TW Groeneweg - Shield Driven Tunnels in UHSC

T.W. Groeneweg

Shield driven tunnels in ultra high strength concrete Reduction of the tunnel lining thickness


T.W. Groeneweg January 2007


T.W. Groeneweg January 2007

Graduation committee Prof.dr.ir. J.C. Walraven Dr.ir. C.B.M. Blom Dr.ir. C.R. Braam Dr.ir. O.M. Heeres Ir. L.J.M. Houben

Preface The research presented in this report is the graduation thesis to obtain my master’s degree in Civil Engineering at Delft University of Technology. Most work for this thesis was performed at the engineering office of Gemeentewerken Rotterdam (Public Works Rotterdam). The objective of the study is to investigate the technical feasibility of reduced lining thicknesses for shield driven tunnels by the application of ultra high strength concrete. The following members took part in the graduation committee: Prof.dr.ir. J.C. Walraven Delft University of Technology Dr.ir. C.B.M. Blom Gemeentewerken Rotterdam / Delft University of Technology Dr.ir. C.R. Braam Delft University of Technology Dr.ir. O.M. Heeres Gemeentewerken Rotterdam / Delft University of Technology Ir. L.J.M. Houben Delft University of Technology     

I would like to thank the engineering office of Gemeentewerken Rotterdam for giving me the opportunity to do the research within their company. My gratitude goes to my colleagues at the office. I am very thankful to Jos, Leon, Tom and Wouter for the nice daily lunch breaks we joined. I would also like to thank my graduation committee for the useful conversations and enjoyable atmosphere during the meetings. Special thanks I would like to address to Kees Blom for his personal guidance and ever lasting enthusiasm that encouraged me to keep on going and get the research to its current level. Finally I thank my family and friends for their help and understanding.

Tom Groeneweg January 2007

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Summary From the history of shield driven tunnelling in soft soil is known that the lining thickness of such a tunnel is of a fixed ratio compared to the tunnel’s diameter. Consequently the thickness equals 1/20 of the diameter. In case of very large tunnel diameters the weight of the tunnel segments increases dramatically. As a result costs strongly increase due to problems in the logistics of construction (the production process, transport to the building site and placing of the segments in the tunnel). However, the wish to construct ever larger tunnel diameters remains. Because the possibility is then created to construct multiple-lane motorways, as for instance motorway A13/16 in the north of Rotterdam, in such a tunnel. This report investigated the feasibility of using new steel fibre reinforced concretes, very high strength concrete C100/115 and ultra high strength concrete C180/210, to reduce the lining thickness of shield driven tunnels with very large diameters. Several mechanisms are known to cause damage in existing shield driven tunnels and therefore may lead to failure. In this report the following four have been studied intensively: 1. Common ring behaviour of the tunnel embedded in soil (serviceability phase) 2. Ring behaviour after grout injection along the tunnel (construction phase) 3. Introduction of thrust jack forces from the tunnel boring machine into the segments (construction phase) 4. Torsion in tunnel segments by deformations due to the grout injection, also known as the trumpet effect (construction phase) Each mechanism resulted in a boundary condition on the required lining thickness. It was shown that the serviceability phase is never governing. However, the strength related conditions by the ring behaviour due to grout injection and by the introduction of thrust jack forces dictate the required lining thickness. In case of ordinary concrete these mechanisms result in the standard thickness of 1/20 D as well. Hence the construction phase should never be excluded in the design of a shield driven tunnel. Torsion by the trumpet effect quickly leads to the formation of cracks in tunnel segments. Very high lining thicknesses are required to prevent this mechanism from happening. These thicknesses are beyond the standard required thickness of 1/20 D. Consequently cracks during construction of a tunnel with such a lining thickness are likely to occur. Indeed in practice these cracks have been observed. Also for very and ultra high strength concrete cracks will occur if the lining thickness is based on the mentioned strength related conditions. Temporary measures such as adjuster (trusses placed in the ring to prevent in from deforming) or the application of tunnel boring machines with longer shields help to reduce cracks by the trumpet effect. Very thin tunnel linings can be used if conventional reinforcement bars are added to a tunnel lining in ultra high strength concrete. The required amount of reinforcement strongly depends on the tunnel’s depth projection. However at each considered depth a thickness of only 1/58 D is possible. A reduction of governing behaviour for the grout injection can be realised by the use of additional mass (for instance sand fill) in the tunnel tube during construction only. Such temporary measures during the construction phase make thinner linings, even below 1/60 D, achievable.

