Probabilistic Failure Models for Masonry Piers Subjected to In-plane Forces

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U

NIVERSITÀ DI

P

ISA

Scuola di Ingegneria

Dipartimento di Ingegneria Civile ed Industriale Corso di Laurea Magistrale in

INGEGNERIA EDILE E DELLE COSTRUZIONI CIVILI Curriculum: Costruzioni Civili

TESI DI LAUREA

Probabilistic Capacity Models for

Masonry Walls Subjected to

In-Plane Loads

Candidato: Relatori:

Leandro Iannacone Prof. Mauro Sassu

Prof. Paolo Gardoni Dr. Marco Andreini

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A Gio

per non aver potuto

mantenere la promessa

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Abstract

Over the past few decades, masonry buildings have proven to be particularly vulnerable to seismic actions. This is due to the incapability of the mortar that holds the bricks together to resist horizontal forces. The current regulations try to address this problem with approximate models that roughly take into account the different parameters that might play a big role in the collapse of a masonry pier. The purpose of this thesis is to assess the seismic fragility of masonry walls subjected to horizontal in-plane forces. This is done by adjusting the already existing deterministic models with correction terms that account for the inherent bias and uncertainty in those models. In fact, deterministic models are most of the times developed with simplified mechanics rules and tend to be on the conservative side. The performance-based design that has become more and more common during the last few years requires a new level of detail and new models that explicitly account for the prevailing uncertainties. A Bayesian framework will be applied to a set of data from tests that were carried out in different laboratories around Europe. This is an advanced updating technique that allows us to improve our knowledge on a certain model by including the results from experimental data.

This research has been made possible in the context of a collaboration between the Università di Pisa and the University of Illinois at Urbana-Champaign. Both Italy and Illinois can benefit from the results of this work; the whole Italian territory is classified as a seismic region and masonry buildings constitute the overwhelming majority of constructions. Illinois and the Midwest of the United States, on the other hand, are also subjected to seismic hazard; this hazard is usually overlooked due to the fact that last major seismic event was almost two centuries ago. Also, Unreinforced Masonry (URM) buildings constitute nearly one third of the essential facilities (i.e. firehouses, police stations, emergency management centers), which is above the average of the rest of the United States.

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Riassunto

Negli corso degli ultimi anni gli edifici in muratura hanno spesso mostrato i proprio limiti nel resistere le sollecitazioni sismiche. La normativa attuale cerca di affrontare il problema ma spesso si rivela poco efficace nel modellarlo e nel tener adeguatamente in conto di tutti i fattori in gioco nel collasso di un maschio murario.

Lo scopo di questa tesi è quello di valutare la vulnerabilità di maschi murari soggetti ad azioni orizzontali nel piano. Per raggiungere tale obiettivo, si è deciso di partire dalle relazioni disponibili nella normativa attuale (in particolare la normativa italiana) e correggerle mediante metodi statistici in grado di tener conto dell’incertezza e dell’errore sistematico insito nei modelli stessi. Infatti, i modelli deterministici disponibili sono stati sviluppati sulla base di modelli semplificati e poco dettagliati e tendono a fornire risultati estremamente conservativi. Il sempre più diffuso “Performance-Based design” richiede un livello di dettaglio e precisione che queste formule non sono in grado di fornire.

La teoria alla base di questo lavoro è fondata sul concetto di updating Bayesiano; tale teoria si basa sull’assunzione che sia possibile migliorare la nostra conoscenza di un determinato fenomeno inglobando i risultati di prove sperimentali in seguito ad una modellazione iniziale approssimativa. Sulla base dei risultati di una moltitudine di prove distruttive portate a termine in vari laboratori sparsi per l’Europa, è quindi possibile rivedere i modelli già disponibili in letteratura aggiungendo una serie di termini correttivi.

Questo progetto è stato reso possibile nell’ambito di una collaborazione tra l’Università di Pisa e la University of Illinois at Urbana-Champaign. I risultati di questa tesi potrebbero trovare utili applicazioni sia in Italia che in Illinois; infatti, se da un lato l’Italia è un territorio ad alta sismicità in cui il sistema costruttivo maggiormente diffuso è appunto la muratura non rinforzata, l’Illinois e il Midwest degli Stati Uniti presentano un simile rischio sismico, spesso sottovalutato a causa della mancanza di eventi significativi negli ultimi due secoli. Inoltre, gli edifici in muratura non rinforzata costituiscono in questa zona un terzo delle infrastrutture essenziali (ad esempio caserme, stazioni di polizia o centri di gestione delle emergenze), un valore ben al di sopra della media nel resto degli USA.

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Figures

Fig. 1.1. Seismic hazard in Europe according to the SHARE project: PGA, 475 years return

period ... 11

Fig. 1.2. Seismic classification of the Italian territory before (left) and after (right) 2003 .... 13

Fig. 1.3. Pictures from the 2009 Aquila earthquake ... 14

Fig. 1.4. Seismic hazard map of the United States according to USGS ... 15

Fig. 1.5. The New Madrid Seismic Zone according to USGS ... 16

Fig. 2.1. Graphic depiction of Rocking, Diagonal Cracking and Sliding ... 20

Fig. 2.2. Dependence of the type of failure on the vertical load acting on the wall ... 21

Fig. 2.3. Some example of walls failing in diagonal cracking ... 21

Fig. 2.4. Diagonal cracking with weak joints (left) and resisting joints (right) ... 22

Fig. 2.5. Shear and normal stress at the mid-section of a wall subjected to σ and V ... 22

Fig. 2.6. Mohr circle for the stress condition at the mid-section of the wall ... 23

Fig. 2.7. Typical hysteretic cycle for a masonry wall failing in diagonal cracking ... 25

Fig. 2.8. Distinction between rocking and toe crushing... 25

Fig. 2.9. Scheme for the development of the flexure/rocking capacity model ... 26

Fig. 2.10. Typical hysteretic cycle for a masonry wall failing in flexure/rocking ... 27

Fig. 2.11. Some examples of walls failing in sliding ... 28

Fig. 2.12. Graphical representation of a Mohr-Coulomb failure criterion... 28

Fig. 2.13. Graphical representation of dimensions for computing D’ ... 29

Fig. 3.1. Flowchart of the step-wise deletion process ... 49

Fig. 3.2 Representation of data types for a bivariate model ... 51

Fig. 4.1. Loading history in cyclic lateral tests ... 55

Fig. 4.2. Example of Units for Mortar Pocket masonry (ZAG laboratory, Lubjana) ... 56

Fig. 4.3. Example of Units for Tongue and Groove masonry (ZAG laboratory, Lubjana and University of Pavia) ... 56

Fig. 4.4. Hysteresis cycles for a wall failing in flexure (left) and diagonal cracking (right), IUSS report ... 59

Fig. 4.5. Example of a Diagonal Cracking collapse with corresponding hysteresis cycle (IUSS report, 2009) ... 60

