)

between the two surfaces of a crack (Scotta, et al., 2001) (further explanations in chapter 4.2).

The inelastic parameters (Table 2) have been chosen following the results of several authors

(A. Zucchini, 2007), (Guineaa, et al., 2000), (Selby & Vecchio, 1997) and (Šejnoha, et al., s.d.).

Very little literature can be found with meaningful values for the fracture energy and even less

for the post-yielding shear behaviour, due to the lack of experimental data. In spite of existing

procedures for its determination (Šejnoha, et al., s.d.), the heterogeneity of (historic) masonry

and difficulties in the homogenization approaches for the masonry unit-cell based on the

properties of the individual properties of mortar and unit, cause a slower progress in

investigation concerning the mechanical parameters than computational models.

Table 1: Elastic mechanical Parameters adopted according to Ref. (Pelà, et al., 2009), (Pelà, et al., 2012) and (Leprotti, et al., 2010)

Material γ E ν φ c

[kg/m3] [MPa] [-] [deg] [MPa]

Masonry of stone and lime mortar (buttress, spandrel walls, abutments, parapets)

2200 5000 0,2 61 0,58

Masonry of stone and concrete mortar (arch cornice)

2200 6000 0,2 61 0,58

Masonry of bricks and concrete mortar (vaults)

1800 5000 0,2 55 0,35

Backfill 1800 500 0,2 20 0,05

Table 2: Inelastic parameters studied

Material Gc Gf ß

[N/m] [N/m] [-]

All 1500 25 50

100

0,01 0,1




For the identification of the natural frequencies and mode shapes, a dynamic modulus has

been applied, which was set to be double of the static Young´s modulus (Pelà, et al., 2009). As

explained in Ref. (Mockovčiaková & Pandula, 2003) and (Olsen, et al., 2004) this difference is

based on the fact that the elastic modulus is determined from a long-term point of view,

applying a relatively high stress during a time of several minutes during its evaluation in a

laboratory experiment, whereas the dynamic modulus, important for short-term loading like

earthquakes or other dynamic impact, is governed by low stresses lasting for only

microseconds in a non-destructive test.

3.4. NATURAL FREQUENCIES AND MODE SHAPES

The first six mode shapes for Model01 (fully restraint for abutments and pillars) with its natural

frequencies and mass contributions are shown in Table 3. The first two Modes are transversal

bending, Mode 3 has a longitudinal shape, Mode 4 involves torsional bending of the deck,

Mode 5 is a vertical movement of the deck and Mode 6 shows a torsional behaviour

implicating the whole bridge. Model02 and Model03 (with beam elements between the piers

and the soil) exhibit slightly lower frequencies due to the reduced stiffness (Mode 1 then has a

natural frequency of 2,39Hz instead of 4,00Hz).

As explained by other case studies of masonry arch bridges - Sevim et al (Sevim, et al., 2011);

Sevim et al (Sevim, et al., 2011); Bayraktar et al (Bayraktar, et al., 2010); Pérez-Gracia et al

(Pérez-Gracia, et al., 2011) - dynamic identification and model updating are an indispensable

task within whatever kind of structural analysis. Vibration tests on the bridge provide the

necessary dynamic parameters for the correct implementation of characteristics in the

numeric model. Model updating consists of constantly tuning the model parameters (material

properties and boundary conditions) until the dynamic response of the numeric model fits the

experimental results. For the present study dynamic identification results and model updating

has been provided by Pelà et al (Pelà, et al., 2009), (Pelà, et al., 2012) and Leprotti et al

(Leprotti, et al., 2010). The results in Table 3 correspond to the experimentally obtained

frequencies.




Table 3: Model01 - Frequencies, Modal Shapes and Participating Mass (Numbers in brackets

correspond to the results obtained by Ref. [30] applying STRAUS7 software; x indicates

longitudinal, y vertical and z transversal direction)

MODE 1 MODE 2

4,00 Hz (4,00) 7,59 Hz (7,59)

Participating mass in x: ≈0%; in y: ≈0% Participating mass in x: ≈0%

in z: 48,03% in y: ≈0%; in z: ≈0%

MODE 3 MODE 4

12,00 Hz (11,99) 13,31 Hz (13,27)

Participating mass in x: 66,91% Participating mass in x: ≈0%; in y: ≈0%

in y: ≈0%; in z: ≈0% in z: 15,66%

MODE 5 MODE 6

13,93 Hz (13,91) 16,94 Hz (16,87)

Participating mass in x: ≈0% Participating mass in x: ≈0%

in y: 13,19%; in z: ≈0% in y: ≈0%; in z: 2,06%




4. CONSTITUTIVE MATERIAL MODELS

Continuum models for (historic) masonry correctly representing its complex mechanical

behaviour have to combine the challenging task of both stabile numerical description,

overcoming algebraic difficulties like singularities in the yield surface and comprehensive

experimental data.

Concerning the numerical description, a model addressing the materials anisotropic nature

should be available. Anisotropy is given by the geometric arrangement of bricks and mortar,

although these components are isotropic from an individual point of view. Furthermore

different yield criteria have to be established for tension and compression, where tensile

yielding leads to a localized cracking whereas a compressive failure is characterized by more

dispersed crushing and shear failure is dominated by cohesion; hence the anisotropy is even

more amplified in the post-peak regime.

Lourenço (Lourenço, 1996) developed an orthotropic yield surface for plane stress combining a

Hill-type criterion in compression with a Rankine tension cut-off (see Figure 10 for 2D

representation and Figure 11 for 3D showing the two failure surfaces combining similarly an

orthotropic Rankine with an orthotropic Faria criteria (Pelà, et al., 2011)) applying the post-

peak behaviour explained before (tension softening, compressive hardening/softening), which

agrees with experimental results obtained. The material axes are defined parallel and

perpendicular to the bed-joints. Apart from the elastic (anisotropic) parameters, seven

strength parameters (ftx, fty, fmx, fmy, α, β, γ) and five inelastic parameters (Gfx, Gfy, Gfcx, Gfcy, κp)

are necessary for complete description of Lourenço’s model. The first four are the strength in

tension and compression in both directions, α determines the shear stress contribution to

tensile failure, β couples the normal compressive stresses, γ controls the shear stress

contribution to compressive failure, Gf is denoted the fracture energy in tension, Gfc in

compression and κp specifies the equivalent plastic strain corresponding to the peak

compressive strength; detailed explanation in Ref. (Lourenço, 1996). This means that a variety

of different tests are necessary which have to be performed under displacement control in

order to obtain the inelastic parameters.

The model developed by Lourenço (Lourenço, 1996) is introduced into DIANA referred to as

Rankine-Hill Anisotropic and is cited within the DIANA User Manual (TNO DIANA BV ©, 2010)

to be the most appropriate model for masonry. However within the scope of this study its




application has been precluded due to the lack of experimental data of the materials in the

bridge.

Figure 10: Composite yield surface with iso-shear stress lines proposed by Lourenço (Lourenço,

1996). Different strength values for tension and compression along each material axis

Figure 11: Composite failure surface combining orthotropic Rankine with orthotropic Faria

(Pelà, et al., 2011)

In the present study four different material models on the macro level were applied: a simple

Drucker-Prager model with associated flow rule (referred to as Drucker-Prager), a Drucker-

Prager model with Rankine tension cut-off and softening in the tensile regime with associated

flow (referred to as “Drucker-Prager with Tension Softening”) and two models based on total

strain (stress is defined as a function of strain) originally proposed by Vecchio & Collins and

implemented in DIANA in its 3D version following Selby & Vecchio (TNO DIANA BV ©, 2010):

one model where the stress-strain relationship is evaluated in the principle direction of the

strain vector (referred to as “Total Strain Crack Rotating”) and one model where the stress-




strain relationship is evaluated in a fixed coordinate system which does not change once

cracking is initiated (referred to as “Total Strain Crack Fixed”). Both of these models were

employed with exponential tensile softening and parabolic compressive hardening/softening.

The last three models (“Drucker-Prager with Tension Softening”, “Total Strain Crack Rotating”

and “Total Strain Crack Fixed”) are so-called smeared-crack models, where localized cracking is

simulated in a dispersed way, taking advantage of the mesh-assembly of the FE model, in order

to facilitate numeric computation (see chapter 4.2). The constitutive relations of the four

models applied are further described in the following chapters.

4.1. DRUCKER-PRAGER

The Drucker-Prager model is widely applied for soils, rocks, concrete and bricks. It takes into

account isotropic behaviour with equal yield parameters in tension and compression and

hydrostatic pressure dependency of yielding. In the three-dimensional stress space the yield

surface has the shape of a circular cone with its apex limiting the – generally low – tensile yield

strength. The yield function reads:

√ α (3)

Where J2 is the second deviatoric stress invariant and I1 is the first stress invariant and k and α

can be expressed by the means of the cohesion c and the internal friction angle φ:

√ ( ) ( ) α

√ ( ) ( )

Hence the model can be described with only two parameters c and φ and can be easily applied

for FEM applications due to the smooth shape which facilitates obtaining a numerical solution.