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Samenvatting Uit de bouwpraktijk van geboorde tunnels in slappe grond is gebleken dat een vaste verhouding tussen de tunnel’s diameter en vereiste liningdikte (wanddikte) bestaat. De dikte is hierdoor gelijk aan 1/20-ste deel van de diameter. Bij zeer grote diameters worden de tunnelsegmenten zodoende zeer zwaar. Dit levert problemen op in de logistiek (het productieproces, vervoer naar de bouwplaats en de plaatsing van segmenten in de tunnel), die de totale kosten sterk opdrijven. De wens blijft echter bestaan om boortunnels met zeer grote diameters te maken. Hierdoor zal het mogelijk worden ook snelwegen, zoals de toekomstige snelweg A13/16 in het noorden van Rotterdam, in zo’n type tunnel aan te leggen. In deze studie is onderzocht of de nieuwe staalvezel versterkte betonsoorten, zeer hogesterkte beton C100/115 en ultra hogesterkte beton C180/210, kunnen bijdragen aan een reductie van de liningdikte voor boortunnels met een zeer grote diameter. Verschillende mechanismen kunnen bij een boortunnel tot schade en daardoor mogelijk tot bezwijken, leiden. In de studie zijn de volgende vier uitvoerig onderzocht: 1. Algemene ringwerking van de tunnel ingebed in grond (gebruiksfase) 2. Ringwerking na injectie van grout rond de tunnel (bouwfase) 3. Introductie van vijzelkrachten vanuit de tunnelboormachine in de segmenten (bouwfase) 4. Torsie in segmenten door vervormingen ten gevolge van de groutinjectie, ook bekend als het trompeteffect (bouwfase) De genoemde mechanismen resulteerden elk in een grenswaarde van de vereiste liningdikte. Het is gebleken dat de gebruiksfase nooit maatgevend wordt. De sterkte-eisen door de ringwerking bij de groutinjectie en de introductie van vijzelkrachten dicteren de vereiste liningdikte. Deze zijn ook verantwoordelijk gebleken voor de liningdikte uit de standaard vuistregel van 1/20 D voor conventioneel beton. De bouwfase mag in het ontwerp van een boortunnel daarom nooit buiten beschouwing worden gelaten. Torsie door het trompeteffect leidt snel tot scheurvorming in de tunnelsegmenten. Zeer grote liningdikten zijn vereist om dit mechanisme te voorkomen. Deze dikten liggen voor conventioneel beton in ieder geval boven de standaarddikte van 1/20 D. Hierdoor zijn scheuren tijdens de bouw te verwachten, wat in de praktijk ook inderdaad is waargenomen. Ook bij zeer en ultra hogesterkte beton zijn scheuren te verwachten als de vereiste liningdikte wordt gebaseerd op de genoemde sterkte-eisen. Tijdelijke maatregelen als het gebruik van een adjuster (vakwerk in de ring om vervormen te voorkomen) of de toepassing van een tunnelboormachine met een langer schild kunnen dit effect gedeeltelijk terugdringen. Door toevoeging van conventionele wapening aan een tunnellining van ultra hoge sterkte beton is het mogelijk gebleken zeer dunne liningdikten te verkrijgen. Het benodigde wapeningspercentage is sterk afhankelijk van de diepteligging van de tunnel. Echter op alle onderzochte diepten is een dikte van slechts 1/58 D mogelijk gebleken. Voor een vermindering van de maatgevendheid van de groutinjectie kan een tijdelijke massa (bijvoorbeeld een zandlichaam) in de tunnel worden aangebracht. Eventueel kan deze massa na de bouwfase probleemloos worden verwijderd. Door het toepassen van zulke tijdelijke maatregelen tijdens de bouwfase kunnen zelfs liningdikten onder de 1/60 D worden gerealiseerd.

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Notations Latin lower case

b c fc fs f'c f'co h ℓ f ℓ segm ℓ t

r u2 x0 xt xu xw

= = = = = = = = = = = = = = = =

Half segmental width Concrete cover reinforcement bars Design value concrete tensile strength Design value strength reinforcement steel Design value concrete compressive strength Permitted compressive stress due to centric perpendicular force introduction Lining thickness Fibre length Length tunnel segment in tangential direction Height contact area longitudinal joint Internal radius tunnel Ovalisation deformation Ground surface relative to mark NAP Top of tunnel relative to mark NAP Height concrete compressive zone Water table relative to mark NAP

Latin upper case

A Bt D Di E Eoed F jack G I K0 M Mu N Nrep R

= = = = = = = = = = = = = = =

Cross-sectional area concrete Width for torsion by trumpet effect External diameter tunnel Internal diameter tunnel Young’s modulus concrete Oedometer stiffness soil Thrust jack force Shear modulus concrete Moment of inertia Neutral soil support coefficient Bending moment Ultimate resisting moment or bending moment capacity Normal ring force Representative normal force on cross-section External radius tunnel

Greek lower case αt γ f εmax ε'c ε'u θ

= = = = = =

Reduction ratio for tangential soil stress Partial safety factor steel fibres Maximum concrete strain in longitudinal Janßen joint Compressive yield strain concrete Ultimate compressive strain concrete Angle at tunnel perimeter

xi

ν ρsd ρsw σh σr σt σv σw σ'h σ'v τt φ ϕ

= = = = = = = = = = = = =

Poisson ratio (lateral contraction) concrete Specific gravity dry soil Specific gravity wet soil Horizontal soil pressure Radial soil pressure Tangential soil pressure Vertical soil pressure Water pressure Effective horizontal grain pressure soil Effective vertical grain pressure soil Shear stress due to torsion trumpet effect Rotation in longitudinal Janßen joint Internal friction angle soil

Abbreviations

TBM UHSC ULS

xii

Tunnel boring machine Ultra high strength concrete Ultimate limit state

Table of contents Preface

v

Summary

vii

Notations

xi

Chapter 1

Introduction

1

1.1

Introduction to shield driven tunnels

1

1.2

Problem description

2

1.3

Problem definition

2

1.4

Objective

2

1.5

Solution approach and arrangement of this report

2

Chapter 2

State of the art

5

2.1

Introduction

5

2.2

Shield driven tunnels 2.2.1 Assembly process 2.2.2 Tunnel segments 2.2.3 Thrust jack configurations

5 5 6 6

2.3

Ultra high strength concrete

7

2.4

Case study: Tunnel motorway A13/16

9

Chapter 3

Ring behaviour embedded lining

13

3.1

Introduction

13

3.2

Modelling of ring behaviour 3.2.1 Concrete tunnel segments 3.2.2 Longitudinal joints 3.2.3 Ring joints 3.2.4 Soil interaction 3.2.5 Validation of the model

14 14 15 18 20 24

3.3

Model for the case study 3.3.1 Thrust jack configuration 3.3.2 Number of tunnel segments per ring 3.3.3 Dimensions of the tunnel segments 3.3.4 Soil properties

