Advanced computational modelling of Taq Kisra, Iraq

Advanced Computational Modelling of Taq-Kisra, Iraq

Erasmus Mundus Programme

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i

DECLARATION

Name: Nirvan Chandra Makoond

Email: [email protected]

Title of the

Msc Dissertation:


Supervisor(s): Milan Jirásek, Jan Zeman

Year: 2015

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

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

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

I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural Analysis

of Monuments and Historical Constructions is allowed to store and make available electronically the

present MSc Dissertation.

University: Czech Technical University in Prague

Date: 14 / 07 / 2015

Signature:

___________________________



ii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS

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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii

ACKNOWLEDGEMENTS

The author would like to thank Professors Milan Jirásek and Jan Zeman for their kind supervision,

solicitous availability and valuable guidance throughout the preparation of this thesis.

Sincere thanks also go to Dr. Miroslav Zeman from ProjektyZeman.cz for his time, accounts of

observations and the invaluable data and information he provided.

The author would also like to thank Michal Hlobil, Václav Nežerka and Dr. Tomáš Plachý for their help

in estimating the properties of the gypsum mortar.

The development of this thesis would not have been possible without the knowledge shared by all the

lecturers of the Advanced Masters in Structural Analysis of Monuments and Historical Constructions

and the support of classmates.

The author would also like to thank the SAHC consortium for the major financial support granted in the

form of a scholarship, without which this fulfilling experience would not have been possible.

Finally, thanks go to the author’s parents, friends and family for their unwavering support.



iv ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS




ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v

ABSTRACT

The Taq-Kisra monument, built in the 6th Century A.D. and located in Iraq is considered to be the

largest brick vault in the world. In March 2013, a Czech-based consulting company ProjektyZeman.cz

was contracted to perform an intensive investigation of the structure and to propose a strategy for its

strengthening and rehabilitation, which eventually took place from late 2013 until the end of 2014.

Because of the limited three-week time provided by the investor, the remedial measures were

designed on the basis of a two-dimensional linear elastic structural analysis using equivalent beam

elements. The present contribution aims to reconcile the remedial measures in the light of more

accurate non-linear two and three-dimensional macro-scale finite element simulations.

Comparisons of cracks observed on the actual structure were used for validation of the models used

for the purpose of this thesis.

Simulations were carried out to investigate the effects of the self-weight and environmental factors

such as wind loading, rainwater ingress in the vault and temperature variations. The reconstruction of

part of the vault, which formed an important part of the strengthening strategy recommended by

ProjektyZeman.cz, was also investigated. The most vulnerable areas were identified as well as the

effect of the various environmental factors.

Since simulations suggest that it is very likely that rainwater ingress can contribute to the propagation

of cracks, the roofing solution proposed by ProjektyZeman.cz should help limit further deterioration.

It was found that although the reconstruction of part of the vault would initially impose greater strains

in the previous existing structure due to additional weight, it would also result in an improved three-

dimensional structural integrity which should help the structure better resist additional loads. This was

verified for the most critical wind loading scenario.



vi ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS




ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii

ABSTRAKTNÍ

Monument Taq - Kisra v Iráku, pocházející z 6. století našeho letopočtu, je považován za největší

zděnou klenbu na světě. V březnu 2013 byl českou firmou ProjektyZeman.cz proveden detailní

průzkum památky s cílem navrhnout její rekonstrukci, která byla následně realizována v letech 2013 a

2014. Kvůli omezenému třítýdennímu termínu poskytnutému investorem byl návrh rekonstrukce

založen na zjednodušené analýze pomocí ekvivalentního rámového modelu s lineárně pružným

chováním. Cílem této práce je zhodnotit navržená opatření pomocí přesnějších dvoj- a trojrozměrných

nelineárních výpočtů. Výstižnost modelu je prokázána porovnáním rozložení trhlin předpovězených

výpočtem a pozorovaných na místě.

Provedené simulace zohledňovaly vlastní tíhu konstrukce a vlivy prostředí jako je zatížení větrem,

průnik dešťové vody do klenby a zatížení změnou teploty. Byl uvažován jak původní stav konstrukce

před rekonstrukcí, tak i chování rekonstruované konstrukce dle návrhu firmy ProjektyZeman.cz. Na

základě výpočtů byla identifikována nejcitlivější místa konstrukce v závislosti na vnějších vlivech.

Protože simulace prokazují, že nejpravděpodobnějším vlivem, který přispívá k šíření trhlin, je zatékání

deště, zastřešení navržené firmou ProjektyZeman.cz nepochybně přispěje k zamezení dalšího

poškozování konstrukce. Dále bylo zjištěno, že část konstrukce přidaná při rekonstrukci způsobí větší

deformaci vlivem přitížení, ale zároveň přispěje k její větší prostorové stabilitě, která umožní konstrukci

lépe přenášet další zatížení. Toto tvrzení bylo prokázáno výpočtem pro nejnepříznivější zatěžovací

stav.



viii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS




ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS ix

RESUMÉ

La modélisation avancée de Taq-Kisra en Irak par la méthode des éléments finis

Le Taq-Kisra, situé en Irak et construit au 6ème siècle après JC, est considérée comme la plus

grande voûte en brique dans le monde. En Mars 2013, la société de conseil en ingénierie

ProjektyZeman.cz, basée en République tchèque, est engagée pour effectuer une enquête

approfondie de la structure en vue de proposer une stratégie pour sa réhabilitation. Celle-ci prit fin en

2014. Les objectifs imposés par l’investisseur ont contraint l’étude structurelle à une analyse élastique

linéaire en deux dimensions utilisant des éléments équivalents de poutre. La présente contribution

vise à réconcilier les mesures correctives en utilisant des méthodes plus précises en deux et trois

dimensions.

Les modèles utilisés ont été validé au travers de comparaisons de fissures observées sur la structure

réelle.

Des simulations ont été réalisées afin d’étudier l’effet du poids de la structure en fonction des facteurs

environnementaux tels que le vent, l’infiltration de l'eau et les changements de température. La

reconstruction d'une partie de la voûte, qui forme une partie importante de la stratégie de réhabilitation

recommandée par ProjektyZeman.cz, a également été étudiée. Les zones les plus vulnérables de la

structure ont ainsi été identifiées.

Les résultats des simulations suggèrent que l’infiltration de l'eau de pluie peut contribuer à la

propagation des fissures. La solution de toiture proposée par ProjektyZeman.cz devrait ainsi

permettre de limiter sa détérioration.

On constate à travers la recherche, qu'indépendamment des contraintes imposées sur la structure

existante par la reconstruction, celle-ci génère une meilleure cohérence structurelle permettant de

mieux résister aux charges supplémentaires. Cette hypothèse a été vérifiée dans un scénario

considérant le vent le plus critique.



x ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS




ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xi

Table of Contents

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

2. LITERATURE REVIEW .................................................................................................................. 3

2.1 History of Taq-Kisra ................................................................................................. 3

2.2 Project on Reconnaissance and Restoration of Structure ........................................ 5

2.2.1 Overview of Structure ....................................................................................... 5

2.2.2 Materials Report ............................................................................................... 5

2.2.3 Structural Analysis ............................................................................................ 7

2.2.4 Main Problems Identified .................................................................................. 9

2.2.5 Selected Remedial Actions ............................................................................... 9

2.3 Previous structural analysis of Taq-Kisra ............................................................... 10

2.4 Behaviour and properties of masonry .................................................................... 11

2.4.1 Non-linear behaviour of constituents ............................................................... 11

2.4.2 Non-linear behaviour of unit-mortar interface .................................................. 12

2.4.3 Properties of the composite material ............................................................... 13

2.5 Strategies for the numerical modelling of masonry structures ................................ 14

3. GEOMETRICAL MODEL .............................................................................................................. 17

4. ELEMENT SELECTION AND MESH ........................................................................................... 21

4.1 Two-dimensional model ......................................................................................... 21

4.2 Three-dimensional models ..................................................................................... 22

4.2.1 Previous Geometry ......................................................................................... 23

4.2.2 Geometry after reconstruction as a single entity ............................................. 24

4.2.3 Geometry after reconstruction ........................................................................ 25

5. MATERIAL CHARACTERISATION ............................................................................................. 27

5.1 Properties of constituents ...................................................................................... 28

5.1.1 Bricks ............................................................................................................. 28

5.1.2 Mortar ............................................................................................................. 29

5.2 Homogenised properties ........................................................................................ 30

6. SELF-WEIGHT .............................................................................................................................. 35

6.1 Results from two-dimensional plane strain analysis ............................................... 36

6.2 Results from three-dimensional model ................................................................... 37

7. RECONSTRUCTION OF PART OF VAULT ................................................................................ 41

7.1 Results from model considering geometry after reconstruction as a single entity ... 42



xii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS

7.2 Results from model of reconstruction in two load steps .......................................... 43

8. WIND LOADING ............................................................................................................................ 47

8.1 Wind load case 1 ................................................................................................... 49

8.2 Wind load case 2 ................................................................................................... 51

8.2.1 Determination of safety factor against wind load case 2 .................................. 53

8.3 Wind load case 3 ................................................................................................... 56

8.4 Wind load case 4 ................................................................................................... 57

8.5 Wind load case 5 ................................................................................................... 58

9. RAINWATER INGRESS IN VAULT .............................................................................................. 61

10. TEMPERATURE EFFECTS .......................................................................................................... 65

11. SEISMIC HAZARD ........................................................................................................................ 71

12. COMPARISON OF LOAD CASES CONSIDERED ...................................................................... 73

13. COMPARISON WITH RESULTS FROM PREVIOUS STRUCTURAL ANALYSES .................... 75

14. CONCLUSIONS ............................................................................................................................ 77

15. REFERENCES .............................................................................................................................. 79


Erasmus Mundus Programme. 2014/2015 Edition June 2015

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1

1. INTRODUCTION

The Taq-Kisra monument located east of the Tigris River, approximately 35 km south of the modern

city of Baghdad, is the only remaining structure from the royal palace of the ancient Sassanian capital

city of Ctesiphon dating back to the 6th Century A.D. The remaining structure consists predominantly of

a part of the eastern wall and an immense parabolic barrel vault spanning a length of 25.5 m and

reaching a height of 30.3 m at its peak (Figure 1). It is thought to be the largest unreinforced brick

vault in the world. The structure is made of masonry comprising clay-fired bricks and mortar

consisting mostly of gypsum. The vault is severely degraded and in a poor condition.

Figure 1: View of remaining part of barrel vault from the east [1].

In 2013, the Czech based company AVERS was engaged as the main contractor for the restoration

and salvage of the monument. Subsequently, another Czech company, ProjektyZeman.cz, was also

requested to participate in the restoration design. The main tasks carried out by ProjektyZeman.cz

involved surveys and preliminary in-situ tests, collection of samples for determining strength

characteristics, mapping of faults and making an overall assessment of the load-bearing condition of

the structure as well as designing alterations necessary to ensure the stability and extend the service

life of the structure.

Naturally, some form of structural analysis was required to be able to assess the structural state of the

vault and design any necessary alterations. Because of the limited time available, this was done using

two-dimensional equivalent beam elements. This is not the most suitable form of analysis due to the

nature of the construction as well as the complex post-peak behaviour of masonry and the results

therefore need to be interpreted very carefully. Furthermore, three-dimensional analysis could prove to

be particularly insightful due to the unsymmetrical nature of the remaining construction (caused by

collapse of part of the vault).



2 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS

Hence, this thesis will aim to re-evaluate the structural condition of the Taq-Kisra vault as well as the

effectiveness of the repairs using more accurate two-dimensional and three-dimensional non-linear

finite element analyses. This will be achieved by employing the following methodology:

First a literature review will be carried out on the Projektyzeman.cz project, available

information on Taq-Kisra as well as on relevant aspects of non-linear modelling of masonry

structures.

Consequently, a two-dimensional non-linear finite element analysis will be carried out and the

results will be compared with conclusions made from the analysis carried out by

Projektyzeman.cz.

Subsequently, three-dimensional finite element models of the structure before and after

restoration works will be created.

Finally, these models will be used to evaluate the structural state of the monument both before

and after the restoration works.




2. LITERATURE REVIEW

2.1 History of Taq-Kisra

This section aims to provide a summary of the historical evolution of the structure, with particular

emphasis on past events that have influenced its structural integrity.