In case a non-associated flow rule a third parameter for the dilatancy is required. Applications

are found extensively in literature; e.g. Ref. (Pelà, et al., 2009), (Genna, et al., 1998),

(Bayraktar, et al., 2012) and (Sevim, et al., 2011).

The Drucker-Prager model entails some intrinsic simplifications if it is applied to masonry. The

model assumes isotropic behaviour before and after yielding, it fails to describe the brittle

response of masonry in tension, it overestimates the tensile strength of masonry and neither

takes into account the correct decrease of the modulus in compression until no stiffness is left




when the maximum deformation is reached. On the other hand the hydrostatic pressure

dependency of the model is suitable for quasi-brittle materials like concrete or brick, as the

material cannot be damaged at high confinements. However, very high hydrostatic pressures

introduce micro-damages in the material, not detected by the Drucker-Prager model, which

lead to stiffness losses after the structure is liberated from the hydrostatic stress state. Here,

the Drucker-Prager model neglects to incorporate the damage caused by high stress states and

additional conditions need to be defined see (Pelà, et al., 2009) and chapter 4.2 and 4.3.

Some authors prefer to model only unbounded materials and soil, like the filling material of a

masonry bridge, with the constitutive law according to Drucker-Prager (Fanning & Boothby,

2001).

The analysis included in the present study yields a Drucker-Prager model with associated flow

rule, introducing volume increases through dilatancy during loading. As explained by Chen

(Chen & Han, 1988) any yield surface open in the negative direction of the hydrostatic axis

(Drucker-Prager or von Mises) implies a volume increase accompanied by plastic flow with the

associated flow rule (Figure 12). This can lead to an exaggerated volume increase. If the

material model should not entail any volume increase at all, a non-associated flow rule with

angle of dilatancy equal to 0 must be adopted.

Figure 12: Plastic volume expansion associated with Drucker-Prager yield surface

(representation in Haigh-Westergaard stress space) (Chen & Han, 1988)




4.2. DRUCKER-PRAGER WITH TENSION SOFTENING

With the objective of overcoming the deficiencies of the Drucker-Prager model in the tensile

regime, the second model employed in this study combines a Drucker-Prager law in

compression and a tension cut-off with linear softening in the post-peak regime. In DIANA this

model software is called multidirectional fixed crack and cracking is defined for the isotropic

material as tension cut-off, tension softening and shear retention. Therefore additional

information is introduced into the model: constant tension cut-off, constant shear retention,

the softening function (within the scope of this study a linear function was applied) and three

material parameters: the tensile strength, the fracture energy in mode I (tension failure) and a

shear retention factor for failure in mode II (shear). For the tensile strength the predefined

tensile strength involved by applying the Drucker-Prager parameters was used (ft=0,3MPa). For

comparison reasons, additional analysis were carried out applying ft=0,15MPa and

ft=0,075MPa. For the fracture energy the parameters applied were 25, 50 and 100N/m, for the

shear retention factor values of β = 0,01 and 0,1 were analysed (Table 2).

The tension softening model (Figure 13) takes into account the damage produced after

exceeding the yield stress in tension. In the post-peak regime the stiffness decreases gradually:

After every load-step the crack stiffness is computed by (TNO DIANA BV ©, 2010)

[

(

)

] ( )

Where DI is the crack stiffness, fnn the (linear) tension softening function and εcrnn is the crack

strain, which is defined to be (TNO DIANA BV ©, 2010):

(6)

The unloading-reloading path moves along the secant of the function, simulating a closing and

reopening of the crack, in case of further opening the stiffness will decrease.

The ultimate crack strain is supposed to be a material constant and is given by (TNO DIANA BV

©, 2010)

( )




Figure 13: Multidirectional fixed crack model (TNO DIANA BV ©, 2010) with Drucker-Prager

and Tension Softening - Behaviour of material with softening under uniaxial tension (ft denotes

tensile strength; Gf is the fracture energy in mode I)

Where Gf is the tensile fracture energy and h is the crack bandwidth. As explained by Lourenço

(Lourenço, 1996) using a smeared-crack model in a FE calculation means that the material

fracture energy Gf has to be normalized according to an equivalent length h in order to obtain

mesh-objective results regarding the mesh refinement. In a FE model the equivalent length

(=crack bandwidth h) is related to the mesh adopted and depends on the elements type, size,

shape, integration scheme, etc. For the models applied within this study, DIANA assumes the

default crack bandwidth h to be the cubic root of the volume (all of the elements are solid

elements).

The inelastic work gf, which corresponds to the area under the stress-strain diagram for

uniaxial loading (see Figure 13) (Lourenço, 1996):

∫ ∫ ( )

Is then related to the fracture energy by (Lourenço, 1996)1:

( )

The failure surface for mode II (shear - Figure 14) is computed in a similar way as for mode I:

after exceeding the elastic regime the decrease of stiffness depends on the prior load-history.

In case of unloading the previous secant-stiffness is applied, in case of loading the stiffness

1 The explanation concerning the concept of energy dissipation and equivalent length (“crack

bandwidth”) is also valid for the models “Total Strain Crack Rotating” and “Total Strain Crack Fixed” explained in chapter 4.3.




decreases according to the determinate stress-strain function. Although according to Lourenço

(Lourenço, 1996) experimental results show an exponential shear softening law, a linear

decrease was applied in the FEM model.

Figure 14: Multidirectional fixed crack model (TNO DIANA BV ©, 2010) with Drucker-Prager

and Tension Softening - Behaviour of material with shear retention under uniaxial shear (ft

denotes tensile strength; β the shear retention factor)

The residual strength in the τ-γ diagram is the dry friction angle (Lourenço, 1996) and (Scotta,

et al., 2001), which causes again an ascending slope of the τ-γ function beyond the minimum at

the end of the softening branch (see Figure 14). Because of the dry fiction angle, the shear

behaviour cannot be implemented into the FE code with the fracture energy in mode II (GfII),

which is the area underneath the stress-displacement area including the area of the dry

friction, and instead the factor β is applied.

For a plane stress situation the constitutive relation can finally be written as (TNO DIANA BV

©, 2010)

(10)

Figure 15 shows how μ and β are related to the stiffness parameters DIsecant and DII

secant by (TNO

DIANA BV ©, 2010)

( )

( )




Figure 15: Composition of elastic and cracking paramenters to a complete stress-strain space

Although the Drucker-Prager with added Tension and Shear Softening features improvements

regarding the pure Drucker-Prager model by taking into account the loading history and

stiffness decrease after crack initiation, it still lacks to describe more adequately the behaviour

of masonry under compression. Neither does the exponential stiffness decrease in mode II find

consideration.

Furthermore, as explained by Chen (Chen & Han, 1988), chapter 7.4: “the stress-space

formulation” – as the one presented in this chapter – “presents difficulties in distinguishing a

reduction of stress which causes additional plastic deformation and one due to elastic

unloading”, which can be overcome by the formulation in strain space as achieved by the

following Total Strain Crack models (see explanation in Figure 16).

(a) in stress space (b) in strain space

Figure 16: Loading surfaces in stress and strain space (Chen & Han, 1988)




4.3. TOTAL STRAIN CRACK FIXED/ROTATING ACC. TO DIANA (DIANA ©, 2010)

The constitutive models named TOTAL STRAIN CRACK FIXED (TSCF) and TOTAL STRAIN CRACK

ROTATING (TSCR) are based on total strain where the stress is described as a function of the

strain. Like the multi-directional fixed crack model with Drucker-Prager and linear tension

softening explained before, the total strain based crack models follow a smeared crack

approach. Loading and unloading is modelled differently with secant unloading.

TSCF and TSCR models assume the possibility of forming two orthogonal cracks in each

integration point. The basic concept of the Total Strain crack models is that the stress is

evaluated in the directions which are given by the crack directions. The Rotating Crack model

(TSCR), in which the stress-strain relationships are evaluated in the principal directions of the

strain vector, has shown to be well suited for reinforced concrete structures. However more

appropriate for most engineering purposes is the fixed stress-strain concept in which the

stress-strain relationships are evaluated in a fixed coordinate system which is fixed upon

cracking (Lourenço, 2011). Within the scope of this study both models have been analysed.

Following the explanations of Chen (Chen & Han, 1988) and also Lourenço (Lourenço, 1996),

the exponential tension softening applied in the TSCF and TSCR models reflect the post-peak

tension behaviour of masonry or concrete much better than the linear relation used in the

multi-directional fixed crack model with Drucker-Prager and tension softening. The model

shown in Figure 17 respects the fact that quasi-brittle materials have a very short interval of

stable crack propagation and a fast unstable crack-propagation after exceeding the failure

strength.

