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From Research to Practice in Construction

Heft/Vol. 8

UNDER EARTHQUAKE ACTIONS

Tammam Bakeer

Bauforschung und Baupraxis

From Research to Practice in Construction

Herausgegeben von

Prof. Dr.-Ing. Wolfram Jäger und

Doz. Dr.-Ing. Todor Vassilev

Lehrstuhl Tragwerksplanung

Fakultät Architektur

Technische Universität Dresden

01062 Dresden

Fax.: +49 (0) 351 / 46 33 77 13

E-Mail: Lehrstuhl.Tragwerksplanung@mailbox.tu-dresden.de

ISBN: 978-3-86780-130-0

In der Schriftenreihe “Bauforschung und Baupraxis” werden Arbeiten und Beiträge des Lehrstuhls

Tragwerksplanung der TU Dresden ebenso wie solche von Wissenschaftlern und Praktikern, die

mit dem Lehrstuhl in Verbindung stehen, veröffentlicht. Anliegen ist es, neue Ergebnisse aus der

Forschung und Lehre vorzustellen und andererseits Vertretern aus der Praxis die Möglichkeit zu

geben, über interessante Vorhaben, Planungen und Techniken oder Technologien zu berichten. Es

soll damit der Informationsaustausch zwischen den Wissenschaftlern, Ingenieuren und Studenten

gefördert sowie Ergebnisse und Erfahrungen für die weitere Anwendung dokumentiert und

bereitgestellt werden.

COLLAPSE ANALYSIS OF MASONRY STRUCTURES

UNDER EARTHQUAKE ACTIONS

Dissertation

Technischen Universität Dresden

zur Erlangung des akademischen Grades eines

(Dr.-Ing.)

von

geb. am 01.01.1980 in Homs, Syrien

Gutachter:

Prof. Dr.-Ing. Wolfram Jäger Technische Universität Dresden

Prof. Dr.-Ing. habil. Peter Ruge Technische Universität Dresden

Prof. Dr.-Ing. Ghassan Nader Al-Baath Universität, Syrien

To my Rama and Rana

I dedicate this work

Acknowledgements 5

Acknowledgements

The first interest to develop this work originates from the need to preserve masonry

monuments in my homeland Syria, the land that is well known as the cradle of

civilizations.

The research report in this thesis is carried out at the chair of structural design, Faculty

of Architecture, Dresden University of Technology. The experimental data of large scale

structures in this thesis have been provided by the European research project

ESECMaSE. The major tool that used to develop the numerical models is LS-DYNA

software ver. 971, as well as it is used to implement the proposed algorithms. The

calculations of large scale structures have been carried out in the high performance

computing centre of Dresden University of Technology.

I consider my few years in Dresden to be a crucial period in my life, where I introduced

to many challenging problems, in one of the most attractive place to study masonry

structures. First and foremost I would like to express my sincere and deep gratitude

towards my supervisor Prof. Dr.-Ing. Wolfram Jäger for his generosity, wisdom,

outstanding support throughout my research at Dresden University of Technology.

I would like to record my thanks to the team of ZIH (high performance computing center

of Dresden University of Technology) for their fast help and great attention. As well as

my thank goes to the team of ELSA (the European Laboratory for Structural Assessment

in JRC the Joint research centre, Ispra-Italy) and particularly to Dr. Armelle Anthoine for

providing the necessary experimental data which used in this thesis.

I am very grateful for the fruitful collaboration I had with Dipl.-Ing. Peter Schöps. His

encouragement, valuable discussions and great attention are never forgotten. Besides, I

owe my gratitude to Dr.-Ing. Song Ha Nguyen for his suggestions and the valuable time

gave to me. I am deeply indebted to my colleagues at the chair of structural design for

their great support and continuous scientific discussions for providing a stimulating and

fun environment in the chair. I also wish to express my sincere appreciation to Prof.

Gassan Nader & Prof. Eido Shannat for their unlimited support and endless

encouragement. They are of those professors well remembered by students.

Lastly and most importantly, I am extremely grateful to my parents who have been the

source of encouragement and inspiration throughout my life.

Abstract 7

Abstract

Earthquake actions are by far the major risk that causes the collapse of masonry

structures. The assessment of the seismic performance of large scale masonry

structures remains a challenging task despite the progress achieved by means of

experimental studies. Due to the capacity limitation of shaking tables and other dynamic

testing devices, the large scale computer modelling appears to be a very efficient

alternative. The modern calculation methods, the high performance computers and the

advanced software packages are the major research tools in this concern.

The present thesis concerns to develop and investigate computer-aided techniques to

simulate the behaviour of large scale masonry structures under earthquake actions

starting from elastic linear behaviour to the progression of damage up to the point of

collapse.

The numerical models have been developed by means of explicit finite element code LS-

DYNA. The numerical techniques which allow the emergence of discontinuities have

been adopted and the combined Finite-Discrete Element Method, capable of processing

large deformations and discontinuities has been applied for the purpose of collapse

analysis. The ability of mesh free methods, like smoothed particles hydrodynamic, have

been examined for simulation the failure modes and the fragmentation of the material.

The necessary background of the constitutive models of masonry constituents has been

given and the major features, shortcomings and challenging problems have been

highlighted. An interface cohesive model based on the smoothness of the yield surface

has been proposed and implemented. The adopted models have been validated by

means of the dynamic test results of two full scale masonry structures.

The developed approach has been used to study the performance of heritage masonry

buildings, which involve members of different geometries, against the fundamental

earthquake characteristics. The influences of earthquake characteristics on the

vulnerability of collapse have been explored by the collapse analysis of a case study of

full heritage masonry building.

A theory for the shear failure behaviour of vertically reinforced masonry shear walls has

been developed, which gives an idealization for the behaviour after the initial failure of a

composite material with a high level of inhomogeneity like masonry. Besides, collapse

analysis has been employed as a tool to develop and verify the seismic retrofitting of the

reconstructed heritage buildings. The approach has been applied on a case study of

adobe masonry building from Bam citadel which collapsed after the earthquake of 2003.

Table of contents 9

Table of contents

1 Introduction ................................................................................................13

1.1 Masonry in architecture history......................................................................... 13

1.2 The scope and objectives ................................................................................. 18

1.3 Outline of contents ............................................................................................ 18

2.1 Modelling strategies of masonry....................................................................... 21

2.2 Methods of analysis .......................................................................................... 22

2.3 Continuum methods.......................................................................................... 23

2.4 Discrete methods .............................................................................................. 26

2.4.1 Rigid Bodies Spring Method (RBSM) ........................................................... 28

2.4.2 Discontinuous Deformation Analysis (DDA)................................................. 28

2.4.3 Non-Smooth Contact Dynamics (NSCD) ..................................................... 29

2.4.4 Modified Discrete Element Method (MDEM)................................................ 30

2.4.5 Combined Finite-Discrete Elements (DEM/FEM) ........................................ 30

2.4.6 Limit Analysis Models (LAM) ........................................................................ 31

2.4.7 Applied Element Method (AEM) ................................................................... 32

2.5 Concluding remarks .......................................................................................... 33

3.1 Masonry constituents ........................................................................................ 35

3.2 Failure behaviour of masonry ........................................................................... 36

3.2.1 Uniaxial failure behaviour of masonry unit ................................................... 36

3.2.2 Triaxial failure behaviour............................................................................... 44

3.2.3 Failure behaviour of unit mortar interface .................................................... 45

3.2.4 Failure theories of masonry as composite ................................................... 50

3.3 Concluding remarks .......................................................................................... 54

4.1 Governing equations......................................................................................... 55

4.1.1 Conservation laws......................................................................................... 56

4.1.2 Boundary conditions ..................................................................................... 57

4.2 Finite element formulation ................................................................................ 58

Table of contents 10

4.4 Finite element codes and solution strategies ................................................... 64

4.5 Techniques of crack formation ......................................................................... 68

4.5.1 Undetermined crack techniques ................................................................... 69

4.5.2 Predetermined crack techniques .................................................................. 71

4.6 Modelling strategies for collapse simulation..................................................... 74

4.7 Concluding remarks .......................................................................................... 77

5.1 The basics of plasticity theory .......................................................................... 79

5.1.1 Non-smooth multi-surfaces plasticity............................................................ 81

5.1.2 Implementation into LS-DYNA...................................................................... 83

5.2 Constitutive models of the interfaces ............................................................... 85

5.3 Implementation of cohesive interface material model...................................... 88

5.3.1 The smooth yield surface.............................................................................. 88

5.3.2 Return mapping ............................................................................................ 91

5.3.3 The damage functions .................................................................................. 93

5.3.4 Simulation the fragmentation using interface elements ............................... 95

5.4 Constitutive models of masonry constituents................................................... 96

5.4.1 General shape of yield surface for geo-materials ........................................ 96

5.4.2 Geo-materials constitutive models in LS-DYNA......................................... 100

5.5 Concluding remarks ........................................................................................ 105

6.1 The basic approximations of SPH .................................................................. 107

6.1.1 Kernel approximation .................................................................................. 107

6.1.2 Particle approximation ................................................................................ 110

6.2 SPH formulation for solid mechanics ............................................................. 111

6.3 SPH modelling of masonry in LS-DYNA ........................................................ 112

6.4 Concluding remarks ........................................................................................ 113

7.1 Dynamic testing methods ............................................................................... 116

7.1.1 Shaking table testing method (STT)........................................................... 116

7.1.2 Pseudo-dynamic testing method (PSD) ..................................................... 116

Table of contents 11

7.1.3 Real time dynamic hybrid testing method (RTPSD) .................................. 118

7.2 Experimental tests .......................................................................................... 118

7.2.1 Shaking table test ....................................................................................... 118

7.2.2 Pseudo dynamic test .................................................................................. 123

7.3 Numerical analysis.......................................................................................... 130

7.3.1 Finite element modelling............................................................................. 130

7.3.2 Material models........................................................................................... 131

7.3.3 Initialization prior to earthquake loading..................................................... 132

7.3.4 Run time...................................................................................................... 132

7.4 Numerical results for the model of Athens ..................................................... 133

7.4.1 Analysis of numerical results ...................................................................... 133

7.4.2 Collapse of the structure............................................................................. 138

7.4.3 Comparison with shaking table results....................................................... 139

7.4.4 Sensitivity of collapse to the bonding strength with slabs.......................... 141

7.5 Numerical results for the model of Ispra......................................................... 141

7.5.1 Analysis of results for low earthquake intensity ......................................... 141

7.5.2 Comparison with experimental results ....................................................... 144

7.5.3 Analysis of results for moderate earthquake intensities ............................ 146

7.5.4 Analysis of results for strong earthquake intensities.................................. 148

7.5.5 Sensitivity of collapse process to the bond strength .................................. 149

7.5.6 Influence of vertical ground motion ............................................................ 150

7.6 Concluding remarks ........................................................................................ 151

8.1 Selecting the case study................................................................................. 153

8.2 Mosque of Takiyya al-Sulaymaniyya.............................................................. 155

8.2.1 Historical background and the layout of Takiyya ....................................... 155

8.2.2 Architecture of the mosque......................................................................... 157

8.2.3 Construction of the geometry and finite element modelling....................... 161

8.3 Gravity loading ................................................................................................ 164

8.4 Earthquake modelling ..................................................................................... 164

8.4.1 Seismicity of the region............................................................................... 164

8.4.2 The response spectrum .............................................................................. 166

8.4.3 Synthesis of artificial accelerograms .......................................................... 167

Table of contents 12

8.6 Effect of earthquake characteristics ............................................................... 169

8.7 The direction of the earthquake...................................................................... 169

8.8 The frequency content of the earthquake ...................................................... 171

8.9 Concluding remarks ........................................................................................ 173

9 Reinforced masonry.................................................................................175

9.1 Ductile behaviour ............................................................................................ 175

9.2 Reinforcement-masonry bond ........................................................................ 177

9.3 Reinforced masonry shear walls .................................................................... 179

9.4 A shear failure theory for vertically reinforced masonry................................. 184

9.4.1 Initial failure surface .................................................................................... 188

9.4.2 Results of the finite element model ............................................................ 189

9.4.3 Tensile crack opening of one side of the bed joint- Case I ........................ 191

9.4.4 Shear cohesion failure of one side of the bed joint- Case II ...................... 193

9.4.5 The failure surface ...................................................................................... 195

9.5 Modelling strategies of reinforced masonry ................................................... 195

9.5.1 Discrete modelling ...................................................................................... 196

9.5.2 Smeared modelling ..................................................................................... 198

9.6 Verification of retrofitting measures by collapse analysis .............................. 198

9.6.1 The case study............................................................................................ 198

9.6.2 Description of the model ............................................................................. 200

9.6.3 The earthquake........................................................................................... 202

9.6.4 Collapse analysis of unreinforced structure ............................................... 203

9.6.5 Verification of antiseismic reinforcement by collapse analysis .................. 203

9.7 Concluding remarks ........................................................................................ 206

References .................................................................................................................. 213

Appendixes: Numerical Simulation Results................................................................ 227

1 Introduction 13

1 Introduction

Masonry is one of the most primitive building materials known to mankind since the

beginning of earliest civilizations. Masonry has been used in the construction of the most

long-lasting exciting ancient monuments, artifacts, cathedrals and cities in a vast variety

of cultures. Furthermore, Masonry material is used widely in today’s structures due to

the simplicity of building technique and the attracted features that characterize this

material.

Many actions are leading to the collapse

of masonry structures and one of the

major sources of destruction is the

seismic action. The ruins of many

masonry monuments are the evidence for

poor performance of masonry structures

against earthquakes, Figure 1.

A few decades ago, masonry structures

received comprehensive studies, and

attracted a considerable volume of

research either in experimental field or in

numerical modelling.

In spite of the great progress achieved,

the earthquake performance of large Figure 1 Ruins of Palmyra in Syria

scale masonry structures is still

challenging.

Many experimental methods have been showed technical limitations in dynamic testing

of large scale structures due to the high cost of such tests. On the other hand, the

computer modelling has been showed great efficiency and simplicity due to the

availability and growing advance of fast computers and software packages.

However, the creation of an elaborate model that represents the behaviour of the

structure with all stages from initial elastic linear behaviour to the plastic non-linear, the

cracking, the separation and then the collapse is still fraught with difficulties.

Perhaps, the first used masonry material was stonemasonry, Lourenço [109].

Archaeological excavations have revealed one of the earliest examples of the first

permanent stonemasonry houses near Hullen Lake (c. 9,000-8,000 BC), where dry-

stone huts, circular and semi-subterranean constructions were found, Lourenço [109]

and Oliveira [138]. Stone was difficult to shape and due its weight, transporting was

difficult.

Mud brick started in use as an alternative masonry material in dray climate regions

where clay mud is available. The first mud brick constructions probably goes back to

1.1 Masonry in architecture history 14

Jericho1, Palestine (c. 8,350-7,350 BC), where many mud brick houses have been

founded in the site. Old shaped bricks have been founded in Çayönü, a place located in

the upper Tigris area in south east Anatolia close to Diyarbakir. Indus Valley Civilization

also used mud brick extensively, as can be seen in the ruins of Mohen-jor-Daro2 and

Harappa.

In Egypt, from pre-dynastic times (5,000 BC) until the Roman occupation (50 AD) the

basic material to build houses was sun dried brick, commonly of Nile mud, as can be

seen in the ruins of Buhen. The pure Nile mud shrinks over 30% in the drying process

while the addition of chopped straw and sand to the mud prevented the formation of

cracks.

The invention of the burnt brick (as opposed to the considerably earlier sun-dried mud

brick) enabled the construction of permanent buildings in regions where the harsher

climate precluded the use of mud bricks.

Masonry was widely used in the Plain of Shinar where famous ziggurats as hexahedral

towers were erected. They were pyramidal, stepped temple towers that has an

architectural and religious structure characteristic of the major cities of Mesopotamia

"The land between the rivers”. The structural form of the pyramids, which represents one

of the most stable structural shapes, was a logical development of the initial stone piles.

The stacking of large blocks of stone in pyramid form allows reaching great heights. The

most famous pyramids are undoubtedly the Egyptian pyramids at Giza3.

The understanding of the structural behaviour started to play an important role in the

construction of temples with the use of stone lintels to support the masonry above

openings in walls. A famous example in Lion Gate at Mycenae, Greece (1,300 BC.) used

a stone lintel for a span of 3 m and was loaded by 25-30 tons. The lintel idea shows the

beginning of the arched behaviour that would dominate the following millennium, Oliveira

[138].

The arch was first developed in the Indus Valley civilization and subsequently in

Mesopotamia, Egypt, Assyria, Greek and Persian civilizations for underground structures

such as drains and vaults. However, ancient Romans were the first to use them widely

above ground.

The Greeks played an important role in the use of structural elements, namely: columns

and beams to build their temples. The most famous one is Parthenon4 which

represented an important role for the use of limestone in building the structural elements.

The Romans contributed significantly to the construction of buildings. They built roads,

bridges, aqueducts and harbors. Also, they introduced notable innovations in materials,

structural concepts and construction process.

1

The Bronze Age city of Jericho was destroyed about 1,500 BC by the Egyptians, The retaining wall was

some four to five meters high. On top of that was a mudbrick wall two meters thick and about six to eight

meters high

2

Mohen-jor-Daro, 80 km southwest of Sukkur was center of Indus Valley Civilization 2600 BC-1700 BC.

3

The Great Pyramids of Giza (1,220-1,288), stones up to 60-80 tons have been used and transported a

distance of over 500 miles.

4

It is the most famous surviving building of ancient Greece, and has been praised as the finest achievement

of Greek architecture. Its decorative sculptures are considered one of the high points of Greek art. It was

built between 447 and 438 BC. The acropolis of Athens and its sculptural decoration was completed in 432

BC.

1 Introduction 15

1.1 Masonry in architecture history 16

1 Introduction 17

Remarkable examples of the use of dry stone blocks in buildings by Romans are the

Colosseum5 in Roma and Segovia’s aqueduct in Spain. Romans exploited the structural

form of arches to construct magnificent bridges and aqueducts which are surviving until

today, such as Segovia's aqueduct6 in Spain and Pont du Gard aqueduct7 in France,

Figure 2.

Vaults and domes played a great role in the construction of large-span roofs, a good

example are the elegant geometry of vaults and the domes that used in Hagia Sophia in

Istanbul8 (6th century A.D.), Ozkul et al. [145]. Another interesting structural form is the

castles, which were spread from Europe to the Middle East. The Crac des Chevaliers9

and Citadel of Salah Ed-Din10 are good examples of crusaders castles in the Middle

East.

Gothic architecture, which was originated in the 13th century, represented remarkable

improvements in reducing the heaviness of Roman constructions by using framing

elements (columns, arch ribs, flying buttresses and buttress wall or tower) working in

compression. Two of the finest examples of the Gothic architecture are the Cathedral of

Chartres11 and the Cathedral of Amiens12. The history of Gothic architecture with its

pioneering construction is also marked by failures, cracks and permanent deformations.

The Renaissance architecture was initiated in Florence and aroused a new concepts

and forms. Buildings were characterized by regular forms and geometrical symmetry in

plane and elevation. The church of St. Maria del Fiore in Florence (built in 15th century)

and the church of St. Peter in Rome (16th century) are remarkable examples of

Renaissance architecture.

During the Baroque period, no significant or innovative solutions concerning the

structural conception were developed. Important examples in Europe are St. Paul’s

Cathedral in London (17th century) and the Panthéon in Paris (18th century), Figure 3.

5

The Colosseum, is a giant amphitheatre in the centre of the city of Rome. It was built on a site just east to

the Roman Forum. The construction started between 70 and 72 AD and completed in 80 AD, with further

modifications being made during Domitian's reign.

6

Segovia's most unique feature is its still-functioning ancient aqueduct. It was built by the Romans around

50 AD, and is designed to make water flow uphill. Even more amazing, Segovia's aqueduct was built without

mortar. The pillars and the arches of the structure were built simply by stacking large stones.

7

The Pont du Gard is an aqueduct in the south of France constructed by the Roman Empire, and located

near Remoulins, in the Gard département. It was long thought that the Pont du Gard was built around the

year 19 BC. Newer excavations, however, suggest the construction took place in the middle of the first

century A.D.

8

Hagia Sophia, (the Church of) Holy Wisdom, now known as the Ayasofya Museum, is a former Eastern

Orthodox church converted to a mosque in 1453 by the Turks, and converted into a museum in 1935. It is

located in Istanbul, Turkey. It is traditionally considered one of the great buildings in history.

9

Crac des Chevaliers is a famous castle in Syria, which was the headquarters of the Knights Hospitaller

during the Crusades. The name is a mixture of Arabic and French, meaning "Fortress of the Knights", it is

one of the few sites where Crusader art (in the form of frescoes) has been preserved.

10

The Citadel of Salah Ed-Din once known as Saone, also known as Saladdin Castle.

11

The Cathedral of Chartres ("Cathedral of Our Lady of Chartres) is considered one of the finest examples

of the Gothic style of architecture in all France. Construction was begun in 1145.

12

The Cathedral of Our Lady in Amiens (1220-1288) is the tallest complete cathedral in France, with the

greatest interior volume.

1.2 The scope and objectives 18

The following study deals with both historical and today’s masonry structures. It focuses

primarily on earthquake actions. However, the developed models allowed to be used for

another loading regime. Finite element method is the fundamental numerical tool in this

study, and the advantages of using other mesh free methods also have been discussed.

The main concern of the following research focuses to develop and investigate

computer-aided techniques that are able to simulate the collapse of large scale masonry

structures under earthquake actions, which investigation can not be carried out in

laboratory conditions.

The most relevant specific objectives of this study are:

- to develop computer models that can simulate the progress of damage in

masonry structures in all stages, starting from elastic linear behaviour, to plastic

nonlinear behaviour, cracking, separation up to the collapse

- to develop constitutive models for masonry constitutions, capable of mapping the

real behaviour and incorporate the computation efficiency

- to verify and evaluate the developed models in comparison versus experimental

results of full scale dynamic tests

- to investigate the effect of the different earthquake characteristics on the collapse

mechanisms of large scale historical masonry structure

- to develop a theory based on the variation of damage states for the failure

behaviour of vertically reinforced masonry walls

- to adopt the collapse analysis technique for developing the reinforcement of

historical masonry structures.

The text in this thesis is organized into ten chapters and appendixes for the results of

numerical calculation.

The first chapter serves as a motivation with brief overview of the architectural use of

masonry, specifies the scope of thesis and outlines the objectives and the applied

methodology.

The second chapter is dedicated to the state of the art of the modelling techniques of

masonry, gives comprehensive study for the progress of research in numerical analysis,

with focus on the works in discrete methods. A classification of the available techniques

and the application area is achieved. The features and shortcoming of each technique is

discussed, as well.

The third chapter investigates in depth the material properties of masonry and its

constituents and the failure mechanisms. Different failure theories for masonry as a

composite are described and discussed. The experimental setups for the determination

of the basic material parameters of masonry are given and the corresponding failure

modes are described.

1 Introduction 19

The fourth chapter introduces the finite element method as a basic numerical tool for

modelling masonry. The combination of finite element method with discrete element

method, contact analysis and solution techniques are presented. Different numerical

simulation techniques for crack formation reported in the literature are discussed.

Modelling strategies for masonry based on combined finite-discrete element method are

proposed.

This followed by an illustration of plasticity theory and the implementation of a material

model into the explicit solver of LS-DYNA in the fifth chapter. The constitutive models

that serve for the proposed modelling strategies are described and discussed. A

constitutive model for a cohesive interface element based on smooth yield surface is

developed and validated.

The features of applying mesh free methods are presented in the sixth chapter, where

the smoothed particle hydrodynamic is employed.

The seventh chapter reports the experimental results of two dynamic tests. Two

numerical models correspond to each test have been created. The obtained numerical

results compared with that from experiments. Furthermore, additional studies concerning

the simulation of the collapse of the structure under higher earthquake intensities are

performed, where such studies can not be achieved in laboratory.

In the eighth chapter, a large scale historical masonry structure is used as a case study

to explore the performance for different earthquake characteristics. The effect of

earthquake direction, as well as the frequency content of the accelerogram is studied.

The ninth chapter devoted for reinforced masonry structures. A review of the current

state of research is reported and a novel failure theory for vertically reinforced masonry

is presented. The possible modelling strategies for reinforcement modelling are

discussed, and the collapse analysis is used to develop the reinforcement for a case

study of historical masonry structure

The tenth chapter summarizes the major conclusions obtained in this research, and also

suggests recommendations for future research works. The 14 appendices present in

detail the results of numerical simulations.

2 The state of research in masonry modelling 21

Masonry achieved a great progress in the last few decades. Many research efforts for

masonry structures were carried out all over the world to understand the structural

behaviour, either through laboratory tests or by using validated numerical models.

Masonry researches are benefited from the enormous progress in other building

materials. However, the methodologies that are used for the analysis and design of other

building materials are still lacking for masonry. Furthermore, numerical modelling

methods are still fraught with difficulties. This is due to the complexity of the behaviour of

masonry structure, which is determined by the interaction of the individual behaviours of

several parts of the structure, often with different material characteristics.

Several models and methods have been proposed in the literature in order to study the

mechanical behaviour of masonry. This chapter is devoted to give an overview insight

the latest up-to-date progress in modelling of masonry and numerical simulation of

collapse.

Masonry is a composite material, and its overall behaviour is strictly dependent on the

arrangement and properties of its different constituents. The engineering literature on

modelling of masonry is enormous and the research in the field is widespread.

Numerical models may be based on two methodologies, Massart [124]. First,

mesoscopic detailed descriptions consider masonry as a heterogeneous structure with

separate descriptions of each constituent. Second, models intended for large-scale

structural calculations are generally of a phenomenological nature, and represent the

collective behaviour of constituents by closed-form macroscopic constitutive equations.

The three principle modelling strategies are correspond to three different scales of

complexity which have been identified by Lourenço [109] and Rots [157], Figure 4:

1- Micro modelling or two phase material model: starting from the knowledge of

single constituents. Each component of the masonry structure has its own

behaviour which might be complex. This modelling strategy is categorized into:

(a) Detailed micro-modelling whereby units and mortar represented as

continuum, with the unit/mortar interfaces modelled using discontinuous interface

elements as potential crack, slip and crushing planes;

(b) Simplified micro-modelling through the adoption of "geometrically expanded"

masonry units with a single "averaged" interface representing the mortar and the

two mortar/unit interfaces. This model requires the material model of the

expanded unit and masonry joints.

2- Macro-modelling or single phase material model, the quasi-periodic nature of

masonry has prompted to investigate the use of homogenization techniques,

where all masonry components are smeared by an equivalent homogenized

continuum. One-phase material models are treating masonry as an ideal

homogeneous material with constitutive equations that differ from those of the

components.

2.2 Methods of analysis 22

Homogeneous

Unit Unit material

mortar

Smeared

Unit-mortar Joints

Interface

The decision about the suitable technique depends on required accuracy and the size of

the model.

Micro modelling gives more realistic representation of the structural behaviour, but it is

relatively prohibitive to be used due to the great number of the degrees of freedom,

require more input data, and their failure criterion has an elaborated form due to the unit-

mortar interaction. The constitutive equations of the components have normally a simple

form, and they are appropriate for the study of the local behaviour of masonry.

The constitutive models on macro level are relatively simple to use, require less input

data, and the failure criterion has normally a simple form. It requires a priori definitions of

constitutive prescriptions. The constitutive equations are relatively complicated and are

suitable for the study of the overall behaviour of the entire masonry structure to reduce

the numerical calculation.

A large variety of numerical modelling frameworks have been employed to analyse the

mechanical behaviour of masonry. Despite many numerical methods have the ability to

analyse the mechanical behaviour for the early stages of failure, the ability to study the

performance near and after the collapse is still limited and presents challenging

modelling problems. As witnessed by the extensive study of the literature two numerical

2 The state of research in masonry modelling 23

approaches are widely described for analysing the mechanical behaviour of masonry,

Figure 5:

1) Continuum methods: the model is based on continuum material equations. The

finite element method (FEM) is the typical example of this approach. Smeared

crack approach can be adopted in zones where separation occurs between

structural elements.

2) Discrete methods (or distinct methods): assumes that the geometry of cracks

propagation is predefined before the analysis.

Numerical Methods

Modelling Strategies

Continuum approach has been widely used for many application areas, but the use of

this approach is not applicable to collapse analysis. Nevertheless, combining continuum

approaches with discrete approaches produces more powerful and accurate methods.

Moreover, continuum methods are more capable of simulating the behaviour before the

collapse. A review of these two methods is given in the following sections.

There are a wide range of applications involve materials or systems that showing

discontinuity at some level. Despite that some systems are intrinsically discontinuous

they are well approximated by a continuum. This approximation is possible if the scale of

the objects of interest is large enough. Finite element method (FEM) and boundary

element method (BEM) are well suited for representing of continuum media. The interest

to develop continuous model for the discrete structure of masonry is due to computation

efficiencies gained by this model while the discrete type of analyses is very computer

time consuming. Furthermore, masonry often has periodic nature where the application

of the homogenized continuum model would allow for more elegant and efficient

solution, Cerrolaza et al. [41] and Sulem et al. [177].

2.3 Continuum methods 24

An attempt has been made to take into account the characteristics of masonry materials

of micro-polar continuum, such as Cosserat continuum, instead of classical Cauchy

continuum, in order to get better representation for the effect of particle rotation. Masiani

[122] and Masiani [123] present procedures to develop a Cosserat continuum to provide

a description of the mechanical behaviour of masonry with regular texture.

The plasticity theory has been employed to develop macro material models for in plane

behaviour of masonry. Lourenço [109] proposed an anisotropic model of two surfaces

Rankine/Hill. Massart [124] developed two-dimensional anisotropic damage model in a

“multi-plane” framework. Schlegel [164] showed an implementation of the material model

of Ganz theory (Ganz [62]) in ANSYS software. Mistler [132] used the shear failure

theory of Mann & Müller ([117] and [116]) to implement a material model for masonry

panels in ANSYS.

Attempts also have been made to understand the behaviour of rubble and cyclopean

masonry from natural stones (Mann [118], Warnecke [196] and Schlegel [164]) based on

Mohr-Coulomb criteria. The behaviour of multi-leaf masonry has been considered as

well, Egermann [51] and Schlegel [164].

(a) (b)

Figure 6 Continuum modelling for (a) Göltzschtal bridge, Schlegel et al. [163] (b)

Church of our ladies (Frauenkirche), Dresden, Stoll. et al. [176].

Macro-modelling strategy has been often used in literature for continuum models. Some

continuum models were built using micro-modelling strategy (Schlegel [164]), where

material models for mortar and units considered separately with continuous finite

element mesh on unit-mortar interfaces.

In continuum method the softening and local cracking of material considered by the

smeared crack approach, Rots [158]. The smeared crack approach was first developed

for use in concrete structures and has been extended to masonry structures, Lofti et al.

[106]. In this approach cracks are modelled in an average sense by modifying the

material properties at the integration points of finite elements. Smeared cracks are

convenient when the crack orientations are not known beforehand, because the

formation of a crack involves no remeshing or new degrees of freedom. However, the

2 The state of research in masonry modelling 25

smeared crack models can not be able to simulate the final stage of softening process in

masonry material, i.e. the full separation of the continuum can not be accomplished by

means of smeared crack models.

Although, the finite element method has been used extensively in literature for

continuum modelling, less attention was paid for using boundary element method in

modelling of masonry. Rashed et al. [149] employed BEM to model the non-linear

behaviour of masonry where cracking, debonding and crushing failure modes were

considered in the model.

The modelling of masonry structures as continuum is far from being a good

representation of their real behaviour due to the great number of discontinuities. While

this approach suffers a computationally tractable problem, a continuum model offers a

quite crude approximation of what is really a micro-mechanical phenomenon.

Many complications arise with continuum approach for the highly nonlinear behaviours,

either from material or geometrical perspectives. For instance, it is very difficult or

unfeasible to use the continuum approach to study the behaviour of materials or

structures that change their status from continuum state to entirely discrete state, like

behaviour of structures before and during collapse.

Lourenço et.al. [107] used FEM model with interface elements for simulation of uniaxial

compression tests of masonry prisms. A fictitious micro-structure composed of linear

elastic particles separated by non-linear interfaces was adapted to model units and

mortar in a quarter of the basic masonry cell. Cavicchi et al. [40] used limit analysis with

interface elements to determine initial failure of masonry bridge considering arch-fill

interaction.

Interface elements were introduced to consider the discontinuity at planes of failures.

However, with this technique it is only possible to show small displacements before the

failure. The interface elements have limitations to simulate the large displacements at

the collapse of the structure.

(a) (b)

Figure 7 FEM model with interface elements for simulation of uniaxial (a) tensile and

(b) compression tests of masonry prisms, Lourenço et.al. [107]

2.4 Discrete methods 26

Discrete methods are relatively new discipline of numerical methods in computational

mechanics. It deals with discontinuous media where continuous assumptions impossible

to be applied. These methods have the capability to model an inherently discontinuous

medium. However, it has been successfully applied to problems where the transition

from continuum to discontinuum is important. Surface interaction laws between bodies

are invoked instead of a homogenized continuum constitutive law.

Discrete element method goes back to the pioneering work of Cundall et al. [46], where

it was originally used to model jointed and fractured rock masses. The proposed method

showed high efficiency to describe discontinuous phenomena and dynamical problems

of large deformations. Occurrence and propagation of fracture have been naturally taken

into account with a discrete model. Later, this approach was extended to others fields of

engineering, where the elaborate study of joints is required, e.g. soils and other granular

materials, Ghaboussi and Barbosa [64]. This numerical technique is also used for the

modelling of masonry structures, Lemos et al. [99], Sincraian [174] and Lemos [98].

The early formulation of the discrete element method has been originally termed distinct

element method DEM (Cundall [46]) and has been invented for rigid circular bodies in

two dimensions with deformable contacts. The overall solution scheme for the DEM has

been formulated in an explicit time-stepping format. Movements of bodies have been

driven by external forces and varying contact forces. The method considers each body in

turn and at any given time determines all forces (external forces or contact forces) that

are acting on it. Out of balance forces induce accelerations which then determine the

movement of that body during the next time step. The discrete element method

comprises different techniques which are proper for the simulation of dynamic behaviour

of systems of multiple separated bodies. These bodies will be subjected to continuous

changes in contact status and varying contact forces, which in turn influence the

subsequent movement of the bodies. Such problems are non-smooth in space (separate

bodies) and in time (jumps in velocities upon collisions) and the unilateral constraints

(non-penetrability) must be considered.

In case of rigid bodies, the constitutive law of contact interaction is only needed, while

the continuum constitutive law (e.g. elasticity, plasticity, damage and fracturing) must be

included for deformable bodies. Computational modelling of multi-body contacts

represents the dominant feature in discrete element methods as the number of bodies

considered might be very large. According to the nature of the problem and level of

accuracy, the bodies of the system can be considered as rigid, simply deformable

(pseudo-rigid) or fully deformable.