26 26 26 27 27

3.4

Relation with the lining thickness 3.4.1 Maximum bending moment 3.4.2 Bending moment capacity of the lining 3.4.3 Retrieving the required lining thickness

28 29 31 38

3.5

Conclusions

40

Chapter 4

Grouting phase

43

4.1

Introduction

43

4.2

Modelling of the grout phase 4.2.1 Background of the uplift loading case model

45 45

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4.2.2

Complete and incomplete grouting

48

4.3

Relation with the lining thickness 4.3.1 Maximum bending moment 4.3.2 Bending moment capacity 4.3.3 Retrieving the required lining thickness

48 48 51 51

4.4

Conclusions

52

Chapter 5

Additional structural mechanisms

55

5.1

Introduction

55

5.2

Introduction of thrust jack forces 5.2.1 Magnitude of the thrust jack force 5.2.2 Compressive stresses beneath the thrust jack plates 5.2.3 Tensile bursting stresses

55 57 59 62

5.3

Torsion in tunnel segments by the trumpet effect 5.3.1 Torsion capacity of the lining 5.3.2 Torsion displacements in the uplift loading case

66 66 70

5.4

Conclusions

72

Chapter 6

Evaluation lining thickness reduction

75

6.1

Introduction

75

6.2

Importance of the construction phase 6.2.1 Torsion by trumpet effect governing 6.2.2 Grouting phase and introduction of thrust jack forces governing 6.2.3 Ring behaviour of embedded tunnel

75 77 77 78

6.3

Improvement of behaviour steel fibre reinforced concrete in tunnels 6.3.1 Addition of steel bar reinforcement 6.3.2 Reduction of uplift force in grouting phase by ballast 6.3.3 Reduction of thrust jack force

79 79 81 83

6.4

Conclusions

83

Chapter 7

Conclusions and recommendations

85

References

87

Table of figures

89

Appendices

xiv

93

Appendix A Derivations A.1 Janßen joint A.2 Transformation of coordinate systems A.3 Uplift of embedded tunnel

95 97 105 107

Appendix B

Ultimate resisting moment for steel fibre reinforced concrete

109

Appendix C

Safety factors in shield driven tunnels

117

Chapter 1

Introduction 1.1

Introduction to shield driven tunnels

Construction of shield driven tunnels became popular in the Netherlands since the late 1990’s. This type of structure is thought to be a good way to cover up roads and railways. Hindrance at ground surface for both humans and nature is reduced to a minimum level during construction and use of the infrastructural connection. Doubts about the actual feasibility of constructing such a tunnel in the soft Dutch soil with a high water table avoided construction for a long time. Shield driven tunnels should be water tight and during the construction process floating should be prevented. Abroad however more experience was gained with the construction of these tunnels in likewise soft soil. In Germany, France and Japan for instance several projects were completed successfully. The first excavated tunnel in soft soil was finished in 1843. The tunnel connected both sides of the river Thames in London and was designed by Marc Brunel and his son Isambard. Brunel used a shield to keep the open area of the tunnel and the surrounding soil apart. Through the front of the shield miners removed soil through small cavities. Behind the forward moving shield a brickwork tunnel lining was erected. Basically this method is still used in the construction of today’s shield driven tunnels. However miners have been replaced by a fully automated tunnel boring machine (TBM) and the brickwork has been replaced by a prefabricated segmented concrete tunnel lining. Foreign successes resulted in a pilot project in the Netherlands as well. In 1995 construction of the first Dutch shield driven tunnel, the Second Heinenoord Tunnel, commenced. Subsequently nine tunnels have been finished or are still under construction: 1. Second Heinenoord Tunnel 2. Westerschelde Tunnel 3. Sophia Rail Tunnel 4. Botlek Rail Tunnel 5. Tunnel Pannerdensch Canal 6. Green Heart Tunnel 7. North/South Metro Line Amsterdam 8. RandstadRail Staten Tunnel Rotterdam 9. Hubertus Tunnel The Hague During and after construction of these tunnels several research projects studied the behaviour and forces on the lining. As a result of the growing awareness of the mechanisms a shield driven tunnel is subjected to, the diameter of the tunnel increased from 7,6 m for the Second Heinenoord Tunnel to 13,3 m for the Green Heart Tunnel.

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1.2

Problem description

From the foreign history of shield driven tunnel construction in soft soil it is known that the required thickness of concrete tunnel linings has a directly proportional relation with the tunnel diameter. The thickness should be approximately 1/20 of the diameter (1/20 D) [4]. Obviously the circumference of the tunnel also increases proportionally to the diameter. Hence their product, which represents the amount of concrete needed to construct the lining, increases quadratic proportional to the tunnel diameter (see Figure 1). For tunnels with large diameters this results in high costs as a consequence of the material volume, heavy transport and the assembly process. V The mechanism that leads to the required lining thickness has been e m u and still is an intensively discussed subject. In his dissertation [5] l o D D π ² v V ~ 2π -· = D Blom described several mechanisms which lead to observed cracks 2 20 20 e t e r (damage) in existing shield driven tunnels. By considering cracks as c n o an early warning system for failure caused by overloading, these c mechanisms might lead to failure at some point. Despite the ongoing discussion the demand for larger tunnel diameters remains. And preferably these larger diameters should be tunnel diameter D achieved using equal or even smaller lining thicknesses than 1 | Volume of concrete increases applied today to achieve an optimal economic design, hence save quadratic proportional to tunnel diameter costs.