Figure 17: TSCF, TSCR - Behaviour of material with softening under uniaxial tension (ft denotes

tensile strength; Gf is the fracture energy in mode I)




The compressive uniaxial behaviour is characterized by a linear stress-strain relation until one

third of the compressive strength, followed by a parabolic relation for the hardening regime

until reaching the compressive strength and another parabolic branch for the post-peak

softening (Figure 18). As explained by Chen (Chen & Han, 1988), the hardening regime

describes the increase of the isotropic yield surface until it reaches the final failure surface,

which for materials like concrete and masonry is proportionally bigger than the yield surface.

However the surface furthermore changes its shape, depending on the lateral confinement.

The DIANA FE-Code takes this aspect into account by introducing the four-parameter Hsieh-

Ting-Chen failure surface (see (TNO DIANA BV ©, 2010) – chapter 18.2.7).

An interesting argument concerning the specimen height influence on the compression

softening branch is given by Chen (Chen & Han, 1988), chapter 7.1.2.5: when plotting the

stress-stain curves from experiments, the slope of the softening branch decreases with

increasing specimen height. However, hardly any differences can be observed when the stress

is plotted against the displacement. This is explained by the fact that the post-peak strain is

localized in a small region of the specimen, resulting in the same post-peak displacement for

all specimens regardless of their size. As the strains are calculated rather than measured

results, the different heights will results in different strains. Hence Chen (Chen & Han, 1988)

considers the softening a structural property rather than inherent to the material.

Figure 18: TSCF, TSCR - Behaviour of material with hardening/softening under uniaxial

compression (fc denotes compressive strength; Gc is the fracture energy in compression)

Concerning the shear behaviour (Figure 19) modelling is “only necessary in the fixed crack

concept where the shear stiffness is usually reduced after cracking.” The cracked shear stiffness

is then computed as




(12)

For the rotating crack concept the shear retention factor β is assumed equal to one and does

not have any effect on the results.

Similar to the multi-directional fixed crack model, high values of β are unrealistic, leading to an

inadequate increase of the residual frictional strength after vanishing of the normal stresses

and strains introduced after starting of the shear damaging.

TSCF and TSCF models were implemented into the FE model of the bridge making use of an

exponential softening behaviour in tension (Figure 17), applying fracture energies of 25, 50 and

100 N/m and a parabolic hardening/softening behaviour in compression (Figure 18) with a

single value for the compressive fracture energy of 1500 N/m determined following indications

in specific literature (A. Zucchini, 2007), (Guineaa, et al., 2000), (Selby & Vecchio, 1997) and

(Šejnoha, et al., s.d.). The stress-strain relation for shear follows an exponential path (Figure

19) employing shear retention factors β of 0,01 and 0,1.

Figure 19: TSCF, TSCR - Behaviour of material with shear retention under uniaxial shear (ft

denotes tensile strength; β the shear retention factor)

For the two-dimensional stress-strain state the constitutive relations of the cracked material

(Figure 20) are established in the local axes (n,t) as (Lourenço, 2011):

(13)




Figure 20: TSC constitutive model established in the local crack-axes (Lourenço, 2011)

The constitutive matrix for the cracked material is obtained from the tangent stiffness matrix

and the strain transformation matrix (Lourenço, 2011):

(14)

Note that α in the transformation matrix is the same angle as θ in Figure 20.

4.4. SOIL-BRIDGE INTERFACE ELEMENTS

As explained before, the soil-bridge interface has been modelled with beam elements with

tension and compression cut-off in order to avoid the nearly infinite strength at high

hydrostatic stress states according to the Drucker-Prager model without compression cap. The

different results obtained by applying a brittle or ductile stress cut-off are included in chapter

5.2.

4.5. CYCLIC BEHAVIOUR

The Drucker-Prager model and the Drucker-Prager with combined Tension cut-off and

softening have been analysed concerning its behaviour for cyclic loading. The results are

shown in Figure 22 - Figure 25. The Drucker-Prager model shows a perfectly elastoplastic

hardening for normal stresses, the Drucker-Prager with combined tension cut-off and

softening shows perfectly plastic behaviour in the compressive regime and linear softening in

the tensile regime. In tension the first load cycle follows the failure surface for static response

under monotonic loading, the crack closes upon unloading, the material then shows perfectly




plastic behaviour in compressive un- and reloading and remains anelastic strains (Mazars, et

al., 2006) upon reloading in tension. The crack can be reopened again in a following cycle if the

loading in compression has been sufficiently small; otherwise (like in Figure 22) reopening is

not possible anymore. Stiffness degradation and energy dissipation is demonstrated only in the

tensile regime.

Figure 21: Drucker-Prager under uniaxial cyclic loading in tension/compression

Figure 22: DP+TSoft under uniaxial cyclic loading in tension/compression




Figure 23: Zoom of the tensile regime of Figure 22

Concerning the shear behaviour, the Drucker-Prager model (Figure 24) follows a rigid path for

unloading and plastic reloading until the maximum compressive strength is reached. Beyond

that point the failure surface continues increasing by kinematic hardening (τnt is increasing

while -τnt is decreasing). For the Drucker-Prager with combined Tension cut-off and softening

the cyclic loading in shear follows the failure surface of the static response under monotonic

loading too, closing and reopening cracks upon unloading and reloading respectively (Figure

25). After exceeding the minimum residual shear strength, unloading/reloading follows the

residual dry friction level without suffering changes in the stiffness.

Figure 24: Drucker-Prager under uniaxial cyclic loading in shear

Figure 24 and Figure 25 show a phenomenon which has already been revealed in Figure 14 and

Figure 19 (chapter 4.2 and 4.3): For the Drucker-Prager model in DIANA the failure surface




continues increasing after reaching the initial yield surface. For the cracking model too, the

residual dry friction level is increasing linearly after exceeding the minimum residual shear

strength in the unloading path. This does not at all reflect the experimentally documented

behaviour of masonry (see examples in Figure 26) and will be commented further in detail in

chapter 6.

Figure 25: DP+TSoft under uniaxial cyclic loading in shear

(a) Analysis of masonry panels with different diagonal

tension behaviour according to Giassi et al (Ghiassi, et

al., 2012)

(b) Experimentally obtained

hysteretic loops for masonry shear

walls (Tena-Colunga, et al., (2009))

Figure 26: Documented (cyclic) shear behaviour of masonry walls




5. SEISMIC ASSESSMENT

5.1. NONLINEAR STATIC ANALYSIS

Nonlinear static analysis (NSA), also denominated pushover analysis, when employed to

estimate seismic demands for structures, consists of applying a gradually increasing horizontal

load under constant gravity loading. The resultant base shear –displacement pushover curve

can then be converted into spectral acceleration and displacements in order to allow

comparison to a spectral demand. The concept is based on the assumption that the response

of the structure can be defined by the response of an equivalent SDOF system. As explained be

Krawinkler (Krawinkler & Seneviratna, 1998) this implicates that the structure´s response is

characterized by a single mode and that the shape of this mode remains constant, thus

introducing strong simplifications in the analysis. However in case the structure is dominated

by one mode the analysis has proven to give acceptable results. Further improvements can be

achieved by applying a variety of load patterns as explained below.

When it comes to the necessity of combining a variety of load patterns, similar conclusions

have been drawn by Kalkan (Kalkan & Kunnath, 2007) or Chopra and Goel (Chopra & Goel,

2002) amongst others; the later developed a modal pushover analysis including all modes of

vibration with significant contribution to the seismic demand (normally the first two-three

modes). Posterior a modified modal pushover analysis has been presented by Chopra, Goel

and Chintanapakdee (Chopra, et al., s.d.), where the structure is assumed to be linearly elastic

when computing the response contributions of higher modes. The authors suggest that under

intense excitation the inelastic response of the structure is only essential in the first mode

however, can be neglected for the higher modes. Still the error introduced can be significant

for structures with high ductility.

Regarding the purpose of application, NSA is especially useful for demand predictions at low

performance levels when the structure is supposed to undergo severe non-elastic

displacements. Hence the results are helpful for safety assessment and collapse prevention

(Krawinkler & Seneviratna, 1998). It should be taken into account that the NSA bears

uncertainties introduced by the model, the applied load pattern and the demand spectra.

Furthermore the arbitrary control node selection affects the results considerably as explained

by (Pelà, et al., 2009), (Pelà, et al., 2012), and (Leprotti, et al., 2010). This effect can be




overcome by expressing the results in terms of an energy-displacement curve and comparing

them with the energy demand spectra as described by Mezzi (Mezzi, et al., 2006).