Many computer codes are formulated for discrete element methods. The first code was

originally formulated by Cundall [46] to simulate the response of discontinuous media

subjected to either static or dynamic loading and has been further developed by Lemos

et al. [100]. Nowadays, a vast range of open source, non-commercial and commercial

software are available such as: UDEC, BALL & TRUBAL, QUAKE & DAMSEL, NESSI,

FRIP, FLAC, 3DEC, PFC3D, REBOP, EDEM, GROMOS, ELFEN, MIMES,

PASSAGE/DEM, and TRIDEC.

Many existing methods belong to the discrete element computational formwork could

appear under different names and each of them has been developed in its own right.

2 The state of research in masonry modelling 27

Cundall & Hart [45] were defined four classes of discrete element methods: Distinct

Element Methods (DEM), Modal methods, Discontinuous deformation analysis (DDA),

Momentum-exchange methods. The other classifications of the discrete methods are

based on the manner these methods address, Bićanić [20], i.e.: detection of contacts,

treatment of contacts (rigid, deformable, deformability, (constitutive law) of bodies in

contact (rigid, deformable, elastic, elasto-plastic, etc.), large displacements and large

rotations, number (small or large) and/or distribution (loose or dense packing) of

interacting bodies considered, consideration of the model boundaries, possible

subsequent fracturing or fragmentation and time-stepping integration schemes (explicit,

implicit).

The heterogeneous nature of masonry and the discontinuity at block interfaces can be

well described by discrete element approach. This approach is well suited for collapse

simulation of masonry structures, where good quality results have been achieved.

Azevedo et al. [11] was used UDEC (Universal Distinct Element Code) to simulate the

collapse of monumental masonry structures due to seismic actions, Figure 8. Further

examples can be seen in the works of Roberti et al. [155], Psycharis et al. [148].

Extensive research work was devoted to analysis masonry structures using other

versions of DEM such as: Rigid Bodies Spring Method RBSM, discontinuous

deformation analysis DDA, combined discrete-finite elements, non-smooth contact

dynamics NSCD and Modified Distinct Element Method (MDEM).

(d)

Figure 8 Collapse analysis of masonry structures using the discrete element method:

(a) masonry arch bridge (Lemos [97]); (b) Simulation of an aqueduct pillar,

(Lemos [98]); (c) dry stone masonry pedestal sustaining a statue (Sincraian

[174]); (d) Collapse sequence for the S. Giorgio bell tower in Trignano,

Azevedo, et al. [11]

2.4 Discrete methods 28

The Rigid Bodies Spring Method or (rigid block spring model) RBSM was proposed early

as a generalized limit plastic analysis framework. Solid structures are assumed to be

assemblies of rigid blocks interconnected by discrete deformable interfaces with

distributed (elastic) normal and tangential springs.

RBSM has been used for studying the seismic behaviour of masonry walls. Casolo et al.

[39] was proposed a computational model using RBSM for evaluating the dynamical

response and the damage of large masonry walls subjected to out of plane seismic

actions. Besides, Casolo [38] was proposed a rigid block spring model for in-plane

behaviour of masonry walls made of regular textures, where quadrilateral plane rigid

elements were used and connected by normal springs and one shear spring on each

side. In the work of Casolo et al. [37] the RBSM has been adopted for seismic analysis

of in plane dynamic behaviour of masonry walls considering hysteretic energy

dissipation and mechanical deterioration.

An experimental and analytical studies were made by Nerio et al. [137] for earthquake

resistance of confined concrete block masonry structures. The wall specimens made of

concrete blocks have been tested under cyclic lateral load and simulated by a RBSM.

The non-linear behaviour has been modelled by using rigid bodies and boundary

springs. As a result of the study, RBSM has been showed good conformity for analysing

this type of structures. However, analysis using the RBSM is unattainable up to complete

collapse of the structure.

Figure 9 Scheme of an irregular masonry and the ‘unit cell’ defined by four rigid

elements in RBSM, Casolo et al. [37]

The method of discontinuous deformation analysis (DDA) is based on discrete element

approach which uses implicit integration scheme. DDA is a displacement-based method

developed during the 1980’s, Shi et al. [171], Shi [173] and Shi [172] for solving stress-

displacement problems of a jointed rock mass. Jun et. al [91] was extended the original

2D DDA formulation of Shi and Goodman to 3D DDA formulation. DDA has been

typically formulated as a work-energy method. It can be derived using the principle of

minimum potential energy (Jing [88]) or by Hamilton's principle. Once the equations of

motion are discretized, a step-wise linear time marching scheme in the Newmark family

can be used for the solution of the equations of motion. Step-wise linear implicit time

marching allows the so-called quasi-static solution, where step-wise velocities are never

2 The state of research in masonry modelling 29

used. Quasi-static analysis is useful for examining slow or creeping failures. The relation

between adjacent blocks is governed by equations of contact interpenetration and

friction.

This method is capable of analysing a system of discrete, discontinuous blocks under

general static or dynamic loading, with block deformations and rigid body movements

occurring simultaneously. The original DDA framework was based on simply deformable

blocks and the technique was further developed and used for several applications

including masonry structures. In algorithmic terms, the method has been seen as an

alternative way of introducing solid deformability into discrete element framework where

block sliding and separation is considered along predetermined discontinuity planes,

Bićanić et al. [21]. The DDA has been further employed in Bićanić et al. [23] and Bićanić

et al. [22] for modelling of masonry arch bridges.

Figure 10 Edinburgh arch bridge deformed shape following DDA, with simply

deformable Blocks, Bićanić et al. [23].

Later, the DDA has been extended to the Lagrangian discontinuous deformation

analysis (LDDA) and used to simulate of dynamic process of earthquakes Cai et al. [33].

Doolin et al. [49] developed DDA markup language (DDAML) to provide a practical

engineering platform for discontinuous deformation analysis.

The Non Smooth Contact Dynamics method or shortly contact dynamics (NSCD) was

initiated by Jean et al. [86] and developed within FORTRAN software LMGC. In NSCD,

Signorini relation for unilateral conditions and Coulomb law as a dry friction law has been

adopted together with an implicit algorithm scheme for the dynamical equation. NSCD

uses few large time steps, deals with numerous simultaneous contacts and needs much

iteration at each time step.

NSCD was used to simulate masonry as a large collection of bodies under unilateral

constraints and frictional contact, Chetouane et al. [42], Acary et al. [3] and Acary et al.

[4].

2.4 Discrete methods 30

(a) (b)

Figure 11 (a) Cumulated shear at the end of the dynamic loading of masonry wall

modelled by NSCD using the LMGC90 code, Chetouane et al. [42]; (b)

Stresses in masonry arch bridge after a settlement of ground modelled using

NSCD, Acary et al. [3]

The original DEM was considered the material as an assembly of particles at which no

resistant forces exist against traction. Elastic springs and dashpots were added by

Williams et al. [198] to give continuity to the discrete numerical model. It has been

showed that DEM can be viewed as a generalized finite element method. This method is

called the modified DEM or extended DEM. The model behaves as a continuous

medium while the springs are intact. After the breakage of some springs, it is possible to

trace the movement of the individual parts which separated from each other to destroy

the structure’s unity. Using this method, it becomes possible to analyse the fracture-

developing processes.

The MDEM is capable to follow the structural behaviour from initial loading up to the

complete collapse. However, the accuracy of EDEM in the range of small deformation is

less than that in FEM.

For the problems, where the state of stresses and transition from continuum to

discontinuum are important, the Combined Finite-Discrete Element (FEM/DEM)

technique has been introduced to combine the advantages of the FEM and DEM.

In the early 1990s, the combined finite-discrete element method was mostly an

academic subject. In the last ten years, the first commercial codes have been developed

and many commercial finite element packages have been increasingly adopted the

combined finite-discrete element method. The heart of the DEM concerns the automatic

contact detection between surfaces of separate block components. Global search

algorithms have been used to provide short lists of potential contacts. Local search

algorithms have been then implemented to identify the actual contact potential. Finally,

using the penalty method and the defined interface properties, the normal and tangential

forces between the blocks have been resolved.

By combining FEM and DEM, the homogeneous material within each discrete body can

be modelled, facilitating elastic and non-linear material behaviour. During the

deformation of the bodies, failure criterion can be applied to detect if the stress/strain

state reaches the defined limit at which the fracture may occur. When these thresholds

2 The state of research in masonry modelling 31

are exceeded the FEM/DEM technique allows the fracture of discrete bodies (see Owen

et al. [144], Munjiza et al. [135], Munjiza [134], Frangin et al. [61] and Komodromos [93]).

(a) Maximum applied load (b) Onset of collapse (c) Total collapse

FEM/DEM, Brookes et al. [28]

Gifford have been developed the application of the FEM/DEM technique, available in the

explicit dynamic version of ELFEN (Rockfield Software Limited), for the analysis of

masonry arches.

The Method of Limit Analysis Models is originally based on the rigid-perfectly plastic

material modelling in order to evaluate the load bearing capacity and the failure

mechanism of the structure. The applicability of limit analysis to masonry structures that

are modelled as assemblages of rigid blocks depends on some basic hypotheses,

Orduña et al.[141], [142], [140] and [139].

The limit analysis can be regarded as a practical computational tool since only requires a

reduced number of material parameters and it can provide a good insight into the failure

pattern and limit load. Orduña [143] presents an investigation about the capabilities of

limit analysis of rigid block assemblages in structural assessment of ancient masonry

constructions. Ferris and Tin-Loi [57] suggested simple numerical scheme for solving

limit analysis problems for large-scale block structures.

Figure 13 (a) Out of plane loaded wall supported at one edge, Orduña et al. [141]; (b)

Masonry pile (c) 2D Bridgemill with the spandrel wall, Orduña et al. [143]

2.4 Discrete methods 32

Applied Element Method (AEM) was developed at the University of Tokyo by Meguro et

al. [127] and Meguro et al. [130] for analysing and visualizing the response of structures

under extreme loading conditions. The research on AEM is started since 1995. Many

research efforts and validation tests have been conducted and published to introduce

the AEM's breakthroughs: auto-element separation, auto-element contact detection,

realistic element modelling and simplified super-element meshing. AEM is currently

utilized in numerous industries where the analysis and visualization of structures under

extreme loading conditions is crucial: seismic engineering, vulnerability assessment,

demolition, blast analysis, rockfall engineering. (Meguro et al. [127], [129], [128], [130]

and Mayorca et al. [126]).

The major advantages of the AEM are the simple modelling and programming, and high

accuracy of the results with relatively short CPU time. Using AEM, the structural

behaviour can be carried out from initial loading, to crack initiation and propagation,

separation of structural members and up to complete collapse in reasonable time with

reliable accuracy and relatively simple material models.

By this method the structure is modelled as an assemblage of small elements that are

made by dividing of the structure virtually, as shown in Figure 14, where the two

elements are assumed to be connected by normal and shear springs that are placed at

interfaces and distributed around the element surfaces. The springs totally represent the

stresses and deformations of a certain volume of the studied elements.

For modelling of masonry structures, each unit is represented by a set of elements,

where mortar joints are placed at the corresponding contact surfaces, Meguro et al.

[127] and Guragain et al. [72]. One shortcoming of this method is that, there is no

flexibility to model irregular geometries like in finite element method.

Volume represented

by a normal spring

and 2 shear springs

B: Spring distribution and area of influence of each pair of springs

Figure 14 Masonry discretization and AEM modelling, the AEM illustration to the right

from Meguro et al. [128]

2 The state of research in masonry modelling 33

Modelling of masonry has been attracted a great amount of research works in the few

last decades. Most of research activities in this field were focused on representing

masonry as continuum using homogenization theory. The plasticity theory has been

played a big role in developing material models for continuum models of masonry.

Attempts also have been devoted in the field of discrete modelling. However, the works

in this direction were less than that in continuum modelling, and this were limited to small

scale structures.

The reasons for applying continuum models in many research works are:

- Easy handling of the problem for macro models by plasticity theory

- Most of commercial finite element codes provide a possibility to implement a

material model into their codes, whereas the implementation of discrete

modelling techniques into these codes is often not possible

- The limitation in computer resources, where the discrete models need

considerable amount of resources for large scale structures

- The generation of discrete models sometimes is complicated like in case of

rubble masonry.

The discrete models which developed for masonry are based on using specific

numerical techniques like contacts or springs to define the interfaces between the

discrete elements. The failures of the interfaces signify that the discrete elements go on

large displacements. The simulation of large displacements is limited for some discrete

methods, as well as the discrete methods still challenging for large scale structures.

3 Mechanical behaviour and failure of masonry 35

Masonry is a composite material that consists of units and mortar. Its overall behaviour

is strictly dependent on the arrangement and mechanical properties of its different

constituents.

It is quite significant before studying the collapse behaviour of masonry structure to

understand the failure behaviour of masonry material.

The present chapter focuses on the experimental research works which carried out to

explore the mechanical characteristics of masonry material and its failure behaviour.

A wide variety of materials were used in past centuries for building masonry

constructions. Materials that are available in the vicinity were conceivably the most

common in construction work. When civilizations developed in river plains, the alluvial

deposits were used to produce brick constructions and when civilization existed in the

area of mountains, rocky outcrops and stones were used, Hamid et al. [76].

Stones produced by nature were the first units used to build masonry structures. Stones

were widely available from natural rocks like igneous rocks, sedimentary and

metamorphic rocks, Table 1.

Igneous

stones Extrusive stones Porphyry, Basalt, Volcanic tuff

sandstone, greywacke

from fragments of pre-existing rock

Sedimentary

stones Precipitation stones limestone, dolomite

stones and/or high temperature marble

The first stone masonry structure was built using crude units. As proficiency improved,

stone units were shaped into polygonal or squared units so that closefitting joints were

obtained.

Clay bricks were in use for at least 10,000 years. Sun-dried bricks (adobe masonry13)

were widely used in Babylon, Egypt, Spain, South America, the Indian lands of the

13

Wide usage is illustrated by the word "adobe," which is now incorporated in the English language but is a

Spanish word based on the Arabic word "atob," meaning sun-dried brick.

3.2 Failure behaviour of masonry 36

United States and elsewhere. The earliest bricks were made by pressing mud or clay

into small lumps, sometimes cigar-shaped, and letting them to dry in the air or the sun.

Mortar is the material linking the units in masonry that closes the gaps and makes

masonry monolithic. In historical masonry, it is usually composed of washed sand and

other aggregates with a binder to protect from erosion by the wind and rain.

The first mortars were made from clay, bitumen or clay-straw mixtures. It was primarily

used to fill cracks and to afford uniform bedding for masonry units. In addition, the usage

of the thin mortar joints was improved the durability.

The forerunners of modern mortars date to the use of calcined gypsum, lime and natural

pozzolans. The Egyptians utilized calcined gypsum mortars to lubricate the beds of large

stones when they were being moved into position. It was discovered that limestone

when burnt and combined with water, produced a material that would harden with age.

The earliest documented usage of lime as a construction material was approximately

4000 BC when was used in Egypt for plastering the pyramids. The mixing of ground lime

with volcanic ash was produced what became known as pozzolanic cement. The

Coliseum in Rome is an example of a Roman structure that pozzolanic cement mortar

used as bonder and it has been well survived over many centuries, Hamid et al. [76].

No significant developments in cements and mortars took place until the eighteenth

century when John Smeaton in the reconstruction of the Eddystone Lighthouse in

England, mixed pozzolana with limestone containing a high proportion of clay to produce

a durable mortar that would set and harden under water.

The next important development was the manufacture and patenting of Portland cement

by Joseph Aspdin in England in 1824. The combination of Portland cement with sand,

lime and water produces much stronger mortar than what previously possible and this

mortar would also set and harden under water.

In most historical masonry structures lime was used as mortar (Non-hydraulic lime, lime

putty, dry-slaked lime, bag lime, hydraulic lime, Pozzolanic lime). Other mortars also

used as natural pozzolana and brick powder, Mathews [125]. Lime mortar creates good

bonding strength for masonry units. Besides, it increases the load bearing capacity

particularly in flexural loading like what occurs during an earthquake.

The failure behaviour of masonry is fundamentally dominated by the properties of its

components and the arrangement of units and their interaction together. It is therefore

crucial to realize the mechanical behaviour of the individual masonry components, units

and mortar.

When behaviour of masonry component needs to be investigated, the first though is to

examine the constituents disjointedly. However, this is possible only for masonry units,

because the properties of mortar are influenced considerably by the interaction between

mortar and unit during hardening, Schubert [168] and Vermeltfoort et al. [192].

The earlier research works on the behaviour of masonry units were greatly learned from

the studies in rock mechanics and concrete material that were widely examined.

3 Mechanical behaviour and failure of masonry 37

Masonry units are heterogeneous materials belong to quasi-brittle materials which have

a disordered internal structure. It contains a large number of randomly oriented zones of

potential failure in the form of grain boundaries. The designation “quasi-brittle” behaviour

refer to the transferred force which does not immediately drop back to zero, other than

gradually decreases. Such behaviour is often denoted with softening. The softening

causes localization of deformations that causes quick growth of microcracks into

macrocracks and finally to fully open cracks.

In recent decades, many experimental studies were carried out to understand the

mechanical behaviour of masonry units. The modern masonry units received great

studies in literature, for standardization and classification purposes. The experimental

records for the characteristic values of different masonry units can be found in Marzahn

[120], Schubert [167], Schubert [165], Mann [118]. Some mechanical properties for

different natural stones are given in Table 2, Huster [83]:

strength (Mpa) strength (Mpa) Elasticity (Mpa)

Type of natural stones

(1) (2) (1) (2) (1) (2)

(1=Schubert [166], 2=Warnecke [196]), from Huster [83].

In general, mechanical properties of historical masonry units which made from natural

rocks are influenced by a wide range of factors, (Vasconcelos [191]):

- The internal structure and the degree of anisotropy associated with the

arrangement and preferential orientation of minerals (foliation, flow structures

and rift plane)

- The mechanical properties of rocks, like compressive strength or elastic modulus

are greatly dependent on physical properties of the rocks such as porosity and

density

- Weathering conditions

- Petrography, mineralogical characteristics and grain size

3.2 Failure behaviour of masonry 38

water. Strength decrease of the saturated rocks would be attributed on one hand,

to the chemical or physical alteration of its inherent properties and on the other

hand, to the increase of the pore and fissure water pressure.

In the case of concrete and other geo-materials, some experimental tests are necessary

to describe the material law of masonry units, principally tensile and compression tests.

(a) The behaviour under tension

The tensile failure of quasi-brittle materials from which masonry units is made, results in

localization and propagation of micro cracks. The tensile behaviour of such materials

can be well described by the cohesive crack model proposed by Hillerborg et al. [79]. It

includes the tension softening process zone through a fictitious crack ahead of the pre-

existing crack. In the Hillerborg crack it is possible to distinguish between two zones: a

real crack that no more stresses are transferred and a damaged zone extended in the

fracture process zone, in which stresses are still transferred.

Cohesive crack model has been widely used to describe the nonlinear fracture

mechanics of quasi-brittle materials, Bažant [15], Carpinteri et al. [34], Carpinteri et al.

[35], Elices et al. [53] and Guinea [71].

According to cohesive crack model, the tensile behaviour is characterized by two

constitutive laws associated with different stages of the material during the loading

process:

with assumed stress

tensile

stress

ft

δeu δnlu distribution according to

C

ft Hillerborg et al. [79]

(3)

Tensile stress

crack crack

measured curve

σ B (4) (5)

I I

(1) G pre G f D

+

(2)

k0 E estimated curve

σ

δft 0.1- 0.15 ft F

A

(2) Microcracking process (3) Macrocracking δ

(1) Linear behavior growth (4) Bridging (5) Tensile failure

Pre-Peak Post-Peak

Elasto-Plastic Softening

3 Mechanical behaviour and failure of masonry 39

which is valid until reaching the peak load. The nonlinearity prior to the peak

occurs due to micro cracking process. The micro cracks are stable and they grow

only when the load increases.

2. Post-peak stage (from C to F) is characterized by the softening behaviour at the

fracture process zone. The stress in this stage starts to decrease gradually from

its maximum value to approach zero value. This corresponds to increasing in the

distance between the two lips of the crack to the critical value of opening wc

which indicates the failure. The reason for such behaviour is the acceleration of

crack formation around the peak load. The micro cracks start to bridge forming

visible macro cracks. The macro cracks are unstable, i.e. the load has to be

decreased to avoid an uncontrolled growth. The stress-transfer mechanism, due

to bridging effect, is responsible for the long tail of the softening stage. The stress

displacement relationship, which characterizes this stage, is denoted the

softening diagram. The softening diagram assumes a fundamental role in the

description of the fracture process and is characterized by the tensile strength f t

and the fracture energy G If which is given by the area under the softening

diagram.

Hordijk and Reinhardt (Hordijk et al. [81]) were proposed the following formula for the

relationship between stress and crack opening in plain concrete under tension:

w

σ w − c2 ⋅

w

= [1 + (c1 ⋅ ) 3 ] ⋅ e wc − ⋅ (1 + c13 ) ⋅ e −c2 (1)

ft wc wc

where:

c1 , c2 dimensionless constants: 3.0 and 6.93, respectively

wc theoretical critical crack opening at which no stresses are being transferred any

G If

more and it is given by: wc = 5.14

ft

∫ σ ⋅ du )

I

G f Mode I fracture energy (here defined with

ft Uniaxial tensile strength

There are several experimental methods used to measure the fracture properties (tensile

strength, fracture energy and the critical crack opening) which allow the definition of a

constitutive law for the material behaviour in tension, namely direct tensile tests defined

by Van Mier et al. [188], indirect tensile test as the three load test and Brazilian splitting

test.

The pre-peak behaviour is quantitatively characterized by means of the following values:

- The initial stiffness k 0 is calculated as the slope of the linear adjustment to the

stress-strain relationship from zero value up to around 20% of the peak stress

- The displacement at the peak strength δ ft

3.2 Failure behaviour of masonry 40

- The tensile strength f t

- The fracture energy G If is identified with the needed work for complete

separation of the two faces of the macrocrack per unit of area.

Due to long tail of softening, it is not possible to determine the critical crack opening at

which the transferred stress value becomes zero. However, it is possible to estimate the

critical crack opening by linear adjustment of the original diagram following the value

0.1 − 0.15 f t by the least-square method. Therefore, the value of the fracture energy has

been calculated as the result of the sum of two quantities. The first one G If , meas is directly

computed as the area under the measured diagram up to 0.1 - 0.15 of the peak strength.

The second one G If ,est is calculated as the area under the linear estimated curve.

The mechanical properties of masonry units obviously show irregularity according to the

type of material. Stone units usually are stronger and stiffer in contrast to brick units, but

there is also disparity in mechanical properties for the units of the same material.

Vasconcelos [191] was attempted to find correlation between the mechanical properties

of granite. Figure 16 shows that the higher values of tensile strength are associated with

stiffer granites, and this is the general thought for quasi-brittle materials. As well as, the

increase of the deformation at peak stress is associated with a decrease of the tensile

strength value, and there is linear correlation between displacement at peak load and

critical crack opening.

Vasconcelos [191]

In contrast to masonry units of stiffer natural rocks, the units of adobe masonry show

relatively quite brittle failure behaviour under tension, Jäger et al. [85].

3 Mechanical behaviour and failure of masonry 41

0.8

0.7

0.6

L-4

0.5 L-5

Stress (N/mm²)

L-6

0.4 L-7

L-8

L-15

0.3

0.2

0.1

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Displacement (mm)

Figure 17 The failure of adobe masonry specimens under tension, Jäger et al. [85]

In general, the materials of masonry units like stones and bricks principally, can bear

compressive stresses more than tensile stresses. This observation is the principle

feature for geo-materials. Compressive strength tests give a good indication of the

general quality of the used materials. Many studies were performed in order to clarify the

fracture mechanisms that better describe the compressive behaviour of quasi-brittle

materials. Coulomb [43] was pioneered his well known criterion by the investigation of

the fracture of the stones in compression. Griffith [70] was postulated that brittle fracture

is initiated through tensile stress concentration at the tips of small cracks randomly

distributed in the isotropic material. The Griffith theory or at least the basic assumption

from which the fracture initiates is fundamental to all investigations in quasi-brittle

fracture. Many other studies have been followed in order to obtain a better insight into

the fracture behaviour of brittle materials regarding the mechanisms of microcrack

initiation, crack interaction, propagation and coalescence, Kranz [94], Wang et al., [195],

Bobet et al. [24], Tang et al. [181] and Lajtai et al. [96].

The general behaviour of quasi-brittle materials under compression can be defined in the

following stages, Schlegel [164], Liniers [102] and Huster [83], ( Figure 18 and Figure

19):

1. Closure of microcracks and pores (A to B): starts at the beginning of loading up

to the stress level f cc at which the pre-existing microcracks or pore spaces are

closed. This phase is characterized by a nonlinear behaviour of the stress-axial

strain diagram with an increase in the stiffness of the material.

2. Linear elastic behaviour (B to C): after the closure of the pre-existing

microcracks, it exhibits linear elastic behaviour up to a certain stress level of

about 30-40% of the conventional strength. In this stage a linear relationship of

both axial and lateral stress-strain diagrams can be observed.

3.2 Failure behaviour of masonry 42

3. Crack initiation and stable crack growth (C to D): this stage starts with

microcracking at stress level f ci . Microcracks are mainly tensile cracks, Lajtai et

al. [96]. The formation and growing of microcracks in axial direction are

responsible for the nonlinear increase on the lateral strain, as well as on the

volumetric strain.

4. Crack damage and unstable crack growth (D to E): The unstable microcracking

occurs at the crack damage stress level f cd . It is associated to the point of

reversal in the total volumetric strain diagram Vr , which refers to the maximum

compaction of the specimen and to the beginning of dilation, since the increase

of volume generated by the cracking process is larger than the standard

volumetric decrease due to the axial load. For this stage, a rapid and significant

increase of the lateral strains has been observed, as a result of the volume

increase. The microcracking spreading is no longer independent. It starts

bridging to form fracture surfaces parallel to maximum principle stresses until

reaching the maximum compressive strength.

5. The softening and macrocracks growth (E to F): It is characterized with the

gradual weakness of material due to macrocracking growth as strain localization

occurs. Examples of failure patterns that illustrate strain localization are

displayed in Figure 20 for Granite. Macrocracks are the result of bridging the

microcracks when the peak load is reached. The inclination of the stress strain

curve at this stage gives an indication of the brittleness of the material. At the end

of this stage, macrocracks become unstable and compression crushing is

produced under constant level of stresses.

6.

σ bridging of

microcracks

E

fc

Linear

elastic

Compression Stress

D crack damage

macrocracking

fcd and unstable

growth

crack growth

C σ

fci

the failure

F

B closure of crack initiation

fcc

microcracks and stable Gc

and pores crack growth σ

A

Pre-Peak Post-Peak ε

Elasto-Plastic Softening

3 Mechanical behaviour and failure of masonry 43

Uniaxial Stress

Stress - axial strain

Stress - volumetric strain

E

fc

Peak

strength

D

Vr fcd Crack damage

C

Crack initiation fci

B

fcc Crack closure

A

Strain ε

peak load, Eberhart et al. [50]

4.5

mean value

4

max value

3.5 min value

3

Stress (N/mm²)

2.5

1.5

Gc=20.07 Nmm/mm2

1

0.5

0

0 2 4 6 8 10 12 14 16 18 20

displacement (mm)

Figure 21 Compression failure of adobe specimen under compression, Jäger et al. [85]

3.2 Failure behaviour of masonry 44

Many experimental studies have been carried out on strain localization detection in

rocks, Vonk [194] and Haied et al. [74] and for adobe masonry, Jäger et al. [85], see

Figure 21.

Although uniaxial tests are the basic tests that give good knowledge for the material

strength and the damage behaviour, they are lacking when general failure model of the

material is important, practically for geo-material which masonry constituents belong to.

Therefore, the common triaxial laboratory tests can be used to characterize the failure

behaviour of such materials.

σ1 − σ3

TXC

σ1 (3)

(1)

(1) Uniaxial compression

(2) Uniaxial tension

σ3 (3) Tri-axial compression

(5) Hydrostatic

ft fc

− σ1 = σ3 P

3 3 (5)

(2)

TXE

(4)

Figure 22 The relation between von Mieses stress and the hydrostatic pressure

obtained from the triaxial tests

cylinder specimen loaded with axial P

stress σ 1 and lateral stress σ 3 . The

failure curve that describes the relation

between von Mieses stress (3)

P2

(2)

pressure p = (σ 1 + 2σ 3 ) / 3 can be P1

characterize the material failure P0 εv

behaviour of many geo-materials.

Another important relation can be

obtained from triaxial test when Figure 23 Typical response of geo-

hydrostatic compression is applied, i.e. materials under hydrostatic

σ 1 = σ 3 = σ . This curve describes the compression, redrawn from

Schwer [170]

nonlinear relation between hydrostatic

3 Mechanical behaviour and failure of masonry 45

pressure and volumetric strain. For typical geo-material the following phases

characterize this relation, Schwer [170].

- The 1st phase: p0 < p < p1 the material is still elastic, and the slope equals the

bulk modulus K

- The 2nd phase: p1 < p < p2 the pores in the material are compressed

- The 3rd phase: p2 < p the material is fully compacted and no more pores exist

- The 4th phase for removing the pressure.

The unit-mortar interfaces are the weakest linkage in masonry assemblage especially in

historical masonry. The weak bond between unit and mortar has significant influence on

the overall behaviour of masonry structures.

The bonding at unit-mortar interface gets its strength mainly through the absorbency of

the units, the water retention capacity of the mortar, the porosity of the mortar, the

amount of binder and curing conditions.

Significant efforts were achieved to characterize the behaviour of unit-mortar interface of

masonry, Lourenço [107], van der Pluijm [186] and Almeida et al. [5]. According to van

der Pluijm [186], two basic failure modes might be occurred at the level of the unit-mortar

interface:

1. Tensile failure (mode I) is associated with stresses acting normal to joints and

leads to the separation of the interface

2. Shear failure (mode II) corresponds to a sliding mechanism of the units or shear

failure of the mortar joint.

Different experimental methods were developed to characterize the tensile strength and

failure of unit-mortar interface, Figure 24, Almeida et al. [5].

The experimental tests which carried out by Van der Pluijm [186] have been ended up to

an exponential decay tension softening curve, as a result of the coalescence of

microcracks towards a macroscopic crack Figure 25. Van der Pluijm [186] has observed

that no clear correlation between the bond strength and the fracture energy exists, but

the increasing of bond strength is always associated with the increasing of fracture

energy.

The tensile strength is influenced by the failure of the interfaces between units and

mortar and the bulk failure of mortar. The failure mode depends mainly on the quality of

the mortar, the quality of the bonding between both materials, and actual bonding area

between brick and mortar. The cracked specimens have showed a bond area in the

inner part, smaller than the cross sectional area of the specimen. This is a combined

result of the shrinkage of mortar and the process of laying the units in the mortar bed.

The area under the stress-displacement curve is associated with the mode I fracture

energy as illustrated by the shaded area in Figure 25.

3.2 Failure behaviour of masonry 46

F

Clamping Steel Plate

F F F

bolts 2 2 Bolts passing

through units

Steel end Hanger bars

Unit

Clamps Tightening mortar

couplet Bolts Upper

clamp Unit

Lower

clamp F

F

(a) (b) (c)

F F F

4 4 F F 2

Steel 2 2 F

Mild Steel rod 4 F

Unit 4

Steel cross Unit

bar Adhesive

Mortar

Unit Mortar

Unit F

2 F

F F F 4

4 4 F F 4

2 2

(d) (e) (f)

Figure 24 Different Types of tensile tests; (a) Couplet test using special clamps, (b)

Couplet test using clamps, (c) Couplet test using holes and bolts, (d)

Sheffield test, (e) Steel and plates glued with adhesive, Van der Pluijm [186]

(1993), (f) Crossed brick couplet, Almeida et al. [5]

Tensile stress σ (MPa)

Figure 25 Stress-crack displacement behaviour for unit mortar interface, failure Mode I.

The shaded area represents the envelope of three tests, Van der Pluijm

[186]

3 Mechanical behaviour and failure of masonry 47

Several studies with different types of shear tests were carried out to characterize the

shear failure of unit-mortar interface, Figure 26 (van der Pluijm [186], Atkinson et al. [10]

and Amadio [7]). The stress of confinement plays the major role in the shear behaviour

of masonry joints, Hamid [77]. The increase of compression normal to the bed joint leads

to an increase of the shear strength as has been widely reported by Atkinson et al. [10]

and Riddington [153]. For pre-compression stresses above a certain level, the shear

strength decreases and a combined shear-splitting failure or splitting of the units can be

occurred.

Fs

Fn2 Fs Fs

Fn1 2 2 Mortar

Mortar

Fn Fn Fn Fn

Mortar

Fs Unit

Unit

Unit

Fs Fs

2 2

Figure 26 Different types of shear tests;(a) Couplet Hoffmann/Stöckl test; (b) Van der

Pluijm test; (c) Triplet test

The shear test results obtained by van der Pluijm [186] are showed a great similarity with

the behaviour under tension, Figure 27, except that the tail of softening does not fall

back to zero but it becomes stable at a certain shear stress level. This level corresponds

to the dry friction of the two surfaces without cohesion.

Shear stress τ (MPa)

mode II, Van der Pluijm [186]

3.2 Failure behaviour of masonry 48

The shear strength was analysed as a function of the normal stresses on the basis of

coulomb’s friction failure criterion:

τ u = c0 − tan ϕ ⋅ σ (2)

where:

τu shear strength

c0 cohesion or the shear bond strength at σ = 0

ϕ The angle of internal friction, not necessarily equal to dry friction

Lourenço [109] has used the mode II fracture energy G IIf to define the descending

branch beyond the peak via softening of the cohesion in equation (2) by replacing c0

with the following equation:

c0

− ⋅v pl

G IIf (3)

cr = c0 ⋅ e

where:

cr residual cohesion

c0 initial cohesion

G IIf mode II fracture energy

v pl plastic shear displacement

Softening

Failure of cohesion

µ ⋅σ + c

σ

τ

Shear stress

c σ >0 µ ⋅σ

τ

σ

Frictional behaviour

GIIf σ =0 Deformation

ε

Figure 28 Shear behaviour of unit-mortar interface

Van Zijl [190] has explored the usability of Hordijk equation (1) by changing the mode I

parameters:

3 Mechanical behaviour and failure of masonry 49

v

cr v pl 3 −c2 ⋅ vnonlin

pl

v pl

= [1 + (c1 ⋅ ) ]⋅ e − ⋅ (1 + c13 ) ⋅ e −c2 (4)

c0 vnonlin vnonlin

where:

vnonlin shear displacement over which the cohesion can be reduced to zero

c1 , c2 constants

An important issue associated to shear test is the dilatant behaviour which occurs at/

and beyond peak. Dilatancy is the result of crack surface that is not perfectly smooth, it

represents the difference between the normal displacements of the upper and the lower

unit u pl as a result of the shear displacement v pl , Figure 29. The opening of the joint is

associated to positive dilation, whereas negative values of dilatancy represent the

compaction of the joint.

