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EXTENDED ABSTRACT - EN

Mixed buildings constitute a building typology to which we usually refer to indicate that part of the existing

architectural heritage, mostly belonging to the first part of the 20th century, characterized by the simultaneous

presence of structural elements realized with traditional and natural materials such as masonry or timber, and

other innovative (for those times) techniques like steel or reinforced concrete.

Despite the rather widespread use of such structures, weather for residential buildings or for strategic ones such

as schools or hospitals, very little research has been carried out on this topic and some issues are still open,

such as the effect of the interaction between reinforced concrete and masonry walls on the global structural

behaviour of mixed systems. There’s a lack or inadequacy of experimental data, and of numerical simulations.

Moreover, it has been noticed that the results of numerical analyses can vary and they are sensitive to modelling

approaches and to the hypotheses on which they’re based. Codes, either at a national or at an international

level, provide little support for the seismic design and retrofitting of mixed structures in general and masonry-

reinforced concrete structures in particular. As a consequence, practitioners are inclined to design these mixed

systems with simplified assumptions that can lead to misleading provisions of the response of these structures.

The objective of the study is essentially to evaluate the reliability of simplified models for the analysis of mixed

masonry-reinforced concrete buildings, to assess the validity of current Code prescriptions, and to evaluate the

effectiveness of retrofitting techniques.

The aim is to provide a base for future developments, especially for what concerns the Code prescriptions and

to provide a comparison between refined and simplified models, to assess the eligibility of the latter, though in

a still small extent of cases, to analyse, within a tolerable compromise level, this category of buildings.

Case studies have been selected choosing some building for which extensive sets of information were available;

the analyses have been conducted using the code Tremuri and some considerations have been made comparing

some results with the finite element code Diana, deriving information on critical aspects of the behaviour of

these structures that need to be studied more in detail in order to achieve a more comprehensive understanding

of the behaviour of these structures.

Keywords: Mixed structures, Masonry, Reinforced concrete frames, Non-linear analysis, Seismic

behaviour, Modelling approaches

III

EXTENDED ABSTRACT - IT

Gli edifici misti costituiscono un’ampia classe di edifici a cui si è soliti riferirsi per indicare quella parte del

costruito storico, risalente per lo più alla prima metà del Ventesimo Secolo, caratterizzato dalla presenza

simultanea di elementi realizzati con tecniche tradizionali come muratura o legno, ed altre tecnologie più

moderne come acciaio o calcestruzzo armato. Nonostante la diffusione di queste strutture, utilizzate sia in

edilizia residenziale, sia nel caso di alcuni edifici strategici come scuole od ospedali, la ricerca scientifica

sull’argomento è ad oggi molto limitata e alcune questioni sono ancora irrisolte, come la definizione della

effettiva collaborazione tra muratura ed elementi in calcestruzzo armato e del suo effetto sul comportamento

globale dell’intero edificio misto. Inoltre, si registra una scarsità o inadeguatezza sia di dati sperimentali, sia di

simulazioni numeriche.

Le normative, sia in ambito nazionale, sia in ambito internazionale, danno un piccolissimo supporto alla

progettazione di nuove strutture o alla valutazione di edifici esistenti appartenenti a questa categoria; pertanto,

spesso vengono adottate forti semplificazioni che possono portare ad un’interpretazione fuorviante del

comportamento strutturale.

L’obiettivo di questo studio è, essenzialmente, di valutare l’affidabilità di alcuni strumenti di analisi per questa

tipologia di edifici, di stabilire la validità delle indicazioni date dalla vigente Normativa e di dare una stima, se

pur preliminare, sull’efficacia di alcune tecniche di riabilitazione strutturale.

Lo scopo è di fornire una base di dati per futuri sviluppo, specialmente per ciò che riguarda l’ adeguatezza degli

strumenti normativi e di modellazione.

Per condurre queste analisi sono stati selezionati alcuni casi di studio e ci si è serviti del codice Tremuri e del

codice multi-purpose agli elementi finiti DIANA, derivandone alcune considerazioni sulle criticità del

comportamento di questa categoria di edifici e sugli aspetti che meritano di essere trattati ancora più in dettaglio

per una più ampia comprensione del comportamento sismico di tali strutture.

Parole chiave: Strutture miste, muratura, calcestruzzo armato, analisi non lineari,

comportamento sismico, approcci di modellazione

V

EXTENDED ABSTRACT - PT

Os edifícios mistos são uma tipologia construtiva designada usualmente a edificado pertencente ao património

cultural, a maioria relativo à primeira parte do século XX, que se caracteriza por apresentar elementos

estruturais realizados com materiais naturais e tradicionais como alvenaria ou madeira, bem como outros

elementos e técnicas inovadores (para aquela época) como o aço e o betão armado.

Apesar da larga disseminação destas estruturas, para fins residenciais ou fins estratégicos como escolas e

hospitais, pouca investigação foi realizada sobre este tipo de edificado e algumas questões estão ainda por

responder, como o efeito da interação entre o betão armado e as paredes de alvenaria no comportamento

estrutural global do sistema misto. Existe pouca informação experimental adequada, bem como poucos

resultados de simulações numéricas. Por outro lado, sabe-se que os resultados de análises numéricas podem

variar, uma vez que estão dependentes das abordagens e opções definidas na fase da modelação. Os códigos

normativos, quer sejam a nível nacional ou internacional, fornecem poucos pressupostos para o

dimensionamento sísmico e reforço de sistemas mistos em geral e estruturas de alvenaria-betão armado em

particular. Como consequência, os profissionais de engenharia tendem a dimensionar estes sistemas mistos

com base em abordagens teóricas simplificadas que podem levar a erros na avaliação da resposta dos mesmos.

Assim, o objetivo do presente estudo é, essencialmente, o de avaliar a eficácia de modelos simplificados para

a análise de edifícios mistos de alvenaria-betão armado, analisar a validade dos requisitos do código normativo

atual, e avaliar a eficácia de técnicas de reforço.

Pretende-se assegurar uma base para futuros desenvolvimentos, especialmente no que se refere aos

pressupostos normativos e na comparação entre modelos complexos e simplificados, tanto na elegibilidade e

validade dos últimos para que, mesmo apenas num pequeno número de casos, analisar dentro de um

determinado nível de compromisso o edificado pertencente a esta categoria.

Alguns casos de estudo foram selecionados, tendo em conta os edifícios em que um conjunto largo de

informação está disponível; as análises foram realizadas usando o software Tremuri e algumas considerações

definidas pela comparação com os resultados do software de elementos finitos DIANA, no que se refere a

informação de aspetos essenciais do comportamento destas estruturas que necessitam de ser estudados com

mais detalhe, para que se atinga um nível de conhecimento maior sobre o seu comportamento.

Palavras-chave: Estruturas mistas, Alvenaria, Pórticos de betão armado, Análises não-lineares,

Comportamento sísmico, Abordagens de modelação

1

INDEX

EXTENDED ABSTRACT - EN ........................................................................................................... I

EXTENDED ABSTRACT - IT .......................................................................................................... III

EXTENDED ABSTRACT - PT .......................................................................................................... V

INDEX ......................................................................................................................................... 1

CHAPTER 1: INTRODUCTION ...................................................................................................... 3 1.1 MOTIVATIONS AND DESCRIPTION OF THE THEME ................................................................. 3 1.2 PROBLEM STATEMENT .................................................................................................................. 9 1.3 OBJECTIVES ................................................................................................................................. 11

CHAPTER 2: STATE OF THE ART ............................................................................................... 13 2.1 TYPOLOGICAL IDENTIFICATION AND CLASSIFICATION OF THE BUILDINGS .................................. 13 2.2 SEISMIC BEHAVIOUR OF MIXED MASONRY-REINFORCED CONCRETE BUILDINGS: POST-EARTHQUAKE DAMAGE DESCRIPTION ............................................................................................... 18 2.3 SEISMIC BEHAVIOUR OF MIXED MASONRY-REINFORCED CONCRETE BUILDINGS: NUMERICAL AND EXPERIMENTAL STUDIES ................................................................................................................... 23 2.4 REGULATORY FRAMEWORK FOR MIXED MASONRY-REINFORCED CONCRETE BUILDINGS: NATIONAL AND INTERNATIONAL CODES ............................................................................................................. 28

CHAPTER 3: SEISMIC ASESSMENT OF MASONRY STRUCTURES AND MODELLING CRITERIA ... 39 3.1 MODELLING MASONRY BUILDINGS: A DEFINITION OF DIFFERENT MODELLING APPROACHES .... 40 3.2 DIANA .......................................................................................................................................... 51 3.3 TREMURI ..................................................................................................................................... 69 3.4 SEISMIC ANALYSIS OF MASONRY BUILDINGS .............................................................................. 80

CHAPTER 4: CASE STUDY – CAPRI BUILDING .......................................................................... 95 4.1 DESCRIPTION OF THE BUILDING: GEOMETRY .............................................................................. 96 4.2 DESCRIPTION OF THE BUILDING: MATERIALS ........................................................................... 108 4.3 LOADS AND ANALYSES .............................................................................................................. 113 4.4 EQUIVALENT FRAME APPROACH: DESCRIPTION OF THE ADOPTED MODEL .............................. 115 4.5 NON-LINEAR STATIC ANALYSES ................................................................................................. 121

CHAPTER 5: CASE STUDY – Ex-SLAUGHTERHOUSE (Rome) .................................................... 133 5.1 DESCRIPTION OF THE BUILDING: GEOMETRY ............................................................................ 134 5.2 DESCRIPTION OF THE BUILDING: MATERIALS ........................................................................... 136 5.3 LOADS AND ANALYSES .............................................................................................................. 139 5.4 NON-LINEAR STATIC ANALYSES ................................................................................................. 141

CHAPTER 6: SOME CONSIDERATIONS ON FINITE ELEMENT MODELLING OF MIXED MASONRY-R.C. BUILDINGS ...................................................................................................................... 147

6.1 DESCRIPTION OF THE MODEL ................................................................................................... 147 6.2 DESCRIPTION OF THE RESULTS ................................................................................................ 151

CONCLUSIONS ....................................................................................................................... 157

AKNOWLEDGMENTS ............................................................................................................... 161

REFERENCES ......................................................................................................................... 163

CURRICULUM ......................................................................................................................... 171

3

CHAPTER 1: INTRODUCTION

1.1�MOTIVATIONS AND DESCRIPTION OF THE THEME

Mixed buildings constitute a building typology to which we usually refer to indicate that part of the

existing architectural heritage, mostly belonging to the first part of the 20th century, characterized by

the simultaneous presence of structural elements realized with traditional and natural materials such

as masonry or timber, and other innovative (for those times) techniques like steel or reinforced

concrete.

They represent indeed a transition between the continuous structure and the framed structure; in

these buildings the Regola dell’ Arte meets the results of a strong experimental excitement, powered

by the many progresses achieved during the industrial revolution.

The historical period in which a major diffusion of these new building practices is noticed in Europe

is undoubtedly characterized by a large diffusion of information, goods, and new achievements in

technological fields; all these factors have contributed to an efficient collaboration between science,

allowing the use of new energy resources and their optimization, the labour force and raw materials,

coming to the mass production processes which led to a rapid diffusion of new technologies.

4

Fig. 1.1 Hennebique System

From the beginning of the 20th Century, reinforced concrete technology has been spreading in

European countries and, also in Italy, it brought a substantial transformation of theories and

construction approaches, not only from a practical point of view, but also in terms of architectural-

aesthetic perception.

The Italian environment, quite differently from what happened in other European countries, has been

characterized by a long tradition of reinforced concrete used in combination with unreinforced load

bearing masonry walls, both in case of new constructions, and in the retrofitting of damaged or

vulnerable unreinforced existing masonry buildings.

The use of “new materials” is often encouraged by economical reasons and some observations on

their higher mechanical properties, especially if compared to the usual natural materials that were

used and diffused at those times, and because of several experimental campaigns that assessed

their better response in case of fire and against deterioration.

The fact that reinforced concrete can withstand a tensile strength higher than that sustainable by

common conglomerates or stones, having however a higher stiffness than steel structures, allows in

all common cases in which a really high strength is not required, to reduce a lot the dimensions of

structural elements

Thanks to this aspect, the versatility in the use of the space increases a lot and also the construction

phases become much more efficient, resulting in a series of advantages in terms of material needs

and their installation processes.

Some of these building trends were clearly addressed in certain codes, especially in areas with high

seismic hazard: the building norms, issued after the Messina earthquake in 1909, suggest the

5

adoption of mixed masonry-r.c. solutions, with different configurations but with the common feature

of taking advantage of the ductility improvements coming from the use of reinforced concrete frames

to confine masonry panels.

During the reconstruction after the Second World War, in the 1950s, urban spaces were modified,

and sometimes reinvented, thanks to the presence of r.c. buildings, where the use of load-bearing

masonry progressively turned into the use of lightweight infills not meant to have a bearing function

but only intended as enclosures.

As will be demonstrated later on by numerous numerical and experimental studies, however, their

presence is other than irrelevant into the global resisting mechanism and mostly for the energy

balance of the building as a whole (Fiore, 2012; Porco, 2014).

The evolution in the building solutions that make use of both unreinforced masonry and r.c. elements

is characterized by a high heterogeneity: from r.c. structures in which masonry panels are meant as

an infill, so they are inserted in the frames after they have been cast in place, to mixed structures

where horizontal and vertical r.c. elements are in contact with masonry, and that is the case of

confined masonry which differs from the previous type of construction since in this case masonry is

used as a sort of mould (only after the wall has been realized than the frame is cast all around it), to

structures in which the vertical bearing elements are simultaneously composed of masonry walls,

frames and r.c. walls, isolated or in contact with the masonry, to structures with floors made of

different technologies, widespread consequences of uprising original masonry building with new

floors realized with different technologies, either reinforced concrete or steel or wood, to masonry

structures with bracing r.c. elements.

The variety of mixed masonry-reinforced concrete structures is even wider and more complex taking

into account all the possible interventions on existing buildings, especially due to extensions in height

or plan, or the substitution of internal masonry walls with reinforced concrete frames or, quite rarely,

reinforced concrete walls occurred most of the times without any engineering guidance.

6

Fig. 1.2 Aerial view and internal spaces of the Manifattura dei Tabacchi, a mixed building located in Bari

Despite the spread of this typology in Italy and in many other countries in the Mediterranean area

and in Latin America, the scientific research on mixed structures is extremely limited.

This is attributable to the variability in the typologies spread all among the Italian territory and to the

fact that this construction typology is not completely well known and, mostly, not standardised. The

fact that mixed solutions arise most of the times from interventions on previous buildings without

any engineering sense, but only according to functional/architectural needs, makes the topic even

more knotty.

Code prescriptions, either in national or in European and international context, do not give sufficient

and robust indications on analysis and assessment methodologies and criteria to be used in practice

are scarce or absent, especially for what concerns the mixed solutions as such (the one in which

masonry and reinforced concrete do not work in adherence; the following sections will clarify this

concept), while for confined masonry solutions mostly in Latin America countries, where this

technology is widely used because of its simplicity and cost-efficiency, many manuals are now

available, though their slant is mostly practical without precise indications on methodologies and

analysis approaches.

Recent codes, such as EC8 or the Italian D.M. 2008 provide very scarce operative indications to the

designers, mostly limited to general principles for what concerns the design of new mixed masonry-

reinforced concrete buildings and few indications on the analyses to be used in case of the

assessment of the vulnerability of existing ones.

7

The international scientific literature on the seismic behaviour of structural masonry with isolated r.c.

columns and masonry coupled with r.c. walls is extremely poor: in particular for what concerns the

former only few tests were performed in the Nineties, while the latter class has been the object of a

recent experimental campaign performed in the École Polytechnique Fédérale de Lausanne (Paparo

& Beyer, 2014), though this survey has been centred mostly on mixed masonry-reinforced concrete

wall structures.

Mixed structures composed of floors with different technologies have received very low attention from

the scientific community, since this structural typology presents in most cases several drawbacks for

newly designed structures. Nevertheless, the high number of existing buildings to which storeys have

been added in different phases calls for an accurate focus on this typology, in which the irregularity

in elevation, as a consequence of the combined effect of the extension in height, along with the

increase of the masses, can induce a higher seismic vulnerability compared to the one of the original

configuration.

A larger number of studies, including experimental ones, is available for mixed structures composed

of masonry with r.c. members in contact, mainly with reference to the “confined” masonry technique,

in which the function of the confining elements is to provide dissipative capacity and ductility.

However the majority of literature studies is devoted to framed r.c. structures with masonry infills, in

which the masonry is introduced primarily as a closing element and therefore it’s contribution is

meant to be non-structural (circumstance that has been proven to be not always respected), while

the bearing function is assigned only to reinforced concrete frames.

In past ten years, a notable advancement of models for the non-linear analysis of masonry structures

and r.c. frames with masonry infills was observed. Several different nonlinear macro-models, which

permit seismic nonlinear analysis of entire buildings with satisfactory results, have been developed.

These macro-models with appropriate adaptations and improvements can be perceived as a starting

point for the development of models for nonlinear analysis of mixed masonry-reinforced concrete

structures, mainly with the aim of investigating their seismic global behaviour, taking into account

the different deformation capacities of the structural elements.

8

Dealing with the assessment of the vulnerability of existing buildings, obviously even in the case of

mixed masonry-r.c. ones, it’s possible to asses the logical process that starts for the diagnostic phase,

based on many different investigations to identify the weak structural situations and, if necessary, to

formulate a solution aimed at recovering the structure by the elimination of the source of damage,

through appropriate strengthening and retrofit interventions (however, works related to criteria and

techniques of seismic strengthening and retrofit specific to mixed structures are missing).

The literature on strengthening interventions considers the masonry structure separately from the

r.c. structure. Besides classical manuals and textbooks, the main contributions are derived from

post-earthquake experiences, after the events that influenced both building construction scenario in

post-war Italy and the regulatory framework, in which also mixed solutions find some, though limited,

space. These contributions are specifically significant due to their strong relation with the national

construction scene, which is a very relevant aspect (Magenes, 2006).

In the assessment of existing mixed masonry-reinforced concrete structures not only the correct

estimation of material properties and the geometrical features of all the bearing elements is

important, but also the contributions that only non-linear analyses can give.

It has been confirmed, in fact, that for dual systems nonlinear analyses provide quite different results

from linear analyses, as a consequence of the different stiffness, strength, deformation capacity of

masonry and r.c. members.

In case of mixed buildings in which unreinforced masonry is coupled with reinforced concrete frames,

and floor diaphragms are stiff, the frame contribution to the lateral resistance is generally much lower

than the contribution of masonry, and neglecting it is on the safe side.

Care should be taken, instead, when flexible diaphragms are present, since only limited transfer of

horizontal forces may be possible from the frame to the walls, and the frames should be able to

withstand the seismic actions that derive from the vertical loads carried by the frames themselves

(Liberatore & Tocci, 2008).

When both r.c. walls and unreinforced masonry walls are present and rigid diaphragm action can be

assumed, both systems give an important contribution to the seismic resistance and neglecting one

9

of the two systems can produce misleading and unrealistic results. In these cases, the difference

that can be found in the distribution of seismic shear comparing elastic linear and non-linear analyses

can be huge, and that’s why non-linear methods become imperative.

1.2 PROBLEM STATEMENT

Mixed masonry-reinforced concrete structures form a very heterogeneous category, since they can

also vary significantly from region to region (Magenes, 2006; Cattari & Lagomarsino, 2013). In this

thesis, however, the analysis is limited to typical configurations of mixed masonry-reinforced concrete

buildings belonging to the first half of the 20th century with masonry panels and reinforced concrete

elements not in adherence.

Despite the rather widespread use of such structures, weather for residential buildings or for strategic

ones such as schools or hospitals, very little research has been carried out on this topic and many

issues are still open or need to be addressed more in detail:

•� The effect of the interaction between reinforced concrete and masonry walls on the global

structural behaviour of mixed systems: in the case of unreinforced masonry walls and

reinforced concrete frames acting in parallel on the same level and both carrying either

vertical (gravitational) and horizontal (seismic) loads, the global behaviour will be clearly

different from the one of the building with only masonry walls or reinforced concrete frames.

•� Lack or inadequacy of experimental data: experimental data on mixed structures are few

and, as it will be more widely explained in the following chapter, only three experimental

campaigns on mixed masonry-r.c. buildings are reported in literature. They campaigns were

addressed to the investigation of several retrofitting techniques, one of which consisted in

the addition of a central RC wall pinned to the foundation.

Despite the fact that they highlight some relevant aspects, the problems they’re addressed

to are quite distant from the purpose of this research and couldn’t be taken as a reference

to validate the models that have been proposed in literature to study this category of

buildings, mainly because some information, useful to represent the test conditions, are

missing and moreover because in all the specimens the reinforced concrete part doesn’t

give a real contribution to the overall behaviour in terms of shear absorbed by the elements.

10

In fact peripheral masonry walls and a single internal r.c. frame constituted the samples,

with a prevailing contribution of masonry to support seismic loads, as confirmed also by

the damage occurred.

•� Lack of numerical studies simulations: few non-linear numerical investigations were carried

out on the topic. Casoli (2007) examined an existing mixed masonry-r.c. structure and its

retrofitting. More recently, non-linear analyses have been carried out by Cattari and

Lagomarsino (2013) on different models, simulating a part of the variety of interventions

that can involve unreinforced masonry buildings with the insertion of reinforced concrete

elements such as frames or walls. A mixed masonry-r.c. building has been analysed by

Gattulli et al. (2015) though the case-study is characterized by a high complexity and

irregularity in plan and then it’s not suitable for a set of general considerations on the

typology, since the behaviour of the entire structure is certainly affected by the marked

irregularity.

•� The results of numerical analyses can vary and they are sensitive to modelling approaches

and to the hypotheses on which they’re based: mixed masonry-reinforced concrete

structures can be modelled by a macro-modelling approach (equivalent frame), which

appears as particularly suitable for this kind of structure, given the frame nature of at least

one part of the building, or by a more accurate finite element modelling. The first modelling

approach is usually preferred in engineering practice because it represents a balance

between the accuracy of results and the computational effort.

•� Insufficient code indications: Codes, either at a national or at an international level, provide

little support for the seismic design and retrofitting of mixed structures in general and

masonry-reinforced concrete structures in particular. As a consequence, practitioners are

inclined to design these mixed systems with simplified assumptions that can lead to

misleading provisions of the response of these structures.

11

1.3 OBJECTIVES

All the aforementioned “issues” have lead to the study of this typology and the main objectives of

this investigation can be outlined as follows:

�� Give a series of numerical references for a typology that is the most common among the

mixed masonry-reinforced concrete structures.

�� Provide insights concerning the criteria for repartition of the seismic actions between

elements of different technologies.

�� Evaluate the reliability of code prescriptions for the analysis of mixed masonry-reinforced

concrete buildings.

�� Assess the performance of different types of existing masonry-reinforced concrete buildings,

in order to evaluate how the choice of rehabilitating masonry buildings by the insertion of

reinforced concrete elements can affect (and eventually worsen) the performance of the

existing building.

�� Provide a benchmarking set of data for two different case-studies, which can constitute a

small but significant contribution to the understanding of the behaviour of this class of

buildings and on the suitability and effectiveness of simplified approaches.

13

CHAPTER 2: STATE OF THE ART The chapter introduces some topics, relevant to understand the class of building to which this thesis

is addressed and some insights on the studies that have already been carried out on this category.

Section 1 contains the typological identification of the different possible configurations of mixed

masonry-reinforced concrete buildings. In section 2, mainly thanks to the work carried out by

Decanini et al. in the wider framework of ReLUIS program, some evidences collected in several post-

earthquake reports have been resumed in order to better understand the behavior of mixed

structures during past earthquakes.

In section 3 the past experimental and numerical studies on mixed masonry-reinforced concrete

structures are summarized and in the last section (the fourth one) the codes in which some reference

to mixed masonry-reinforced concrete buildings are made are reported.

2.1 TYPOLOGICAL IDENTIFICATION AND CLASSIFICATION OF THE BUILDINGS

It’s a fairly spread idea that masonry buildings do not show a high performance when subjected to

horizontal dynamic loads. Behind this (not completely right) idea many factors have contributed,

especially the fact that masonry buildings have exhibited a relatively high vulnerability during seismic

events.

But most of the times the bad response of masonry structures to dynamic actions has to be related

to the lack of construction detailing and the bad masonry execution more than to the building typology

as such.

Masonry, however, is one of the oldest and most widely used construction techniques ever; in fact

evidences of its use date back to the Mesolithic Era (9000-8000 BC). The in-situ availability of raw

14

material necessary to the building process, the relative simplicity in the manufacturing and the

construction processes, the resistance and high durability of masonry structures are just few of the

factors that have contributed to the spread and success of this construction technique that survived

till today, being still adopted in many countries.