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5 Fig. 4.6. Example of a Flexural/Rocking collapse with corresponding hysteresis cycle (IUSS report, 2009) ... 60 Fig. 4.7. Predicted capacity vs Measured capacity – Deterministic model for diagonal cracking ... 66 Fig. 4.8. Predicted capacity vs Measured capacity – Deterministic model (log

transformation) for diag. cracking ... 66 Fig. 4.9. Step-wise deletion process for the diagonal cracking capacity model ... 67 Fig. 4.10. Predicted capacity vs Measured capacity – Probabilistic model (log

transformation) for diag. cracking ... 68 Fig. 4.11. Predicted capacity vs Measured capacity – Probabilistic model for diagonal cracking ... 69 Fig. 4.12. Measured vs Predicted capacities based on the deterministic (left) and

probabilistic model (right)... 69 Fig. 4.13. Predicted capacity vs Measured capacity – Deterministic model for

flexure/rocking capacity ... 70 Fig. 4.14. Predicted capacity vs Measured capacity – Deterministic model (logarithmic) for flexure capacity ... 71 Fig. 4.15. Step-wise deletion process for the flexure/rocking capacity model ... 71 Fig. 4.16. Predicted vs Measured capacity – Probabilistic model (log transformation) for flexure/rocking ... 72 Fig. 4.17. Predicted capacity vs Measured capacity – Probabilistic model for flexure/rocking ... 73 Fig. 4.18. Measured vs Predicted capacities based on the deterministic (left) and

probabilistic model (right)... 73 Fig. 4.19. Predicted capacity vs Measured capacity – Bivariate model ... 74 Fig. 4.20. Predicted capacity vs Measured capacity – Deterministic bivariate model

(logarithmic) ... 75 Fig. 4.21. Step-wise deletion process for the bivariate capacity model ... 76 Fig. 4.22. Predicted vs Measured capacity – Probabilistic bivariate model (log

transformation) ... 78 Fig. 4.23. Measured vs Predicted capacities based on the deterministic (left) and

probabilistic model (right)... 79 Fig. 4.24. Predicted vs Measured capacity – Probabilistic bivariate model ... 79 Fig. 5.1. Interpretation of failure probability integral for independent C and D ... 81

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6 Fig. 5.2. Graphical representation of the component reliability problem (Courtesy of

Gardoni) ... 83

Fig. 5.3. Transformation of the pdf and the limit state function when moving from the original space (left) to the standard normal space (right) ... 84

Fig. 5.4. Graphical representation of the properties of the standard normal space ... 85

Fig. 5.5. The FORM approximation for a component problem (Gardoni) ... 88

Fig. 5.6. Flow-chart for the HL-RF algorithm (Gardoni) ... 91

Fig. 5.7. Visualization of a parallel system ... 92

Fig. 5.8. Failure domain of a parallel system in the original space and the standard normal space ... 93

Fig. 5.9. Visualization of a series system ... 93

Fig. 5.10. Failure domain of a series system in the original space and the standard normal space ... 94

Fig. 5.11. Bounds on fragility curve (Gardoni, 2002) ... 96

Fig. 6.1. Fragility curve for diagonal cracking failure ... 98

Fig. 6.2. Fragility curve for diagonal cracking failure with bounds ... 99

Fig. 6.3. Fragility curve for flexure/rocking failure with bounds ... 100

Fig. 6.4. Fragility curve for flexure/rocking and diagonal cracking failure (using results from bivariate analysis)... 100

Fig. 6.5. Fragility curves for diag. cracking and flexure and with the fragility curve for the system (N=350 kN) ... 102

Fig. 6.6. Effect of the input parameters uncertainties on the fragility curve for diagonal cracking ... 103

Fig. 6.7. Effect of the vertical load on the fragility curves ... 104

Fig. 6.8. The formation of compressed struts limits the tensile stresses and prevents flexure/rocking ... 105

Fig. 6.9. Effect of the slenderness ration on the fragility curves ... 106

Fig. 7.1. Measure vs predicted capacity for deterministic model (left) and probabilistic model (right) ... 108

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Tables

Table 2.1. Different regulations for OPCM3431 (Italy), Eurocode 6 (Europe), FEMA 356 and

ACI 530-05 (USA) ... 31

Table 3.1 Empirical formulations for F for given marginal of the initial variables X ... 47

Table 3.2 Probability Terms for Bivariate Capacity Model with Lower Bounds and Failure Data ... 52

Table 4.1. Values for the initial shear strength of masonry, fvk0 ... 58

Table 4.2. Classification of the 55 tests ... 62

Table 4.3. Perforated bricks according to the Italian code ... 63

Table 4.4. Posterior statistics for Θ of the diagonal cracking capacity model ... 67

Table 4.5. Posterior statistics for Θ of the flexure/rocking capacity model ... 72

Table 4.6. Posterior statistics for Θ of the bivariate capacity model ... 77

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Chapter 1 Introduction

The main purpose of this introductory chapter is to assess the need for the probabilistic models that will be hereby developed. The objective and scope of the thesis is underlined with a quick review of the common practices together with their limitations and how the theory developed can help overcome them. The first two paragraphs offer a brief history of how structural reliability has influenced both research and design over the past century. The exact specifications for masonry are left to be analyzed in the following chapter in more detail. We will then provide a general framework of the seismic hazard in Europe (with a focus on Italy) and the United States (with a focus on the state of Illinois), where the main contributions to this work were developed.

1.1 Objectives and scope

Being the most common building system around the world, practices in masonry have been perfected over time, but never really optimized up until recently. The enormous difficulties in modeling a heterogeneous material such as masonry have somehow prevented the development of a comprehensive theory that specifically took into account the inherent uncertainties in the design. Predictive capacity models in current structural engineering practice are typically deterministic and on the conservative side, and this is particularly true for masonry. Due to its long history, the practices were so incepted into the minds of builders and designers that following them always constituted an easier solution than conceptually studying its behavior, especially in light of the fact that other materials (e.g. concrete and steel) were getting all the attention with the early development of solid mechanics. This is the approach of the traditional design codes, where common practices and prescriptions are usually preferred over an optimized design. Unfortunately, recent earthquakes have emphasized the vulnerability of most of the masonry buildings in seismic zones and the need to mitigate the risk consequent to the failure of these buildings. The assessment and prediction of damage from an earthquake and the estimation of consequent losses provide valuable information for the adaptation of existing buildings and the design of new ones.