--

The application of new concrete materials such as very high strength concrete and ultra high strength concrete might be a solution to this controversy. The use of these steel fibre reinforced concretes already resulted in some very slender structures for bridges and roofs. Simultaneously in [16] it is stated that by the addition of steel fibres to tunnel segments of ordinary strength concrete ‘cost savings can be achieved in two ways: either by reducing the thickness of the elements or by reducing the amount of traditional reinforcement’.

1.3

Problem definition

The amount of concrete in shield driven tunnel linings increases quadratic proportional to the tunnel diameter. Consequently shield driven tunnels with large diameters are very costly. New steel fibre reinforced concrete materials with significantly higher compressive strengths – very high strength and ultra high strength concrete – might be a solution to reduce the lining’s thickness.

1.4

Objective

The objective of this study is to investigate the technical feasibility of reduced lining thicknesses for shield driven tunnels by applying very high strength and ultra high strength steel fibre reinforced concrete in stead of ordinary concrete with steel bar reinforcement. The research will be applied on a shield driven tunnel for the future motorway A13/16 in the North of Rotterdam.

1.5

Solution approach and arrangement of this report

The subjects of shield driven tunnels, ultra high strength concrete and the mentioned case study of motorway A13/16 will be explained briefly in Chapter 2 to gain a better understanding of the issues discussed in this report. Several mechanisms which are assumed to possibly result in the required lining thickness of 1/20 D have been presented by Blom [5]. In other studies as well the behaviour of these mechanisms has been investigated:

2

Shield driven tunnels in ultra high strength concrete

 



Ring behaviour of a tunnel embedded in soil [17] Uplift loading case: ring behaviour of a tunnel during construction in the semi-liquid grout [5] Introduction of thrust jack forces from the TBM into the tunnel lining [7, 12]

In this report the relations between those failure mechanisms and the required lining thickness will be studied for three concrete strength classes: 1. Ordinary concrete with steel bar reinforcement C35/45 2. Very high strength steel fibre reinforced concrete C100/115 3. Ultra high strength steel fibre reinforced concrete C180/210 Each mechanism will return a boundary condition of the required lining thickness for the case that particular mechanism would be governing. For that purpose Chapter 3 and Chapter 4 discuss the calculation methods and resulting lining thickness boundary conditions for the tunnel’s ring behaviour when embedded in soil and in the uplift loading case respectively. From literature [3] it’s known that shear forces are of minor influence in a circular shield driven tunnel and will not turn out to be governing. Therefore the focus of these calculations will be on the generated bending moments and normal ring forces. Chapter 5 discusses the influence of two additional mechanisms from the tunnel’s construction process. One of them is the already mentioned introduction of thrust jack forces, the other is the socalled trumpet effect. The latter has been described in [5] and deals with observed cracking due to the distortion of tunnel segments in the uplift loading case. Cracks resulting from this imposed deformation do not result in a failure mechanism to occur, but ‘only’ cause damage. Nevertheless this mechanism will be studied for the concrete strength classes presented in order to investigate any possible variations in the sensitiveness. All boundary conditions regarding the lining thickness will be combined in Chapter 6. Then it will become clear which mechanisms actually dominate the required lining thickness for the case study’s tunnel. The possible reduction of the lining thickness by the use of very and ultra high strength concrete will then become visible as well. Since the governing mechanism is then known, alterations to the tunnel design might reduce the lining thickness even more. Some examples are given in Chapter 6. Conclusions about the research presented in this report and recommendations for additional research are described in Chapter 7. Figure 2 schematically shows the structure of this report.

Reduction of the tunnel lining thickness

3

5 4 / 5 3 C

5 1 1 / 0 0 1 C

0 1 2 / 0 8 1 C

Concrete material Tunnel depth

Ring behaviour

Uplift loading case (grouting)

Layout tunnel design

Introduction thrust jack forces

Torsion in segments (trumpet effect)

Lining capacities Required lining thickness Chapter 3

Required lining thickness Chapter 4



Chapter 6 Governing failure mechanism Required lining thickness(es)

Adaptation of design to reduce effect of governing mechansim Chapter 6

Economy

Tunnel design, required thickness, depth projection and concrete material

2 | Visualisation of this report’s structure

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Chapter 2

State of the art 2.1

Introduction

From practice it is observed that despite all effort put in analyses and engineering of shield driven tunnels, the lining thickness is still of a constant ratio to the tunnel diameter: 1/20 D. However the wish remains to construct ever larger tunnel diameters with similar or even smaller lining thicknesses compared to the ones used today. The newly developed ultra high strength concrete might be a solution to this problem. Concrete strength classes in existing shield driven tunnels did never exceed regular strengths like for instance C30/40. In section 2.3 the rise of the new type of concrete will be clarified briefly. The subject of shield driven tunnels will be discussed in section 2.2. The construction process of the rings and the tunnel segments themselves are dealt with. In the north of Rotterdam a new motorway is planned, the A13/16. This motorway has to reduce the amount of traffic on the heavily loaded motorways in Rotterdam’s urban areas. Section 2.4 describes the location and environment of the new motorway. The option to construct this motorway in a shield driven tunnel is presented. This project is used as a case study in this report.