The need for a more ductile behaviour, but mainly the need to comply with structural requirements

and to achieve a certain architectural versatility, together with the spread of new technology, such

as reinforced concrete, led in the first years of the 20th century, to the combined use of masonry and

other materials in order to overcome all the limits to which masonry had been tied for a long time.

To improve the seismic behaviour of existing masonry buildings, acting on ductility and dissipation

of energy, the most innovative techniques were based on the employment of other materials,

combined with masonry; so, for example, reinforced concrete frames or walls started to replace

masonry walls in order to give a higher versatility to the spaces in the building and a certain ductility

to the entire structure.

This second aim is attained, though, only if the reinforced concrete elements are properly designed.

Moreover, the insertion of reinforced concrete walls, for example in order to rehabilitate staircases

or in case of the insertion of elevators in existing buildings, may lead to a significant change in the

distribution of stiffness, affecting then the regularity in plan of the entire structure (Cattari and

Lagomarsino 2007).

Figure 2. 1 Masonry wall with timber frames in an urban expansion area in Madrid, XIX Century

15

Many other solutions, not specifically addressed in this study, but that will be mentioned also in the

upcoming sections since they’re recognized and referred to as mixed, are the one in which the

ductility properties of reinforced concrete, timber, steel frames, have been used to confine masonry

panels, improving their properties and changing somehow the failure mechanisms, that are certainly

affected by the presence of different technology elements.

All these solutions have followed on the one hand the technological progress in materials, on the

other the evolution in the constructive theory and technique evolution. For example, the “Pombalina

gaiola” or the “Casa baraccata” systems (mainly spread in Portugal and Italy, respectively) are based

on the idea of reinforcing masonry through timber-framed walls; then, the progressive and rapid

spread of reinforced concrete has led to the adoption of mixed masonry-reinforced concrete

solutions, such as the confined masonry.

In Italy, after the 1908 Messina earthquake, confined masonry (with reinforced concrete frames) was

suggested as an anti-seismic solution and this technology has been widely used to rebuild destroyed

parts of several cities in Sicily. Nevertheless, in Italy, this proposal (the use of confined masonry for

the reconstruction of destroyed historic centres) doesn’t appear in following codes, including the

current ones: thus its use has remained restricted to a very limited area and to a precise time range.

On the contrary, in other countries such as the South American ones (e.g. Argentina, Chile, Mexico),

confined masonry finds nowadays wide application as an anti-seismic solution and during last

decades, many regulations and, most of all, practical manuals, encourage its use as an efficient

building technology, easy to realize and economically efficient.

Though with the term mixed one can refer to a wide variety of different structural types, a first

classification, that is also the one which the Italian code refers to, has been proposed by Liberatore

et al. (2007).

It is possible to identify mixed buildings with reinforced concrete and masonry elements in adherence

(that constitute a first typological grouping) and buildings with masonry and different technology

elements not in adherence. However, in the case of different types of elements not in adherence, it’s

possible to identify other sub-classes, such as series systems (different elements located at different

levels) or parallel systems (when the elements are placed at the same level).

16

Mixed masonry-reinforced concrete buildings working as a series system could derive from raising

up by means of structures with different technologies such as RC frames, while combined masonry-

reinforced concrete buildings working as a parallel system could derive from masonry structures

subjected to plan enlargements or in many other cases consist of external masonry walls and internal

isolated reinforced concrete columns or frames (solution frequently used not only in existing buildings

but also in new constructions).

This great variety, involving either structures conceived with a specific role for earthquake behaviour,

and existing structures that, during the years, with the spread of new technology, have undergone

changes in plan and/or height, has been lead more by functional needs than by structural/seismic

ones. It poses, of course, many issues concerning a proper typological classification and a structural

standardization.

This aspect is also worsened by the fact that this category which now constitutes a noticeable part

of the architectural heritage mainly in Italy but also in some other countries, especially in the

Mediterranean area, includes not only residential buildings, but also a series of strategic buildings

such as hospitals or schools.

The variety of all these cases poses difficulties not only for their typological classification but also for

their structural scheme standardization. Moreover, since these interventions usually rise from a

spontaneous building tradition, any capacity design or ductility concepts are neglected and they are

designed most of the times only to bear vertical loads.

In addition it’s worth noticing that these interventions on existing masonry buildings, modifying their

structure by enlargements, uprising, internal walls demolition, have involved during the years not

only residential buildings but also strategic buildings such as schools and hospitals.

The current Italian code D.M. Infrastrutture (2008) suggests for combined masonry-reinforced

concrete buildings, a classification similar to that proposed by Liberatore et al. (2007) for combined

masonry-reinforced concrete buildings with different technology elements not in adherence while

does not do the same with the masonry-reinforced concrete technologies in adherence.

Despite the spreading in building practice, very few references to experimental campaigns and

numerical studies specifically addressed to this class of mixed masonry-reinforced concrete buildings

may be found in the literature.

17

Considering what has been highlighted up to this point, a first classification may involve the

aforementioned typology as follows:

i.� buildings in which masonry and reinforced concrete elements are adherent leading to a

direct interaction between these two materials and most of the times to a modification of the

damage patterns in masonry panels

ii.� buildings in which masonry and RC do not directly interact.

Several typologies belong to class i, since the cooperation in adherence of masonry and reinforced

concrete can be detected also in case of the presence of reinforced concrete ring beams at the level

of the floors, and in case of confined masonry (the interaction effects in the first case are quite local

and may be properly taken into account by modifying only the strength criteria of spandrels keeping

the other hypotheses usually adopted for masonry piers unchanged).

This typology, if correctly realized, is obviously intended to improve the behaviour of the original

unreinforced and unconfined masonry building for which a reduction of vulnerability is expected. This

has been proved not only by a series of experimental campaigns, but also by the after earthquake

damage detection.

Within class ii, it’s really hard to identify all the different typologies that rise from that

architectural/structural trend, which was, as mentioned, quite spontaneous and sometimes didn’t

follow seismic prescriptions.

However, some recurrent configurations can be taken as representatives of three different sub-

classes. The first consists of all the buildings characterized by peripheral masonry walls and internal

reinforced concrete columns or frames; the second includes masonry buildings in which reinforced

concrete walls have been inserted in order to realize staircases or elevator units; the third one which

is the category of parallel mixed masonry- r.c. buildings, in which entire floors have been realized in

reinforced concrete (Cattari, 2013).

The interaction between the two structural types in case of parallel systems is a direct consequence

of their differences in terms of strength and stiffness. In particular, difference in stiffness leads to a

different nonlinear response during the different stages. These structural elements should intervene

18

in subsequent phases of the global response; moreover, it stresses the need for a proper design not

only of connections but also of floors in order to guarantee their role of transferring actions among

the various structural elements until building collapse.

The problem of classification becomes even more troublesome since starting from these typologies

many other hybrid solutions are possible.

2.2 SEISMIC BEHAVIOUR OF MIXED MASONRY-REINFORCED CONCRETE BUILDINGS:

POST-EARTHQUAKE DAMAGE DESCRIPTION

Observing the damages occurred in buildings after severe earthquakes it is possible to outline how

different typologies lead to a complete different behaviour under seismic loadings.

Damage mechanisms (essentially for what concerns masonry buildings) can be substantially

categorized in two main groups, as suggested by Giuffré (1993): first mode mechanisms and second

mode mechanisms. First mode mechanisms are addressed to when the collapse occurs because of

rocking phenomena and mainly out of plane kinematic.

Second mode mechanisms are mainly related to the in-plane response of the panels.

The occurrence of one of these two categories is related to the global behaviour of the building,

strongly influenced by its typological and technological features.

The identification of the main resisting elements and their mutual connections becomes of great

importance, in order to evaluate their influence on the global behaviour.

The most simplified way to intend the building as a group of different elements, is to distinguish

between horizontal and vertical elements; before analysing the contribution of the elements, however,

it’s worth pointing out that the global behaviour strongly depends on the mutual connection between

them.

The virtuous behaviour of a masonry building mainly relies on the so called box behaviour: resisting

elements, that can be easily identified with the vertical walls oriented in both directions in the plan

and with the horizontal slabs must be connected so to ensure a global behaviour which guarantees

the withstanding of the horizontal actions no matter if they’re parallel to one direction or

simultaneously acting in both directions. Thanks to this attitude that has been shown in some cases

during the past, also masonry buildings, if properly designed, can show a satisfying seismic

behaviour.

19

A difference in terms or stiffness, or the lack of a proper connection between elements, can be the

cause of the occurrence of local or global mechanisms; that’s why it’s important to focus the attention

to discontinuities or singularities because they represent a weak point in which the seismic action

can cause severe damages that, especially when the building has not been conceived as a box, can

lead to the global collapse of the entire structure.

To avoid first mode mechanisms, some punctual interventions can be done, mainly in order to

transfer the out of plane action on a wall to the orthogonal ones, for which the response becomes

mainly in their own plane.

Figure 2. 2 First mode mechanisms

Since the vulnerability of masonry buildings depends on their global box behaviour, it’s fundamental

to have a modelling tool able to represent the three-dimensional features of the phenomena involved

in their response to seismic actions.

These general remarks are applicable also to mixed masonry-reinforced concrete structures, in the

case of elements made of different technologies not in adherence. The importance of connections

and of floors able to give to the structure a high level of collaboration, seems to be even more felt in

these cases, since different technologies have to behave as a whole and, sometimes, the differences

in terms of mechanical properties and stiffness can enhance the concentration of actions on

structural elements or torsional effects.

The problem of non-uniform stiffness distribution is particularly evident in case the of insertion of

reinforced concrete walls (for example in order to realize staircases or elevators) or in case of uprising

with a reinforced concrete floor; the non-uniform distribution of masses can also cause severe

damages; moreover the irregularity of modes, especially in case of discontinuities in mass

20

distribution along the height of the building, render also inappropriate, sometimes, the use of some

of the load distribution suggested in the codes.

However, as remarked throughout this and the previous chapter, the typology to which this study is

referred is not easily and clearly identified since it is mainly constituted by all that masonry buildings

that, during the first half of the 20th Century together with the spread of new technologies such as

the reinforced concrete one, have undergone modifications of their structure, sometimes even

without following any code prescription, but as a spontaneous need of functional more than

architectural nature.

For what concerns post-earthquake damage detection, the main contribution has been given by

Decanini et al. (2007), who analysed more than 80 post-earthquakes reports.

Their study shows the undoubted difficulty in recognizing a well-defined typology; that’s why

references to mixed buildings with masonry and reinforced concrete not in adherence are scarce or

absent, except for what concerns the report after the Tangshan ear thquake (China) occurred in

July 28th 1976 (Huixian, 2004).

This report describes quite accurately the damages occurred in mixed masonry-reinforced concrete

structures constituted by peripheral masonry walls and internal reinforced concrete elements. The

use of these solutions (there are many and some of them are shown in Figure) that started to be

spread in China in the second part of the 20th Century was driven mostly by the chance to have more

space and flexibility in order to realize, especially in the lower floors, shops, garages, warehouses.

Figure 2. 3 Huixian [2004]. Failure of the external walls at the top of a building (left), collapse of the external wall and subsequent failure of beams

and floor slab.

21

In the city of Tangshan 59 buildings have been inspected and above them only 12 didn’t collapse,

though they suffered heavy damages.

Since that region was not considered to be a seismic one, no particular attention had been paid to

seismic actions in the design of structures, so the fact that some of the buildings did not experience

complete loss of their capacity is more a circumstance related to some particular aspects regarding

the presence of confining r.c. elements placed with a certain spacing in the peripheral walls, or the

high stiffness of some r.c. elements or the properties of the soil under the foundations (most of the

buildings that did not collapse were founded on rock).

In other areas, where the intensity of the earthquake was not so high (VIII-IX), some of the buildings

collapsed, while in other areas where the intensity was even lower (VII) only moderate damage has

been detected.

Referring to the typologies mentioned before it can be concluded that the most damaged ones are

those in which the first floor is realized in reinforced concrete while the upper floors are realized with

unreinforced masonry.

When reinforced concrete frames are spread throughout the height of the building, the ones with

only one column in the centre are quite more damaged than the ones with two or more columns

inside.

One interesting aspect is that it has been possible to detect some kind of sequence in the damage:

masonry panels are usually the first one that loose their capacity, followed by a tendency of the

beams to behave as cantilevers, their subsequent failure, followed by the failure of slabs and internal

frames, leads then to the collapse of the entire building.

Some additional information have been found in the report after the San Ferdinando earthquake,

occurred in February 1971, in which the so-called box-system is widely described. It is a system

made of a single story box realized with brick masonry or reinforced concrete precast panels, internal

columns made of reinforced concrete or, sometimes, of steel. The roof is generally a wooden one

and it’s supported by steel beams. Under horizontal loads, the behaviour of this category of buildings

is mainly influenced by the effect of the collaboration between the roof and the external walls, no

considerable contribution comes from the internal columns and the damage detected has been

22

addressed to the lack of connection between the different structural elements and the distribution of

openings in the peripheral walls.

In the report also the data recorded after the Chilean earthquake occurred in 1985 are included

(Flores et al. 1986); the data are taken by a statistical analysis made on several buildings with

different technologies. In particular, some of the buildings were realized with peripheral reinforced

masonry walls and the internal structure made of confined masonry panels or reinforced concrete

frames; some others were confined masonry buildings with an internal nucleus made of reinforced

concrete walls. Other situations were those of buildings with a lower level realized with reinforced

concrete and upper storeys in concrete block masonry.

The average level of damage is pretty high, measured by a defined index, especially in the last case.

Since it’s a statistical analysis based of a great number of buildings, additional detailed information

concerning the kind, extent of damage and collapse mechanisms are not given in this study.

The case of a building very similar, in terms of bearing structure, to the ones analysed in this thesis,

is reported in the document by Johnstone & Potangoroa (1993) describing the damage subsequent

to the Weber earthquake, occurred in May 1990 in New Zeeland.

It’s a building belonging to the first years of the 20th century with external masonry walls and internal

reinforced concrete structure. It has undergone heavy damages either because of the bad quality of

the construction, and because it had been already damaged during the earthquake occurred in that

same year during February. However that report doesn’t give more detailed information on the extent

and features of the damage.

A report by Baratta (1908) is mentioned because in that document a clear reference is made to the

fact that even after severe earthquakes, in some Italian regions people continued to realize additional

floors made of different technology on pre-existing masonry buildings. These bad habits, together

with the absence of any kind of reinforcement of the lower part, which most of the times presented

inadequate wall thickness or foundations, led to disastrous collapse of buildings.

The work of Coburn et al. (1982) is mentioned because, concerning the Irpinia earthquake (1980),

it addresses some of the damage detected to the fact that during the years there was a tendency to

build additional floors on existing buildings without any engineering guidance.

23

The study conducted by Decanini stresses how difficult is to analyse this category of buildings, on

the one hand because of the great variety of typologies, on the other because of the scarce

experimental references. This survey on past earthquake damage reports has also shown that

sometimes the collapse or the damage occurred can be related to particular factors that to not deal

specifically to the building typology, such as the soil characteristics.

2.3 SEISMIC BEHAVIOUR OF MIXED MASONRY-REINFORCED CONCRETE BUILDINGS:

NUMERICAL AND EXPERIMENTAL STUDIES

Since the interest for mixed masonry-reinforced concrete structure has been growing only in recent

times, only few experimental campaigns can be found in literature and the case studies are very

limited both in numerical and in experimental studies.

Concerning experimental campaigns, apart from reinforced concrete frames with masonry infills,

only few tests on mixed masonry-r.c. structures were conducted in the past.

Tomaževič and co-workers (Tomaževič et al., 1990) carried out a shaking table test campaign on

unreinforced masonry buildings. One of the tested models consisted of a simple squared plan

structure with an internal RC column and two RC beams (Figure 2.4). However, since the masonry

walls were much stiffer than the r.c. column, the latter exhibited almost negligible contribution on

the overall seismic behaviour of the system.

Four scaled models (1/5) representatives of three-floor buildings entirely realized with peripheral

masonry walls but changing the internal structure. Two of the four models have an internal column,

while in the other two models the internal column is substituted by two perpendicular masonry walls.

24

Figure 2. 4 Ground floor plan and elevation of the models tested by Tomaževič et al. (1990). (left): masonry building with and internal RC column.

(right): plain masonry building (dimensions in mm).

The variation within the first and the second couple of models relies in the type of masonry, since

both unreinforced and reinforced masonry have been used.

Shaking table tests have been performed on the models using spectrum compatible ground motions,

with a peak ground acceleration scaled to 0.1 g in order to analyse the dynamic features of the models

in the linear phase.

Then, to study also the nonlinear behaviour and the collapse mechanisms an accelerogram recorded

during an earthquake occurred in Montenegro during 1979 has been used. The motion has been

applied along the short side of the specimens.

In the mixed building with a single central column, both in case of unreinforced and for the reinforced

masonry specimen the damage is mainly concentrated at the ground floor and the first modal shape

is more or less the same at the end of the elastic branch and when the maximum resistance is

reached. What differs between the two models is the natural frequency, since the reinforced masonry

25

specimen is characterized by a higher stiffness. The unreinforced masonry specimen collapsed soon

after the maximum resistance point, showing a 2% drift at the ground level, while the reinforced

masonry one appeared to be more ductile, since the collapse is attained with a drift of 3.4 %. It’s

worth noticing that in the building entirely in masonry the damage is quite uniformly distributed along

the height of the building; the resistance exhibited is slightly higher and a better post peak behaviour,

with a more gradual degradation of strength in the post-peak phase. The ultimate state shows a 3.8

% drift at the ground floor.

Jurukovsky et al. conducted shaking table tests on 1/3 scaled models up to the near collapse limit

state (Figure 2.5). They investigated the seismic behaviour of a masonry structure with one

r.c. frame at the ground floor (Jurukovsky et al. 1989a) and examined several strengthening

solutions (Jurukovsky et al. 1989b; Jurukovsky et al. 1991a). One of these solutions consisted

in the addition of one pin-based reinforced concrete wall, which was continuous from the

foundation up to the roof (Jurukovsky et al., 1991b; Jurukovsky et al., 1992).

A set of five accelerograms was applied as an input along the direction of the bigger side of the plan.

Each of them has been scaled in order to obtain different levels of excitation and at the end of each

test some dynamic features of the model were measured; for example the decrease in the natural

frequency of the model (which stands for a decrease in stiffness) and a variation of the modal shape.

The damages were originally flexural ones and then they turned to be more similar to shear

phenomena, but they were always concentrated at the ground and first floor. The reinforced concrete

frames didn’t show any damage, despite the consistent value of drift reached by the building (more

than 1,5%).

26

Figure 2. 5 Plan of the second floor and elevation of the models tested by Jurukovsky et al. (1992), dimensions in cm.

After each test the cracks were filled with grouting injection and then the test have been repeated on

the retrofitted structure.

In the second set of tests the cracks appeared mainly on the first floor, together with a lower value for

the displacement of the ground floor, which is in good agreement with expectations since the grouting

injection give to the model a higher stiffness.

From that study also some conclusions can be inferred concerning the damage pattern of the

retrofitted configuration with the added r.c. wall, which is quite different from the one of the original

structure.

As typical of URM structures, in the original model the damage in the masonry members is

concentrated in the first floor, whereas in the retrofitted configuration plastic deformations are more

distributed all over the height of the walls. The retrofitting intervention also resulted in a considerable

increase of the global strength, in terms of PGA. In fact, the maximum applied PGA, leading the

specimens to the near collapse limit state, was 0.51g for the original masonry model and 1.07g (so

more than doubled) for the retrofitted configuration.

Alessi et al. (1990) provided some results of experimental studies on the seismic behaviour of mixed

masonry-r.c. buildings, but the high plan and elevation irregularity of the system led the research

interests towards different topics from those studied in this thesis.

Some numerical studies have been conducted, especially in the last 10 years, on mixed masonry-

reinforced concrete structures, especially by Italian researchers.

27

The assessment of one existing masonry structure and its retrofitting with the addition of two

retrofitting walls have been carried out by Casoli (2007). In his research it is outlined that the addition

of the RC walls affects the global response of the structure and, as a consequence, in numerical

simulations the assumption of the stiffness of the reinforced concrete walls influences the global

response of the retrofitted building.

The results achieved in this study also show that the insertion of reinforced concrete walls, if they’re

properly designed according to performance based design principles, can lead to a higher global top

displacement of the retrofitted building if compared to the original mixed masonry-reinforced concrete

(frame) structure.

Cattari and Lagomarsino (2013) carried out non-linear analyses on mixed masonry-reinforced

concrete constructions to simulate a series of interventions that could have occurred in an original

unreinforced masonry building.

Figure 2. 6 Ground floor plan (dimensions in cm) and 3D model of the URM structure before interventions, from Cattari and Lagomarsino (2013).

One of the investigated models consisted in the demolition of the internal masonry walls and their

replacement with reinforced concrete frames and walls. All the reinforced concrete members were

non capacity-designed and they exhibited smaller displacement capacities than the unreinforced

masonry walls, so their introduction lead to a decrease of the displacement capacity of the entire

mixed system, when compared to the original structure. The study has however suggested some

remarks on this typology:

•� The addition of reinforced concrete members changes the global behaviour of the entire

building and their presence must be taken into account for a correct estimate of strength

and displacement capacity;

28

•� The insertion of reinforced concrete elements can improve the seismic behaviour of an

unreinforced masonry building only if the reinforced concrete members exhibit larger

displacement capacities than those of the unreinforced concrete walls.

Augenti and Parisi (2008; 2009) studied the problem of predicting the seismic behaviour and the

distribution of horizontal forces among the elements of torsionally and non-torsionally eccentric

buildings composed of unreinforced masonry walls and reinforced concrete elements. They also

studied the distribution of the internal forces among the members over the height, pointing out the

importance of taking into account the interaction between the various structural elements.

This procedures, however, were developed only for linear elastic analyses and the application in the

plastic range has not yet been carried out.

Figure 2. 7 Isometric view of a mixed RC-URM building (Augenti and Parisi, 2009).

The studies conducted by De Felice et al. (2009) are mainly focused on the analysis of the behaviour

of mixed buildings (the case-studies that have been considered are the same included in the ReLUIS

project), taking into account the contribution of the out of plane failure mechanism of the walls.

2.4 REGULATORY FRAMEWORK FOR MIXED MASONRY-REINFORCED CONCRETE

BUILDINGS: NATIONAL AND INTERNATIONAL CODES

In this paragraph the attention is focused on the code prescriptions that, especially concerning the

seismic behaviour of mixed masonry-reinforced concrete, have been introduced, mainly in the Italian

context, but also with some reference to European and international codes in general.

29

The following documents, issued after two notable seismic events seem to be remarkable since they

firstly recognize the use of another technology (in that period mainly wood was used to these

purposes) in combination with masonry in order to strengthen it and to improve its seismic behaviour.

As one of the first regulations concerning seismic behaviour and prescriptions, the “Norme tecniche

ed edilizie per ricostruire le case distrutte”, issued soon after an earthquake occurred in 1783, firstly

promoted the adoption of the typology called casa baraccata.

The regulations issued by the State of the Church after another earthquake occurred in the neighbour

of Norcia (PG) in 1859 and those consecutive to the one occurred in the Isle of Ischia in 1883 refer

to the same kind of buildings.

Some other relevant documents regarding seismic prescriptions or indications for the retrofit of

buildings damaged during earthquakes have to be mentioned: the RD (standing for Regio Decreto)

issued on April 18th 1909, n°193 “Norme tecniche e igieniche obbligatorie per la riparazione,

ricostruzione e nuove costruzioni degli edifici pubblici e privati nei Comuni colpiti dal terremoto del

28 dicembre 1908 o da altri anteriori”, and another document issued soon after the terrible

earthquake that struck the province of Messina in 1909.

It has to be noticed that in these documents a first understanding of the dynamic features of the

earthquake and the response of the buildings seems to be present, though it is pretty clear that

knowledge and tools were not so much developed as to treat the problem in an accurate way.

Nevertheless, for the first time, the problem of the analysis of the effects of the earthquake on the

buildings is treated according to an embryonic form of the Equivalent static method, applying to the

structure a set of horizontal forces, meant to be the equivalent of the inertia forces developed during

the seismic event.

Concerning the mixed masonry-reinforced concrete technology, the Regio Decreto represents a sort

of key-point for the introduction of the confined masonry as a proper construction technique in the

Italian context. However, though a precise reference to the typology as it is meant in more modern

documents is not present, some of the prescriptions can be seen as a consistent set of rules on the

basis of which the technique has been developing during the years.