Taq-Kisra was built as a royal palace in the 6th Century A.D. in the capital of the Sassanian Empire,

Ctesiphon. The vault that can still be observed today served as the audience hall. The structure was

eventually captured by the Arabs in 637 A.D.

In the early 19th century, it can be deduced that the structure had already experienced significant

degradation with only the eastern façade, the great arch and the iwan (rectangular space, usually

vaulted, walled on three sides with one end entirely open) still standing, (Figure 2).

Figure 2: Drawing of Taq-Kisra made by Captain Hart in 1824 [2].

It is presumed that the front arch collapsed in 1887 and that a flood of the Tigris in 1888 caused the

northern part of the eastern façade to collapse. The remaining structure can be seen in Figure 3.

Figure 3: Picture of Taq-Kisra taken in 1940 by Roald Dahl [3].




In the first half of the 20th century, stabilisation works were carried out to ensure the safety of the

southern part of the eastern façade which was tilting outwards. This included adding a concrete base

in 1922 and the construction of a trapezoidal shaped masonry buttress adjacent to the wall in 1942

(Figure 4).

Figure 4: Picture of buttress built to stabilise southern part of eastern façade [1].

Restoration works were also carried out between August 1963 and March 1964 which included

reinforcement of the foundations with concrete, filling of the gaps in the masonry up to a height of 2m

above ground level with bricks and mortar made of salt-resistant cement, reconstruction of the

northern part of the eastern façade using concrete cladded by bricks, restoration of the upper part of

the monument and mending of cracks.

In the 1970s, further works were carried out including adding a concrete membrane on the extrados of

the vault and continuing the reconstruction of the northern part of the eastern façade. More works

were planned at the time but they were halted between 1980 and 1988 during the Iran-Iraq war and

also during the embargo in the 1990s that followed the First Gulf War.




2.2 Project on Reconnaissance and Restoration of Structure

This section aims to provide an overview of all the relevant information found in the project

documentation of the works carried out by ProjektyZeman.cz on Taq-Kisra.

2.2.1 Overview of Structure

A brief description of the general dimensions and materials comprising the structure is given here.

The hall under the vault has approximate dimensions of 25.5 x 47.4 m. The vault, which is completely

exposed to the environment, has a peak elevation of 30.3 m, with a thickness which increases from

1.35 m at the apex to approximately 2.10 m at the springing level. The vault is smoothly connected to

the walls which increase to a depth of 7.5 m at ground level.

The vault and walls are made of masonry consisting of clay fired brick and gypsum mortar. The units

have dimensions of 300 x 300 x 70 mm and are arranged in horizontal layers in the walls whilst those

in the vault are arranged in vertical layers. Although there are significant variations in the thickness of

the joints, the average width of the mortar joints can be considered to be 35 mm. Pieces of wood have

been found which suggests that there were probably floors made of wooden beams, but these

structures no longer exist.

There was a concrete membrane of relatively poor quality over the vault which was removed as part of

the ProjektyZeman.cz restoration project. The thickness of the concrete membrane varied from

150 mm at the peak of the vault to 350 mm at the springing level.

Many parts of the vault have collapsed, the most significant of which is a large segment on the

western side of the vault.

2.2.2 Materials Report

The investigation carried out by ProjektyZeman.cz also included the collection of samples of units and

mortar for laboratory testing. The samples (Figure 5) were then tested at the Klokner Institute of the

Czech Technical University in Prague to determine certain material characteristics and important

information on the composition and porosity of the constituent materials which could prove very useful

when estimating the material properties of the masonry composite. The main findings are summarised

in the following sub-sections.

Figure 5: Set of 9 samples that were tested (5 brick joists, 3 mortar cubes and 1 mortar joist) [4].




2.2.2.1 Bricks

It was found that the main component of the brick element is augite and that the units are generally

very porous and of relatively poor quality. The relevant properties that were determined experimentally

are summarised in Table 1.

Table 1: Properties of bricks determined experimentally.

Name Symbol Units Value

Bulk density γ kg/m3 1160

Elastic Modulus E MPa 2100

Tensile strength - dry ft (dry) MPa 1.4

Compressive strength -dry fc (dry) MPa 2.8

Tensile strength - saturated ft (sat.) MPa 1.2

Compressive strength - saturated fc (sat.) MPa 2.4

Coefficient of thermal expansion α 1/K 7.80E-06

2.2.2.2 Mortar

Thermal analysis, infrared spectrometry and microscopic analysis of the samples were used to

determine that the mortar consists primarily of gypsum (about 67% of the weight) with a smaller

degree of calcium carbonate (5.5% of weight). Small chips of bricks up to the size of 5mm were also

found in the samples. Due to the limited availability of samples, the testing program effectuated on the

mortar was less extensive than the one conducted for the bricks and hence fewer properties could be

determined. The relevant properties found are summarised in Table 2.

Table 2: Properties of mortar determined experimentally.





coefficient of thermal expansion α 1/K 1.65E-05




2.2.3 Structural Analysis

The structural analysis undertaken by ProjektyZeman.cz can be subdivided in two main sections. The

first consists of a two-dimensional evaluation of the structural state of the vault using equivalent beam

elements with different thicknesses (Figure 6). The second consists of an evaluation of the bearing

capacity of the foundation based on the European Eurocode 7. Since it was found that the bearing

capacity of the foundations are satisfactory for the loads considered, this section will focus on the

procedure, loading and findings of the two-dimensional analysis of the vault that was carried out.

Figure 6: Beam elements and location of supports used for 2D analysis by ProjektyZeman.cz [1].

The location of supports used for the analysis is shown in Figure 6. Different beams were assigned

different elastic section moduli in different directions based on the cross-sections and this was used to

evaluate the deformations and stresses caused by different loading combinations.

The dead loads considered were estimated using results from laboratory tests on units and mortar

and using combination equations from ČSN EN 1991-1. Four combinations of wind loads, calculated

according to ČSN EN 1991-1-4, were also considered as climatic load. The basic wind speed

considered for this calculation was 25.00 m/s (based on a wind speed hazard distribution map and a

fifty year return period). The four wind load cases considered are listed below and shown

schematically in Figure 7.

Wind Case 1: Wind from the left, with pressure on the left side of vault

Wind Case 2: Wind from the right, with pressure on the right side of the vault

Wind Case 3: Wind from the left, with suction on the whole vault

Wind Case 4: Wind from the right, with suction on the whole vault




Figure 7: Wind load cases considered by ProjektyZeman.cz in structural analysis [1].

The main conclusion from this structural analysis was that the vault in general has enough strength to

support its own weight despite the asymmetrical deformed geometry predicted by the structural

analysis (Figure 8). However, in certain regions, it was deemed that the vault locally cannot withstand

some loads particularly if faults arise due to overloading from the wind. This is particularly true for

areas which experience tensile stresses as shown in Figure 9. Hence, stainless steel reinforcement

was suggested for areas experiencing tensile stresses.

Figure 8: Deformed geometry of vault under self-weight based on structural analysis carried out by ProjektyZeman.cz

[1].

Wind Case 1 Wind Case 2

Wind Case 3 Wind Case 4




Figure 9: Tensile stresses under self-weight based on analysis carried out by ProjektyZeman.cz

[1].

2.2.4 Main Problems Identified

ProjektyZeman.cz identified salinity and high moisture content in the foundation and walls as one of

the most noteworthy issues. The fact that the water table is just higher than the base of the foundation

is probably one of the main causes for the high level of moisture present at the base of all the walls.

Although the foundations were considered adequate after the structural analysis, ProjektyZeman.cz

has specified that the salinity and moisture are problems that need to be addressed.

Another problem identified is the big temperature differences that are experienced by different parts of

the structure. This was detected by ProjektyZeman.cz through a survey conducted using a

thermographic camera. It is highly probable that humidity and temperature cycles give rise to a

degradation process in the material. These problems have most probably been aggravated by the lack

of roofing and waterproofing for the protection of the vault.

Damages and cracks in the vault were also documented. Remaining segments of the vault after the

collapse of other segments were identified as a significant problem since they could no longer achieve

equilibrium and were therefore unstable.

2.2.5 Selected Remedial Actions

ProjektyZeman.cz has proposed an Electro-osmosis method for reducing moisture and salinity in the

walls up to a height of 2m.

A double-skin ventilated roof with a ventilated air gap was also proposed to reduce the effect of

temperature and humidity cycles. This should also help to prevent ingress of water in the vaults and

walls.

Another remedial action suggested by ProjektyZeman.cz involved the stabilisation of the cracks in the

masonry vault with reinforcement. Finally, the reconstruction of part of the vault that has collapsed on

the western side for static activation resulting in less unstable segments was also proposed. It is

important to note that it was specified that this is to be built using reproduced materials with similar

characteristics as the original material.




2.3 Previous structural analysis of Taq-Kisra

Another form of structural analysis of the Taq-Kisra vault was performed as part of a case study by

J.F.D Dahmen and J. A. Ochsendorf. Since high self-weight and low compressive stresses usually

characterise unreinforced masonry vaults, they can be regarded primarily as problems of stability

rather than strength. Hence a structural analysis technique based on plastic theory; more particularly

on the lower bound theorem of limit analysis was used to investigate the structural stability of the Taq-

Kisra vault. The analysis is based on three underlying fundamental assumptions [5]:

Masonry has no tensile strength

Masonry effectively has unlimited compressive strength due to stresses being so low

No failure due to sliding

Static graphics can then be used to find multiple paths of compressive only forces known as thrust

lines. The lower bound theorem states that if one thrust line can be found to lie entirely within the

masonry, the structure can be demonstrated to be safe. The result from this analysis for the Taq-Kisra

vault is shown in Figure 10.

Figure 10: Static graphic analysis of Taq-Kisra vault [6].

As can be seen from Figure 10, a thrust line can be found to lie within the masonry in the case of the

Taq-Kisra vault. This leads to suggest that the geometry alone is not responsible for the collapsed

segments and that material degradation along with other events in the past have had an important

influence in causing the damages that can be observed today. Nevertheless, it can be seen that the

thrust line is particularly eccentric in different parts of the vault. These eccentricities could suggest that

the structure is in a somewhat precarious state of equilibrium and that some areas of the vault could

be particularly susceptible to damage when experiencing external forces.




2.4 Behaviour and properties of masonry

Masonry is one of the oldest building materials to have been used, with earliest uses dating back more

than 10,000 years. It is a heterogeneous material consisting of units and joints. Units are typically

made of stone, clay-fired bricks, earth bricks or concrete blocks held together with some form of

mortar (except in the case of dry stone masonry where no mortar is used).

Naturally, due to its widespread use over a long period of time, there is a strong variability in the

different arrangements and materials used for masonry structures. This has made the prediction of

material properties of masonry a very difficult task. Furthermore, masonry usually exhibits distinct

directional properties due to its heterogeneous nature. These two aspects of masonry prove

particularly problematic for modern structural analysis using the finite element method since this

requires a mathematical description of the relation between the stress and strain tensor in a material

point of the body, known as a constitutive model. Hence, accurate numerical modelling of masonry

structures often requires an extensive testing campaign. This poses many difficulties, particularly in

the case of historic structures where regular sampling methods and standards cannot always be

applied.

2.4.1 Non-linear behaviour of constituents

The properties of masonry are strongly dependent upon the properties of its constituents. In most

cases, these are quasi-brittle materials exhibiting non-linear behaviour that is characterised by

softening. Softening is the gradual decrease of mechanical resistance under a continuous increase of

deformation forced upon a material specimen or structure [7]. Softening behaviour can be attributed to

the process of progressive crack growth which leads to failure in quasi-brittle materials.

For tensile failure, the phenomenon of softening has been well-identified and a characteristic stress-

displacement diagram for quasi-brittle materials is shown in Figure 11. The figure also shows the

tensile strength (ft) and the definition of fracture energy as the integral of the stress-displacement

diagram.

Figure 11: Typical behaviour of quasi-brittle materials under uniaxial tension [7].

For shear failure, a softening process is also observed as a degradation of the cohesion in Coulomb

friction models [7].




Softening behaviour is also present for compressive failure; however, the softening behaviour is highly

dependent on the boundary conditions. A characteristic stress-displacement curve for a quasi-brittle

material under uniaxial compression is shown in Figure 12.

Figure 12: Typical behaviour of quasi-brittle material under uniaxial compression [7].