0.10

[mm]

∆upl

tanψ =

upl

0.08

C ∆vpl

Plastic normal displacement

upl

0.06

ψ v pl

0.04

0.02

[mm]

0 0.25 0.50 0.75

Plastic shear displacement vpl

displacement v pl beyond the peak of a test carried out with normal pre-

compression, Van der Pluijm [186]

As has been pointed out by Lourenço [109], dilatancy in masonry wall structures leads to

a significant increase of the shear strength in case of confinement.

Vasconcelos [191] showed different shear failures at different confining stresses for

granite:

- For low level of confining stresses: shear failure occurs at the unit-mortar

interface along one face of the unit or more frequently divided between two unit

faces, Figure 30.

- For high level of confining stresses: the failure only occurs in the mortar.

3.2 Failure behaviour of masonry 50

(a) (b)

Figure 30 Typical shear failure modes, no visible damage in units; (a) specimen

submitted to low level confining stresses, (b) specimen submitted to high

level of confining stresses, a large amount of small mortar particles

detached, Vasconcelos [191]

In adobe masonry, where the strength of masonry constitutes are weak, the failure

occurs in the mortar and will be extended to the units under high confinement, because

the mortar strength relatively is close to the strength of the units, Figure 31.

0.275

L-10

0.25 L-11

L-12

0.225

L-09

0.2

Shear stress (N/mm²)

0.175

0.15

0.125

0.075

0.05

0.025

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

displacement (mm)

In terms of the composite behaviour of masonry, all possible failure modes of the units

and mortar are necessary to be considered. Many research attempts have been devoted

to derive failure theories of masonry based on the fundamental properties of the

component materials. The strength and the interaction of the masonry components and

their differing in deformation characteristics are dominated the failure of masonry.

3 Mechanical behaviour and failure of masonry 51

developed a failure theory based on the N

elastic analysis by considering the

interaction of masonry components,

Hilsdorf [80] has proposed an

alternative approach based on an

assumed linear relationship between V -σ y

compressive stress multiplied by a non- σx

uniformity factor.

Mann and Müller ([116] and [117]), have

developed a failure theory based on the

equilibrium and strength considerations y

σx

of a unit within a wall, Figure 32. The x

τyx

no shear stresses can be transferred in -σ y

Figure 33. Figure 32 Shear stressed masonry wall

the normal stresses and the shear

stresses, it follows that -σ y

∆σ y 2h -σ

- ∆ σ /2

⋅τ yx

2y

− = (5) -σ 1y

y

2 l - ∆σ /2

y

where:

σ y1 + σ y 2 τyx b

τyx

∆σ y = σ y1 − σ y 2 and σ y = l/2

2 h

∆σ y ∆σ y

σ y1 = σ y + and σ y 2 = σ y −

2 2

The following failure modes have to be

considered:

τyx τyx

-σ - ∆ σ /2 y

(1) Tensile cracking of the bed joint 1y

-σ 2y

- ∆ σ /2

y

f tmy − σ y

τ yx ≤ (6) -σ y

c yx − µ yx ⋅ σ y

τ yx ≤ (7) theory

1 + r ⋅ µ yx

3.2 Failure behaviour of masonry 52

f bt σy

(3) Tensile cracking of the units τ yx ≤ 1− (8)

2.3 f bt

f cmy + σ y

(4) Compression failure of masonry τ yx ≤ (9)

r

where

τ yx average shear stress on bed joint

σy average normal stress

l, h length and height of the unit respectively

µ yx friction coefficient on bed joint

c yx cohesion on bed joint

f tmy tensile strength of bed joint mortar

f bt tensile strength of unit

f cmy compressive strength of masonry

r 2X ratio of height / length of unit

τ yx

(2) shear failure (3) cracking of the units

of the bed joint

(1) tensile cracking arctan( µ′)

of the bed joints

c'yx

σ y

fbt

ftmy

fcmy

Dialer [48] has developed the approach of Mann and Müller to take into account the

stresses acting on the head joints:

3 Mechanical behaviour and failure of masonry 53

c yx − µ yxσ y + rµ yx (c xy − µ xyσ x )

τ yx ≤ (10)

1 + rµ yx

(2) Tensile cracking of the units

1 f bt σ x + σ y σ x ⋅σ y

τ yx ≤ (c xy − µ xyσ x ) + 1− + (11)

2 2. 3 f bt f bt2

(3) Compressive failure of the masonry

f cmy + σ y

τ yx ≤ + c xy − σ x µ xy (12)

r

where:

c yx , c xy cohesions on bed and head faces, respectively

µ yx , µ xy coefficients of friction on bed and head faces, respectively

More generalized approach to study

the shear capacity of masonry is

based on considering the biaxial

loading of masonry. Page [146] has

proposed failure surfaces for

masonry stressed in orthogonal

tension-compression directions by

applying the normal stresses to small

specimens, where the bed joints

were inclined at various angels to the

axes of the applied stresses.

Ganz [62] proposed a set of Figure 35 Shear failure envelop of masonry

equations that defines the failure panel according to Ganz [62]

surface of masonry, but the effect of

unit dimensions was missing in the

equations:

(1) Failure of the units

3.3 Concluding remarks 54

⎡ π ϕ ⎤

Tensile failure in the bed joints τ xy2 + σ y ⋅ ⎢σ y + 2c tan( + )⎥ ≤ 0 (17)

⎣ 4 2 ⎦

The typical standard experimental tests are given in this chapter and the failure

behaviours are discussed. Masonry constituents show a big variation in material

characteristic behaviours. It is not easy to give a general failure description for the

material to which masonry belongs.

Masonry materials have a feature of increasing their shear strength by increasing the

confined pressure. This feature characterizes the general behaviour of geo-materials.

The standard tests of tension, compression and shear are widely described in literature.

However, triaxial tests are absent for the most masonry materials. The need to get an

accurate failure model for masonry requires also an experimental determination for

biaxial and triaxial behaviour parameters. The porosity of masonry materials causes an

inelastic behaviour under hydrostatic compression, and such behaviour can only

captured by triaxial tests.

4 Finite element modelling of masonry 55

In the present chapter, the mathematical formulation of the finite element method is

described, with treatment for the contact problem between the discrete bodies. By

involving the contact within finite element procedure, more powerful method can be

obtained, which is well known as finite-discrete element method.

Different integration schemes have been widely implemented into finite element

packages. A comparison between these solution techniques is made, and the

preference of the explicit solvers is explained. The possible numerical techniques for

crack formation are described briefly and discussed. Different levels of modelling

strategies for collapse analysis are suggested as well.

Let us consider a single discrete element that represents a deformable body in an initial

state (reference configuration) at time t = 0 as shown in Figure 36. The domain of the

body is denoted by Ω 0 and its boundaries by Γ0 . The current configuration of the body

at any instance time t is occupied by domain Ω and its boundaries Γ .

When the body moves from domain Ω 0 to domain Ω , the point X ∈ Ω 0 moves to point

x∈Ω .

X 3 , x3

n

Γ

x Ω

N

dΩ

u Current configuration, t

X

dΩ 0 X 2 , x2

Ω0 Γ0 x = ϕ ( X, t )

X 1 , x1

Initial or reference configuration, t=0

The fundamental equations which govern the problems of solid mechanics can be

obtained from the following conservation or balance laws: Belytschko et al. [19], Johnson

[89], Wriggers [200], Bathe [14], Hughes [82], Hallquist [75] and Munjiza [134].

4.1 Governing equations 56

(1) Conservation of linear momentum

The momentum conservation is the statement of Newton’s second law of motion.

σ ij , j + ρ ⋅ f i = ρ ⋅ &x&i (18)

where:

σ ij Cauchy stress tensor

ρ mass density

ρ ⋅ fi body forces, where f i a force per unit mass

&x&i accelerations

(2) Conservation of angular momentum

The angular momentum provides additional equations which govern Cauchy stress

tensors:

σ ij = σ ji (19)

Mass conservation requires the mass of a material subdomain to be constant.

∫ ρ ⋅ dΩ = ∫ ρ

Ω Ω0

0 ⋅ dΩ 0 = constant (20)

ρ det(J ) = ρ 0 (21)

where:

ρ0 reference density

∂xi

determinant of the vector mapping function ϕ ( X, t ) which is J ij =

∂X j

The conservation of energy requires that, the rate of change of the total energy in the

body, which includes both internal energy and kinetic energy, equals to the work of the

applied forces

4 Finite element modelling of masonry 57

d

P int = ∫

dt Ω

ρ wint dΩ (23)

The disposition of the internal work depends on the material. In an elastic material it

stored as elastic internal energy which is fully recoverable upon unloading, while in the

elasto-plastic material some of the work is dissipated as heat. In the present study, a

purely mechanical process is supposed. Hence, the heat energy vanishes, whereas

some of energy is irretrievably dissipated in changes of the internal structure of the

material.

The rate change in kinetic energy given as

d 1

dt Ω∫ 2

P kin = ρ v ⋅ v dΩ (24)

P ext = ∫ v ⋅ ρ b dΩ + ∫ v ⋅ t dΓ (25)

Ω Γ

When heat flux and heat sources vanish as in the present study, the statement of the

conservation of energy can be written as

d 1

∫

dt Ω

( ρ wint + ρ v ⋅ v)dΩ = ∫ v ⋅ ρ b dΩ + ∫ v ⋅ t dΓ

2 Ω Γ

(26)

d int

ρ w = σ ij Dij (27)

dt

where:

1 ∂vi ∂v j

Dij rate of deformations Dij = ( + )

2 ∂x j ∂xi

(1) Traction boundary condition

On the boundary Γ f the traction boundary condition can be described as:

σ ij n j = ti (t ) on Γ f (28)

where:

nj outward normal to the boundary

4.2 Finite element formulation 58

On the boundary Γd the displacement boundary condition is

xi ( X, t ) = xi (t ) on Γd (29)

the contact discontinuity can be defined by the following equation

When a virtual displacement δxi is given to the body that satisfied all boundary

conditions on Γd , it leads to the following virtual work

∫ ( ρ &x& − σ

Ω

i ij , j − ρ fi ) δxi dΩ + ∫ (σ ij n j − ti ) δxi dΓ

Γf

(31)

+ ∫ (σ − σ ij− ) n j δxi dΓ = 0

+

ij

Γc

transformations, the weak form of equilibrium equations can be obtained, Hallquist [75]:

Ω Ω Ω Γf

(32)

− ∫t δxi dΓ = 0

c

i

Γc

1424

3

Contact contribution

where:

ti contact tractions applied on contact surface Γc

Let us superimpose a mesh of finite elements interconnected at nodal points on a

reference configuration and track particles through the time, i.e.

4 Finite element modelling of masonry 59

m

xi ( X, t ) = xi ( X(ξ ,η , ζ ), t ) = ∑ φ j (ξ ,η , ζ ) xij (t ) (33)

j =1

where:

φj shape functions of the parametric coordinates (ξ ,η , ζ )

Ωe Ωe Ωe Γef

en

δπ = ∑ δπ e = 0 (35)

e =1

en

∑ ( ∫ ρ &x& Φ

e =1 Ωe

i

e

i dΩ + ∫ σ ij Φ ie, j dΩ − ∫ ρ f i Φ ie dΩ − ∫ ti Φ ie dΓ) = 0 (36)

Ωe Ωe Γef

en

∑( ∫ ρ N

e =1 Ωe

t

N a e dΩ + ∫ B t σ dΩ − ∫ ρ N t b dΩ − ∫ N t t dΓ) = 0 (37)

Ωe Ωe Γef

where:

N interpolation matrix

σ stress vector σ t = (σ xx , σ yy , σ zz , σ xy , σ yz , σ zx )

B strain-displacement matrix

4.3 Contact analysis 60

The size of equation (37) can be reduced by introducing the following assumptions

en en

M a = ∑ M e a e = ∑ ∫ ρ N t N a e dΩ (38)

e =1 e =1 Ω e

and

Feext = ∫ ρ N b dΩ + ∫ N t dΓ

t t

(39)

Ωe Γef

Feint = ∫ B t σ dΩ (40)

Ωe

where

en

M a − ∑ (Feext − Feint ) = 0 (41)

e =1

In equation (32) the last term gives the effect of contact forces as a contribution in

externally applied tractions.

The concept of contact analysis can be described by considering two bodies i and j

which are in contact at time t, Figure 37, but the given concept below can be generalized

for multiple-body contact.

4 Finite element modelling of masonry 61

ϕi

Ωi0 Ωi

contact surface pair

Γc

Xi

ϕ (Xi , t ) = ϕ (X j , t )

Ω0j Ωj

Xj

ϕj

reference configuration t with contact

The two bodies are assumed to be supported, so that, without contact no rigid body

motion is possible. The contact traction t c which acts on point x i of the contact surface

Γci can be decomposed into normal and tangential components corresponding to n and

s on Γci , Figure 38.

tc = λ n +η s (42)

x i − x j (x i , t ) = minj x i − x (43)

2 x∈Γc 2

g ( x i , t ) = ( x i − x j )T n (44)

With these definitions, the conditions for normal contact can be, therefore, written as

g ≥ 0; λ ≥ 0; gλ =0 (45)

which are known as Hertz-Signorini-Moreau conditions for frictionless contact. The last

equation implies that if g > 0 , it must be λ = 0 and vice versa.

4.3 Contact analysis 62

Ωi

Γci

master surface

x i ηs

λn tc = λ n +η s

n

xj

s

Γcj

Ωj

slave surface

manner through Coulomb’s law of friction on the contact surface. Consider that µ is the

coefficient of friction. The magnitude of the relative tangential velocity is

η

τ= (47)

µλ

With these definitions Coulomb’s law of friction states

| τ |≤ 1

These interface conditions are illustrated in Figure 39. The last conditions must be valid

for the solution of the virtual work equation (32).

4 Finite element modelling of masonry 63

λ τ

| τ |≤ 1

| τ |< 1 implies u& = 0

g≥0

| τ |= 1 implies sign(u& ) = sign(τ )

λ≥0

gλ =0

g u&

Normal conditions Tangential conditions

Various algorithmic procedures have been proposed to solve contact problems, most of

these procedures are based on penalty and Lagrange multiplier techniques for enforcing

the contact constraints, Belytschko et al. [19], Wriggers [200], Bathe [14], Hallquist [75]

and Munjiza [134]. In the following the contact algorithm of LS-DYNA software is

explained.

The node-segment pair is the root level of all contact types in LS-DYNA. The node is a

point with mass and is usually named as slave node. The segment is either 3-noded or

4-noded connectivity information and is usually named as master segment. The contact

algorithm consists of the following steps, Bala [13]:

(a) The slave node-master segment pair is assembled so that, the projection of the

slave node onto the master segment, along the master segment normal must lie

within the area enclosed by the 3 or 4 nodes of the segment. The projection point

called contact point and the distance from slave node to contact point called

projection distance. In order to collect the nodes which may lie near the edges, it

is necessary to use a small increase in the area of the segment. LS-DYNA uses

an additional 2% increase to the master segment.

segment normal

η

projection distance

transformation to

2 (-1,1) 1 (1,1) isoparametric

slave node

coordinates of the master segment

master segment

ξ

scaled segment

contact point

3 (-1,-1) 4 (1,-1)

(b) Determining of the contact point in the isoparametric coordinates of the master

segment.

4.4 Finite element codes and solution strategies 64

(c) Computing the projection distance in the local coordinate system which is

embedded in the master segment.

(d) When the projection distance found to be negative, its absolute value indicates

the depth of the penetration. The slave nodal force is calculated according to the

following equation

f

{s

= K

{c

⋅ δ{

(49)

contact force contact stiffness penetration depth

(e) Distributing the contact force to the master segment nodes. Each master node

gets a fraction of the slave force based on the contact point location by using the

isoparametric shape functions

⎧ 1

⎪ N1 = 4 (1 + ξ )(1 + η )

⎪

⎪ N = 1 (1 − ξ )(1 + η )

⎪ 2 4

f mi = N i (ξ ,η ) ⋅ f s where ⎨ (50)

⎪ N = 1 (1 − ξ )(1 − η )

⎪ 3 4

⎪ 1

⎪ N 4 = (1 + ξ )(1 − η )

⎩ 4

during the solution occur in the attached elements but not at the interface itself. For

penalty based approach in LS-DYNA the stiffness K c is calculated using the following

equation:

f s ⋅ A2 ⋅ k

Kc = (51)

Ve

where:

fs penalty factor

Finite element codes have received a significant progress in last decades. Most of FE

codes since the early of 1990s have focused on static and dynamic solutions by implicit

4 Finite element modelling of masonry 65

methods. ANSYS14, ABAQUS15 and ADINA16 are examples of the first commercial FE

codes.

In 1966 Costantino has developed what is probably the first explicit finite element

program. It was limited to linear material and small deformations. Later, the element-by-

element technique has been first implemented in SAMSON code in 1969, this technique

has had the advantage of the computation of nodal forces without use of the stiffness

matrix. SAMSON code was extended in 1972 to WRECKNER which was developed for

fully nonlinear three dimensional transient analysis, Belytschko et al. [18] and Belytschko

et al. [19].

The pioneer work in explicit finite element codes is John Hallquist’s work at Lawrence

Livemore Laboratories in the mid-seventies. The major feature of Hallquist’s finite

element code was the development of contact-impact interfaces.

In the recent version of LS-DYNA, numerous capabilities and options have been added

in terms of elements, material modelling, and contact interfaces to model non-linear

dynamic events accurately and inexpensively. LS-DYNA software is not limited to any

particular type of simulation, other than it is primarily limited by the storage capacity of

the computer on which it is being run.

An illustration for the key features of explicit and implicit methods of time integration will

be given in the following. The central difference method (explicit time integration) and the

Newmark β-method (implicit time integration) are compared with close focus to the major

features and drawbacks of each method.

(a) Explicit time integration (Central difference method)

Many explicit time integration methods have been proposed in literature (Munjiza [134]),

namely: central difference, leap frog, T-1/12, D-1/12, Gear’s predictor-corrector, CHIN,

and Forest & Ruth. The central difference method is the most popular among the explicit

methods in computational mechanics and physics, and it has been widely demonstrated

in many finite element codes. LS-DYNA uses the explicit central difference scheme to

integrate the equations of motion. The central difference method has been developed

from central difference formulas for velocity and accelerations.

The time of the simulation 0 ≤ t ≤ t E is subdivided into time steps

∆t n = t n − t n −1 (52)

1

n+ ∆t n + ∆t n+1

∆t 2

= (53)

2

14

ANSYS was developed first by John Swanson at Westinghouse as nonlinear finite element program for

nuclear applications.

15

David Hibbitt worked with Pedro Marcal until 1972 and then co-founded HKS, which markets ABAQUS.

16

ADINA lunched by Klaus-Jürgen Bathe.

4.4 Finite element codes and solution strategies 66

1 1 1 1

n+ n+ n+ n+

v 2

= (u n +1 − u n ) / ∆t 2

⇒ u n +1 = u n + ∆t 2

v 2 (54)

1 1 1 1

n+ n− n+ n−

an = (v 2

−v 2

) / ∆t n ⇒ v 2

=v 2

+ ∆t na n (56)

Let us consider the semi-discrete equations of undamped equation of motion for rate-

independent materials at time t n , equation (41)

M an = Fn (57)

where

en

F n = ∑ (Feext − Feint ) n (58)

e =1

a n = M −1F n (59)

The displacements u n are known at any time step n. The nodal forces F n can be

determined by applying in sequence the strain-displacement equations, the constitutive

equations and the relation of the nodal internal forces. Therefore, by applying equations

(58), (55) and (54), the displacement u n +1 at time step n+1 can be determined. In such

way it is clear that the entire update can be achieved without solving any system

equations provided that the mass matrix M is diagonal, and this is the salient

characteristic of the explicit method, Belytschko et al. [19].

Explicit time integration is easy to implement and very robust. The explicit procedure

seldom aborts due to failure of numerical algorithm. The salient disadvantage of explicit

integration is the conditional stability. If the time step exceeds a critical value, the

solution may grow unboundedly.

For the stability of central difference scheme, the following criterion should be satisfied

∆t ≤ ∆t crit (60)

The time step size is bounded by the largest natural frequency of the structure which is

in turn bounded by the highest frequency of any individual element in the finite element

mesh.

The critical time step for a model depends on the mesh and material properties and it is

given by Courant-Friedrichs-Levy criterion

4 Finite element modelling of masonry 67

le

∆t ≤ ∆tcrit = min (61)

ce

where:

ce wavespeed in the finite element e and is given by

Ee

ce =

ρe

It is clear from equation (61) that, the critical time step decreases by the mesh

refinement and increasing the stiffness of the material, Belytschko et al. [19] and

Hallquist [75].

For stable computations, the time step is chosen by the computer code so that the time

step is less than the time required for a stress wave to travel through the shortest

element, and therefore this would result in excessive run times as the level of

discretization increases.

(2) Implicit time integration (Newmark β-method)

For implicit time integration formulation the discrete momentum equation must be

considered at time step n+1

M a n+1 − F n+1 = M a n+1 − F ext (u n+1 , t n+1 ) + F int (u n+1 ) = R (u n+1 , t n+1 ) = 0 (62)

In order to obtain F ext and F int , the assemble of Feext and Feint is needed for all finite

elements of the structure, and it must be allocated in the computer memory.

By employing the Newmark β-method to integrate the equation of motion, the updated

displacements and velocities can be given by: (Belytschko et al. [19])

~ n + β ∆t 2 a n +1

u n +1 = u (63)

~ n = u n + ∆t v n + ∆t (1 − 2β) a n

2

u (64)

2

v n +1 = ~

v n + γ∆t a n +1 (65)

~v n = v n + (1 − γ) ∆t a n (66)

where u~ n , ~v n are the history values of nodal displacements and velocities, respectively

at the time step n. β and γ are parameters.

4.5 Techniques of crack formation 68

Equation (63) can be solved for the updated accelerations for β > 0 . By substituting

Equation (63) in equation (62) it gives

1 ~ n ) − F ext (u n +1 , t n +1 ) + F int (u n +1 ) = R (u n +1 , t n +1 ) = 0

M (u n +1 − u (67)

β ∆t 2

equations need an iterative algorithm for solution. The widely used and most robust

method for that is Newton-Raphson method, Belytschko et al. [19].

There are also some restrictions on the size of time steps in implicit methods arising

from accuracy requirements and the decreasing robustness of the Newton-Raphson

procedure as the time step increases. The latter is particularly marked in problems with

very rough response, such as contact-impact. With a large time step, the initial iteration

might be far from the solution, therefore, the possibility of failure of the Newton-Raphson

method to converge increases. Small time steps are often used to improve the

robustness of the algorithm. The major disadvantage of implicit solver is that, the cost

per step is unknown since the speed depends mostly on the convergence behaviour of

the equilibrium iteration which can vary widely from problem to problem.

Explicit time integration is the ideal approach for highly non-linear short duration

transient events, including contact and impact, Bala [13]. On the other hand the implicit

approach is more efficient for static problems or problems which have long time loading,

where this kind of problems will be not feasible and expensive to be solved by explicit

approach.

In the explicit approach the solution can be achieved on an element-by-element basis

and therefore the assemblage, the memory storage of huge matrices and the iterative

solution of the nonlinear equations are not required, like in implicit approach.

Consequently, explicit methods are able to treat large three-dimensional models with

comparatively modest computer storage requirements. The other advantages include

easy implementation and accurate treatment of general nonlinearities in a relatively

simple way. However, the price paid for this advantage is the conditional stability, where

the time step in solution process should not exceed a critical value. Thus, the solution

needs a large number of cycles to cover the solution time.

The relatively short duration and high degree of non-linearity related to contact impact

problems destined that, it is well-situated to employ explicit approach for collapse

simulation problems. LS-DYNA code has a robust explicit central difference solver which

is employed throughout developing this study.

For some problems the combination of both implicit and explicit solvers offers high

computational efficiency. For instance, it is possible to utilize the implicit solver just to

initiate the stress state in the model under static loading and then applying the dynamic

actions by means of explicit solver. This way is frequently used in the present study to

reduce the computation time.

One of the basic tasks behind the simulation of the collapse is to permit the formation of

discontinuities in continuum material.

4 Finite element modelling of masonry 69

As the real cracks locations can be well determined, the accuracy of collapse simulation

can be enhancing. The accurate simulation of post failure behaviour is of a great

significance after cracking and separation, as well. The post failure behaviour includes

the contact and impact of the fragments that causes the dissipation of kinetic energy.

The finite element method is rooted in the concepts of continuum mechanics, it has the

capability for accurate simulation of the pre-failure behaviour but it is not suited to

general fracture propagation and fragmentation problems. In contrast, the discrete

element method is specifically designed to solve problems that exhibit strong

discontinuities. It has the capability to simulate the post-failure behaviour. The

combination of finite element method with discrete element method results in powerful

method capable to simulate the pre-failure and the post-failure behaviours. The full

simulation of the behaviour by combined FEM-DEM requires a transition from continuum

to discontinuum. The transition from continuum to discontinuum by finite element is

fraught with troubles. Different techniques have been proposed in literature for this

purpose, which have been implemented in many finite element codes. However each

technique has some shortcomings. According to the crack formation, these techniques

can be classified into: undetermined crack techniques and Pre-determined crack

techniques, and according to mesh dependency it can be classified into: mesh

dependent techniques and mesh independent techniques.

In the following, the techniques of crack formation are explained with a focus on the

features and drawbacks of each one.

By using undetermined crack techniques, it is not necessary to determine the crack path

priori. This is the basic feature of these techniques, i.e. the path of crack formation is

determined during the solution process. In the following the finite element removal and

the finite element-splitting techniques are explained:

This technique is well known as kill element, element deletion or element erosion

technique. It is the most popular method used in today’s finite element codes. In this

technique the finite element will being removed from the mesh as soon as the strength

vanishes at the end of strain softening process. Therefore, the discontinuity in finite

element mesh will be simply introduced during the solution process.

During a single time step the algorithm is often divided into a series of phases, Ren et al.

[152]:

(a) A check at the sampling point level: if the material at the sampling point has

completely lost the load carrying capacity, then a flag is set up marking the

sampling point as failed

(b) A check at the element level: if all sampling points which belong to one element

are failed, then the element and consequently its sampling points will be

removed from the mesh

(c) A topological check: if any higher-order elements are removed or when all

elements which linked to one node are removed, then the node has to be

removed from the mesh

(d) Regeneration of the contact surfaces after the element removal

4.5 Techniques of crack formation 70

(e) All new vectors and matrixes must be regenerated for the remaining elements at

the next step, where any computations concerning the removed elements, nodes

and sampling points are omitted.

Element removal technique has been widely used for high velocity impact simulations in

literature, Vignjevic et al. [193] and Hayhurst et al. [78]. It has been showed fairly good

results when used with small elements and accurate failure models.

Element removal technique is easy to implement in finite element codes and it does not

contribute to any additional calculation time. There is no need to any change in the

topology after removing the elements, besides, the related data to the removed

elements, nodes and sampling points are no longer needed. By using this technique it

can be avoid the excessive distortion of the elements which may causes termination

during the solution.

In addition to being the element removal technique highly mesh dependant, the main

shortcoming is that the mass and momentum are not conserved in the crack area. This

would be severe particularly for large scale models with large elements, Alsos [6].

reomved elements

Figure 41 Formation of crack and separation using finite element removal technique

The element-splitting technique is mesh-independent technique for simulating the crack

initiation and propagation in continuum media, Johnson et al. [90].

This technique uses an efficient local adaptive remeshing algorithm in which the

elements are divided along the emerging discontinuity using piece-wise planar

segments. The algorithm can be divided into a series of phases as given in the following:

(a) Determining the failed sampling points using the employed constitutive model

(b) Determining the crack planes at the failed sampling points by using the

information which is provided by adjacent elements

(c) If the crack plane falls near an existing node, two options are possible, either the

crack plane snapped to that node or the node can be moved to the crack plane

by using a simple local adaptive mapping algorithm

4 Finite element modelling of masonry 71

(d) The cracked elements separated using the defined crack plane in last step. This

has been achieved by inserting new nodes at the intersection points of element

edges with crack plane

(e) Updating the topology of the cracked elements after inserting the new nodes

(f) The new created outer element faces are added onto the list of contacting

entities.

crack front

new contact surfaces

decision is made to node duplicated and element the element after seplitting

start from this node topologies reconnected

Figure 42 Illustration of the element splitting technique for tetrahedral element, redrawn

from Johnson et al. [90]

The update of the finite element mesh needs a local adapting of the result from the last

mesh to the new mesh. The accuracy depends here on the adapting algorithm. The local

generation of the finite element mesh increases the number of finite elements and

contact surfaces. This technique is time consuming for full collapse simulation of large

models. It is also not easy to be handled in commercial finite element codes. It has been

employed only for special cases of small model sizes.

In predetermined crack techniques, the crack trajectory is known priori. The

predetermination of crack paths must be identified before the solution process. It would

affect the accuracy, but this is the consequences paid to gain more efficiency and

flexibility.

This technique depends on defining a number of possible potential cracks within the

geometry. The number of potential cracks depends on the required accuracy, and the

pre-knowledge of possible cracks. For instance, in masonry structures, it is well known

that, the cracks mainly take place in unit-mortar interfaces. During the solution process,

the crack propagates only along the predetermined potential cracks. For defining the

potential cracks, the following modelling techniques would be possible.

4.5 Techniques of crack formation 72

Tied contact has been proposed to achieve the continuity between the discrete elements

on the contact interfaces, where the master surface and the slave surface are glued. The

effect of tied contact is that, when the master surfaces are deforming, the slave nodes

are forced to follow that deformation.

In combined discrete-finite element method there is a discontinuity in the finite element

mesh along the contact interface between the discrete elements. The finite element

mesh of the slave surface may not coincide with the mesh of the master surface. The

achievement of complete displacement compatibility on the contact interface is only

possible when each master node coincided with a slave node. By using tied contact type

in LS-DYNA, it is not possible to include nodes that are involved in a tied interface in

another tied interface. Therefore, care should be taken to avoid conflicting constraints in

modelling.

The transition from continuum to discontinuum on the tied contact surface is possible if

the tied contact has been failed according to specific failure criteria. This can be

achieved by pinning the slave nodes to the master surface using penalty stiffness. After

the failure criterion is exceeded, the slave nodes will be separated from the master

surface. The tied contact algorithm in LS-DYNA differs from the normal contact algorithm

described in section 4.3 in the following:

(a) For every slave node, a unique master segment based on the smallest projected

normal distance is located

(b) The contact point is calculated only at the beginning for all tied and tiebreak

contacts, while it is computed at every cycle for all other contacts

(c) A contact spring is internally created between the slave node and the contact

point on the master segment

(d) For any subsequent incremental change in the projected distance, a force

proportional to the incremental change in the projected distance is applied to the

slave node. The distribution of this force to the master nodes is based on the

contact point.

slave surface

segment normal

slave node

master segment

contact point

scaled segment

4 Finite element modelling of masonry 73

Interface elements are widely used in crack growth analysis, Xie et al. [204]. Interface

elements are embedded along the potential crack path and used in conjunction with

cohesive-zone models (CZMs). The cohesive zone constitutive relationship is included in

the formulation of the interface element stiffness. It can be use element removal

technique to remove the failed interface elements from the simulation.

Interface element

Interface elements

predetermined crack

This approach also is known as nodal release approach. In this context two element

nodes, initially constrained to identical displacements, are allowed to be separated by

releasing the constraints and nodal forces which hold the elements together, Vignjevic et

al. [193].

The nodes are linked by springs in normal and tangential directions to avoid sudden

energy release. After the failure of the link, the springs will be removed.

spring linkage

tied nodes

predetermined crack

4.6 Modelling strategies for collapse simulation 74

There are some options available in LS-DYNA for breakable tied nodes like: constrained

tied nodes failure, constrained tie-break, constrained spot weld and constrained

generalized weld spot, LSTC [112] and LSTC [111].

The main shortcoming of the finite element removal technique is the severe loss of mass

and momentum when removing the large elements. It is possible to reduce this effect by

the refinement of the finite element mesh along the predefined crack trajectory, Figure

46. However, care should be taken in the refined mesh, because the use of elements

with high aspect ratio might interrupt the solution due to negative volume error in explicit

solver.

refinment mesh

predetermined crack

The simulation of collapse can be achieved by the implementation of the transition from

continuum to discontinuum using one of the crack formation techniques, as mentioned in

section 4.5. In masonry structures where the unit mortar interface is the weakest linkage

in the system, the predetermined crack formation techniques are more suited and

efficient.

According to the required accuracy, the following models can be proposed:

(1) Detailed discrete micro model

In this model, units as well as mortars are discretized into discrete elements. The overall

system therefore, contains two types of discrete elements, namely mortar discrete

elements and unit discrete elements. In between, three types of interfaces can be

recognized: unit-unit interface, mortar-mortar interface and unit-mortar interface.

The detailed discrete micro strategy is well suited for accurate models when the

fragmentation of units as well as mortars are important, however this is limited to small

specimens due to the large number of discrete elements.

4 Finite element modelling of masonry 75

FE

smeared joint interface

unit-mortar interface

FE unit unit interface

mortar-mortar interface

unit-unit interface

FE

unit discrete element

mortar

unit unit

(a) Discretized detailed micro modelling (b) Discretized simplified micro modelling

for Collapse Simulation

finite element

homoginized masonry

discrete element smeared joint interface

imaginary masonry

interface parallel

to head joints

interface parallel

unit

to bed joints

structures.

In this model, units are only discretized to discrete elements, and the mortar is smeared

at joints between units. Therefore, the overall system holds only the discrete elements of

units. There are two types of interfaces between these discrete elements: the smeared

4.6 Modelling strategies for collapse simulation 76

joint interface which links masonry units, and unit-unit interface which links the discrete

elements of the same unit.

The simplified discrete micro strategy is well suited when the influence of the size of

mortar can be ignored, i.e. thin layer mortars or dry masonry. The fragmentation of

masonry units can be simulated using this strategy as well.

(3) Simplified micro model

This model is very well known in literature (section 2.1). Each unit is represented by a

unique discrete element and the smeared joint interfaces are linked masonry units.

The simplified micro strategy can be used for rigid units of high strength, where the

collapse of the system takes place primarily due to the failure of masonry joints. It can be

also possible to use this strategy for weak units with smeared crack model.