The main purpose of this thesis is to overcome the limitations of the traditional building codes that consistently limit the freedom in the design and develop a new theory that well fits the context of performance-based design. A building constructed in this way is required to meet certain measurable or predictable performance requirements, such as energy efficiency or seismic load, without a specific prescribed method by which to attain those requirements. Such an approach provides the freedom to develop tools and methods for design that do not necessarily follow what has previously been done. The clearest definition of performance based building approach was explained in 1982 by the CIB W60 commission in the report 64, where Gibson stated that first and foremost, the “performance approach

is [...] the practice of thinking and working in terms of ends rather than means.[ …] It is concerned with what a building or building product is required to do, and not with prescribing how it is to be constructed”. This kind of gets back the construction practices to their dawn of times, when builders

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9 had to follow straight-forward statements such as “a house should not collapse and kill anybody” (Hammurabi’s Code).

By taking into account all the prevailing uncertainties, the models hereby developed are introduced as an improvement of the current models. In particular, we will operate by adding correction terms to the existing models in literature in order to correct the bias in them and evaluate the dispersion of the results. Further insight into these models can be gained in chapter 2. The new probabilistic capacity models are used to estimate the fragility of masonry walls subjected to in-plane forces that can be intended as seismic forces, with special attention given to the treatment of aleatory and epistemic uncertainties (we remember that the fragility is defined as the conditional probability of a structural component or system for a given set of demand variables).

Although the methodology described in this work is aimed at developing probabilistic capacity models, the approach is general and can be applied to the assessment of models in many engineering fields.

1.2 Literature review

With the advent of structural reliability in the second half of the past century, there have been several studies on the seismic fragility of structures. Different strategies and approaches have been followed by many authors in this field. Hwang and Huo (1994) have presented an analytical method for generating fragility curves based on numerical simulation of the dynamic behaviour of specific structures. Among other works using Monte Carlo simulations (i.e. simulations based on random generation of numbers) for specific structural models we can list Fukushima et al. (1996), Kai and Fukushima (1996), Shinozuka et al. (2000), and Karim and Yamazaki (2001). The uncertainties in the inputs of the site-structure system are quantified by considering the parameters in the system as random variables and generating values from their distributions. The fact that these approaches based on random generations of numbers were having a boost at the end of the past century is because that period coincides with a huge improvement of the computational capacities. Also methods were developed to save computational time in the Montecarlo simulations. In particular, Fukushima et al. (1996) and Kai and Fukushima (1996) have proposed a fragility analysis method where random vibration theory in the frequency domain is used to evaluate the structural response.

Fragility estimates have also been developed based on expert opinion. In 1985 an advisor Project Engineering Panel has developed the damage probability matrices for 78 different facility types based on consensus estimates. Other authors have developed empirical fragility curves on the basis of the records of damage resulting from past earthquakes. For example, Basoz and Kiremidjan (1997) used the data from damage observations after the Northridge earthquake to develop empirical fragility curves by logistic regressions. They also developed curves for bridges grouped in 11 classes according to their characteristics. In other applications related to seismic fragility assessment (Singhal and Kiremidjian, 1998) the fragility is defined as the conditional probability that the damage index (as defined in Park and Ang, 1985) exceeds a certain threshold for a given ground motion. Singhal and Kiremidjian have assumed that the randomness in the damage index at a specified ground motion level can be represented by a lognormal distribution with unknown median and known constant standard deviation. Observed damage data from past earthquakes have been used to update the distribution of the median of the lognormal distribution of the damage index by using conjugate distributions.

The approaches of fragility analysis have been reviewed in detail by Casciati and Faravelli (1991) with the purpose of summarizing the viable approaches. Among the more original methods we can mention some based on artificial intelligence techniques.

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10 Up until 2002, these approaches were highly sector-based, and the fragility estimate for a specific structural system could not be used to assess the fragility of another structure, unless the two structures were of a similar type (in that case crude approximations could be operated). Also, most of the fragility curves could only be obtained by analyzing the experimental results at the system level, and no use was possible for whatever experimental data was available at the component level. Finally, these approaches did not properly account for all the uncertainties that were involved. By specifically addressing the different uncertainties (epistemic and aleatory) and developing a method for obtaining a system fragility starting from the fragility of the components, Gardoni and Der Kiureghian (2002) managed to overcome most of the limitations in the previous approaches. In particular, most of the theory developed in this thesis is based on the 2002 paper Probabilistic Capacity Models and Fragility

Estimates for Reinforced Concrete Columns based on Experimental observation. In that work, a

Bayesian framework was introduced for the development of multivariate probabilistic capacity models for structural components and, specifically, it was applied to a bridge column under cyclic loading. These theories were furtherly developed in Choe, Gardoni and Rosowsky (2003).

In the context of masonry walls subjected to horizontal in-plane forces, many contribution can be found. Among the most important assessment of the capacity of masonry wall, a fundamental contribution was given by the studies that were carried out in Slovenia by Turnšek and Čačovič (1970) and later on perfected by Turnšek and Sheppard (1980), Mann and Muller (1982) and Tomaževič and Sheppard (1982). The models that were developed in those works, especially the ones for the capacity of walls in diagonal cracking, are still the main reference when dealing with this type of problems. Further developments are still being carried out, and Tomaževič (2008) recently studied how to include these theory into the regulations provided by the Eurocode 6. Italy is also one of the main contributors to the field, with works from Magenes and Calvi (1992, 1997) that analyzed the In-plane Seismic Response of Brick Masonry walls by comparing the models available in literature with non-linear finite element analyses. The foundation of the EUCENTRE (European Centre for Training and Research in Earthquake Engineering) in 2003 was a major step towards a uniform approach in tackling the problems related to masonry. The major activities in the center are relative to research applied in the field of seismic engineering and definition of specific guidelines for the evaluation of vulnerability and risk.

Other works available for further insight in masonry are the ones from Vasconcelos and Lourenҫo (2006, 2009) who developed some simplified models for the assessment of the in-plane shear strength of dry stone masonry walls and, together with Mosele, da Porto, Modena et al. (2005) studied innovative systems for reinforced masonry walls. Also at the University of Illinois studies about the analysis of seismic damage and the design of Unreinforced Masonry Buildings were done by Kwok and Ang (1987). In general, in the past few decades several researchers investigated experimentally the in-plane behavior of URM walls. Researchers examined the effect of the main parameters of URM walls such as aspect ratio and normal force level on the in-plane behavior of URM walls. The walls were tested under monotonic (Magenes and Calvi 1994, Epperson and Abrams 1989 and König et al. 1988), static cyclic (Zilch et al. 2002, Anthoine et al. 1995 and Abrams and Shah 1992), pseudo dynamic (Zilch et al. 2002 and Anthoine et al. 1994) and/or dynamic loading (Magenes and Calvi 1994 and König et al. 1988). The specimens were tested under different boundary conditions. In Italy (Anthoine et al. 1994, Magenes and Calvi 1992 and Magenes and Calvi 1994), specimens were tested under complex loading conditions intended to reproduce those actually affecting piers in a building during an earthquake. These conditions were constant vertical force and double bending moment. In the United

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11 States (Abrams and Shah 1992, Abrams 1992, Epperson and Abrams 1989) and elsewhere (König et al. 1988), cantilever specimens were tested.