2.2

Shield driven tunnels

The lining thickness over diameter ratio for Dutch shield driven tunnels approaches the standard ratio of 1/20 relatively close (Table 1). Most tunnels use a slightly smaller ratio of 1/22 (among others the Westerschelde Tunnel and the Green Heart Tunnel), although in the Rotterdam RandstadRail project the ratio is 1/17. Table 1 | Lining thickness over diameter ratio for Dutch shield driven tunnels Tunnel project

Internal diameter [m]

Lining thickness [m]

Ratio

Second Heinenoord Tunnel

7,6

0,35

1/22

Westerschelde Tunnel

10,1

0,45

1/22

Sophia Rail Tunnel

8,65

0,40

1/22

Botlek Rail Tunnel

8,65

0,40

1/22

Tunnel Pannerdensch Canal

8,65

0,40

1/22

Green Heart Tunnel

13,3

0,60

1/22

North/South Metro Line Amsterdam

5,62

0,30

1/19

RandstadRail Tunnel Rotterdam

5,8

0,35

1/17

Hubertus Tunnel The Hague

9,4

0,45

1/21

2.2.1

Assembly process

During the construction of a shield driven tunnel the tunnel boring machine (TBM) digs through the ground by excavating soil with a cutting wheel. The actual forward movement of the TBM is realised by pushing off at the front of the already constructed tunnel lining using thrust jacks. These jacks are usually combined in groups of two, which share the same so-called thrust jack plate to make contact with the tunnel lining.

5

bolt pocket handle hole width longitudinal joint thickness

bearing pad

ring joint length longitudinal joint

3 | Dimensions and elements of the tunnel lining

As soon as the TBM cleared enough soil, a tunnel ring is assembled by several tunnel segments within the protection of the TBM’s shield. These segments are temporarily bolted together to the previous ring in order to prevent shifting and dropping during the complete construction process. Because the tunnel rings are assembled within the TBM’s shield, the external diameter of this machine is slightly larger than the tunnel’s diameter. Hence a tail void is created between the tunnel lining and the surrounding soil due to the forward movement of the TBM. To prevent settlements of the soil and to embed the tunnel the viscous material grout is injected. That material stiffens over time.

2.2.2

Tunnel segments

The tunnel lining is assembled from rings each containing a certain number of tunnel segments. Most Dutch shield driven tunnel have seven segments per ring. Rings in the Green Heart Tunnel have nine segments. Adjacent rings are rotated by half a segmental length to create a masonry layout and thereby avoid joints to be in line. Besides regular tunnel segments, a small trapezoid key stone is present in each ring. This is the last segment placed in the assembly of the ring to close it. Its trapezoid shape makes it easier to force it in. In the Netherlands prefabricated concrete segments are commonly used. Only the Rotterdam RandstadRail Tunnel includes some steel segments as well. Tunnel segments in one and the same ring make contact in longitudinal joints (see Figure 3). In adjacent rings segments are connected in the ring joint. Deformations occur due to loading of the tunnel rings by the surrounding soil and water pressure. These deformations force the joints to slightly rotate. Damage to the edges of the concrete segments in these rotating joints is prevented by reducing the height of the contact area compared to the full segmental thickness, or lining thickness.

2.2.3

Thrust jack configurations

Ring jonts do not make contact along the full length of the segment; it’s reduced to some predefined locations. The enormous axial normal forces which are introduced in the tunnel segments by the TBM’s thrust jack forces are passed to the other rings via these contact areas as well. Consequently the location of contact areas is defined by the positions of the thrust jack plates on the tunnel segments. Two thrust jack configurations are commonly used in Europe (see Figure 4): 1. German configuration (thrust jacks at both edges and in the middle of segments, hence at 0, ½ and 1 of the segmental length) 2. French configuration (thrust jacks at ¼ and ¾ of the segmental length)

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a

b 4 | Thrust jack configurations: a) German method; b) French method

In the German configuration thrust jacks at the edges of the segment are realised by one thrust jack plate over the longitudinal joint of two adjacent segments. The total number of thrust jack plates per ring is therefore similar for both configurations. A significant disadvantage of the German configuration is the appearance of large tensile forces by the introduction of thrust jack forces at the edge of the segment (see Chapter 5). Because of the presence of three contact areas in the ring joint, this configuration is vulnerable to placing errors in already assembled rings as well. If one out of three contact areas is not able to fully interact with the other ring, the huge thrust jack force at the other front face of the segment has to be diverted to the other remaining supports. The result is the creation of large internal tensile forces. In concrete this implies the formation of cracks, hence damage (see Figure 5). If the axial contact pressure in the ring joints is present, the joints are able to interact in radial and tangential direction as well. The distribution of contact areas along the segmental length of the German configuration, especially the area in the middle, then introduces significant additional peak bending moments in the tunnel lining. All mentioned problems are significantly reduced or diminished by reducing the number of contact areas per tunnel segment to two and by locating these areas as far from the longitudinal joints in adjacent rings as possible. These properties have been realised in the French thrust jack configuration. From the finished Dutch shield driven tunnels only the Green Heart Tunnel made use of this configuration.

a

b

5 | Uneven support of tunnel segments for: a) German configuration; b) French configuration

2.3


Almost simultaneously with the appearance of the first shield driven tunnels in the Netherlands, a new material was introduced in the world of concrete, namely ultra high strength concrete. In Canadian Quebec the Sherbrooke Footbridge was constructed in 1997. This extraordinary bridge has


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been constructed from concrete with a cylindrical compressive strength of 200 MPa. Compared to the highest strength achievable in those days, 90 MPa, the new material was a huge leap forward. However already in 1981 Danish Hans Henrik Bache described the possibility to make concrete with an ultra high compressive strength by using very fine particles in the mixture [19]. Despite the high compressive strength, the material behaved very ductile by the addition of large amounts of steel bar reinforcement. This material is now known as CRC (Compact Reinforced Composite), a product of the Danish company Aalborg Portland. In the materials developed from the late 1990’s on the reinforcement bars were replaced by steel fibres in the concrete mixture itself. Companies like the French Lafarge, with the product Ductal, and the likewise French company Eiffage, with its product BSI-Céracem (Béton Spécial Industriel), introduced their own ultra high strength concrete (UHSC) or ultra high performance concrete (UHPC) on the market. Table 2 shows an enumeration of concrete s trength classifications. Table 2 | Classification of concrete strength classes [19] Term