(4) Discrete macro model

Macro modelling strategy has been extensively studied in literature, where the

discretized masonry system modelled as continuum. As discussed in section 4.5.2, it is

possible to discretize the continuum media using predetermined crack techniques. The

same can be achieved for continuum media of masonry on macro level. The overall

system therefore, consists of homogenized masonry discrete elements.

If masonry system is discretized by interfaces parallel to bed joints and head joints, then

two types of interfaces are linked the discrete elements, namely: imaginary masonry

interfaces parallel to bed joints and imaginary masonry interfaces parallel to head joints.

The discretized macro strategy is well suited for masonry which has strength of mortar

close to the strength of units. However, it is also possible to use it with any type of

masonry if the constitutive models of interfaces and the homogenized material model of

masonry discrete elements are accurately defined. This modelling strategy needs less

computational efforts. It is therefore efficient for big models of large scale masonry

structures.

A particular modelling strategy to be selected depends upon the purpose of the collapse

simulation and the nature of the outcome desired from the simulation. If the overall

behaviour is desired without regard to completely realistic fragmentation and cracking

and local stresses, then, macro level framework represents conceivably the best choice.

However, if an elaborate local behaviour is of interest, micro models are more

appropriate.

It is worth to be mentioned that, modelling of masonry using all these strategies has the

same concept: discretization and interface linkage. The differences are only in the

constitutive models, either for the discrete elements or for the interfaces, Table 3.

In addition to the discretization strategy, the constitutive models of both discrete

elements and interfaces are of high importance, for accuracy and computational

efficiency.

The constitutive models of the discrete elements can be ranged from rigid to deformable

bodies and from elastic to elasto-plastic material model. The decision about the

constitutive models is more depending on the size of discrete elements. It is necessary

to think about the accuracy of constitutive models when big discrete elements are used.

The other key factors of accuracy are the constitutive model of the interfaces, and the

4 Finite element modelling of masonry 77

crack formation technique. Comprehensive study about the constitutive models will be

presented in Chapter 5.

(1) Detailed discrete micro

Ö mortar discrete Ö mortar-mortar interface

modelling strategy

elements Ö unit-mortar interface

Ö unit discrete elements

modelling strategy Ö smeared joint interfaces

Ö units Ö smeared joint interfaces

modelling strategy

(4) Discrete macro Ö homogenized masonry parallel to bed joints

modelling strategy discrete elements Ö imaginary masonry interface

parallel to head joints

masonry structures

In principle, all crack formation techniques explained in 4.5.2 can be employed for

collapse simulation models. However it is important to consider the possibility and the

availability of using these techniques in commercial finite element software.

Collapse analysis of structures threatened by earthquake actions requires a suitable

numerical tool, capable to predict the emergence of discontinuities at different scales.

The combined finite-discrete element method which merges finite element method with

the algorithms of discrete element method allows the transition from continua to

discontinua.

In this chapter, different techniques have been presented and discussed, and the

possible modelling strategies have been proposed. Different techniques can be

employed for crack formation, but the accurate modelling is only possible if the

constitutive models of the interfaces are provided accurately.

There are many undetermined crack formation techniques other than those described in

this chapter, but most of general purpose finite element software packages include

features to support the predetermined crack formation techniques. Developing such

special solution for crack formation might leads to drop the generality of the software.

Therefore, this is often limited to software, which is developed to deal with special

problems.

5 Constitutive models 79

5 Constitutive models

Material constitutive models are the mathematical description of the material behaviour

which yields the relation between the stress and strain tensor in material point of the

body up to failure. The implementation of an accurate material that tracks the empirical

laws is quite essential in any numerical analysis. Therefore, this research discipline has

received a great activity from masonry research communities in past few years, (for

example: Lourenço [109], Schlegel [164] and Mistler [132]). However, modelling of

masonry material is still challenging despite that a great progress already achieved.

Plasticity theory is the basic theoretical tool that well describes the inelastic material

behaviour and the damage phenomena. In the following the basics of the plasticity

theory, the non-smooth multi-surfaces plasticity, and the implementation of a material

model based on plasticity theory are given.

A smooth surface cohesive interface model is proposed and implemented into the

explicit solver of LS-DYNA. The available geo-material models in LS-DYNA which can

be employed for masonry are described as well.

In plasticity region, the total strain increment consists of elastic part and plastic

irreversible part:

dε = dε e + dε p (68)

where equation (68) is Prandtl-Reuss equation:

dε e elastic strain increment

This decomposition is correct for cases of infinitesimal strain only. Assuming that the

plastic deformation is rate insensitive, the stress increment is linearly related to the

elastic strain increment in the plastic region and will be given by Hook’s low

dσ = D ⋅ dε e = D ⋅ (dε − dε p ) (69)

where:

D elastic matrix

The elastic region is limited by the yield surface that separates the plastic region from

elastic region

F (σ, κ ) = 0 (70)

where

κ parameter introduced to measure the softening κ = κ (ε p )

5.1 The basics of plasticity theory 80

The demonstration of the flow surface for isotropic material is possible in principle stress

space. However, for anisotropic material like masonry the reference to a fixed space is

important. For identifying the softening parameter there are two approaches, the first one

would be by considering dκ as a measure of the equivalent plastic strains dεeqp and the

other one by using dκ as a measure of the plastic work dW p . For masonry material,

the first approach has been employed to identify dκ , Schlegel [164].

The general mathematical treatment of the constitutive equation for the flow of plastic

strains has been called plastic potential theory which was proposed by Huber-von Mises

in 1928, Yu et al. [205] and has the following form:

∂Q

dε p = dλ (71)

∂σ

where:

dλ positive proportional scalar factor

Q Q(σ, κ ) = 0 is the plastic potential function which represents a surface in the six-

dimensional stress space

A common approach in plasticity theory is to assume that the plastic potential function is

the same as the yield function Q ≡ F , thus:

∂F

dε p = dλ (72)

∂σ

Consequently, the plastic flow vector is normal to the yield surface, and this is called the

associated flow rule. On other hand, in case of inequality the flow rule is called non-

associated flow rule.

After a stress increment dσ , the state of plastic stresses must be again on the flow

surface

F (σ + dσ, κ + dκ ) = 0 (73)

dF (σ, κ ) = 0 (75)

Therefore, for consistent condition of stress changing and using the chain rule, the

following equation can be obtained, Will [197]:

T T

⎛ ∂F ⎞ ∂F ⎛ ∂κ ⎞

⎟ dσ + ⎟ dε = 0

p

⎜ ⎜ (76)

⎝ ∂σ ⎠ ∂κ ⎝ ∂ε p ⎠

By substituting the plastic strain from flow rule equation (71) it gives

5 Constitutive models 81

T T

⎛ ∂F ⎞ ∂F ⎛ ∂κ ⎞ ∂Q

⎜ ⎟ dσ + ⎜ ⎟ dλ = 0 (77)

⎝ ∂σ ⎠ ∂κ ⎝ ∂ε p ⎠ ∂σ

T

∂F ⎛ ∂κ ⎞ ∂Q

H =− ⎜ ⎟ (78)

∂κ ⎝ ∂ε p ⎠ ∂σ

this yield:

T

⎛ ∂F ⎞

⎜ ⎟ dσ − H ⋅ dλ = 0 (79)

⎝ ∂σ ⎠

⎛ ∂Q ⎞

dσ = D ⋅ ⎜ dε − dλ ⎟ (80)

⎝ ∂σ ⎠

T T

⎛ ∂F ⎞ ⎛ ∂F ⎞ ⎛ ∂Q ⎞

⎜ ⎟ D ⋅ dε − ⎜ ⎟ D⋅⎜ ⎟ ⋅ dλ − H ⋅ dλ = 0 (81)

⎝ ∂σ ⎠ ⎝ ∂σ ⎠ ⎝ ∂σ ⎠

T

⎛ ∂F ⎞

⎜ ⎟ D ⋅ dε

⎝ ∂σ ⎠

dλ = T (82)

⎛ ∂F ⎞ ⎛ ∂Q ⎞

⎜ ⎟ D⋅⎜ ⎟+H

⎝ ∂σ ⎠ ⎝ ∂σ ⎠

The yield surfaces of most engineering materials including masonry are defined by

means of multiple yield criteria, which result in non-smooth multi-surfaces. The

shortcoming of using multi-surfaces yield function is the corners which are the singular

points on the non-smooth yield surface.

Let us consider a material governed by the following yield criteria

By assuming that, the stress state of the point under study is yielded under both criteria,

hence, the plastic strains can be described as a contribution of both yield criteria

according to Koiter’s generalization:

5.1 The basics of plasticity theory 82

∂Q1 ∂Q2

dε p = dλ1 + dλ 2 (84)

∂σ ∂σ

⎛ ∂Q ∂Q ⎞

dσ = D ⋅ ⎜ dε − dλ1 1 − dλ2 2 ⎟ (85)

⎝ ∂σ ∂σ ⎠

The same treatment above will be applied for each yield criterion

The application of the chain rule for the 1st yield function gives

T T

⎛ ∂F1 ⎞ ∂F ⎛ ∂κ ⎞

⎟ dσ + 1 ⎜ p1 ⎟ dε = 0

p

⎜ (87)

⎝ ∂σ ⎠ ∂κ1 ⎝ ∂ε ⎠

T T T

⎛ ∂F1 ⎞ ∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞ ∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞

⎜ ⎟ dσ + 1 ⎜ p1 ⎟ ⎜ 1 ⎟dλ1 + 1 ⎜ p1 ⎟ ⎜ 2 ⎟dλ2 = 0 (88)

⎝ ∂σ ⎠ ∂κ1 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠ ∂κ1 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠

T T

∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞ ∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞

H11 = − 1 ⎜ p1 ⎟ ⎜ 1 ⎟ H12 = − 1 ⎜ p1 ⎟ ⎜ 2 ⎟ (89)

∂κ1 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠ ∂κ1 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠

T T T

⎛ ∂F1 ⎞ ⎛ ∂F ⎞ ∂Q ⎛ ∂F ⎞ ∂Q2

⎜ ⎟ D ⋅ dε = ⎜ 1 ⎟ D ⋅ dλ1 1 + H 11dλ1 + ⎜ 1 ⎟ D ⋅ dλ2 + H12 dλ2 (90)

⎝ ∂σ ⎠ ⎝ ∂σ ⎠ ∂σ ⎝ ∂σ ⎠ ∂σ

By applying the same concept for the 2nd yield criterion gives

T T T

⎛ ∂F2 ⎞ ⎛ ∂F ⎞ ∂Q ⎛ ∂F ⎞ ∂Q2

⎜ ⎟ D ⋅ dε = ⎜ 2 ⎟ D ⋅ dλ1 1 + H 21 dλ1 + ⎜ 2 ⎟ D ⋅ dλ 2 + H 22 dλ 2 (91)

⎝ ∂σ ⎠ ⎝ ∂σ ⎠ ∂σ ⎝ ∂σ ⎠ ∂σ

where

T T

∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞ ∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞

H 21 = − 2 ⎜ p2 ⎟ ⎜ 1 ⎟ H 22 = − 2 ⎜ p2 ⎟ ⎜ 2 ⎟ (92)

∂κ 2 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠ ∂κ 2 ⎝ ∂ε ⎠ ⎝ ∂σ ⎠

dλ1 and dλ2 can be obtained by solving equations (90) and (91).

5 Constitutive models 83

In the following, an explanation for implementing a material model based on plasticity

theory into the explicit solver of LS-DYNA will be given. LS-DYNA has the feature to

implement the material subroutine of the user. The explicit solver of LS-DYNA provide

the stresses σ of the last step, the strain increment ∆ε , and history variables that

includes damage parameters to the material subroutine, then the task of the subroutine

is fundamentally to update the stresses and history variables according to the

implemented material model.

The following simplification can be used for the last equations

T T

⎛ ∂Fi ⎞ ⎛ ∂F ⎞

⎜ ⎟ D ⋅ ∆ε = ⎜ i ⎟ ∆σ ≈ ∆Fi (93)

⎝ ∂σ ⎠ ⎝ ∂σ ⎠

Inside the material subroutine, the first decision is to assume the elastic behaviour of the

material in order to calculate the trial stresses.

yield surfaces then no yielding Region IJ

exists and the material still behaves σ2

linearly i.e. the last trial stresses are σtrial

∂Qj ∆λ

true and the subroutine can be D

∂σ j

returned to the solver, otherwise the

material has been yielded and the

trial stresses must be returned back ∂Qi ∆λ

to the yield surface by the suitable D

∂σ i

Region I

return mapping procedure. For this

purpose, two cases can be σ

Fi (σ ,

κi )

considered:

Region J

Fj

(σ

,κ j

)

been reached

Case II: only two yield criteria have

been reached, Figure 48. σ1

After updating the stresses using

return mapping procedure, it must Figure 48 Return mapping of the trial stresses in

be checked again if they are inside case of two yield criteria

or outside the yield surface and this

procedure will be looped until

getting the stress state point inside

the yield surfaces. Figure 49 shows

the flow chart of the subroutine.

5.1 The basics of plasticity theory 84

(1) strain increment ∆ε and history variables that

represent damage state of material

(2) calculate the trial stresses σ = σ + D ⋅ ∆ε

e

(3) then end (6) then go to case I. (4) go to case II. (4)

(4)

Calculate H i Calculate H mn and amn for m = i, j

T

and n = i, j

∂F ⎛ ∂κ i ⎞ ∂Q i

Hi = − i ⎜ p⎟ T

∂κ i ⎝ ∂ε ⎠ ∂σ ∂F ⎛ ∂κ ⎞ ⎛ ∂Q ⎞

H mn = − m ⎜ mp ⎟ ⎜ n ⎟

∂κ m ⎝ ∂ε ⎠ ⎝ ∂σ ⎠

Calculate ∆λi

T

∆Fi ⎛ ∂F ⎞ ⎛ ∂Q ⎞

amn = ⎜ m ⎟ D ⋅ ⎜ n ⎟ + H mn

∆λ i = T ⎝ ∂σ ⎠ ⎝ ∂σ ⎠

⎛ ∂Fi ⎞ ⎛ ∂Q i ⎞

⎜ ⎟ D⋅⎜ ⎟ + Hi Solve for ∆λi and ∆λ j

⎝ ∂σ ⎠ ⎝ ∂σ ⎠

Calculate ∆ε

e ∆Fi = aii ⋅ ∆λi + aij ⋅ ∆λ j

∂Q i ∆F j = a ji ⋅ ∆λi + a jj ⋅ ∆λ j

∆ε e = ∆ε − ∆λi

∂σ If there is no solution show an error

message

Calculate the elastic strains

∂Q1 ∂Q2

∆ε e = ∆ε − ∆λi − ∆λ j

∂σ ∂σ

e

(5)

(6) END

Figure 49 Flow chart for the subroutine of material model based on plasticity theory

5 Constitutive models 85

As has been already described in section 4.5.2, various numerical approaches can be

employed to simulate the crack formation, tied or adhesive contact surfaces with failure,

interface elements, breakable tied nodes and pre-refinement of the mesh along potential

crack. Those numerical approaches can only give a real representation of the interfaces

in case of masonry if an accurate constitutive model is employed. All these numerical

approaches are already implemented in LS-DYNA with various options, but the

possibility to employ those approaches for masonry with the appropriated options and

the validation is still questioned.

Due to the robustness of contact algorithms in LS-DYNA the first task was to examine

the available contact options for modelling masonry interfaces. The Tiebreak contact in

LS-DYNA allows the modelling of connections which transmits both compressive and

tensile forces with optional failure criterion. The separation of the slave node from the

master is resisted by contact spring for both tensile and compressive forces until failure,

after which the tensile coupling is removed, Bala [13]. The option 6 of contact tiebreak

permits damage modelling by scaling the stress components after failure is met,

Hallquist [75]. The following yield function has been employed:

σ2 τ2

F (σ, κ ) = + − Ω(κ ) (94)

ft 2 c2

where:

After the damage is initiated, the stress is linearly scaled down until the crack width

reaches the critical distance wc at which the interface failure is complete.

Tiebreak contact in LS-DYNA uses penalty method. This produces some relative

displacement between the surfaces before the damage of the contact which results in

deponding. Due to that, the yield criterion is possible to be achieved. This brings out an

unrealistic behaviour because the tractions between the surfaces are suddenly jumped

down. Further options to prevent this behaviour are available in LS-DYNA by increasing

the stiffness scale factor of the contact. However, care should be taken with higher

penalty stiffness, which results in high frequency modes and therefore instability in the

solution.

5.2 Constitutive models of the interfaces 86

Beattie et al. [17], Beattie et al. [16] and Burnett et al. [31] have developed a discrete

crack model in LS-DYNA for modelling masonry joints within a project for study the

performance of masonry parapet walls that subjected to vehicle impact. The yield

function represents the fundamental key features of masonry joints and has the following

form:

{ }⎞⎟ 2 2

⎛ τ − µ ⋅ σ trial

c ⎛ σ trial

t ⎞

F = ⎜⎜ trial + ⎜ ⎟ (95)

κ ⋅c ⎟ ⎜ κ ⋅σ t ⎟

⎝ ⎠ ⎝ f ⎠

where { } Macaulay brackets, τ trial is the trial shear stress at the interface, σ trial is the

trial normal stress, (positive in compression), in tension σ trial

t

= σ trial , σ trial

c

= 0 and in

compression σ trial

c

= σ trial , σ trial

t

= 0 . Figure 50 shows the yield surface.

Shear Stress

initial

shear or tensile strength

limite of softening

displacement

softening c

uf

arctan( µ )

displacement

softening

residual

u >uf normal stress

yield surface, ft

σ

Figure 50 Yield surface for masonry mortar joint, redrawn after Burnett et al. [31]

σt =

σ trial

t

, σ c = σ trial

c

and τ =

{τ trial − µ ⋅ σ trial

c

} + µ ⋅σ c (96)

trial

F f

the global damage parameter κ controls the rate at which the surface can be shrink

κ = {κ I + κ II − 1} (97)

where:

⎧⎪ p ⎫

ut / u I ⎪

f

⎧⎪ p

us / u II

f ⎪

⎫

⎨ log e ( 0.001) ⎬ ⎨ log e ( 0.001) ⎬

⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭

κ I , κ II given by κ I = e , κ II = e

5 Constitutive models 87

u If = − log e (0.001) ⋅ G If / f t , u IIf = − log e (0.001) ⋅ G IIf / c

The yield surface in the proposed model is concave, Figure 50. For this reason the

return mapping has been used parallel to shear axis in compression region instead of

radial return mapping, but in such case the assumption of the plastic potential theory

(Huber-von Mises theory) is not valid.

Interface elements have been used widely in literature to model masonry mortar joints,

Lourenço [108] has proposed an interface element for masonry mortar joints using non-

smooth yield function consists of three parts, Figure 51.

cap-shear corner τ

Shear Stress

tension-shear corner

σ2 (κ2 )

f1 = σ −σ1 (κ1 )

shear mode

tension mode

f 2 = τ +σ ⋅ µ −σ2 (κ2 )

cap mode

f 3 = Cnnσ 2 + Cssτ 2 + Cnσ − (σ3 (κ3 ))2 normal stress

σ

σ3 (κ3 ) σ1 (κ1 )

Figure 51 Yield surface for masonry mortar joint, redrawn after Lourenço [108]

Giambanco et al. [67], Giambanco et al. 5

7 midsurface

[66] and Formica [58]. All those ∆ x84

interfaces models share the feature of ∆ x51 σ

using non-smooth yield function. τ1

∆ x73

6 τ2 τ

Interface element also has been 1 4

composite materials, Jiang et al. [87].

Interface elements are available to be Figure 52 Interface element in LS-DYNA,

used with the following cohesive material redrawn after LS-DYNA [112]

models in LS-DYNA:

(1) Elastic cohesive material model,

(2) Tvergaard and Hutchinson cohesive material model, and

5.3 Implementation of cohesive interface material model 88

Although, various cohesive material models haven been implemented into LS-DYNA,

they do not reflect the desired behaviour of the interfaces for masonry.

To simulate the dynamic events after the failure of the interface elements, it would be

possible to replace the surfaces which linked by the failed interface elements with

frictional contact model. The deletion of the interface element after the failure does not

bring any loss in the mass if the thickness of the interface element has been set to zero.

In such way, the inherent difficulties associated with large displacement after the failure

of the interface element also are avoided. In LS-DYNA ‘contact eroding single surface’

offers a possibility to detect the contact on the eroded surfaces after the failure of the

interface element.

Various aspects of interface models are available in literature, but all are based on non-

smooth yield functions which bring out many numerical difficulties and further

computation times for handling the singularities of corners. Therefore, in the following a

smooth yield surface is proposed and implemented into LS-DYNA. The smooth yield

surface provides economies in coding and CPU’s times, as well as, eliminates the

numerical complexity of treating corner regions.

In order to match the smoothness conditions, the following smooth yield surface has

been proposed

(1) tensile region

F1 = τ 2 + α1 ⋅ σ 2 + 2α 2 ⋅ σ − α 3 (98)

F2 = τ − F f (99)

F3 = τ 2 − Ff2 ⋅ Fc (100)

where:

Ff Mohr-Coulomb yield surface and given by Ff = c ⋅ Ω(κ ) − µ ⋅ σ , where c the

cohesion, µ is the friction ratio and Ω(κ ) is the damage function.

Fc cap function, suggested by Sandler et al. [159] for geo-materials and given by:

5 Constitutive models 89

2

⎡ σ + L(κ c ) ⎤

Fc = 1 − ⎢ ⎥ (101)

⎣ X (κ c ) − L(κ c ) ⎦

to guarantee a smooth transition from tensile yield surface to shear yield surface. To get

a continuity of order C 1 , the following condition must be satisfied:

∂τ

= −µ (104)

∂σ σ = 0 ,τ = c⋅Ω

c 2 ⋅ Ω2 − α3 = 0 (105)

α1 ⋅ ft 2 ⋅ Ω2 + 2α 2 ⋅ ft ⋅ Ω − α 3 = 0 (106)

α1 ⋅ σ + α 2

− = −µ (107)

τ σ = 0 ,τ = c ⋅Ω

The values of α1 , α 2 and α 3 can be obtained by solving equations (105), (106) and

(107)

c2

α1 = (1 − α ) , α2 = µ ⋅ c ⋅ Ω , α 3 = c 2 ⋅ Ω2 (108)

ft 2

where

ft

α = 2µ ⋅ (109)

c

thus,

5.3 Implementation of cohesive interface material model 90

c2

F1 = τ 2 + (1 − α ) ⋅ σ 2 + 2 µ ⋅ c ⋅ Ω ⋅ σ − c 2 ⋅ Ω 2 (110)

ft 2

According to the value of α the following cases can be obtained for the tensile part,

Figure 53:

If 0 < α < 1 elliptic function

If α = 1 parabolic function

τ

Shear Stress

c2

F1 =τ 2 + (1 −α) ⋅σ + 2µ ⋅ c ⋅ Ω⋅σ − c 2 ⋅ Ω

2 2

F2 = τ − c ⋅ Ω(κ) + µ ⋅σ ft 2

f

α = 2µ ⋅ t

c

0 < α <1 elliptic

c Ω (κ) α =1 parabolic

2 > α > 1 hyperbolic

c

µ Ω (κ)

σ

normal stress

f t ⋅ Ω (κ )

For the cap yield function, the difference X (κ c ) − L(κ c ) represents the major axis of the

cap. It can be therefore linked to the minor axis by the cap ellipticity ratio R which is a

characteristic parameter of the material, the continuity of the cap with the shear yield

function yields:

damage function.

fc ⋅ Ωc − R ⋅ c ⋅ Ω

L(κ c ) = (112)

1+ R ⋅ µ

5 Constitutive models 91

µ = µ0 + ( µr − µ0 ) ⋅ (1 − Ω(κ )) (113)

23

τ

µ = µ0 + ( µr − µ0) ⋅ (1 − Ω (κ))

Shear Stress

F2 = τ − c ⋅ Ω (κ) + µ ⋅σ

arctan( µ ) F12

Region 3 Region 2

c2

F3 =τ 2 − Ff2 ⋅ Fc F1 =τ 2 + (1 −α) ⋅σ 2 + 2µ ⋅ c ⋅ Ω⋅σ − c 2 ⋅ Ω2

ft 2

c ⋅ Ω (κ)

2

Softening

⎡ σ + L(κc ) ⎤ arctan( µr )

Fc =1 − ⎢ ⎥

⎣ X (κc ) − L(κc ) ⎦ Region 1

B

Ff = c ⋅ Ω(κ) + µ ⋅σ c

-fc ⋅ Ωc (κc ) -L(κc ) f t ⋅ Ω(κ ) µ

σ

normal stress

A C

ft 2 −α

A= Ω

2 1 −α

c 2 −α

B= Ω

2 1 −α

c α2

C= Ω

Region 2 4µ 1 −α

F12

F23

Figure 54 The proposed smooth yield surface of the cohesive interface model

The followed return mapping procedure is similar to that described in section 5.1.2. but

for cohesive material model, the trial stresses can be calculated from

σ = K ⋅u (114)

where:

T

σ

displacement array u = {u n , u s }

T

u

5.3 Implementation of cohesive interface material model 92

For smeared mortar joints, Lourenço [108] has proposed the following equations to

calculate the stiffness of the interface element

Eu ⋅ Em Gu ⋅ Gm

kn = ks = (115)

t m ( Eu − Em ) t m (Gu − Gm )

where:

Gu , Gm shear modulus of units and mortar, respectively

For return mapping the derivatives with respect to the damage scalar can be calculated

as following:

= , = and = (116)

∂κ ∂Ω ∂κ ∂κ ∂Ω ∂κ ∂κ c ∂Ω c ∂κ c

∂α ∂α ∂µ f

= = −2 t ( µ r − µ 0 ) (117)

∂Ω ∂µ ∂Ω c

Thus,

∂F1 c

= 2 ( µ r − µ 0 ) ⋅ σ 2 + 2c ⋅ σ (µ − ( µ r − µ 0 )Ω ) − 2c 2 Ω (118)

∂Ω ft

∂F2

= −c − ( µ r − µ 0 ) ⋅ σ (119)

∂Ω

∂Fc 2(σ + L) c ⋅ Ω − µ ⋅ σ ∂L fc

=− and = (120)

∂L R2 (c ⋅ Ω + µ ⋅ L ) 3 ∂Ω c 1 + R ⋅ µ

∂F3 ∂F ∂L ∂F3 f c ⋅ F f2 σ + L c ⋅ Ω − µ ⋅σ

= − F f2 c or =2 ⋅ ⋅ (121)

∂Ω c ∂L ∂Ω c ∂Ω c R 2

1 + R ⋅ µ (c ⋅ Ω + µ ⋅ L ) 3

5 Constitutive models 93

∂F1 c2 ∂F1

( F1 ) = 2 2 (1 − α ) ⋅ σ + 2 µ ⋅ c ⋅ Ω = 2τ

∂σ ft ∂τ

∂F2 ∂F2

( F2 ) =µ = sign (τ )

∂σ ∂τ (122)

∂Fc

∂F3 σ + L(κ c ) = 2τ

( F3 ) = 2 µ ⋅ Fc ⋅ F f + 2 F f2 ∂τ

∂σ [X (κ c ) − L(κ c )]2

The damage function Ω(κ ) represents the rate at which the material strength is

degraded once the initiation criterion is reached. Four optional damage functions are

employed in the following to represent the softening, Figure 55:

(1) Linear softening

κ damage scalar 0 ≤ κ < 1

∂Ω

Ω(κ ) = 1 − κ = −1 (123)

∂κ

1 − c2 1 − c2

1− κ 0 ≤ κ < c1 − 0 ≤ κ < c1

c1 ∂Ω c1

Ω(κ ) = = (124)

c2 ∂κ c2

(1 − κ ) c1 ≤ κ < 1 − c1 ≤ κ < 1

1 − c1 1 − c1

∂Ω κ c1

Ω(κ ) = 1 − κ c1

= −c1 (125)

∂κ κ

∂Ω (126)

= 3c13κ 2 e −c2κ − (1 + c13κ 3 )c2 e −c2κ − (1 + c13 )e −c2

∂κ

5.3 Implementation of cohesive interface material model 94

σ σ

1 − c2 w

1− if 0 < w < c1wc

w c1 wc

ft σ 1− if 0 < wc < w ft σ

= wc c

= 2 (1 − )

w

if c1wc < w < wc

ft 0 if 0 < wc < ∞ ft 1 − c1 wc

0 if wc < w < ∞

Gf =

wc f t wc f t c1 = 2 / 9 G = wc f t

2 Gf = c2 = 1 / 3

f

2 /(c1 + c2 )

3.6

c2 ft

crack opening crack opening

0 wc w 0 c1 wc wc w

σ σ

w

w w −c2 ⋅ w

σ = 1 − ( )c if 0 < wc < w σ

= [1 + (c1 ⋅ wc ) ] ⋅ e − ⋅ (1 + c13 ) ⋅ e −c2 if 0 < wc < w

1 3 wc

ft wc ft wc

ft ft

0 if 0 < wc < ∞ 0 if 0 < wc < ∞

c1 = 0.31

c1 = 3

c2 = 6.93

wc f t

Gf = wc f t

4.226 Gf =

crack opening 5.136 crack opening

0 wc w 0 wc w

(c) Nonlinear softening (Moelands & Reinhardt) (d) Nonlinear softening (Hordijk & Reinhardt)

Figure 55 Softening models, (a) Linear softening, (b) Bilinear softening, (c) Nonlinear

softening (Moelands & Reinhardt), (d) Nonlinear softening (Hordijk &

Reinhardt).

for the cap part, Figure 56.

The necessary derivatives for return mapping are:

κ p − κc

(Ω p − Ω i ) for 0 ≤ κ c < κ p

κ p 2κ cκ p − κ c2

∂Ω c κc − κ p

= (Ω m − Ω p ) ⋅ 2 for κ p ≤ κ c < κ m (127)

∂κ c (κ m − κ p ) 2

κ c −κ p

m

m κ −κ

for κ m ≤ κ c

(Ω m − Ω r ) ⋅e m p

κm − κ p

5 Constitutive models 95

Ω κc κc2

(1) Ω c (κc ) = Ωi + (Ωp − Ωi ) 2 −

Ωp

peak κ p κ p2

2

⎛ κc −κ p ⎞

(2) (2) Ω c (κc ) = Ωp + (Ωm − Ωp ) ⋅ ⎜⎜ ⎟

⎟

(1) ⎝ κm −κ p ⎠

κc −κ p

m

(3) Ω c (κc ) = Ωr + (Ωm − Ωr ) ⋅ e κ −κ m p

middle

Ωm

Ωm − Ωp

with m =2

Ωi κm −κ p

initial

(3)

residual

Ωr

κc

κp κm

hardening softening

Figure 56 Hardening softening model for compression, redrawn after Lourenço [109].

In the following the interface elements are employed to simulate the fragmentation of the

material. One masonry unit is assumed to be in impact with rigid surface, and the crush

problem is simulated.

The masonry unit is divided into finite elements and the interface elements introduced at

some planes, Figure 57.

The planes at which the interface elements are introduced represent the planes of failure

or planes of potential cracks. The initiation of cracks along these planes is possible by

deletion of the interface element after failure. In order to avoid the termination during the

5.4 Constitutive models of masonry constituents 96

calculation the ‘contact eroding single surface’ is introduced which offer the detection of

the eroded surface after the failure of the interface element, Figure 58.

Figure 58 Drop test simulation for one masonry unit, crack formation and

fragmentation by deletion of the interface elements

5.4.1 General shape of yield surface for geo-materials

Masonry constituents belong to geo-materials. There are large amounts of experimental

data reported by different researchers over past years. Different multi parameter yield

functions have been proposed to fit with the experiential data, which resulted in a variety

of constitutive models capable to represent various aspects of geo-materials behaviour

such as limestone, granite as well as concrete and ceramics.

The key feature of material model is to identify the relationship between the stress tensor

and strain tensor depending on few parameters that characterize the material behaviour.

The yield surface can be best described in principle stress space. The plane that

contains the hydrostatic axis is the meridian plan, and the plane that perpendicular to the

hydrostatic pressure is the deviatoric plane. The cross-section of the meridian plane and

the deviatoric plane brings out good understanding for the shape of the yield surface.

Figure 59 shows an example of a yield surface, and the cross-sections through the

meridian plane and deviatoric plane.

5 Constitutive models 97

Hydrostatic axis

Compression meridian

Cap

rc

rt = ψ ⋅ rc r

Cap

(a) Three-dimensional view in (b) Section in the yield surface (c) Section in the yield surface

principal stress space across the meridian plane across the deviatoric plane

Figure 59 Example of typical geo-material yield surface, modified after Fossum et al.

[59]

The meridian race of the yield surface defines the relation between:

(1) the first stress invariant J1 or the hydrostatic pressure p , which has direct physical

significance in most applications,

p = (σ 1 + σ 2 + σ 3 ) / 3 J1 = 3 p (128)

(2) and the second invariant of the deviatoric stress J 2 or von Mieses stress σ e which

measures the shear stresses.

1 3

J2 = ∑ (σ i − p) 2

2 i =1

σ e = 3J 2 (129)

The deviatoric race defines the relation between the second invariant of the deviatoric

stress J 2 and the third invariant of the deviatoric stress J 3 where

1 3

J2 = ∑

3 i =1

(σ i − p ) 3 (130)

The deviatoric race can be well represented using Lode angle, which associated with the

stress invariants J 2 and J 3 by the following equation:

3 3 J3

cos 3θ = where 0 o ≤ θ ≤ 60o (131)

2 J 23 / 2

5.4 Constitutive models of masonry constituents 98

Ottosen (1977) has suggested four parameters yield criterion for concrete accounting all

three invariants

The meridian race

J2 J2 J

F ( J1 , J 2 ) = a ⋅ +λ + b 1 −1 = 0 (132)

f c′ 2

f c′ f c′

1

λ (θ ) = k1 cos[ cos −1 (k 2 cos 3θ )] for cos 3θ ≥ 0 (133)

3

The yield surface of this criterion has a curved meridian and noncircular cross sections

on the deviatoric plane. The cross section has convenience geometric characteristic for

many geo-materials like changing from nearly triangular to nearly circular along the

hydrostatic stress axis. By setting a = 0 and b = 0 it leads to Huber-von Mises criterion,

and if a = 0 and b ≠ 0 then Drucker-Prager criterion can be obtained.

William-Wranke have proposed five parameter yield criterion for concrete accounting all

three invariants yield surface, Yu et al. [205]. For the meridian race, two parabolic tensile

and compressive meridians have been defined:

rt = a0 + a1 p + a2 p 2 rc = b0 + b1 p + b2 p 2 (134)

For the deviatoric race, a smooth convex triangular surface based on elliptic equation

has been used to relate the yield surface to Lode angel θ

= (135)

rc 4(1 − ψ 2 ) cos 2 θ + (1 − 2ψ ) 2

where ψ = rt / rc

The boundaries of this surface can be defined by settingψ = rt / rc = 1 then the ellipse

degenerates into a circle “the deviatoric trace of von Mises and Drucker-Prager”, and if

ψ = rt / rc = 1 / 2 then the deviatoric race becomes nearly triangular.