The suitability of the limit states for the analyses of masonry structures in the context of the performance-based earthquake engineering (PBEE) was investigated by Mouyiannou, Rota et al. (2014) by making use of non-linear dynamic analyses. In this work, the damage states were identified through displacement indicators. More recently Andreini, de Falco, Giresini and Sassu (2014) investigated the mechanical characterization of masonry walls with chaotic textures with In-situ tests such as shear-diagonal tests with flat jacks.

Finally, we should mention that reliability and statistics are starting to be applied to the verification of Masonry walls, in particular in the works of Montazerlghaem and Jager (2016) and Bracchi, Rota, Magenes and Penna (2016).

1.3 Seismic hazard in Europe

Fig. 1.1. Seismic hazard in Europe according to the SHARE project: PGA, 475 years return period

Geophysical sources for seismic hazards go beyond geographical national borders and efforts have been done recently to harmonize the probabilistic evaluations of the seismic risk together with the practices to be followed when designing buildings to resist seismic actions, with harmonized data for the creation of models for risk that cover the Euro-Mediterranean region like never before. Collection and interpretation of seismic data can depend mostly on already existing rules coming from different cultures. The idea of “high-risk” zone in a country such as France is different from the same idea in a

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12 different country, such as Italy, more subjected to seismic hazard. An integrated and cooperative approach is needed to harmonize the data and evaluate the risk of earthquakes to develop accurate prediction models that could increase the safety of citizens and infrastructures. The same applies to the models for the failure of structures subjected to seismic hazard, and this work focuses on the ones for masonry and tries to increase their accuracy and incorporate the prevailing uncertainties in them. EU funds have allowed the scientists involved in the SHARE project (“Seismic hazard Harmonization in Europe) to coordinate the resources, the time and the competences of researchers around the European territory, the Maghreb and Turkey (except the areas close to the Middle East and the Red Sea). Collaborators are required to have competences in field such as Geology, Seismology and, of course, Seismic Engineering. With a strategy that encourages a strong collaboration, SHARE has provided publicly available resources like never before. The initial assessment of the requirements from an engineering perspective has guided the creation of an adequate database and the selection of equations for the prediction of soil movements. This has led to the creation of the European catalog of earthquakes (SHEEC), the first pan-European database of active faults and seismic sources, and to the homogenization of the models for the entire territory. This is perfectly integrated in the norm for the constructions, i.e. the Eurocode and their respective National Annexes. In particular, the Eurocode 8 is the one dedicated to the design of structure for seismic performances. This, together with the Eurocode 6 which contemplates the design of masonry structures, has been one of the main sources for the analysis of the existing models in chapter 2.

Deadly earthquakes have hit the continent over the past centuries: among the deadliest outside of Italy, we remember the Granada earthquake in 1884 that killed 800 people, the Greece earthquake in 1881 that killed 3550 people and the Lisbon earthquake in 1755 that killed tens of thousands. If we look at the map in figure 1.1., which represents the most relevant contribution of the SHARE project and visualizes the Peak Ground Acceleration (PGA) with 10% exceedance probability in 50 years, we can get an idea of how the seismic hazard is distributed in the European territory. This map was created by combining the data coming from more than 30000 European earthquakes with magnitude greater or equal to 3.5 degrees on a Richter scale from the year 1000 AD, and by considering the damages caused. From this map we can see how the regions that are the most exposed are Italy, the Balkans, Greece, Bulgaria, Romania and Turkey. Most of these sites (especially Italy, Greece and Romania) have clay masonry among the most common building types. Masonry is also extremely common in other countries belonging to areas that are not as dangerous, such as Spain, France and England. In general, one should not underestimate the prominence of the clay masonry industry in Europe. It is true that the overall market is shifting toward other building types, but in the whole European territory there are still over 1300 sites for the production of clay bricks and tiles, and the sector has invested heavily in product and process innovations over the last decades. Furthermore, the study of the behaviour of masonry buildings still constitutes a topic of predominant importance due to the extraordinary amount of existing constructions built in masonry, especially in Italy. The following paragraph focuses on the classification of the Italian territory.

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1.4 Seismic hazard in Italy

Fig. 1.2. Seismic classification of the Italian territory before (left) and after (right) 2003

Italy is the country in Europe that has experienced the deadliest earthquakes: among the most notable dates, we can recall December 28, 1908, when a massive quake rocked southern Italy and left about 95000 people dead in the Sicilian port city of Messina and across the strait in Reggio Calabria, January 13, 1915, when a quake in the central Abruzzo region killed 30000 people and November 23, 1980, when the southern Campania and Basilicata regions were hit by a quake that left 2916 people dead in the Irpinia region close to Naples. More recent earthquakes, while less disastrous, also play a very important role because they have helped shaping the norm regulating the construction of masonry buildings in the Italian territory (see next chapter for more details). In particular, right after the 2003 earthquake of San Giuliano di Puglia in the Molise region, the code was changed and the whole territory was completely re-classified. The change is evident in figure 1.2. After 2003, the whole peninsula and the two main islands are classified as seismic, with the overwhelming majority of the territory belonging to at least a zone 2 (zone 4 being the most dangerous). Also most territories that were not classified as seismic found themselves in a seismic zone after 2003 and, as a result, the presence of buildings, also quite recent, that do not satisfy the seismic requirements has considerably increased. Before 2003, the first homogeneous classification of the territory dated back to 1984. Italy is probably the country in Europe with the worst combination of seismic hazard and amount of masonry constructions. A secular heritage, the lack of seismic criteria in design (because they were not required at the time), conservation interventions without adequate structural verifications, are some of the factors that make the Italian construction patrimony the one most subjected to seismic hazard. Luckily, and also consequentially, the Italian code is one of the most developed in this field

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14 and, as we will see in more detail in chapter 2, the approaches proposed by the Italian regulations are much more comprehensive than the ones of the Eurocode.

According to the 2001 census, out of 27,291,993 buildings, more than 60% was built before 1971. The distribution is almost homogeneous over the national territory, with peaks of 80% in the Liguria, Piedmont and Tuscany regions and lows of around 50% in Sardinia. Just these percentages can give an idea of how old and inadequate to the modern seismic specifications the Italian residential heritage is. Concerning the typology, the same census reported that 6,903,982 buildings (61.50%) were clay masonry buildings, 2,768,205 (24.66%) were concrete buildings (most of which with clay masonry claddings) and 1,554,408 had other characteristics.

Of course, this does not mean that, after a big earthquake, everything is destined to collapse, because general prescriptions have been used since ancient times. These prescriptions were sort of limiting in the freedom given to the builder (then) and the designer (now). The approach proposed in this work helps overcome the predominant conservativism that has dominated during the past centuries by accounting for the prevailing uncertainties in the models.