Following standard procedures as explained in Eurocode8 (UNI ENV 1998-1 (Eurocode8),

December 2004) or FEMA 440 (FEMA 440, 2005) analysis should be carried out applying at

least two vertical distributions of the lateral loads. According to Eurocode8 a uniform pattern

based on the lateral forces that are proportional to mass and one modal pattern, where the

forces are proportional to the lateral force distribution calculated in the linear-elastic analysis.

Other possible load patterns include concentrated load, triangular, first mode, code

distribution or adaptive load, as explained in FEMA 440 (FEMA 440, 2005).

Within the scope of this study several pushover analyses have been performed making use of

the DIANA nonlinear static solver algorithm. The purpose was to identify the influence of

different constitutive material models, which were explained in chapter 4 on the performance

evaluation carried out by NSA.

The procedure consisted in applying the self-load in a first step (actually it was applied in two

steps each attaching 50% of the self-load, but one would yield the same result) and

subsequently adding incrementally horizontal forces proportional to the mass distribution in z

direction (= transversal direction). No other load pattern has been applied, disregarding the

recommendation in (Krawinkler & Seneviratna, 1998), (UNI ENV 1998-1 (Eurocode8),

December 2004), (FEMA 440, 2005). Concerning the conclusion of Krawinkler (Krawinkler &

Seneviratna, 1998) that structures vibrating in one predominant mode will more likely show

accurate estimates for the global and local inelastic deformation, it should be noticed that the

S. Marcello Pistoiese bridge has a predominant mass participation in transversal direction of

48,03% in the first mode. Consequently the failure mechanism will take place for forces in

transversal direction, justifying the employed load pattern in this direction. However the third

mode exhibits 66,91% in the longitudinal direction and another 15,66% are excited in the

transversal direction in mode 4 (Table 3). As the present study only deals with the response in

transversal direction, i.e. the most vulnerable one, the mass participation in the longitudinal

direction can be neglected, although it should find consideration if the global seismic response

of the bridge must be evaluated. Additionally the 4th mode mass participation could be taken

into account by applying load patterns that account for higher mode effects.




It should be made clear at this point, that the procedure followed within the scope of this

study neglects the requirements of the Italian building code NTC08 (CS.LL.PP., 2008), where

the modal load pattern should take into account the dominant mode in the considered

direction which incorporates an effective modal mass participation of at least 75%. A modal

mass participation of 75% might be achieved with the dominate mode for a structure with

lumped masses like a building with high mass contribution by the slabs but literally no mass

contribution from the elements in between; however, reaching 75% of effective modal mass

will be much more complicated for structures with dispersed mass distribution like a bridge.

For the S. Marcello Pistoiese bridge the highest modal mass participation for the predominant

(1st) mode in transversal direction (z-direction) accounts to 48,03% as explained before.

Regarding the numerical implementation of the NSA, geometrical and material nonlinearities

have been taken into account. The applied load steps varied from 0,1 to 1e-10 multiplied by

the mass distribution; the analysis was stopped if no further solution was found after

decreasing the load steps to 1e-10. The loading was energy, force and displacement

controlled; iterations were performed with regular Newton-Raphson method integrating arc-

length method and Line search. The maximum number of iterations was 150 for the model

with 8 respectively 6-node elements, whereas only 30 iterations were allowed for the 20

respectively 15-node model due to the need of time saving. The average time to operate an

analysis with a 2,3GHz processor was 90 minutes for the Drucker-Prager models with tension

cut off and softening, 5 hours for the TSCF and TSCR models and several days for the analysis

performed with the 20node elements. The later was stopped after the 5th load step (which

took 24 hours of analysis) as convergence for the 6th load step was not achieved after 7 days of

computation. A detailed list of all the different analysis performed is given in Table 4.




Table 4: Material Models and parameters of all the NSA performed

Material Model Gf [Nm/m2] Gc[Nm/m2] β [-] Soil-Bridge transition

stress cut-off

Drucker-Prager - - - Ductile

Brittle

Drucker-Prager with Tension Cut-off (ft=0,3MPa)

and Softening

100

-

0,1 Ductile

Brittle

0,01

Ductile 50

0,1

0,01

25 0,1

0,01

Drucker-Prager with Tension Cut-off

(ft=0,15MPa) and Softening

50

- 0,01 Ductile

25

Total Strain Crack Fixed

100

1500

0,1 Ductile

Brittle

0,01 Ductile

Brittle

50 0,1

Ductile 0,01

25

0,1

0,01 Ductile

Brittle

Total Strain Crack Rotating

100

1500

0,1

Ductile

0,01

50 0,1

0,01

25 0,1

0,01




5.2. NSA RESULTS AND DISCUSSION

The influence of the different constitutive models on the bridges seismic performance

determined by the nonlinear static analysis has foremost been determined by comparing the

pushover curves for arbitrarily chosen control nodes (Figure 27). Instead of showing the typical

displacement - base shear curves, it was chosen to use directly the displacement-acceleration

curves in order to facilitate further comparisons with response spectra and nonlinear dynamic

analysis results (see chapter 0 and 5.5). Node 3027 and 2183 are situated on the keystone at

the longitudinal axis of the side arch and the middle arch respectively. Node 5774 is situated at

the façade on the top of the parapet and the nodes 5447 and 5605 are located at the façade,

the first between the pier and the cornice of the middle arch and the second at the abutment

close to the middle arch, very close to the centre of mass of the structure. Node 5774

describes a local response of the parapet and is not included in the conclusions.

Figure 27: Control Node Position - Solid line around the number indicates node position on the

facade, dashed line indicates a position in the middle of the arch (longitudinal axis of the

bridge)

For all the analysis performed node 5774 showed the highest displacements at the last load-

step applied, followed by node 2186, 3027, 5605 and 5447 in this sequence. Only for the

Drucker-Prager model node 5605 shows a slightly higher displacement than node 3027. It is

worth mentioning that node 5774 shows displacements at the last load-step which are more

than three times higher than the displacements of node 5447, node 2186 bears an increase of

approximately 2,5 regarding to node 5447 and the nodes 3027 and 5605 still show results

5774

5447

2186

3027

5605




which are approximately 50% higher than node 5447. This pattern can be found in all the

analysis performed. The control node sensitivity of the results which has been mentioned

before is therefore proven.

The pushover curves obtained with the Drucker-Prager material model are shown in Figure

28.2 They are perfectly smooth curves following closely a second order polynomial function.

After an acceleration of 5,45m/s2 proportional to the mass distribution, no further

convergence was found, with displacements in the range of 0,034 to 0,0102m.

Figure 29 illustrates the pushover curves for the TSCF material model. Typically for all the

results with exception of the Drucker-Prager model, the curves indicate a moment of

unloading shortly after exceeding the elastic path, followed by a strongly non-linear behaviour

and a second snap-through soon before no further convergence could be reached. In case of

the TSCF model with Gf=50N/m and β=0,01 the displacements for the maximum acceleration

of 2,32m/s2 were in the range of 0,0224 to 0,0641m.

Figure 28: Pushover curves with Drucker-Prager model (ductile soil-bridge interface cut-off)

An interesting phenomenon confirming the suggested failure mechanism commented later in

this chapter, can be observed by examining the direction of the capacity curves of node 5605

and 3027. With exception of the Drucker-Prager model, the curves of node 3027 are always

stiffer than the one of node 5605 until the moment of the first snap-through shortly after the

elastic path. Afterwards the curve of the node above the side-vault (node 3027) is always

2 The pushover curves shown for the Drucker-Prager model were obtained with FEM code STRAUS,

applying self-weight with one load-step followed by 38 force controlled load-steps from 0,5m/s2 to the

final 5,45m/s2




ahead of the curve of the node at the abutment (node 5605). The results summarized in

chapter 5.2 will show that this is caused by intense cracking of the side vaults which finally

determines the failure mechanism.

The individual detailed results can be consulted in the Annex I of this document.

Figure 29: Pushover curves with TSCF model (Gf=50N/m; β=0,01; ductile soil-bridge interface

cut-off)

Figure 30, Figure 31 and Figure 32 show the results of the NSA for control node 5605 for

Drucker-Prager combined with Tension cut off and softening (DP+TSoft), Total Strain Crack

Fixed (TSCF) and the Total Strain Crack Rotating (TSCR) material models. Without adding new

information, Figure 33, Figure 34 and Figure 35 demonstrate supplementary comparisons

relating the different material models with the same tensile and shear parameters. All the

graphs show the Drucker-Prager model results as a reference.

The Drucker-Prager model obviously overestimates the seismic elastic capacity when

compared to the other material models. However, the response is not more ductile than the

others (with exception of the TSCR model and low shear retention factors). The results in

terms of ductility show good agreement with the study from (Pelà, et al., 2009), where a

Drucker-Prager model had been applied. From Figure 30 arises suspicion that the acceleration

(respectively base shear) supported from the Drucker-Prager model and a combined Drucker-

Prager model with tension cut off, softening, high tensile fracture energy and shear retention

factor is similar close before the collapse of the structure. However, the Drucker-Prager model

predicts a much higher collapse load than other models studied (Figure 31 and Figure 32).