Rubin has proposed to relate the radius r of any stress point in the deviatoric race to

the radius rc using the scaling function ℜ , Schwer et al. [169]:

r − b1 + b12 − 4b2b0

=ℜ ℜ( J 1 , β ) = (136)

rc 2b2

where:

5 Constitutive models 99

− a1 + a12 − 4a2 a0

b = (2Q1 + a ) 2 − 3 a=

2a2

Q1 = rt / rc Q2 = rr / rc

β = θ −π / 6 −π / 6 ≤ β ≤ π / 6

σ3

Mises-Schleicher

TXE

Drucker-Prager

at the torsion Mohr-Coulomb

meridian

Modified Tresca

distance

at the tensile

meridian rt

rr

− β rc

σ1 + σ2

TXE TXE

TOR

TXC

distance at the

compressive

meridian

Figure 60 The deviatoric section of the yield surfaces, showing the Rubin angel β

5.4 Constitutive models of masonry constituents 100

Another yield criteria also have been proposed for concrete like Podgorski model,

Kotsovos model, multi-parameter unified yield criterion, Yu et al. [205].

Several material models have been implemented into LS-DYNA code which represent a

wide range of Geo-materials, Davidson [47]. The present section is devoted to give

insight into material models which capable to model masonry.

The first thought would be to find convenient material models capable to simulate the

behaviour of masonry. The accuracy of the material model can be augmented by

increasing the number of material parameters, which require enough experimental data.

Soil and foam material model No.5

This material model has been mainly developed for cases of soils, foams, and concrete,

and is based on the formulation suggested by Key (Sawenson [160]). The pressure is

positive in compression and the volumetric strain is negative in compression. The

volumetric strain is given by the natural logarithm of the relative volume

ε v = ln(V / V0 ) (137)

tensile more than the cutoffs value then the pressure reset to the cutoff value.

The yield function is defined as

F ( p, σ e ) = σ e − 3 a0 + a1 p + a2 p 2 (138)

The variation of 3 J 2 as a function of pressure has three conical forms which depend

on the parameter a2 :

(2) parabolic ( a2 = 0 ) or

These three forms are shown in Figure 61. The elliptic yield function curves back toward

the hydrostatic axis at higher pressure and predict a softening behaviour which is not

often observed in test data. The elliptical yield surface therefore is used up to the point of

maximum 3 J 2 then the yield surface is extended as von Mises surface.

There is no strain hardening in this model, so the yield stress can be completely

determined by the pressure.

5 Constitutive models 101

F ( p, J 2 ) = 3J 2 − 3 a0 + a1 p + a2 p 2

∆σ = 3J 2 > 0

a2 > 0 hyperbolic

pressure a2 = 0 parabolic

cutoff

Von Mises extension

σ3 σ2

-a0 a2 < 0 elliptic

2a1

a1

2a2

p

a12 − 4a0a2

2a2

σ1

This material decouples the volumetric and the deviatoric response where an equation of

state is needed to get the current pressure.

The deviatoric stress limit ∆σ is defined as a linear interpolation between two

independent functions, Malvar et al. [115].

∆σ = η ⋅ ∆σ m + (1 − η ) ⋅ ∆σ r (139)

where

p p

∆σ m = a 0 + ∆σ r = (140)

a1 + a2 p a1 f + a2 f p

ai material parameters

η increases from η y = η (0) at the beginning of yielding to η (λm ) = 1 when

reaching the maximum stress difference and then decreases to zero η (λr ) = 0 .

5.4 Constitutive models of masonry constituents 102

εp

dε p

λ= ∫ (1 + p / f )

0 t

b1

(141)

The full yield surface of this material is obtained by rotating the meridian plane around

the hydrostatic axis that is why the sections of the yield surface in the deviatoric planes

are circles.

p

∆σ = 3J 2 > 0 ∆σm = a0 +

Compressive Meridian a1 + a2 p

ft f c′

−

3 σ3 σ2

pressure

cutoff

f c′

− ft

3 p

ft

ψ ⋅ f c′

tensile

cutoff

σ1

Figure 62 Yield surface of material pseudo tensor in LS-DYNA, the drawing based on

Malvar et al. [115]

Material model 16 has been enhanced to get the concrete material 72. The maximum

yield surface is the same but the residual yield surface has been changed to (Malvar et

al. [115]):

p

∆σ r = (143)

a1 f + a2 f p

p

∆σ y = a 0 y + (144)

a1 y + a2 y p

pseudo tensor, but here η increases from η y = η (0) = 0 at the beginning of yielding to

5 Constitutive models 103

η (λm ) = 1 when reaching the maximum stress difference and then decreases to zero

η ( λr ) = 0 .

If 0 ≤ λ < λm (hardening range) then the actual yield surface is given by:

∆σ = η ⋅ (∆σ m − ∆σ y ) + ∆σ y (145)

If λm ≤ λ < λr (softening range) then the actual yield surface is given by:

∆σ = η ⋅ (∆σ m − ∆σ r ) + ∆σ r (146)

⎧ε p dε p

⎪∫ for p ≥ 0

⎪ 0 rf (1 + p / rf f t )

b1

λ = ⎨ε (147)

⎪ dε p

p

⎪ ∫ r (1 + p / r f )b 2 for p < 0

⎩0 f f t

The equations (145) and (146) define the intersection curve of the yield surface and the

meridian plane in compression, or the distance of the compression meridian from the

hydrostatic axis rc . In order to get the distance from the hydrostatic axis to the tensile

meridian rt , the function ψ ( p ) has been introduced where rt = ψ ( p ) ⋅ rc , ψ ( p ) takes

values between 1/2 at tensile negative hydrostatic pressure to 1 at high confinement.

The smooth convex triangular surface that proposed by Willam and Warnke has been

used in this material to relate the yield surface to the third stress invariant J 3 or Lode

angel θ , equation (135).

p

∆σ = 3J 2 > 0 ∆σm = a0 +

Compressive Meridian a1 + a2 p

f c′

pressure σ3 distance at the

σ2

cutoff rc

compressive

meridian

∆σ = 3 ( p + ft )

f c′

− ft

3 p

ft ft

−

3

rt r

ψ ⋅ f c′ distance at the

tensile

meridian

θ

Lode

Angle

P

tensile

cutoff Tensile Meridian

σ1

5.4 Constitutive models of masonry constituents 104

This material model has been used in conjunction with an equation of state that gives

the current pressure as a function of the current and previous volumetric strain.

p = C (ε v ) (148)

The Winfrith concrete model is a smeared crack model (some time known as pseudo

crack model) implemented in eight nodded single integration point continuum element.

The yield surface for this material is the Ottosen yield surface that described above.

The tensile yielding occurs in this model when the maximum principle stress at yielding

reach half of the current tensile strength, after which a special post-yield treatment is

invoked that decays the cracks-normal tensile strength as the crack develops.

The advantage of this material model over the other models is that, it allows a graphical

representation of the cracks within the finite element in Post-processors of LS-DYNA

(LS-TAURUS or LS-PrePost [203]). Up to three orthogonal cracks can be displayed in

any element. If the yield is indicated in triaxial compression, the concrete deemed to be

crushed, and three instantaneous (closed) cracks can be generated, so that, the material

becomes without tensile capacity on unloading, Broadhouse [26] and Broadhouse [27].

Cap material models No.25, No.145 and No.159

The following material cap models are available in LS-DYNA: Geological cap model

No.25, Schwer Murray cap model No.145, and continuous surface cap model CSCM

Material No.159.

All these material models share similar principles of using the cap model of Sandler

[159], Figure 64 .

J2

J 2 = FF ( J1 )

Failure envelope

J 2 = Fc ( J1 , κ)

cap envelope

tensile

cutoff

− L(κ ) − X (κ) − J1

The recent developed continuous surface cap model will be described in the following.

The yield surface of this material is formulated in terms of the three stress invariants

F ( J1 , J 2 , J 3 , κ ) = J 2 − ℜ 2 F f2 Fc (149)

5 Constitutive models 105

F f ( J1 ) = α − γ ⋅ e −η ⋅J1 + ϕ ⋅ J1 (150)

The function Fc represents the hardening cap and given by: (Sandler [159], Schwer et

al. [169])

⎧1 J1 ≤ L(κ )

⎪

Fc ( J1 , κ ) = ⎨ ⎡ J1 − L(κ ) ⎤ 2 (151)

⎪1 − ⎢ ⎥ J1 > L(κ )

⎩ ⎣ X (κ ) − L(κ ) ⎦

where

this represents the intersection point of the cap and the axis of J1 . The intersection

depends on the cap ellipticity ratio R and

⎧κ 0 κ ≤ κ0

L(κ ) = ⎨ (153)

⎩κ κ > κ0

The expansion and contraction of the cap is based on the hardening rule

2

ε vp = W (1 − e − D ( X − X

1 0 ) − D2 ( X − X 0 )

) (154)

where

The mathematical formulation of the theory of plasticity has been presented in this

chapter. The treatment of non-smooth yield surfaces and the implementation of a

material model into the explicit solver of LS-DYNA have been described as well.

5.5 Concluding remarks 106

The adoption of non-smooth yield surface needs further treatments for corners which

increase the complexity of implementation, and increase the computation time. The

complexity is going to be worse especially for explicit solvers like LS-DYNA, where the

material subroutine has to be called in time steps smaller than those in implicit solvers.

Various constitutive models with different aspects are available in literature to represent

the interfaces of masonry. Some have been developed for contact formulation and

others for interface elements. The drawbacks of those models have been presented and

discussed in section 5.2. However, the available research works have been based on

non-smooth yield surfaces.

A smooth yield surface is proposed for cohesive material model to be used with interface

elements. The proposed model is multi yield surface but does not need any further

treatment of the transition points. It reduces the computation time and avoids the

treatments of corners. The proposed model has been implemented into the explicit

solver of LS-DYNA.

The fragmentation of one masonry unit under impact has been simulated using the

developed interface model. For the large displacements after the failure of the interface

elements a frictional contact has been adopted.

One drawback of the fragmentation using interface elements is the limitation of the

interface elements for small displacements, where a termination of the solution can be

aroused during the calculation. The erosion of the failed element is one possible solution

but the available contact formulations in LS-DYNA are not well developed in this

direction. The models of the interface elements needs a special pre-processor tool to

generate the necessary input data for the discretization and insertion of the interface

elements and this is also absent in LS-DYNA pre-processors. The other drawback is

that, even if the simulated structure is small, the resultant model is very big and will be

time consuming.

Although interface elements give good representation up to the initial failure, but the

adoption of contact formulation is more suitable for post failure behaviour. However, the

lack and absent of implementing a contact model into LS-DYNA is the reason to develop

in direction of interface elements.

LS-DYNA comprises material models that cover a wide range of masonry constituents,

but such materials have been developed basically for concrete and soils. They represent

the general behaviour of many geo-materials. However, the general triaxial empirical

laws of many masonry materials are still lacking in literature and further investigations

are required.

6 Application of mesh free methods 107

The Lagrangian mesh based numerical methods like FEM suffer a lot of difficulties when

are applied to simulate the fracture and fragmentation of material under high dynamic

loading. The combination of finite element with discrete element method brings out a

great enhancement. However, it is still based on mesh connectivity which shows

difficulties at the level of one discrete element.

Contrary to the Lagrangian mesh, the Eulerian mesh is fixed on the space and by time

the materials are flowing across the mesh. Therefore, large deformations in the material

do not cause any deformations in the mesh. By this way the numerical problems in

Lagrangian mesh based methods can be avoided at this point. Nevertheless, Eulerian

methods dominate the area of computational fluid dynamic, and the application of this

method for irregular geometries brings up a lot of difficulties.

The Lagrangian and Eulerian approaches have been combined to overcome the

limitations of each one and to produce more robust numerical approaches, like CEL

coupled Eulerian Lagrangian (CEL), Arbitrary Lagrange Eulerian (ALE), (Liu et al. [103]

and [105]). Those approaches are developed basically to solve problems of solid fluid

interaction.

Despite the great success of mesh based methods, the prerequisite of mesh is the main

reason for the limitation of those methods and at the same time it is the key success.

Several research efforts during the last years are focused to develop mesh independent

methods, which have been driven to mesh free methods. The key idea of mesh free

methods is to represent the domain of the problem using set of nodes or particles

without considering any connectivity in between.

Smoothed particle hydrodynamics17 (SPH) is mesh free Lagrangian method. It has been

originally invented since 1977 by Lucy, Gingold and Monaghan (Liu et al. [104]) for

modelling astrophysical phenomena and later has been extended for the application of

solid and fluid mechanics.

The most attractive features of SPH are the simplicity and adaptability to handle large

deformations without regarding the distribution of particles.

The formulation of SPH is based on two approximations, kernel approximation and

particle approximation.

Let us consider the following identity which gives an integral representation of a function

17

The first term refers to the smoothed approximation by using weighted average over the neighboring

particles for stability, and the third term hydrodynamics refers to the role of the method for hydrodynamic

problems, Liu et al. [104].

6.1 The basic approximations of SPH 108

Ω

where:

⎧1 if x − x′

δ (x − x′) = ⎨ (156)

⎩0 otherwise

integral representation of f (x) is then given by

Ω

function, h is the smoothing length. In SPH convention, the kernel approximation

operator is marked by the angle bracket (e.g., Liu et al. [103], [104], and [105]),

therefore the equation (157) can be rewritten as

Ω

Various options are possible for choosing the kernel function W (x − x′, h) . The

requirements which must be placed on the kernel function are, Liu et al. [104]:

(1) Normalization or unity condition

∫ W (x − x′, h) dΩ = 1

Ω

(159)

h →0

κ is a constant associated with the smoothing function for the point at x , and it

identifies the non-zero area of the smoothing function.

6 Application of mesh free methods 109

Monaghan and Lattanzio (Liu et al. [104]) have developed the following smoothing

function based on the cubic spline functions which well known as B-spline function,

Figure 65-a:

⎧ 23 − R 2 + 12 R 3 0 ≤ R <1

⎪⎪

W ( R, h) = α d × ⎨ 16 (2 − R ) 3 1≤ R < 2 (162)

⎪0 R≥2

⎪⎩

α d is the constant of normalization and equals to 1 / h , 15 / 7πh 2 , 3 / 2πh 3 for one, two,

and three dimensional domains, respectively.

Another important property in the kernel approximation of the spatial derivatives of a

function ∇f (x) can be described in the following:

For the derivative of a function

Ω

The divergence in the integral is taken with respect to the primed coordinate

Ω Ω

By applying the divergence theorem on the first term of the right side of the equation

(165), it gives

v

∫ ∇ ⋅ [ f (x′) ⋅ W (x − x′, h)] dΩ = ∫ f (x′) ⋅ W (x − x′, h) n ⋅ dΓ

Ω Γ

(166)

v

where n is the unit vector normal to the surface Γ of the domain Ω .

Since the kernel function has compact support, therefore the surface integral is equal to

zero when the support domain is located within the problem domain, but if the support

domain overlaps with the boundary then the surface integral will be no longer zero. For

the points which their support domain inside the problem domain the following equation

is obtained

Ω

6.1 The basic approximations of SPH 110

1.00

0.75

0.50

×αd

j R κ

⋅h

i

Function

0.25

0.00 i

support domain

-0.25 of particle i

-0.50

-0.75

Ω

smoothing function

derivative of the smoothing function

R

-1.00

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

If the entire problem domain is represented by a finite number of particles that have

individual mass and occupy individual space, the continuous volume integral in equation

(158) can be converted to the sum over discrete interpolation particles

N

f (x) = ∑ f (x j ) ⋅ W (xi − x j , h) ⋅ ∆Ω j (168)

j =1

where

∆Ω j volume of particle j

Equation (168) can be written with respect to mass m j and density ρ j of particle j

N mj

f ( x) = ∑ f ( x j ) ⋅ W ( x − x j , h) (169)

j =1 ρj

N mj

∇ ⋅ f (xi ) = ∑ f (x j ) ⋅ Wij (170)

j =1 ρj

where

6 Application of mesh free methods 111

Wij = W (x i − x j , h) (171)

The particle approximation for the spatial derivative of the function can be handled in

similar manner

N mj

∇ ⋅ f (x i ) = −∑ f (x j ) ⋅ ∇Wij (172)

j =1 ρj

Conservation equations which govern the solid mechanics can be expressed as

following, Limido et al. [101]:

dρ ∂v β dρ i N m ∂Wij

= ρi ∑ (v βj − viβ ) β

j

= −ρ β (173)

dt ∂x dt j =1 ρ j ∂xi

αβ

dvα 1 ∂σ αβ dviα N

σ αβ σ j ∂Wij

= = ∑ mj ( i 2 − 2 ) β (174)

dt ρ ∂x β dt j =1 ρi ρ j ∂xi

3) Conservation of energy

dt

=

ρ ∂x β dt

= − i2

ρi

∑ m (vα − vα ) ∂x β

j =1

j j i (175)

i

The first research works of SPH have been considered the smoothing length constant

during the simulation. However, the smoothing length may change dramatically from the

initial configuration in case of large deformation. Therefore, in recent works each particle

supposed to has its smoothing length, Lacome [95].

The symmetry and thus conservation of momentum is maintained if the smoothing

length taken to be the mean value between particles i and j, hij = 12 (hi + h j ) . This

approximation might lead to unstable results, Swegle et al. [178]. LS-DYNA uses what is

called gather formulation scheme by defining hij = hi , i.e. the neighbour particles of a

given particle are the particles included in a sphere centred in x i with radius of hi ,

Lacome [95].

6.3 SPH modelling of masonry in LS-DYNA 112

Modelling using SPH needs the initial distribution of particles over the problem domain.

The generation of SPH particles can be easily obtained by employing the same mesh

generation algorithms for finite elements. The preparation of SPH model for LS-DYNA

calculation can be preformed in LS-PrePost [203] by generating the initial configuration

of SPH particles. The initial configurations are the position and the mass for each

particle. The distribution of SPH particles needs to be regular to guarantee the stability of

calculation.

The calculation cycle for SPH in LS-YNA is similar to that for finite element except the

steps of kernel approximation. Kernel approximations are used to calculate the forces

from spatial derivative of stresses. The spatial derivatives of velocity are required to

calculate the strain rates, Lacome [95].

Figure 66 shows 2D SPH modelling for masonry shear wall. Masonry units as well as

mortar have been represented using SPH particles spaced each 1 cm. The whole model

is composed of 5625 SPH particles

The initial SPH density for each particle is considered to be equally distributed over unit

or over mortar. Few geo-materials in LS-DYNA support SPH modelling. Concrete model

No.72 that described in 5.4.2 is employed to model unit and mortar material.

Vertical and horizontal displacements have been applied increasingly on the model, to

simulate the shear failure of the wall.

Horizontal displacement

2D SPH model for Masonry shear wall Plastic strain output results of SPH analysis

SPH analysis of the masonry shear wall has been showed the ability of this method to

represent all failure modes, even the fragmentation due to compressive crushing which

difficult to handle in finite element analysis.

6 Application of mesh free methods 113

The Lagrangian finite element method and other numerical methods which request a

mesh priori need further treatments for collapse simulation. The prerequisite of the mesh

is the major reason for numerical difficulties.

The combination of the finite element method with the discrete element method

overcomes some drawbacks of both methods, but this also need predefinition of

discretization planes. For masonry, in most cases, mortar joints represent the weakest

planes, but the need to represent the cracking inside the unit or even the fragmentation

is of high importance. The numerical techniques which deal with such problems have

discussed in section 4.5.

The other numerical methods based on Eulerian mesh are suitable for high deformable

materials, but also suffer a lot of problems for modelling the complicated geometries like

masonry structures. For these reasons, mesh free methods have been adopted as

alternative solution of the problem. Although The SPH method has been used widely in

literature for many engineering applications, the application on masonry is still lacking.

The obtained results after the simulation of masonry shear wall have been proofed the

ability to represent all failure modes, even the crushing under high compression and

fragmentation of the material without numerical problems. Furthermore, the

implementation of SPH method is simple and does not need elaborate numerical

techniques for handling the tensile cracking of material.

One drawback of this method is the need for large numbers of particles for masonry

simulation, even if the model is small the computation time is relatively big, and the

accuracy is less than that in finite element method.

7 Full scale dynamic testing and computer modelling 115

There is an ever-increasing demand for advanced and more controllable experiments to

support studies for the structure performance under earthquake loading. The studies on

full scale masonry structures are still challenging, even in testing techniques or in

numerical modelling. The intricacy is increasing, when loading the structure by random

earthquake action.

The experimental tests which are able to reflect the actual behaviour of the physical

problem in reality can not be always achieved, due to technical problems and high costs.

In case of earthquakes, the running of the test under strong motion is going to devastate

the instruments, after which the measuring process will be no longer available.

Furthermore, the preparation of many full scale structures to be tested under different

loading conditions is prohibitive. To avoid that, the same specimen is normally used for

different loading conditions, i.e. the structure will get severe damage at the final stage of

loading.

The numerical analysis of masonry structures has been received a great advance in last

decades, and it has been aroused as an alternative research discipline. The numerical

analysis can overcome a lot of difficulties in experimental tests, and it allows more

flexibility to get out the required results. However, it is important to know the capabilities

and limitations of the numerical model to be used for predicting the real physical

behaviour.

This chapter is devoted for studying the seismic behaviour of masonry construction by

means of numerical analysis. A number of experimental tests were carried out within the

scope of the ESECMaSE project, Caballero González et al. [32] for evaluating the

earthquake performance on specific masonry construction. Mainly, two full scale tests

will be considered for the validation and calibration of the numerical models. The first

one is a shaking table test18 that carried out at the National Technical University of

Athens, Carydis et al. [36] and the second one is a pseudo dynamic test19 that carried

out at the Joint Research Centre (JRC) of the European Commission in Ispra, Anthoine

[9]. Both specimens have been built by calcium silicate units.

The main concern in this chapter is to demonstrate the capability of the numerical model

to capture the behaviours observed in the experiments and then to be employed to

investigate further problems.

The comprehensive investigation by means of the numerical analysis will focus on the

following problems:

- to apply strong earthquake actions on the model up to collapse which are

impossible to carry out in laboratory conditions

- to examine the effect of vertical acceleration on the total behaviour of the

structure

- to apply the earthquake action on the undamaged structure to avoid the effect of

the cumulative damage during the testing procedure, whereas the stronger

earthquake actions in the tests have been applied on pre-damaged structure.

18 th

The 7 work package “Static and Dynamic Shear Tests on Structural Members”

19 th

The 8 work package “Large Scale Earthquake Test on Building”

7.1 Dynamic testing methods 116

The decision about the appropriate dynamic testing method to examine the earthquake

performance on structures is of high importance, due to high cost of such tests.

Nowadays, dynamic testing methods are ranging from real dynamic loading tests to

pseudo dynamic tests in real time or in enlarged time, and according to the capacity of

the testing facility they range from tests on full scale large specimens to scaled or small

specimens and from complete structure to sub-structure, Reinhorn et al. [151].

In the shaking table test, the dynamic motion of the table is generated by the computer

in real time and imposed on the actuators of the shaking table. In this way the real

performance of the specimen under actual earthquake acceleration records can be

simulated and the inertial effects and structure assembly issues are well signified. The

size of the tested speicemen is limited to the capacity of the shaking table. The size of

the structure can be scaled down to adjust with the available capacity or the test can be

performed on a part of the structure, Figure 68.

In pseudo-dynamic test the dynamic

effect which results in form of inertial

forces are computed by the computer Input exitation f i

and imposed on the structure by means

of the quasi-static loads. −1

⎛ ∆t ⎞ ⎛ ∆t ⎞

The response of the structure under ai +1 = ⎜ m + ⋅ c ⎟ ⎜ f i +1 − ri +1 − c ⋅ vi c ⋅ ai ⎟

⎝ 2 ⎠ ⎝ 2 ⎠

earthquake motion can be calculated

numerically as shown in Figure 67. The

∆t

computed displacements can be vi +1 = vi + (ai + ai +1 )

2

imposed to a real system or to

convenient structural component by

means of hydraulic actuators. Therefore ∆t 2

d i + 2 = d i +1 + ∆t ⋅ vi +1 + ai +1

the restoring forces which depend on 2

the current stiffness of the structure are

properly applied to the specimen, Figure

Imposed d i +1 on the structure

69.

This technique was originally developed

in Japan and, soon afterwards, in United Increase i

States, Nappi [136]. Facilities for

pseudo-dynamic tests were also set up

in Europe, Geradin et al. [63], where the

largest facility in Ispra, Italy has been Figure 67 The flow diagram of Pseudo-

used for the big pseudo dynamic tests dynamic testing method

of ESECMaSE project.

7 Full scale dynamic testing and computer modelling 117

Actuators

Accelerogram

Shaking table

Imposed

displacements

Servo-

Hydraulic Force Displacement

actuators transducers transducers Numerical Model

frame

Accelerogram

Reaction Wall

Imposed

displacements

x(t )

Measured restoring forces R(t )

Computational Model

Actuator

Computational

substructure

Response feed back x(t )

Reaction Wall

Physical

substructure

Computational

substructure

Shaking table

7.2 Experimental tests 118

The pseudo-dynamic test should be handled with care in case of masonry structures,

due to sensitivity of masonry for load rate effect, since an experimental record of a few

seconds may takes hours or even days to be processed.

The main feature of pseudo-dynamic test is the applicability for large scale full building. It

is possible to use this test only for substructure if the other structure parts are well

understood.

This testing method is based on the combination of shaking table with substructure

techniques. Only part of the structure (the physical model) needs to be constructed and

tested on the shaking table. The rest part of the structure (the numerical model) is

numerically modelled by computer, Figure 70.

The earthquake effect on the physical part can be calculated by computer. The

existence of other nonphysical parts has to be considered. The calculated effect can be

applied to the tested physical part by the actuators (force control based). The size of the

specimen can be large or very large, Reinhorn et al. [151].

A number of full scale masonry specimens, with

different masonry units have been tested within

the scope of the ESECMaSE project using the

shaking table facility of the NTU Athens20. In

the current study the optimized calcium silicate

specimen (test A1) is only considered, which

was tested on October, 2nd 2006.

The specimen is two story building, consists of

T-shaped part and a single wall in the opposite

side, Figure 71. The reinforced concrete slabs

of each story have been prefabricated and have

thickness of 12 cm. The specimen has been

placed on the shaking table using a steel base,

and it has been connected to the shaking table

using 36 bolts M30, Carydis et al. [36].

Additional masses have been added to each

story, 3.50 tons, and 4.0 tons on the first and

second level, respectively, Figure 72.

Figure 71 The tested specimen A,

Carydis et al. [36]

20

National University of Athens, School of Civil Engineering, Laboratory for Earthquake Engineering,

Athens-Greece.

7 Full scale dynamic testing and computer modelling 119

4.0 tons

Masonry Unit

Calcuim Silicate 6DOF

5x

0.625 + m2=

0.375

7.65 tons

-----------

3.5 tons

m1=

9.12 tons

Idealized two

mass system

for the structure

Base Excitation

The base of the specimen has been excited unidirectionally using an artificial

accelerogram. The artificial accelerogram has been generated by the team of NTU

Athens to match EC8 design spectrum. The response spectrum (type I, with ground

acceleration 4%g and soil category: B) has been used to generate the artificial

accelerogram, Figure 73.

In order to adjust with the available displacement capacity of the shaking table, the

artificial accelerogram has been filtered with high pass filter 1 Hz. The resulting

accelerogram is shown in Figure 74.

The artificial accelerogram has been integrated twice to get the displacement excitation

that can be applied on the shaking table, Figure 75.

7.2 Experimental tests 120

1.40

Acceleration (m/sec )

2

1.00

0.80

0.60

0.40

0.20

Period (sec)

Figure 73 Elastic response spectrum which used to generate the artificial accelerogram

0.6

Acceleration (m/sec )

2

0.4

0.2

-0.2

-0.4

-0.6

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (sec)

7 Full scale dynamic testing and computer modelling 121

6

Displacements (m)

-2

-4

-6

-8

-10

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (sec)

The first test was carried out on the specimen for determining its dynamic

characteristics, whereas random signals from DC to 50 Hz were applied prior to

earthquake tests in order to determine the dynamic characteristics of the specimen.

Several tests were subsequently performed along in-plane direction (X- direction) with

step-wise increasing to the base acceleration up to the collapse of the specimen, started

with 4%g until 18%g (2%g increment), Carydis et al. [36].

The dynamic characteristics of the specimen

which obtained from the first test are shown in frequency 3.71 Hz

Table 4.

period 0.27 sec

Subsequently, the tests from 2 to 9 which

correspond to acceleration intensities from 4%g to damping ratio 4.37 %

18% g were performed.

During the test with 14%g, diagonal cracks were Table 4 Dynamic

nd

formed at the shear wall of the 2 level. In the characteristics of the

next test 16%g, these cracks were enlarged and specimen A1.

diagonal cracks emerged in the same wall at the

1st level. Cracks also were occurred on the transversal wall of T-part in the 1st and 2nd

levels. The slabs were opened and moved permanently, especially in the 2nd level, which

were not tightly connected with the walls, Figure 76. The interface between the

transversal and the shear wall of T-shaped part was separated.

Permanent out of plane displacement of transversal walls of both stories of T-shape part

was occurred. Out of plane movement of the transversal walls was also observed in the

upper part of 2nd level, Carydis et al. [36].

7.2 Experimental tests 122

shaped part at 2nd level. level.

moved permanently,

especially in the 2nd level.

Permanent out of plane

the transversal and shear

displacement of transversal

wall of T-shaped part was

walls of both stories of T-

separated.

shape part was occurred.

shaped part at 1st level. level.

Figure 76 Experimental results for the model of Athens, the damage state after applying

16%g earthquake intensity, the photos provided by Carydis et al. [36]

7 Full scale dynamic testing and computer modelling 123

The maximum and minimum values for accelerations, displacements and the base shear

which were achieved in the shaking table test are shown in Table 5.

intensity 1st level 2nd level 1st level 2nd level (KN)

min -0.89 -1.84 -3.5 -8.1 -19.59

08%g

max 1.25 2.34 3.5 6.2 27.67

min -1.44 -2.17 -6.0 -16.2 -22.50

12%g

max 1.77 2.92 4.7 8.1 30.50

min -2.02 -2.14 -7.8 -26.6 -23.56

14%g

max 2.72 3.32 8.1 18.2 38.51

min -2.08 -2.16 -13.5 -35.2 -26.08

16%g

max 2.77 3.62 11.9 26.6 37.44

base shear for specimens A1, Carydis et al. [36]

Two big pseudo dynamic tests

were carried out within the scope

of the ESECMaSE project by

means of the ELSA-JRC21

reaction wall in Ispra. The test

specimens have been chosen to

simulate a real full scale two

storeys terraced house building,

Figure 77. In the first one,

masonry is made from calcium

silicate units and in the other one

from clay bricks. The present (a) (b)

study focuses only on the test of

calcium silicate specimen.

Figure 77 The Plane and the section of the tested

The following assumptions have terraced house, Fehling et al. [56]

been considered in the test,

Anthoine [9]:

1. The rigidity of cellar, thus, two storey structure has been considered

2. The mass of roof and its elements have been considered in the second floor as

additional masses

3. Only the walls around the staircase and corners have been considered.

The geometry of the structure is shown in Figure 78. By considering the last

conceptions, and due to quasi-symmetry of the structure, one half was tested, Figure 79.

21

ELSA is the European Laboratory for Structural Assessment in the Joint research centre JRC of European

commission in Ispra-Italy.

7.2 Experimental tests 124

Calcuim Silicate 6DOF

m2=

25.4 tons

m1=

28.2 tons

system for half of

the structure

Earthquake

direction

under the walls

The connection

between the shear

wall and transversal

wall

Figure 79 Test configuration, and the connection of structure elements, Anthoine [9]

7 Full scale dynamic testing and computer modelling 125

Masonry walls were built from calcium silicate units of type 6DOF (LxWxH =

250x175x250) which were optimised for the project. The units were assembled with thin

layer mortar bed joints, whereas the head joints were unfilled and simply juxtaposed.

The connections between the structure elements (slabs and walls) were carried out as

following:

- Under the walls in each floor thicker layer mortars (cement mortar Z01) were

used

- The concrete slabs were poured directly on the top of the walls without any

mortar joint

- The shear walls were connected to the perpendicular walls through a continuous

vertical mortar joint with masonry connectors (metal strips) inserted at the level of

the bed joints (Z01), Figure 79

The structure has been loaded by additional masses equivalent to 2.05 KN/m2 for each

floor (include floor pavement, bottom covering and lightweight separating plates) and 1.2

KN/m2 on the 2nd floor has been applied on the perimeter of the slab (include roof and

roof supporting structure which are missed in the tested specimen), Figure 80. The

weight of hydraulic actuators, their attachments to the slabs and the additional steel

reinforcement in the slab, and safety supporting frames also have been accounted. The

additional masses are provided to the specimen by water tanks which are distributed

over the concrete slabs in each floor, Anthoine [9]. The imposed base acceleration is the

same artificial accelerogram which used for the last test.

The first tests were performed for determining the system dynamic parameters of the

model including natural frequencies, damping ratios and vibration mode shapes. The

7.2 Experimental tests 126

different positions.

intensity 1%g to check the functionality of the frequency 6 Hz

whole installation, getting information about the

elastic properties of the structure and also to period 0.167 sec

determine the proper control parameters like

testing speed. Subsequently, several tests then damping ratio 2%

were performed along in-plane direction (X-

direction) with step-wise increasing to the base Table 6 Dynamic

acceleration up to the pre-collapse of the characteristics of CS

specimen, started with 2%g until 20%g (2%g specimen

increment), Anthoine [9].

Acc Maximum

7.2.2.2 Test results Test no

intensity dis.

The frequency which resulted from hummer tests

was 6Hz and the damping ratio was 2%. K08 04%g 1.6 mm

For the tests 2%g and 4%g the structure was K09 06%g 2.6 mm

showed no damage and the maximum

displacement measured at the top was 1.6 mm. K10 08%g 5.1 mm

The responses due 2%g and 4%g were not

proportional. The nonlinearity is properly due to K11 10%g 7.3 mm

the rocking behaviour of the slender shear walls.

During the tests 6%g and 8%g and in the first two K12 12%g 17.1 mm

large cycles something were heard, but no cracks

were recognized, the maximum measured K13 14%g 26.0 mm

displacements at the top 2.6 mm and 5.1mm for

6%g and 8%g, respectively. During the tests 12%g K14 16%g 26.0 mm

and 14%g, large stepwise cracks were formed in

the shear walls at the 1st floor, as well as, out of K15 18%g 47.0 mm

plane cracks also were noticed in the long lateral

K16 20%g 67.0 mm

walls, Figure 81.