30

Some interesting indications can be found within the Decreto in few articles, mainly the 33th, which

says:

“Gli edifici lesionati e non costruiti col sistema intelaiato o baraccato, elevantisi oltre il pian terreno (…) devono essere

rafforzati da montanti di legno, di ferro o di cemento armato, infissi solidamente a incastro nelle fondazioni, continui fino

alla sommità dell’edificio e rilegati tra di loro da cinture al piano della risega di fondazione e al piano del solaio e della

gronda, in modo da formare un’armatura a gabbia. I detti montanti devono essere collocati almeno in corrispondenza di

tutti gli spigoli dell’edificio e in ogni caso a distanza non maggiore di 5 m uno dall’altro”.

Tough it is intended to be a document concerning the retrofitting techniques to be used in case of

masonry buildings damaged during the earthquakes, it gives the fundamental idea of the technique

of inserting a spatial frame to improve the behaviour of the masonry, a quite recurrent feature of

confined masonry, though this kind of constructions, as a proper technology, has been conceived

only after an empirically based evolution of the regola dell’artei, which led through the so called

“telaio denso”, widely used during the reconstruction after the Messina Earthquake, to the modern

recognized typology.

With the following Regio Decreto in 1916 some precise indications are given concerning the value of

the equivalent horizontal forces to be applied to the structure to simulate the effect of the earthquake

and how they must be distributed along the height of the building.

In 1935, another document was issued, but the most relevant thing is that, from that point on,

prescriptions addressed to confined masonry are scarce or absent, and this fact may have led to the

decrease in the adoption of this typology.

Indeed, the confined masonry technology as a proper building typology can be identified as a

“reconstruction” technique, which is, moreover, mainly spread in a certain part of the South of Italy

(Abruzzo, Calabria and Sicily) and belonging to a relatively defined period, which is thought to end

approximately in the Fifties (1950).

Many years after, in 1975, the Decreto Ministeriale introduces for the first time the design spectra

as we recognize them in modern design procedure along with the introduction of the use of dynamic

analysis, though these indications do not seem to be meant for the analysis of masonry buildings

with reinforced concrete ring beams, which are the only mixed typology included in the document.

31

In 1981, after another earthquake that hit Basilicata, Campania and Apulia, the Decreto Ministeriale

and mostly the following instructions (released the year after) suggest the use of ring beams and

reinforced concrete columns in order to enhance the ductility of masonry walls, as it can be seen in

these lines:

“l’inserimento di pilastrini in c.a. in breccia è effettuato a distanze regolari (...) Il funzionamento dell’insieme strutturale

si modifica profondamente in senso positivo, solo se gli elementi in cemento armato sono convenientemente organizzati

tra loro ed in rapporto alla muratura, come può ottenersi eseguendo una serie di cordoli verticali ed orizzontali tutti

collegati tra loro”.

Where the concept of confining and framing masonry panels with reinforced concrete is meant to be

strengthening and retrofitting technique for existing buildings, despite it is not recognized as a proper

construction technique and, in fact, it’s no longer used for new buildings, at least among the Italian

territory.

In 1981 some other important and innovative indications are introduced, concerning the modelling

criteria, referring in general to the equivalent frame approach and describing in detail the POR

method, which is one of the first and most simplified modelling techniques.

This method has been firstly introduced according to the hypothesis of Strong Spandrels Weak Piers

(SSWP): floors and masonry beams connecting the piers at the floor level (spandrels) are regarded

as infinite stiff and resistant: so the behaviour of the building is entirely and only affected by the

behaviour of the piers; in this scenario the modelling of spandrels and floors with their own

mechanical and elastic features becomes negligible.

This simplified approach has been widely used, especially by professional engineers, for the

assessment of existing masonry buildings, with a consistent overestimation of the resources of the

analysed buildings; the use of this hypothesis and the need for the buildings to be consistent with it

(whereas a virtuous trend is to mould the model on the real case) led to the substitution of existing

wooden roof with stiffer reinforced concrete ones and to the simultaneous introduction of reinforced

concrete ring beams, in order to enhance the stiffness of spandrels, that in this case behave as

struts.

Another typology of mixed masonry-reinforced concrete buildings can then be identified, though

ignoring how the increase in stiffness could have worsen the global response of the building has led

32

in the past to some severe damages in masonry building that had undergone deep restoration with

the insertion of stringcourse, or floor substitution.

It’s pretty clear that, dealing with interventions on existing buildings, it has to be acknowledged that,

despite in principle they seem to improve the behaviour of the structure or to let it be more suitable

for a certain kind of modelling approach, sometimes they can radically change (and unfortunately

worsen), the global response of the structure.

The prescriptions introduced by more recent Decreto Ministeriale in 1986 do not change what was

stated in the previous documents for the design of new masonry buildings, but introduces the

fundamental distinction between “adaptation” and “improvement” interventions on structures.

In 1996 reinforced masonry appears for the first time in law prescriptions, along with new simplified

verification method for masonry building and limit state concepts, but still no reference is made to

confined masonry as a codified technique.

Some general considerations are made concerning buildings with masonry and reinforced concrete

not in adherence, on which this thesis is focused.

In the D.M. Lavori Pubblici (1996), the first to deal with mixed building typology meant as buildings

with vertical elements made of both masonry walls and elements or frames of different technologies,

the possibility to realize new structures in which the vertical loads are carried by elements of different

technology is addressed, provided that the horizontal seismic actions are entirely carried by masonry

walls, minding connections between different elements, the deformations’ compatibility and the load

distribution.

In case of uprising some more detailed information are given (this is the case of an existing masonry

building in which an additional is realized, mostly many years after its construction, entirely made of

another technology, weather steel or reinforced concrete): for example, the document gives

limitations in terms of height and suggests to increase the horizontal action to be carried by the

“added” part.

In case of existing buildings with masonry and reinforced concrete (or in general another technology)

elements not in adherence the following paragraph can be found in the document (DM ’96):

33

“nel caso di edifici le cui strutture resistenti siano realizzate con combinazioni di elementi in muratura, in calcestruzzo

armato o metallici, si applicano le prescrizioni relative alla tipologia degli elementi strutturali ai quali é prevalentemente

affidato il compito di resistere alle forze orizzontali. Deve essere verificata la compatibilità delle deformazioni dei vari

elementi presenti nonché la validità dei collegamenti fra gli elementi strutturali di diversa tipologia”.

Some additional information and explanations on what already suggested in the previous document,

are given in the Circolare Ministero dei Lavori Pubblici n° 65 10-4-1997, which contains the

instructions on how to apply the principles given in the Decree:

“la trasmissione delle azioni sismiche in una struttura mista può avvenire attraverso un organismo strutturale che

presenti elementi in muratura ed elementi in cemento armato o in acciaio funzionanti in parallelo (ossia disposti

altimetricamente su piani successivi). Nel primo caso le azioni sismiche devono essere integralmente affidate alla

struttura muraria. La prescrizione è riconducibile alla maggiore rigidezza e minore duttilità che le strutture in muratura

tipicamente hanno rispetto alle strutture monodimensionali in cemento armato o in acciaio. La compatibilità tra le

deformazioni subite dai diversi elementi costruttivi deve essere espressamente valutata; in particolare si dovrà controllare

che le azioni sismiche siano effettivamente attribuibili tutte alla scatola muraria e che la presenza di elementi in cemento

armato o in acciaio distribuiti in modo disuniforme sia planimetricamente che altimetricamente non modifichi

significativamente la posizione del centro di rigidezza della sola scatola muraria e la ripartizione delle azioni orizzontali

tra i diversi setti murari. A tal fine, è da considerare con particolare attenzione l'adozione di corpi scala e/o corpi ascensori

realizzati con pareti in cemento armato, per la forte rigidezza alle azioni orizzontali tipica di tali strutture, ed analoga

attenzione deve essere prestata nel caso di elementi verticali in cemento armato o in acciaio dotati di elevata rigidezza

a flessione ed a taglio. Particolare importanza rivestono i collegamenti tra elementi di tecnologia differente

(orizzontamenti, cordoli, travi di ripartizione). Gli orizzontamenti consentono alle diverse pareti in muratura di scambiare

tra loro forze orizzontali nell'ambito di un complessivo comportamento scatolare ed assicurano la trasmissione alla

scatola muraria delle forze d'inerzia di origine sismica di diretta competenza delle masse gravanti sulle strutture in cls

armato o in acciaio. Occorrerà dunque verificare che gli orizzontamenti, sia in termini di rigidezza che in termini di

resistenza a flessione e taglio nel loro piano, consentano il corretto realizzarsi del meccanismo globale di funzionamento

sopra illustrato. Contemporaneamente si dovrà verificare che non si raggiungano tensioni eccessive per effetto delle

azioni concentrate che gli elementi in cemento armato o in acciaio e i solai si scambiano a causa del sisma e dei carichi

(cordoli, travi di ripartizione, ecc.), e con una continua attenzione alla centratura dei carichi verticali sugli elementi

resistenti sottostanti. Quanto alle prescrizioni relative agli edifici costituiti da struttura muraria nella parte inferiore e

sormontati da un piano con struttura in cemento armato o in acciaio, la limitazione sull'altezza massima è riconducibile

all'intento di contenere le tensioni su tali edifici entro gli ambiti propri degli edifici totalmente in muratura, ad essi

assimilandoli; mentre la prescrizione sulle azioni da attribuire alla parte superiore in cemento armato o in acciaio è legata

all'esigenza di evitare per dette strutture plasticizzazioni premature e conseguenti eccessive richieste di duttilità”.

34

Later on, in 2002 another earthquake stroke the south of Italy, causing the death of 27 children and

their teacher in a school in the town of San Giuliano di Puglia (CB). This tragic event led to the

introduction of the Ordinanza del Presidente del Consiglio dei Ministri 3274 in 2003, then revised

and re-edited in 2005 (OPCM 3431/05); these documents marked a turning point mostly for what

concerns analysis methods (non-linear analyses) and verification criteria introduced; this represented

a deep change in the usual practice, though it basically just aligning the Italian scenario to the

international (mostly European) framework.

Regarding mixed buildings, the construction of new buildings where different technologies are used

for elements acting on the same floor is allowed, provided that the verification method to be used is

the one belonging to the prevalent one, then assigning to that part of the structure the seismic

actions; nevertheless, in the code, reference is made to non-linear methods in order to evaluate

(once the designer recognized the need) the effects of the interaction between the different elements.

The importance of the connections is stressed also in this document regarding buildings in which

uprising are realized with different technology; the prescriptions, in fact, regard the action in the joint

between the elements made of different material, which has to be incremented (30%), and the

accuracy, weather for linear and non-linear analyses, of the adoption of a certain load distribution

along the height of the building, since the presence of and additional floor, especially if made of a

different technique, can deeply change the distribution of stiffness and then render inadequate, for

example, a triangular distribution in order to simulate the first mode of vibration of the structure. It

is then much more appropriate in this case to refer to more refined distribution or analysis

approaches which first of all take into account the real modal shape of the building and sometimes

it could be worth to use an adaptive pushover procedure (though scientific contributions on this

aspect are still lacking and adaptive pushover techniques for masonry buildings are still being studied

and developed).

For what concern existing buildings, their assessment is extensively dealt with, though mixed

buildings are referred to in a small extent, especially if compared to the instructions given, in the

same document, for what concerns new mixed masonry-reinforced concrete buildings.

In the OPCM 3431, the theme of the choice of the most appropriate behaviour factor in case of

buildings with floors made of different technologies is treated, especially when the lower part is made

of masonry and the upper part is realized with reinforced concrete or steel. In particular, while

performing linear analysis, the code prescribes to adopt the behaviour factor that would have been

35

adopted in case of simple masonry building, while to analyse the upper part one must refer to the

behaviour factor belonging to that precise typology (however not lower than 2.5).

As a general remark, it can be assessed that in case of mixed buildings for which the seismic action

is meant to be assigned to elements realized with the same technology the behaviour factor can be

assumed as the one belonging to that kind of buildings.

When the different elements are both resisting to seismic actions, the definition of a response

spectrum to be used for the entire building is a much more delicate subject, since one must take

into account the equivalent behaviour of a single degree of freedom system, which should represent

the building as a whole.

In this case, however, it seems to be more appropriate to use different values of the behaviour factor,

depending on the element that is being analysed.

It’s clearly recognized how, along with buildings subjected to uprising with different technologies,

pre-existing masonry buildings may have undergone deep modifications, which could have led to the

substitution of internal masonry walls with reinforced concrete frames or plan enlargements with

different technologies; the code, however, doesn’t give precise indications but underlines the need

to use modelling techniques and types of analysis able to take into account the peculiarities of these

structure, clearly implying non-linearity.

The current D.M. Infrastrutture (2008) provides the same criteria for both new and existing buildings.

For new buildings, the seismic actions could be entirely distributed either only on masonry walls or

only on the elements of different technologies but, if the designer wants to take into account the

collaboration between masonry walls and elements of different technologies in withstanding seismic

actions, the structural behaviour should be assessed by nonlinear analysis.

For existing buildings, it is recommended to perform nonlinear (static or dynamic) analysis

accounting for the collaboration between masonry walls and elements of different technologies. It is

easy to recognize, in fact, that a reliable evaluation of the seismic action sharing between structural

elements characterized by different technologies is not easy to achieve using simplified models based

on linear analysis.

36

Besides Italian code D.M. Infrastrutture (2008) highlights the important role of slabs in distributing

the seismic action; this phenomenon, in fact, depends on the stiffness of the slab in its own plane

and especially on the relative stiffness compared to the one of vertical resistant elements.

The international framework, no matter the spread of the mixed typology also in several other

countries, lacks in giving accurate law prescriptions.

The Argentinean guideline (Normas Antisismicas Argentinas (NAA) 1980) points out the fundamental

role performed by slab in distributing seismic actions between vertical resistant elements realized

with different technologies.

In the hypothesis of rigid slab, these guideline suggests distributing horizontal loads between

structural vertical elements based on their stiffness, while in the hypothesis of a flexible slab, the

seismic load repartition can be performed assuming the slab as a continuous beam on rigid supports.

Three different situations are taken into account:

•� Rigid Floor: in this case horizontal forces will be distributed according to the stiffness of the

vertical elements and the additional shear given by torsional effects will be taken into account

among the different elements proportionally to their stiffness and to the square of the

distance between the element and the centre of stiffness.

•� Flexible floor: Vertical elements act independently and torsion is negligible. Force distribution

can be calculated assuming the behaviour of the floor similar to the one of a continuous

beam on rigid supports. The shear, which is calculated with this approach, however, cannot

be lower than the one resulting from an influence areas approach.

•� Intermediate cases: the action repartition within the elements must take into account the

actual stiffness of the floor. However it’s also possible to consider the maximum value

between the two values obtained using the previous extreme hypotheses.

The Mexican seismic code (Gaceta Oficial del Distrito Federal, 2004) deals with the definition of the

behaviour factor for mixed structures intended as the Italian Code does (with bearing masonry walls

and other elements made of different technologies). The value suggested by the Mexican code is 4,

but provided the following conditions are fulfilled:

37

•� Elements realized with a different technology (reinforced concrete or steel frames, reinforced

concrete walls) are able to sustain at least the 80% of the seismic action

•� All the elements are designed in order to achieve a certain level of ductility

•� The single floor shouldn’t exhibit a global resisting shear different for more than 35% from

the average value of this parameter

If all these prescriptions are not respected the value of the behaviour factor must be taken as 2 or

1.5 depending on the type of masonry used for the bearing walls.

It is important to highlight that, actually, the other main international codes, such as Eurocodes

(CEN) (2001, 2004) and American standards (American Concrete Institute), don’t deal specifically

with combined RC-masonry buildings.

New versions of the code should also account for the fact that real construction might result in mixed

structures, such as the classical reinforced concrete wall-frame structure but also across building

materials, such as reinforced concrete elements combined with steel frames or load-bearing masonry

walls.

It is expected that the portion of such mixed structures within the entire building stock will increase

in the future due to seismic retrofit interventions, and because structures might be altered or

extended rather than completely new structures built. Such mixed structures are more complex to

assess than structures with a single structural system and guidelines for the latter cannot simply be

extrapolated to mixed systems.

Displacement-based procedures are very well suited to deal with mixed structural system (Priestley

et al., 2007). The nonlinear static assessment procedure included in EC8 can therefore be readily

applied to mixed structural systems. However, for the design of new structures it would be also

desirable to develop a force-based design approach for mixed structural systems. This design

approach needs to account for the characteristics of mixed structural systems with regard to the

stiffness and strength of the subsystems that are coupled (Beyer, 2015).

Beyer at al. have approached mixed masonry-reinforced concrete wall buildings with the elastic

shear-flexure cantilever beam model, for which deformed shape and closed form solutions for internal

force distributions exist. This model has been applied to the displacement-based assessment of

38

systems with URM walls and RC walls (Paparo and Beyer, 2015) and current work at EPFL

investigates the development of behaviour factors for mixed systems based on this model.

Confined masonry is not the object of this investigation but some small remarks can be outlined,

especially to underline how, compared to the mixed buildings as such (with masonry and reinforced

concrete elements not in adherence), the regulations in the Countries where this typology is quite

spread (Turkey, China, and some Countries in South America), give some indications concerning

their design and verification.

These prescriptions, indeed, are most of the times empirically based ones, and refer to geometrical

features and construction processes (maximum height od the buildings, maximum distance within

the columns, minimum requirements for the reinforcement in confining elements).

For what concerns the verifications, regulations such as the Mexican code (GNDT) suggest to take

into account the contribution of reinforced concrete in the flexural resistance of the panels neglecting

their influence on the shear mechanism, while Eurocodes (EC6, EC8) assess that the concrete

elements should be taken into account in the evaluation of the shear capacity of the walls, provided

that an appropriate shear reinforcement is present in the former.

39

CHAPTER 3: SEISMIC ASESSMENT OF MASONRY STRUCTURES AND MODELLING CRITERIA

The large population of existing and historical mixed masonry-reinforced concrete buildings, mostly

in Italy but also in many other countries in the Mediterranean area, and their potentially high

vulnerability to earthquakes require to improve the knowledge of their seismic behaviour, and to

assess the suitability of analytical and numerical models for their structural assessment.

The reliability of models represents nowadays one of the most popular and important issues, mostly

in the assessment and strengthening of existing buildings.

This chapter is focused on the modelling techniques and the analyses suitable for masonry,

reinforced concrete and mixed masonry-r.c. buildings, mostly considering a global behaviour. In this

context, the presence of local mechanisms is not taken into account and they are supposed to be

part of specifically addressed verifications, which aren’t, however, part of this thesis in which the

buildings are thought to show a global box behaviour, assuming that proper connections prevent the

activation of local failure modes, mainly associated with the out-of-plane response of walls.

Masonry buildings had exhibited during past earthquakes bad behaviour mainly because of the poor

quality of the materials and the bad construction practice; that is one of the reasons (but of course

not the only one) that have led framed structure, usually in reinforced concrete or steel, to be

nowadays the most used construction systems.

The theme of sustainability, as remarked also by Marques (Marques & Lourenço 2014), is, on the

other hand, a driving principle in societal, economical and environmental aspects and the use of

more economical and environment-friendly construction typologies has become mandatory. In this

40

context, unreinforced masonry can play an important role in the construction of low- and medium-

rise buildings.

That’s the reason why new models and methodologies have become more and more widespread

subjects in the scientific community; they differ either in terms of the scale of the detail or for what

concerns the theoretical bases on which they lay; the great variety of models can be somehow

justified since the different types of constituents and of their arrangement prevent to assess the

suitability of a model for every kind of masonry.

However, in the professional field, the use of simplified models is driven mostly by their cost-efficiency

and by the circumstance that the results are available in a relative short time and with a limited

computational effort.

Indeed, attention must be paid on a conscious use of these models, since they are based on strong

simplifying hypotheses on the geometry and the mechanical behaviour of masonry structures. Thus,

their reliability depends on the consistency between such hypotheses and the actual structure.

3.1 MODELLING MASONRY BUILDINGS: A DEFINITION OF DIFFERENT MODELLING

APPROACHES

Masonry is a composite material, which involves ordered arrangement of units and mortar in

alternate layers; even though, as a material, it involves only two components, since different type of

units (mainly bricks, stones, adobes, blocks) along with different types of mortar (lime, clay, cement,

bitumen) and many possible arrangements can be used in combination, different kinds of structure

can be identified1.

1 This classification doesn’t include all the possible disordered arrangements, which are very common especially among

historical masonry.

41

Figure 3. 1: Different types of arrangement

As a result, the strength of masonry, also depends on the geometrical arrangement of units and

mortar and not only on their mechanical properties; however, even if the mortar has a good quality

and its properties are comparable with those of the blocks, the mortar joints often act as a plane of

weakness both in horizontal and in vertical direction and, since the interface between units and

mortar has weak degree of union, the response is strongly dependent on the orientation of joints.

There are many factors that can then influence the behaviour of masonry. According to Hendry

(1990), in general the stress-strain behaviour depends on:

•� Units: compressive and tensile strength, type and geometry (solid, perforated, hollow etc)

and absorption capacity

•� Mortar: strength, thickness, Poisson’s ratio

•� Unit-mortar interface: bond between the two, direction of stress and local strain.

Masonry can be modelled according to different approaches and scales for the description of its

behaviour.

Depending on the level of accuracy and the simplicity desired, it is possible to use the following

modelling strategies: (a) Detailed micro-modelling, in which units and mortar in the joints are

represented by continuum elements whereas the unit-mortar interface is represented by

discontinuous elements; (b) Simplified micro-modelling, in which expanded units are represented by

42

continuum elements whereas the behaviour of the mortar joints and unit-mortar interface is lumped

in discontinuous elements; (c) Macro-modelling, in which units, mortar and unit-mortar interface are

smeared out in a homogeneous continuum (P.B. Lourenço, Oliveira & Milani).

Figure 3. 2: Modelling strategies for masonry structures (from left to right): detailed micro-modelling, simplified micro-modelling, macro-modelling.

(Lourenço et al.)

Using the first approach, all the properties for units and mortar need to be defined in order to perform

the analysis: Young’s modulus, Poisson’s ratio and, if the non-linear behaviour needs to be taken

into account, also the inelastic properties of the constituents must be included in the input data. The

interface between block and mortar is seen as a potential failure plane and a dummy value of the

stiffness is usually given, in order to avoid interpenetration of the continuum phenomena.

One of the main difficulties is, provided the undoubted computational effort that the numerical model

requires for its solution, the definition of all the parameters for each constituent which can be not

feasible, especially for existing structures and especially when experimental tests’ results are not

available.

In the second approach, which approximates the behaviour of the joint, (which consists of mortar

and the two unit-mortar interfaces), the properties of both mortar and interfaces are grouped into an

average interface while the dimensions of units change since they need to be expanded in order to

keep the geometry of the entire elementary cell unchanged. This process causes a predictable loss

in terms of accuracy since Poisson's effect in the mortar joint is not taken into account.

The third approach, that approximates the masonry as an anisotropic continuum, doesn’t make a

distinction between individual units and joints but assigns to the entire masonry unit homogenized

properties that include the peculiar behaviour of both the constituents and their arrangement.

43

It’s very difficult to assess in absolute terms which of the scales can better interpret the real behaviour

of the material intended as a whole; in fact, this issue is a matter of compromise between the

accuracy required and the efficiency in terms of time and costs needed.

In principle, for a detailed understanding of the problems, especially in a small scale, the micro-

modelling appears more suitable since it gives a large amount of information at a local scale, but

this approach requires large computational efforts and it’s time consuming. Facing more practical

problems and at a larger scale, micro-modelling approaches become inadequate and sometimes the

detail level that can be achieved is not even necessary.

The FEM approach can be used at all these different scales, provided an adequate choice of the

elements and the material properties.

A second approach, which is substantially different from the first and can be included in the models

that describe the material at a macro-scale, consist in idealizing the structure through an “equivalent

frame”.

This approach resulted particularly suitable for the analysis of standard masonry buildings made up

of well connected walls with openings’ patterns that can be defined as regular. Suggested nowadays

by national and international codes (Eurocode 8 and the Italian Code) is based on an idealization of

the structure through a frame in which each resistant wall is discretized by a set of masonry panels

in which the non-linear response is concentrated (Tomaževič M. et al., 1990; Magenes G. et al.,

1998; Galasco A. et al., 2004; Penelis GG, 2006; Roca P. et al. 2005).

The panels are connected by rigid nodes and are discretized into piers, which are the principal

vertical resistant elements for both dead and seismic loads; spandrels, which are secondary

horizontal elements, coupling piers in the case of seismic loads.

44

Figure 3. 3: Different approaches in modelling masonry buildings: Finite element model and Equivalent Frame idealization (Calderini)

Usually, in this approach, only in-plane resistant mechanisms are considered.

By concentrating damage, sliding and rotations in defined sections of the structural elements, these

models enable to perform non linear incremental analyses of entire buildings, provided that local

mechanisms have already been analysed and prevented.