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1.5 Seismic hazard in the United States and Illinois

Fig. 1.4. Seismic hazard map of the United States according to USGS

This thesis was developed in the context of a collaboration between the University of Pisa and the University of Illinois. A brief explanation of how the results coming from this thesis could be useful not only in Europe but also in the Unites States is provided below.

Unreinforced Masonry is one of the most common structural types of low-rise building in the United States. That is why most states that have experienced strong earthquakes in the past have regulated further construction of URM buildings. For example, California prohibited unreinforced masonry in 1933, and a state law enacted in 1986 required seismic retrofitting of existing structures. Despite these precautions, problems still persist in existing structures and the 2006-04 California safety commission report stated that there are still 7800 URM buildings with no retrofitting in the state, 1100 in the city of Los Angeles. Unfortunately, California happens to be the best-case scenario. There are some seismically active regions in the country that have not experienced strong earthquakes in the recent past, where communities have continued to build unaware of the potential loss of a strong seismic event. That is the case with Utah’s most populous metropolitan area, the Wasatch Front, whose earthquake threat was not widely recognized until 1970s. This area has a population of 2 million and contains 200,000 UMBs compared with the entire state of California’s 25,000.

Concern is particularly noteworthy in the Central and Southern US (CSUS), where researches have shown that nearly one third of the essential facilities (i.e. firehouses, police stations, emergency management centers etc.) are low-rise (i.e. two stories or less) URM structures. This region includes the New Madrid Seismic Zone (NMSZ). The Federal Emergency Management Agency has estimated that, in a repeat of the 1811-1812 NMSZ earthquakes, 90% of severe casualties would be due to the 500,000 URMs in the area.

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16 These are only some of the examples that show that the assessment of seismic fragility of masonry buildings subjected to horizontal forces is particularly important even in a country like the US, where masonry (especially clay brick masonry) is not the most common method of construction.

Fig. 1.5. The New Madrid Seismic Zone according to USGS

1.6 Organization of the thesis

Following the general introduction given in this chapter, Chapter 2 will focus on what the current models for masonry walls subjected to horizontal forces are. After providing a general framework for the current regulation and a brief history of the norm in Italy, we will go over the three different types of failure: Diagonal cracking, Rocking/Flexure and Sliding. The models for each one of the capacities are analyzed together with the theory behind them. The chapter ends with a comparison between the different codes and their ways of tackling the problem.

Chapter 3 discusses a Bayesian approach for the statistical analysis. A general framework of the

differences between the frequentist and the Bayesian approach is provided, together with a brief history of the second. The Bayes theorem is introduced and the problem of constructing a non-informative prior that reflects no previous knowledge on the problem is investigated. The application of Bayes’ theory to the correction of existing models is analyzed, and in this context we will introduce the fundamental concepts of bias correction term, explanatory functions and step-wise deletion. Chapter 4 starts with a brief description of the set of experimental data at our disposal and the assumptions that were made to classify the different specimens. Then, the methodology presented in chapter 3 is used on the models introduced in chapter 2 to develop univariate and multivariate probabilistic capacity models for masonry walls subjected to in-plane forces. An objective assessment of the qualities the new models is made.

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17 of the fragility curves at the following chapter. In particular, the structural reliability index β is introduced and the First Order Reliability Method for assessing its value is described in detail. The chapter continues with the assessment of the reliability of a system starting from the probability of failure of the single components. Finally, the concept of fragility curve is introduced and also how to put bounds on the probability of failure for each one of the levels of demand.

Chapter 6 presents the fragility curves for the masonry walls based on the models obtained in chapter

4. Different fragility curves are drawn with different values for the input variables in order to investigate the effect of said variables on the capacity (diagonal cracking and flexure/rocking) on the wall.

Chapter 7 contains the conclusions and some ideas for future developments on this same topic. Annex A contains the MATLAB scripts that were used to obtain the results presented in chapter 4 and

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18

Chapter 2 Current Models for Masonry

2.1 A General Framework

When considering masonry structures, there are two contrasting aspects of the theory and the research underlining this area of study that one must consider. On one side, masonry is probably the most common worldwide building material consumed in residential buildings. This is particularly true in countries such as Italy where its long lasting history has contributed to its diffusion. On the other hand, the approach to masonry construction has always been extremely on the conservative side (for example, the massive structures that still survive nowadays have not been really designed with an optimized perspective in mind). From ancient times masonry building regulations have been based on formal, if not philosophical, criteria (we can think of classical writers such as Vitruvius). The first rationalization of the study of masonry buildings can be traced to the work of De la Hire (1712) and Couplet (1730), which tried to study the behaviour of masonry portals; their approach did not even consider the friction between the units; we will have to wait 50 more years before somebody took that effect into account (Coulomb, 1773). Moreover, masonry is a highly heterogeneous material, and with the development of the solid mechanics theories, the interest was shifted towards other materials (such as concrete) that better fit the models for the “ideal” solids. The common practices in constructing masonry buildings were so eradicated in popular culture that even today people tend to follow the prescribed practice instead of tackling the problems with an optimized design in mind. This last point explains why, in spite of the overwhelming diffusion of this building typology, the reliability studies on masonry structures are few. Moreover, what is currently available in literature is mostly limited to the basic models only (e.g. Ellingwood(1980)). In recent years, especially with the recent uprising of the performance-based design, which is focused on the attainment of a certain performance without imposing prescribed practices, this approach has been starting to show its limits. Also, most of the studies focus on masonry structures subjected to axial loads, while horizontal forces, both in-plane and out-of-plane, have not been tackled prominently. Currently, there is even some contradiction between different codes (e.g. the Eurocode 6 and some national codes) on how to model the effect of horizontal in-plane forces on a wall. Some approaches completely differ in the types of failure they assume to be possible.

In the context of structure reliability works on Unreinforced Masonry Walls (URMW), extensive studies to obtain objective safety values for existing structures has been carried out by Luc Schueremans (2001), where the results were applied to historical buildings subjected to axial forces. Schueremans, together with van Gemert and Maes (1999) also adopted a combined model to evaluate the reliability of structural masonry elements through First Order Reliability Method (FORM). These models were used for the limit state formulations in a probabilistic approach. Stewart and Lawrence (2002) have developed preliminary techniques to estimate the structural reliability of masonry walls for vertical one-way bending and compression loading. We can also cite Brehm (2011), who developed accurate models to evaluate the structural reliability of masonry shear walls. Also Glowienka and Graubner (2008) ran reliability analysis on masonry walls made of large size units and thin layer mortar. The

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19 EUCENTRE (European Centre for Training and Research in Earthquake Engineering) has been working extensively on this topic; in particular we will work on the database of experimental tests collected in an Interpretation of experimental shear tests on clay brick masonry walls and evaluation of q-factors

for seismic design by Frumento, Magenes and Calvi (2009).