As can be seen from Figure 12, typical quasi-brittle materials used in masonry usually experience a

small hardening phase before the onset of the softening behaviour under compression.

A discussion of test methods and empirical equations which can be used to predict the nonlinear

mechanical properties of masonry constituents described above such as strength and fracture energy

will not be given here but can be found in [7]; [8]; [9].

2.4.2 Non-linear behaviour of unit-mortar interface

Under many circumstances, the bond between the unit and mortar has often been found as the

weakest link in masonry assemblages. This makes the non-linear response of the joints one of the

most relevant features of masonry behaviour. In general, two different failure modes can be

associated with the unit-mortar interface: One linked with a tensile failure (mode I) and the other linked

to a shear failure (mode II).

An exponential softening curve is most often associated to the tensile failure mode with a non-linear

response similar to the one depicted in Figure 11.

The other failure mechanism linked to the joints consists of a slip of the unit-mortar interface under

shear loading. The typical shear stress-displacement diagram for this type of failure is shown in

Figure 13.

Figure 13: Behaviour of masonry under shear [7].




It is clear from Figure 13, that the response of masonry joints to shear loading is strongly dependent

on the level of stress normal to the joint. The higher the level of stress normal to the joint, the higher is

the resistance to shear loading. For this type of loading, the bond shear strength or cohesion (see

Figure 13) and the mode II fracture energy, defined as the integral of the τ-δ diagram are parameters

that can be used to describe the non-linear behaviour. A Coulomb friction model based on cohesion

and a friction angle to define the failure envelope in terms of shear and normal stresses is often used

to model this type of failure.

Once again, testing methods and empirical relations which exist to estimate the mechanical properties

described in this section will not be discussed here but can be found in [7]; [8]; [9].

2.4.3 Properties of the composite material

Due to the anisotropic nature of masonry, it is useful to describe its composite behaviour with regard

to the material axes, namely the directions parallel and normal to the bed joints.

For a long time, the compressive strength of masonry in the direction normal to bed joints was

regarded as the only relevant structural material property. Uniaxial compression of masonry results in

a state of tri-axial compression in the mortar and a state of compression and bi-axial tension in the

units. It is commonly accepted that the difference in elastic properties of the unit and mortar is the

precursor of failure under this type of loading. Testing procedures such as the RILEM test or the

stacked bond prism test have been developed to determine the uniaxial compressive strength of

masonry, with results from the RILEM test considered as being more accurate. Nevertheless, it should

be noted that the specimen required for these tests are relatively large and costly to execute.

The uniaxial compressive strength of masonry parallel to bed joints is usually lower than that

perpendicular to bed joints. It has received substantially less attention, despite the inherent anisotropic

nature of regular masonry and the fact that very low longitudinal compressive strengths could have a

significant effect on load bearing capacity. Previous research suggests that the ratio between the

uniaxial compressive strengths of masonry parallel and normal to bed joints range from 0.2 to 0.8 [10].

Failure in tensile loading perpendicular to bed joints is usually caused by failure of the bond between

the bed joint and the unit and the tensile strength in this case can be approximated to the tensile bond

strength. In cases where the units have lower tensile strength than the bond between the units and the

joints, the tensile strength of the masonry assemblage can be estimated as the tensile strength of the

unit.

For tensile loading parallel to the bed joints, a previous experimental program has identified two main

types of failure: stepped cracks running through head and bed joints or cracks running almost

vertically through the units and the head joints. In the first case, the post-peak response is governed

by the tensile behaviour of the head joints and the behaviour of the bed joints under shear. In the

second case, the post-peak response is governed by the tensile behaviour of the head joints and the

units.




Investigations have also been made on the biaxial behaviour of masonry which cannot be completely

described from the constitutive behaviour under uniaxial loading conditions. Due to the anisotropic

nature of masonry its biaxial strength envelope cannot be described simply in terms of principal

stresses. Instead, it has to be described either in terms of the full stress vector in a set of material axes

or in terms of principal stresses and the rotation angle between the principal axes and the material

axes.

2.5 Strategies for the numerical modelling of masonry structures

Due to the distinct directional properties and the inherent complexities of the material behaviour

described in the previous section, different strategies with different levels of sophistication have been

used for the numerical representation of masonry structures. In general, modelling can focus on the

individual components or on the masonry composite as a whole. Depending on the application and

level of accuracy required, one of the following three strategies can be used [8]:

Detailed micro-modelling: units and mortar are modelled using continuum elements whereas

the interface between the two is modelled using discontinuum elements.

Simplified micro-modelling: expanded units are modelled using continuum elements whilst the

behaviour of the mortar and interface are combined in discontinuum elements.

Macro-modelling: In this strategy, no distinction is made between the units and the joints and

masonry is modelled as a homogeneous isotropic or anisotropic continuum.

See also Figure 14.

Figure 14: Different modelling strategies for masonry: (a) detailed micro-modelling, (b) simplified micro-modelling,

(c) macro-modelling [7].

The first strategy allows for a very accurate description of the masonry and the combined action of

unit, mortar and interface can be studied in depth. Some accuracy is lost in the second approach as

the Poisson effect of the mortar is not taken into consideration. Complete micro-models can include all

the failure mechanisms of masonry namely cracking of joints, sliding over head or bead joints,

cracking of the units and crushing of the masonry. Naturally, micro-models require a lot of

computational effort as well as a geometrical description of all the units and joints. Hence, they cannot

be feasibly used to model whole large structures; they are mainly applied to modelling small structural

details or to obtain a better understanding of the composite behaviour.

Subsequently, macro-modelling is the only strategy that can be used in practice to model large

structures. It should be noted that macro-models cannot account for shear failure at the joints since

unit and mortar geometries are not discretised. Therefore, failure has to be linked to tension and




compression modes in principal stress space. Naturally, different material models and failure criteria

accounting for the nonlinear post-peak behaviour of masonry can be used. Defining the homogenised

material parameters which control the material behaviour is therefore of utmost importance in

obtaining reliable results.








3. GEOMETRICAL MODEL

The basic information used as a starting point for the construction of the geometrical models consisted

of surveys conducted by ProjektyZeman.cz. The data were provided in the form of eight cross-sections

from a front view perspective and their positions on a plan of the structure as well as engineering

drawings of the interior view of the two walls supporting the vault both before and after the

reconstruction of part of the vault.

It is important to note that since this thesis aims to evaluate the structural condition of the vault, only

the vault and the side walls supporting it were modelled to save computing time and memory

requirements. The historical picture presented in Figure 3 clearly shows that the façade walls on the

sides at the front of the structure were clearly detached from the side walls of the vault. Furthermore, it

can be seen in Figure 15, that the connection between the sidewalls supporting the vault and the back

wall can be considered as being merely superficial. In fact, the sidewalls appear to be lying next to the

back wall with only a thin line of connection with the back wall. Hence, the vault and the sidewalls can

be considered to behave independently from other parts.

Figure 15: Junction between side walls supporting vault and back wall; (a) northern side, (b) southern side [1].

Determination of the geometry to be used for the two-dimensional analysis was relatively

straightforward as it was simply based on a typical cross-section which best represents the vault. The

geometry is shown in Figure 16.

Figure 16: Geometry used for two-dimensional model (dimensions in metres).

(a) (b)




The three-dimensional geometry was first constructed by extruding eight cross-sections and applying

appropriate cuts according to the profiles of the side walls and of the damaged vault. The resulting

three-dimensional geometry after this process is shown in Figure 17.

Figure 17: three-dimensional geometry before simplification.

As can be seen from Figure 17, there are many small variations in the topography of the walls and of

the vault. Although these variations are unlikely to have any significant impact on the response of the

structure, they would certainly cause many difficulties while creating a mesh and could result in

localised, spurious and mesh-related stresses during the analysis. Hence, it was decided that a

simplified geometry (Figure 18) which eliminates small irregularities would be used in order to obtain a

more computationally efficient model.

Figure 18: Simplified three-dimensional geometry

The next step consisted of partitioning the volume into quasi-homogeneous parts, in order to partially

reproduce the heterogeneities of the structure. The different parts considered are listed below and

shown in Figure 19:

Dry masonry with bricks in horizontal layers

Wet masonry with bricks in horizontal layers

Dry masonry with bricks in vertical layers

South-east view North-west view

South-east view North-west view




Figure 19: Different parts modelled using homogeneous quasi-brittle material models.

The survey undertaken by ProjektyZeman.cz involved measurements of moisture and the height of the

rising damp was recorded at several points. The average value of these measurements (3 m) was

used to define the boundary between the wet masonry and the dry masonry in the model. Since the

materials report also included results on saturated specimens, it was deemed that the loss in strength

of the material due to humidity could be taken into account by reducing the strength and other

properties based on the experimental results. The material models and parameters used are

explained in greater detail in Section 5.

An important part of the restoration works carried out by ProjektyZeman.cz was the reconstruction of

part of the vault using a compatible material. It was important to model this as this thesis also aims to

evaluate the structural state of the structure after restoration. The geometry of the reconstructed part

is presented in Figure 20.

Figure 20: Reconstruction of part of the vault.

It can be seen in Figure 16 as well as in the figures of the simplified three-dimensional geometrical

models that the vault has been divided in 4 sections (surfaces in two dimensions and volumes in three

dimensions). These partitions were made so that the wind load on the vault could be applied as

recommended by the Eurocode 1. This is described in further detail in Section 8.

Bricks in vertical layers

Bricks in horizontal layers (dry)

Bricks in horizontal layers (wet)








4. ELEMENT SELECTION AND MESH

4.1 Two-dimensional model

Linear triangular elements with a one-point integration scheme were chosen to model the structure in

two dimensions. Although these elements tend to be quite stiff and particularly prone to “locking”, the

relatively low computational effort required to run the two-dimensional simulations allowed a very fine

mesh to be used with good quality elements which should produce good quality results (see Figure

21). A plane strain idealisation was used for these elements since the vault cannot be modelled as a

thin body, and it was therefore deemed that assuming all strains normal to the front face being zero

would provide better results. All elements were assigned a thickness of 1 m.

A trial and error procedure was used to obtain the final mesh. A shape quality criterion based on the

likeness of each element to an equilateral triangle was used to compare different meshes. The

mathematical expression of this quality measure is as follows:

𝑞 =4∙√3∙𝐴

∑ 𝑙𝑖23

𝑖=1

[11]

where A is the area of the triangle and li are the lengths of the triangle’s edges.

An equilateral triangular element will have a value of one, with decreasing values as the shape of the

element becomes worse. A negative value indicates that the element has a negative Jacobian at

some point. Hence, a strong requirement of the final mesh was that no elements could have a

negative shape quality criterion.

An unstructured mesh consisting of 16,411 elements, created using the Rsurf surface mesher

available in the GiD pre-processor [11], with elements having an average assigned side length of 0.3m

was chosen for the final model. The mesh and the cumulative distribution of the quality criterion

described above can be seen in Figure 21.

Figure 21: a) Mesh used for two-dimensional model; b) Cumulative distribution of shape quality criteria.

It can be seen that due to the small size of elements used, the software was able to generate very

good quality elements, most of which have a shape quality criterion, q, greater than 0.9.

(b) (a)




4.2 Three-dimensional models

Linear tetrahedral elements with a one-point integration scheme were employed for three-dimensional

problems. Once again, the final mesh used to run the analyses was chosen using a trial and error

procedure in order to obtain a mesh which would provide a good balance between quality of results

and computational effort. A shape quality criterion based on the likeness of each element to a regular

tetrahedron was used to compare different meshes. The mathematical expression of this quality

measure is as follows:

𝑞 =6∙√2∙𝑉

∑ 𝑙𝑖36

𝑖=1

[11]

where V is the volume of the tetrahedron and li are the lengths of its edges.

A regular tetrahedral element will have a value of one, with decreasing values as the shape of the

element becomes worse. Once again, no elements were allowed to have a negative shape quality

criterion as this would be indicative of a negative Jacobian at some point.

The three-dimensional shape contained many small corners and boundaries between volumes,

particularly in the case of the previous geometry before the reconstruction of the vault, which could

prove particularly difficult to mesh. This meant that the model could be susceptible to inaccurate

mesh-related results in certain areas. This was also taken into consideration and a mesh which did not

exhibit such spurious results was chosen for the final analyses.