Within the range of parameters studied, the softening in shear has a much higher influence on

the seismic capacity than the tensile fracture energy. For all the models applied, an increase of

the shear retention factor β from 0,01 to 0,1 lead to higher capacities than the increase of the

tensile fracture energy from 25N/m to 100N/m. Evidently β has no influence in the TSCR

results as the stress-strain relationships in this material model are evaluated in the direction of

the principle strains, changing the direction of the cracks after each evaluation. Consequently

application of the shear retention factor does not lead to any changes in the results, q.e.d.

Figure 33, Figure 34 and Figure 35 show that the TSCR model shows similar capacity curves as

the TSCF model when the shear retention factor of the later is set to be 0,1 but comes closer to

the results of the DP+TSoft model when β=0,01. However, for all the results obtained the

capacity curves of the TSCR model are much too short to allow coherent evaluation.

Figure 30: Influence of tensile fracture energy (Gf in N/m) and shear retention factor (beta) on

the pushover results for DP+TSoft model (ductile soil-bridge interface cut-off) – Node 5605





the pushover results for the TSCF model (ductile soil-bridge interface cut-off) – Node 5605


the pushover results for the TSCR model (ductile soil-bridge interface cut-off) – Node 5605




β=0,1 β=0,01

Figure 33: Influence of the constitutive model3 on the pushover results with Gf=100N/m

β=0,1 β=0,01


β=0,1 β=0,01


The smoothness of the curves is governed by the material model applied and the parameters

designated, leading to a higher amount of snap-throughs in the TSCF and TSCR than the

DP+TSoft model. The lower the fracture energy and the lower the shear retention factor, more

snap-through incidents occur during the loading.

3 Drucker-Prager without nonlinear parameters added for comparison reasons




With the aim of assessing the effect of the bridge-soil interface, additional NSAs have been

performed, modelling a brittle stress cut-off of the beams simulating the interface. The

corresponding pushover curves are shown in Figure 36 together with its corresponding

counterparts with ductile stress cut-off. The curves show a similar path, however the models

with brittle behaviour do not find further solution as soon as the first beam loses its stiffness.

Figure 36: Influence of the bridge-soil interface stress cut-off on the pushover results – Node

5774

In order to understand the following graphs, it must be mentioned that “Abutment” refers to

the abutment supporting the main arch; “Abutment 2” refers to the two small side abutments

holding the side arches. “Vault” together with the outer stone “Cornice” is the ring of the main

arch, whereas “Vault 2” and “Cornice 2” form together the two smaller side arches. “Filling”

refers to the filling material close to the spandrel walls, whereas “Filling 2” is the same

material in the core of the “Filling” (Figure 37).

Figure 37: Structural components denomination




Explanation of the internal degradation of the structure during the loading process is given in

Figure 38, Figure 39 and Figure 40. The graphs show the percentage of cracked elements in

relation to the total amount of elements in direction 1 and 2 (direction 3 has been omitted as

the graphs show a similar trend like for direction 3) at different load-steps: first after applying

only the self-load, second at the load-step (LS) exactly before the first snap-through (LS 13 in

Figure 38, LS 7 in Figure 39 and for the TSCR model LS 10 was chosen for comparison reason

because no snap-through occurs in the corresponding pushover curve), third the load-step

exactly after the first snap-through (LS 14 in Figure 38, LS 8 in Figure 39 and for the TSCR

model LS 13 was chosen for comparison), for the TSCF model there follow two more load-steps

(LS 29 and LS 30 in Figure 39) for the second snap-through observed close to the final loading

(see Figure 29) and finally the last converged load-step is shown. The amount of cracked

elements includes cracks and closed cracks, direction 1 means the direction of the first crack,

direction 2 is orthogonal to direction 1 and shows a crack as soon as he angle between the

existing crack in direction 1 and the principal tensile stress exceeds the value of a certain

threshold angle; Diana User´s manual (TNO DIANA BV ©, 2010); a total of three crack

directions is possible

It should be noted at this point that the principal stresses obtained are sometimes higher than

the strength of the material, which is physically impossible. The author could not find an

explicit explanation for this, however the Diana User´s manual states that with the criteria

applied for crack initiation in the FE algorithm it “is possible that the tensile stress temporarily

becomes greater than three times the tensile strength while the threshold angle condition was

still not violated” (TNO DIANA BV ©, 2010) and further refers to the PhD Thesis of Jan Gerrit

Rots concerning Computational Modeling of Concrete Fracture presented at Delft University of

Technology in 1988.

Presenting the history of the internal degradation in terms of strains was rejected due to the

fact that the ultimate strain depends on the fracture energy which itself depends among

others on the size of the elements which change throughout the structure.

Application of the self-weight already leads to the exceeding of the tensile strength in the

spandrel walls, although in only four single elements situated in the corner of the spandrel

where the bridge touches the soil of the valley. Subsequently the bridge shows a similar failure

pattern for the DP+TSoft and for the TSCF model. However the TSCR model differs from the

others.




% of cracked elements – direction 1 (Piers,

Spandrel walls, Abutments, Cornice) % of cracked elements – direction 1 (Vault) % of cracked elements – direction 1 (Filling)



Figure 38: % of cracked elements (including closed cracks) - DP+TSoft (Gf=50N/m; β=0,01)








Figure 39: % of cracked elements (including closed cracks) - TSCF (Gf=50N/m; β=0,01)




% of cracked elements – direction 1 (Piers, Spandrel walls, Abutments, Cornice)

% of cracked elements – direction 1 (Vault) % of cracked elements – direction 1 (Filling)



Figure 40: % of cracked elements (including closed cracks) – TSCR (Gf=50N/m; β=0,01)




For the TSCF model, at the end of the 7th load-step (Figure 39), before the first snap-through,

all the structural components show some cracked elements in direction 1. Still only a small

amount of elements has exceeded the tensile strength (Figure 41). At this point the

component with the highest damage is Vault 2 with 35% cracked elements. The following load-

step (total acceleration at this point is 1,32m/s2) results in an exceeding of the compressive

strength of the filling material and of Vault 2. At this moment the structure initiates the non-

linear behaviour. At load-step 29 (total acceleration at this point is 2,27m/s2), though most

parts are still sound in compression, wide parts of the structure are cracked. Especially the

small side arches (Vault 2 and Cornice 2) are already heavily damaged (Figure 42). Also the

filling material (Filling and Filling 2) has exceeded its tensile and compressive limits in the area

above the side arches. From load-step 29 to 30 the second snap-through occurs, leading to an

unloading branch in all the elements of the structure. Further loading leads to further cracking

especially in the area of the side arches (Vault 2, Cornice 2, Filling and Filling 2 above the side

arches). Additionally a sudden increase of the principal stresses in the side abutments

(“abutment 2”) and the piers leads to the final failure. The most severe cracking had then

suffered the cornice and the vaults of the side arches (“cornice 2” and “vault 2”) with 78%

(“cornice 2” and “vault 2”) of cracked elements in one direction and 63% (“cornice 2”),

respectively 68% (“vault 2”) additional cracking in direction 2. The principal stress situation for

the last converged load-step is documented in Figure 43 and Figure 44.

Figure 41: Contour Plot - principle tensile stresses after load-step 7 (1,43m/s2). TSCF -

Gf=50N/m; β=0,01 (blue: <0,3MPa; red: ≥0,3MPa)




blue: <0,3MPa; red: ≥0,3MPa Vaults - blue: <0,221MPa; red:

≥0,221MPa

Figure 42: Contour Plot - principle tensile stresses after load-step 29 (2,27m/s2). TSCF - Gf=50

N/m; β=0,01


≥0,221MPa

Figure 43: Contour Plot - principle tensile stresses after last converged load-step (2,32m/s2).

TSCF - Gf=50N/m; β=0,01

principle tensile stresses - blue: <0,07MPa;

red: ≥0,07MPa principle compressive stresses - red: >-

0,143MPa; blue: ≤-0,143MPa

Figure 44: Contour Plot - principle stresses in the filling after last converged load-step

(2,32m/s2). TSCF - Gf=50N/m; β=0,01




Concerning the TSCR model, the amount of cracked elements is much lower in all the

structural components (Figure 40). It has to be taken into account that convergence was only

reached until the final “loading-acceleration” of 1,99m/s2 and that the cracks continuously

rotate during the loading, hence the percentage of cracked elements is much smaller than for

the DP+TSoft or the TSCF model. Generally the TSCR model shows about 50% less cracks than

the TSCF model in direction 1, however this is not true for the vaults and the cornices, where

the amount of cracked elements is nearly the same (hardly any cracks for “vault” and “cornice”

and high amount of cracks for “vault 2” and “cornice 2”). This proves that for all the models

the predominant cause of failure is given by severe cracking of the side arches. The principal

stress situation for the last converged load-step is documented in Figure 45 and Figure 46.