2.2 A brief history of regulations in Italy

Due to the fact that masonry buildings constitute the overwhelming majority of the constructions in Italy, most of the regulations have been focused on how to preserve and consolidate the existing structures.

At the end of the 18th_{century there were few or no prescriptions on how to construct masonry} buildings, and studies on the topic were experiencing a hiatus due to the development of new techniques and theories such as concrete behavior. Regulations were modified sporadically, especially in the aftermath of catastrophic events such as the earthquakes in Calabria (1783), Liguria (1887) and Sicily (1908), but most of the times these modifications were none other than new prescriptions on how to build, such as forbidding the use of wooden or metallic floors particularly vulnerable to horizontal forces, or imposing limits to the height of the buildings. Numeric designs and verifications for masonry buildings subjected to horizontal forces were mostly ignored. It will be only in 1981, after an earthquake that completely devastated the Italian region of Irpinia (1980) that a substantial change was made to the code by introducing some of these aspects, although limited to maintenance and reinforcing. An administrative order specifically designed for masonry had already been issued in 1974 (Legge 64), but it would not be published until 1987 (D.M. 20.11.87).

It was again after another catastrophic event, the Molise earthquake of 2002, that the norm was revised (Ordinanza 3274, March 2003). Due to some formal issues, these new regulations were subjected to a lot of revisions before their final version was actually published in the OPCM (Ordinanza del Presidente del Consiglio dei Ministri) 3431. This was the norm that introduced substantial innovations in this field. It imposed the Limit Design and a fourth seismic zone (making the whole Italian peninsula subjected to seismic hazard, see figure 1.2).

What is currently being used is the NTC2008 (Norme Tecniche per le Costruzioni), issued with the D.M. (Decreto Ministeriale) 14.01.2008 with the main goal of reorganizing what was introduced in the previous years and introducing the new prescriptions from the Eurocode 6. The chapters of this code dedicated to masonry are:

• Chapter 4, Costruzioni civili ed industriali (residential and industrial buildings), paragraph 4.5 • Chapter 7, Progettazione per azioni sismiche (Seismic design), paragraph 7.8

• Chapter 8, Costruzioni esistenti (Existing structures)

• Chapter 11, Materiali e prodotti per uso strutturale (Construction materials), paragraph 11.10

These regulations must be integrated with the CE (Circolare Esplicativa) 02.02.2009, especially for the parts concerning the existing structures. Finally, in case of buildings belonging to the historic heritage (quite common in Italy), one could use the guidelines in the DPCM (Direttiva del Presidente del Consiglio dei Ministri) “Valutazione e riduzione del rischio sismico del patrimonio culturale con riferimento alle Norme Tecniche per le Costruzioni” (Evaluation and reduction of the seismic risk for the cultural heritage).

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20 Before we analyze how the problem of the failure of a certain wall subjected to in-plane forces has been tackled by the Italian code and the Eurocode, we should first analyze the three different typologies of failure that are commonly identified in these cases.

2.3 Modes of failure

According to the geometric properties of the wall, its boundary conditions and the vertical load it is subjected to, three different types of failure are usually identified:

1. Diagonal Cracking: the failure is governed by the formation of diagonal cracks that can either

pass through the bricks or follow a bed- and head-joints path, according to the relative resistance of the mortar and the elements. A combination of the two phenomena is also very common.

2. Rocking Failure: the increasing in load or displacement demand causes the bedjoints to crack

in tension. As a consequence, the shear is only carried by the compressed masonry; the final failure is obtained by overturning of the wall and simultaneous crushing of the compressed corner. This type of failure is sometimes identified as flexure failure. In fact, a pretty obvious parallel can be drawn between this failure and the flexure failure of a beam (in particular in light of the fact that both start with the cracking of elements on the tension side). This explains why, in this text, the terms “flexure” and “rocking” are sometimes used interchangeably. 3. Sliding: Potential sliding plane can form along cracked bedjoints as a consequence of the

formation of tensile horizontal cracks due to the reversed seismic action. This type of failure is very common in case of low levels of vertical load or low friction coefficient. This could happen in buildings for the piers constituting the walls of the upper stories but, as we will see, it is not very common in a laboratory environment.

Fig. 2.1. Graphic depiction of Rocking, Diagonal Cracking and Sliding

It must be noted that, given the geometric properties of a certain wall and its boundary conditions, one of the main factors controlling the type of failure is the vertical load acting on that wall. In the general case, especially when analyzing existing structures, one needs to extrapolate the single panel from the complete structure in order to study it. At that point, the actions on the wall will be determined from how the panel itself is interacting with the rest of the structure; in particular, most of the normal vertical action will be due to the gravity loads acting above the panel object of study. For example, on a building façade, while the different panels might be assimilated in terms of geometric and material properties, the same does not apply in terms of vertical load acting on them, because the lower stories’ walls will be subjected to greater loads than the upper stories’. For the same building then, panels with apparently the same properties could experience different types of failure. The following figure qualitatively shows the shape of the different failure domains for a generic

Sub-vertical cracks Tensile Flexural cracking Diagonal crack

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21 wall. We can observe how sliding controls the capacity of the wall for lower demands, and how this shifts to diagonal cracking and flexure/rocking when we increase the load.

Fig. 2.2. Dependence of the type of failure on the vertical load acting on the wall

We will now go in-depth for each one of the three failure types, introducing the models that are most common among the different codes and how they have been obtained. According to the definition given in Gardoni, Der Kiureghian and Mosalam (2002) a “model” is defined as a mathematical expression in relation to one or more basic variables ( , , … , ). The main purpose of the model is to provide means for predicting the quantities of interest for the given deterministic or random values of the basic variables. This definition perfectly fits our case and it will be used to refer to any mathematical relationship of shear capacity prediction of Unreinforced Masonry Walls.

2.4 Diagonal Cracking

Fig. 2.3. Some example of walls failing in diagonal cracking

Probably the most common type of failure observed after seismic events on historical buildings, it usually happens for intermediate values of the vertical loads. Together with the sliding failure, it is usually classified as “Shear failure”, as it is caused by the shear stresses τ that are present in the panels as a direct consequence of the horizontal forces acting on the wall. If the mortar joints are the weakest

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22 part of the masonry ensemble, it usually manifests itself as a stepped crack passing through the bedjoints and the headhoints. More rarely, the cracks can go through the units. Also a combination of the two phenomena can be possible.

Fig. 2.4. Diagonal cracking with weak joints (left) and resisting joints (right)

This phenomenon is the consequence of a multitude of different factors like the properties of the single components of the masonry (mortar and units) and the way those components are arranged in the wall (layout). Differently from other failures, this one is deeply dependant on the heterogeneity of the system, and it has always been a difficult task to come up with a closed form formula that could represent the actual behaviour. Some approximation had to be done to obtain a model that could estimate the capacity in the best way possible. The most common references when dealing with the type of failure are the works from Turnšek and Čačovic (1970) and Mann and Muller (1980). In the standard interpretation, the diagonal cracking capacity of the wall is obtained by assuming an isotropic linearly elastic model and that the panel collapses when the principal tensile stress at its center attains its maximum value. Brignola et al. (2009) have recently assessed that this interpretation of the test is reliable, since in non-linear range the stress redistribution occurring in the panel does not significantly affect the value of computed by the elastic isotropic solution.