Three different meshes were used for the final analyses. The first one was used to model the previous

geometry before the reconstruction of part of the vault. The second mesh was used to model the

geometry of the vault after the reconstruction as a single entity. Naturally, the behaviour of this model

would be unrealistic as the reconstruction was added on the already loaded and deformed previous

geometry of the vault. However, it provides some qualitative insight on how the reconstructed

geometry would behave under additional loading such as wind loads and was therefore included as a

part of this thesis. Finally, a third mesh was used to model the reconstruction in two loading steps in

order to obtain a more realistic understanding of the behaviour of the structure after the reconstruction.

Each of the three meshes used are described briefly and shown in the following sub-sections.




4.2.1 Previous Geometry

An unstructured mesh consisting of 180,913 elements, created using the Advancing front volume

mesher available in the GiD pre-processor [11], was used to model the previous geometry before the

reconstruction of part of the vault. Elements making up the side walls were assigned an average side

length of 1 m. Since most of the damage was expected in the vault, a smaller average side length of

0.5 m was assigned to elements used to construct it. This was done for all the meshes of three

dimensional models. The mesh is shown in Figure 22 while the cumulative distribution of the shape

quality criterion appears in Figure 23.

Figure 22: Mesh used for three-dimensional model of geometry before reconstruction of part of vault.

Figure 23: Cumulative distribution of shape quality criterion for three-dimensional model of geometry before

reconstruction of part of vault.




4.2.2 Geometry after reconstruction as a single entity

This geometrical model allowed a mesh with much fewer elements to be used for satisfactory results

without compromising the shape quality of the elements, since there were fewer irregularities in the

volumes. The resulting unstructured mesh consisted of 92,404 elements and can be seen in Figure

24. The cumulative distribution of the shape quality criterion is shown in Figure 25.

Figure 24: Mesh used for three-dimensional model of geometry with reconstruction considered as a single entity.

Figure 25: Cumulative distribution of shape quality criterion for three-dimensional model of geometry with

reconstruction considered as a single entity.




4.2.3 Geometry after reconstruction

Finally, an unstructured mesh consisting of 185,190 elements was used to simulate the reconstruction

on the deformed previous geometry after its self-weight had already been accounted for. The mesh is

shown in Figure 26 and the cumulative distribution of the shape quality of elements making up this

mesh is presented in Figure 27.

Figure 26: Mesh used to model reconstruction of part of vault.

Figure 27: Cumulative distribution of shape quality criterion for three-dimensional model of geometry with

reconstruction.








5. MATERIAL CHARACTERISATION

As previously discussed, the structure consists broadly of two mesoscopic heterogeneous patterns,

namely bricks in horizontal layers and vertical layers (Figure 19). In addition to this, a different material

was assigned to areas below the level of rising damp in order to account for the loss of strength of the

material due to moisture. The non-linear mechanical behaviour of the masonry in all regions were all

modelled using the CC3DNonLinCementitious2 plastic fracturing material model available in the

ATENA commercial code. [12].

This model assumes initial isotropy and adopts a quasi-brittle constitutive law with tensile behaviour

governed by the Rankine failure criteria with exponential softening while the compressive behaviour

makes use of the Menétrey-Willam failure surface with hardening and softening phases. The

equivalent uniaxial stress-strain diagram employed by this model is shown in Figure 28(a) and the

biaxial failure criterion is shown in two-dimensional principal stress space in Figure 28(b). Fracture is

modelled using the orthotropic smeared crack formulation and the fixed crack model with a mesh

adjusted softening modulus. It is important to note that one of the main drawbacks of this material

model is that it does not account for the anisotropic nature of masonry. However, this is partially taken

into account indirectly by using different material models for regions where bricks are in horizontal and

vertical layers.

Figure 28: (a) Uniaxial stress-strain law for CC3DNonLinCementitious2; (b) Biaxial failure criteria [12].

In order to represent a specific material with this model, the following parameters have to be defined:

Tensile strength, ft

Compressive strength, fc

Elastic modulus, E

Poisson’s ratio, ν

Tensile fracture energy, Gf

Limit compressive crack opening, wd

Studies on Poisson’s ratio of masonry assemblages have found an average initial value of 0.17 for

clay brickwork [10] and hence this value was used in all the different material models.

(a) (b)




The limit compressive crack opening is the plastic displacement which defines the end point of the

softening curve in compression. Since there is very little literature on this parameter, the default value

of 0.5mm, based on the experiments of Van MIER on normal concrete [12], was used for all the

material models. It should be noted that this parameter only comes into play when defining localised

damage after the peak compressive stress. Due to the large thickness of the walls at the base, it can

be expected that the compressive stresses being experienced by the material will be very low and

hence the model should still give satisfactory results. An examination of the highest compressive

stresses and strains after modelling the structure under its own self-weight was used to confirm this

hypothesis.

The constitutive model automatically calculates the onset of crushing (shown by f0 in Figure 28) as

2.1∙ft, but it also provides the option of entering it manually. Since experimental results have

suggested the linear-parabolic stress-strain relationship shown in Figure 29, the value of f0 was set to

0.75∙fc for all the material models.

Figure 29: Proposed stress-strain relationship for masonry assemblage [13].

The remaining material parameters listed above were estimated using results from experiments made

on the constituents (bricks and mortar) and existing empirical relations.

Before this was done for the homogenised materials, the nonlinear properties of the constituents were

estimated after gathering the properties available from testing. Although these values were not all

used directly within the scope of this thesis, they could be used for micro-modelling of the masonry.

This would allow both the bricks and the mortar to be modelled as quasi-brittle material. By defining a

suitable periodic cell with appropriate interface elements and the application of periodic boundary

conditions, a computational homogenisation procedure could be used to estimate the nonlinear

properties of the different masonry assemblages and these could be compared to the estimates made

as part of this thesis.

5.1 Properties of constituents

5.1.1 Bricks

The material parameters of the bricks available from testing were: bulk density, elastic modulus,

tensile strength (dry and saturated condition) and compressive strength (dry and saturated condition).

Hence only the Poisson’s ratio and the tensile fracture energy had to be estimated to be able to define

the non-linear behaviour of dry units. A Poisson’s ratio of 0.17 was assumed (Section 4). The fracture




energy was estimated using the ductility index, du, given by the ratio between the fracture energy and

the tensile strength [9]:

𝐺𝑓 = 𝑑𝑢 ∙ 𝑓𝑡

A value of du = 0.029mm was used as recommended in [9] for brick and mortar in the absence of more

information. This results in the fracture energy of 40.6 Nm-1

.

The material parameters of the dry bricks are summarised in the table below along with properties

obtained experimentally on saturated samples.

Table 3: Summary of brick material properties.



Elastic Modulus E MPa 2100

Tensile strength - dry ft (dry) MPa 1.4


Poisson’s ratio* ν - 0.17

Fracture Energy* Gf Nm-1

40.6

Tensile strength - saturated ft (sat.) MPa 1.2


*estimated values

5.1.2 Mortar

Due to a more limited range of mortar samples, there were less experimentally obtained material

parameters available. Only the bulk density and the compressive strength (both for dry and saturated

samples) were obtained from testing. The Poisson’s ratio was once again estimated as 0.17.

The elastic modulus of the mortar proved particularly difficult to estimate since only a limited amount of

research has been carried out on gypsum mortars and there are considerable variations in the moduli

of different mortars of this type. Nevertheless, information from the materials report such as porosity

helped indicate what range of values was reasonable. Subsequently, a trial and error procedure using

different empirical relations developed for concrete was used. The formulation recommended by the

Architectural Institute of Japan [14], which relates the modulus of elasticity to the compressive strength

and the bulk density, provided reasonable results:

𝐸 = 21000 (𝛾

2300)

1.5

(𝑓𝑐

20)0.5

with E and fc in MPa and γ in kg/m3.

The elastic modulus was estimated to be 4690 MPa.




The relationship suggested by the CEB-FIP Model Code 90 was used to estimate the tensile strength

(in MPa) of the mortar from the compressive strength. The relationship suggested is [12]:

𝑓𝑡 = 0.24 ∙ 𝑓𝑐𝑢

23

with fcu representing the compressive strength, in MPa, obtained by laboratory testing on cubic

samples.

The tensile strength was estimated as 0.6MPa.

The tensile fracture energy as well as the Poisson’s ratio were estimated using exactly the same

approach as that described in Section 5.1.1 for the bricks.

The material parameters of the dry mortar are summarised in the table below along with the

compressive strength of saturated samples obtained experimentally.

Table 4: Summary of mortar material properties.



Elastic Modulus* E MPa 4690

Tensile strength – dry* ft (dry) MPa 0.6


Poisson’s ratio* ν - 0.17

Fracture Energy* Gf Nm-1

17.2


*estimated values

5.2 Homogenised properties

Material properties for four different material models were estimated:

Dry masonry with bricks in horizontal layers

Wet masonry with bricks in horizontal layers

Dry masonry with bricks in vertical layers

Wet masonry with bricks in vertical layers

The first three material models listed are used in every model. However, the fourth material model (wet

masonry with bricks in vertical layers) has only been used to investigate the effect of rainwater ingress

in the vault through a reduction in its strength.

The first material property that was estimated was the compressive strength of the masonry

assemblage. This was done using the relationship suggested in Eurocode 6 between the

characteristic compressive strength of the masonry assemblage, the normalised mean compressive

strength of the units (fb) and the mortar strength (fm) [15]:

𝑓𝑘 = 𝐾 ∙ 𝑓𝑏0.7 ∙ 𝑓𝑚

0.3




K is a constant whose value can be approximated using a table available in the Eurocode, it was set

as 0.55. Both the compressive strengths of the mortar and the units have been obtained

experimentally. However, the compressive strength of the units (derived from tests on cubes) had to

be reduced as specified by the Eurocode. The bricks in horizontal layers are likely to experience

compressive loads mostly perpendicular to the bed joints, which is the most common scenario in

masonry structures. Hence the Eurocode formulation described above was used to estimate the

homogenised compressive strength for this particular arrangement of bricks. Since the compressive

strength of the mortar and units had also been obtained experimentally on saturated samples, the

same procedure could be used to approximate the characteristic strength of masonry with horizontal

brick layers under damp conditions. The calculated values are shown in Table 5.

In the absence of more information, the tensile strength of masonry was estimated as 1/10th of the

compressive strength as suggested in [16].

In order to calculate the tensile fracture energy, the modified recommendation from the CEB-FIB

Model Code 90 was used. This is based on the assumption that the relation between tensile and

compressive strength of concrete is 5%. The expression is as follows:

𝐺𝑓 = 0.025 ∙ (2𝑓𝑡)0.7 [9]

Although this relation has been recommended for bricks or for mortar, in the absence of more

information, it has been used to estimate the fracture energy of the different masonry arrangements

considered (Table 5).

It can be expected that the masonry with bricks in vertical layers would be experiencing the greatest

compressive loads in the direction parallel to bed joints. Since the resistance to compressive loads

parallel to bed joints is usually less than that normal to bed joints, an attempt was made to reduce the

characteristic compressive strength defining the material model representing this arrangement. Very

little information is available for masonry strengths under this arrangement and the only indication

found was that the ratio between the uniaxial compressive strength parallel and normal to bed joints

ranges from 0.2 to 0.8 [7].

Hence, a small parametric study was carried out to investigate the effect of changing the ratio between

compressive strength parallel and normal to bed joints from 0.2 to 0.8. The remaining material

parameters, namely tensile strength, fracture energy and elastic modulus were estimated from the

compressive strength in the same way they were for the masonry with bricks in horizontal layers.

Subsequently, a full three-dimensional analysis of the structure under its self-weight was carried out

for three different ratios within the suggested range (0.2, 0.6 and 0.8). The main cracks resulting from

these simulations as well as a comparison of the maximum displacement for each case is shown in

Figure 30.




Figure 30: Simulations run with different ratios between compressive strength parallel and normal to bed joints;

(a) cracks for ratio of 0.2, (b) cracks for ratio of 0.6, (c) cracks for ratio of 0.8, (d) comparison of maximum displacements

As can be seen from Figure 30(d), the maximum displacement obtained as a result of using a ratio of

0.2 is much greater than the displacements obtained with ratios of 0.6 and 0.8. Furthermore, the

cracks are much more localised in the case of a ratio of 0.2 in spite of the greater displacement. In

order to be able to study the response of the structure as a whole, the distribution of cracks exhibited

by the cases where ratios of 0.6 and 0.8 were used are more desirable since they allow for a better

understanding of vulnerable areas. Local irregularities are likely to be more prominent in the case of a

ratio of 0.2 and results could be less representative of the structural response. Areas experiencing

damage can be visualised better in the case of a ratio of 0.6 when compared to a ratio of 0.8. Hence,

a ratio of 0.6 was chosen to define the compressive strength of masonry parallel to bed joints.