≥0,221MPa


TSCR - Gf=50N/m; β=0,01

principle tensile stresses - blue: <0,07MPa;

red: ≥0,07MPa principle compressive stresses - red: >-

0,143MPa; blue: ≤-0,143MPa

Figure 46: Contour Plot - principle stresses in the filling after last converged load-step

(1,99m/s2). TSCR - Gf=50N/m; β=0,01




Figure 47 shows the deformed mesh (TSCF) at the last load-step, clearly indicating mode 1

vibration. All the models show a similar deformation, proving that the way of loading applied

in this study excites the dominant mode shape. The shape already gives a hint about the failure

mechanisms of the bridge, which is even better indicated observing the crack-strains at the last

load-step.

Figure 47: Shape Plot – total deformation (qualitative) after last converged load-step

(2,32m/s2). TSCF - Gf=50N/m; β=0,01

Figure 48, Figure 49 and Figure 50 exhibit the crack strains of the TSCF and TSCR model (as

explained before, the DP+TSoft model shows a similar behaviour as the TSCF model and is

therefore not further described). The strains are approximately 5 times higher in the TSCF than

in the TSCR model. Due to the lowest tensile strength of this material, the highest strains

appear in the filling material. For the TSCF model predominant cracking occurs at two radial

lines close to the keystone of the side arches (which coincides with the elevated stresses in this

area) and at the bottom of the piers, indicating out-of-plane overturning of the separated part.

However the TSCR model shows one damage line at each side arch (together with a slight toe-

crushing at the bottom of the piers), close to the abutment which probably caused the lack of

further convergence in the analysis. The failure mechanism for this model seems to be less

“dramatic” than for the TSCF model showing a cut-off of the bridge at this damage lines.




(a) TSCF - 2,32m/s2 (b) TSCR - 1,99m/s2

Figure 48: Vector-Plot - Crack-Strain at the last converged load-step (Gf=50N/m; β=0,01)

Figure 49: Deformed Contour-Plot - Crack-Strain of the TSCF model (Gf=50N/m; β=0,01) at the

last converged load-step (2,32m/s2) - (maximum crack strain in the filling: 4,13E-02)

Figure 50: Deformed Contour-Plot - Crack-Strain of the TSCR model (Gf=50N/m; β=0,01) at the

last converged load-step (1,99m/s2) - (maximum crack strain in the filling: 8,16E-03)




In order to examine the influence of the tensile strength, additional analyses have been

performed applying ft=0,15MPa and ft=0,075MPa (which equals to ½, ¼ respectively of the

tensile strength entailed by the Drucker-Prager parameters employed – 0,3MPa). In one case

the fracture energy was set to be the same as for the analysis with ft=0,3MPa, decreasing the

strength but maintaining the ductility, and in another case the fracture energy was set to

decrease so that the ultimate strain will be the same as for the analysis with ft=0,3MPa.

For the model with ft=0,075MPa the NSA could not be performed, as the bridge will fail before

applying 100% of the self-load. The results for the models with ft=0,15MPa are shown in Figure

51 in comparison with the pure Drucker-Prager results and the pushover curve with ft=0,3MPa.

Figure 51: Influence of tensile strength on the pushover results for the DP+TSoft model (ductile

soil-bridge interface cut-off; β=0,01) – Node 5605

The model with ft=0,15MPa and Gf=25Nm/m2 will fail soon after appearance of the first cracks,

the model with ft=0,15MPa and Gf=50Nm/m2 has still a lower strength but an increased

ductility, leading to improved performance.

Figure 52 shows the principle tensile stresses at the last converged load-step for the model

with DP+TSoft, ft=0,15MPa and Gf=25N/m. When compared to the same pictures for ft=0,3MPa

and Gf=50N/m (see Figure 43 for the principal tensile stresses concerning the TSCF model with

ft=0,3MPa and Gf=50N/m; as mentioned before, the stress and strain distribution for the TSCF

and the DP+TSoft model are nearly identical) the arches behave in a similar way, where the

middle vault does not show failure, whereas the side vaults are heavily damaged. However the




reduced strength in the arches provokes prior failure when strength and fracture energy are

decreased proportionally. Hence hardly any damage can be observed in the spandrel walls for

ft=0,15MPa and Gf=25N/m whereas they are severely affected after the last load step for

ft=0,3MPa and Gf=50N/m.


≥0,1105MPa


DP+TSoft – ft=0,15MPa; Gf=25N/m; β=0,01

For all the results described so far, the failure of the bridge is induced by cracking of/above the

side vaults mainly, whereas the main vault hardly showed any damage even at the last load-

step. Also the piers had suffered little damage so far. This is completely changed if the relation

between the strength and the fracture energy is changed. Remaining the strength and

increasing the fracture energy results in a more ductile behaviour of the bridge and damage is

widened to other elements of the bridge. Figure 53 shows damaged in the main vault and

cracking of the piers.

back side front side

Figure 53: Deformed Contour Plot - principle tensile stresses after last converged load-step

(2,06m/s2). DP+TSoft (ft=0,15MPa; Gf=50N/m; β=0,01) - blue: <0,15MPa; red: ≥0,15MPa

Figure 54 and Figure 55 show the crack strain distribution for the last converged load-step of

the DP+TSoft model with Gf=25Nm/m2 and Gf=50Nm/m2 respectively. The higher ductility of




the model with higher fracture energy can be observed clearly, however, the main cracking

remains in the two side vaults.

Figure 54: Deformed Contour-Plot - Crack-Strain of the DP+TSoft (ft=0,15MPa; Gf=25N/m;

β=0,01) model at the last converged load-step (acceleration: 1,31m/s2)

Figure 55: Deformed Contour-Plot - Crack-Strain of the DP+TSoft (ft=0,15MPa; Gf=50N/m;

β=0,01) model at the last converged load-step (acceleration: 2,06m/s2)




5.3. SEISMIC PERFORMANCE RESULTS AND DISCUSSION

San Marcello Pistoiese (longitude 10,7929; latitude 44,0574) is situated in a seismic zone with

peak ground acceleration values of 0,15-0,2 (Figure 56). The response spectra corresponding

to the Italian Code (CS.LL.PP., 2008) applied for the assessment of the performance points was

determined applying SIMQE software (Vanmarcke, et al., s.d.). Two response spectra (Figure

57) have been selected for soil type A (rock-like geological formation) and topographic

category T3: one specifying limit state of life safety (SLV) and a second one specifying limit

state of damage (SLD), both for a return period of 475 years (see details in the Italian Building

code [5]).

Figure 56: Seismic hazard maps for a return period of 475 years in terms of (a)

macroseismic intensity on the MCS scale; (b) peak ground acceleration (Crowley & et al,

2008)

Figure 57: Elastic Response Spectrum for PGA=0,2 according to the Italian code (Sa(e) denotes

elastic spectral acceleration; SLV denotes limit state of life safety; SLD limit state of damage) –

The grey dashed lines mark the range of the dominant periods of the bridge




In order to provide quantitative results, the displacement-acceleration curves have been

transformed into idealized elastic-perfectly plastic relationships applying the procedure

described in Annex B of the Eurocode 8 (UNI ENV 1998-1 (Eurocode8), December 2004) based

on the equal energy principle. The results in Figure 58 - Figure 61 are examplified shown for

the Drucker-Prager model and one parameter combination for each of the other models

(Gf=50Nm/m2; β=0,01) together with the performance points evaluated for ultimate limit state

of life safety (SLV) and ultimate limit state of damage (SLD).

The equivalent bilinear representation stresses the overestimation of the elastic capacity when

applying the simpler Drucker-Prager model. The ultimate “yield acceleration” for the Drucker-

Prager model is approximately 150% higher than for the TSCR and TSCF with β=0,01 models

and still approx. 80% higher than for the TSCF with β=0,1. The difference decreases for the

DP+TSoft model, especially for high tensile fracture energy and high shear retention, where

the difference exhibits approximately 25-50% overestimation using Gf=50 or 100 and β=0,1.

Still, again the determining effect of the shear retention can be observed, showing an increase

of the estimated “yield acceleration” of approximately 100-170% when applying the pure

Drucker-Prager model in comparison to the DP+TSoft with β=0,01.