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23 For a masonry wall that can be approximated with a Saint-Venant solid subjected to a vertical distributed load σ (not necessarily uniform) and an horizontal force V, shear and normal stresses at the midsection can be approximated like in figure 2.5. In particular, the vertical stress can be assumed to be equal to the average normal stress ; if the wall is subjected to a normal stress N this value can be expressed as

=

Where D is the width of the wall and t is its thickness. The stress condition at the mid-section can be represented with the Mohr circle in the following figure. In the Turnšek-Čačovic model, the attainment of the maximum shear stress τM corresponds to the attainment of the tensile strength of the masonry ft.

Fig. 2.6. Mohr circle for the stress condition at the mid-section of the wall

By imposing the equality between the length of the segments OB and OA we obtain

+ 2 = 4 +

The maximum shear stress can be expressed as a multiple of the average shear stress τm, via a factor b that is usually assumed to be ranging from 1.0 to 1.5 according to the slenderness ratio of the

parameter. = = 1.0 !" < 1.0 !" 1.0 < !" < 1.5 1.5 !" > 1.5 (2.1) (2.2) (2.3) & ( , ) (0, − )

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24 And the average shear stress can be expressed as a function of the horizontal in-plane load and the dimensions of the section through the formula

=()

Where Vd is the diagonal cracking capacity of the wall. The formula (2.2) can then be rewritten as

+1₂ = 1_{4 * + + *} ()+

Which can be rearranged to obtain the well-known formula by Turnšek and Čačovic that will be our starting point for the development of a diagonal cracking capacity model

Where we recall the meaning of each one of the involved terms:

• is the tensile strength of the masonry, computed as explained in paragraph 4.1.1 • is the width of the panel

• is the thickness of the panel

• is a coefficient dependent of the slenderness ratio of the panel • is the vertical stress on the wall

It is important to underline how there is little agreement about the value of the coefficient b. Some works by Turnšek himself (Turnšek and Sheppard, 1980) and more recent researches (Andreini et al., 2014) completely disregard the b coefficient as a conservative approach to the diagonal cracking failure. We have decided to drop the coefficient out for the same reasons, together with the fact that the effect of the slenderness coefficient could be investigated in more detail by our Bayesian updating process. Although the simplification of idealizing masonry as an equivalent isotropic homogeneous continuum is rather drastic, this approach has been historically used with good results and it is currently contemplated in the OPCM 3431 and most of the national annexes to the Eurocode 6. Failure in diagonal cracking is considered, together with failure for sliding, a shear-type failure; it is associated with a brittle behavior. Before the first cracks start to appear, the degradation of strength and stiffness is negligible; the post-peak response is generally characterized by higher energy dissipation but by rapid strength and stiffness degradation. As a result, the graphs for the hysteretic cycles (force/displacement) for walls that manifested this type of failure are rather large compared to hysteretic cycles for walls that had a flexure-type of failure.

() = 1 +

(2.4)

(2.5)

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25

Fig. 2.7. Typical hysteretic cycle for a masonry wall failing in diagonal cracking

2.5 Flexure/Rocking failure

This type of failure is typical for masonry piers that are subjected to higher levels of vertical load, e.g. lower stories of buildings. It usually starts with the cracking of the bedjoints on the side of the wall subject to tension. As a consequence, the section of the wall resisting the vertical load is reduced, and the stress on the resisting part gradually increases. This eventually leads to the failure in compression of the toe on the compressed side of the wall. In literature, the distinction between flexure failure,

rocking failure and compressed toe crushing is very ambiguous. Some tend to distinguish between the

overturning of the wall as a single block due to failure in tension of bedjoints (rocking) and the crushing due to the failure in compression (compressed toe crushing). The following figure tries to highlight this distinction.

Fig. 2.8. Distinction between rocking and toe crushing

Although toe crushing and rocking are very different phenomena if considered separately, they are a manifestation of the same typology of failure. We will not operate the previous distinction and identify the two aspects under the same umbrella term flexure/rocking failure. The way this problem has usually been tackled is very close to what is usually done for the flexural failure of concrete beams. In particular, the tensile strength of the masonry is completely disregarded (= 0) and the distribution of the compressive stress on the toe is approximated to a constant rectangular distribution (stress block).

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26

Fig. 2.9. Scheme for the development of the flexure/rocking capacity model

If we define σm as the average compressive stress on the wall = ⁄ , where t is the thickness of

the wall, we can impose the equilibrium of the wall of the vertical axis and obtain = - ./

Where k is the coefficient due to the approximation of the general compressive stress distribution into the stress block and it is usually taken equal to 0.85 (just like the usual practice for the concrete). From the equilibrium to the rotation around the geometric center of the wall we obtain

(!0= 1

We can express the eccentricity e of the load in terms of the width of the compressive block

1 = 2 −/2

By combining (2.7), (2.8) and (2.9) in the case when the wall is attaining its maximum horizontal in-plane capacity (V = Vf) we can obtain the well-known formula for the flexure/rocking capacity

(2= !0 _{2 *1 −} -.+

Where H0 is the effective wall height (distance from zero moment) which depends on the boundary

conditions of the wall and is related to the shear ratio 34

34=_{( =}5 !0=6′! (2.7) (2.8) (2.9) (2.10) (2.11)

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27 Where ψ’ is the coefficient depending on the boundary conditions of the wall. In other words, the effective pier height can be rewritten as ψ’H where ψ’ assumes a value of 1 when the pier is in a fixed-free boundary condition and 0.5 when it is in a fixed-fixed boundary condition. The final relationship that we are going to use as starting point for our Bayesian updating process is

Where each term has previously been introduced in this paragraph.

The flexure/rocking failure is a typology usually characterized by a high ductility; very large displacements are theoretically possible and can be obtained without significant loss in strength, especially when the mean axial load is low compared to the compressive strength of masonry. Like other types of failure, its hysteretic cycle presents a high level of non-linearity due to the fact that the resisting section is continuously changing. The displacement is usually limited by the occurrence of other types of failure (usually diagonal cracking) or to second order effects (P – Δ) associated to overturning. Particularly the first case is very common, and the collapse of a wall due to a flexure failure is not usually recognizable only by pictures of the collapsed wall, because the diagonal cracks tend to hide the initial cracks. In a laboratory environment, though, the process that leads to failure can be followed step-by-step and the beginning of a flexure/failure collapse can be easily recognized. Differently from a shear-type of failure, the hysteretic cycle of a flexural collapse is usually characterized by a narrow, S-shaped graph. This is an index of the low dissipation of energy associated with this process.