Range strength classes

Ordinary strength concrete

up to C53/65

High strength concrete

C53/65 to C90/105

Very high strength concrete

C90/105 to C150/170


C150/170 to C200/230 1)

Super high strength concrete

from C200/230

1)

Level not included in source [19]

By now a number of projects have been constructed in the new steel fibre reinforced concretes. These are mainly bridges and roofs in both France and Japan. All were accomplished with very slender structures. Figure 7 shows the Bridge of Peace in South Korea. The creation of ultra high strength concrete is possible by making some changes in the design of the concrete mixture [10, 6]: Reduce the water-cement ratio Increase the packing density by optimising the grading curve (see Figure 6) Improve the homogeneity by using small sized particles only Add steel fibres to the mixture to resist minor internal tensile bursting stresses and to bridge minor cracks    

C27/35

C65/85

C130/150

C170/200

B180/210 (BSI)

m c 1

6 | Concrete materials: significant increase of density and homogeneity for BSI

Both types of ultra high strength concrete, with steel bar or steel fibre reinforcement, are extremely dense. This results in outstanding durability properties. Ductal for instance claims a carbonation depth of only 2 mm in 500 years [9]. For the construction of road traffic bridges in ultra high strength concrete in the French town of Bourg-Lès-Valence design recommendations for the material used were required. As a result the French organisations AFGC (Association Française Génie Civil) and Setra (Service d’études

8


7 | Pedestrian Bridge of Peace in Seonyu, South Korea

techniques des routes et autoroutes) published a document with some interim recommendations in 2002. At the moment of publication of this report the French recommendations were the only recommendations present; hence calculations in this study were based on their findings. Prices of ultra high strength concrete are yet very high. A cubic metre of BSI-Céracem (C180/210) costs approximately 400-800 euro. The same amount of ordinary concrete costs only 150-200 euro. If the production of the material increases because more engineers and contractors dare to apply it in their structures, the costs are supposed to show a significant decrease.

2.4

Case study: Tunnel motorway A13/16

In the entire Randstad area in the Netherlands’ coastal region severe traffic jams on motorways occur daily. An obvious bottleneck is the low-capacity junction Kleinpolderplein in the north of Rotterdam. This junction causes long jams on the urban motorways A13, A20 and the connected A16 (see Figure 8). A significant amount of traffic however only uses the A20 and the junction to get from the A13 to the A16 and vice versa. Therefore the Dutch Rijkswaterstaat (Ministry of Transport and Water) appealed for the design of a new motorway to interconnect both motorways A13 and A16 [14]. This motorway A13/16 will significantly decrease the amount of traffic on the existing urban motorways A20 and A13 and the junction Kleinpolderplein. The route Rijkswaterstaat suggests runs close to (future) housing an existing and future green zones, among which the picturesque river Rotte and park Lage Bergsche Bos. Construction of the motorway in a shield driven tunnel appears to be a sympathetic solution to diminish hindrance during its construction and use in those areas.


9

A4

Prins Clausplein

Zoetermeer Rijswijk

A12

Nootdorp Ypenburg

Pijnacker

Delft

Bleiswijk Delfgauw N209

Berkel en Rodenrijs A4

Bergschenhoek N470 A13 Hoge Bergsche Bos Lage Bergsche Bos

A13/16

Rotterdam Airport

Rotte

Terbregseplein A20 Overschie

Kleinpolderplein Kethelplein

Rotterdam

Schiedam Vlaardingen

A16

Nieuwe Maas

A4

8 | Surroundings of future motorway A13/16 in the north of Rotterdam Urban area (future)

Motorway (future)

Industrial area (future)

Main road

Green zone (future)

Local road

Water

Railway (industrial)

© 2006, TW GROENEWEG

N

1 km

Tunnel diameter

The maximum allowed speed at motorways connecting to the new road A13/16 is restricted to 80 km/h nowadays. By allowing this speed in the tunnel as well, a serious reduction of the required lane widths is possible (holds for motorways with a maximum speed of 90 km/h and below according to Dutch guidelines for tunnel design SATO [13]). In the Netherlands multi-lane motorways have not been constructed in shield driven tunnel yet. The width required for all lanes in one direction would lead to very large tunnel diameters. The same might hold for this motorway. Rijkswaterstaat demands two lanes in each direction plus a safety strip. The width of the safety strip should be sufficient to construct a third lane in future. The required free cross-sectional surface for the motorway is a rectangle with a width significantly larger than its height. Hence fitting of this rectangle in a circular tunnel would result in huge useless free areas above and below the motorway (Figure 9a). By stacking both traffic directions on top of each other, the construction of a second tunnel is omitted (Figure 9b) and less free area is present. A diameter increase of only approximately 10 % is sufficient. Now the required internal diameter for the tunnel of motorway A13/16 is 16,7 m. Compared to the present world record diameter for shield driven tunnels in soft soil – 13,3 m of the Green Heart Tunnel – this implies a very significant increase. For now this increase of 25 % is expected to be too large for the goal of this study. More measures are therefore required to reduce the height and most of all the width of the traffic lanes. The most effective solution is to ban trucks and lorries from the tunnel by permitting a maximum vehicle height of 2,6 m only. Trucks, being a small part of the total traffic only, should take the traditional route over the junction Kleinpolderplein. The internal diameter of the tunnel is now reduced by 1,8 m to 14,9 m (see Figure 10). Plans for a new motorway A6/9 near Amsterdam (eliminated by politics now) included a similar tunnel with similar diameter as well [15].