It is worth mentioning that the observed damage in Table 7 Max. displacements

the long lateral walls would probably cause a reached at the 2nd

partial or total failure of these walls under real level during the tests

dynamic loading. Further test were carried out to

assess the remaining capacity of the shear walls.

The test 16%g was caused no additional damages. The displacements were more or

less than in the previous test (26mm), thus, the test 18%g was carried out. This test was

caused many additional damages to the structure (new cracks and further opening of

existing cracks) and the displacements at the top were reached 47mm. Some of the

LVDT’s also were damaged, Figure 82. Nevertheless, a further test 20%g was carried

out. The maximum top displacement was 67mm. This test was very damaging to the

structure, Figure 83. Many of the LVDT’s were reached their saturation. Therefore, a

decision was made to stop testing at this stage because the structure is not safe any

more, and further test would not provide much additional information, other than could

cause a partial/total collapse which might damage the equipments.

7 Full scale dynamic testing and computer modelling 127

Figure 81 Pseudo dynamic test results- The first visible cracks after testing with 14%g,

photos provided by ELSA

7.2 Experimental tests 128

Figure 82 Pseudo dynamic test results- The visible cracks after testing with 18%g,

photos provided by ELSA

7 Full scale dynamic testing and computer modelling 129

Figure 83 Pseudo dynamic test results- The visible cracks after testing with 20%g,

photos provided by ELSA

7.3 Numerical analysis 130

Both tests (shaking table and pseudo dynamic) have been simulated by means of

numerical analysis. The intension is to get real model as far as possible. For that reason

a big size and elaborated models have been built using LS-DYNA software in order to

achieve accurate results.

The shaking table test has been performed in real time of the applied earthquake, which

is 10 seconds in our test. The calculation of the pseudo dynamic test through the testing

time, which was some hours, is highly consuming for explicit solvers. Therefore, the

calculations were preformed in the real time of the earthquake for both tests. The

simplifications in numerical part of the pseudo dynamic test were not considered in this

context, and this could gives explanation about the validity of both models.

The shaking table test was simulated for the whole testing time and further simulations

were carried out until reaching the collapse. For the pseudo dynamic test, the first test

was used to calibrate the dynamic characteristic of the structure. Further calculations

were performed on undamaged structure to examine the performance under different

acceleration intensities, and to investigate about the influence of vertical accelerations

as well.

Simplified micro modelling strategy has been adopted for masonry walls due to thin layer

mortar. The modelling concept is based on the creation of discrete parts of the structure

(concrete slabs and units) and then assembling them using the appropriate contact

models.

Each unit is meshed by 3x3x3 brick elements. Constant stress eight-nodded brick

element with single integration point (reduced integration RI) has been employed to build

the models. This element is the default option in LS-DYNA and it is computationally fast.

Figure 84 shows the meshed geometry of each specimen.

Single-point integration element has a shortcoming of zero energy modes which are

termed hourglassing modes. Undesirable hourglass modes tend to have periods

typically much shorter than the periods of the structural response and they are often

observed to be oscillatory. One way of resisting undesirable hourglassing is with viscous

damping or small elastic stiffness able to stop the formulation of the anomalous modes

with negligible affect on the global modes, Hallquist [75].

In order to reduce the effect of hourglassing, LS-DYNA provides several hourglass

control types. In the present study Flangan-Belytschko stiffness form has been

employed, Hallquist [75]. This type of hourglass control has an advantage to reduce the

probability of negative volumes occurrence during calculation. Negative volumes are

highly coming up when materials undergo large deformations near failure. The

occurrence of negative volume causes the calculation to be terminated.

7 Full scale dynamic testing and computer modelling 131

34173 Nodes 76672 Nodes

Figure 84 Finite element mesh for the model of Athens and the model of Ispra

The material of calcium silicate units CS20-

Modulus of elasticity 9090 Mpa

1.8–249x175x258 has been examined

experimentally in Brameshuber et al. [25]. Compression strength 18.6 Mpa

The material properties of Calcium silicate

units are given in Table 8. Tensile strength 2.12 Mpa

DYNA for modelling the behaviour of such

material. In the present study, Winfrith Fracture energy 60 N/mm

concrete material model has been adopted

for calcium silicate units, section 5.4.2. Table 8 Calcium-silicate properties

Thin layer mortar has been used to

assemble calcium silicate units. The

mechanical properties of masonry joints

Tensile cohesion 0.30 Mpa

have been investigated experimentally in

Brameshuber et al. [25], Table 9. LS-DYNA Fracture energy 2.5 N/mm

tiebreak contact is employed for modelling

masonry joints. Tiebreak contact allows Initial shear adhesion c 0.24 Mpa

modelling of connections which transmit

both compressive and tensile forces with Sliding friction coefficient 0.7

optional failure criteria. The separation of

the slave node from the master is resisted Shear friction coefficient 0.55

by a contact spring for both tensile and

compressive forces until failure, after which Table 9 Masonry joints properties

the tensile coupling is removed, Bala [13].

7.3 Numerical analysis 132

The option 6 of contact tiebreak is used in the present study which permits damage by

scaling the stress components after failure, section 5.2.

It is essential, before imposing the earthquake action on the model, to initialize the

stresses and deformation state in the structure which can be developed from gravity

loads. The application of gravity loads immediately together with earthquake loads

causes further unwanted vertical vibrations to the structure at the beginning of

simulation. Therefore, in order to eliminate the dynamic effect of gravity loads, it must be

applied (statically) through enough time. In our case, the gravity acceleration has been

increased linearly from 0 to 9.81 m/sec2 through 5 seconds. The calculation time that

needed for initialization is considerable for big models like the model of Ispra. An

alterative method therefore is used which is well known as dynamic relaxation. This

method is based on damping the structure to reach the relaxation state (zero velocities)

in short time. The available method in LS-DYNA follows the work of Underwood and

Papadrakakis, (Hallquist [75]). The deformation state of the structure for both models

after initialization with gravity loads are shown in Figure 85.

Figure 85 Deformation state of walls under gravity loads, the displacements are scaled

500 times

Due to the big number of elements (13515 elements for the model of Athens and 34173

elements for the model of Ispra), the calculation using single core processor will be

highly consuming. Therefore, parallel processing has been adopted.

The calculation was performed using 10 parallel Intel Itanium processors “SGI Altix

4700” in the centre of High Performance Computing of TU Dresden.

The calculation for the model of Athens has been carried out along 56 hours for 120

seconds of loading with initial time step 7.42E-06, and one calculation for the model of

7 Full scale dynamic testing and computer modelling 133

Ispra has been performed through 36 hours for 16 seconds of loading with initial time

step 4.72E-06 second.

Max dis.1 Max dis.2

Athens has been calculated through the whole Acc intensity

testing procedure described in section 7.2.1.2 to (mm) (mm)

capture the affect of damage for each successive

test. The measured period of the structure at the 04%g 0.7 1.8

beginning of calculation was 0.23 sec. The

06%g 1.1 2.8

structure was behaved well and showed no

damages until reaching 10%g, where the first

08%g 1.7 4.1

cracks have been emerged at the connection

between slabs and walls, and started to grow up. 10%g 5.7 12.2

Out of plane cracking, opening of slabs and

rocking of the shear wall were the main observed 12%g 10.8 18.6

behaviours in this stage.

For 12%g, the shear wall at the 1st level has been 14%g 12.2 20.4

established a sudden damage. The bonding

between the shear wall and the transversal wall 16%g 13.6 24.3

was vanished and the bonding with ground has

18%g 23.9 34.2

been lost as a result of rocking of the shear wall,

as well. 20%g 33.9 large

For 18%, diagonal cracks in the shear wall of the

1st level, as well as out of plane cracks in the Table 10 Max. displacements

transversal walls have been observed, Figure 86, attained in numerical

Figure 87 and Figure 88. analysis

Owing to the observed behaviours, a comprehensive study has been carried out for the

normal forces in each wall during the earthquake. The vertical reactions under each wall

are not steady due to rocking behaviour, but they change by time and vary around the

relaxation value. The sum of vertical reactions is roughly equal to the weight of the

structure. These little variations during earthquake are conceivably due to dynamic effect

of rocking and debonding between units. As a result of increasing the earthquake

intensity, tensile forces have been aroused due to the variation in vertical reactions.

During the earthquake process, these tensile forces have been leaded to lose the

connection between walls and slabs, or between units along bed joints. Figure 89 shows

the time at which the tensile bonding strength in the wall W2 has been completely

vanished. Following the tensile failure in wall w2, the force transmitting has been

changed in the whole structure, where W3 assists the weak wall w2 to transmit the

tensile vertical forces. This resulted in a sudden variation in vertical reaction forces of

wall w3, Figure 89. Following that, the reaction value of wall w3 has been attained lower

values. This behaviour reduces the capacity of the shear wall w3 which is highly

dependent on the compressive forces. Furthermore, it will be responsible to initiate the

cracks in the shear wall around 12%g. After the tensile failure, the variation in vertical

reaction forces has been reduced, and a great amount of energy has been released.

7.4 Numerical results for the model of Athens 134

Displacements (mm)

-50

-40

-30

-20

-10

10

20

30

40

50

0

0

5

4%g

10

15

6%g

20

25

30

8%g

35

40

10%g

45

50

55

12%g

Tim e (sec)

60

and rotation of the shear wall were the main

65

14%g

70

75

16%g

80

85

90

18%g

95

100

105

20%g

110

115

120

7 Full scale dynamic testing and computer modelling 135

Displacements (mm)

-50

-40

-30

-20

-10

10

20

30

40

50

0

0

5

4%g

10

15

6%g

20

25

30

8%g

35

40

10%g

45

50

55

12%g

Tim e (sec)

60

structure was received a

65

14%g

70

75

16%g

80

85

90

18%g

95

100

105

20%g

110

115

120

7.4 Numerical results for the model of Athens 136

Displacements (mm)

-50

-40

-30

-20

-10

10

20

30

40

50

0

0

5

4%g

10

15

6%g

20

25

30

8%g

35

40

10%g

45

50

55

12%g

Tim e (sec)

60

65

14%g

70

75

16%g

80

85

90

18%g

95

100

105

20%g

110

115

final accelerogram 20%g

120

7 Full scale dynamic testing and computer modelling 137

50.0

40.0 W1

20.0 W3

10.0 Structure

0.0

-10.0

-20.0

Vertical Reaction (KN)

-30.0

-40.0

-50.0

-60.0

-70.0

-80.0

-90.0

-100.0

-110.0

-120.0

-130.0

-140.0

-150.0

-160.0

-170.0

-180.0

-190.0

-200.0

3 13 23 33 43 53 63 73 83 93 103 113

Time (sec)

7.4 Numerical results for the model of Athens 138

50.0

W1 W2 W3

40.0

30.0

Horizontal Reaction (KN)

20.0

10.0

0.0

-10.0

-20.0

-30.0

-40.0

-50.0

3 8 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123

Time (sec)

By applying 20%g earthquake intensity, the collapse of the structure has been initiated

by opening of the slab at the 2nd floor on the right side. This opening and the horizontal

inertia forces have been caused rotating to the right transversal wall around its base.

The traversal wall was impacted the shear wall and the whole system was undergone

progressive collapse process, Figure 91 and Figure 92, (Appendix A1).

at 112.70 sec state at 113.10 sec

7 Full scale dynamic testing and computer modelling 139

The overall behaviour which resulted in numerical simulation has been showed a good

conformity with the experimental evidences. The maximum displacements reached in

the numerical calculation as well as in the shaking table test are plotted via the applied

earthquake intensities as shown in Figure 93.

20

18

16

14

Earthquake intesity %g

12

10

6

d1 Numerical

4 d2 Numerical

d1 Experimental

2

d2 Experimental

0

0 5 10 15 20 25 30 35

numerical and experimental results

The comparison between the story drift histories of numerical calculation and the

shaking table test have been showed satisfied agreement, Figure 94 and Figure 95.

12.0

6.0

Story drift ( /00)

0

0.0

-6.0

1st level story drift

2nd level story drift

-12.0

0 2 4 6 8 10 12 14

Time (sec)

Figure 94 Story drifts histories for 16%g acceleration intensity, obtained from the

shaking table test

7.4 Numerical results for the model of Athens 140

-12.0

-6.0

Story drift (0/00)

0.0

6.0

2nd level story drift

12.0

0 2 4 6 8 10 12 14

Time (sec)

Figure 95 Story drifts histories for 16%g acceleration intensity, obtained from the

numerical calculation

The Hysteresis loops are plotted for the base shear force via the 2nd level relative

displacement as shown in Figure 96. The maximum base shear values as well as the

maximum displacement attained in the test and numerical calculation are similar to each

other.

-50 40

04%g 04%g

-40 08%g 08%g

12%g 12%g

-30

14%g 14%g

20

16%g 16%g

Base Shear (KN)

-20

Base Shear (KN)

-10

0 0

10

20

-20

30

40

50 -40

40.0 20.0 0.0 -20.0 -40.0 -40 -20 0 20 40

Figure 96 Hysteresis loops for base shear force via 2nd level relative displacement

7 Full scale dynamic testing and computer modelling 141

bonding strength with slabs

The cracking of the structure and the collapse

process are highly influenced with the initiation of

cracks. The variation of the material properties

and construction quality from one part to other in

the structure are of significant importance for

advanced damage state in the structure.

Therefore, in order to get an idea about the size

of influence, another model has been built

excluding the initial bonding strength between

the slabs and walls. The behaviour of the

generated model is mainly influenced with sliding

of slabs of the 1st level, Figure 97.

of Ispra

Figure 97 Model of Athens, sliding

As mentioned earlier, the model of Ispra has the slab of the 1st level

been calculated for different earthquake

intensities without regarding the cumulative damage to reduce the computation efforts.

To be convinced that the following steps are right, the numerical results for earthquake

intensity 4%g is validated with the results of pseudo dynamic test, where the structure at

this stage has been received no pre-loading. Afterwards, the model putted through

earthquake intensity 20%g, where the testing procedure has been ended at this stage.

4%g earthquake intensity has been applied to the model of Ispra in order to capture the

dynamic characteristics of the undamaged structure. The analysis of the transmitted

normal forces to each wall has been showed bigger values for the walls w3 and w4 as a

result of their relatively long length.

The maximum attained displacements were 0.8 mm at the first floor and 2 mm at the

second floor, and the period of the structure was 0.18 sec.

The vertical reaction under the wall w4 was received the biggest variation during the

earthquake. This variation has been caused reduction in the compression value of the

vertical reaction which changed finally to tensile force. The big variation of normal forces

in w4 was conceivably due to the rocking of the adjacent shear wall w5, Figure 98.

Under higher earthquake intensities the values of tensile force reaches bigger values

and may causes tensile cracking along the wall w4.

The last behaviour indicates that the adjacent shear wall to the transversal wall was the

reason behind increasing the variation of the normal force value, which may causes

tensile cracking in transversal wall. The longest of the adjacent shear wall the increase

of the variation in normal forces. This causes tensile cracking in the transversal walls

which are adjacent to the longer shear walls sooner than others.

7.5 Numerical results for the model of Ispra 142

Vertical Reactions (KN)

100

from compression to tension

0

-100

-200

-300

-400

Structure W1= 6.20% W2= 8.18% W3= 39.64% W4= 26.08% W5= 19.89

-500

-600

-700

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (sec)

7 Full scale dynamic testing and computer modelling 143

50

W1

Horizontal Reaction (KN)

40 W2

W3

30

W4

W5

20

10

-10

-20

-30

-40

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (Sec)

60

W1

Moment Reaction (KN.m)

50 W2

W3

40 W4

W5

30

20

10

-10

-20

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (Sec)

1.000

W1

0.800

W2

0.600

Eccentricity (m)

W5

0.400

0.200

0.000

-0.200

-0.400

-0.600

-0.800

-1.000

-1.200

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (Sec)

7.5 Numerical results for the model of Ispra 144

The transversal walls have less participation than shear walls to resist the horizontal

earthquakes. However, this participation is considerable for such weak wall and might

causes out of plane failure. Figure 99 demonstrates the horizontal reactions attained in

each wall during the earthquake. It is quit evident, the importance of considering out of

plane failure for the transversal walls, which possibly appear in high earthquake

intensities.

Moreover, what is notable from the numerical calculation is the higher moment reaction

of the longest shear wall if compared with other shear walls. The source of such moment

could be the big rocking of the shear wall which subsequently results in big variation of

the normal forces in the adjacent transversal wall, Figure 100.

The eccentricity seems to have approximately the same values for each shear wall

during the earthquake. This indicates that there is proportionality between the moment

and normal forces attained in each shear wall, Figure 101.

The cracks of the structure during the earthquake in numerical analysis are similar to the

cracks attained through pseudo dynamic test, Figure 106.

The numerical results for 4%g have been compared with the results of pseudo dynamic

test. Figure 102 and Figure 103 show the obtained displacements on each floor from the

numerical calculation and from the test, respectively. It is quite evidence the similarity of

the maximum displacement from both plots. The period of the structure from numerical

and experimental results is identical as well.

2.0

Displacments (mm)

0.5

0.0

-0.5

-1.0

-1.5

-2.0

-2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (sec)

7 Full scale dynamic testing and computer modelling 145

2.0

Displacments (mm)

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (sec)

Figure 103 Floor displacements histories obtained from pseudo dynamic test

In addition, restoring forces which resulted in numerical analysis are compared with

those attained in the test. Figure 104 and Figure 105 show the numerical and

experimental results for restoring forces histories of both stories, respectively.

60

1st floor

Restoring forces (KN)

2nd floor

40

20

-20

-40

-60

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (Sec)

60

Restoring forces (KN)

1st floor

2nd floor

40

20

-20

-40

-60

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

Time (Sec)

Figure 105 Restoring forces histories obtained from pseudo dynamic test

7.5 Numerical results for the model of Ispra 146

Time (Sec)

Figure 106 Comparison the crack patterns from numerical calculation and the pseudo

dynamic test

By applying 20%g earthquake intensity the structure was showed cracks in different

positions of the walls, namely, tensile cracks along the bed joints in the long transversal

walls, diagonal cracks in the long shear wall, horizontal cracks in the slender shear

walls, partial separation between the shear walls and transversal walls, and uplift of the

transversal wall which is adjacent to the longest shear wall, Figure 107.

Figure 107 Visible crack pattern resulted from 20%g excitation, the deformations are

scaled 10 times, time=7.7 sec

7 Full scale dynamic testing and computer modelling 147

By applying 24%g and 26%g earthquake intensities, the structure was showed more

cracks at the same positions like for 20%g. The longest shear wall was showed

inclination to buckling if the earthquake duration was longer and it is more noticeable for

26%g earthquake intensity, Figure 108 and Figure 109.

Figure 108 Visible crack pattern resulted from 24%g excitation, the deformations are

scaled 5 times, time=10.85 sec

Figure 109 Visible crack pattern resulted from 26%g excitation, the deformations are

scaled 5 times, time=10.85 sec

By applying 28%g and 30%g earthquake intensities, the structure was behaved the

same like in the last loading, but the longest shear wall was less willing to bucking than

that observed under lesser earthquake intensities. Alternatively, diagonal cracks were

increased in the longest shear wall, Figure 110 and Figure 111.

7.5 Numerical results for the model of Ispra 148

Figure 110 Visible crack pattern resulted from 28%g excitation, the deformations are

scaled 5 times, time=14.55 sec

Figure 111 Visible crack pattern resulted from 30%g excitation, the deformations are

scaled 5 times, time=10.25 sec

34%g earthquake intensity has been applied to the model of Ispra. The initial achieved

crack patterns were approximately the same for smaller intensities. However, the longer

shear wall was showed complete buckling collapse. The buckling collapse of this wall

would cause the total collapse of the structure if the duration of earthquake was longer,

Figure 112, (Appendix A2).

7 Full scale dynamic testing and computer modelling 149

Figure 112 The state of the structure at time=14.95 sec for 34%g excitation

The structure was subjected to 40% earthquake intensity, the initial achieved crack

patterns were like for lower earthquake intensities but no buckling was attained and

alternatively out of plane failure was occurred in the 2nd floor in the transversal wall

perpendicular to the longer shear wall. This kind of failure was aroused conceivably due

to increasing of the inertial forces of this wall after getting tensile failure, after which the

structure lost the stability and underwent progressive collapse, which caused the total

collapse of the structure, Figure 113, (Appendix A3).

In order to understand the effect of bonding strength between units on the capacity of

the structure, the shear and tensile initial cohesions were set to zero in the model and

subjected to earthquake intensity 20%g. As a result, the overall structure was destroyed

under this intensity, which indicates the high influence of bonding strength between the

units to the collapse mechanism, Figure 114, Appendix A4.

7.5 Numerical results for the model of Ispra 150

Figure 114 The state of the structure at time 6.55 sec, no initial bonding at the bed joints

In order to understand the influence of vertical accelerations on the structure, 40%g

vertical acceleration intensity has been applied to the structure together with 26%g

horizontal acceleration. The same earthquake accelerogram in Figure 74 has been used

for vertical acceleration. In order to take into account the whole terraced house in Figure

77, the symmetry of the structure has been considered in calculation. The simulation

results were showed no big influence of the vertical acceleration on the collapse of the

structure in comparison with the applied 40%g vertical intensity. The collapse is once

more due to out of plane failure. It is similar to the collapse under only 40%g horizontal

acceleration, Figure 115. Appendix A5 shows the collapse sequence.

7 Full scale dynamic testing and computer modelling 151

Numerical investigation of earthquake performance on masonry structures have been

carried out and the generated models were validated by the results of two full scale

experimental tests.

Besides the accuracy of the numerical model, this work has been focused on the

behaviour of the structure which is difficult to capture through the experiments, due to

limitation of shaking table capacity, damaging of testing equipments and measuring

techniques.

Throughout the observations and the obtained results, the following remarks worth to be

pointed out:

- The bonding strength between units has high influence on the capacity of the

structure

- The bonding strength between structure elements, slabs and walls, is

significantly influence the collapse mechanism

- For some buildings, the tensile failure in the transversal walls is the main reason

that causes the collapse, not the low capacity of the shear wall

- The increasing of the possibility to get earlier out of plane failures in the

transversal walls for strong earthquake intensities

- The structure does not show the same collapse mechanism under different

unidirectional earthquake intensities

- The vertical earthquake actions do not have big influence on the structure for

moderate earthquakes

- The variation of normal forces in the walls during the earthquake may causes

buckling collapse due to increasing and decreasing of the compression force on

the walls

It is significant to prevent the tensile failure in the transversal walls by adding vertical

reinforcement and by making strong connections between the structure elements to

allow the shear walls to carry the horizontal earthquake actions with their maximum

capacities.

For further works, due to the high computation efforts using micro modelling strategy, it

would be recommended to establish macro model that can comprise all failure types

including out of plane failure together with shear failure.

8 Earthquake characteristics and collapse behaviour 153

Different factors influence the collapse behaviour of masonry structures, and one of the

major factors is the characteristic of the earthquake itself. This effect is going to be more

complicated for historical constructions, which involve different structure members with

different geometries. The study in this chapter focuses on the effect of earthquake

characteristics on the collapse mechanisms of masonry structures.

A full historical masonry building has been selected for the present study that comprises

different structural members (domes, vaults, minaret, and arched walls, with different

sizes) and located in hazardous earthquake region.

Many parameters have been introduced in earthquake engineering to characterize the

earthquakes, namely: horizontal peak ground acceleration, vertical peak ground

acceleration, incremental velocities and response spectra which describe the frequency

content. The effect of earthquake direction and the frequency content of the earthquake

will be the major investigated issues of this study.

A brief background of the selected case study and modelling process are given first, and

the geometry of the whole structure is created where micro modelling strategy is

employed. The collapse analysis of the structure is performed under an artificial model

based on the earthquake characteristics of the site.

Unidirectional earthquake are applied to the structure from different directions in order to

investigate the weakest situation. Finally, different earthquake models are generated

with different frequency contents according to considered soil profiles and then applied

consequently on the structure to explore the worst situation.

The architecture of historical masonry structures shows a wide disparity through

centuries, and the structure members are formed in various geometries (pillars, arches,

vaults, domes and minarets). However, the variation in the geometries of the structure

elements also results in different performances against earthquakes, and their

vulnerability to collapse.

The study of the behaviour of a single structural element may give an indication about its

individual earthquake vulnerability. However, the success of such study is associated

with considering the other existing structure members that shape the entire geometry.

For example, domes are intrinsically much stronger against earthquakes than other

members and their possible weakness is essentially associated with the stiffness and

the strength of the supporting members. In the case of Hagia Sophia, the deformability

of the main pillars and supporting arches were caused sometimes the damage and the

collapse, Croci [44].

8.1 Selecting the case study 154

(7)

(6)

(5)

(8)

(4)

(9)

(3)

(1)

(10)

(11)

(2)

(12)

Figure 116 The mosque of Takiyya al-Sulaymaniyya, and other works of Mimar Sian

8 Earthquake characteristics and collapse behaviour 155

For these reasons, the collapse of historical masonry structures will be demonstrated in

this chapter throughout a case study of a real full masonry structure that comprises the

disparities in structure elements (pillars, arches, vaults, domes and minarets). Indeed,

many masonry structures exist in the world that has such variety in elements, but those

which are in earthquake hazardous regions are more preferable. The good example of

such buildings are conceivably those build by the medieval architect of Ottoman empire

Mimar Sinan22 Figure 116 23, which was a landmark in the history of architecture.

Furthermore, most of those buildings are constructed in regions that experience seismic

activity like Turkey, Syria, Greece, Cyprus, Ukraine and Bulgaria.

Among the big number of the works of Sinan the beauty mosque of Takiyya al-

Sulaymaniyya24 in Damascus, the capital city of Syria, has been chosen for the present

study.

8.2.1 Historical background and the layout of Takiyya

Takiyya al-Sulaymaniyya is a complex in Damascus, Syria, considered as the most

important Ottoman cultural building in the city. It was build by the Sultan Süleyman I or

Sulayman al-Qanuni (1520-1566) between 1554 and 1560. The major complex is

located on the bank of Barada River. It was built on the ruins of Mamluk sultan Baybars

palace “Qasr al-Ablaq” which was destroyed by Tamerlane. The same stones of “Qasr

al-Ablaq” have been used to build Takiyya, Rihawi [154].

Takiyya was designed by Sinan and supervised by architects sent from the imperial

architectural office and finished during the period of city’s governor Kheder pasha 1559.

In 1566, a Madrasa has been added to the southeast of this complex by Sultan Selim II

and linked to them with an Arasta (Souk).

The complex layout of the original core of Takiyya is symmetric. The mosque and the

public kitchen are situated at the two ends of the main axis of rectangular enclosure.

To the east and west of the mosque there are two rows of six arcaded cells used as

guestrooms, equipped with fireplaces and covered with domes higher than the domes of

the mosque portico.

22

Ḳoca Mi‘mār Sinān Āġā or in Ottoman Turkish language: "( "ﺧﻮﺟﻪ ﻣﻌﻤﺎر ﺳﻨﺎن ﺁﻏﺎApril 15, 1489 - April 09,

1588) was the chief Ottoman architect and civil engineer for sultans Suleiman I, Selim II and Murad III. He

was responsible for the construction or the supervision of every major building in the Ottoman Empire. More

than three hundred structures are credited to his name. (English Wikipedia)

23

(1) Takiyya al-Sulaymaniyya, Damascus, Syria (1554-1560), (2) Sokollu Mehmed Pasa Mosque at

Azapkapi, Istanbul, Turkey (c.1573-1577/1578), (3) Mesih Mehmed Pasa Mosque, Istanbul, Turkey (1584-

1585/1586 ), (4) Kiliç Ali Pasa Complex, Istanbul, Turkey (1578-1580/1581), (5) Sultaniye Complex,

Karapinar, Turkey (1560-1563/1564), (6) Selimiye Complex, Edirne, Turkey (1568-1574), (7) Süleymaniye

Complex, Istanbul, Turkey (1548-1559), (8) Tatar Khan Mosque, Yevpatoriya, Ukraine (c. 1552), (9)

Mehmed Aga Complex, Istanbul, Turkey (1584-1585), (10) Bosnali Mehmed Pasha Mosque, Sofia, Bulgaria

(1547-1548), (11) Haseki Hürrem Baths, Istanbul, Turkey (1556-1557), (12) Atik Valide Complex, Istanbul,

Turkey (1571-1583)

24

locally known in Arabic language " "ﺟﺎﻣﻊ اﻟﺘﻜﻴﺔ اﻟﺴﻠﻴﻤﺎﻧﻴﺔother names: Sulayman I Complex, Süleymaniye

Camii, Sultan Süleyman Mosque, Tekkiye Mosque, Tekke Mosque, Takiya al-Sulaymaniyya, Süleymaniye

Tekkesi, Süleymaniye Külliyesi, Complex of Kanuni Sultan Süleyman (Suleiman the Magnificent) (ArchNet

website)

8.2 Mosque of Takiyya al-Sulaymaniyya 156

Figure 117 Isometric view of Takiyya al-Sulaymaniyya, the mosque marked in red,

ArchNet [201]

8

2

2 3

7

6

0 10 20 m

5 5

N

4

Figure 118 Plane of Takiyya al-Sulaymaniyya and Madrasa showing:(1) the mosque, (2)

mosque portico (3) guestrooms, (4) public kitchen and hospice, (5)

caravanserais with stables, (6) ablution pool, (7) latrines, (8) madrasa and (9)

bazaar (Arasta), corrected and redrawn from ArchNet [201]

8 Earthquake characteristics and collapse behaviour 157

The public kitchen consists of a line of six equal-size cells, enlarged into a room at the

centre with two vaulted bays. It faces the courtyard with a portico of twelve small domed

bays. Two caravanserais are located on the two sides of the public kitchen, composed of

fourteen domed cells arranged in two rows. The caravanserais and the public kitchen

share a private courtyard behind the public kitchen that is accessed with two gates from

the main Takiyya courtyard. The entire complex was restored in the 1960s by

Directorate General of Antiquities of Syria.

The mosque is the largest and the major part of the complex, located on the southern

end of the courtyard. The architecture of the mosque is similar to the prototypical forms

used by Sinan a cubic mass crowned by a vast hemispherical dome rising over

pendentives, with a portico in front and twin minarets.

8.2 Mosque of Takiyya al-Sulaymaniyya 158

8 Earthquake characteristics and collapse behaviour 159

The hall of the mosque is based on a square plane of 16 by 16 meters. The dome is

suited on a square of 14.3 by 14.3 meters and rises to a height of 7.4 meters on a

circular base supported by four pendentives and four large arches with diameter of 13.1

meter which stand at the corners of the square bases, Figure 122. The diameter of the

hypothetical sphere which includes both the dome and the pendentives (the diagonal of

the square) is about 20.1 meters.

Dome D=14.7 m

Pendentive

Arches D=13.1 m

Columns

L=14.3 m

The pendentive25 permits placing the circular dome over the square room, so that,

provides transition between the dome and the square base. The pendentives receive the

weight of the dome and transmit it to the four corners that can be received by the piers

beneath. The pendentives are triangular segments of a sphere, tapered to points at the

bottom and spread at the top to establish the continuous circular base needed for the

dome.

25

This constructive device was commonly used in Byzantine, Renaissance and baroque churches, and in

Ottoman constructions. The first attempts were made by the Romans, but full achievement of the form was

th

reached only by the Byzantines in Hagia Sophia at Constantinople (6 cent.).

8.2 Mosque of Takiyya al-Sulaymaniyya 160

the outside, and displays a ring of 24 windows

pierced into the drum of the dome, and braced

with four pairs of buttresses on the exterior. The

hall of the mosque is flanked by white and black

stone walls that cover the whole supporting

system of the dome.

Three-bay interior portico is adjacent to the north

side of the mosque hall. Two domes and one

vault are roofed the interior portico.

The domes of the portico rest on pendentives,

whereas the central bay over the mosque

entrance is roofed by a vault higher than the

domes. The domes and vault are carried on

pointed arches that rest on four marble columns

with carved stalactite capitals.

The exterior portico is covered by a shed roof

and has pointed arches carried on columns

thinner than the columns of the interior portico.

The pointed arches in exterior and interior Figure 123 The interior portico of the

porticos are braced using steel bars. This kind of mosque, ArchNet [201]

bracing system is commonly used in the most

works of Sinan.

Pendentive

Pointed

Arches White

and black

stones

Columns

Supported Bracing

on the wall of system

mosque

Figure 124 The interior portico in front of the mosque with domes and the vault

8 Earthquake characteristics and collapse behaviour 161

The mosque of Takiyya al-Sulaymaniyya has

twin high polygonal minarets rise atop the east

and west corners of the mosque.

Each minaret is 35.4 m high and has a single

balcony. The first high storey of the minaret has

a height of 26 m and ends with a balcony

supported by stone muqarnas. The second

short storey is 9.4 m height and covered by

typical conical crown, sheltered by lead.

The minarets are accessed from the interior

portico and the balconies are reached by spiral

masonry staircase installed on the walls of

minarets.

The variation of such structural elements in the

mosque of Takiyya al-Sulaymaniyya might

result in very sophisticated behaviour under

earthquake actions. The existence of many

other similar constructions in such seismic

hazardous regions would provide a great

importance to consider such building in the

present research as a case study.

finite element modelling Figure 125 Minaret of the mosque of

The construction of the geometry of the Takiyya al-Sulaymaniyya

mosque of Takiyya al-Sulaymaniyya is based

on the drawings26 used for retrofitting process in 1960s which are achieved by

Directorate General of Antiquities and Museums in Damascus.

The simplified micro modelling strategy has been followed in the construction of the

geometry. Therefore, the whole building has been modelled stone by stone. The stone

cuts have been considered through modelling. The construction of the 3D volumetric

objects of the geometry has been created in AutoCAD. Due to the large number of 3D

objects needed for modelling, the construction achieved part by part to allow for more

control and easier verification of the resultant parts and to avoid dealing with huge sizes.

After the creation of geometrical objects for each part of the structure, the meshing has

been performed in ANSYS, and a file of elements has been created for each part.

The mesh of each stone has been generated carefully, so that, stones of simple

geometries meshed by 3x3x3 elements. Finer meshes lead to a large number of

elements in the overall structure and courser meshes possibly cause numerical

instabilities in contact treatments using LS-DYNA code. Constant stress eight-nodded

brick element with a single integration point has been employed with a Flangan-

Belytschko stiffness form to control the hourglass effect, Hallquist [75].

26

The Author is greatly acknowledged to Eng. Ayman Hamuk and Eng. Kassem Taffour from Directorate

General of Antiquities and Museums in Damascus for providing the necessary documents of the mosque.