Other properties estimated with volumetric averages were the bulk density (used to define the self-

weight of the structure) and the coefficient of thermal expansion. Those properties had been

determined experimentally for the constituents. Due to the periodic nature of the brick arrangements, a

representative volume element (RVE) could be used to estimate the volume fraction that each of the

constituents occupies. The RVE (Figure 31) was constructed according to the requirements listed in

[17]. It was estimated that the masonry consists of 54% brick and 46% mortar.

Figure 31: Representative volume element.

All the homogenised material parameters estimated are summarised in Table 5.

(a) (b)

(c) (d)




Table 5: Summary of homogenised material properties for different masonry arrangements.

a) Eurocode6: fk=K∙fb0.7

fm0.3

b) Ratio between compressive strength parallel and perpendicular to bed joints taken as 0.6

c) Taken as approximately 1/10th of compressive strength

d) Eurocode6: E=1000fk

e) Approximation adopted from CEB-FIB Model Code 90: Gf=0.025∙(2ft)0.7

Cracks predicted by the models under different loading scenarios were compared with cracks

observed on the actual structure for validation of the model. These comparisons are described in parts

of the following sections which explain how the different load cases were modelled and describe the

most relevant results.

Horizontal

layers

Horizontal

layers

(wet)

Vertical

layers

Vertical

layers

(wet)

Name Symbol Units Value Value Value Value

Compressive strength fc MPa 1.56a 0.93

a 0.93

b 0.56

b

Tensile strength ft MPa 0.16c 0.10

c 0.10

c 0.06

c

Elastic Modulus E MPa 1555d 928

d 933

d 557

d

Poisson's ratio ν - 0.17 0.17 0.17 0.17

Fracture Energy Gf N/m 11.0e 8.1

e 8.1

e 5.7

e

Limit compressive crack opening wd m 0.0005

Bulk density

kN/m3 12.73

Coefficient of thermal expansion α 1/K 1.2E-05








6. SELF-WEIGHT

The self-weight of the structure was modelled by simply prescribing the bulk density of the material

(estimated as 12.73 kN/m3) as a load. For the two-dimensional model, this was done by assigning a

weight for two-dimensional surface component to all the surfaces making up the geometry of the vault

whilst weights were assigned to volumes making up the geometry of the three-dimensional model.

Fixed boundary conditions restraining movement in any direction at the base of the structure were

prescribed using line constraints for the two-dimensional model and surface constraints for the three-

dimensional model. This was considered appropriate since the investigation carried out by

ProjektyZeman.cz shows that the foundations are adequate to support the structure which is most

likely not susceptible to experiencing differential settlements. All the prescribed loading and boundary

conditions for the two-dimensional and three-dimensional model are shown schematically in Figure 32

and Figure 33 respectively.

Figure 32: Schematic representation of boundary conditions to model self-weight with two-dimensional model.

Figure 33: Schematic representation of boundary conditions to model self-weight with three-dimensional model.




6.1 Results from two-dimensional plane strain analysis

The deformed shape predicted by the two-dimensional plane strain analysis is shown in Figure 34 with

a deformation scale of 1:500. It is clear that the geometry of the vault leads to a greatly asymmetric

response under its own self-weight. A similar asymmetrical deformed shape was predicted by the

structural analysis carried out by ProjektyZeman.cz. This shows that despite appearing symmetrical at

first sight, the geometry itself results in a skewed loading distribution under self-weight which results in

certain areas being more prone to experiencing damage.

Figure 34: Deformed shape under self-weight predicted by two-dimensional plane strain analysis.

Nevertheless, the nonlinear two-dimensional plane strain analysis predicts no cracks in spite of the

low strength properties of the material. It is important to note that the finite element package used

considers tensile stresses and strains as positive whilst compressive stresses and strains are

considered negative. As can be expected due to the large thickness of the walls, the highest

compressive strains and stresses indicate that the material is still exhibiting elastic behaviour in

compression in all parts of the structure.

An examination of the tensile strains indicates that the material has not yet experienced softening at

any point in the structure. Furthermore, no reduction in tensile strength due to stresses was yet to be

experienced by the material model in any part of the structure. Hence, the maximum principal

stresses, shown in Figure 35, would still give a good idea of what parts of the structure are most

susceptible to experiencing damage.




Figure 35: Maximum principal stress under self-weight based on two-dimensional plane strain analysis

(values shown in MPa).

Figure 35 shows a region closer to the intrados on the right side of the vault where the material is

experiencing tensile stresses close to its tensile strength. This is the same region where the structural

two-dimensional analysis conducted by ProjektyZeman.cz predicts there will be tensile stresses.

Moreover, the project documentation from ProjektyZeman.cz states that cracks have been observed in

this region of the structure.

6.2 Results from three-dimensional model

Similarly to the two-dimensional model, the three-dimensional model of the geometry before the

reconstruction of part of the vault also displays an asymmetrical deformed shape under its self-weight.

This is shown using a deformation scale of 1:500 in Figure 36.

Figure 36: Deformed shape predicted by three-dimensional model under self-weight.




Furthermore, the three-dimensional model also predicts cracks in the intrados of the vault on the right

hand side and very small cracks on part of the extrados as shown in Figure 37.

Figure 37: Crack width distribution predicted by three-dimensional model of previous geometry under self-weight

(values shown in metres).

It is important to note that the project documentation from ProjektyZeman.cz reports finding cracks on

the structure in the region of the intrados where the three-dimensional model forecasts damage.

Further investigation has revealed that it was observed that these cracks were mostly horizontal in the

longitudinal direction of the vault as the smeared cracks of the model seem to suggest. This was used

for validation of the model.

In addition to this, the region where very small cracks have been predicted by the model on the

extrados appears to be where one of the most prominent cracks on the extrados of the actual

structure is located, as can be seen in Figure 38 which shows the back of the vault.

Figure 38: Crack on vault extrados [1].




It is obvious that the crack that can be observed in Figure 38 is certainly more prominent than the type

of damage predicted by the model under the self-weight only, however, some other load cases

considered trigger the propagation of this crack in the model. This will be discussed further in

subsequent sections of the thesis.

The two-dimensional model failed to predict any cracks most probably because it cannot take into

account the significant loss of material that the western end of the vault has experienced.

Therefore, this model suggests that the geometry itself and the low strength characteristics of the

material are partially responsible for some of the damage that can be observed on the structure. The

vault experiences high strains resulting in some areas that are particularly prone to cracking.








7. RECONSTRUCTION OF PART OF VAULT

As previously mentioned, an important part of the work carried out by ProjektyZeman.cz involved the

reconstruction of part of the vault on the western side that had collapsed. Naturally, this will help

consolidate crumbling unstable parts at the edge of the vault on the western side. Nevertheless, an

investigation into the more global effects of the reconstruction on the structure was carried out as a

part of this thesis, particularly to find out if the reconstruction also resulted in additional benefits

relating to the overall structural behaviour.

Obviously, this could not be modelled in two dimensions and required the use of a three-dimensional

model. Since it was specified that the reconstruction was to be executed using compatible materials

made to exhibit similar characteristics as the original material, the same parameters used to describe

the material with bricks in vertical layers were employed for the reconstructed part.

Two different modelling strategies were used to investigate the effect of the reconstruction.

The first involved modelling the whole geometry after the reconstruction as a single entity. Naturally,

this is an unrealistic model as the previous geometry would have already deformed and experienced

damage under its own self-weight before the addition. Hence this model should not be used to draw

direct conclusions on the effect of the reconstruction. Nevertheless, it could still provide some insight

into how the structure would respond to additional loads and was therefore included as a part of this

thesis. Moreover, since the vault used to consist of a more integral structure before the collapse of a

significant part of it on the western side, this model could also provide some insight on the state of the

structure before this collapse had occurred fully. The most important results from this modelling

strategy are described in sub-section 7.1.

The second modelling strategy consisted firstly of simulating the previously erected existing structure

under its own self-weight, and then adding the undeformed reconstructed part on the already

deformed previous geometry in a second load step. This was done by removing all the elements of

the reconstructed part from the first load step and adding them in the second load step. It was

important to ensure mesh compatibility at the boundary between the reconstructed part and the

previous geometry. It is important to note that a separate material had to be assigned to the

reconstructed part, even though it had the same properties as the remaining part of the vault, since

ATENA effectively deletes the whole material model associated with the element group when

removing elements. This model should provide a more realistic understanding of how the structure

would behave after the reconstruction. The most important results from this investigation are

described in sub-section 7.2.




7.1 Results from model considering geometry after reconstruction as a single entity

As can be expected, modelling the vault with the reconstructed part as a single entity predicts a much

stronger vault better able to resist the loading from its own self-weight. Much less cracks are caused to

form under self-weight as can be seen in Figure 39 (it should be noted that a different scale is used for

displaying the crack widths of the reconstructed vault considered as one entity in Figure 39 since the

cracks were much smaller and could not be visualised using the same scale).

Figure 39: Crack width distribution predicted for three-dimensional models of the previous geometry and of the

reconstructed vault considered as a single entity (values shown in metres).

Contrarily to the model of the previous geometry, no cracks can be seen to form on the extrados of the

reconstructed vault considered as a single entity.

These results clearly indicate that a more integral structure is better able to distribute and withstand

loads. Much fewer areas are left susceptible to damage under this scenario. This indicates that the

collapse of a big part of the vault in the past, and the resulting geometry it has left the vault in, is partly

responsible for making some areas of the remaining vault more vulnerable. This loss of three-

dimensional integrity is one of the main reasons why the vault is not well represented by a two-

dimensional model. In fact, the predictions of stresses and strains differ less between the two-

dimensional model and the three-dimensional model considering the reconstructed vault as a single

entity as it does between the two-dimensional model and the three-dimensional model of the previous

geometry.

Naturally, this is an over-prediction of the actual ability of the new structure to resist loads.

Nevertheless, it could be suggestive of a stronger resistance to additional loads other than the self-

weight due to the effect of the three-dimensional structural integrity. This had to be verified using a

model which provides a more realistic representation of the reconstruction of part of the vault.




7.2 Results from model of reconstruction in two load steps

This model presents very different results under the self-weight when compared to the model

considering the reconstructed vault as a single entity. Obviously, the previous geometry of the vault

deforms and cracks in exactly the same way as described in section 6.2 under self-weight before the

addition of the reconstructed part of the vault. However, contrarily to the model considering the

reconstructed vault as a single entity, this model suggests that addition of the reconstructed part

causes further propagation of the existing cracks since slight increases in crack widths can be

observed, particularly towards the front of the structure (further away from the reconstructed part

which is stiffer). This is most likely due to the additional weight that the reconstructed part entails.

Since certain regions in the previous geometry have already experienced damage, they are vulnerable

locations and addition of the weight of the reconstructed part results in those areas being affected first.

Figure 40: Crack width distribution predicted by the model before and after reconstruction of part of vault

(values sown in metres).

Nevertheless, comparisons of damages and strains before and after the reconstruction reveal that this

progression of damage is very small. This is illustrated in Figure 41, which shows the maximum

principal tensile strains before and after the reconstruction was added.




Figure 41: Maximum principal tensile strains before and after reconstruction of part of vault

(values shown in MPa).

Hence, the damage progression is not necessarily synonymous with the vault being in a more

vulnerable state overall. It would therefore be interesting to investigate whether or not the beneficial

effect of three-dimensional integrity observed when the reconstruction was considered as a single

entity would come into play when the structure experiences additional loading greater than the self-

weight.

In order to investigate this, the design wind load case which caused the most damage was applied on

both the model of the previous geometry and the model of the structure after reconstruction and

increased gradually until failure was observed. Although it is unlikely that any wind loads much greater

than the design loads will be experienced by the structure, at least in the near future, this approach

allows a safety factor to be determined allowing us to quantify in some way the ability of the structure

to resist certain additional loads.