Figure 58: Performance points for the Drucker-Prager model (ductile soil-bridge interface cut-

off) – Node 5605




Figure 59: Performance points for the DP+TSoft model (Gf=50Nm/m2; β=0,01; ductile soil-

bridge interface cut-off) – Node 5605

Figure 60: Performance points for the TSCF model (Gf=50Nm/m2; β=0,01; ductile soil-bridge

interface cut-off) – Node 5605

Figure 61: Performance points for the TSCR model (Gf=50Nm/m2; β=0,01; ductile soil-bridge

interface cut-off) – Node 5605




The spectral demand was compared with the bilinear elastoplastic idealization of the capacity

curves from in order to determine the performance points by applying the N2 method (Fajfar,

2000) formulated in the acceleration-displacement format. Therefore an inelastic demand

spectrum with constant ductility is determined from the elastic design spectrum for SLV and

SLD. For periods T ≥ Tc, where T is the period of the equivalent SDOF system and Tc is the

characteristic period of the ground motion (in our case 0,32 sec for SLV and 0,28 sec for SLD)

and which is always true for the results within this study, the inelastic accelerations are

determined by dividing the elastic acceleration with the ductility factor, which itself is defined

to be the ration of the maximum displacement divided by the yield displacement. The

computed inelastic demand spectra is then compared with the bilinear elastoplastic

idealization of the pushover curves which are already expressed in terms of spectral

acceleration and spectral displacements resulting in the so-called performance point.

In case of the Drucker-Prager model (Figure 58), the performance point for SLD is still in the

elastic branch of the capacity curve, whereas for SLV it situates its capacity in the plastic

branch. The same accounts for the TSCF model (Figure 60) with Gf=50Nm/m2 and β=0,01,

although SLD is very close to the plastic branch already. The same parameters applied to the

DP+TSoft model (Figure 59) reveal location on the (ultimate part) of the plastic branch, wheras

the performance points computed for the TSCR model (Figure 61) exceed the ultimate

displacements of the capacity, viz. failure of the structure.

The information concerning the performance points for the other models studied is

summarized in Table 5, where a safety factor is calculated defined as the ratio of the ultimate

available displacement divided by the needed displacement given by the performance point;

Ref. (Pelà, et al., 2009). The results show that the bridge is safe for both events (SLV and SLD)

for the models Drucker-Prager, DP+TSoft and TSCF if β=0,1, with exception of DP+TSoft

Gf=25Nm/m2. In case of SLD the only non-safe situation if given with DP+TSoft - Gf=25Nm/m2

and β=0,01, obviously the parameters applied introduce an exaggerated fragile behaviour

which can lead to convergence problems during the analysis. The fact that TSCF -

Gf=100Nm/m2 and β=0,01 shows a safety factor of <1 whereas TSCF - Gf=50Nm/m2 and β=0,01

remains safe for SLV lacks of explanation. The TSCR model does not withstand the seismic

demand whatever parameters are applied. This is caused by the fact that TSCR introduces a

very fragile behaviour which again leads to convergence problems. The final displacements

achieved for these models were approximately 1cm, which would not lead to a collapse.




Noteworthy is the very high safety factor for DP+TSoft with Gf=100N/m; β=0,1. Obviously the

ductility introduced by applying a high tensile fracture energy and high shear retention lead to

this result. Especially the increase of the shear retention factor leads to a non-conservative

overestimation of the capacity in all the models (with exception of TSCR, where β has no

influence at all according to the definition of the constitutive model).

Table 5: NSA ultimate displacements, performance points and safety factors

Model Gf

[N/m] β [-]

Performance point [m]

Ultimate Displacement

[m]

Safety factor [-]

SLV SLD SLV SLD

DP - - 3,21E-02 1,67E-02 4,53E-02 1,41 2,71

DP+Tsoft

100 0,1 4,83E-02 2,51E-02 1,85E-01 3,83 7,35

0,01 3,05E-02 1,59E-02 2,13E-02 < 1 1,34

50 0,1 3,86E-02 2,01E-02 5,49E-02 1,42 2,73

0,01 3,54E-02 1,84E-02 3,35E-02 < 1 1,82

25 0,1 3,36E-02 1,75E-02 2,29E-02 < 1 1,31

0,01 3,12E-02 1,63E-02 1,42E-02 < 1 < 1

TSCF

100 0,1 3,59E-02 1,87E-02 4,44E-02 1,24 2,38

0,01 3,60E-02 1,87E-02 3,55E-02 < 1 1,89

50 0,1 3,96E-02 2,06E-02 4,49E-02 1,13 2,17

0,01 5,60E-02 2,91E-02 6,41E-02 1,15 2,20

25 0,1 3,33E-02 1,74E-02 4,22E-02 1,26 2,43

0,01 3,92E-02 2,04E-02 3,31E-02 < 1 1,62

TSCR

100 0,1

2,97E-02 1,54E-02 1,30E-02 < 1 < 1 0,01

50 0,1

2,94E-02 1,53E-02 9,66E-03 < 1 < 1 0,01

25 0,1

2,94E-02 1,54E-02 8,07E-03 < 1 < 1 0,01

DP+TSoft and TSCF behave similarly; the pure Drucker-Prager model shows a high final base

shear, however the response is not more ductile exhibiting performance points similar to the

ones obtained with DP+TSoft or TSCF with Gf=50N/m2 and β=0,1.




5.4. NONLINEAR DYNAMIC ANALYSIS

Nonlinear Dynamic Analysis (NDA) consists of exciting the numerical model of the structure

with a combination of spectrum matching ground motion records. As explained in chapter 2,

these records are either real records, simulated ones or artificial accelerograms, giving priority

to the first, as the artificially calculated accelerograms can lead to unrealistic results (Pelà, et

al., 2012). As explained in FEMA 440 (FEMA 440, 2005), NDA can give results less uncertain

than other seismic assessment methods. Still the degree of uncertainty depends on the

variability of the ground motion records and on the uncertainties related to the definition of

the structural model, which should describe correctly the complex hysteretic nonlinear

behaviour of members.

Following the requirements of the Eurocode 8 (UNI ENV 1998-1 (Eurocode8), December 2004),

a minimum of seven earthquake recordings have to be found in order to use the average of

the response quantities as the design value, otherwise the most unfavourable response is

used. For artificial accelerograms a minimum of three should be used, where the mean of the

zero period spectral response acceleration values should not be smaller than the value of ag*S

for the site in question, where ag is the design ground acceleration on type A ground and S is a

soil factor. Furthermore a lower-bound tolerance is defined for in the range of periods

between 0,2T1 and 2T1, where T1 is the fundamental period of the structure (0,15 - 2sec and

0,15T1 - 2T1 in the Italian code for ultimate limit state), indicating that the elastic spectrum

calculated from the time histories should not be less than 90% of the corresponding value

from the spectrum in the code.

Within the scope of this study only one time history has been applied. However, the spectrum

compatibility of the adopted record has been carefully verified, with the aim of providing a

general notion for the differences between NSA and NDA assessment.

A single matching real record could not be found with REXEL (Iervolino, et al., 2010); therefore

an artificial accelerogram was computed using SIMQKE software (Vanmarcke, et al., s.d.). The

software already incorporates the specifications of the Italian code and an artificial time

history of 30 seconds was created by selection of the corresponding geographic area of the

bridge (see Figure 56 – PGA=0,2), specifying life safety requirement (SLV) for a return period of

475 years, soil type A and topographic category T3 (see chapter 0). The artificial time history

employed and its corresponding spectrum are shown in Figure 62 and Figure 63. Comparison




between Figure 57 and Figure 63 proofs that artificially created time histories match the based

spectra perfectly.

Figure 62: Time Histories for SLV (ag=0,342)

Figure 63: Elastic Response Spectra corresponding to the Time History in Figure 62 (Sa(e)

denotes elastic spectral acceleration; SLV limit state of life safety)

Concerning the damping of the structure, classical damping, which is an appropriate

idealization for practical applications if similar damping mechanisms are distributed

throughout the whole structure (Chopra, 2007), has been used applying the Rayleigh damping

matrix, where damping is assumed to be a linear combination of the stiffness and the mass

matrix of the initially elastic system (Chopra, 2007):

α ( )

(

α

) ( )

α

( ) ( )

( ) ( )




where C is the damping matrix, M the mass matrix, K the stiffness matrix, ξ the damping ratio,

which within the purpose of this study is assumed to be constant 5% and wi and wj the highest

and the lowest frequency supposed to cover all the frequencies of interest (G+D Computing ©,

1999). The purpose of this procedure is to decrease the influence of physically incorrect modes

(see Figure 64).

Figure 64: Variation of the modal damping ratios with natural frequencies (Máca & Oliveira,

2012)

Following the specification by Pelà et al (Pelà, et al., 2012), the natural frequencies of the 1st

and the 11th mode with a cumulative effective mass participation of 66% in transversal

direction were chosen for wi and wj resulting in the mass respectively stiffness proportional

coefficients α=0,206 and β=0,005799.

For the time integration method the accelerogram was divided into 3000 time steps of 0,01sec

each, the Hilber-Hughes-Taylor (HHT) method was used, applying α=-0,1. The computational

cost with a 2,93GHz – 4 GB RAM processor amounted to approximately 108hours.