9 | Fitting two directions in one tunnel tube. a) Two tunnels with large useless free areas; b) One tunnel with less free space

Alignement

Rijkswaterstaat already suggested two routes for the new motorway. Because no hindrance of a motorway in a shield driven tunnel is recorded on the ground surface, the shortest route can be applied. Every possible route forces the motorway to cross the high speed railway line HSL-Zuid. At the crossing location the railway is situated at a slab founded on a significant number of poles. These piles require an underground crossing at a depth of at least -24,00 m + NAP (soil overburden of 19,26 m). The research project of this report does not deal with the complete design of the tunnel. The effects of the application of new concrete materials on the reduction of the lining thickness of this tunnel are investigated. A vertical alignment of the motorway will be assumed only. For now it’s supposed to be horizontal at a depth of -24,00 m + NAP. Only at the ends the tunnel will reach smaller overburdens. At the engineering office of Gemeentewerken Rotterdam soil data of the tunnel’s route was available. It shows that the tunnel is fully embedded in packed sand, the so-called Layer of Kedichem. Figure 11 shows both the alignment and soil conditions of the tunnel.


11

traffic signs and lighting extra height for safety e t u o r e p a c s e

BD 35 JR

safety strip

RW 13 16

T rans rt p e r o

IG WR19

VTUD 06

safety strip

10 | Cross-section of tunnel for motorway A13/16 with personal cars only (scale 1:150)

A13

1 km

N470

HSL

N209

Rotte

A16 NAP 0 m -10 m

t f a h s

t f a h s

Peat

shield driven tunnel

Klay

-20 m -30 m -40 m

Silt Sand

-50 m

Unknown

11 | Schematic representation alignment of the tunnel for motorway A13/16

12


Chapter 3

Ring behaviour embedded lining 3.1

Introduction

Reaction forces and deflections in the serviceability stage of a shield driven tunnel are described by calculations on ring behaviour for the embedded lining. Calculations are made on the cross-sectional face of the tunnel perpendicular to its axis. Soil surrounding the tunnel has certain stiffness, just like the concrete lining itself. Stiff parts attract bending moments. Consequently the tunnel and soil will cooperate to bear all loads. These loads result from the soil’s mass and ground water pressure s urrounding the shield driven tunnel. In the introduction of this report it was stated that shear forces in circular tunnels are not governing [3]. On the other hand the bending moments, in combination with a normal force in ring direction, are governing. Hence the scope of calculations in this chapter will be on those re action forces. In 1964 Schulze and Duddeck described ring behaviour of shield driven tunnels by a collection of graphs. By means of those graphs bending moments and normal forces could be retrieved for various depth projections of the tunnel and various ratios between the tunnel s tiffness and soil stiffness. When computers developed and the time needed for more comprehensive calculations decreased, the creation of models specifically designed for one tunnelling project grew popular. The main difference in the models created by now is the modelling of the soil. In finite element models soil is normally introduced as a continuum around the tunnel lining. In more uncomplicated framework analyses the soil has been reduced to springs and loads representing the supporting and loading effects of the soil on the tunnel lining. This model focuses on the tunnel structure only; the developments of deformations and stresses in the surrounding soil are omitted. Finite element models however are able to return these soil results as well. Modelling of the tunnel lining itself can be realized by reducing the ring to a homogenous ring beam, a segmented single ring beam or a segmented double ring beam. The homogeneous ring beam is most simplified, but ignores peak moments which develop in the lining due to the presence of longitudinal joints and ring joints. The segmented single ring beam model takes care of the longitudinal joints as well. This model is valid if no axial normal forces are present; hence no interaction between rings occurs via the contact areas in the ring joints. The segmented double ring beam model introduces the effects of both longitudinal longitudinal and interacting ring joints in the calculation. calculation. All models can be created in a 2-dimensional or 3-dimensional environment. A 3-dimensional model however is a waste if no serious changes in loading, support or geometry occur in the third dimension, the tunnel’s axial direction. The only significant defect in this direction is the ring joint. Such a joint however can be modelled in a 2-dimensional environment as well, which is then referred to as 2½-dimensional as well. Hence a 2-dimensional environment is sufficient. In section 3.2 the creation and validation of a segmented double ring beam model with soil interaction represented by springs and loads will be described. This type of model is a reliable representation of reality [5] focussing on the tunnel structure only with relatively short calculation

13

times. Therefore this model is widely used in practise and will be used in this research project as well. Section 3.3 abstracts parameters from the case study’s tunnel for use in the yet created model. In order to reduce the lining thickness of a shield driven tunnel, knowledge about the relation between the thickness and safety level of the tunnel is required. Section 3.4 concludes this relation for the case study tunnel of this research project, motorway A13/16. It will turn out that the depth projection of the tunnel has a serious influence on the safety level. Therefore this chapter will not focus on the depth projection as suggested in Chapter 2 only, but includes a wider range. Comparison of the actual safety level with a required safety level generates the possibility to abstract a boundary condition on the required minimum lining thickness. In section 3.4 the requirement by the ring behaviour of the embedded tunnel lining will be concluded for the considered concrete materials, namely C35/45, C100/115 and C180/210. Conclusions concerning this chapter will be described in section 3.5.