8.2 Mosque of Takiyya al-Sulaymaniyya 162

8 Earthquake characteristics and collapse behaviour 163

Figure 128 The performed meshes for some parts of the structure

separate file that contains the finite Property Granite Limestone

elements of this part. Later, the whole

parts are assembled in LS-PrePost Density 2.7 ton/m3 2.6 ton/m3

[203] where LS-PrePost designed well

Modulus of elasticity 25 GPa 20 GPa

to deal with huge models. A verification

step for the contact between the Compression strength 100 MPa 70 MPa

adjoining parts has been performed in

LS-PrePost to avoid the initial Tensile strength 10 MPa 8 MPa

penetrations.

Poisson ration 0.2 0.25

Elastic material model is adopted in the

following study, due to relatively high Bulk modulus 50 GPa 65 GPa

materials strength. The elastic material

model guarantees a smooth running Table 11 Material properties of masonry

through the calculation and avoids the units

termination due to negative volumes

which arises with soft materials. The

material properties for white stones

(limestone) and black stones (Granite) Tensile strength 0.10 Mpa

are shown in Table 11.

Fracture energy 2.0 N/mm

Tiebreak contact model is employed to

represent the interface between the Initial shear strength c 0.14 Mpa

units, the properties of masonry joints

are given in Table 12. Sliding friction coefficient 0.65

The resultant model is involved with Shear friction coefficient 0.6

768887 nodes and 363567 elements.

Therefore, parallel processing is Table 12 Masonry joints properties

adopted.

8.3 Gravity loading 164

Each calculation is performed using parallel 40 Intel Itanium processors “SGI Altix 4700”

in the centre of High Performance Computing of TU-Dresden. The calculation for the

model was carried out along 7 days for 20 seconds of loading.

The stress and deformation state of the structure must be initialized before applying the

earthquake action. The application of gravity load is similar to that described in section

7.3.3. The gravity load is increased linearly to explore the safety margin of the structure

without the bracing system that shown in Figure 124. After increasing the dead loads 1.3

times the portico was collapsed, Figure 129. This indicates that the bracing bars were

added at the beginning of construction, otherwise the interior portico in front of the

mosque will be unsafe.

Figure 129 Collapse of the portico without the bracing system of the arches

8.4.1 Seismicity of the region

Syria lies in historically active region on the northern slope of the Arabian plate. The

region has experienced several destroyable earthquakes through the past centuries. The

region of Syria is one of the few places worldwide where the historical strong earthquake

events are well documented, Figure 130, Sbeinati et al.[161] and Malkawi et al. [114].

Syrian engineering code provides PGA distribution map for minimum 10% probability of

being exceeded in life time of 50 years which corresponds to return periods of 475

years. This PGA distribution map has been used for the design of engineering

structures. However, for historical monuments a longer life time up to 200 years must be

considered.

8 Earthquake characteristics and collapse behaviour 165

(EAF) Eastern Anatolian fault system,

(EFS) Euphrates Graben fault system,

(GF) Al-Ghab fault,

(RSF) Ar-Rassafeh fault,

(RF) Raum fault,

(SF) Serghaya fault,

(SPF) Southern Palmyride fault,

(YF) Al-Yammouneh fault

Figure 130 The distribution of epicentres of the historical earthquakes in Syria, Sbeinati

et al. [161]

Figure 131 Maximum peak ground acceleration (cm/sec2) with 10% probability

exceeding in a life time of 200 years (Return period=1898 years)

8.4 Earthquake modelling 166

Malkawi et al. [114] employed a probabilistic model that able to estimate the probability

of occurrence of forthcoming earthquakes in Syria, based on the available information on

Seismicity, geo-tectonics setting and attenuation characteristics of peak ground

acceleration. Several PGA distribution maps have been proposed by Malkawi et al.

[114]. The map of PGA distribution for 10% probability exceeding in life time 200 years is

plotted in Figure 131.

In addition to the regional seismicity, further information can be obtained from the design

response spectra provided in engineering standards. The spectral analysis of

engineering buildings in Syria usually follows the response spectrum provided in UBC97

[185], Figure 132.

Se

Spectral Acceleration (g/s)

T

Ts =

Cv ( 1) Se (T ) = Ca (1 +1.5 )

2.5Ca T0

( 1) ( 2) Se (T ) = 2.5 Ca

T0 = 0.2Ts

C

( 3) Se (T ) = v

T

Ca ( 3) Cv

T

T

T0 Ts Periods (s)

The response spectrum in UBC97 [185] is related to the site specific values of Ca , Cv

are showed in Table 13:

Type description (m/sec)

SC very dense soil and soft rock 360 to 760 0.40 0.56

Table 13 The values of Ca , Cv according to the soil profile type, UBC97 [185]

8 Earthquake characteristics and collapse behaviour 167

Nowadays, synthetic accelerograms which compatible with response spectra are

increasingly being used in earthquake engineering, where the generation of artificial

accelerogram does not require too much information and only few parameters such as

geological conditions of the site, distance from the source and fault mechanism are

needed. However, the generated accelerogram must have real duration, frequency

content, and intensity representing the physical conditions of the site.

It is possible to predict the ground motion using the accelerograms of past earthquakes

which recorded at an appropriate distance and have suitable intensities or by the scaling

of existing accelerograms, but in our case there is no earthquake records for strong

intensities, therefore the generation of an artificial one is needed.

Many methods have been proposed in literature to generate the artificial accelerogram

for regions lacking earthquake records, namely: the sums of harmonic functions, filtering

of white noise, Meskouris [131] and Thiele [182], spectral density model, Rofooei [156]

or finite element modelling of the fault system, Aagaard [1].

employed in the following study to generate the

artificial accelerograms. This method is based on

1

representing the earthquake accelerogram as a sum

of harmonic functions

t

n t1 t2

a (t ) = I (t ) ⋅ ∑ Ai ⋅ sin(ωi t + φi )

(a) Trapezoid form

(176) I (t )

i =1

I (t ) = A t B exp(−C t )

1

t

represents the change of intensity through the

(b) Exponential form

earthquake duration, Figure 133. I (t )

1 t

generate n frequency sampling points distributed ( )n

t1 I (t ) = exp(−c(t − t2 ))

must satisfy the following condition ∆f ≤ 2ξ ⋅ f in

t

t1 t2

(c) Mixed form

order to consider all frequencies of the response

spectrum, where ξ is the damping ratio of the target

Figure 133 Different possible

accelerogram.

intensity functions

For each sampling point, the amplitude Ai can be

obtained from the response spectrum at the frequency ωi to calculate the acceleration

a (t ) in equation (176). Following that, the resultant accelerogram must be checked out

with the response spectrum from which generated. Therefore, the response spectrum of

the generated accelerogram should be determined and compared with the original one.

In case of big differences, the amplitudes Ai must be scaled by the ratio of old response

spectrum to the new one. The earthquake accelerograms in the present study have

8.5 Collapse analysis of the structure 168

been generated using the above described method, where MATLAB program has been

used for this purpose.

Three earthquake components have been generated which are compatible with the

response spectra described in section 8.4.2. There is no information available about the

soil profile of the site. However, the complex of Takiyya al-Sulaymaniyya is placed on

the bank of the Barada River, which indicates that soft soil profile (SE) exemplifies the

site. The generated accelerogram for X and Z directions are scaled to fit with the peak

value of 0.4g, which represents the maximum peak ground acceleration that

characterizes the site, being obtained from the map in Figure 131 for Damascus. The

generated accelerogram for Y direction which represents the vertical component is

scaled to fit with the peak value of 2/3 of the peak acceleration in horizontal direction. 10

seconds earthquake period have been only considered in the study to reduce the

calculation time. The gravity acceleration has been first applied increasingly on the

structure to initialize the stress state under the dead loads, and then the structure has

been subjected to the generated earthquake components. The calculation has been

performed for further 10 seconds after the end of the earthquake in order to allow

relaxation of the collapsed elements. Figure 134 shows the collapse state of the

structure after 9 seconds of earthquake initiation, where the complete collapse progress

is shown in Appendix B1.

4

3

2

1

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

4

3

2 Y

1

X

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

Z

The acclerogram in Y direction reduced by 2/3

4

3

2

1

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

Figure 134 Collapse analysis of the structure under the generated earthquake model

8 Earthquake characteristics and collapse behaviour 169

the portico was the first collapsed part

of the structure and later followed by

the dome and minarets. Due to high

intensity of the earthquake, the

pendentive supporting system was

destabilized and caused the collapse

of the domed hall. The minarets were

the last collapsed parts due to

relatively their big natural period of

vibration. More time was needed for

developing the collapse in minarets

than other structure parts. However,

this study confirms that, the portico is

the weaker part in the structure and Figure 135 The collapsed portico in “Koursoum

received an earlier collapse. This Camii” or Osman Shah Mosque,

result fits with the evidences observed Trikala, Greece built in 1567-68,

in the collapse of similar structures ArchNet [201]

like Osman Shah Mosque in Greece,

Figure 135.

The collapse analysis in pervious section has been performed for specific earthquake

motion which gives a very crude estimation for the behaviour of prospective

earthquakes.

The collapse analysis under specific actions could emphasize the weak parts of the

structure, but the weak situation is also associated with loading conditions. In case of

earthquakes, there are high uncertainties of loading conditions. However, several

parameters were provided in quantitative form to characterize the random motion of

earthquakes that might influence the behaviour of the structure, like Peak ground

acceleration and incremental velocities, Syed [179]. The frequency content of the ground

motion also is of high importance. The response of a structure to an applied ground

motion could amplify the most when the dominant frequency content of the motion and

the fundamental natural frequency of the structure are close to each other.

In the following, the collapse analysis of the structure is performed in order to

understand the response of the structure for different earthquake directions, as well as to

explore the effect of frequency content of the earthquake.

The earthquake motion comprises vertical and horizontal components. The vertical

component of the earthquake has less effect on the structure, due to the safety margins

against the static gravity acceleration which has a high value if compared to the possible

vertical earthquake accelerations. However the peak vertical acceleration is often

assumed to be 2/3 of the peak horizontal acceleration, Wilson [199].

To study the earthquake component in horizontal plane, two principle directions can be

considered. The first is associated with the structure ‘The principle direction of the

8.7 The direction of the earthquake 170

direction of the structure, and the

other is associated with the Y

earthquake ‘the principle direction of

X

the earthquake’ and corresponds to

the direction at which the horizontal

Z

ground acceleration amplifies the

maximum. The worst case occurs

when the principle direction of the

structure and the earthquake are

identical.

θ=0o

The principle direction of the structure

can be estimated in some

engineering method, but it is not easy

to estimate the principle direction of

the earthquake for most geographical

locations, although there are some Y

studies in this direction based on X

geological facts and the records of

past earthquakes. Z

earthquake direction and in order to

get well assessment, the structure

should be capable of equally resisting

earthquake motions from all possible θ=45o

directions.

In some of the existing engineering’s

standards the structure should be

assessed for “100 % of the prescribed

Y

seismic forces in one direction plus

30-40 % of the prescribed forces in X

the perpendicular direction”, Wilson

Z

[199]. However, no suggestions have

given on how the directions have to

be determined for complex structures.

In order to understand the collapse

behaviour of the structure for different

earthquake directions, the generated θ=90o

accelerogram in section 8.5 for x

direction has been employed. The

generated accelerogram has been

applied on the structure in horizontal

plane for different angels with respect Figure 136 The collapse states of the structure

to x axis, namely: θ=0o, 45o and 90o, for different earthquake angels at

as well as, the same accelerogram time 9 seconds

has been applied in vertical direction.

8 Earthquake characteristics and collapse behaviour 171

Figure 136, shows the collapse states of the structure which corresponds to the angels

of θ=0o, 45o and 90o after 9 seconds of applying the earthquake. The complete collapse

progress is shown in Appendix B2.

It is quite evident that the weakest case of the structure corresponds to θ=0o, where at

this angel, the earthquake is much destructive than other directions. The application of

the same accelerogram in vertical direction does not bring out any collapse to the

structure.

The earthquake actions span a broad range of frequencies. The frequency content

describes how the amplitude of the ground motion is distributed among different

frequencies. The well description of this relation for an earthquake can be obtained from

the corresponding response spectrum.

Due to the significant influence of frequency content of the earthquake ground motion on

the structure, it has been subjected to different earthquakes with different frequency

contents. The geological properties of the site are highly influencing the frequency

content of the earthquake motion that the structure receives. Therefore, three

earthquake motions were generated for different soil profiles, namely: SA, SC and SE,

Figure 137. The aim of considering several soil profiles in this study is to understand the

influence of the site characteristics on the collapse behaviour of the structure.

Furthermore, many structures in several countries have similar architecture to the

mosque of Takiyya al-Sulaymaniyya.

1.20

Spectral acceleration (g/s)

1.00 Soil profile type C

Soil profile type E

0.80

0.60

0.40

0.20

0.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Periods (s)

Figure 137 Response spectra for different soil types A ,C and E for Z=0.4, the thick lines

are the original response spectra, whereas the thin lines refer to the

response spectra of the generated accelerograms

8.8 The frequency content of the earthquake 172

4

3

2

Acc X

1

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

4

Y

3

2

Acc Y

0

-1

-2

X

-3

-4 Soil profile

SA

0 1 2 3 4 5 6 7 8 9 10

4 Z

3

Acc Z

2

1

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

3

2

1

0

Acc X

-1

-2

-3 Soil profile

-4

0 1 2 3 4 5 6 7 8 9 10 Y SC

4

3

2

1

X

0

Acc Y

-1

-2

-3

Z

-4

0 1 2 3 4 5 6 7 8 9 10

4

3

2

1

Acc Z

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

3

2 Soil profile

1

SE

Acc X

0

-1

-2

-3

Y

-4

0 1 2 3 4 5 6 7 8 9 10

4

3

X

2

Z

Acc Y

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

4

3

2

1

Acc Z

0

-1

-2

-3

-4

0 1 2 3 4 5 6 7 8 9 10

Figure 138 The collapse states of the structure for different soil profiles at time 7

seconds

8 Earthquake characteristics and collapse behaviour 173

Figure 138 shows the collapse states of the structure which corresponds to the soil

profiles SA, SC and SE after 7 seconds. The complete collapse sequence is shown in

Appendix B3.

It is quite evident that the weakest case of the structure corresponds to the soft soil

profile SE. It means the low frequency of the earthquake is much destructive. This effect

has significant influence on the collapse of the structure at which the soil profile

dominates the frequency contents of the earthquake. However, each element of the

structure has showed different responses to the applied frequencies, which indicates the

necessity of considering the particular response of each element and its participation in

the global response of the whole building.

Collapse analysis of large scale historical masonry structure is performed with an aim to

explore the effect of different earthquake characteristics on the structure.

The study is performed on the Mosque of Takiyya al-Sulaymaniyya, one of the finest

works of Mimar Sinan in Damascus since the fifteenth century. The different types of

structural members and the different geometries in this historical building might result in

different responses and therefore different collapse mechanisms. The interactions

between those members are also of high importance for the whole response of the

structure, as well.

The model of the geometry has been created in LS-PrePost [203], and the analysis has

been performed using the explicit solver of LS-DYNA. An artificial earthquake

accelerogram compatible with a response spectrum has been generated. The necessary

data for generation the earthquake model has been obtained from the site information

and the hazard map zonation for life time of 200 years.

The relatively high deformations of the pendentives were the major reason for the

collapse of the dome. The minarets were needed more time to be collapsed than other

structure member.

Collapse analysis of the structure under unidirectional earthquake actions was

performed, to explore the weakest direction of the structure. According to the geological

map and the seismic history of the region, the major earthquake direction can be

estimated. However if the direction of the earthquake is not well predicted, the structure

must be strengthened in the weakest direction.

The present study shows that, the earthquakes in regions of soft soils are more

destroyable to the structure. However, the soft soils dissipate a great amount of the

kinetic energy that cased by earthquake. Furthermore, the other phenomena that might

occur due to the failure of soil and liquefaction should be considered for more detailed

study.

This study is primarily focused on the effect of frequency content of the earthquake, and

the other factors related to soil-structure interactions are not considered.

9 Reinforced masonry 175

9 Reinforced masonry

The reinforcement is a supplementary material can be added to the original masonry

system in specific positions in order to gain an enhancement in load bearing capacity

and/or to guarantee ductile failure mechanism.

During the recent years, the performance of reinforced masonry structures received

much attention and attracted a considerable volume of research. Various strengthening

measures have been identified and studied: strengthening by means of grouted anchors,

Van Gemert et al. [187], Polymer grids and reinforcement in bed layers, Sofronie [175]

and FRP sheets, Marfia et al. [119]. Brookes et al. [28], Owen [144] and Mabon [113],

have made numerical and experimental investigation for strengthening masonry arch

bridges in UK, where stainless steel reinforcing bars were inserted and grouted into

masonry. Brookes et al. [30] have studied the effect of various reinforcement measures

on the collapse of reinforced masonry walls subjected to explosive loads.

Nowadays, there is an increasing demand to get an optimal design for the modern

masonry constructions, as well as to develop robust retrofitting measures to preserve

our cultural heritage for future generations.

The previous chapters have demonstrated the assessment of seismic resistance of

unreinforced masonry buildings by means of collapse analysis. In case, the load bearing

capacity of the structure is found to be insufficient, the structure must be reinforced.

Thus, it is necessary to think about the amount and the location of reinforcement in order

to get an optimal design.

The present chapter gives an insight into a novel understanding of the failure behaviour

of reinforced masonry shear walls, as well as, the retrofitting measures are going to be

assessed and the evaluated based on collapse analysis.

The Ductility of the structure describes its ability to deform beyond the elastic limit

without excessive strength decay or collapse. Increasing the ductility gives the structure

more time before going to collapse. The ductile behaviour is highly recommended for

seismic design as it demonstrates of being far from a sudden brittle collapse, Figure 139.

Masonry materials show somehow brittle behaviour under seismic actions. However, the

insertion of reinforcement into masonry increases its ductility. The displacement ductility

µ ∆ is introduced to describe the overall ductility of the reinforced system. It can be

defined as the ratio of the ultimate displacement ∆ u to the yield displacement ∆ y :

∆u

µ∆ = (177)

∆y

characterize the behaviour of the structure. The response factor is the ratio of elastic

response Fel to the response at yielding Fy :

9.1 Ductile behaviour 176

Fel

q= (178)

Fy

unreinforced masonry of reinforced masonry

Figure 139 Comparison the brittle and ductile failure of masonry wall, Tomaževič [183]

Fel equivalent Fel

the response

the response

perfectly plastic perfectly plastic

Fy Fy

∆y ∆u ∆y ∆u

The following two principles have been employed to determine the q factor, Figure 140,

Paulay et al. [147] and Schermer [162]:

(1) Principle of equivalent deformation work

9 Reinforced masonry 177

1 F F

( Fel − Fy ) ⋅ ( el ∆ y − ∆ y ) = Fy ⋅ (∆ u − el ∆ y ) (179)

2 Fy Fy

q = 2µ ∆ − 1 (180)

Fel ∆ u

q= = = µ∆ (181)

Fy ∆ y

Grouted anchors are often engaged for many application areas of strengthening

masonry structures. They enable to increase the transverse tensile strength of existing

masonry. Steel, stainless steel or reinforced polymer rods are inserted in drilled holes

and bonded to the masonry with an appropriate grout.

The enhancement in load bearing capacity of the reinforced masonry is highly affected

by the bonding strength between the reinforcement and masonry. The bonding strength

is, therefore, the fundamental issue for strengthening using grouted anchors.

The issue of bonding strength was early questioned for reinforced concrete and it has

been widely studied in literature Figure 141.

(a) (b).

Figure 141 Bond stress versus slip behaviour for reinforcement, (a) monotonic loading,

(b) cyclic loading, after Eligehausen et al. (Lowes [110])

Many failure modes have been recognized for bonding, which are associated with the

mechanical properties of the materials. Gigla [68] has identified two types of failure for

the pull-out testing of an injected anchor, one is the failure at the interface between the

reinforcement element and the grout material, and the other failure is at the interface

between the drill hole and the grout material.

In his experimental study, Gigla [69] has made a series of experimental tests to

determine the capacity of boding between the reinforcement and masonry, Figure 142.

9.2 Reinforcement-masonry bond 178

plate

Free end Loaded end

tension jack and

tensile

load cell

ancor bar

clamp

Injected

material in

drill hole

testing

load F

parallel measure

of displacements

overlap L0

bond length Ltb free steel length Lfs

Figure 142 The added injection anchor with measurement setup, after Gigla [69]

The following empirical formula has been proposed for bonding strength X A, d :

φ J f G ,c

2

X A, d = ⋅( + X B ,w ) (182)

γ M 500

where

1.9 ⋅ f B ,t ⋅ Lb ⋅ π ⋅ d B ⋅ (hs2 − d B2 )

F≤ (183)

γ m ⋅ tan ϕ ⋅ (hs2 + d B2 )

where

Lb bonding length

9 Reinforced masonry 179

partial safety factor for tensile strength of the stone, γ m ≥ 1.5 has been

γm

proposed

The key feature of reinforcing masonry shear walls is not only to increase the shear

capacity but also to get better ductile behaviour under high seismic intensities. The

vertical reinforcement enhances the capacity of masonry to carry tensile forces

perpendicular to the bed joints. The insertion of reinforcement bars requires special

openings in the blocks where frequently used with hollow units being filled with grout,

Figure 143.

Several studies have been devoted to determine the shear capacity of reinforced

masonry. Ganz [62] has extended the shear theory of unreinforced masonry that

described in section 3.2.4 to include the reinforcement, where the following assumptions

have been adopted

- The reinforcement is arranged in bed joints or perpendicular to the bed joints in

unit holes

- The reinforcement works only in direction of the reinforcement bar and has rigid

bonding with masonry

- The compressive strength of reinforcement is neglected.

Reinforcement bar

Grout material

Hollow units

(a) (b) (c) (d) (e)

Figure 143 Different types of masonry shear walls, (a) reinforced masonry (RM), (b)

wide spaced reinforced masonry (WSRM), (c) grouted masonry (GM), (d)

partially grouted masonry (PGM), (e) unreinforced masonry (URM)

The shear failure criteria in stress space σ x , σ y and τ xy are shown in Figure 144 and

given by the following set of equations

9.3 Reinforced masonry shear walls 180

τ xy2 − (ω y ⋅ f my − σ y ) ⋅ (ω x ⋅ f mx − σ x ) ≤ 0 (I)

τ xy2 − (σ y + f my ) ⋅ (σ x + f mx ) ≤ 0 (II)

τ xy2 + (σ x + f mx ⋅ ( 1 2 − ω x ) )2 − ( 1 2 ⋅ f mx )2 ≤ 0 (IIIa)

τ xy2 − (c − (σ y − ω y ⋅ f my ) ⋅ µ )2 ≤ 0 (IVa)

τ xy2 − ( 1 2 ⋅ f mx )2 ≤ 0 (V)

τ xy2 + (σ y + f my − 1 2 ⋅ f mx )2 − ( 1 2 ⋅ f mx )2 ≤ 0 (VI)

where

asx f sy asy f sy

ωx = ωy = (185)

b f mx b f my

asx , asy are the reinforcement’s areas per unit length in x, y directions, respectively.

σx

Spacing of contour lines 0.1 ⋅ f my

ωx ⋅ f mx

I

τxy σy

VI

IIIa

f my

0.4

II 0.3 IIIb

0.2

0.1

f my ωy ⋅ f my

Figure 144 Shear failure criteria in stress space for reinforced masonry after Ganz

The shear behaviour of masonry joints under pure shear or confining pressure has been

studied by Marzahn [121] through a triple shear test, Figure 145.

9 Reinforced masonry 181

Figure 145 Experimental triple test setup for grout-dowelled masonry, after Marzahn

[121]

It has been observed that the grout material enhances the shear capacity but the

contribution of reinforcement is very small. However, the reinforcement offers high

deformation ability prior to the failure, which indicates an enhancement in the ductility,

Figure 146. Marzahn [121] has considered that the total shear cohesion is a contribution

of the following:

cvr = ci + cG + cR (186)

Where ci is the initial cohesion, cG is the shear strength of the grouted core and cR is

the shear strength of the reinforcement.

The shear strength of the grout core is reduced to 85% due to the shrinkage of the

material:

cG = 0.85 ⋅ f Gs ⋅ aG (187)

The value of cG is limited due to the local compression failure in unit. Marzahn has

proposed to use 0.75% reduction of the local compression area φG ⋅ h

n ⋅ φG ⋅ h

cG = 0.85 ⋅ f Gs ⋅ aG ≤ 0.75 ⋅ σ c (188)

b ⋅l

9.3 Reinforced masonry shear walls 182

(Mpa) 0.80

τShear Stress

0.60

0.50

Reinforced

0.40 Grouted masonry masonry

Unreinforced

0.30 masonry

0.20

0.10

0.00

0.00 1.00 2.00 3.00 4.00 5.00

displacements w (mm)

grouted and reinforced masonry, redrawn after Marzahn [121].

diameter φ to the diameter φ14 = 14 mm using the following value

φ

cR = n ⋅ 0.1 (189)

φ14

The contribution of the initial cohesion of the bed joint is limited to the earlier failure.

Therefore it should be neglected after failure

c0 (1 − aG − aR ) if c0 (1 − aG − aR ) > cG

ci = (190)

0 otherwise

By excluding the area of the grouted core from the cohesion area of the bed joint, then

c0 = c ⋅ (1 − aG − aR ) (191)

where c0 is the pure cohesion of unit mortar joint, aG is the ratio of the grouted area

excluding the reinforcement area to the bed joint area of the unit and aR is the ratio of

the reinforcement area to the bed joint area of the unit.

According to the proposed cohesion formula by Marzahn [121], it can be determine the

shear failure mode in reinforced masonry wall by employing the shear failure theory of

Mann/Müller [116] using the following equation:

9 Reinforced masonry 183

cvr − µ ⋅ (1 − aG − aR ) ⋅ σ

τ= (192)

1+ µ ⋅ r

Ernst et al. [55] and Ernst [54] have proposed empirical formulas to estimate the load

bearing capacity of horizontally and vertically reinforced clay brick masonry, Figure 147.

Ernst et al. [55] have showed that the horizontal bed joint reinforcement could reduce the

tensile cracking due to confinement, which enhances the compression strength. Ernst

also has identified that, the failure has been caused by the absent of bonding before the

yield of reinforcement.

Figure 147 The failure envelope for unreinforced clay brick masonry (1a) tensile failure

of bed joints, (2a) shear failure of bed joints, (3a) the tensile cracking of

units, (4) compression failure of masonry, while failure envelope for

reinforced clay brick masonry is given by (2b) shear failure of bed joints, (3b)

tensile cracking of units and (4).compression failure, after Ernst et al. [55]

Haider [73] has carried out numerical and experimental investigation on wide spaced

reinforced masonry WSRM walls that contain vertical reinforced cores at horizontal

spacing up to 2 m.

9.4 A shear failure theory for vertically reinforced masonry 184

The interaction between the unreinforced masonry panels and vertical reinforced cores

were determined using an elastic finite element analysis. A series of experimental tests

on WSRM under monotonic and cyclic loading has been carried out as well, Figure 148.

Let us consider the vertically reinforced masonry shear wall with grouted cores subjected

to in-plane vertical and horizontal loads, Figure 149. The vertical reinforcement is

distributed in a consistent way with Mann/Müller shear theory. Therefore, the

compressive stresses that act at the head joints were ignored due to their low negligible

value. Furthermore, the cracking of the units was ignored in order to get more flexibility

in handling the problem.

In what follows, a failure theory based

on force interaction between the

N

different constituents will be given. The

failure theory should determine the

coupled values of σ and τ at which

the failure take place.

In addition to the described failure V σ

criteria by Mann/Müller for

τ

unreinforced masonry, other failure

criteria correspond to the

reinforcement and the grout material

must be introduced.

In order to get an idea about the

initiation of the failure, the elastic

behaviour is adopted with complete

combined work between all

constituents, i.e. units, mortar, grout τ

material and reinforcement. This σ

assumption is true prior to the initial

failure and after which the nonlinear Figure 149 Vertically reinforced masonry

behaviour should be considered. shear wall

9 Reinforced masonry 185

σ2

σ1 ∆σ/2

∆σ/2

τ m

τ max

τ

r r

r

τ b

TG

FR2

φ τ

l/4 FR1

TG h τ r

Reinforcement Rebar

Grout material

o

τ r

o

TG

FR1

TG

FR2

d

pull-out shear stress

τ τ

σ1 ∆ σ/2

distribution

σ2

∆σ/2

Figure 150 The calculation diagram of the stress state for one masonry unit

Therefore, according to the compatibility of strains between units, mortar, grout material

and reinforcement, the following equation is valid, Figure 150:

σ M 1, 2 σ G1, 2 σ R1, 2

ε1, 2 = = = (193)

EM EG ER

where

1,2 indicates that the equation is valid for both loaded sides of the bed joint

in masonry unit

EM , EG , E R Elastic Modulus for masonry units, grout material and reinforcement bars

respectively

σ G1, 2 , σ R1, 2 normal stresses acting on the grout material and reinforcement,

respectively

σ M 1, 2 normal stresses acting on both sides of the bed joint of masonry unit

this yields,

9.4 A shear failure theory for vertically reinforced masonry 186

where

EG ER

mG = mR = (195)

EM EM

A A − 2 AG − 2 AR

⋅ σ 1, 2 = ⋅ σ M 1, 2 + AG ⋅ σ G1, 2 + AR ⋅ σ R1, 2 (196)

2 2

where

A area of the bed interface of the unit including the grouted core and reinforcement

areas

AR area of reinforcement

and they are given in the following for circular grouted core

π ⋅ φR2 π

A = b ⋅l AR =

4

AG =

4

(

⋅ φG2 − φ R2 ) (197)

where

This yield

where

2 AG 2 AR

aG = and a R = (199)

A A

By rewriting equation (198) with respect to the stresses on masonry unit σ M 1, 2 and

employing equations (194), it gives:

9 Reinforced masonry 187

By assuming that

1

r1 = (201)

1 + aG ⋅ (mG − 1) + a R ⋅ (mR − 1)

it yields

σ M 1, 2 = r1 ⋅ σ 1, 2 and ∆σ M = r1 ⋅ ∆σ (202)

The same treatment can be also applied to obtain the shear stresses on the unit, the

grouted core and the reinforcement

τ M = r2 ⋅τ τ G = r2 ⋅ nG ⋅τ τ R = r2 ⋅ n R ⋅τ (204)

where

1

r2 = (205)

1 + aG ⋅ (nG − 1) + a R ⋅ (n R − 1)

GG GR

nG = nR = (206)

GM GM

GM , GG and GR are shear moduli for masonry units, grout material and reinforcement

bars, respectively.

The stress difference and the mean stress will be used for prospective formulations

σ1 + σ 2

∆σ = σ 1 − σ 2 and σ = (207)

2

Thus,

∆σ ∆σ

σ1 = σ + and σ 2 = σ − (208)

2 2

By applying the equilibrium of moments around the centre of the unit, it gives:

∆σ

= r ⋅τ (209)

2

9.4 A shear failure theory for vertically reinforced masonry 188

where r = 2h / l

The contact forces which act on the grout material and reinforcement can be obtained

with respect to σ 1, 2 using the following equations:

Thus, the forces differences which represent the contact forces between the units and

grout material and between the grout material and reinforcement can be written in this

form:

The failure can be initiated using the following failure criteria

(1) Tensile cracking of the bed joint

σ M 2 = r1 ⋅ σ 2 ≥ − f tm (212)

Thus,

f tm + r1 ⋅ σ

τ≤ (213)

r ⋅ r1

τ M ≤ c + µ ⋅σ M 2 (214)

Thus,

c + µ ⋅ r1 ⋅ σ

τ≤ (215)

r2 + µ ⋅ r1 ⋅ r

σ M 1 = r1 ⋅ σ 1 ≤ f m (216)

Thus,

f m − r1 ⋅ σ

τ≤ (217)

r ⋅ r1

To avoid the pull out failure between the reinforcement and the grouting, it must

guarantee that

9 Reinforced masonry 189

∆FR

τr = ≤ τ = α ⋅τ r ,max (218)

π ⋅ φ R ⋅ h r ,m

where τ r ,m = α ⋅ τ r ,max is the mean value for shear bonding stress distribution at failure,

τ r ,max is the maximum shear bonding stress, and α is the transition factor.

By substituting ∆FR from equation (211) into equation (218), it gives

1 l

τ≤ ⋅ ⋅ τ r ,m (219)

mR ⋅ r1 φR

The shear force that acts on the grout material is

TG = r2 ⋅ nG ⋅ AG ⋅τ ≤ τ G ⋅ AG (220)

Thus,

τG

τ≤ (221)

r2 ⋅ nG

The shear force that acts on the unit generates high local compression on the supporting

area of the grouted core in units

TG = r2 ⋅ nG ⋅ AG ⋅τ (222)

compression failure, it must be:

h h

TG ⋅ h ≤ σ c ⋅ φG ⋅ ⋅ (223)

2 3

Therefore,

2 1 h

τ≤ ⋅ ⋅ ⋅σ c (224)

3π r2 ⋅ nR φG

Due to the lack of experiments for the reinforced masonry shear walls, a finite element

model has been created in ANSYS software [8] to get deeper insight into the problem.

Figure 151-a shows the results of analysis for pull-out failure of reinforcement, while

Figure 148–b shows the local compression that might causes the failure.

9.4 A shear failure theory for vertically reinforced masonry 190

The existing reinforcement gives more additional tensile bearing capacity for the shear

wall after the tensile cracking of one unit side of the bed joint. The overall system has the

capacity to bear additional shear loads after the failure of the cohesion on one unit side

of the bed joint.

1 1

NODAL SOLUTION NODAL SOLUTION

JUN 3 2007 JUN 3 2007

STEP=1 19:01:41 STEP=1

21:31:11

SUB =1 SUB =1

TIME=.05 TIME=.05

S1 (AVG) S1 (AVG)

DMX =.898E-04 DMX =.435E-04

SMN =-688.535 SMN =-717.435

MX

SMX =3830 SMX =4284

MN

MX

MN

Y

X

Z

-688.535 315.685 1320 2324 3328 -2800 -1556 -311.111 933.333 2178

-186.425 817.795 1822 2826 3830 -2178 -933.333 311.111 1556 2800

Figure 151 FE model in ANSYS for reinforced masonry shear wall, the reinforcement

smeared with grout material

Therefore, different additional cases must be considered in order to get the failure

surface of reinforced masonry after the initial degradation in tensile strength or shear

cohesion of one unit side of the bed joint.