5.5. NDA RESULTS AND DISCUSSION

The artificial time history applied is shown in Figure 62 (SLV - ag=0,342). The structure with

applied Drucker-Prager model shows a maximum displacement of 0,058m after 10,48 seconds

(0,019m in the control node 5605 – see Figure 65). Comparison between the result of NSA and

NDA is shown in Figure 66, confirming the affirmation of Pelà et al (Pelà, et al., 2009), (Pelà, et

al., 2012) that the NSA generally overestimates the displacements in a conservative way

compared to NDA.




Figure 65: Displacement History of Drucker-Prager model obtained with the artificial

accelerogram for SLV (ag=0,342)

Figure 66: Comparison between NSA capacity curve and NDA maximum




6. CONCLUSIONS

The present work determines the dependency of the results of seismic safety evaluation for a

three-vaulted masonry arch bridge on different constitutive laws which were applied for the

material. The bridge geometry has been introduced in DIANA FE software and static pushover

analysis in transversal direction have been carried out by applying

- a Drucker-Prager model

- Drucker-Prager in compression combined with a smeared cracking model in tension

with tension cut-off and linear softening in tension and shear (DP+TSoft)

- a so-called Total Strain Crack Fixed model (TSCF), a smeared cracking model which is

formulated in the strain space and combines compressive parabolic

hardening/softening and exponential softening in tension and shear; as soon as a crack

appears it is fixed in space

- a so-called Total Strain Crack Rotating model (TSCR) with the same characteristics as

the TSCF model but with new evaluation of the crack direction after every load-step

Each model (with exception of the Drucker-Prager model, which does not make use of these

parameters) has been studied concerning their behaviour for different tensile fracture

energies (Gf=25, 50 and 100N/m) and shear retention factors (β=0,01 and 0,1). Furthermore

the effect of decreasing the tensile strength by ½ has been shown.

Additionally non-linear dynamic (time-history) analysis has been conducted for the Drucker-

Prager model.

All the results have been evaluated concerning their performance for the seismic demand

specified for the geographic region in the Italian building code (ag=0,2), therefor the seismic

demand was scaled so that it fits the requirements of life safety limit state (SLV - ag=0,342) and

damage limit state (SLD - ag=0,2) according to the code.

Following the major conclusions are summarized:

- In case no experimental data concerning the material properties in the post-peak

regime is available, a notion of safety can be provided by varying the mechanical

properties within a range of parameters found in other case studies.




- The type of material model and non-elastic parameters applied fundamentally change

the seismic performance of the bridge, affirming the importance of careful selection of

these aspects.

- The Drucker-Prager model shows the highest base shear force with over-estimation of

approximately 150% regarding to the other material models, however the ultimate

displacements are similar to the ones obtained with DP+TSoft or TSCF. DP+TSoft with

Gf=100 or 50Nm/m2 and β=0,1 shows the highest ductility.

- The DP+TSoft and the TSCF model show similar results with high ductility especially for

high shear retention (β=0,1), viz. the introduction of compressive crushing with the

TSCF model does have hardly any impact on the capacity of the bridge.

- The lowest capacity showed the TSCR model, where no convergence was reached soon

after passing the elastic path due to the fact that this model does not show the

residual friction like the DP+TSoft or TSCF model, but instead dissipates the same

fracture energy for cracking like the tensile stresses. No further convergence can be

achieved for the TSCR model as soon as the softening branch in shear reaches its

minimum. The TSCR model is used widely for concrete structures, where it has been

shown (Parka & Kimb, 2005) that rotating crack damages are better suited than the

TSCF in order to describe the deviation from the initial crack during progressive tensile

cracking (during loading). However these cracks can be independent from each other

which is not described by TSCR.

- The increase of the shear retention factor β from 0,01 to 0,1 has a more determining

impact on the seismic capacity than the increase of the tensile fracture energy from

25N/m to 100N/m.

- The shear behaviour is unrealistic for the models studied. In case of pure Drucker-

Prager and DP+TSoft and TSCF with high shear retention (e.g. β=0,9) the shear

strength keeps growing after reaching the maximum compressive strength, although

with a decreasing stiffness and in case of DP+TSoft and TSCF with lower shear

retention (e.g. β=0,1 of β=0,01) the residual friction is increasing after passing a

minimum. No additional parameters for simulation of strength increase due to lateral

confinement had been applied. Figure 67 shows the uniaxial behaviour of combined




tension and shear for the smeared crack models, revealing that the maximum principle

stresses are the compressive and the tensile strength assigned.

- No clear explanation of the shear behaviour, especially for cyclic loading with the

Drucker-Prager model, could be found by the author but should be a future task of

urgent matter.

(a) shear response (b) response in normal direction

Figure 67: Response for uniaxial combined shear and tension of the smeared crack models

- It the tensile strength is decreased from 0,3MPa to 0,075MPa (and proportionally for

all the other materials) the bridge cannot bear its self-load. A tensile strength of

0,15MPa maintaining the ultimate crack strain leads to a failure soon after appearance

of the first cracks, however a tensile strength of 0,15MPa maintaining the fracture

energy (=doubling the ultimate strain) shows an enhanced performance allowing wide

cracked areas also in the middle vault and the front side of the piers which had not

been observed with the other parameters applied.

- The DP+TSoft model which resembles somehow the Rankine-Hill criterion but

neglecting anisotropy, is the most appropriate model for masonry according to the

DIANA user’s manual if the later cannot be applied. The failure with DP+TSoft is similar

to the one with TSCF. The failure mechanism deduced from all three cracking models

follows the movement of the 1st vibration mode and shows heavy cracking of the

lateral vaults leading to an overturning of the separated middle part of the bridge.

However the final crack strains for the TSCR model are 5 times smaller than for

DP+TSoft or TSCF.




- The assemblage of the finite elements applied in this study may not be appropriate for

addressing local failure mechanisms documented in literature. The internal

morphology of the bridge had not been studied and might have misled the results

(Brencich & Colla, s.d.). Furthermore it had been shown elsewhere (Rota, 2004) that a

typical failure mechanism of this type of bridge consists of local out-of-plane failure of

the spandrel walls due to the interaction with the filling, which cannot be simulated

with the FE model applied in this study.

- Most of the models do withstand seismic demand specified in the Italian code for limit

state of damage (SLD - ag=0,200) for a return period of 475 years with exception for

TSCR and DP+TSoft with Gf=25N/m - β=0,01, where the introduced brittleness leads to

convergence problems in the analysis. The models with sufficiently high ductility in

mode I and II also bear the demand for ultimate life safety (SLV - ag=0,342). The

models failing this demand are again the TSCR for whichever fracture energy applied

and the TSCR and DP+TSoft with tensile fracture energies of Gf=25N/m and shear

retention factor β=0,01. The final displacements achieved with the brittle models

(TSCR with whatever parameter, DP+TSoft and TSCF Gf=25N/m or β=0,01) are only in

the range of 1-2cm, which will unlikely lead to a collapse of the structure, however the

fragile simulation of the material leads to convergence difficulties in the analysis.

DP+TSoft with Gf=100N/m; β=0,1 obviously introduces very high ductility in the

structure and leads to a safety factor of 7,35 for SLD. The fact that the DP+TSoft model

with Gf=100N/m; β=0,01 shows a lower performance than the one with Gf=50N/m;

β=0,01 lacks of explanation and should be revised. Probably the rigorous analysis-stop

at load-steps as small as 1*10e-10 is the cause for this incoherency.

- The results obtained for the Drucker-Prager model with static nonlinear analysis could

be confirmed to be slightly conservative in comparison to applying nonlinear dynamic

analysis. After applying a SLV spectrum matching artificial record the structure shows a

maximum displacement of 0,058m after 10,48 seconds (0,019m in the control node

5605).




Following a list with suggestions for further investigation:

- Changing the Finite element model so that local failure can be simulated by

introducing interface elements in between the different structural elements of the

bridge, especially in between the materials with low (filling) and high (spandrel walls,

vaults) cohesion.

- Computing a full set of code matching real ground motions records.

- Determining through a comprehensive study of material parameters the lower and

upper range of the materials non-elastic parameters for each material separately and

computing a probabilistic output for safety evaluation.

- Rigorous assessment of the (DIANA) material models concerning shear, their feasibility

for masonry and impact on the safety evaluation.

- Further analysis of the structure by leading the pushover curve through a definite

“pushover” by applying decreased load-steps and displacement controlled iterations,

in order to analyse the behaviour in the post-peak range.




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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS A-1

A. ANNEX I

Note: the following graphs were computed with a maximum tensile strenght of 0,3MPa and a

maximum compressive strenght of 4,5MPa, otherwise it is indicated.

nonlinear seismic analysis of a masonry arch bridge · nonlinear seismic analysis of a masonry arch...

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