The model of the entire structure is obtained by assembling masonry walls (idealized as 2D frames)

and horizontal floors (not necessarily assumed as rigid). The Equivalent Frame idealization in those

cases in which walls do not have regular opening patterns, the proper prediction of the in-plane load

bearing capacity of panels, the interaction between orthogonal walls, may imply a series of

hindrances.

Figure 3. 4 Damage detected in masonry buildings (Lagomarsino)

In this thesis the study of two different case-studies of buildings with mixed masonry-reinforced

concrete structures has been conducted referring both to simplified equivalent frame models, and

to more detailed finite element procedures, adopting a macro-scale for the definition of the

mechanical parameters characterizing the behaviour of the masonry.

45

It seems then appropriate, in the following paragraphs, to give an insight on both the modelling

approaches.

Among the big amount of different models that can be used to analyse masonry buildings, it’s

possible to identify some classification criteria.

A first distinction is possible among the models that are based on the limit analysis and a kinematic

approach, in which the collapse load multiplier is calculated basing on equilibrium principles and the

identification of a collapse mechanism, considering masonry portions as rigid bodies.

As an alternative to these methods, it’s possible to refer to a wide set of different models that consider

deformations in the elastic branch followed by inelastic deformations.

Figure 3. 5 Influence of the assumed stiffness of spandrel elements

Most of these simplified models that have been used to study the behaviour of masonry, have been

derived from the observation of damage patterns in existing buildings after past earthquakes, studied

and correlated to their behaviour by the concepts of structural mechanics.

When they originally began to be introduced and developed, these methods were based on two-

dimensional macro-elements and they were supposed to be used to perform planar wall analysis,

and assuming for example ‘‘no-tension’’ hypothesis.

Indeed the need to consider a global response of buildings led many researchers to idealize one-

dimensional macro-elements to simulate a similar response to that of framed structures, and to apply

then conventional methods of structural mechanics.

46

Among these models, some categories can be identified: the ones which keep a bi-dimensional

approach to the problem, in which it is possible to isolate the elements (piers and spandrels), that

can be idealized as squat beams with a marked non-linear behaviour or with struts.

In the case of bi-dimensional elements a fundamental aspect of the modelling is the assumption of

no tension behaviour for the material, circumstance that provides a different stiffness to the element,

according to the stress state.

The mono-lateral behaviour is intended when the tensile strength of the material is limited or

negligible; this behaviour can be generalized in the sense that the tensile strength is supposed to be

null in every direction, or limited to some particular directions, usually parallel to mortar joints.

The implementation of this condition is made through techniques that manage to modify the

geometry of the elements, not considering those regions subjected to tensile stresses, as in the

models proposed by D’Asdia e Viskovic (1994) or through a specific formulation of the stress

distribution among the panel as in the model proposed by Braga e Liberatore (1990)

Considering the previous two models, in the compressed areas linear elastic stress-strain relations

are kept and it’s necessary to introduce checks on the maximum value of the reached compressive

stress to take into account crushing failure modes.

The same kind of checks is required also in case of shear failure, since the no tension hypothesis

it’s not necessarily on the safe side.

Usually, once the resistance criteria are defined, the analysis is stopped as soon as the strength

threshold is reached.

Another class of models is the one of mono-dimensional models, in which the panel is idealized as

a beam or a strut element: Calderoni et al.(1987 and 1989) originally thought to model the acting

portion of the wall with a strut whose inclination and stiffness are able to reproduce the average

behaviour of the panel.

The partialization of the section is followed by a variation of the geometrical properties of the

equivalent strut (inclination, dimensions), and that’s why these methods are usually referred to as

variable geometry methods.

The collapse of the single panel is achieved once a precise limit equilibrium configuration is reached

or the limit stress for compression is attained in a section.

47

Another category of elements is the one which includes beam elements that, however, take into

account shear deformation; among these models there are many formulations that can be used:

some of them are modelled with a changing stiffness, and are based on the partialized sections

Braga and Dolce (1982), some others are characterized by a constant stiffness in the elastic phase,

followed by a plastic deformation (Tomaževic, 1978, Dolce, 1989, Tomaževic e Weiss, 1990).

In the last case the non-linearity which characterizes the behaviour is triggered by a limit state defined

in terms of strength. Most of the analysis methods which are based on the floor mechanism

(including the well-known POR method) are included in this latter group.

The POR method takes into account a storey failure mechanism, in which the global response of

each storey in terms of base shear-storey displacement can be computed as the sum of the individual

response of each wall. However real buildings damaged by earthquakes and experimental testing

programs showed that many other mechanisms are possible since the geometrical and mechanical

features can differ a lot and cause mixed failure mechanisms. In fact, masonry materials, geometry

of the piers mainly in terms of slenderness and the existing normal pre-compression acting on the

panels allow to obtain flexural, diagonal shear and sliding shear mechanisms and most of the times

all these phenomena act together among the same wall.

It can be worth noticing that, during the last years, some limitation of POR method have lead to his

progressive disuse; they were related mainly to:

•� the assumption that masonry piers are the only parts of the structure that can be damaged,

without considering the possibility for the spandrels to be damaged as well (SSWP

hypothesis);

•� only one failure mechanism is possible for piers (shear failure for diagonal cracking) not

considering the flexural/crushing mechanisms

Different versions of the POR model were aimed to an improvement, especially for what concerns

the latter aspect among the two aforementioned ones(Dolce, 1989, Tomaževic e Weiss, 1990),

introducing additional failure criteria.

48

The first limitation has not been overcome completely, since it’s a direct consequence of one of the

fundamental hypotheses of the model (the floor mechanism which characterize the overall

behaviour), according to which the analysis is made separately for each floor and in this context the

behaviour of the spandrels can be considered only in an approximate way.

The model proposed by the research group in the University of Genova, coordinated by Professor

Lagomarsino (Gambarotta e Lagomarsino, 1996; Brencich e Lagomarsino, 1997 e 1998) is different

from the ones presented above; though it gives a good estimate of the non-linear behaviour of

masonry panels in non-linear static analyses, it has been conceived to study the cyclic behaviour of

masonry structures.

It is a macro-element model and, as such, it can be used to perform dynamic analyses with a

reasonable computational effort.

The cinematic quantities involved in the model are mainly displacements and rotations in the

extremities of the elements and the resultant forces in terms of axial stress, bending and shear.

However, since some internal variables and some considerations on rocking or shear-sliding are

included in the model, a bi-dimensional nature of the model can be slightly identified; it is able to

give, then, a complete view of the most important features of the non-linear behaviour of masonry

panels.

One of the main limitations of the models that, however, have been gradually overcome during last

years, is the ex-post calibration of the mechanical parameters of the material in order to obtain more

reliable results, especially when compared to experimental or more refined numerical modelling, that

sometimes is needed.

Figure 3. 6 Macro-elements of variable geometry, multiple fans, three layers, equivalent frame, multiple springs

49

Despite this, the great advantage brought by this model is that, since it is able to reproduce the cyclic

behaviour of masonry, it can be used to simulate the processes of dissipation of the energy in case

of non-linear dynamic analysis and, since the model is still more simplified that a rigorous finite

element model, this can be done with a reasonable computational effort, then suitable also for more

practical applications.

Software codes TreMuri and SAM II provide examples of this kind of macro-models, whose

introduction as computational tools was initially motivated by the introduction of the new Italian code

OPCM 3274/2003 and its revision OPCM 3431/2005 which represented a significant change in

modelling and analysis procedures, more aligned to European standards, and they have been then

developed in the commercial codes 3Muri and AndilWall respectively.

Both programs use an equivalent frame model according to which each wall of the building is

subdivided into piers and spandrels (each modelled by a macro-element), which are then connected

by rigid nodes.

These models have been used and continue to represent one of the most used approaches, mainly

in the professional practice. More recently, research group coordinated by Caliò and Vanin and

Foraboschi assessed the presence of some inaccuracies in the use of beam-type macro-elements in

order to simulate the behaviour of an actual bi-dimensional portion of a wall, mainly because the

simulation of the interaction between macro-elements is not accurate and because the cracked

conditions and the evolution of the phenomenon are involved in the problem in a too simplified way.

These authors proposed to analyse the problem with two-dimensional elements, using a pinned

quadrilateral made with four rigid edges, in which two diagonal springs are connected to the corners

to simulate the shear behaviour and some other springs with a unidirectional constitutive law govern

the bending behaviour: the interaction between two adjacent panels in the first case, and a strut and

tie model in the latter.

One of the most important advantages of the aforementioned models, maybe the one that leads to

heir widespread together with the relative simplicity of the implementation phase, which is often

managed by user-friendly interfaces, is that these approaches require only few parameters to

describe the behaviour of the material: elastic modulus, shear modulus, compressive strength and

50

pure tangential shear strength (this represents a not negligible advantage especially because in case

of existing masonry buildings the actual regulations give for the material, intended as a whole, the

exact same parameters in order to characterize the behaviour.

However, the tensile strength needs to be defined as well, implicitly or explicitly, since it influences

the flexural behaviour in the model proposed by Caliò and the diagonal shear domain in the one-

dimensional models.

The approaches described above are based on experimental tests mainly performed on single

panels, but the results from modelling an entire building are thought to be much more complex and

the interaction between panels becomes non-negligible while considering a problem at a building

scale.

In the latest Italian Building Code (NTC 2008. Decreto Ministeriale 14/1/2008. Norme tecniche per

le costruzioni. Ministry of Infrastructures and Transportations. G.U. S.O. n.30 on 4/2/2008; 2008)

the WSSP hypothesis is assumed for the simplest allowed modelling technique (cantilever models)

in case of the analysis of the so-called simple buildings; the SSWP hypothesis (storey mechanism)

is, on the contrary, no more allowed for the assessment of multi-storey masonry buildings.

It’s undoubtedly true that the adoption of such simplified and manageable models reduces

consistently the amount of time required to study the seismic behaviour of an entire building;

however, it’s worth noticing that the use of these fast solutions requires some a priori choices to be

made.

Furthermore, some walls may show a behaviour that can’t be approximated by one of the two

extreme conditions because it is a combination of the two previous models and can show both types

of response in different regions or which can change in a different behaviour with the increase of

nonlinear response and the progressive failure of some of the elements.

However, sometimes the presence of certain constructive details, like reinforced concrete lintels or

ring beams, though in some cases not supported by a quantitative evaluation of their effectiveness,

can guarantee the achievement of most of the hypotheses on which these simplified models are

based (Cattari S, Lagomarsino S. 2009).

51

The use of equivalent frame models is allowed by latest regulations which define the cases in which

spandrels have to be taken into account as coupling elements in the structural model, basing the

assumption on the conditions in terms of bonding to the adjoining walls, quality of the connection

both to the floor tie beam and to the lintels.

Once the wall is idealized by its structural components and their geometrical arrangement in the

local and global coordinate systems have been defined, a representation of the characteristics of

each structural member is crucial for a reliable prediction of its overall behaviour.

A shear-displacement analysis requires also some assumptions on the constraints of the piers’

extremities; the type of constraint depends on the stiffness and the resistance of the horizontal

elements that are supposed to couple the piers, such as spandrels or reinforced concrete ring beams,

that are increasingly stressed, while the horizontal load applied on the structure increases and, as a

consequence, subjected to cracking or failure mechanisms.

It is clear that these phenomena can be analysed with an acceptable accuracy only if a global analysis

is carried out.

The global analysis is moreover the only way to analyse the building and guarantee the overall

equilibrium; the analysis which is conducted for each floor separately doesn’t take into account the

variation of normal action on the piers, though this aspect can strongly influence the stiffness and

the resistance of these elements.

3.2 DIANA

As mentioned in the previous sections, the case studies presented in the following chapters will be

analysed using both a simplified equivalent frame approach, and a finite element approach, through

general purpose finite element code DIANA; a brief description of some of the fundamental features

of the software may be useful to highlight the peculiarities of the approach and the variables involved

in the problem.

DIANA is a multi-purpose finite element code, based on the displacement method. The program’s

functionalities include wide element, material and procedure libraries based on advanced database

52

techniques, linear and non-linear capabilities, full 2D and 3D modelling features and tools for CAD

interoperability.

For what concerns the element types, DIANA offers a great variety of elements, such as beams (three

classes of beams, either straight or curved are available), solids, membranes, axisymmetric and

plane strain elements, plates, shells, springs, and interface elements.

Concerning the material models, a great variety is available and it’s not possible to refer here to all

of them; all these features can be implemented, such as elasticity (linear isotropic and orthotropic

elasticity, nonlinear elasticity, hyper-elasticity, visco-elasticity, regular plasticity, orthotropic plasticity,

visco-plasticity), cracking (smeared crack, total strain fixed and rotating crack), interface

nonlinearities (discrete cracking, crack dilatancy, bond-slip, friction, nonlinear elasticity, and a

general user-supplied interface model) as well as user-supplied models, in order to include particular

and customized aspects of the behaviour.

The analyses that can be performed are also many and they can be linear static, nonlinear, dynamic.

Euler stability analysis, potential flow analysis, coupled flow-stress analysis, phased analysis,

parameter estimation and lattice analysis are also possible.

DIANA offers a wide range of material models for the analysis of the non-linear behaviour of concrete,

which comprises cracking, crushing and shearing effects in cracks and joints, special techniques for

modelling reinforcement and pre-stressed cables, determination and integration of creep and

shrinkage and advanced solutions for the analysis of young hardening concrete. Moreover, special

elements may be used to model embedded reinforcement in concrete structures: bars, grids and

pre-stressed a built-in pre-processor (FX+) in which reinforcement can be defined globally.

3.2.1 Structural Elements, Deformations and Strains

Different elements to model the geometry are offered in the software package DIANA: beam

elements, truss elements, plane stress elements, plane strain elements, axisymmetric elements,

plate bending elements, flat shell elements, curved shell elements, solid elements, interface

elements, embedded reinforcements and other special elements.

53

In the models that have been analysed, beam elements have been used to model the concrete parts

(beams and columns), embedded reinforcements to represent the steel bars and curved shell

elements for masonry walls.

Beam elements are characterized by a mono-dimensional geometry, which requires that one

dimension (longitudinal one) prevails on the other two dimensions, perpendicular to it.

Beam elements may have axial deformation � shear deformation �, curvature � and torsion:

therefore, the strength components that can be described for this category of elements are axial

force, shear force and moment.

The element library, however, offers three classes of beams:

•� Class I: classical beam elements with directly integrated cross-sections. These elements may

be used in linear and in geometric nonlinear analysis;

•� Class II: fully numerically integrated classical beam elements. These elements may be used

in linear and in geometric and physic nonlinear analysis;

•� Class III: fully numerically integrated Mindlin beam elements. These elements may be used

in linear and in geometric and physic nonlinear analysis.

Figure 3. 7 Beam Elements, characteristics

Main variables for the elements are the displacements in the nodes and the orientation of the

displacements depends on the beam class and on the dimensionality.

54

Figure 3. 8: Displacements for class-I and class-II beams. Left: two-dimensional, Right: three dimensional.

Figure 3. 9 Displacement for class-III beams. Left: two-dimensional, Right: three-dimensional

DIANA can calculate strains and stresses in so-called stress points of beam elements.

For the numerically integrated beam elements (class-II and class-III) the stress points are equivalent

with the integration points. For the directly integrated beam elements, stress points must be specified

explicitly. The following figure shows the element axes in an infinitesimal part dx of a beam element.

Point P is a stress point in a cross-section.

For the fully numerically integrated beam elements (class-II and class-III), DIANA derives the

deformations for an infinitesimal part from the displacements in the nodes. The deformations that

55

can be derived depending on the dimensionality of the beam element. From these deformations,

DIANA derives the primary strains as described for the three classes of beam elements. The sign

convention for strains is that an elongation yields a positive strain.

Figure 3. 10: Deformation for two-dimensional beams

Figure 3. 11: Deformation for three-dimensional beams

For beam elements, Diana can calculate forces and moments in nodes and cross-sections and

Cauchy stresses in stress points. The set of forces, moments and stresses depends on the

dimensionality of the element.

� ��

��

��

Figure 3. 12: Moments and forces for two-dimensional beams

56

Figure 3. 13: Moment and forces for three-dimensional class-III beams

For all beam elements, Diana performs a numerical integration along the bar axis (in ξ

direction). The class-II and class-III beam elements are integrated in the area of cross-section as well.

Reinforcements

Discussing about modelling the reinforcements, DIANA offers appropriate elements precisely meant

to this purpose: embedded reinforcements to model steel bars and grid reinforcements to model

shear reinforcements: they have the characteristic to add stiffness to the finite element model.

57

Reinforcements are, most of the times and especially in the case of common reinforced concrete,

modelled as embedded in other structural elements; the peculiarity of this elements is that the code

ignores the space occupied by an embedded reinforcement and the host element doesn’t loose

stiffness or weight; consequently reinforcement doesn’t contribute to the weight (mass) of the

element, and is not described by specific degrees of freedom.

Figure 3. 14: Reinforcement Bar, topology and stress

Reinforcement strains are, by default, computed from the displacement field of the element in which

they’re embedded; this implies a perfect bond between the reinforcement and the surrounding

material. However, it is possible to specify that the reinforcement is not bonded to the embedding

elements and slipping phenomena can be taken into account.

The variables for a bar reinforcement are the strains �� and the stresses ��. The strains and

stresses are coupled to the degrees of freedom of the surrounding element. It has to be specified

that reinforcement can be embedded only in beam elements of class-II and class-III.

Curved Shell elements

The curved shell elements in Diana are based on isoparametric degenerated-solid approach by

introducing two shell hypotheses:

•� Straight-normals: assumes that normals remain straight, but not necessarily normal to the

reference surface. Transverse shear deformation is included according to the Mindlin–

Reissner theory.

•� Zero-normal-stress: assumes that the normal stress component in the normal direction of a

lamina basis is forced to zero: �� . The element tangent plane is spanned by a

lamina basis which corresponds to a local Cartesian coordinate system �� defined at

each point of the shell with �� and �� tangent to the �� plane and ��perpendicular to it.

58

Figure 3. 15: Curved Shell elements, characteristics

The in-plane lamina strains � �� vary linearly in the thickness direction. The transverse

shear strains � �and �� are forced to be constant in the thickness direction.

Since the actual transverse shearing stresses and strains vary parabolically over the thickness, the

shearing strains are an equivalent constant strain on a corresponding area. A shear correction factor

is applied using the condition that a constant transverse shear stress yields approximately the same

shear strain energy as the actual shearing stress.

Five degrees of freedom have been defined in every element node: three translations and two

rotations. Further characteristics of curved shells are the following. They must be thin and force loads

F may act in any direction between perpendicular to the surface and in the surface. Moment loads

M should act around an axis which is in the element face.

The basic variables in the nodes of the curved shell elements are the translations �� in the

global ��directions and the rotations �� and �� respectively

around the local and �� axes in the tangent plane.

59

Figure 3. 16: Displacements and translations in curved shell elements

The displacements in the nodes yield the deformations �� of an infinitesimal part

�� and the deformations �� of an infinitesimal part � ��. From these deformations

DIANA derives the Green-Lagrange strains in the local �� axes. These Green–Lagrange strains are

derived for all integration points. The sign convention for strains is that an elongation yields a positive

strain and that a positive curvature has the convex side in �� direction.

From the basic strains of Equation Diana derives the Cauchy stresses of Equation in the integration

points and bending moment and forces.

Figure 3. 17: Cauchy stresses

60

Figure 3. 18: Generalized Forces

3.2.2 Material Properties

Concrete and masonry

To model concrete structures, or in general structures made of brittle and quasi-brittle materials,

DIANA offers a wide range of element types. The constitutive behaviour of quasi-brittle material is

characterized by tensile cracking and compressive crushing, and by long-term effects like

shrinkage and creep.

Cracking phenomena can be modeled with a multi-directional fixed crack model with tension

softening and shear retention. Brittle cracking, linear tension softening, multi-linear softening, and

nonlinear softening according to Moelands et al. and Hordijk et al. is available. Also a plasticity-based

formulation for cracking is available: the principal stress criterion of Rankine which shows much

resemblance with the rotating crack model.

In multi-axial stress states the compressive stress can exceed the compressive strength of the

material. In this case the crack model can be combined with a plasticity model which describes the

crushing of the material. Especially the Mohr-Coulomb and Drucker-Prager model are applicable for

quasi-brittle structures.

The combination of tensile and compressive stresses can also be modeled with a multi-surface

plasticity model, available for biaxial stress states. However, this model too is only applicable for

plane stress, plane strain and axisymmetric elements.

The Maekawa concrete model, modified for DIANA, combines a multi-axial damage plasticity model

for the compressive regime with a crack model based on total strain for the tensile regime. This

model also describes hysteresis effects.

61

Masonry structures are analyzed on two different levels: the macro level where the global behaviour

is simulated, and the meso-level where the behaviour is analysed in more detail. For the first case,

DIANA offers the multi-directional fixed crack model and the plasticity models to simulate cracking

and crushing respectively. However, the orthotropic nature of masonry cannot be modelled with the

fixed crack and standard plasticity models, because these models involve isotropic elasticity and do

not allow combination with orthotropic elasticity. The anisotropic Rankine-Hill plasticity model is

appropriate for modeling masonry, because it allows orthotropic elasticity to be employed and

incorporates different strength and degradation parameters to simulate the different behavior parallel

and perpendicular to bed joints in masonry.

The orthotropy of masonry can be modelled via the meso-level approach, where the bricks are

modelled by continuum elements and the joints by interface elements. For this type of modelling

various models to describe the interface behaviour are available: a discrete crack model, a Coulomb

friction model, and a combined Coulomb friction/tension cut-off/compression cap model.

A smeared cracking model for tension Is usually combined in DIANA with a plasticity model for

compression, such as Mohr-Coulomb or Drucker-Pragar, both of which consider strain hardening. As

an alternative to the specification of two separate models, the user can choose one of the following

three special concrete plasticity models which can handle both tension and compression: 1) Rankine;

2) Rankine/Von Mises; and 3) Rankine/Drucker-Prager. In each case, the Rankine criterion bounds

the tensile stresses and the latter two cases, either the Von Mises or Drucker-Prager criterion is

applicable in the compression region.

Figure 3. 19: DIANA Rankine Plasticity Models

Each of these models can be combined with hardening/softening models to better predict response.

The available hardening/softening models can be seen in the figure below:

62

Figure 3. 20: DIANA Hardening/softening models

The available cracking models in DIANA are: smeared cracking, and total strain crack models based

on fixed and rotating crack concepts.

Smeared Cracking, also called multi-directional cracking, is fundamentally based on strain

decomposition in which the total strain is decomposed into elastic strain and cracking strain as well

as three parameters; tension cut-off, tension softening, and shear retention. There are two tension

cut-off models for which crack initiation are defined: constant and linear.

Figure 3. 21: DIANA smeared cracking tension cut-off in twodimensional principal stress space

In the constant tension cut-off model, a crack arises if the major principal stress exceeds the tensile

strength of the concrete. In the linear tension cut-off model, a crack arises if the major principal

tensile stress exceeds a minimum of two values, the tensile strength or a formula that accounts for

63

lateral principal stress. The available brittle, linear, multilinear, and nonlinear tension softening

models for use with the smeared cracking are shown in the following figure:

Figure 3. 22 Tension Softening – DIANA Smeared Cracking

The constitutive models based on total strain describe both the tensile and compressive response

and are “developed along the lines of the Modified Compression Field Theory” (1999). Three types

of cracking models are available including Fixed crack model (constitutive relations are evaluated in

a coordinate system that is fixed upon cracking), Rotating crack model (constitutive relations are

evaluated in the principal directions of the strain vector), and Non-orthogonal model (unlike the

previous two, crack directions are not assumed to be orthogonal).

The available pre-defined compressive behavioural models for use with the total strain crack models

are shown in the figure below. In addition, these models can be enhanced by adding an increase in

compressive strength due to lateral confinement as proposed by Vecchio and Selby or a reduction

duel to lateral cracking as proposed by Vecchio and Collins in 1993.

Compression functions can also be customized by the user.

64

Figure 3. 23: Tension softening – DIANA total strain crack model

The tension and compression stiffness degradation resulting from internal damage to the concrete

are accounted for separately in the loading-unloading-reloading curves as shown below. Also, the

user can define the hysteretic behaviour for use with the Non-orthogonal model.

Masonry can be described as a quasi-brittle, heterogeneous material, of many different typologies,

regarding the geometry, consistency and construction process. Despite its high diversity, masonry

elements have in common a very low tensile strength. The weak and decisive point for the structural

behaviour and gradual failure of masonry is the bond between the unit and the mortar (Lourenço

1998).

The objective of an accurate simulation, through numerical models, of the quasi brittle behaviour of

masonry is to represent the transition from the elastic stage to the quasi brittle behaviour that involves

cracking, leading eventually to failure. This can be defined as the softening behaviour of masonry.