A detailed description of how the wind loads and cases were determined and the main results

obtained by simulating the different cases is given in the next section of this thesis (Section 8). It was

found that wind coming from the right hand side of the structure (or north) and exerting pressure on

the right side of the vault whilst causing suction forces on the remainder of the vault (wind load case 2)

resulted in the most critical damage progression when compared to other wind load cases. It was

therefore decided that this load case would be used to investigate if the vault is better able to resist

additional loads after the reconstruction.

It is important to note that in simulations of the structure under design loads, the Newton-Raphson

method was used to solve the numerical problem. This method applies a constant load increment and

determines the iterative increment of the displacement vector. However, for simulations of the

structure at ultimate loads close to failure, it is important to observe the complete load-displacement

relationship rather than applying a constant loading increment. Hence, the arc-length solution method

was employed to solve the numerical problem when increasing the loads up to failure, since it fixes not

only the loading, but also the displacement conditions at the end of each step. Both models before and

after the reconstruction predicted that the vault would collapse from a typical four-hinge mechanism.

This is shown in Figure 42, with the cracks causing the failure mechanism labelled according to the




order in which they appear. The figure also shows whether the cracks first appeared at the intrados or

the extrados (cracks appearing first at the intrados would be indicative of a hinge at the extrados and

vice versa for this particular mechanism). Failure was defined as the point when the crack defining the

fourth hinge was able to form completely throughout the length of the vault.

Figure 42: Failure mechanism predicted by models before and after reconstruction (values shown in metres).

Based on these simulations, the safety factor against the most critical wind load scenario can be said

to have improved from 6.6 to 8 after the reconstruction. Hence, these results suggest that even though

the reconstruction could initially contribute to further propagation of existing cracks and damages due

to additional weight, it would also improve the integrity of the structure and therefore enable it to better

resist additional loading.








8. WIND LOADING

The wind loads to be applied on the structure were determined according to Eurocode 1, Part 1-4 [18],

in a similar way as they were determined for the structural analysis conducted by ProjektyZeman.cz.

All the wind load cases considered were applied incrementally on models after the effect of the self-

weight had already been computed. Naturally, since the design wind loads are of a much smaller

magnitude than the self-weight, the incremental effect of the wind loads is not as prominent as the

initial effect caused by the self-weight. Hence, wind loads were applied incrementally up to twice the

design value for all wind load cases, in order to better understand trends from which conclusions could

be made on the effect of the wind loading scenario in question. Nevertheless, all discussions made in

the subsequent sub-sections refer directly to the design wind loads.

The basic wind velocity, vb, of 25.0 m/s, suggested by ProjektyZeman.cz for a mean return period of

50 years (equivalent to an annual probability of exceedance of 0.02), was used for the determination

of wind actions. This basic wind velocity corresponds to a basic velocity pressure, qb, of 390 Pa,

computed according to:

𝑞𝑏 =1

2∙ 𝜌 ∙ 𝑣𝑏

2 [18]

where ρ is the wind density, taken as 1.25 kg/m3.

This basic pressure was multiplied by a coefficient related to a reference height for different parts of

the structure to obtain peak velocity pressures, qp, for each of the respective parts. Subsequently,

different coefficients were used for different parts of the structure under the different cases considered

as recommended in the Eurocode.

For all the wind load cases considered, the displacements at the base of the structure were

constrained in the same way they were for the models simulating the structure under its own self-

weight.

The first four load cases considered are equivalent to the load cases considered by ProjektyZeman.cz,

namely:

Wind load case 1: Wind from the left (or south), with pressure on the left side of vault

Wind load case 2: Wind from the right (or north), with pressure on the right side of the vault

Wind load case 3: Wind from the left, with suction on the whole vault

Wind load case 4: Wind from the right, with suction on the whole vault

The actual design pressure acting on the sidewalls for these four load cases were based on the

recommended coefficients for walls of a rectangular plan building with pressure applied on walls

directly facing the incoming wind and suction forces applied to walls on the leeward side. The design

pressures acting on the vault were calculated using coefficients determined according to the

recommendations of the Eurocode for vaulted roofs. The Eurocode suggests that the first quarter of




the vault on the windward side experiences pressure whilst the remainder of the vault experiences

suction. This recommended load distribution corresponds to wind load cases one and two, considering

wind coming from the south and north respectively. In addition to this, two other cases considering

suction forces acting on the whole vault while the forces on the sidewalls remain unchanged from

cases one and two were also modelled. It should be noted that in order to be able to assign different

loading conditions on quarter sections of the vault, the macro-elements representing the vault had to

be divided accordingly. This can be seen in figures of the geometry for two-dimensional and three-

dimensional models in Section 3.

These four wind design load cases were applied on the two-dimensional model as well as on the

three-dimensional models of the geometry before reconstruction, the reconstructed vault considered

as a single entity and the reconstruction simulated using two distinct load steps. For the two-

dimensional models, the forces from the wind were simulated using a static approach applying the

wind pressure and suction loads as distributed line loads (in kN/m) on the external boundaries. For

pressure and suction forces, the force should be acting perpendicular to the surface. Since the finite

element package used required assigning components of the loads in orthogonal directions (x and y

directions), the equivalent components of the resulting loads were computed and applied accordingly.

For curved surfaces, the normal of the midpoint of the surface was used to determine the x and y

components. The same approach was used for imposing loads on the three-dimensional models, but

the wind loads were applied as distributed loads on the external surfaces (in kN/m2). Once again, the

components of these loads in the orthogonal x and y directions had to be specified and they were

computed using the same approach as for the two-dimensional models. The prescribed boundary

conditions on the two-dimensional and three-dimensional models are shown in the relevant sub-

sections of this chapter followed by a presentation of the most relevant results.

In addition to these four wind load combinations, the three-dimensional models enabled a fifth wind

load case to be considered that could not be modelled in two dimensions:

Wind load case 5: Wind from the front (or east), with frictional forces on the interior and

exterior surfaces of the side walls and vault as well as pressure and suction forces applied on

windward and leeward oriented faces.

How the design load values were determined for this particular scenario is discussed further in sub-

section 8.5. This wind load case was applied on the three-dimensional models of the geometry before

reconstruction, the reconstructed vault considered as a single entity and the reconstruction simulated

using two distinct load steps. The principal outcomes of these simulations are also presented in sub-

section 8.5. It should be noted that friction forces were disregarded for the first four wind load cases as

suggested in Eurocode 1 for situations when the total area of all surfaces parallel to the wind is much

smaller than the total area of all external surfaces perpendicular to the wind.




8.1 Wind load case 1

The first wind load case considered wind coming from the left hand side of the structure with pressure

on the first quarter of the vault facing the windward direction and suction on the remainder of the vault.

How this scenario was implemented is shown schematically and with the corresponding load

assignments for both two-dimensional and three-dimensional models in Figures 43 and 44. Only the

three-dimensional model of the previous geometry is shown here, but the loads for this wind load case

were also applied on the three-dimensional models of the reconstructed geometry considered as a

single entity and on the model of the reconstruction in two load steps in exactly the same way.

Figure 43: Two-dimensional model for wind load case 1: (a) Schematic representation of loading,

(b) Assignment of distributed line loads.

Figure 44: Three-dimensional model of previous geometry for wind load case 1: (a) Schematic representation of

loading, (b) Assignment of distributed surface loads.

Results from the simulations indicate that the greatest resultant displacements experienced by

the structure under only the self-weight decrease as the design loads of the first wind load case are

applied. As described in Section 6, the geometry of the vault results in an asymmetric deformed shape

under the self-weight. The main positive horizontal displacement components as a result of this

asymmetry are towards the left of the structure (see Figures 34 and 36). Since the first wind load case

considers wind from the left, the forces effectively counteract some of the displacements being caused

by the self-weight of the structure. Naturally, since the two-dimensional model did not predict any

damage under self-weight, design loads from the first wind load case only result in a decrease in the

maximum principal tensile stresses at the vulnerable location on the intrados of the vault on the right

hand side. However, the three-dimensional model of the previous geometry predicts a reduction in the

crack widths formed on the intrados of the right hand side of the vault under self-weight as the wind




loads from the first scenario are increased, as shown in Figure 45. This suggests that wind coming

from the south could contribute to the closing of some cracks formed under the self-weight.

Figure 45: Progression of intrados crack width distribution as loads from wind load case 1 are increased


On the other hand, the simulations predict an increase in the tensile strains on the extrados in the

region where some minor damage was forecasted by the model considering only the self-weight.

Subsequently, this results in further propagation of the cracks on the extrados as can be seen in

Figure 46.

Figure 46: Progression of extrados crack width distribution as loads from wind load case 1 are increased


This is most likely caused by the accentuated differential movements imposed on the vault as a result

of having pressure and suction forces acting next to each other on the vault’s surface.

The three-dimensional models of the reconstructed geometry considered as a single entity and of the

reconstruction in two load steps also reflected the same trend discussed above. However, as

presented in Section 7, the model of the reconstructed geometry constructed as a single entity

experiences much less damage and hence these trends are not as evident.





The second wind load case considers wind coming from the right hand side of the structure with

pressure on the first quarter of the vault facing the windward direction and suction on the remainder of

the vault. The way in which this was implemented for two-dimensional and three-dimensional models

is illustrated in Figures 47 and 48.

Figure 47: Two dimensional model for wind load case 2: (a) Schematic representation of loading,




Contrarily to the loads considered in the first wind load case, the loads corresponding to the second

case no longer counteract the asymmetric deformation caused by the self-weight of the structure. In

fact, the pressure applied on the first quarter of the vault on the right hand side effectively contributes

to increasing the damage already suffered due to the self-weight at the intrados of the vault on the

right hand side. The two-dimensional model predicts some very small cracks at this location under the

design wind load as can be seen in the figure below showing part of the vault.




Figure 49: Crack width distribution predicted by two-dimensional model due to design wind loads for wind load case 2


Naturally, the three-dimensional model of the previous geometry predicts a propagation of the cracks

already formed at the intrados due to the self-weight as displayed in Figure 50. This clearly indicates

that wind coming from the right hand side of the structure is more likely to induce damage on the

intrados of the structure compared to wind coming from the left hand side.

Figure 50: Progression of intrados crack width distribution as loads from wind load case 2 are increased


Similarly to wind load case 1, the three-dimensional model of the previous geometry also forecasts the

development of more cracks on the extrados under the second design wind load case in the same

location as some had begun to form under the self-weight. Once again, this is most likely due to the

accumulated effect of pressure forces on one side of the vault and suction forces on the other.

However, as can be seen in Figure 51, the damage caused by the wind loads for this case is more

severe than that caused by the first wind load case.

Figure 51: Progression of extrados crack width distribution as loads from wind load case 2 are increased


Hence, it can be concluded that wind coming from both the left and the right of the structure could

cause cracking on the extrados, in the location that can be seen in the figure above. As previously

discussed, this is the location where one of the most prominent cracks on the extrados of the vault can

be observed (see Figure 38). Therefore, it is likely that wind loads could have contributed to the

propagation of this crack.




8.2.1 Determination of safety factor against wind load case 2

Since results from all the simulations taking design values of wind loads into consideration reveal that

the second wind load case causes the most damage, it can be considered as being the most critical

wind loading scenario (results from wind load case 3, 4 and 5 are presented in subsequent sub-

sections). Hence, it was chosen for a more in-depth investigation on how well the structure is able to

withstand wind loads.

In order to quantify the ability of the structure to resist these loads, the design values of the wind loads

were increased incrementally until failure was observed. The arc-length solution method was used to

solve the numerical problem, and 10 times the design load was applied in 50 load steps (as can be

expected, the problem failed to converge for some load steps at a point after failure). This procedure

was implemented after the effect of the self-weight had been computed on a two-dimensional model of

the vault, on a three-dimensional model of the previous geometry and on a three-dimensional model

simulating the addition of the reconstructed part of the vault. One of the principal objectives of this

investigation was to assess whether or not the reconstruction of part of the vault better equips the

structure to resist additional loads. The main outcomes relating to this have already been discussed in

Section 7, and will not be covered here. This sub-section provides a more in-depth analysis on the

failure mechanism observed as a result of the simulations.

All three models considered predicted a similar crack progression. Analysis of the crack progressions

and deformations revealed that the failure eventually predicted appeared to be in accordance with

theorems of plasticity on which limit analysis is based, as stipulated by [5]. The most important cracks

leading to failure could therefore be associated with the formation of hinges. Subsequently, it can be

said that failure will occur when a sufficient number of hinges have formed to turn the structure into a

mechanism [5], four in this case. The cracks corresponding to these hinges are shown on the

deformed shapes of the three models at the ultimate limit state in Figure 52.