Fig. 2.10. Typical hysteretic cycle for a masonry wall failing in flexure/rocking

2.6 Sliding failure

The last typology of failure due to in-plane forces acting on masonry walls is the sliding failure. It usually happens when the vertical load acting on the wall is not high enough to contribute to the friction of the mortar. As a consequence, we assist to the creation of a sliding plane, usually in one of the low-part bedjoints of the wall, and high displacements are possible without the loss of integrity of the wall. It often happens for the upper stories of residential buildings, but it is not very common in a laboratory enviroment such as the one that has been studied in this work, where higher loads are usually applied to the specimens.

(2= 6′! 2 *1− 0.85

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28

Fig. 2.11. Some examples of walls failing in sliding

Like in most of the types of failure in which friction plays a dominant role, it is usually modeled starting from a Mohr-Coulomb type criterion; we remember here how, according to the Mohr-Coulomb theory, the failure domain can be represented in the σ-τ space as a line whose angle with the σ-axis is the friction angle 9. At that point a sample is assumed to fail if the Mohr circle representing its stress condition is tangent to the failure plane. In other words, the Mohr-Coulomb failure criterion can be written as the equation for the line that represents the border for the failure domain.

:= ; + < = ; + =(9)

Fig. 2.12. Graphical representation of a Mohr-Coulomb failure criterion

Where τ is the shear strength, σ is the normal stress, c is the intercept of the failure envelope with the

τ axis and it is often referred to as cohesion, and 9 is the slope of the failure domain and, as already

mentioned, corresponds to the angle of internal friction in the general case. Sometimes the coefficient =(9) takes the name of friction coefficient and it is identified with the letter μ. This approach has been largely adopted in design and assessment of masonry structures but different interpretations have been given for its practical use in the evaluation of the ultimate load Vs (Magenes, 1997).

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29 Generally the cohesion term of the formula is identified with the characteristic initial shear strength under zero compressive stress _>?0, the vertical stress is assimilated to the average stress over the compressed part of the wall that is providing shear resistance (either the whole cross-section or just the uncracked part), and the friction coefficient is usually taken equal to 0.4 (EN.1996). If _: is taken as the characteristic strength of the masonry fvk, then the previous relation can be transformed into

the well-known formula available in the Eurocode 6

>?= >?0+ 0.4

The first approach is to consider _>? as the expression of an average ultimate shear stress in the total horizontal section of the wall, and that leads to the following formulation for Vs

(@= >?= ( >?0+ < ) = * >?0+ < +

In this approach the parameters _>?, < and _>?0 have the meaning of global strength parameters, and due to the non-uniformity of the real stress distribution, they can only be related to the material properties (such as the strength of the mortar) via empirical formulations (for example, see table 4.1 for the value of _>?0).

The second, more accurate, approach is similar to the first one but refers to the effective uncracked section length. This approach is adopted for instance by the Eurocode 6 on masonry structures. To calculate the length of the effective uncracked section we neglect the tensile strength of bedjoints and we assume a simplified distribution of compression stresses, most commonly constant or linear. If a triangular distribution of stresses is assumed, the geometric relationships in the following figure apply

Fig. 2.13. Graphical representation of dimensions for computing D’

(2.14)

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30 In this case, we can compute the length of the uncracked section of the wall as

A

= B = 3 = 3 *2 −(!0+ And the ultimate shear capacity of a wall can then be calculated as

(@ = ′ * >?0+ < ′ +

By combining (2.16) and (2.17) we obtain the final formula which is available in the Eurocode and the Italian annex for the computation of the ultimate sliding capacity.

When sliding on horizontal bedjonts occurs, a very stable mechanism is involved, and very high displacements are theoretically possible without the wall losing its integrity. This means that both damage and dissipation are concentrated in the bedjoint that fails, the response is very close to an indefinitely elastic perfectly plastic response and high energy dissipation is still possible but cyclic shear tests on bedjoints are needed to interpret the hysteretic cycles. In general, defining an ultimate displacement limit for sliding has little or no meaning, since it would be so high that the occurrence of other failures would in practice determine the real displacement limit.

Again, we want to underline how the conditions that usually lead to a sliding type failure (i.e. low levels of vertical stress) were not replicated in the laboratory environment object of study. Further researches are necessary to extend the theory developed in the following chapters to the case of sliding shear.

The following paragraph is dedicated to how the models that have been introduced so far are contemplated in the current regulations around the world.

2.7 Failure of masonry walls in the current code

There is no common agreement by the regulations that are currently in use in the different countries on how to model the failure of a masonry wall subjected to in-plane forces. The very difficult task of modeling a heterogeneous material such as masonry and the differences in the common practices among different cultures are the main cause for the lack of a common ground between codes. In particular, the formulations that have been introduced for diagonal cracking, flexure/rocking and sliding are a direct re-adaptation of what is now available in the OPCM 3431, proposed in 2003 and soon after incorporated into the NTC2008, one of the most important stepping stones in a clear understanding on seismic actions for masonry. Although its use is limited to the Italian territory, this modeling was considered to be the most comprehensive and was thus used as a starting point for our updating. In particular, the national code has proven to be much more specific than the Eurocode 6, which only considers the Mohr-Coulomb critierion when dealing with seismic action; this is a direct

(@ =1.5 >?0 + < 1 + 3 >?06′!

(2.16)

(2.17)

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31 consequence of a much stronger prencence of seismic hazard in Italy compared to the rest of the continent, which has forced the code to change and adapt in a better way.

The following table shows how this problem is dealt with in 4 different codes: • N.T.C. (Norme Tecniche per le Costruzioni) 2008 (from the OPCM 3431) • Eurocode 6 (EN 1996-1-1)

• F.E.M.A. (Federal Emergency Management Agency) 356 • A.C.I. (American Concrete Institute) 530-05

The FEMA 356 and the ACI 530-05 provide regulations for masonry buildings in the United states. It can be noted how the first follows a more detailed approach comparable, to the certain extent, to the OPCM 3431, while the latter provides more empirical formulations.

Diagonal Cracking Sliding Flexure/Rocking

NTC 2008 (₎= 1 + . (@ = ′ >?0 (2 = 6′! 2 *1 − 0.85 .+ Eurocode 6 - (_D)= >? E - FEMA 356 (₎ = )′F G H ℎJ22K 1 + ) ′ (LM@= ( JF (.= 3NOG H ℎJ22K *1 − 0.7 A+ ACI 530-05 3.8F Q A 56F + 0.45 : 90F + 0.45 : 23F 300F

Table 2.1. Different regulations for OPCM3431 (Italy), Eurocode 6 (Europe), FEMA 356 and ACI 530-05 (USA)

For a deeper insights into the various formulas and a complete understanding of the different terms, please refer to the different codes.