Mechanical resistance decreases under a continuous increase of deformation, by the formation of

cracks that progress gradually. The process evolves from the state of a diffused pattern of micro-

cracks to localized macro-cracks, and the accumulated tensile or compressive stresses are released,

as presented in Figure 6-4 (Lourenço 1998).

65

Figure 3. 24: Typical behaviour of quasi-brittle materials under uniaxial tension and definition of fracture energy (left), Typical behaviour of quasi-brittle

materials under uniaxial compression and definition of fracture energy (right)

The above inelastic quasi-brittle behaviour is quantified by the integral of the � � � diagram,

denoted as fracture energy �� for tension and �� for compression, quantities that are considered

material properties and describe the inelastic response of the material, usually considered as mode

I. In masonry, an additional failure mechanism is present that depends on the shear resistance of

the unit-mortar interface, denoted as mode II fracture energy �� and is the integral of the � �

diagram with no confinig loading (Lourenco 1996).

Figure 3. 25: Shear stress-displacement diagram of quasi-brittle materials

For the numerical determination of the fracture energy values, different kind of tests associated with

the unit-mortar interface are required that measure the strains in terms of displacement versus

stresses.

The tensile and shear stresses diminish exponentially, while compression combines a hardening and

a softening phase (Lourenço 1998). The recommended fracture energy values for all materials are

presented in Table (Lourenco 2014).

In the static nonlinearity section, a “total strain rotating crack” behaviour has been selected. In a

total strain approach, the stresses are described as a function of the strains: in the fixed crack

approach, the stress-strain relationship is evaluated in a fixed coordinate system which is fixed upon

66

cracking, whereas in a rotating crack approach is evaluated in the principal direction of the strain

vector: however, the basic idea of the total strain approach is that the stresses are evaluated in

directions given by the cracks. The tensile behaviour can be modelled using a quite wide variety of

softening curves (usually with exponential one), based on fracture energy and related to the crack

bandwidth, as in usual smeared crack model, in order to avoid mesh-dependency.

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, 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.

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

exponential tension softening applied in the Total Strain Rotating Crack models reflects 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 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.

The compressive behaviour in concrete elements could be influenced by the lateral confinement

(strength and ductility increase with increasing isotropic stress) and by the lateral cracking (peak

stress and peak strain are reduced if the material is cracked in the lateral direction) but these

phenomena are not taken into account; the compressive behaviour is just modelled with a parabolic

function based on fracture energy and crack bandwidth.

The parameters that rule the behaviour in tension and compression are, apart from the tensile and

the compressive strength the tensile fracture energy, the compressive fracture energy and the crack

bandwidth; the European Standards don’t give references to determine these parameters.

67

Only the Model Code (CEB-FIP, Model Code 1990 –2.1.3.3.2) gives some indications to determine

the tensile fracture energy for concrete, as a function of the concrete class and of the maximum

aggregate size (��= 20 mm in this case):

��

� ��

��

For what concerns the compressive fracture energy, regulations don’t give indications: then, a

formula proposed by Lourenço (2008), depending on the compressive strength has been used:

��

provided � ��

For the crack bandwidth, a value corresponding to the length of the element in the mesh

discretization (this is a default value assigned in DIANA) is usually used.

Steel

Three material models are available in DIANA for the modelling of the bar and grid embedded

reinforcement.

1. Linear Elasticity: This model is based on Young’s model and includes the influence of temperature

variations.

2. Plasticity: Includes Von Mises yield criterion, strain hardening. This model also includes the

influence of temperature variations and corrosion.

3. User Supplied: A general user-supplied nonlinear material behaviour model can be specified for

both bar and grid type embedded reinforcement.

If truss members are used to model the reinforcement, a plasticity model combined with a strain

hardening model can be implemented (including plastic offset hysteretic response).

For nonlinear static purposes, a Von Mises plasticity is available for embedded reinforcements: it’s

possible to define a hardening diagram or an ideal plasticity; the last one was chosen in this work to

68

represent the inelastic behaviour of the steel. In this case just the yielding strength has to be defined

according to the code and depends on the class of reinforcement that is being used in the analysis.

3.2.3 Nonlinear structural analysis procedure

Numerical integration is based on the calculation of a function to be integrated in a

number of specific points, the so called integration points. These values in the integration points are

then weighted and summed to obtain the value of the integral.

The weight function depends on the method of integration. For finite element integration usually the

Gauss integration scheme is applied, as this method requires the least number of integration points.

The integration formula of a function can be numerically written as:

� � ��

�

��

�

�

where �� describes the weight function of the applied method for the specific integration interval,

�� the number of integration points and � the coordinate of the integration point.

A minimum number of integration points is required by the numerical integration method and it

depends on the order of the polynomial interpolation. In order to integrate all terms in the integrand

a full integration scheme is necessary.

For the integration along the axis of line elements, i.e. in the iso-parametric x direction, DIANA offers

integration rules according to Gauss, Simpson, Newton-Cotes, and Lobatto.

Figure 3. 26 Integration schemes in quadrilateral zone

69

Referring to beam elements, class-II and class-III types are not only integrated along the bar x axis,

but also in the cross-section area. The integration in the cross-section area depends on its shape and

also on the dimensions of the beam element. For a two-dimensional beam element, the zones are

integrated in h direction only, for a three-dimensional beam element in h and z directions.

Available rules on the cross-section are Gauss and Simpson.

DIANA default integration schemes are 2-point Gauss and 3-point Simpson.

For embedded reinforcements (that can be integrated only along the element axis) DIANA default

integration scheme (2-point Gauss), is thought to be appropriate.

3.3 TREMURI

The method implemented in the computer program TREMURI is based on a masonry macro-element

in which axial, flexural, and shear behaviours are taken into account (Brencich and Lagomarsino,

1998).

The model of the entire building is obtained by the assemblage of plane walls with rigid offsets at the

end sections of macro-elements, according to the aforementioned idealization of the entire wall in

piers, spandrels and rigid nodes made according to the experimental evidences collected after past

earthquakes (Brencich et al., 1998).

The analysis method was assessed through experimental tests carried out on a brick masonry

building by Magenes et al. (1995) at the University of Pavia, Italy. Both the macro-element and

analysis procedure were improved by Penna (2002) and Cattari et al. (2004). More recently, Cattari

and Lagomarsino (2007) extended the analysis procedure to mixed masonry-RC buildings, while

Galasco et al. (2006) proposed an adaptive variant of the method.

The matrix analysis procedure implemented within TREMURI allows to include also in-plane flexibility

of floor diaphragms in the capacity model, and hence to analyse existing buildings with wooden,

metallic, or vaulted diaphragms.

In the case of mixed masonry-reinforced concrete structures, one can separately define: macro-

elements for both RC shear walls and masonry panels; membrane elements for floor diaphragms

(three or four nodes elements; rod elements for ties, RC ring beams, columns, and beams). Internal

70

variables are included in the macro-element to simulate not only flexural mechanisms including toe

crushing effects, but also shear mechanisms and their evolution in terms of both strength and

stiffness degradation.

The method was originally conceived basing on a macro-element characterized by a central rigid part

and two extremities in which flexural deformability was concentrated. This macro-element was

described by six degrees of freedom, three for each flexible part (two translations and a rotation).

The flexible parts were also the areas where the plasticity was concentrated, then the model can be

assumed to be characterized by a lumped plasticity.

Once the masonry has been discretized into different elements, the overall behaviour depends on

the single response of piers and spandrels; moreover also other elements can be modelled and this

aspect is particularly useful in case of existing and mixed buildings, since it makes the method very

versatile.

In the last century, as remarked in several previous sections, the growth of interest in other

technologies, rather than only masonry or timber structures, lead to the adoption of mixed solutions,

or to the modification of existing building with other technologies.

A simplified formulation consists in the nonlinear beam element with lumped plasticity at the

extremities of the element. The element Is analysed in terms of global parameters and with a proper

force-displacement relationship.

The method is approximated but it has the following main advantages:

1.� It requires a reasonable computational effort in nonlinear analysis

2.� Few mechanical parameters need to be defined.

The formulation can be assumed to be in line with the latest code recommendations; in fact, the

ultimate bending and shear force can be calculated according to the indication s given by the code.

The characterization of the different masonry elements’ behaviour depends on the different failure

mechanisms that may occur; they are usually classified in two groups: failure for shear, failure for

flexure; it’s obvious that in a complex structure also mixed modes are possible.

71

It has to be remarked that this classification has been meant and is valid for piers elements since

more information are available also concerning experimental campaigns, while for spandrels it’s

usually more difficult to assess their behaviour since it is strongly affected by the normal force acting

on the element.

The behaviour is conditioned also by the presence of tie rods or ring beams and crushing phenomena

in these elements are quite rare since because the normal force acting on the spandrels is usually

very small and, due to interlocking phenomena the shear sliding is not likely to happen.

All the failure phenomena can be conditioned by some parameters; in case of piers they are:

•� The geometry of the element

•� The boundary conditions

•� The material mechanical properties

•� Masonry properties (arrangement of blocks, type of masonry, presence of strengthening

solutions)

In the case of spandrels, other parameters may be important like:

•� Interlocking phenomena

•� Type of lintels

•� Interaction with other structural elements

The failure modes are based on the calculation of the local and mean stress produced by the external

actions in some definite points or sections; always referring to the actual normal force acting on the

element, the reference value is the minimum calculated with bending and/or shear failure

hypothesis.

During the non-linear static analysis, the normal action on the building may change as the horizontal

load increases and, according to this, also the shear and flexural response changes:

Then, failure conditions are supposed to be reached when a drift limit is reached because of one of

the two above-mentioned failure modes; this limit is usually called �� and the limit value is set by

the code.

As said before, masonry panels are discretized among “piers” and “spandrels”, which are modelled

as 2D elements. The constitutive model is bi-linear and there’s a cut-off in strength without hardening,

for non monotonic actions there’s a stiffness degradation.

72

Kinematic variables and generalized forces are the following, for a six-degrees of freedom element:

��

��

Loads are applied only at the nodes of the elements.

The elastic branch of the force-displacement relation is related to the geometry and the material

characteristics; in fact, in the relation:

� � � �

the stiffness matrix is given by:

and contains the parameter � which is linked to stiffness degradation. Conversely to what happens

with more refined models used in the finite element approach, here the stiffness is reduced since

the beginning, taking into account its effect also in the initial steps. The codes usually suggest a

reduction of the 50% but, as mentioned in several studies (Calderini), the results can be very different

from the more accurate numerical formulation and from experimental evidences.

However, the reduction factor can’t be univocal and it depends on the normal actions acting on the

element. So, Lagomarsino has proposed a relation to take into account the compressive acting

stress, normalized to the normal compressive strength of the material ��.

Rigid offsets are used to transfer kinematic and static variables.

73

Non-linear correction of elastic predictions is made through a comparison with the limit strength; the

re-distribution of internal forces is made according to the element’s equilibrium.

The ultimate shear and bending are computed according to simplified crit4eria which are function of

the failure modes.

The criteria implemented in Tremuri, comprehend the evaluation of the axial load variation and

update each step of the analysis with a new value of the ultimate strength, provided that a limit has

been stated for the normal compressive strength equal to � � �� , where is the

masonry compressive strength.

Failure modes criteria to be implemented have been distinguished according to the type of masonry

to be studied. The following parameters can condition the choice:

•� Regularity of the masonry

•� Ratio between strength and stiffness of mortar and blocks

Most of the numerical predictions for spandrels have been validated using experimental tests that

were originally made on piers, and few formulations have been conceived for their behaviour. Italian

Code gives different options which depend on the actual axial force acting on the element; if the axial

force is known then the criterion is the same that is used in case of piers.

If the acting axial axial load is not known and the spandrels are coupled with a tensile resistant

element or in case of infinite stiff floor, a strut and tie mechanism is assumed and the maximum

compressive force acting on the spandrel is the maximum tensile strength of the coupling element.

However, codes usually lead to a very conservative estimate of the spandrels’ flexural strength and,

as a consequence, the failure of spandrels usually occurs because of flexure rather than for shear.

The formulation proposed by Cattari and Lagomarsino (CITE) for spandrels’ behaviour is based on

the assumption that it’s possible to refer to an equivalent tensile strength in the horizontal direction,

parallel to bed joints, which is influenced by the effect of interlocking with adjacent masonry portions.

In the program, one can choose between two different formulations, the one proposed by Cattari and

the one proposed by the current Italian Code, in which the maximum value of axial load is taken into

account and taken as reference, since most of the times the axial force that the software calculates

is an underestimate of the actual value.

74

The strength criteria proposed for unreinforced masonry panels may be distinguished in

Rocking/Crushing criteria and Shear criteria and among them a further distinction is made

considering if they are applied to piers, to spandrels or if they can be applied to both of the structural

elements.

Rocking/Crushing criteria:

Criterion adopted in case of piers:

��

��

��

Criterion adopted in case of spandrels:

��

��

��

��

where:

�� is the masonry compressive strength

� is the length of the panel

� is the thickness of the panel

�� is the maximum value between the normal force acting on the spandrels and the tensile strength,

which is the minimum value between the tensile strength of the coupling element (tie rods or ring

beam) and the ��

�� is the compressive strength in the horizontal direction.

The limit domain is assumed as an elastic-perfectly plastic relation with limited ductility in

compression �� and in tension ��.

The equivalent tensile strength for spandrels �� is determined knowing: �� the tensile strength of

the bricks, the friction � the cohesion � in the mortar joints, ��the interlocking parameter, �� the

compressive action at the extremities of the element.

In the end the ultimate bending moment and the tensile resistance are given as functions of several

parameters, as follows:

75

! � ��"

�� " � ��

�" ! ��"

� # ��$�

Shear Failure

Two different mechanisms can take place in case of shear failure, the bed joint sliding, which is

described by the following expression in piers:

! �%� # �� & ��'(�)$

and by the following in case of spandrels:

! ��

Diagonal cracking is described by the following expressions in case of both piers and spandrels:

�� ! ��

��#

�

��

�� !�

�*�� # ��+ & ��'(�)$

The first expression is a Coulomb criterion in which: �% is the length of the compressed part of the

cross section, ��'(�)$ is a limit to take into account the possibility of block failure, � is the height

of spandrels’ transversal section, �� is the masonry shear strength, � is a function of the slenderness

and takes into account the load distribution.

In the second mechanism a coulomb-type criterion is used where is the equivalent cohesion and

� the equivalent friction parameter.

The panel’s collapse is checked basing on the value of the drift:

� !�, - ��

�#

�, # ��

�& �

whose value needs to be compared with the ultimate value � , which is defined by the codes and is

different in case of flexural or shear failure, and in case of new or existing buildings.

Codes (Italian and European ones) suggest, in case of new buildings:

76

� ��

��

while in case of existing buildings:

� ��

��

For flexural and shear failure respectively.

In case of spandrels this limit is higher but once the collapse is reached, the element turns to be a

strut, so no flexure or shear can be sustained; only �, provided it’s � � ��.

Nonlinear r.c. elements are modelled as 2D or 3D elements in case of beams or columns

respectively. Plasticity is concentrated at the ends of the elements; the elastic branch is related only

to geometrical and mechanical features, like in the case of masonry piers, and reinforcement doesn’t

give a further contribution.

Also in this case a reduction of the stiffness due to cracking phenomena can be taken into account,

analogously to masonry elements, by the � coefficient, which is kept constant during the analysis.

Shear and compressive/tensile failures are assumed as brittle failures while combined axial-bending

moment, modelled by plastic hinges at the end of element, are regarded as ductile failure.

Shear strength is evaluated according to current Italian regulations either in case of transversal

reinforcement placed in the elements or when it’s absent; if the transversal reinforcement is present,

then the element is studied with the equivalent truss model.

In case of combined normal force �, bending moment �the domain is built on the hypothesis of

plane sections, perfect bonding between reinforcement and concrete, stress block distribution.

In case of walls or reinforced concrete columns, the element is three-dimensional and the domain is

a �� one.

�� and �� are considered separately on each plane according to the actual normal action; then a

linear interpolation is thought, but also more refined models can be adopted.

Once activated, the plastic hinge involves both the planes � and �. Ultimate limit state is identified

in case of ductile failure modes by the chord rotation �, when it reaches the ultimate value ��

defined by the code.

77

Once the failure conditions are reached, the element is a truss and instability and second order

effects are not taken into account.

The three-dimensional model

From the 2D plane wall to three-dimensional model, the assemblage relies on these hypotheses:

•� The bearing structure is made of walls and horizontal diaphragms

•� Walls are the ones bearing and diaphragms are responsible for the distribution of load among

the walls

•� Flexural behaviour of horizontal floors and out of plane response of walls are not taken into

account, or better, they are thought to be negligible compared to the overall behaviour and

the in-plane contribution of walls.

Once the two-dimensional model has been defined, one has to understand how to assemble the

elements in a three-dimensional model.

� � � ��

3D nodes can be obtained assembling 2d rigid nodes on each panel, projecting the local degrees of

freedom in the global reference system and then considering a full coupling among orthogonal

connected walls.

This solution is particularly efficient to reduce the degrees of freedom but since 2D nodes haven’t

degrees of freedom in the direction orthogonal to the wall’s plane the nodal mass needs to be

distributed to the three-dimensional nodes placed in the corners, taking into account the distance of

the 2D nodes from the 3D ones:

��

� � � ��

�

��

� � � ��

�

the same criteria are used for the distribution also in the other node and this to allow to perform

three-dimensional analyses.

78

Diaphragm, as clear from the considerations above and in the previous chapters, play an important

role in the load distribution and affect the overall behaviour, since in case of flexible diaphragms

there’s no possible transfer to stiffer and undamaged walls, when the other walls are undamaged.

In case of rigid diaphragm hypothesis, the effect of re-distribution can be overestimated. Moreover,

in case of existing buildings, this hypothesis usually assumed in models such as POR or SAM, can

give unrealistic results.

In Tremuri program, specific floor elements are implemented with three or four nodes; they are

identified by a principal orientation characterized by stiffness ��, ��is the Young modulus in the

orthogonal direction and �� is the shear modulus that is responsible for the repartition of horizontal

actions among walls in linear and non-linear range.

In case of three-nodes membrane, the matrix that correlates stress and strains is the following:

�

��

� � ��

� � ��

� ��

� � ��

� � ��

� � ��

with � ��

��.

Applying a rotation matrix � to take into account the actual orientation of the floor, the matrix can

be transformed in � .

The stiffness matrix is then assembled starting from � by adopting linear shape functions. For

each node ��of the 3-node element, the matrix �� can be defined as:

��

��

��

� ��

where ��, ��,�� and �� are the coordinates of nodes � and��, and � is the area of the triangle.

Starting from the matrices �� and� � , the stiffness matrix of the three-node membrane element

can be assembled as

�

��

79

where �� , with s equivalent thickness assumed for the membrane element.

In the case of 4-nodes elements, the stiffness matrix is obtained as the averaged contribution of the

two possible meshes of 3-node elements that can be obtained, observing the geometry. In some

cases the definition of stiffness is easy, as for example, in the case of a reinforced concrete floor with

beams and slab: in this case the shear stiffness is mainly given by the slab, whereas the beam axial

stiffness influences the value of ��.

Seismic analysis procedures

In Tremuri, a series of procedures have been implemented, in order to perform incremental static

analysis with force or displacement control, three-dimensional pushover analysis and three-

dimensional time history analysis.

The pushover procedure implemented transforms the problem of pushing a structure maintaining

constant ratios between the applied forces into an equivalent incremental static analysis with

displacement control at only one degree of freedom.

The general formulation of the pushover problem is represented by the following equations:

��

��

��

�

��

where � is the control degree of freedom and �� is the coefficient vector of the applied load pattern.

The system of equations can be transformed by subtracting the �-th row, multiplied by a proper

factor, from the first � � � rows.

After this transformation, the �-th equation then becomes:

��

��

��

��

��

��

And the new system of equations, with a modified stiffness matrix, becomes:

80

��

� ��

��

��

It’s equivalent to a displacement control one, in which the �-th d.o.f. �� is the imposed one.

This formulation has been rewritten by introducing the nonlinear contribution in incremental form, in

order to be implemented in the nonlinear procedure. The algorithm results quite effective and robust,

so allowing pushover analyses on very complex models.

Horizontal forces, proportional to tributary masses and the assumed mode shape, are applied to

each node at the level of each floor. This aspect represents a crucial issue, especially referring to the

estimation of the axial load that acts on the spandrel elements. In case of flexible floors, if the forces

are applied at the corners of the building, the axial forces in the spandrels are underestimated.

To summarize somehow the main features of this modelling technique, the distinctive aspects can

be envisaged as follows:

•� The model allows the creation of a three-dimensional frame based on the assemblage of

two-dimensional walls, condensing degrees of freedom and reducing the computational cost

of the analysis

•� The implementation of specific elements allowing the representation of non-linear behaviour

also of other elements with different technologies (reinforced concrete, timber elements)

•� Possibility to model the orthotropic behaviour of membrane elements as non rigid

diaphragms

•� The algorithm that has been developed and included in the code to perform non-linear static

(pushover) analyses is able to catch the deterioration in the elements and so it’s also possible

to catch the softening part of the capacity curve.

3.4 SEISMIC ANALYSIS OF MASONRY BUILDINGS

The first methodologies to design buildings were based on concepts and rules that implicitly

considered all the various kinds of actions that a structure could have experienced during his whole

life cycle. These approaches were meant to guarantee a certain safety level against gravitational and

81

seismic loads at the same time, but the previsions of these approaches were, most of the times, too

conservative, and it has to be remarked that, sometimes, the needs for “gravitational resistance”

may be in contrast with the principles of seismic design (Maques, 2012).

For example, concerning masonry structures, the increment in the dimension of the walls (of

thickness, in the specific case) will enhance the resistance to vertical actions but, at the same time,

the increment in the mass of the structure leads to an increment of the inertia forces that will act on

the structure.

The first seismic regulations issued in the Italian territory mainly suggested to adopt anti-seismic

solution to improve the behaviour of masonry structure introducing timber frames to confine walls or

portions of them, and the evolution of the codes mainly during the last Century, though briefly

reported, has been included in the previous chapter with a particular focus on the mixed masonry-

reinforced concrete structures.

It is well known that the earthquake is a dynamic action as such and it’s also a dissipative one;

nevertheless, all the difficulties related to the correct simulation of this action have led, in the past,

to the adoption of many different simplified methods to replicate the effect of dynamic actions by the

application of equivalent static load patterns, also taking into account the non-linearity in the response

of the structure.

From a quantitative point of view, the intensity of a seismic event can be evaluated through a

spectrum or, in a more complete and accurate way, from a natural recording in the form of an

accelerogram.

Concerning the response of the structure, provided the non-linear nature of the phenomena involved

in the problem, it can be evaluated through different approaches, that are dependent on the adopted

analysis method.

82

3.4.1 Linear and Non-Linear Analyses

When a structure has to be designed, one of the first steps to be performed, is the identification of

the forces that will act on the building and the effect that these forces (both gravitational and seismic)

cause on the elements.

Only for this purpose it’s possible to assume a linear behaviour, provided a certain ductility backup

of the structure.

According to this approach, many national and international codes (Eurocode 8 ant the national

Italian Code for example) suggest an equivalent linear multimodal procedure to be used in case of

new structures, of which the linear elastic method is a simplified version, since it considers only a

deformed shape for the structure in a 2D analysis, which is however allowed only for building that

show a certain regularity in elevation.

When the building is not regular in elevation the analysis should take into account the different mode

shapes of a bi-dimensional analysis or, in particular cases characterized by a certain irregularity in

plan, a specific tri-dimensional model might be analysed. The different scenarios that can be found

are summarized in the following table.

Regularity Allowed simplifications Behaviour factor

(for linear analysis) Plan Elevation Model Linear elastic

YES YES 2D Lateral Force Reference value

YES NO 2D Modal Reduced value

NO YES 3D Lateral Force Reference value

NO NO 3D Modal Reduced value

Table 3.1: Indications on the regularity and the type of analysis that needs to be performed according to Eurocode 8.

In the dimensioning for the ultimate limit state, the Eurocode 8 (but also the Italian Code) defines a

design response spectrum, which is obtained starting from an elastic response spectrum for a period

of 475 years, scaled to take into account a reduction factor q that should represent, though in a very

approximate way, the inelastic response of a structure at its ultimate limit state.

According to Magenes (2006) the security verification is made in correspondence to two performance

levels, an ultimate limit state (near collapse conditions) and a damage control level (operability level);

83

in the former case the verification is made in terms of drift, while in the latter is made in terms of

displacement, which is identified in the specific case with the limit force.

Basing on the type of verification that has to be performed and on the performance level required,

the seismic action is different since it is related to a different probability of occurrence. Then, for

each limit state that has to be checked, the equivalent static actions are applied to the structure and

the internal forces are evaluated.