Figure 52: Crack width distribution on deformed shape (deformation scale of 1:70) of three models at failure





One observation that can be made from the results is the fact that the two-dimensional model and the

model simulating the reconstruction of part of the vault both predict the same ultimate load with

respect to overloading from the wind. This is particularly interesting since the two-dimensional model

experiences no damage under only the self-weight, whilst out of the three models considered for this

investigation, the model of the reconstruction is the one that experiences the most damage under self-

weight once the reconstructed part is added to the model. In order to better understand this

phenomenon, a more in-depth analysis on the formation of each crack, as the design loads were

increased incrementally, was carried out.

Figure 53, which can be found on the following page, shows the propagation of each crack

contributing to the formation of hinges eventually leading to failure. Both three-dimensional models

predict that the crack related to the first hinge appears under the self-weight itself, whereas the two-

dimensional model only predicts the onset of this crack when 1.6 times the design wind load has been

applied. Some damage had also been observed on the three-dimensional models under the self-

weight in the region where the crack related to the second hinge eventually forms. Therefore, Figure

53(b) shows the point at which these damages are first seen to progress for the three-dimensional

models, whilst showing the point at which the two-dimensional model predicts the formation of this

crack as the wind loads are increased gradually. The loads at which the formation of the third crack

and fourth crack can be observed in all three models are also shown in Figure 53.

It is clear that the two-dimensional model does not give a good representation of the previous

geometry when compared to the three-dimensional model due to its inability to account for the

irregularities present along the length of the vault. Furthermore, it appears to underestimate damage

at the design load values when compared to the three-dimensional model of the vault after

reconstruction. This is once again, because it cannot predict the damage already experienced by the

structure before the reconstruction. However, it can be seen that as the loads are increased, the

reconstruction plays a bigger part in the response of the structure, and the behaviours predicted by the

two-dimensional model and the three-dimensional model of the vault after reconstruction tend to be in

better agreement. This suggests that although the two-dimensional model does not give a good

representation of damage under design loads for the previous geometry or even the geometry after

reconstruction, it could be useful to provide indications on the ultimate limit state of the structure after

reconstruction.




Figure 53: Progression of cracks leading to failure mechanism: (a) Formation of first crack; (b) Onset of propagation of

second crack; (c) Formation of third crack; (d)Formation of fourth crack.





This wind load case is very similar to wind load case 1 since it also considers wind coming from the

left. However, in this case, only suction forces are applied on the entire vault, as can be seen in

Figure 54 in two dimensions and Figure 55 in three dimensions.





Contrarily to the wind loading scenario considered in wind load case 1, results from these simulations

show almost no changes in the crack width distribution when compared to the structure under self-

weight. No significant crack closing phenomenon can be observed at the intrados and only very minor

progression of the damages on the extrados can be visualised. This indicates that since the vault can

be considered as the most vulnerable part of the structure, pressure imposed on part of it plays an

important role in causing the trends described in section 8.1.





This wind load case is analogous to wind load case 3, but with wind coming from the right hand side of

the structure. Hence, the values of the pressure to be applied on the external surfaces were

determined in exactly the same way as they were for wind load case 2 but with suction instead of

pressure applied on the first quarter of the vault facing the windward side. This applied loading is

shown in two and three dimensions in the figures below.





Once again, the simulations of this wind loading scenario predict almost no changes to the damages

observed as a result of applying the self-weight. This reinforces the conclusion that pressure imposed

directly on the vault itself as a result of wind loading contributes significantly to the opening or closing

of cracks (depending on the wind direction) that could have formed at the intrados under the self-

weight of the structure. Furthermore, results from wind load cases 3 and 4 strengthen the hypothesis

that the propagation of the cracks on the extrados are due to the combination of pressure on one side

and suction on the other since much more significant propagation is predicted when this is the case.





This wind loading scenario considers wind coming from the front of the structure. Since, in this case,

large areas of the structure will be swept by the wind, it was deemed necessary to account for friction

forces acting tangentially to surfaces parallel to the wind direction. Hence, aside from pressure forces

on the front and suction forces on the back of the structure, computed and applied in a similar way as

for the first four wind load cases, this wind load case also involved the application of tangential

frictional surface loads on both the inner and outer surfaces of the structure (since the front of the

structure is open, leaving the inner surfaces completely exposed to frictional wind forces). This was

implemented by applying distributed surface loads as shown in Figure 58. The frictional component of

the wind loads to be applied on the surfaces was found by multiplying a friction coefficient to the peak

velocity pressure. This friction coefficient is based on the roughness of the surface. In the absence of

more information, this was taken as 0.02 as recommended in the Eurocode for rough surfaces such as

rough concrete or tar-boards.



In addition to these loads, wind coming from the front would also exert pressure on the back wall. It is

difficult to say to what extent this would influence other parts of the structure, due to the limited

connectivity which exists between the back wall and the side walls (previously described in Section 3

and shown in Figure 15). Nevertheless, a worst case scenario whereby all the pressure experienced

by the back wall is transferred to the edges of the sidewalls was taken into consideration in

simulations of wind load case 5.




The net equivalent pressure acting on this wall was determined according to the recommendations of

the Eurocode for free-standing walls with returning corners on either side. This pressure was then

multiplied by the windward facing area of the back wall resulting in a force of 1214 kN. In order to

distribute this force to the edges of the sidewalls, half of the force was assigned to each sidewall and

this was divided by the length of the corresponding edge in contact with the back wall to be applied as

distributed line loads as shown in Figure 59.

Figure 59: (a) Distribution of pressure acting on back wall to edges of side walls; (b) Application of distributed line

loads on three-dimensional models.

Results from all three models simulating this wind loading scenario reveal no significant changes from

the state of the structure under its self-weight only. This seems to indicate that wind coming from the

right or left and imposing direct pressure and suction forces on the sides of the vault are much more

critical scenarios with respect to the safety of the vault when compared to wind coming from the front

of the structure. This is most likely due to the fact that most surfaces are only experiencing friction

forces in this case, which are of a much smaller magnitude than the pressure and suction forces they

experience with winds coming from the sides of the structure. Since this would also be true for winds

coming from the back of the structure, no investigation was carried out on the possible effects of such

a scenario.








9. RAINWATER INGRESS IN VAULT

The project documentation provided by ProjektyZeman.cz clearly identified the lack of waterproofing

before the reconstruction works as one of the most noteworthy issues. This left the vault exposed to

the environment and made it particularly susceptible to rainwater ingress. It should be noted that

although the structure is found in a region with a relatively dry climate, precipitation values of up to

30mm have been recorded during peak rainfall periods [19], and hence rainwater ingress could

definitely affect the structure. In order to gain a better understanding of what implications this could

have for the structure, simulations of the structure under self-weight which account for the effect of

rainwater ingress through a reduction of the strength and material properties of the vault were carried

out. A different material model was created for wet masonry with bricks in vertical layers. This was

done by estimating the parameters defining the material based on laboratory tests carried out on

saturated samples of units and mortar, as previously described in section 5. The two-dimensional

geometrical model and the three-dimensional geometrical model of the previous geometry were used

for the execution of this simulation. The material assignments of these two models appear in

Figure 60.

Figure 60: Material assignment for models considering rainwater ingress in vault; (a) two-dimensional model,

(b) three-dimensional model of previous geometry.

The two-dimensional model taking rainwater ingress into consideration predicts only very slight

changes in the way the structure is able to resist its self-weight when compared to the two-

dimensional model with estimated parameters of dry masonry used to define the material of the vault.

Higher tensile strains can be observed in the same vulnerable location on the intrados resulting in the

onset of cracking under the self-weight as can be seen in Figure 61.

Figure 61: Crack width distribution for part of two-dimensional model considering wet material for the vault





On the other hand, the three-dimensional model of the previous geometry presents considerably more

prevalent differences from predictions of the model considering dry masonry in the vault. The widest

cracks form in the same area on the intrados and on the extrados as predicted by the model with a dry

condition of the vault. However, the extent of the damage predicted in those areas is greater if

rainwater ingress in the vault is accounted for as can be seen in Figure 62.

Figure 62: Crack width distribution shown from the bottom and top for three-dimensional models considering dry and

wet material for the vault (values shown in metres).

Furthermore, small signs of damage can also be observed on the opposite surface in the same

location as the most prominent cracks (damages can be seen on the extrados directly opposite to

where the most prominent cracks have formed on the intrados and vice versa). This seems to indicate

that cracks are able to propagate more easily through the thickness of the vault. However, since the

material model does not take into consideration the initial anisotropy, which is characteristic of

masonry, this particular phenomenon might not be truly representative of the response of the actual

structure. Instead, a more realistic interpretation of these results could simply be that the reduced

strength of the material as a consequence of rainwater ingress results in a greater extent of damage in

the vault. Another remark that can be made based on the results is that the cracks which form under

the self-weight have a particular tendency to propagate in the longitudinal direction of the vault. This is

very much in line with some of the cracks that could be observed over the extrados of the real

structure on the poor quality concrete membrane, as can be seen in Figure 63. It is possible that

rainwater ingress has facilitated the propagation of these cracks.




Figure 63: Cracks on the extrados of the structure [1].

It should be noted that since part of the restoration works carried out by ProjektyZeman.cz involved

implementing a new roofing system which would improve the waterproofing of the vault, the effect of

rainwater ingress in the vault should be substantially reduced after the restoration.








10. TEMPERATURE EFFECTS

One of the possible causes of damage and deterioration identified by ProjektyZeman.cz was the big

temperature changes experienced by the structure. In order to confirm this hypothesis, a simulation of

the mechanical response of the structure to temperature changes was carried out.

Thermal loads can be considered in the mechanical analysis by prescribing different temperature

changes at the integration points of every element. The thermal expansion of each element and the

associated initial strain load can subsequently be computed based on the thermal expansion

coefficient, estimated using results from experiments carried out on the masonry components as

described in section 5.

One of the main difficulties therefore lies in estimating the temperature changes experienced at each

integration point, which requires modelling heat transport inside the structure. For the purposes of this

thesis, a fully uncoupled approach was used whereby the transport problem was solved completely

independently and the resulting temperature histories were then imported into a different module for

the execution of the mechanical analysis. The temperature loads were only considered with the self-

weight of the structure in the analysis. To be exact, both the transport and static analysis should be

executed simultaneously but as temperature transport does not depend significantly on structural

deformations, the implemented “staggered” solution yields sufficiently accurate results [12].

Ideally, the transport analysis should be based on data collected through an extensive monitoring

campaign covering at least a full temperature cycle. However, since such results were not available in

this case, an approximate method was employed in an attempt to encompass some of the

temperature changes that the structure is likely to experience during a temperature cycle. The only

available information relating to temperature was provided in the form of a set of thermograms, such

as those shown in Figure 64, produced by a thermographic camera.

Figure 64: Sample thermograms showing: (a) the intrados of the vault, (b) the exterior of the southern wall.

Sixty-five representative readings of temperature on the inner and outer surface of the structure shown

on different thermograms were collected to quantify the temperature changes experienced. The

temperature on the inner surface was found to vary from 19.1°C to 36.1°C (corresponding to a change

of 17°C) whilst varying from 22.2°C to 50.0°C on the outer surface (corresponding to a change of

approximately 28°C). The initial temperature of elements making up the structure was set as the




average of the minimum temperature values recorded on the inner and outer surface (20.7°C). The

temperatures on the inner and outer surfaces were then increased by the respective temperature

changes described above by prescribing Dirichlet boundary conditions on appropriate surfaces. The

temperature of these surfaces was then kept constant and then finally decreased by the same

temperature change, in an attempt to approximately model heating and cooling phases of a

temperature cycle.