σΜ2

σΜ1 σΜ1

τ Μ2

τ Μ1 τ Μ1

b b

T

φ T

φ

l/4 l/4

T h T h

∆ FR ∆ FR

o o

∆ FR ∆ FR

T T

T T

τ Μ1

τ Μ1

τ Μ2

σΜ1 σΜ1

σΜ2

Case I: tensile crack opening Case II: shear cohesion failure

of one side of the bed joint of one side of the bed joint

Figure 152 The stress state of one unit after the initial failure

9 Reinforced masonry 191

Figure 152 shows the stress state of one unit in Case I: after tensile cracking of one side

of the lower and upper bed joints and in Case II after cohesion failure of one side of the

lower and upper bed joints. Both cases provide new failure criteria that differ from those

obtained for initial failure surface.

9.4.3 Tensile crack opening of one side of the bed joint- Case I

On the cracked side of the bed joint σ M 2 = 0 , consequently

σ 2 = aG ⋅ σ G 2 + aR ⋅ σ R 2 (225)

By assuming

1

r3 = (226)

aG ⋅ mG + a R ⋅ mR

it becomes

σ G 2 = mG ⋅ r3 ⋅ σ 2 σ R 2 = mR ⋅ r3 ⋅ σ 2 (227)

σ G1 = r1 ⋅ mG ⋅ σ 1 , σ R1 = r1 ⋅ mR ⋅ σ 1 (228)

1

τ= (1 − aG − aR ) ⋅τ M + aG ⋅τ G + aR ⋅τ R (229)

2

1

τ= (1 + aG ⋅ (2nG − 1) + aR ⋅ (2nR − 1) ) ⋅τ M (230)

2

By assuming

2

r4 = (231)

1 + aG ⋅ (2nG − 1) + a R ⋅ (2nR − 1)

it yields,

τ M = r4 ⋅τ τ G = r4 ⋅ nG ⋅ τ τ R = r4 ⋅ nR ⋅ τ (232)

By applying the equilibrium of moments around the centre of the unit, the same equation

(209) will be still valid

9.4 A shear failure theory for vertically reinforced masonry 192

τ M1 ≤ c + µ ⋅σ M1 (233)

thus,

c + µ ⋅ r1 ⋅ σ

τ≤ (234)

r4 − µ ⋅ r1 ⋅ r

π

∆FR = φ 2 ⋅ mR ⋅ (r1 ⋅ σ 1 − r3 ⋅ σ 2 ) (235)

4

π

∆FR = φ 2 ⋅ mR ⋅ [(r1 − r3 ) ⋅ σ + (r1 + r3 ) ⋅ r ⋅τ ] (236)

4

l 2τ l r3 − r1

τ≤ ⋅ r ,m + ⋅ ⋅σ (237)

φ ⋅ mR r1 + r3 2h r1 + r3

The shear force that acts on the grout material is

TG = AG ⋅ r4 ⋅ nG ⋅τ ≤ AG ⋅τ G (238)

thus,

τG

τ≤ (239)

r4 ⋅ nG

The force that produces the local compression in supporting area of the grouted core is:

TG = r4 ⋅ nG ⋅ AG ⋅ τ (240)

To avoid the local compression failure, the following criterion should be guaranteed:

h h

TG ⋅ h ≤ σ c ⋅ φ ⋅ ⋅ (241)

2 3

9 Reinforced masonry 193

consequently,

2 1 h

τ≤ ⋅ ⋅ ⋅σ c (242)

3π r4 ⋅ nG φG

9.4.4 Shear cohesion failure of one side of the bed joint- Case II

Figure 152 shows the stress state of one unit after the cohesion failure of one side of the

lower and upper bed joints. On the failed side of the bed joint the cohesion vanishes with

τ M 2 = µ ⋅σ M 2

The mean shear stress on bed joint is given by

1

τ= (1 − aG − aR ) ⋅τ M 1 + 1 (1 − aG − aR ) ⋅ µ ⋅ σ M 2 + aG ⋅τ G + aR ⋅τ R (243)

2 2

It gives

τ M 1 = r4 ⋅τ − r5 ⋅ µ ⋅ σ M 2 (244)

where

r4

r5 = (1 − aG − aR ) (245)

2

τ M1 ≤ c + µ ⋅σ M1 (246)

thus,

r4 ⋅ τ − r5 ⋅ µ ⋅ r1 ⋅ σ 2 ≤ c + µ ⋅ r1 ⋅ σ 1 (247)

r4 ⋅τ − r5 ⋅ µ ⋅ r1 ⋅ (σ − r ⋅τ ) ≤ c + µ ⋅ r1 ⋅ (σ + r ⋅τ ) (248)

It gives

c µ ⋅ r1 ⋅ (r5 + 1)

τ≤ + σ (250)

r4 − r ⋅ µ ⋅ r1 ⋅ (r5 + 1) r4 − r ⋅ µ ⋅ r1 ⋅ (r5 + 1)

This failure mode can be described according to equation (219), the same like the case

of initial failure.

9.4 A shear failure theory for vertically reinforced masonry 194

To prevent this failure mode it must be guaranteed that:

TG

= nG ⋅ τ M 1 ≤ τ G (251)

AG

nG ⋅ r4 ⋅ τ − nG ⋅ r5 ⋅ µ ⋅ σ M 2 ≤ τ G (252)

thus,

σ M 2 = r1 ⋅ (σ − r ⋅ τ ) (253)

nG ⋅ r4 ⋅τ − nG ⋅ r5 ⋅ µ ⋅ r1 ⋅ (σ − r ⋅τ ) ≤ τ G (254)

nG ⋅ (r4 + r5 ⋅ µ ⋅ r1 ⋅ r ) ⋅τ ≤ τ G + nG ⋅ r5 ⋅ µ ⋅ r1 ⋅ σ (255)

Consequently,

τG r5 ⋅ r1 ⋅ µ

τ≤ + ⋅σ (256)

nG ⋅ r4 + nG ⋅ r5 ⋅ r1 ⋅ µ ⋅ r r4 + r5 ⋅ r1 ⋅ µ ⋅ r

The shear force on the grouted core is:

TG = AG ⋅ nG ⋅ τ M 1 (257)

h2

τ M1 ≤ σ c ⋅ (258)

φG 3π ⋅ nG

h 2

(r4 + r5 ⋅ r1 ⋅ µ ⋅ r ) ⋅ τ ≤ σ c ⋅ + r ⋅ r ⋅ µ ⋅σ (259)

φG 3π ⋅ nG 5 1

Consequently,

1 ⎡ h 2 ⎤

τ≤ ⎢σ c ⋅ + r5 ⋅ r1 ⋅ µ ⋅ σ ⎥ (260)

r4 + r5 ⋅ r1 ⋅ µ ⋅ r ⎣ φG 3π ⋅ nG ⎦

9 Reinforced masonry 195

As has been presented in previous sections, one failure surface gives no real description

for all possible failure modes in the reinforced masonry system. However, the failure

surface varies according to the state of damage. Case I and case II bring out additional

critical information for the state of failure surface. Figure 153 shows plots of the resultant

failure surfaces from initial failure, from case I and from case II. The stress space can be

divided into domains. Each one is controlled by specific failure surface. The transition

from one failure surface to another is possible by introducing the damage law after the

initial failure. However, if case I dominates the failure then any increase in the tensile

capacity will be accompanied with a decrease in shear capacity.

τ

Shear Stress

hardening

softening

area of enhancement

failure surface of unreinforced

masonry from Mann/Müller

normal stress

σ

Case I Case II

Figure 153 Variation of the failure surface of vertically reinforced masonry shear wall

Additional cases can be also considered like the complete tensile failure of the bed

joints, and the complete shear failure of the bed joints. The above described theory does

not give distinct determination for the limit of failure. Nevertheless, this theory gives the

current state of failure surface under the current damage conditions.

An important result of this theory is that, the plastic potential theory (Huber-von Mises

theory) is not valid for masonry and this theory gives no real description of the

homogenized material of composites.

Various modelling approaches have been proposed for masonry. Nevertheless, the

inclusion of the reinforcement into the model is still challenging, and fraught with

difficulties, consequently, reinforced masonry is still lacking in literature.

In analogue with modelling strategies that have been proposed for reinforced concrete

models, the following modelling strategies can be employed for the reinforced masonry:

discrete modelling and smeared modelling.

9.5 Modelling strategies of reinforced masonry 196

In discrete modelling, the reinforcement can be modelled by means of bar elements, and

masonry can be modelled using solid elements (2D or 3D). The nodes of reinforcement

bars must be merged with masonry elements through the shared nodes, Figure 154-a.

The restriction to create shared nodes might result in some inflexibility in mesh

generation. However, it is not quite accurate to apply full bonding between the

reinforcement and masonry mesh. The bonding model can be represented by dummy

spring elements that connect the duplicated nodes from reinforcement and masonry.

The spring element has no dimension and serves only as a breakable linkage between

reinforcement and masonry. Therefore, the failure model is the most important part of

spring element.

reinforcement and unit

points of compatible

displacements between

reinforced bar element

reinforcement and unit

reinforced bar element

In order to avoid the restriction of node sharing between reinforcement and masonry, an

embedded formulation can be introduced. In the embedded formulation, the intersection

points of reinforcement bar with the segments of masonry elements are first identified

and then used to create the nodal locations of the reinforcement elements, Figure 154-b.

Brookes et al. [29], have utilized the partially constrained spar formulation to model

reinforcement independently from masonry. The connection between the reinforcement

and masonry meshes was achieved through a non-linear bond element. The

arrangement of reinforcement is automated without the need for topologically consistent

element meshes.

In LS-DYNA [112], (also Hallquist [75]) the following methods can be employed for

modelling the reinforcement of masonry:

9 Reinforced masonry 197

The 1D Contact was originally developed to offer bond slip failure for modelling

reinforced concrete. In addition, it is possible to employ this feature for reinforced

masonry. The principle of this contact model is to allow the sliding of reinforcement

nodes along masonry nodes, where the sliding initiates after the rebar debonds. The

bond model is assumed to be elastic perfectly plastic. The maximum allowable slip strain

is given as:

u = γ max ⋅ eα ⋅κ (261)

Where

κ damage parameter κ n +1 = κ n + ∆u

The shear force that acts on the area As of the reinforcement at time n + 1 is given as:

This constrained method has been developed for modelling the fluid structure interaction

and frequently used to embed the reinforcement rebar inside concrete element, Abu

Odeh [2]. The reinforcement mesh maintained to be fixed within the solid elements.

However, the bond slip failure has not been considered in this formulation.

For masonry, the reinforcement can be treated as a slave material that is linked to the

master material of masonry by means of ‘constrained Lagrange in solid’. Both masonry

and reinforcement mesh must be Lagrangian.

(3) Constrained spotweld

The spotweld provides a breakable connection for the nodal points of the nodal pairs.

The failure force at which the spotweld is failing can be regarded as the pull-out force of

reinforcement, LS-DYNA [112].

(4) Discrete beam elements with nonlinear plastic discrete beam material to simulate

failure of the beams.

Although several methods are available in LS-DYNA to build embedded discrete models

of reinforcement concrete, care should be taken when applying these methods on

reinforced masonry by considering the correct bonding behaviour between reinforcement

and masonry.

9.6 Verification of retrofitting measures by collapse analysis 198

The approach of smeared

smeared unreinforced

representation of composite materials masonry model

in one homogenous material has been

widely used in many engineering fields,

smeared reinforced

Figure 155. masonry model

reinforcement with masonry in one finite

element. The resultant element has to

be constructed from the individual

properties of masonry and

reinforcement using composite theory.

This technique has been often applied

to large structures, where the reinforced

details are not essential to capture the

overall response of the structure.

Haider [73] has developed numerical

tool in ABAQUS for smeared modelling

of WSRM shear walls.

Figure 155 Homogenized modelling of

In LS-DYNA, many material models reinforced masonry

which can represent the reinforced

concrete include an option to represent

the reinforcement in a smeared fashion. However, the material models that represent

reinforced masonry or even unreinforced masonry in smeared approach are missing in

LS-DYNA.

9.6.1 The case study

Bam Citadel was the largest adobe building in the world, located in bam, a city in

Kerman province of southeastern Iran. It is listed by UNESCO as a part of the World

Heritage Site ‘Bam and its Cultural Landscape’. This enormous citadel was built some

time before 500 BC and remained in use until 1850 AD.

On December, 26th 2003, the citadel was severely destroyed by 6.5 Richter scale

earthquake, where more than 90 % of the (sun-dried) adobe masonry structures have

been collapsed, leaving ruins with only a few remaining walls and piers, Figure 156.

Several efforts were devoted by the Iranian authorities and the international community

to reconstruct the collapsed citadel, however, the adobe material that the citadel was

built from, is very weak material to survive the forthcoming earthquakes. Therefore, a

special reinforcement technique must be incorporated with the original adobe masonry.

Within the frame of the reconstruction of Bam citadel by international experts, the

German team took part in Sistani’s House. In order to develop a strengthening

methodology for Sistani’s House, two rooms R 0.11 and R 0.12 in the northwest corner

were selected as a case study, Figure 157.

9 Reinforced masonry 199

Figure 156 Bam citadel before and after the earthquake of 2003, Jäger et al. [85]

(a) (b)

(c)

Figure 157 Sistani’s House, (a) the ground plan, selected part for the pilot project is

marked in red, Jäger et al.[85], (b) 3D view, Einifar [52], (c) after the

earthquake, the debris are removed, Jäger et al.[85]

In the following study, collapse analysis has been employed, first to explore the weakest

points in the structure, and then to develop the reinforcement which makes the structure

sustainable for the forthcoming earthquakes.

9.6 Verification of retrofitting measures by collapse analysis 200

The structure elements (thick walls,

vaults and piers) comprise a big

number of bricks. Therefore, the

generation of geometry brick by brick

will be highly consuming, rather than

the geometry of the bricks is not

known before. As a result, the micro

modelling strategy is impossible to be

applied in such case, Bakeer et al.

[12]. An alternative remedy that can

overcome the problem is to employ

macro modelling, which needs few

modelling efforts for large structures. Figure 158 Construction of adobe masonry

It reduces the model size and

wall, Jäger et al.[85]

calculation time as well.

been used to assemble the mud

bricks, Figure 158. This point planes of

toward getting material discretization

continuation in between the units

and mortar, and to get

contact

approximately similar mechanical interface

characteristic for masonry in

orthogonal directions as well. Discrete

element

Since the structure has been

destroyed by the earthquake of

2003, the creation of the geometry

will be based on the available data Figure 159 discretization of adobe masonry wall

in the site for the remaining parts,

and the available pictures before the collapse.

(a) (b)

Figure 160 The generated model (a) the discretized geometry (b) the finite element

mesh

9 Reinforced masonry 201

The whole geometry of the structure is discretized into discrete elements by means of

CAD tools, Figure 159. Each two adjacent discrete elements are sharing a contact

interface which ties both interface elements prior to the failure. This intends to get

separation of the tied discrete elements after the failure of contact take place. Those

interfaces are representing the locations of potential cracks. Without those interfaces the

calculation process terminates, due to large deformations of finite elements at failure.

LS- DYNA tiebreak contact model

which described in section 5.2 has

mass density ρ = 1.8 T / m3

been used to model the planes of

failure. Tiebreak contact is active for

elastic modulus E = 1800 MPa

nodes which are initially in contact.

The slave nodes are sticking

Poisson ratio ν = 0.25

permanently until reaching the failure

criterion, after which, this contact G = 720 MPa

shear modulus

option behaves as frictional contact.

unloading bulk modulus k = 5400 MPa

The discretization was performed in a

regular manner, so that, to get regular

p c = 0.35 MPa

geometrical shapes, and to avoid cut u. tensile strength

angles, Figure 160-a. The bad mesh

f c = 1.5 MPa

causes problems with handling the u. compressive strength

contact. In addition, the finite elements

have been generated in a mapped u. shear strength τ u = 0.35 MPa

manner to the discrete elements,

where, eight-nodded brick finite friction coefficient µ = 0.6

element with a single integration point

has been used, Figure 160-b. This Table 14 Material properties

avoids getting early negative volumes

problems that terminate the calculation in explicit solvers.

The soil and foam constitutive model which described in section 5.4.2 has been

employed for modelling adobe masonry. The material properties of adobe masonry that

used in our model are based on Taheri [180] , Kiyono et al. [92] and Jäger et al.[85] and

given in Table 14.

To find the values of a0 , a1 and a2 of the yield surface equation (138), from the

available material parameters, the following simplifications have been introduced.

From the cohesion c and the friction angel φ of the direct shear test, the following

equation will be valid by considering Mohr-Coulomb failure criterion

σ1 − σ 3 σ1 + σ 3

= sin φ + c ⋅ cos φ (263)

2 2

Therefore, by rewriting equation (263) with respect to ∆σ and p and by equating with

equation (138) at p = 0 , it gives

3 cos φ

∆σ = 3 a0 + a1 p + a2 p 2 = 2c ⋅ (264)

p =0

p =0 3 + sin φ

9.6 Verification of retrofitting measures by collapse analysis 202

= 3 = (265)

∂p p =0 2 a0 + a1 p + a2 p 2 3 + sin φ

p =0

obtained by setting

σ1 + σ 2 + σ 3 f cm

∆σ = σ 1 − σ 2 = f cm at p = = (266)

3 3

The data for the accelerogram

of Bam earthquake (05:26 on 26

December 2003 in Bam city,

Kerman State, Iran) was taken

from the records of the Bam

accelerograph station record

No.: 3168/02, BHRC [202]. The

epicentre was located at 29.01

N and 58.26 E.

The total duration of the

earthquake action was 66.55

sec, and had a magnitude of

MW 6.5 (Ms 6.7 USGS). The

peak accelerations of

longitudinal, transversal, and

vertical components were 778.2,

623.4, and 979.9 (gal),

respectively, Kiyono et al. [92].

The up and down motion of the

vertical component has been

showed a very large amplitude.

The direction of the

accelerometer of the component

(L) was N278E, which was

showing vibration approximately

in east-west direction.

To reduce the solution time, the

Figure 161 Records No.: 3168/02 from Bam

calculations have been

accelerograph station for earthquake of

concerned in the period from

15.7 sec to 30 sec, where the 26/12/ 2003, BHRC [202]

maximum acceleration values

are located in this period, Figure 161.

9 Reinforced masonry 203

In order to explore the performance of the unreinforced structure which collapsed under

the earthquake of 2003, the collapse analysis technique has been carried out on the

unreinforced structure.

The collapse of the structure is presented in Figure 162, Appendix C1. The collapse

analysis of the structure shows that the collapse initiates in the longitudinal direction of

the earthquake which is approximately in our model x axis. This was due to the relatively

big acceleration at the beginning of earthquake. A few seconds later, the walls, which

are perpendicular to the transversal earthquake direction, were collapsed due to the

increase of earthquake intensities at this time along transversal direction. This order of

collapse sequence agrees with the finding on the site, where the falling debris point out

that the direction of the collapse is mainly along east-west direction.

The collapse goes on due to out of plane failure of the walls, this agrees with the failure

mode that described in Kiyono et al. [92] for the adobe masonry buildings in Bam. The

relatively high value of friction coefficient 0.54~0.62 causes this mode of collapse.

The collapse analysis of unreinforced structure shows that the collapse is mainly initiated

due to out of plane failure of the walls. This kind of failure is possible in such form of

structures due to the existing of vaults. Therefore, the capacity of the structure can be

enhanced to resist out of plane actions by adding ring reinforcement. The vertical

reinforcement enhances the integrity and the load bearing capacity as well.

Glass fibre nets with clay-cement grout are supposed to be used as reinforcement,

where a series of experimental pull-out tests was carried out in order to determine the

proper reinforcement and the grout material to be used with adobe masonry, Jäger et al.

[85]. The reinforcement built in 36 cm into the wall, which is the tested anchor length. In

case of standard glass fibre net built into drill hole d=30 mm, the average maximum pull-

out force for clay-cement grout was 9 KN, Figure 163.

9.6 Verification of retrofitting measures by collapse analysis 204

(a) (b)

Figure 163 Pull out test of reinforcement (a) test setup, (b) pull-out force versus

displacement plot for glass fibre nets using different grout material, Jäger et

al. [85]

In order to simulate the pull-out failure of reinforcement, breakable bars that can be

linked to masonry elements via the nodes along the reinforcement locations are utilized.

Although, LS-DYNA provides various possibilities for modelling the reinforcement, each

has its drawback, section 9.5.1. However the spotweld linkage has been defined along a

string of nodes which belong to each reinforced discrete element. The distance between

the pair nodes linked by a spotweld is 0.5 m.

Vault

reinforcement

Horizontal

reinforcement

Vertical

reinforcement

The reinforcement has been calculated step by step, several reinforcement trials were

tested by collapse analysis on Room11. Figure 164 shows the reinforcement

arrangement (horizontal ring reinforcement, vertical reinforcement, and vault

9 Reinforced masonry 205

where the spotweld failure force which Maximum bonding force (KN / 50 cm)

produces pull out of the reinforcement is

Trial Horizontal Vertical Vault Reinf.

given in Table 15. No. Reinf. Reinf.

In the first trial, a simple reinforcement

has been added at corners just to #1 10 only vertical at corners

examine the behaviour of reinforced parts,

#2 10 10 -

Figure 165, (Appendix C2).

In the second trial vertical and horizontal #3 50 10 -

reinforcement were added to the walls,

the final remains were showed that some #4 100 25 -

walls were remained at the end of

#5 100 25 25

earthquake, other walls were collapsed

partially and the vaults were collapsed

Table 15 Reinforcement bonding

due to progressive collapse, Figure 166,

capacity

(Appendix C3). The reinforcement was

enough for some walls, and it must be

increased for others. The horizontal ring reinforcement has been increased in the third

trial and it was found to be not enough, where the ring reinforcement was failed, Figure

167, (Appendix C4).

Figure 165 Reinforcement trial #1, the Figure 166 Reinforcement trial #2, the

collapse state at T=17.89 Sec collapse state at T=22.71 Sec

Figure 167 Reinforcement trial #3, the Figure 168 Reinforcement trial #4, the

collapse state at T=19.61 Sec collapse state at T=24.23 Sec

9.7 Concluding remarks 206

showed good stability but the vaults were

partially collapsed and showed large

deformations, Figure 168, (Appendix C5).

Thus, one task must be achieved for the

next step which was to add reinforcement

into the vaults, Figure 169, (Appendix C6).

The reinforcement positions in the last

models are the centres of the glass fibre

bars. For the design of the glass fibre

reinforcement the average maximum pull-

out force per 0.5 m is considered which Figure 169 Reinforcement trial #5, the

equals to 9 × 0.5 / 0.36 = 12.5 KN . collapse state at T=24.19 Sec

The other room 12 was handled in similar

manner. The reinforcement distribution can be seen in Figure 170.

The combination of reinforcement with masonry enhances the load bearing capacity and

produces better ductile failure behaviour under the higher seismic intensities. Several

issues related to reinforced masonry have been presented and discussed in this chapter.

A novel theory of the shear failure behaviour of vertically reinforced masonry shear walls

has been presented. The proposed theory is based on the force interaction between the

different constituents. The elastic behaviour has been adopted with complete combined

work between all constituents. In contrast to the failure theories of reinforced masonry in

9 Reinforced masonry 207

literature, the proposed theory gives an idealization for the behaviour after the initial

failure. The previous works, attempt to describe the progress of damage in masonry or

reinforced masonry by means of plasticity theory based on plastic potential assumption

(Huber-von Mises theory). This assumption is revealed to be used with low level of

inhomogeneity, but for composite materials like masonry or reinforced masonry there are

big inhomogeneity. Therefore, the material models based on this assumption perhaps

give prediction for the behaviour near initial failure, but after which the model ceases to

describe this phase. The proposed theory is an attempt to describe the behaviour of

reinforced masonry shear walls after initial failure and confirms that, the order of damage

occurrence is not possible to be described only by one failure surface.

The modelling of reinforcement masonry in discrete or smeared fashion has been

discussed as well. A real case study from the earthquake of Bam, 2003 has been

presented. The collapse analysis of two rooms in the adobe citadel of Bam has been

preformed and the necessary reinforcement of the structure has been developed by

means of collapse analysis.

Modelling of reinforcement is still an absent subject in finite element packages. The

current tools in use have been developed primarily for modelling problems other than

reinforcement.

10 Conclusions and recommendations 209

In this study, attempts have been made to develop numerical models capable to

simulate the collapse of large scale masonry structures under earthquake actions, which

can not be achieved experimentally. Most of research activities in few recent years were

adopted plasticity and homogenization theories for simulating the performance of large

scale masonry structures and fewer attempts were made in the field of discrete

modelling. The reasons for that have been described in section 2.5.

One of the challenging problems in collapse simulation using finite element method is

that, the model undergoes large deformations during collapse. Such problem causes the

termination of calculation in the finite element codes. The numerical techniques which

allow the emergence of discontinuities have been adopted for this purpose. The

combined finite-discrete element method which merges finite element method with the

algorithms of discrete element method allows the transition from continua to discontinua.

This method has been employed for simulation the collapse through LS-DYNA code.

The constitutive models of masonry constituents have been studied and a smooth yield

surface has been proposed for the cohesive material model to be used with interface

elements. The proposed model is multi yield surface but does not need any further

treatment of the transition points. It reduces the computation time and avoids the

treatments of corners. The proposed model has been implemented into the explicit

solver of LS-DYNA.

Attention has been given to mesh free methods as well, where SPH method has been

adopted for masonry. The obtained results for simulating a masonry shear wall have

been proofed the ability to represent all failure modes even the crushing under high

compression and the fragmentation of the material without any numerical problems. One

drawback of this method is the need to large numbers of particles for simulating

masonry, even if the model is small the computation time will be relatively big, and the

accuracy is less than that in finite element method.

A validation of the adopted models has been performed through the results of dynamic

tests of two large scale masonry structures. The developed numerical models thus, have

been offered investigations for the performance of large scale masonry structure under

the seismic actions. The failure in masonry structure has been initiated by tensile

cracking of the walls perpendicular to the shear walls, which behaviour is responsible for

the collapse.

Collapse analysis of large scale historical masonry structure has been performed with an

aim to explore the effect of different earthquake characteristics on the structure.

Unidirectional earthquakes have been applied from many directions in order to explore

the weakest direction of the structure. The effect of frequency content of the earthquake

has been explored, where the structure was showed better performance for higher

frequency contents.

A novel theory for the shear failure behaviour of vertically reinforced masonry shear

walls has been presented. In contrast to the failure theories of reinforced masonry in

literature, the proposed theory gives an idealization for the behaviour after the initial

failure. The previous works attempt to describe the progress of damage in masonry or

reinforced masonry by means of plasticity theory which based on plastic potential

assumption (Huber-von Mises theory). This assumption has been revealed to be used

9.7 Concluding remarks 210

with low level of inhomogeneity, but for composite materials like masonry or reinforced

masonry there are big inhomogeneity.

A real case study from the earthquake of Bam, 2003 has been presented. The collapse

analysis of two rooms in the adobe citadel of Bam has been preformed and the

necessary reinforcement of the structure has been developed by means of collapse

analysis.

Recommendations for prospective works

Further research can be pursued in the following directions:

(1) Solution strategies

The contact formulation is more appropriate for representing the post failure

behaviour than interface elements. It emphasizes, therefore, to offer possibility

for implementing user contact models into LS-DYNA in next versions. This allows

the researchers from different research fields to develop their own constitutive

models.

The progressive crack growth methods in fracture mechanics like Virtual Crack

Closure Technique VCCT and Discrete Cohesive Zone Models DCZM, are highly

recommended for further research work, where such methods are not embedded

in most of finite element codes. In order to study the effect of dynamic events that

cause high local distortion, failure or fragmentation, the features of the mesh free

methods are urged.

(2) Constitutive models

Masonry materials have a feature of increasing their shear strength by increasing

the confined pressure. This feature characterizes the general behaviour of geo-

materials. Several finite element codes comprise material models that cover a

wide range of geo-materials, like concrete, soils. The possibility to employ these

material models for masonry is related to the availability of enough experimental

data to get the material parameters. For example, the general triaxial empirical

laws of many masonry materials are absent in literature, therefore it is

recommended to do further research in this direction.

When developing material models at macro level, care should be taken for the

use of plasticity theory. The plastic strain flow rule of plastic potential theory

(Huber-von Mises theory) is not valid for masonry for post failure behaviour.

So far, the developed material models at macro level were for special cases, in-

plane shear loading, or out-of-plane loading. The combination of both load

regimes is absent in literature and further research works are needed in this

direction.

It has been proofed that the smooth yield surfaces are computationally stable

and efficient for implementation. It is recommended thus to apply the smooth

functions to represent the yield surface, which is especially recommended for

explicit solvers like LS-DYNA, where the material subroutine will be called in time

steps smaller than those in implicit solvers.

(3) The seismic performance of masonry structures

There is lack in design guidelines in standards for considering the effect of the

combined work of masonry shear walls with the transversal walls. The numerical

10 Conclusions and recommendations 211

models have been demonstrated that the failure of the structure can be initiated

due to the tensile cracking of the transversal walls but not due to the low capacity

of the shear walls.

The effect of earthquake direction on the structure is of great importance. There

are already some engineering methods in this direction. It is not easy to

determine the direction of the prospective earthquakes. However, it is

recommended to introduce some parameters based on the available data of the

geology, the fault system and the seismic history in the region to develop design

guidelines that consider the earthquake direction.

This study has been primarily focused on the effect of frequency content of the

earthquake and the other factors, which are related to soil-structure interactions,

are not considered. The soft soil dissipates a great amount of the kinetic energy

of the earthquake. Furthermore, the other phenomena which occur due to the

failure of soil and liquefaction should be considered for more detailed study.

(4) Reinforced masonry

The discrete modelling of reinforcement or reinforcement-masonry bonding

models are lacking in literature. The available tools in finite element packages

have been developed primarily for modelling problems other than reinforcement.

It is recommended therefore to consider this issue for further research in this

direction.

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Appendixes 227

Appendixes

Numerical Simulation Results

Appendix A2- Ispra model under earthquake of 34%g .............................................. 232

Appendix A3- Ispra model under earthquake of 40%g .............................................. 235

Appendix A4- Ispra model under earthquake of 26%g, no mortar............................. 239

Appendix A5- Ispra model under earthquake of Horz. =26%g and Ver. =40%g ....... 242

Appendix B1- Collapse simulation of the Mosque of Takiyya al-Sulaymaniyya ........ 245

Appendix B2- The effect of the earthquake direction ................................................. 247

Appendix B3- The effect of frequency contents of the earthquake ............................ 250

Appendix C1- Part of Sistai’s House under the earthquake of Bam 2003................. 253

Appendix C2- Reinforcement trial 1............................................................................ 255

Appendix C3- Reinforcement trial 2............................................................................ 256

Appendix C4- Reinforcement trial 3............................................................................ 257

Appendix C5- Reinforcement trial 4............................................................................ 258

Appendix C6- Reinforcement trial 5............................................................................ 259

Appendixes 229

Appendixes 230

Appendixes 231

Appendixes 232

The state of the structure at time=3.85 sec, the deformations are scaled 10 times,

The state of the structure at time=4.55 sec, the deformations are scaled 10 times

The state of the structure at time=4.80 sec, the deformations are scaled 10 times

Appendixes 233

The state of the structure at time=6.05 sec, the deformations are scaled 10 times

Appendixes 234

Appendixes 235

The state of the structure at time 3.3 sec. The displacements are scaled 10 times. The

collapse initiated primarily due to the tensile failure in the transversal walls.

The state of the structure at time 3.5 sec for 40%g acceleration.

The state of the structure at time 4.5 sec for 40%g acceleration.

Appendixes 236

The state of the structure at time 4.85 sec. Out of plan failure of the transversal wall

Appendixes 237

Appendixes 238

The state of the structure at time 11.5 sec for 40%g acceleration.

Appendixes 239

Appendixes 240

Appendixes 241

Appendixes 242

Ver.=40%g

Appendixes 243

Appendixes 244

Appendixes 245

Sulaymaniyya

The state of the structure at time=4 sec The state of the structure at time=5 sec

The state of the structure at time=6 sec The state of the structure at time=7 sec

The state of the structure at time=8 sec The state of the structure at time=9 sec

Appendixes 246

The state of the structure at time=10 sec The state of the structure at time=11 sec

The state of the structure at time=12 sec The state of the structure at time=13 sec

The state of the structure at time=14 sec The state of the structure at time=15 sec

Appendixes 247

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

Appendixes 248

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

Appendixes 249

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

o o o

θ=0 θ=90 θ=45

Appendixes 250

SA Sc SE

SA Sc SE

SA Sc SE

Appendixes 251

SA Sc SE

SA Sc SE

SA Sc SE

Appendixes 252

SA Sc SE

SA Sc SE

SA Sc SE

Appendixes 253

Bam 2003.

Appendixes 254

Appendixes 255

Appendixes 256

Appendixes 257

Appendixes 258

Appendixes 259

261

Tammam Bakeer

1991 – 1994 Preparatory School, Homs, Syria

1994 – 1997 Secondary School, Homs, Syria

1997 – 2002 Bsc, Civil Engineering, Al-Baath University, Homs, Syria

2002 – 2003 Diploma, Structural Engineering, Al-Baath University, Homs, Syria

2003 – 2006 Scientific Assistant at Al-Baath University, Faculty of Civil Engineering

2006 – 2008 PhD candidate at Dresden University of Technology, Faculty of

Architecture, Chair of Structural Design

Since 2008 Research assistant at Dresden University of Technology, Faculty of

Architecture, Chair of Structural Design

262

Publication Series of the Chair of Structural Design, TU Dresden

Bauforschung und Baupraxis

From Research to Practice in Construction

Previously published in this series:

Heft 1: Burkert, T.

Untersuchungen zur baukonstruktiven Ausbildung und zum

Verwitterungsverhalten der Kuppeldeckschicht beim Wiederaufbau der

Frauenkirche zu Dresden

Juni 2002

Traditional and Innovative Structures in Architecture

Februar 2004

Verwendung modifizierter Siliciumdioxid-Nanosole zum Schutz und zur

Konsolidierung von umweltgeschädigten Kulturgütern aus sächsichem

Elbsandstein am Beispiel der Skulpturen der Fasanerie Moritzburg

August 2004

Heft 4: Scheidig, K.

Die Berechnung von Maß- und Toleranzketten im Bauwesen

Dezember 2005

Heft 5: Pflücke, T.

Traglastbestimmung druckbeanspruchter Mauerwerkswände am

Ersatzstabmodell unter wirklichkeitsnaher Berücksichtigung des

Materialverhaltens

Januar 2006

Heft 6: Müller, H.

Zur mechanischen Verhaltensanalyse von Tragwerken: SATRA-DGL, STATRA-

FEM und FALT-FEM – war da noch was? Eine grobe Übersicht mit Beispielen

April 2006

Heft 7: Baier, G.

Der Wand-Decken-Knoten im Mauerwerksbau: Verfahren zur realistischen

Bestimmung der Lastexzentrizität in den Wänden

Februar 2007