Usually the ultimate limit state is the most punitive one, which means that as the resistance in terms

of shear or bending is reached in a single structural element, the safety of the entire structure is not

satisfied. In this regard some considerations have been made by Marques (2012): sometimes the

use of conservative hypotheses, such as the possibility of a certain redistribution even after that a

single element has reached the ultimate state, or the use of a certain value for the behaviour factor,

may result in misleading security checks.

Specifically, according to Magenes and Morandi (2008), using a q factor of 1.5-2.0 it’s almost

impossible to accomplish the security check for some configurations of unreinforced masonry

buildings with two or three floors, for a maximum ground acceleration (agS) greater than 0.1g. This

conclusion contradicts the experimental results and the evidences of nonlinear analyses (Morandi,

2006; Lourenço, 2009) so the criteria to define the value of the behaviour factor q must be re-

considered.

The behaviour factor is conventionally interpreted as an approximation of the ratio between the

seismic actions that the structure can stand if the response is purely elastic and the minimum

seismic actions that can be used in the dimensioning with a conventional model of elastic analysis,

ensuring a satisfying response of the structure. Many experimental studies have been performed in

Europe and mainly in Slovenia (Tomaževič e Weiss, 1994; Tomaževič et al.,2004) and Italy

(Benedetti et al., 1998; Benedetti, 2004) in order to asses the value of the behaviour factor for

unreinforced masonry structures.

Referring to Figure 3.28, the response of the structure is represented by a capacity curve in terms

of force and displacement (of a control node) which can be seen as similar to the experimental

response of a building subjected to a seismic action; so the typical criterion according to which the

behaviour factor for an unreinforced masonry building is defined is as the ratio between the maximum

elastic shear Fel,max and the resistance of an equivalent bilinear system.

84

Figure 3. 27: Parameters for the definition of the behaviour factor q (F is the base shear, d is the displacement at a control point, e.g. the roof).

However, according to Magenes, once the maximum resistance has been reached (for shear or

flexure) in one element, the displacement capacity in the nonlinear field, despite in some cases it

can be limited, is most of the times enough to allow a redistribution of the internal forces, so that

the structure is still able to sustain an increasing horizontal load, though coupled with an increase of

the action in the other elements.

This possibility for redistribution is considered in most of the cases in which the seismicity is high,

like America, New Zeland and also in the European countries, since it’s considered both in Eurocode

8 and in the Italian Code.

For every analysis method, regulations give the spectrum for the seismic design of the structure, on

the basis of which the seismic forces are calculated. In the case of linear elastic analysis, it’s possible

to define an equivalent horizontal force which can be calculated on the basis of the spectrum as:

��

where Sd (T1) is the ordinate of the spectrum and represents the acceleration that corresponds to an

equivalent SDOF system that is characterized by a first period of vibration T1; m is the mass of the

entire building, λ is a corrective factor that is equal to 0.85 if T1≤ 2 TC and the building has more

than two floors, or 1 in all the other cases; TC is the limit value for the period that represents the end

of the constant acceleration part of the spectrum.

85

Figure 3. 28: Basic shape of the elastic response spectrum in EN 1998-1

For the determination of the fundamental period of vibration it is possible to use to structural

dynamics (Rayleigh) expressions or simplified expressions such as:

��

which is valid for a building with a maximum height of 40 m, and the coefficient Ct may assume the

value of 0.05 for masonry structures, H is the height of the building from the foundation to the

highest rigid floor. The method refers to a reduced spectrum (reducing the ordinates of the elastic

one) that allows to implicitly consider in the analysis the inelastic properties of the building and its

capability to dissipate energy through deformation and induced damage.

This reduction of the elastic spectrum is made by the q factor, which takes into account in a synthetic

way the inelastic properties in the model.

The distribution of horizontal forces along the height is made according to the fundamental vibration

modes in the horizontal directions that can be calculated using structural dynamics principles or can

be approximated by a linear (inverse triangular) distribution, varying along the height of the building

according to the following expression:

� ��

��

86

Where �� is the force acting in correspondence of the floor i; �� is the total base shear which can be

calculated according to the expression �� ; � and � represent the elevation of,

respectively mass � and �, from the ground level that has been chosen in the model, while �

and � are the mass that can be intended as placed at the level of the horizontal floors.

Another aspect that needs to be considered in this modelling approach is the evaluation of torsional

effects that, in a building, are the consequence of the fact that the centre of stiffness and the centre

of mass are not coincident; this happens all the times in which the vertical resisting elements are

not uniformly distributed.

An estimate of the accidental torsional effects can be done, provided that the vertical resisting

elements are almost uniformly distributed among the plan of the building, referring to a simplified

formulation, according to which the torsion is evaluated multiplying the force acting in each element

for a factor δ obtained as:

� � � ��

��

where � is the distance from the centre of mass of the considered element, measured in the direction

orthogonal to the seismic action while �� is the distance between the two extreme bracing elements.

However, in the case of a marked asymmetry in the plan, in terms of mass or of stiffness, it’s not

possible to take into account the eccentricity in a too simplified way, but a more refined method is

needed, also referring, if necessary, to a three-dimensional model.

3.4.2 Non-linear static analysis

Different typologies of non-linear analysis, or pushover, as it is commonly referred to, enable to

analyse the effect of non-linearity with a simplified and less time-consuming procedure than the,

though more complete, non-linear dynamic analysis with time integration.

The different methods that can be used are:

-� N2 Method (Fajfar)

-� Capacity spectrum method (Freeman)

87

-� Coefficient method (ASCE)

These procedures are all based on the deformation control, while de displacement demand is

calculated through a spectral analysis of an equivalent structure with a single degree of freedom,

which will then be compared with the capacity of the structure.

These procedures have been implemented in most of European codes, including the most recent

Italian ones.

The deformation capacity has been calculated starting from the capacity curve, which is peculiar of

each building, which represents the relationship between the displacement of a control point and a

the base shear force. This curve is calculated applying to the structure a load distribution which is

characterized by a progressive increment during the analysis, that however does not alter the ratio

between al the components of the lateral distribution.

Usually the shape of these load distributions are defined by the codes and they suggest two main

groups among which, according to some criteria, the distributions have to be chosen.

Usually the two distributions are the one proportional to the inertia forces which, in the case of a

building with a uniform distribution of masses along the height corresponds to a uniform one, and

the one proportional to the first mode that characterize the structure, and that in case of regular

buildings can be considered well approximated by an inverse triangular one.

Aa mentioned in several studies, the two distributions represent the limit cases of the response of

the structure, in the sense that the first distribution enhances the floor mechanisms and more brittle

failure modes, while the second aims at representing a more ductile behaviour and generally leads

to a more diffused damage along the height od the building.

For the security check, the N2 method, which is the one to which the actual italian regulation refers

to, requires the definition of an equivalent capacity curve for a single degree of freedom system. The

main steps of the procedure can be synthetized as follows:

-� First of al the model of the structure has to be implemented, defining the mechanical

features not only in the elastic range but also in the plastic one. So in addition to the data

needed for the usual elastic analysis, the nonlinear force - deformation relationships for

structural elements under monotonic loading are also required.

88

Seismic demand is traditionally defined in the form of an elastic (pseudo)-acceleration

spectrum �� in which spectral accelerations are given as a function of the natural period

of the structure �. The specified damping coefficient is taken into account in the spectrum.

-� Starting from the acceleration spectrum, the inelastic spectra in acceleration-displacement

format needs to be determined. In case of an elastic SDOF system, the following relations

can be applied:

��

��

where �� and �� are the values in the elastic acceleration and displacement spectrum,

respectively, corresponding to the period T and a fixed viscous damping ratio. A typical

smooth elastic acceleration spectrum for 5% damping, normalized to a peak ground

acceleration of 1.0 g, and the corresponding elastic displacement spectrum, are shown in

the following figure. Both spectra can be then plotted in the acceleration-displacement

format, which is reported in the same figure, on the right.

Figure 3. 29: Typical elastic acceleration and displacement spectrum for 5% damping normalized to 1.0g peak ground acceleration. Traditional format

(left), AD format (right).

For an inelastic SDOF system with a bilinear force - deformation relationship, the acceleration

spectrum (��) and the displacement spectrum (��) can be determined as (Vidic et al. 1994).

��

��

��

��

��

��

��

89

where is the ductility factor defined as the ratio between the maximum displacement and

the yield displacement, and �� is the reduction factor due to ductility, i.e., due to the

hysteretic energy dissipation of ductile structures.

��

� ��

��

Where � is the characteristic period of the ground motion and is usually defined as a

transition period between the constant acceleration and the constant velocity periods. In the

medium and long period range the displacement of the inelastic system is equal to the

displacement of the corresponding elastic system with the same period.

In the following figure a set of reduced spectra in the acceleration-displacement format are

represented, all normalized to a 1.0 g peak ground acceleration.

Figure 3. 30 Demand spectra for constant ductility in AD format normalized to 1.0 g peak ground acceleration.

-� The pushover analysis is performed applying a monotonically increasing pattern of lateral

forces, representing the inertial forces which would be experienced by the structure when

subjected to ground shaking. Using a pushover analysis, a characteristic nonlinear force -

displacement relationship of the MDOF system can be determined.

90

The selection of an appropriate lateral load distribution is an important task within the

pushover analysis and there isn’t a univocal solution. One practical possibility is to use two

different displacement shapes (load patterns) and to envelope the results.

-� In the N2 method, seismic demand is determined by using response spectra. Inelastic

behavior is taken into account explicitly. Consequently, the structure should, in principle, be

modeled as a SDOF system. Different procedures have been used to determine the

characteristics of an equivalent SDOF system.

Starting from the equation of motion of a multiple degree of freedom system:

��

where U and R represent displacements and internal forces, 1 is the unit vector and a is the

ground acceleration in the time domain (damping is not included in this formulation).

If the displacement shape is assumed as constant, the displacement vector during the

analysis is represented by the following expression:

� � �

where � is the time-dependent top displacement, is the displacement vector normalized

to the displacement of the control point, whose value is 1.

Since, from the statics:

� � �

that’s to say that the internal forces � are equal to the statically applied external loads �.

Once � has been defined as

� � � � �

the equation of motion can be re-written in the following form:

��

91

after a series of transformations, the equation can be written in the form:

��

where � is the equivalent mass of the single degree of freedom system, expressed as:

� � ��

while �� and �� are respectively the force and the displacement of the equivalent single

degree of freedom system, calculated with the following expressions:

��

�

��

�

in which � is the base shear of the multiple degree of freedom model:

� � ��

and the constant � is defined as:

� ��

��

��

��

�

��

And usually called modal participation factor.

The elastic period of the idealized bilinear system �� can be determined as:

��

�

��

92

in which ��and ��

� are the displacement and the force at the yielding point. In the end the

capacity diagram in acceleration-displacement format is obtained dividing the forces in the

force-deformation diagram by the equivalent mass ��:

��

��

-� The seismic demand for the equivalent SDOF system can be determined by using a graphical

procedure. The intersection of the radial line corresponding to the elastic period of the

idealized bilinear system T* with the elastic capacity diagram have been plotted in the same

graph. The intersection of the radial line, corresponding to the elastic period of the idealized

bilinear system �, with the elastic demand spectrum �� defines the acceleration demand

(strength) required for elastic behavior and the corresponding elastic displacement demand.

The yield acceleration �� represents both the acceleration demand and the capacity of the

inelastic system. The reduction factor �� can be determined as the ratio between the

accelerations corresponding to the elastic and inelastic systems:

��

��

��

This reduction factor, however, doesn’t take into account the overstrength effect, but only

the energy dissipation effect.

If the elastic period � is greater than the characteristic period � , the inelastic displacement

demand �� is equal to the elastic displacement ��.

Figure 3. 31 Determination of the displacement demand

The ductility demand is then obtained as � � �� and it is equal to ��.

93

While if the elastic period of the system is smaller than �� , the ductility demand and the

displacement demand can be calculated as:

� � ��

��

��

��

��

�

�

In both cases the inelastic demand in terms of accelerations and displacements corresponds

to the intersection point of the capacity curve for the equivalent single degree of freedom

system and the demand spectrum corresponding to the ductility demand �.

-� The displacement demand for the single degree of freedom model �� is transformed into

the equivalent displacement of the MDOF system ��, defined as the target displacement.

The local seismic demand (joint rotations or storey drift) can be determined using the

pushover analysis; under monotonically increasing lateral loads with a different pattern, the

structure is pushed to its target top displacement. It is assumed that the distribution of

deformations throughout the structure in the static analysis approximately corresponds to

the one that would have been obtained using a dynamic analysis.

It’s worth noticing that the estimated target displacement doesn’t represent a precise value,

but an average of all the possible values for the same seismic demand, intended as the

same spectrum taken to represent the earthquake that can occur in that location with that

probability according to the limit state that is being investigated. However, some Codes, for

example the FEMA 273, suggest to carry out the analysis to at least 150% of the calculated

top displacement.

-� In the last step of the procedure the performance can be assessed comparing the seismic

demands, determined in the above described step, with the capacities for the relevant

performance level.

Non-linear dynamic analysis outcomes, with increasing excitation level, often show an intermediate

structural behaviour between two bounding states, first soft storey mechanism and global one typical,

respectively, of uniform loading profile and modal distribution of N2 analysis; the two pushover curves

define two boundary structural behaviours. Therefore, from a design code point of view, the worst

situation may be assumed to guarantee the structural safety.

95

CHAPTER 4: CASE STUDY – CAPRI BUILDING As mentioned in the previous pages, masonry is a complex material to model because of its

anisotropy and variability of properties. Only in few cases constitutive nonlinear models able to

consider different strength and deformation capacity along the material axes have been

implemented: in particular, Lourenço (2000) and Calderini and Lagomarsino (2008) for finite

elements, Milani et al. (2007) for limit analysis, and Peña et al. (2007) for rigid block analysis. These

models, however, require a deep knowledge of material properties, also concerning the internal

features of arrangement of block and that’s why they are not widely used; it can be difficult to use

these kinds of models for existing and traditional masonry buildings, since sometimes it can be

difficult to characterize the fabric with a high level of detail.

During the last decades, however, as an alternative to highly complex models, a simpler solution is

to adopt simple geometrical indices, e.g., Lourenço and Roque (2006), to make a first, non binding,

screening of seismic assessment.

In this chapter, taking as an example a literature case study called “Capri Building”, some analysis

have been performed with the aim of giving firstly an insight of the implication of the presence of

reinforced concrete frames that act in parallel with masonry walls and the influence of their

geometrical features on the overall behaviour; moreover, the effectiveness of extreme simplifications

in the analysis given in equivalent frame approaches, particularly suitable for this kind of building,

has been evaluated combining this approach to a more refined standard isotropic smeared crack

approach.

The case study that has been chosen to perform some evaluations on the suitability of regulatory

indications, to evaluate the influence of the dimensions of reinforced concrete elements on the overall

behaviour of the buildings and to compare the results obtained using two different analysis

methodologies, namely Equivalent Frame Model and Finite Element approach, through the use of

96

the commercial software 3Muri (and its scientific version Tremuri) and the general purpose code

DIANA.

The case study has been widely used in an Italian research project that has been completed in 2008

and it has been studied by different research groups, giving some first results on the overall capacity

of the building.

4.1 DESCRIPTION OF THE BUILDING: GEOMETRY

Within the framework of the study made by ReLUIS (Progetto esecutivo 2005-2008 Linea 1) on the

vulnerability of mixed masonry-reinforced concrete buildings, one of the case-studies that have been

analysed is the “Capri Building”.

It’s a building characterized by peripheral walls and internal reinforced concrete frames; it’s mainly

meant for residential use and it has been built at the beginning of the 20th century. It has been

assumed as a reference for an entire typology of buildings, quite spread especially in the South of

Italy and belonging to the first decades of the 20th Century.

In this chapter five structural models are examined. Three of them are mixed type buildings (Model

1, Model 2 and Model 5), and they are conceived on the bases of the building named “Capri”, in

which perimeter masonry walls are combined with internal r.c. frames. The other two models (Model

3 and Model 4) have the same arrangement and dimensions of the former ones, but internal frames

have been replaced by masonry walls. The models are three stories buildings with a rectangular plan

of about 17.0 m x 20.0 m and a height of 13.0 m. Masonry walls taper form the ground to the upper

floors, with thicknesses of respectively 70 cm, 60 cm, 50 cm; four pilasters are placed at the edges

of each r.c. frame, at each floor.

Internal frames are characterized by three spans, with beams supported by two intermediate

columns and the pilasters at the edges; slabs are mixed masonry-reinforced concrete (usual floor

typology used in Italy) with unidirectional orientation of the T-shaped beams parallel to the longer

direction and lightening blocks.

In the masonry building the walls replacing the inner frames have a thickness of 50 cm at each floor.

97

Concerning the r.c. elements, a cylindrical compressive strength of 25 MPa for the concrete and a

yield strength of 380 MPa for the steel have been considered. Reinforcement details for beams and

columns are reported in the following tables.

The vertical applied loads are those due to the structure’s weight (weight for unit volume equal to 20

kN/m3 for the masonry and 25 kN/m3 for the r.c. elements), and to the slabs (4.3 kN/m2). The

horizontal forces are applied parallel to the shorter side as mass proportional loads and with a

distribution proportional to the first mode, according to code prescriptions.

As mentioned, three different mixed masonry-r.c. models have been analysed, the first one is the

one studied within the ReLUIS project, whose characteristics in terms of materials have been updated

in order to follow the most recent regulations on existing buildings (in the following it will be referred

to as Model 1).

The second one in characterized by the same material and overall geometry and distribution of

reinforced concrete elements in the plan of the building, but beams and columns are characterized

by dimensions, amount of reinforcement and type of detailing which have been determined on the

basis of the most recent prescriptions on seismic design, following resistance hierarchy principles.

This model will be referred to as Model 2.

The third mixed model, which will be called in the following Model 5 has been meant to represent all

those buildings in which the seismic prescriptions are neglected. All the columns (which are,

however, the effective bearing elements) are designed only to carry vertical loads and no seismic

detailing concept is followed.

Model 3 and Model 4 are realized entirely with masonry and no reinforced concrete elements are

present. The difference between the two models lay in the presence (in the case of Model 3) of a

reinforced concrete ring beam.

Model 1

Reinforced concrete frames are constituted by beams and columns of rectangular shape; the

dimensions and the amount of reinforcement are representative of an entire category of buildings

98

widespread especially in the southern part of Italy, characterized by the absence, in their design, of

any particular seismic prescription.

In fact, the internal frames have been conceived to sustain essentially gravity loads and their

reinforcement has been calculated only referring to gravity loads. In the following tables, most of the

geometrical data relative to the reinforced concrete elements are summarized.

Figure 4. 1 Plan view of Capri Building, Model 1

Figure 4. 2 Section view of Capri Building, Model 1

99

It seemed worth to calculate some indexes in order to characterize the models and to identify some

parameters basing on which the comparisons between the different analysed models can be done.

In Table 4.2 and in Table 4.3, in fact, the compressive axial force ratio on the columns at each level

and the percentage of reinforced concrete area (always calculated on the basis of the columns’ area)

compared to the masonry and overall area of the building have respectively been calculated.

Table 4. 1 Capri Building, Model 1. Geometrical details of reinforced concrete elements.

Table 4. 2 Area of reinforced concrete elements, vertical load ratio

100

Table 4. 3 Reinforced concrete-masonry ratios

The figure that follows represents the sections of the beams and the columns at each floor such as

they are described in the tables above.

Figure 4. 3 Details, reinforcement in beams and columns. Model 1

101

Model 2

As mentioned above, this model has been conceived in order to represent a vistuous design, following

the most modern code prescriptions in terms of seismic design of reinforced concrete buildings,

either referring to their geometrical features, and to the amount of reinforcement in beams and

columns.


It seems wort to refer to the recommended amount of reinforcement that are reported in the Italian

Code NTC2008, that have been respected in the design of the elements in this Model.

In Chapter 7, limitations on geometry and reinforcement for beams, columns and their connections

are reported, in particular:

•� Dimensions of beams should be such that �

� ratio exceeds 0.25 and the critical zones should

have a length of h in CDB structures and 1.5 h in CDA structures.

102

•� At least 2φ14 longitudinal bars should be present both in compressed and tensile zone of

beams, and the geometrical ratio of the tensile reinforcement should respect the following

condition:

��

��

��

��

•� For what concerns stirrups, they must be placed with a span that is not greater than the

minimum of the following dimensions:

−� 0.25 h (where h is the height of the beam

−� 175 or 225 mm in case of CDA and CDB structures respectively

−� 6 φlongitudinal or 8 φlongitudinal in case of CDA and CDB structures respectively

−� 24 φtransversal

•� Dimensions of columns (however bmin must be at least 250 mm)should prevent second order

effects and the critical zone is defined as the greater value among the height of the section,

1/6 of the height of the element, 45 cm.

•� For what concerns longitudinal reinforcement, the distance between bars shouldn’t exceed

25 cm and the geometrical ratio ρ can vary between 1 and 4 %. Stirrups have to be placed

in order to prevent instability phenomena for longitudinal bars.

•� The diameter of stirrups should be greater than 6mm and their span has to be less than the

minimum among the following dimensions:

−� 1/3 or 1/2 of the minimum dimension of the transversal section in case of CDA

and CDB respectively

−� 125mm or 175mm in case of CDA and CDB respectively

−� 6 φlongitudinal or 8 φlongitudinal in case of CDA and CDB structures respectively

Some other limitations concerning the amount od reinforcement and the detailing especially in the

connections are reported in the code and were taken into account for the design of the element, but

are not entirely reported for the sake of brevity of this section.

103



Table 4. 6 Reinforced concrete-masonry ratios

104


Model 3 and Model 4

As mentioned, two buildings entirely realized in masonry have also been studied in order to compare

the results obtained from the non-linear static analysis, with the ones with mixed structure. Internal

masonry walls have a constant thickness of 50 cm and the difference between Model 3 and Model

4 is essentially that in the former reinforced concrete ring beams are present; they have the same

features of the one in mixed buildings: they are characterized by a height of 25 cm and a width equal

to the thickness of the walls at each floor. The reinforcement is constituted by 4 longitudinal bars

(14 mm diameter) and stirrups (8mm diameter) equally spaced (20 cm) along the beams.

105

Figure 4. 6 Plan view of Capri Building, Model 3 and Model 4

Figure 4. 7 Section view of Capri Building, Model 3 and 4

Model 5

The last model (Model 5) has been conceived in order to represent a consistent part of the

interventions occurred during the past on masonry buildings, whose structure has been modified

most of the times without any engineering guidance.

106

In fact, all the elements have been designed in order to sustain only the dead loads and no particular

attention has been paid to the seismic detailing.

Both the dimensions of the elements and the reinforcement, despite the widespread of these

arrangements in existing buildings, denote a scarce attention to modern seismic design concepts.



107



108

4.2 DESCRIPTION OF THE BUILDING: MATERIALS

Structural analysis can be seen as the development of a mathematical representation of the

structure. The model usually includes, in the case of existing buildings, geometry, structural systems,

observed damages, morphology and material characteristics.

The results of the analysis may be used, together with qualitative and quantitative results to

understand the structural behaviour, to define a safety level for the entire structure, to check the

causes of observed damages and predict their development, identify critical regions that can undergo

further damage, analyse the effects of interventions already occurred or foresee the effect of

upcoming interventions.

As already mentioned, the structural analysis of existing buildings requires an approach that is

radically different from the one followed during the design of new constructions. In fact, in the design

of new buildings geometry and mechanical properties of materials are clearly defined and controlled

and code recommendations are readily available for the design of each structural element.

In case of existing buildings, a fundamental role is held by the geometric survey and, in general, by

the process of knowledge of the structure. Geometry, function of the structural systems and material

properties are quantities that can vary a lot, not only between structures of similar typology but within

the structure itself. In addition, alterations occurred during the years (sometimes not accompanied

by a proper documentation), interventions and damage can significantly affect the strength and

effectiveness of structural elements.

All these reasons imply that the selection of appropriate structural analysis techniques must account

for these uncertainties, and most modern codes give precise indications on the way of taking in

account the “quality” of the survey and the level of knowledge of the construction that has been

achieved.

To perform the assessment of an existing structure, the following data need to be collected:

•� Information concerning the code that has been used in the original design of the structure

•� Identification of the soil category for the site in which the building is placed

•� Foundations characteristics

109

•� Identification of all the bearing elements

•� Geometry and dimensions of structural elements, percentage of reinforcement, mechanical

properties of materials, connections with non-structural elements

•� Identification of possible imperfections in materials and in the detailing of the elements

•� Nature of previous (if present) restoration interventions

The Eurocode 8 and, according to it, also the Italian code (NTC08), define the levels of knowledge,

whose definition is based on the following aspects:

•� Geometry: geometrical features of structural elements

•� Structural detailing: amount and distribution of reinforcement, quality of connections

•� Materials: Mechanical properties of materials.