The ATENA Transport module [12] was used for the transport analysis. It should be noted that

although this module allows the analysis of coupled heat and moisture transfer, moisture parameters

of the material were deactivated and humidity was kept constant since the only unknown field variable

of interest in this case is temperature. Two input parameters are required to define a particular

material for the transport analysis in the ATENA Transport module, namely the thermal conductivity, λ,

of the material (in W∙m-1

∙°C-1

) and a coefficient to define the material heat capacity (in J∙m-3

∙°C-1

). This

coefficient can be found by multiplying the specific heat capacity, c, of the material (in J∙kg-1

∙°C-1

) by

the material’s density, ρ (in kg/m3). As previously described the bulk density of the homogenised

masonry was estimated as 12.73 kN/m3 using volumetric averages. This corresponds to a density of

1298 kg/m3 which was used to evaluate the coefficient defining heat capacity. In the absence of more

information, the following values, suggested for fired clay bricks in the ASHRAE Handbook [20], were

adopted to describe the thermal properties of the material:

Thermal conductivity, 𝝀 = 𝟏𝑾

𝒎∙ °𝑪

Specific heat capacity, 𝒄 = 𝟖𝟐𝟖𝑱

𝒌𝒈∙ °𝑪

It was also important to decide on the duration of each phase of the cycle being considered. Since the

temperatures used were recorded in the summer, the approximate duration for which the structure

would be exposed to sunlight on an average day in summer was used to estimate this. The average

length of a day (from sunrise to sunset) in Iraq is 14 hours in the summer [21]. Hence, the duration of

the cooling phase was set as the duration for which the structure would not be exposed to sunlight

(10 hours). The remaining 14 hours were divided into 5 hours of gradual heating up to the maximum

temperatures recorded and 9 hours for which the temperatures on the surfaces were kept constant.

The resulting temperature distribution predicted by the transport analysis after these heating and

steady phases is shown in Figure 65. It can be observed that the temperature variation is mainly

confined to layers close to the boundary. After the cooling phase, the temperatures at almost all

integration points have returned to the initial temperature of 20.7 °C as can be seen in Figure 66.




Figure 65: Element temperatures after heating and steady phases.

Figure 66: Element temperatures after full cycle considered.

The crack width distribution predicted by the static analysis of the effect of self-weight considering

temperature changes after the heating and steady phases is shown in Figure 67. It can be seen that

many small cracks form close to the boundary of the side walls, where the greatest temperature

variations are being experienced. This indicates that temperature changes could have contributed to

the degradation of material that can be seen on the outer surfaces of the sidewalls of the structure.

Furthermore, the model predicts the formation of many small cracks on the extrados in the same

region as some were previously forecast under the self-weight without considering temperature

changes. Hence, it appears that extreme temperature changes could also have contributed to the

propagation of the crack that has been observed on the structure in a similar region.




Figure 67: Crack width distribution predicted by three-dimensional model after heating phase


However, it can clearly be seen that the values of crack widths in the vulnerable location in the

intrados are smaller when compared to those predicted by the model which does not take into

consideration any temperature changes. This is most likely due to the volumetric strains resulting in

the elements as a result of the positive temperature changes. This could be interpreted as expansion

of the material at the borders of the cracks hence resulting in smaller cracks.

Although many changes in the crack widths can be observed after the extreme positive temperature

change when compared to results from the model not considering temperature changes, an

examination of the crack widths predicted after both the heating and cooling phases considered

(shown in Figure 68) reveals that the collective effect of the temperature changes results mostly in the

closing of previously opened cracks, as well as the re-opening of cracks in the vulnerable location in

the intrados. In fact the crack width distribution observed after the full cycle is almost identical to the

crack width distribution predicted by the model which considers only the self-weight with no thermal

effects.




Figure 68: Crack width distribution after complete temperature cycle considered (values shown in metres).

However, it should be noted that a more realistic analysis of the temperature transport based on actual

data from temperature monitoring will most likely result in a better representation of accumulated

effects which could arise as a result of the temperature cycle.








11. SEISMIC HAZARD

In order to examine the potential risks of earthquakes affecting the structure and to determine if a

more in-depth analysis of its response to seismic loads was required, an assessment of the seismic

hazard associated to the region was carried out. Figure 69 below presents the seismic hazard map of

the region with the location of the structure and the distribution of expected peak ground accelerations

with a 10% probability of exceedance in 50 years.

Figure 69: Seismic hazard map showing location of Taq-Kisra [22].

It is clear from the figure that the seismic hazard associated to the area where the building is located is

very low, with peak ground accelerations only expected to range from 0 to 0.2. Hence, it was deemed

that further investigation into the response of the structure to seismic loads was not strictly necessary

since it is unlikely that the building would experience such loads, at least in the foreseeable future.








12. COMPARISON OF LOAD CASES CONSIDERED

The simulations carried out indicate that aside from the fact that certain areas of the vault experience

high strains with respect to the low strength characteristics of the material under only the self-weight,

environmental and climatic factors such as wind loads, temperature effects and rainwater ingress in

the vault also contribute to further damage and deterioration of the material.

A comparison of the different wind loading scenarios considered reveals that wind coming from the

north or the south and acting directly on the sides of the structure have a greater impact on the

structure when compared to wind coming from the front. More surfaces experience direct pressure or

suction in the case of wind coming from the sides. The magnitudes of loads exerted by such forces

are much greater than those of friction forces, experienced by most surfaces in the case of wind

coming from the front. This could explain why wind coming from the sides lead to more significant

changes in the crack width distribution. Furthermore, it was found that wind load cases resulting in a

pressure component acting directly on part of the vault on the windward side are more likely to

influence the crack width distribution. Both wind coming from the right and left hand side of the

structure can cause further propagation of small cracks that could have formed under only the self-

weight in a particular region on the extrados. However, wind coming from the right hand side of the

structure can be considered as more critical, since it also induces further cracking in the vulnerable

area on the intrados whereas wind coming from the left appears to cause a reduction in crack widths

in that area by counteracting the asymmetric loading which arises as a result of just the self-weight of

the structure.

A comparison of the maximum displacements predicted by all the models simulating different

conditions is shown in Figure 70. This should not be used to draw conclusions on the condition or

safety of the structure since it provides no indication on the extent of damage. For example, the model

taking into consideration the extreme positive temperature changes after the heating phase of the

temperature cycle predicts almost the same maximum displacement as the model which does not

consider temperature changes, since the greatest deformations are mostly a result of the self-weight.

However, there are considerable differences between the damage predicted by the two models, with

more cracks forming in areas experiencing big positive temperature changes. Nevertheless, these

comparisons can prove useful when assessing the serviceability limit states of the structure under

different loading conditions. For instance, it is interesting to note that although results from simulations

carried out as part of this thesis indicate that the reconstruction of part of the vault better equips the

structure for resisting additional loads, the additional weight it imposes results in larger displacements

under design loading conditions.




Figure 70: Comparison of maximum displacements predicted by different models under design loading conditions.




13. COMPARISON WITH RESULTS FROM PREVIOUS STRUCTURAL

ANALYSES

The finite element analyses conducted as a part of this thesis predicts similar areas susceptible to

tensile stresses under self-weight and wind loading when compared to results from the two-

dimensional analyses carried out by ProjektyZeman.cz. However, more factors related to the

behaviour of the material, the three-dimensional geometry and the environment could be taken into

consideration. This has allowed more conclusions to be drawn on observed damages and on the

possible effects of the reconstruction of part of the vault.

It is also interesting to note that areas identified as prone to experiencing damage and high tensile

stresses under self-weight as a result of simulations carried out for the purposes of this thesis appear

to correspond to areas where the thrust line is most eccentric in the static graphic analysis presented

in Section 2.3. Naturally, the thrust line shown is only one of many possible solutions, it does not give

any indication on the actual state of the structure and can only be used to evaluate whether or not a

structure is safe. However, in this case, the regions where the thrust line is most eccentric are most

likely where it would first leave the boundary of the structure upon overloading and can therefore be

identified as being vulnerable. This is very much in line with findings from simulations carried out for

the purpose of this thesis.








14. CONCLUSIONS

Based on the analyses carried out as a part of this thesis, the Taq-Kisra monument can be considered

as being safe globally, since it can resist all the design loads considered without reaching an ultimate

limit state. However, the existing geometry and self-weight of the structure itself result in an

asymmetric deformed shape causing certain areas of the vault to experience high strains, particularly

with respect to the low strength properties that characterise the material. This combined with

environmental factors such as exposure to the wind, rainwater ingress in the vault and big temperature

changes results in certain local areas being particularly prone to damage. Moreover, the significant

loss of material that has occurred in the past towards the western side of the vault has left these areas

in an even more vulnerable state, since the loss of structural integrity results in a less efficient

distribution of stresses in the vault. The cumulative effect of such local damages, particularly over long

periods of time, definitely has serious implications for the overall safety of the structure.

It has been demonstrated that the most vulnerable region is most likely on the right hand side of the

intrados of the vault, since the greatest crack widths predicted under almost all loading combinations

considered tend to be localised in this area. Another area which has been identified as prone to

damage, particularly when environmental factors are taken into consideration, is found on the extrados

of the structure, just to the left of the apex. Not only do simulations predict the onset of cracking in

both these areas under the self-weight alone, but some of the other environmental factors considered

also contribute to further propagation of these cracks. Reports from observations of the real structure

confirm the presence of cracks on the real structure in these locations.

It has also been found that winds coming from either side of the structure (north and south) are more

likely to affect its stability when compared to winds coming from the front or back (east and west). Both

wind coming from the north and south appear to cause further propagation of cracks in the vulnerable

location on the extrados. However, only wind coming from the north induces further cracking in the

vulnerable area on the intrados whilst wind coming from the south appears to cause a reduction in

crack widths in that area by counteracting the asymmetric loading which arises as a result of the self-

weight of the structure. Hence, wind coming from the north and exerting pressure on part of the vault

facing the windward side can be considered as the most critical wind loading scenario.

Another environmental factor whose possible effects were investigated as a part of this thesis is

temperature changes. Simulation of the temperature transport inside the structure seems to indicate

that temperature variations are mainly confined to layers close to the boundary. Although many cracks

form particularly at the boundary of the sidewalls after the heating phase, most of these cracks close

as a result of a subsequent cooling phase. This could suggest that the collective effect of a

temperature cycle would result in the closing of previously opened cracks. However, it should be noted

that the temperature transport was modelled on a very approximate basis, only considering possible

temperature changes during a single day, and it is likely that a more accurate representation of the

temperature cycle based on monitoring data could yield different results. Nevertheless, the crack width

distribution predicted after the heating phase gives an indication of the possible effects that




temperature variations could have and also suggests that temperature variations could have

contributed to the degradation of material on the outer surface of the side walls of the vault.

It has also been demonstrated that rainwater ingress could significantly contribute to damage in the

vault, particularly in the two vulnerable locations identified on the intrados and on the extrados. The

roofing solution proposed by ProjektyZeman.cz should definitely help reduce this effect since it would

improve the waterproofing of the vault. It should also help reduce temperature variations in the vault.

Another part of the restoration works suggested by ProjektyZeman.cz involved the reconstruction of

part of the vault. Results from simulations indicate that this would initially exert slightly greater strains

in the previous structure and induce more cracking due to the additional weight. However, it does

create a structure with better three-dimensional structural integrity, and hence should improve its

ability to resist additional loads. This was verified only for the worst case wind scenario by finding a

safety factor to quantify the structure’s ability to resist this additional wind loading before and after the

reconstruction. It was found that the structure would be able to resist 8 times the design wind load

after the reconstruction as opposed to 6.6 times before.

The two-dimensional models underestimate damage under design loads when compared to three-

dimensional models, most likely because it cannot take into account the significant loss of material

that has occurred on the western side before the reconstruction. However, it is interesting to note that

when investigating the ultimate limit state related to the most critical wind loading scenario, it predicts

a similar four-hinge failure mechanism at the same load as the three-dimensional model after

reconstruction. Hence, it is possible that the two-dimensional model could be used to give an

indication on the ultimate limit states of the structure after reconstruction.

Further works on the structural condition of the Taq-Kisra monument could involve a more accurate

analysis of the temperature transport based on data from temperature monitoring. This would allow

the actual effect of the temperature variations to be better represented by the model. Furthermore, the

same methodology as employed for the worst case wind scenario in this thesis could be used to

evaluate the safety factors related to the ultimate limit states of the structure under all the wind load

combinations, taking temperature and rainwater ingress into consideration, both before and after the

reconstruction of the vault. This would provide a quantitative representation of the structure’s safety

under particular conditions, which would allow the effect of the reconstruction to be evaluated and

more precise conclusions to be made on the effect of different conditions.




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Documents

Advanced computational modelling of Taq Kisra, Iraq