The obtained Knowledge Level implies then:

•� The analysis method to be used

•� The confidence factors to apply in the definition of the design mechanical properties

Table 4. 9

Geometry Structural Details Material

Properties Analysis Methods

Confidence

Factor

LC1

From original

outline construction

drawings with

sample visual

survey or from full

survey

Simulated design

according to relevant

practice and from

limited in-situ

inspection

Default values

according to

standards of the

time of

construction and

from limited in-

situ testing

Linear Static or

Dynamic 1.35

LC2

From incomplete

original detailed

contruction drawings

with limited in-situ

inspection or from

From original

design

specifications

with limited in-

situ testing or

All 1.20

110

extended in-situ

inspections

from extended in-

situ testing

LC3

From original detailed

construction drawings


inspection or from

comprehensive in-situ

inspections

From original test

reports with

limited in-situ

testing or from

comprehensive

in-situ testing.

All 1.00

The confidence factors are applied to obtain the design value of the mechanical parameters:

��

��

Where the value �� represents the average value obtained by the investigations.

For what concerns the mechanical properties assigned to the structural elements, they have been

derived directly from the Italian code, for uncut stone masonry with facing walls of limited thickness

and infill core.

It has been assumed that a Level of Knowledge �� had been achieved and consequently the

parameters are calculated as follows:

•� Elastic parameters are taken as the average value of the interval given by the code

•� Resistance parameters are taken as the average value of the interval given by the code

Since we’re dealing with an historical masonry, some coefficient can be applied in order to take into

account the positive effects provided by some possible improvements (note: it has to be remarked

that, for historical masonries, the properties are given under the assumption of: poor mortar, thick

joints, no transversal connection).

In this case the only coefficient that has been applied is that according to the hypothesis of good

transversal connection, but it has to be applied only to resistance parameters. (fm and �).

The appropriate value of the confidence factor has been applied to calculate the design value of the

resistance (for LC2 the value of the confidence factor is 1.2).

111

The �� depends on the kind of action that is involved in the analysis and on the type of analysis that

is being performed; since we’re dealing with seismic actions and with non-linear analyses the value

of the material partial security factor is 1 (so no further reduction is applied)

Table 4. 10 Mechanical Parameters for existing masonry (Table C8A.2.1, NTC2008)

Table 4. 11 Corrective coefficient to take into account improvement solutions in historical masonry

112

For what concerns concrete and steel, the mechanical parameters are those belonging to a C25/30

class for concrete and FeB38K for steel.

113

Table 4. 12

Table 4. 13

4.3 LOADS AND ANALYSES

Taking into account the structural elements and the other loads applied on every floor, the value of

permanent and live loads for every level has been derived, distinguishing the internal floors from the

roof level.

The load combination used to evaluate the masses present on each level is the one given by the

Italian code (NTC08)

� � ��

�

�

114

Where the value of the �� coefficient, introduced in order to take into account the possibility that

variable loads cannot occur at the same time and with the full intensity on the structure, changes

according to the use for which the floor is intended.

Two different distributions of forces have been used during the non linear analyses, performed in

both directions, according to the indications given by the Italian and the European codes.

The non-linear static analysis procedure adopted in the Eurocode 8 and in the Italian seismic code

(NTC 2008), both for design and assessment, is based on the evaluation of a maximum

displacement, which depends on the parameters of an equivalent elastic perfectly plastic single

degree of freedom (S.D.O.F.) system, derived from a capacity curve (which originally refers to the

real structure, so to the multi degree of freedom as such) obtained by a pushover analysis.

This kind of analysis requires a predefined pattern of horizontal forces to be applied to the structure

and, keeping constant the relative force ratios, the horizontal displacement of a control node is

incremented (in case the displacement control approach is used). However, the choice of control

node and force distribution is not univocal and results may depend on it.

In this case the control node has been chosen as correspondent to node 24 (as it can be seen in the

following figure) and an average of the displacements at that level (third one) has been automatically

calculated and taken as representative of the displacement of the whole structure.

Figure 4. 10 Three-dimensional view of the model.

Control Node

X

115

The seismic action is simulated by the application of horizontal loads at the level of the floors. The

load distribution aims at representing the distribution of inertia forces induced by the seismic event,

for which two profiles are commonly considered, one proportional to the mass of each storey and

another proportional to the mass and the height of the floor, which is usually referred to as inverse

triangular and is assumed as representative of the first modal shape, at least in case of regular

buildings. These two distributions may be assumed as limiting conditions for seismic analyses, in

the sense that the actual response of the building when subjected to a real seismic excitation is

supposed to be included in the previous ones.

It’s worth noticing that the choice of load distribution to be used is a complex issue for masonry

structures without box behaviour (Marques & Lourenço, 2011)

4.4 EQUIVALENT FRAME APPROACH: DESCRIPTION OF THE ADOPTED MODEL

A first numerical model, whose geometry and mechanical parameters have been described in the

previous paragraphs, has been created within the code Tremuri, that implements a theoretical

formulation developed by Gambarotta & Lagomarsino (1996), Brenchic & Lagomarsino (1997,

1998) and later refined by (Cattari et al. 2004).

Within the adopted approach, the masonry walls have been idealized by a frame, in which deformable

elements, where the inelastic response is concentrated, connect rigid nodal portions, which remain

usually undamaged; in this structure also reinforced concrete elements have been easily

implemented.

To model the entire masonry wall as a set of structural elements, they need to be firstly identified.

Many criteria for the identification of the elements’ geometry are often assumed in literature as based

on damage survey and experimental tests (Dolce).

In the case of perforated walls with regularly distributed openings, the identification of masonry piers

and spandrels may result rather trivial but it becomes more difficult and uncertain when openings

are irregularly arranged. Moreover, for existing buildings, the patterns of pre-existing cracks should

be taken into account for properly defining the geometry of the structural components (Lagomarsino

et al. 2013).

116

As it has been widely stressed in the previous sections of this work, the widespread of mixed

typologies is combined with a relative lack of prescriptions and experimental/numerical studies on

the subject.

Masonry panel’s behaviour is given by a relation characterized by a cut-off in strength and stiffness

decay in the nonlinear phase, while the initial stiffness matrix takes into account the geometrical and

elastic properties of the elements.

The ultimate and the yielding strength have the same value, meaning that no hardening phenomena

are taken into account; the failure mechanism that can cause the collapse of a masonry element

can be:

•� Bending-axial force behaviour

•� Shear with sliding on bed joints

•� Shear with diagonal cracking

•� Compression-tension

Usually these mechanisms are classified as ductile in case of bending-axial force behaviour, or brittle

in all the other cases and the first three ones depend on the acting normal axial stress on the element.

For what concerns lintels, the limit values of the resistance are the maximum obtained from the

criteria above mentioned and the ones obtained as:

•� Cohesion for Shear

•� Strut mechanism for flexural behaviour

This is due mainly to the fact that the normal force in the lintels can be underestimated.

The expressions with which it is possible to calculate the limit value of the internal force that defines

the behaviour of the elements are the ones presented in the previous chapter and are synthetically

reported in the following:

•� Shear sliding mechanism:

��

��

with � as friction coefficient and �� as cohesion.

•� Shear diagonal cracking mechanism:

117

��

��

��

�� is the shear resistance and is the reduction factor that takes into account the slenderness of

the wall.

•� Bending with axial force:

��

��

�

��

� is the compressive strength of the masonry.

•� Shear in Lintels:

� � ��!

� is the height of the element.

•� Bending with axial force in lintels:

�� "�

��

��"

��

The element failure is determined when a limit drift is reached; this limit is 0.4 % of the effective

height in case of shear failure and 0.6 % of the effective height for bending failure in existing buildings,

provided that once an element collapses, the capacity is only related to the vertical loads.

This formulation may be, however, too simplified since coupling effects between vertical

displacements and rotations are neglected; they are related to compressive damage or toe-crushing.

Reinforced concrete elements are modelled as three degrees of freedom elements in case of beams

or as five degrees of freedom elements in case of columns and walls.

The initial elastic branch of the constitutive law is computed on the basis of the geometry and the

material properties. In the estimation of the elastic stiffness matrix we refer only to the reinforced

concrete part, neglecting the effect of the reinforcement and because of damage, the cracked

118

conditions are simulated in a simplified way using a coefficient that remains constant during the

analysis.

Determination of the plastic hinge is made referring to the bending-axial force interaction domains,

calculated on equilibrium hypotheses. In case of columns, the behaviour can be the effect of the

combination of bi-axial bending-axial action � �� , but to simplify the conditions the

bending moments ��and��are calculated separately and they’ve a linear relationship.

The collapse of the reinforced concrete section is determined correlating the chord rotation calculated

on the basis of the shear length �� with the ultimate value �, determined in codes on empirical

basis.

Once the collapse is reached, the element is only able to carry gravitational loads and instability

phenomena or second order effects are not taken into account.

The effectiveness of this modelling strategy, especially for what concerns the approximation of the

behaviour of reinforced concrete elements with lumped plasticity, Cattari and Lagomarsino analysed

a benchmark model; it is constituted by beams and columns with 30.0 cm x 40.0 m and 30.0 cm x

40.0 cm respectively. The material used are a �� concrete and a steel with ��=400 MPa, the

longitudinal reinforcement is 8φ16 and stirrups φ8 with a spacing of 20 cm in critical regions.

Figure 4. 11 Comparison between results of a pushover analysis (with a load pattern proportional to the mass-height product) on a 2D r.c. frame from

Tremuri and Seismostruct programs: pushover curves(left) and damage patterns (right)

The interaction effects are derived in terms of stiffness, and to study that two simplified models have

been compared: the first one is a masonry building with an internal reinforced concrete columns

connected to the external walls by mean of reinforced concrete beams and the second is the same

building but with reinforced concrete walls inside.

119

Figure 4. 12 Non-linear analysis results on a simple “box” building (Cattari, 2006)

In the first case the masonry walls are, at the beginning, the main resisting elements; then, the

reinforced concrete elements start to carry the load and they’re responsible for the increase of the

global resisting shear.

In the second case the reinforced concrete walls absorb the actions till they reach their maximum

resistance; then, the masonry walls contribute to the global resistance and can be responsible for

the increase of the base shear.

Modal Analysis

In order to assess the applicability of the pushover analysis both in case of mixed and in case of

masonry building, and to have a first idea of the behaviour and the dynamic features of the buildings,

modal analyses have been performed.

As it can be seen from the following tables, in both cases of mixed and URM building, the first mode

is characterized by a pure translational motion, that is always characterized by a high ratio of

participating mass (about 90% in case of mixed buildings and 80% in case of masonry ones). These

circumstances allow to apply the non-linear static analysis for both the classes of buildings. It’s also

worth noticing that in the case of mixed building the first mode is characterized by a translational

motion in X direction, which is the less stiff, while in case of masonry buildings, because of the

arrangement of the internal masonry walls, the first mode implies a translation in the Y direction,

that is in this case the less stiff direction.

120

Table 4. 14 Modal Analysis. Results for Model 1




121


4.5 NON-LINEAR STATIC ANALYSES

In the following the results of nonlinear static analyses conducted on the previously described models

are discussed.

The most recent Italian Code (D.M. Infrastrutture 2008) for the design of new combined masonry-

reinforced concrete buildings would lead to distribute the seismic actions either only on masonry

walls or only on the elements of different technologies, that would mean that one could assign entirely

the horizontal action either on the peripheral walls or on the internal r.c. frames.

However, if the collaboration between masonry walls and elements of different technologies in

withstanding seismic actions needs to be taken into account, a non-linear analysis (static or dynamic)

should be conducted.

The presence of reinforced concrete ring beams inserted at the level of the connection with reinforced

concrete horizontal floors is very common in case of new mixed masonry reinforced concrete

buildings.

The capacity curves that have been obtained are expressed in terms of base shear and displacement

of the control node located at the top of the building.

The obtained results seem to confirm the criteria given by the current Italian code D.M. 2008. In

fact, if the RC frames are designed to carry only gravity loads, it’s possible to distribute the horizontal

actions only on masonry walls. The same cannot be done if RC frames are seismically designed; in

this case a non- linear analysis is necessary to determine the level of collaboration between masonry

walls and RC frames, which in some cases could lead to a ratio of about 30 % of base shear on

frames.

122

Figure 4. 13 Model 1. Pushover Curve. +X Direction. Distribution proportional to masses (left). Distribution proportional to the first mode (Right)

The description of damage patterns is made analysing three different phases, which correspond to

the yielding step, to the step corresponding to the achievement of the maximum base shear and in

correspondence with the ultimate displacement.

The word plastic is intended to define the condition in which the element reaches the maximum

resistance (either for shear or for bending) while the word collapsed indicates the condition in which

the element has reached the ultimate drift.

Damage pattern in Model 1. Distribution proportional to the masses (uniform)

1)� Yielding: Spandrels at ground and first floor are damaged because of shear while only some

piers are damaged because of flexure (bending plastic deformation). Deformation and

damage are localized in the extreme piers and due to interlocking phenomena, also the

orthogonal piers are damaged. Internal frames in X direction are still in the elastic phase.

2)� Maximum Shear: Spandrels at all levels are in the plastic phase because of shear

mechanisms; piers at the ground floor are in plastic phase because of shear deformation.

The pier at the opposite side of the load at the ground floor has still only a damage because

of bending and piers at upper floors are either undamaged or damaged because of flexure.

First flexural plastic damage begin to appear at ground floor for both columns at the base

and beams at the first level.

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Base

She

ar [

kN]

Displacement [cm]

Shear TotalShear R.C. Shear Masonry

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Base

She

ar [

kN]

Displacement [cm]


123

3)� Ultimate Displacement: At the ultimate displacement level, the spandrels fail because of

shear at the first elevation, while at second and third elevations they’re still in the plastic

phase bus have not yet reached the limit deformation. Piers are mostly damaged because

of flexure, except at the ground floor, where also shear damage is present. Most of the

columns are damaged and also some of the beams, because of bending and at the first floor

are damaged.

Damage pattern in Model 1. Distribution proportional to the first mode (inverse triangular)

1)� Yielding: At this level of deformation all the spandrels are damaged because of shear in both

the walls parallel to the X direction. Only some piers are damaged on the loaded side and at

all the floors, not only at the bottom level. In one of the two walls only the higher pier is

damaged because of bending. Reinforced concrete are all still in the elastic phase, and then

undamaged.

2)� Maximum Shear: All the spandrels are damaged because of shear and piers are mostly

damaged because of bending, except for two piers damaged for shear at the lower floor.

Damage in piers is distributed all over the height of the building but mainly on the side on

which the load is applied. Some of the beams are damaged for bending and only four

columns, on the loaded side of the structure, are in the plastic phase.

3)� Ultimate Displacement: almost all the spandrels fail because of shear while piers are

damaged mostly for flexure, except for the most squat ones, damaged for shear. The beams

of the second and the third elevation are damaged because of bending

In both cases the collapse mechanism is a floor mechanism at the ground floor. However, while in

the case of the distribution proportional to masses also the spandrel failure is mainly concentrated

at the ground level, in the case of inverse triangular distribution the damage is much more diffused

in the upper floors and at the collapse point also the spandrels at the second and third floor have

reached collapse.

124

Damage pattern in Model 2. Distribution proportional to the masses (uniform)

1)� Yielding: all the spandrels are damaged because of shear and only in one of the two walls

parallel to the X direction the spandrels at the roof level are not damaged. Extreme piers at

the ground floor are damaged in both the walls parallel to the loading direction. Reinforced

concrete elements are not damaged at this point.

2)� Maximum Shear: at ground and first floor piers are damaged because of shear while at the

highest floor the piers (even if squat) are damaged because of bending. The damage in the

beams is concentrated at the first two floors, while only the base sections of the columns at

the ground floor are damaged for bending.

3)� Ultimate Displacement: at the first elevation, spandrels are collapsed because of shear in

both the walls parallel to X direction, while in the other floors they’re mostly only damaged

because of shear. Reinforced concrete elements are, again, only damaged for bending.

Damage pattern in Model 2. Distribution proportional to the first mode (inverse triangular)

1)� Yielding: all the spandrels are damaged because of shear while piers are damaged mostly

for bending and in the decompressed part of the model, that is the side on which the load

is applied. Reinforced concrete elements are all undamaged.

2)� Maximum Shear: the spandrels are still all damaged because of shear at all the levels. Piers

start to be damaged at almost all the levels and only the most squat ones are damaged

because of shear. Only beams at the first two floors are damaged because of bending.

3)� Ultimate Displacement: most of the spandrels at all levels are collapsed because of shear

while piers are all damaged because of bending at all the levels (only the most squat are

damaged because of shear). Damage in reinforced concrete elements is diffused but no

more damaged elements are present.

125


Damage patterns for Model 3 and model 4

1)� Yielding: most of the spandrels are damaged because of shear while few piers are damaged,

mostly for bending and in the decompressed part of the model, that is the side on which the

load is applied.

2)� Maximum Shear: the spandrels are still all damaged because of shear at all the levels. Piers

are damaged because of bending, mainly at the ground floor.

3)� Ultimate Displacement: most of the spandrels at all levels are collapsed because of shear

while piers are all damaged because of bending mostly at the ground floor

In case of distribution proportional to the first mode, the configuration of the damaged structure is

more or less the same, though differently from the uniform distribution, in the triangular one the

damage is much more diffused along the height of the building.

The presence of RC bond beams inserted into the masonry walls mainly affects the behaviour of

lintels, causing the increase of maximum base shear and changing the resistant mechanism of lintels

in an equivalent strut mechanism.

In fact, the damage pattern of the building named Model 4, characterized by the same geometry and

materials of model 4 but without the presence of reinforced concrete ring beams is characterized by

a more ductile behaviour, which is a direct consequence of the fact that, even in the case of uniform

mass-proportional distribution, the damage and the failure mechanism in spandrels are governed by

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Base

She

ar [

kN]

Displacement [cm]


0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Base

She

ar [

kN]

Displacement [cm]


126

bending. In case of distribution proportional to masses, as it is easy to imagine, the damage is mainly

concentrated ant the ground floor. However, as a consequence, model 4 is characterized by a higher

ductility and a lower strength.



Damage pattern in Model 5 Distribution proportional to the masses (uniform)

1)� Yielding: most of the spandrels are collapsed because of shear. Only two piers in one of the

walls parallel to X direction are damaged because of bending. The beams at the first floor

are damaged because of shear from the first steps of the analysis. All the columns are still

in the elastic phase.

2)� Maximum Shear: in correspondence of the maximum shear, the piers at the ground level

are damaged because of shear while the ones on the loaded side (decompressed) are

damaged at the second and third elevation. More beams are collapsed because of shear

and still no column is in the plastic phase.

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 1.0 2.0 3.0 4.0 5.0

Base

She

ar [

kN]

Displacement [cm]

Shear Total

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 1.0 2.0 3.0 4.0 5.0

Base

She

ar [

kN]

Displacement [cm]

Shear Total

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Base

She

ar [

kN]

Displacement [cm]

Shear Total

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Base

She

ar [

kN]

Displacement [cm]

Shear Total

127

3)� Ultimate Displacement: piers at the ground floor are damaged because of shear while at the

second and third elevation they are damaged because of bending. Spandrels at the first

elevation are collapsed because of shear. More beams are collapsed for shear and no more

columns are in plastic phase

Damage pattern in Model 5 Distribution proportional to the first mode (inverse triangular)

1)� Yielding: many spandrels are damaged because of shear. Beams on the first elevation and

on the side opposite to the one loaded fail because of shear. Piers are mostly undamaged,

except for the ones on the loaded side at all the elevations.

2)� Maximum Shear: all the spandrels are damaged because of shear; in case of piers, the lower

central and the squattest are damaged because of shear, the others are damaged as well,

mostly because of bending and on the loaded side. Beams are, once again, collapsed

because of shear and columns are still in the elastic phase.

3)� Ultimate Displacement: Among all the spandrels that are damaged because of shear, most

of them are collapsed at this step of the analysis. More beams are collapsed for shear though

no column seems to be damaged.


The results of seismic action repartition are reported in the following tables for both triangular and

uniform load patterns and in two different steps of the analysis, when the maximum global shear is

reached and when the ratio of the shear absorbed by reinforced concrete elements is maximum.

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Base

She

ar [

kN]

Displacement [cm]

Shear Total

Shear R.C.

Shear Masonry

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5

Base

She

ar [

kN]

Displacement [cm]

Shear Total

Shear R.C.

Shear Masonry

128

Shear distribution between elements

129

Figure 4. 18 Shear distribution between elements when the base shear elements reaches the maximum value

Figure 4. 19 Shear distribution between elements when the shear ratio in reinforced concrete elements reaches the maximum value

130

In the following a synthetic representation of the results of the nonlinear static analyses is reported.

In particular, the values of ultimate displacement and demand displacement are reported, together

with the value of the behaviour factor q*.

Table 4. 19 Non-linear static analyses results

It is worth noticing that in two cases, namely for model 4 and model 5, verifications in Y and X

direction respectively are not satisfied.

In the first case, for the particular configuration of the building, whose conception had been thought

essentially to perform a comparison in the X direction, the results underline a certain weakness in Y

direction that in this case is not mitigated by the presence of reinforced concrete ring beams.

For model 5, the fact that verification is not checked in the X direction underlines an important

aspect; in fact, in this model all the elements were not designed according to seismic prescriptions,

but conceived only to sustain vertical loads; the modification of existing buildings through the

insertion of r.c. frames must then be designed with attention since, as in this case, to a considerable

decrease in the base shear, doesn’t correspond an increase in the ductility.

The results obtained from the analyses have validated the approach suggested by the new Italian

code.

In fact, for existing buildings, it is necessary to perform nonlinear analysis in order to evaluate the

seismic action sharing between structural elements characterized by different technologies since the

131

seismic action rate sustained by external masonry walls can strongly change depending on the type

of analysis (linear or nonlinear), and depending on the dimensions of the reinforced concrete

elements. In fact, in case of elements designed only to sustain vertical loads, the amount of shear

absorbed by columns doesn’t exceed 7% even in the non-linear field.

In the case of mixed masonry-reinforced concrete buildings in which frames have dimensions typical

of seismically designed r.c. structures, neglecting their contribution withstanding horizontal actions

would lead to overestimate the actual portions acting on masonry walls, since the amount of shear

that can be absorbed is about the 20-30% of the global value, depending on the dimensions of the

elements.

Indeed, the relative percentages shared between masonry and RC elements should be assessed by

means of a nonlinear static analysis.

These conclusions are in line with the current Italian code prescriptions for new RC-masonry

buildings.

On the other side, the analyses also allow to make some consideration on the circumstances of

retrofitting existing masonry buildings by the insertion of reinforced concrete elements. The

simultaneous observation of the results of modal analyses and non-linear static ones leads to the

consideration that, when analysing the behaviour of existing masonry buildings that have been

transformed into mixed r.c.-masonry ones, substitution of internal masonry walls with RC frames can

increase the elastic period of the building and the displacement capacity, considering in both case

the presence of reinforced concrete ring beams, since these elements can strongly change the

behaviour of spandrels causing a combination of increase of resistance and decrease of the ultimate

displacement since they leads spandrels to collapse for more brittle shear mechanisms.

Nevertheless, the insertion of reinforced concrete frames after the demolition of internal walls could

also strongly decrease the base shear capacity.

Most of the buildings that have been analysed, however, show that the combination of the decrease

in strength and the increase in period determines that the seismic checks are satisfied with

capacity/demand ratios that don’t differ significantly.

133

CHAPTER 5: CASE STUDY – Ex-SLAUGHTERHOUSE (Rome) The Slaughterhouse in Rome was built in 1891, designed by Gioacchino Ersoch, a student of

Giuseppe Valadier.

It is an architectural structure placed at the burden of the historical centre of Rome; between the

Tiber and Monte Testaccio. It is characterized by rectangular pavilions that are organized in an

orthogonal pattern.

The structure shows an advanced functional organization, considering the period in which it was

built. It is characterized by the simultaneous presence of different materials such as bricks, iron,

reinforced concrete and travertine, at a time when these new building technologies were still in an

embryonic state, in terms of expressive forms.

The transformations occurred during the years were carried out to adapt the structures of the original

buildings to the new slaughter systems that developed over time (the Mattatoio was used in this way

until about 1960) and to the changes of the surrounding urban fabric. Building interventions following

the completion by Ersoch, can be traced back to 1911, 1919, 1923, 1932 and 1953. New buildings,

the refrigerating establishment, with a reinforced steel structure, date back to 1911. Changes with

regards to the pavilions – the demolition of the internal partitions - and the subsequent consolidation

of new buildings and the inclusion of reinforced concrete and steel elements, such as internal frames

and overhead rails, transformed the structure's architectural aspect.

The external aspect of the buildings did not undergo major changes aside from the amount and size

of the openings compared to the original wall structure.

134

Nowadays, the structure has different uses, amongst which the main one is dedicated to the offices

of the Department of Architecture of the University of Studies of Rome “Tre”, to which the perimetral

pavilions towards the Tiber River and Testaccio district were assigned.

5.1 DESCRIPTION OF THE BUILDING: GEOMETRY

The part of the building that has been analysed is the one which belonged to the refrigerator factory.

The building is very regular both in plan and in elevation and can be classified as a mixed masonry-

reinforced concrete structure with peripheral masonry walls and internal r.c. frames. Both systems

have been designed to sustain vertical and horizontal action, so the entire structure can be defined

as coupled or, according to the classification mentioned in the first chapter, as a parallel system.

The entire building is composed also by two lateral parts that have smaller dimensions and that were

used as storage, but since the aim is to analyse the behaviour of mixed structures, according to what

has also been done in the past, only the central part has been analysed.

The internal structure is composed by four reinforced concrete frames, with columns that taper from

ground to second floor, whose dimensions are respectively 50cmx50cm at the ground floor,

45cmx45cm at the first floor and 40cmx40cm at the second floor, on which a net of reinforced

concrete beams is placed, parallel to both directions. This net is constituted by principal beams

(25cmx65cm) and stiffening beams whose section is 25cmx45cm in some cases, 25cmx55cm in

some others.

Both groups of beams are supported by the peripheral walls and a continuous reinforced concrete

slab is placed above them, whose thickness is about 10 cm and that contains reinforcement only in

the short direction. There are no ring beams along the perimeter, and the connection is provided

only by the 15 cm penetration of the concrete slab into the walls.

The entire complex is enclosed by masonry walls 75 cm thick, in which small openings are present

(most of the information about this building have been provided by Professor De Felice and Dr.

Malena, to whom the author is sincerely thankful).

135

Figure 5. 1 Plan of the building

Figure 5. 2 Sections of the building

136

5.2 DESCRIPTION OF THE BUILDING: MATERIALS

Taking into account all the peculiar aspects concerning the determination of mechanical properties

for existing masonry structures that have been widely explained in the previous chapters, the material

has been defined according to the Italian code and to the indications that are given in particular in

the Circolare (2009).

In 2006 an extensive experimental campaign has been conducted by the structural department of

University Roma Tre on the building; the campaign comprehended both in situ and laboratory testing

and involved the entire building complex that, as mentioned before, includes several structures.

However, starting from the reports that the university has kindly provided and taking into account the

level of knowledge that has been reached, mechanical parameters have been determined and are

summarized in the following.

(The level of knowledge that has been reached is the LC1 according to the following table that has

been recalled in this section as well, which implies a high value of confidence factor, hence a strong

penalization of material properties).

Geometry Structural Details Material Properties Analysis

Methods

Confidence

Factor

LC1

From original

outline

construction

drawings with

sample visual

survey or from

full survey

Simulated design

according to relevant

practice and from

limited in-situ

inspection

Default values according

to standards of the time

of construction and from

limited in-situ testing

Linear Static or

Dynamic 1.35

LC2

From incomplete

original detailed

contruction drawings


inspection or from

extended in-situ

inspections

From original design

specifications with

limited in-situ testing or

from extended in-situ

testing

All 1.20

LC3

From original detailed

construction drawings


inspection or from

comprehensive in-situ

inspections

From original test reports

with limited in-situ testing

or from comprehensive

in-situ testing.

All 1.00

137

According to the data collected it is possible to identify the masonry which constitutes the peripheral

walls as a solid brick masonry with lime mortar.

Table 5. 1 Mechanical Parameters for existing masonry (Table C8A.2.1, NTC2008)

Table 5. 2 Masonry mechanical parameters

138

In the table above the final values of mechanical properties are reported.

For what concerns concrete and steel, the mechanical parameters are those belonging to a C20/25

class for concrete and FeB32K for steel.

Table 5. 3 Concrete mechanical parameters

The features of the reinforced concrete elements have been determined on the basis of the original

drawings that have been collected. In particular, the system of beams that support the floors consists

in four principal beams, in the shortest direction (Y direction in the analyses) and with a higher depth,

and five secondary beams, in the orthogonal direction (X direction), three of which are supported

directly by the primary beams. The concrete slab contains reinforcement in the shortest direction of

the building.

Five different sections of beams have been identified while columns sections just change at each

storey, since they taper from ground to second floors.

Figure 5. 3 Beams, longitudinal section

139

Figure 5. 4 Beams, transversal section

The choice of this case study seemed appropriate because of its relative simplicity, that allows to

make general consideration on the overall behaviour. It consists in a regular rectangle, without any

geometrical articulation, neither in plan, nor along the height.

The openings pattern is regular, the wall thickness and the masonry properties are constant, and the

interstorey heights are almost the same at each floor.

Reinforced concrete elements are placed in a regular configuration and a net of very stiff concrete

beams, with 10 cm thick slab, provides the connection among the vertical elements.

5.3 LOADS AND ANALYSES

The seismic action is defined according to Italian Code taking into account the position and the

category of the building.

It is located in Rome, on a soil belonging to category B; moreover, since offices opened to public are

meant to be inside the building, a use class III has been identified.

The parameters that define the spectrum are the following:

140

and according to the formulation indicated in the code, the following spectra can be determined for

different limit states.

On the slabs, a permanent load of 3.25 kN/m^2 has been applied, together with a live load, properly

combined, of 5 kN/m^2 on the internal floors and of 2 kN/m^2 at the roof level.

Non-linear static analyses have been performed on the structure according to which the building is

subjected to a force distribution proportional to the distribution of masses along the height of the

building and to the first mode shape.

Since the procedure that has been followed is the same that has been described in the previous

chapters, in this section doesn’t seem worth to repeat the basic concept on which this method relies.

Moreover, the resisting system is easily identified within this building; it satisfies the requirements

concerning the regularity of walls geometry, the regularity and the alignment of openings; connection

between different structural elements seems to be guaranteed by the slab penetration into the walls;

this circumstance is considered to be sufficient to avoid local mechanism. In this case, then, the

“frame type” modelling is considered to be realistic and to be able to give reliable estimate of the

overall behaviour.

141

Modal Analysis

In order to assess the applicability of the pushover analysis, and to have a first idea of the behaviour

and the dynamic features of the buildings, modal analysis has been performed.

As it can be seen from the following table, the first mode is characterized by a pure translational

motion, that involves a high portion of participating mass (about 90%). These circumstances allow to

apply the non-linear static analysis and, in this case, the weakest direction can be identified with the

Y one.

5.4 NON-LINEAR STATIC ANALYSES

In this case, the same considerations made for the case-study "Capri" have been made. The damage

pattern in case of non-linear static analyses both in X and Y direction for both distribution of loads

(proportional to masses and proportional to the first mode, which is assumed to be coincident with

a linear inverse triangular distribution, which is allowable in the case of regular buildings in plan and

elevation) is described in the following.

X Direction. Distribution proportional to masses.

In correspondence of the yielding point, the walls parallel to X direction present a diffused damage

in all the spandrels that are damaged because of shear. Reinforced concrete frames are undamaged

and still in the elastic phase.

When the maximum strength is reached, also piers at the ground level are damaged because of

shear while reinforced concrete frames are still undamaged.

142

To the evident decay in strength corresponds a diffused collapse of spandrels, due to shear while

damage in piers still remains localized at the ground floor, with exception of some piers in the

compressed part of the building.

At the ultimate displacement step, frames present bending damage, especially in columns and in

some beams of the first floor, while walls do not present any particular increase in damage.

A slight increase in the shear denotes the circumstances according to which it is possible, in mixed

buildings, to assist to a phase, beyond the ultimate limit for masonry, in which the reinforced concrete

elements can give a slight but unneglectable contribution to the overall resistance.

Figure 5. 5 Pushover curve. Distribution proportional to Masses. X Direction

Figure 5. 6 Pushover curve with load repartition. Distribution proportional to Masses. X Direction

143

X Direction. Distribution proportional to the first mode

Also in this case, at the yielding point, the overall damage is dominated by shear damage in lintels,

though for a distribution proportional to the first mode, a more diffused damage along the height of

the building is clearly detectable. in fact, in correspondence of the maximum shear reached, all the

piers at the ground floor are damage, but also some piers at the second floor present shear damage,

while, as in the previous case, reinforced concrete frames are still undamaged.

The decay in strength is accompanied by diffused collapse of spandrels and bending damage of piers

at all the floors. Reinforced concrete frames are still in the elastic phase. The increase of strength

has to be attributed to frames since the damage in the walls parallel to X direction is more diffused.

The collapse is reached when all the spandrels are collapsed for shear and the bending damage in

piers is diffused in all the floors.

Figure 5. 7 Pushover curve. Distribution proportional to the first mode. X Direction

Figure 5. 8 Pushover curve with load repartition. Distribution proportional to the first mode. X Direction

144

Y Direction. Distribution proportional to masses.

At the yielding point, the piers at the ground floor are damaged because of shear, as well as all the

spandrels of the walls parallel to Y direction.

At the ultimate displacement step, walls parallel to Y direction present shear damage at all the floors

while reinforced concrete frames only at this stage of the loading present bending damage, mainly

in columns at the ground floor.

Figure 5. 9 Pushover curve. Distribution proportional to masses. Y Direction

Figure 5. 10 Pushover curve with load repartition. Distribution proportional to Masses. Y Direction

Y Direction. Distribution proportional to the first mode

In case of triangular distribution, the shear damage in spandrels and in piers of the walls parallel to

Y direction start to appear since the yielding step. The ultimate displacement is reached when

145

spandrels at all floors are collapsed for shear and the piers at all the floors are damaged, mainly

for shear.

Bending damage in the columns of reinforced concrete frames is, in this case, diffused also in the

second floor.

Figure 5. 11 Pushover curve. Distribution proportional to the first mode. Y Direction

Figure 5. 12 Pushover curve with load repartition. Distribution proportional to the first mode. Y Direction

146

The performance of this building, according to the analyses that have been conducted, is respectful

of the requirements of the latest Italian code, according to the location of the building and the use

for which the structure is meant, since offices are thought to be placed in that building.

Despite the fact that the verifications on this building have given positive results, it’s important to

underline that in this model the out of plane mechanisms are not taken into account, but are meant

to be the object of specific verification that, however, in this case are thought to be fulfilled because

of the particular arrangement of the internal structure.

Some researchers (De Felice et al. 2009), in fact, have studied the effect of the coupling between

frames and portions of the walls that act out of their plane, in order to evaluate the contribution of

the confining effect provided by these walls to the performance of the frame.

Figure 5. 13 Pushover curves considering the out-of-plane contribution of external walls (De Felice, Malena)

It has been found that, the confining effect increases the shear capacity of the frame and reduced

the horizontal displacement of the frame. A fundamental role is played by the geometry of the walls

and, of course, by the connections between beams, floors and walls.

147

CHAPTER 6: SOME CONSIDERATIONS ON FINITE ELEMENT MODELLING OF MIXED MASONRY-R.C. BUILDINGS

In order to understand the critical aspects of the analysis of mixed masonry-reinforced concrete

structures, one of the models that have been analysed in chapter 4 has been implemented in the

general purpose code DIANA, and some preliminary non-linear static analyses have been performed.

Though a complete understanding of the phenomena involved and a proper calibration of the model

still needs to be object of further and more detailed studies, the first results give some interesting

insights on peculiar aspects of this category of structures. In particular, the original configuration

(Model 1, as mentioned in chapter 4) has been taken as reference for the performed analyses.

6.1 DESCRIPTION OF THE MODEL

Since in the previous chapter the building (or, better, the group of building that derive from the

original case-study referred to as Model 1) has been widely described in terms of geometry, in this

section the geometrical description of the model is not reported, pointing out that all the information

about the dimensions of reinforced concrete elements, wall thickness and reinforcement amount

can be found in the previous pages.

It’s on the other hand important to clarify some of the simplifications that have been adopted in order

to reduce the computational cost of the analysis that, however, takes about four hours to be

completed (with high performance personal computer).

148

First of all, modelling live loads, in this case, didn’t seem to be worth since the combination of vertical

loads that has to be considered while performing a nonlinear static analysis for seismic actions leads

to a reduction of this quantity; in particular, the live load on the internal floors would be less than

10% of the permanent load, while at the roof level the live load doesn’t have to be considered,

according to code prescriptions; so, permanent load is the only vertical load considered acting on

the structure.

6.1.1 Mesh and discretization of the model

The construction of the geometry and the meshing process have been made through the pre-

processor Midas FX+, through which it is also possible to post-process the output data generated by

DIANA.

The mesh has been constructed using curved shell elements to model masonry walls and floors,

beam elements to model reinforced concrete frames; however, it seemed appropriate to consider

the nonlinear behaviour concentrated only in the column elements to which a non-linear constitutive

model for concrete has been assigned. Beams and floors are supposed to remain elastic. Floors

have been modelled, as mentioned, with curved shell elements and equivalent thickness and an

equivalent self weight have been assigned, in order to reproduce load and stiffness of the actual

floors that are realized with T-Shaped concrete beams and lightening blocks.

Masonry walls are discretized with a mesh of curved shell elements and with a total strain based

crack model in order to reproduce the non-linear behaviour of the material.

149

Figure 6. 1 3D view of the model

Figure 6. 2 Finite element model of the building, front view (on the right)

The final mesh has 66626 nodes and 44087 elements, since the maximum dimension allowed for

the elements has been chosen as 20 cm.

The total mass of the model is almost 1550 tons.

6.1.2 Materials

The nonlinear behaviour of the masonry is modelled through the adoption of a total strain crack

model [TNO, 2009], as addressed before. The stress-strain diagrams follow an exponential

relationship to simulate tensile inelastic behaviour and a parabolic relationship to simulate

compressive inelastic behaviour.

150

Figure 6. 3 Stress-strain diagrams to define masonry behaviour in tension and compression [TNO, 2009].

Mechanical parameters that have been used in the model are briefly summarized in the following

tables, for masonry, concrete and reinforcement.

151

The procedure for the pushover analysis that has been implemented in the code refers to a

distribution proportional to masses and uses the regular Newton-Raphson method with an energy

convergence criterion with a tolerance of 10−3.

6.2 DESCRIPTION OF THE RESULTS

The nonlinear static (pushover) analysis was selected for evaluation of the seismic response of the

building and to compare the results obtained with this modelling approach with the ones previously

obtained by the equivalent frame modelling.

The analysis is made using an incremental-iterative procedure, which allows to predict the base

shear-displacement response (capacity curve) and to simulate the damage evolution in the individual

elements. The structure is in a first stage submitted to the vertical loading (one non-linear step), and

then the analysis proceeds with horizontal loading replicating the seismic load, for which a mass-

proportional pattern is assumed, as recommend in Lourenço et al. (2011) for the analysis of masonry

structures without box behaviour.

The horizontal load has been applied in X direction, parallel to the shortest side of the building. bNo

particular convergence problems have been envisaged and the results are briefly summarized in the

following.

Figure 6. 4 Pushover curve

152

Figure 6. 5

Figure 6. 14 Horizontal Displacement. Step 20



153


In the case examined is not possible to detect a clear horizontal branch in the pushover curve and

this is attributable to the fact that it’s possible to assume that the structure is involved in local failure

mechanisms that are still far to prevent the overall capacity of the building.

The crack pattern shows a diffused damage mainly in correspondence of the first floor, as can be

expected from the load pattern applied, with cracks that start at the edge of the openings, in the

walls parallel to the horizontal loads (in this case).

During the load process the cracks extend also at the base of the central piers, and the collapse is

assumed to be reached when these cracks reach a high extent.

Figure 6. 9 Crack width. Step 20

154




One of the most interesting aspects, that needs to be deepened, is the appropriate definition of the

connections between reinforced concrete frames and the external walls. In fact, from the analysis of

the crack pattern, it’s possible to locate damaged zoned in correspondence of the support of the

beams on the walls orthogonal to the load direction.

155

Some analysis conducted on masonry walls and masonry walls coupled with reinforced concrete

frames have shown that the definition of the tensile strength can influence the results of the analysis,

which are, on the other hand, almost independent from the compressive strength assumed (but not

from the compressive constitutive law).

Moreover, the numerical model strongly depends on the dimension of the elements, coupled with

the definition of a proper non-linear behaviour, especially in the tensile part.

The results obtained from these analyses that, however, are meant to be at a preliminary stage and

need to be refined in terms of meshing and material characterization, give some interesting hints on

the aspects that need to be studied more in detail.

The definition of an appropriate constitutive model for existing masonry is a crucial point, since

especially the tensile behaviour can strongly influence the results of the analyses.

The finite element model is able to catch some critical aspects, though requiring a much heavier

computational cost even for simple geometries, as the delicate role of the connection areas between

elements of different technologies, fact that in the equivalent frame model is not taken into account

or, however, doesn’t seem to influence the overall behaviour because of the hypothesis at the base

of the formulation of the model.

157

CONCLUSIONS

In this work the seismic behaviour of mixed masonry-reinforced concrete structures has been

studied.

An extensive bibliographic analysis has first been performed in order to identify a typological

classification for this complex and wide category of structures; it has been possible to understand

the critical aspects of the behaviour that these buildings have exhibited during past earthquakes and,

despite the lack of documents on this topic, also some data on past numerical studies have been

collected.

The regulatory framework, both in the national and in the international context, has been analysed

and it has been also possible to reconstruct an historical evolution of code prescriptions, mainly in

the Italian scenario, regarding mixed buildings.

The fundamental concepts of seismic assessment and modelling approaches for masonry building

have been explained in order to clarify the approaches that have been followed in the analysis of the

case studies presented.

Two case studies have been selected, for which information, though not always complete, have been

collected in literature and thanks to the help of other institutions.

First the case study named Capri Building has been analysed with an equivalent frame approach,

through the code Tremuri; on the basis of the original model proposed in literature, some variations

concerning the geometry and the structural systems have been made in order to understand the

contribution of the presence of reinforced concrete frames and the influence of their dimensions on

158

the global behaviour of the structures, especially referring to the shear repartition between structural

elements.

These analyses have led to some considerations on the adequacy of current Code prescription both

in case of new mixed constructions and in case of existing ones.

The results obtained have validated the approach suggested by the new Italian code.

In fact, for existing buildings, it is necessary to perform nonlinear analysis in order to evaluate the

seismic action sharing between structural elements characterized by different technologies since the

seismic action rate sustained by external masonry walls can strongly change depending on the type

of analysis (linear or nonlinear), and depending on the dimensions of the reinforced concrete

elements.

In the case of mixed masonry-reinforced concrete buildings in which frames have dimensions typical

of seismically designed r.c. structures, neglecting their contribution withstanding horizontal actions

would lead to overestimate the actual portions acting on masonry walls, since the amount of shear

that can be absorbed is about the 20-30% of the global value, depending on the dimensions of the

elements.

Indeed, the relative percentages shared between masonry and RC elements should be assessed by

means of a nonlinear static analysis.

The analyses also allow to make some consideration on the circumstances of retrofitting existing

masonry buildings by the insertion of reinforced concrete elements. The simultaneous observation

of the results of modal analyses and non-linear static ones leads to the consideration that, when

analyzing the behavior of existing masonry buildings that have been transformed into mixed r.c.-

masonry ones, substitution of internal masonry walls with RC frames can increase the elastic period

of the building and the displacement capacity. Though the insertion of reinforced concrete frames

after the demolition of internal walls could also strongly decrease the base shear capacity, the

combination of the decrease in strength and the increase in period determines that the seismic

checks are satisfied with capacity/demand ratios that don’t differ significantly.

159

The second case-study that has been analysed is the Ex-Slaughterhouse in Rome; actually, only a

part of this large building complex has been analysed but, despite the strong simplifications, the

regularity both in plan and in elevation of the structure allowed to infer some general conclusions.

In particular, the non-linear analyses performed on this building show an important collaboration of

the reinforced concrete frames that withstand in the non-linear phase a consistent part of the base

shear (almost 30% of the global shear), fact that causes, once masonry elements collapse, the

presence of a ductility backup and a slight resistance recovery.

In both cases, the out of plane mechanisms are assumed to be part of preliminary checks that do

not affect the presented assessment, moreover, the connections between elements of different

technologies are not analyzed in detail and are meant to provide a perfect collaboration during all

the steps of the analysis.

The implementation of one of the case studies (the original configuration of the Capri Building) in the

finite element code DIANA, though more refined analyses and calibrations are required, has

highlighted some critical aspects that need to be taken into account to deepen the understanding of

the actual behavior of these structures that, however, have shown a high vulnerability during past

earthquakes.

The results in terms of crack patterns, in fact, seem to confirm what had been already observed

about the out of plane contribution of masonry walls that act out of their plane and, consequently,

about the chance of taking advantage of a confining and stiffening contribution given by walls

perpendicular to load direction, when connections between elements are effective and properly

designed.

Future developments of this study will surely include these aspects; at first sight, in fact, they seem

to have a primary role, together with the recognized one (Code recommendations) of floors, in the

repartition of loads and the overall behaviour.

161

AKNOWLEDGMENTS

A sincere thank you to my supervisors, for having given me the chance to work on this topic and give

my (small) contribution in the scientific community. To all the people that helped me with suggestions

and useful tips, dedicating to me part of their time.

Thank you to all the people that have surrounded me in Bari and have supported me during these

three years, to all the friends I met in Guimarães, where a piece of my heart will always be.

Thank you to my family, for having made me feel that, even thousands miles away, I would have

never been alone.

[…] Domani andrò giù al porto

e gli dirò che sono pronto a partire

getterò i bagagli in mare

studierò le carte

e aspetterò di sapere per dove si parte

quando si parte

e quando passerà il monsone

dirò levate l'ancora

diritta

avanti tutta

questa è la rotta

questa è la direzione

questa è la decisione.

163

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171

CURRICULUM PERSONAL Name: MARIALUIGIA Surname: SANGIRARDI Date of Birth: 15/01/1988 Nationality: ITALIAN Contacts: [email protected],

[email protected] EDUCATION 2013 to present: PhD student at Politecnico di Bari, Bari, Italy 2012: Licenced to practice civil engineering in Italy (“Esame di stato di Ingegnere sezione A”). 2006-2012: MS and BS in Civil Engineering, (Curriculum Structures). Politecnico di Bari, Bari, Italy 2001-2006: Liceo Classico Socrate, Bari, Italy TEACHING EXPERIENCE 2013: Teaching Assistant for the Master level course Earthquake Engineering. 2014: Teaching Assistant for the Bachelor level course Building Construction; Reinforced Concrete Structures PROFESSIONAL AFFILIATION 2014 to present: Ordine degli Ingegneri della Provincia di Bari

172

LANGUAGES Italian: native language English: full professional proficiency French: intermediate level Portuguese: intermediate level JOURNAL PAPERS

-� Porco F, Uva G, Sangirardi M, Casolo S (2013). About the reliability of punching verifications in reinforced concrete flat slabs. THE OPEN CONSTRUCTION & BUILDING TECHNOLOGY JOURNAL, vol. 7, p. 74-87, ISSN: 1874-8368, doi: 10.2174/1874836801307010074

-� Porco F, Porco G, Uva G, Sangirardi M (2013). Experimental characterization of "non-engineered" masonry systems in a highly seismic prone area. CONSTRUCTION AND BUILDING MATERIALS, vol. 48, p. 406-416, ISSN: 0950-0618, doi: 10.1016/j.conbuildmat.2013.07.028

-� Fiore A, Porco F, Uva G, Sangirardi M (2014). The influence of uncertainties of infill panels relative to the seismic response of RC existing buildings. WIT Transactions on the Built Environment, Volume 141, 2014, Pages 479-490, doi: 10.2495/SUSI140411.

-� Sangirardi M, Porco F, Uva, G, Fiore, A. (2014). Numerical modelling techniques for the evaluation of the dynamic effects induced by excavation in existing structures. WIT Transactions on the Built Environment, Volume 141, Pages 323-334, DOI: 10.2495/SUSI140281

CONFERENCE PAPERS

-� Sangirardi, M., Reliability of Non-linear Models for the Seismic Vulnerability Assessment of Mixed Masonry-R.C. Structures, Proceedings of the First Score@Poliba, Track D, Dec 2014

-� Sangirardi, M., Uva, G., Lourenço P.B., Reliability of non-linear models for the seismic vulnerability assessment of mixed masonry-r.c. structures: a case study. Proceedings of COMPDYN Conference 2015, Crete Island, May 2015.

BOOKS

-� Sollazzo, M. Mezzina, M. Sangirardi – Chapter 2: "Teoria delle strutture" in Fondamenti di Tecnica delle Costruzioni, De Agostini Scuola Editore

˘ ˇˆ˙ - universidade do minho · to provide a comparison between refined and simplified models,...

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