3.2

Modelling of ring behaviour

A segmented double ring beam model with soil interaction represented by springs and loads on the tunnel will be created and validated in this section. Thanks to the relative simplicity of this type of model, a framework model, only a few choices and calculations are required prior to the actual framework analysis. Contact behaviour in the longitudinal joints and ring joints for instance will be reduced to springs, which are more common and require much less calculation time. For the lining’s ring joint, interaction between two adjoining rings is included in this model. In reality both rings are situated on a different level in the third dimension (the tunnel’s axis). The framework model however is 2-dimensional, which saves a lot of calculation time. Therefore the modelled rings are located at the same spot, but are connected by springs in the original contact areas only to imitate the third dimension. Hence it is called a 2½-dimensional model. Modelling of the interaction forces in one ring joint requires that the influence zone of this particular joint is included in the model only. This implies that only half the width (dimension in axial direction) of each tunnel segment is used. If this fact is considered in all calculations and checks of the embedded ring behaviour, no difficulties occur. Calculations in this study will not include the key stone. The narrow width of this segment results in a small decrease of the length (tangential dimension) of adjoining segments. By excluding the stone their width is normal again and their common longitudinal joint is positioned at the centre of the actual key stone. The validation of the model will prove that no significant changes occur due to the key stone. Section 3.2.1 describes the reduction of the concrete tunnel segments to elements for the framework analysis. Modelling of the longitudinal joints and ring joints is described in sections 3.2.2 and 3.2.3 respectively. In section 3.2.4 the modelling of soil is taken into consideration, which will be represented by springs and loads as described before. The final model will be validated by comparing its resulting bending moments to the ones of two other studies in section 3.2.5. The software application ANSYS 6.1 is used to process the framework analysis from this study.

3.2.1

Concrete tunnel segments

In the cross-sectional plane of a ring tunnel segments are a kind of beams. The “span” of these beams is projected in between the longitudinal joints at both ends. In a framework model these beams are represented by so-called beam elements. If the concrete material is assumed to behave linear elastic, elastic straight beam elements are applied in the gravity point of the beams (for rectangular tunnel segments in the centre point). For these framework elements are straight by definition, multiple

14


elements connected in nodes are necessary to simulate the curved shape of the tunnel segments. This results in a more accurate image of the forces and deflections. First of all forces and deflections are defined at the ends of beam elements or at the nodes only, so local peaks in the bending moment development are better observed at a higher accuracy. Second the representation of the hydrostatic water pressure and forces from the soil s oil (defined at the nodes) is improved in a high accuracy model. A certain number of nodes were needed any way. Connections between adjoining rings, representing the ring interaction, can be attached to nodes only. For the French thrust jack configuration now at least two nodes are required in the segment’s “span”. By adding the nodes at the ends of the segment (in the longitudinal joints) to this number, at least four nodes (hence three beam elements) have to be used. To keep the length of each beam element equal (for output handling reasons), another node is added to the middle of the segment. So at the end at least five nodes and four beam elements make a tunnel segment with the French thrust jack configuration. More detailed output data is achieved by multiplying this required number of elements by any integer. In this study 12 beam elements will be used for each tunnel segment. In ANSYS five properties have to be defined for each elastic beam element: 1. Cross-sectional surface in axial-radial plane A = bh 2. Moment of inertia I = 121 bh 3 3. 4. 5.

Height h Young’s modulus E Poisson ratio ν

Where b is half the original width of the tunnel segment. One probably noticed that no property has been assigned to the segmental mass. It will not be added as a loading of the structure either. Because the weight of the tunnel lining is very small compared to the mass of the excavated soil, the high pressure on the tunnel’s external surface will hardly be influenced by the tunnel’s own mass. Consequently this load will not be introduced to the calculation. The circular shape of shield driven tunnels generates an ideal opportunity to use a cylindrical system of coordinates. In this system each node of the beam elements is defined by a fixed radius and an angle. Of course this assumes a fully circular tunnel ring prior to loading.

3.2.2

Longitudinal joints

Longitudinal joints are under pressure due to a tangential normal force in the lining, the normal ring force. This normal force prevents opening of the joint by applied bending moments to some extent. If however the rotation reached a certain relatively high level, opening will occur at one side of the joint. As a result the stiffness of the joint is reduced. Janßen developed an often applied method to include the non-linear rotational behaviour of the longitudinal joint in a rotational spring. This non-linear spring connects the end nodes of two adjacent tunnel segments in the same ring. The so-called Janßen joint assumes that the contact area can be represented by a concrete beam with a depth equal to the joint contact area’s width (segmental width) and its height and width equalling the joint’s contact height. Opening of the joint is included by the fact that the concrete beam is unable to bear any tensile stresses at all. Now it is possible to find a relation between the rotation in, bending moment in and normal force on the beam. Next this relation is translated into a spring stiffness of the rotational spring. At an increasing rotation in the joint or beam, three stages are distinguished:


15

2N ) Ebl t

1.

Closed joint (rotation φ ≤

2.

Opened joint (rotation φ >

3.

Opened joint with plastic concrete behaviour (maximum strain ε' c < ε max ≤ ε' u )

2N and maximum strain in joint ε max ≤ ε' c ) Ebl t

The enumeration also shows the boundary conditions for the rotation or maximum strain in the joint to be valid for that particular stage. The parameters represent: b = Width contact area longitudinal joint (segmental width) E = Young’s modulus lt = Height contact area longitudinal joint N φ ε'c ε'u

= = = =

Normal ring force Rotation Compressive yield strain of concrete Ultimate compressive strain of concrete

In Appendix A.1 the equations for the Janßen rotational spring stiffness k r have been derived. Here the final equations are given for all three stages: bl 2 E 1. Closed joint: k r = t 12

2. 3. Where: M f'c

⎛ 2M ⎞ 9 bl t EM⎜⎜ − 1⎟⎟ N l ⎝ t ⎠ Opened joint: k r = 8N

2

Opened joint plastic concrete behaviour: kr 1 , 2 =

= =

(6N ±

6(− 2Mf 'c b + l t Nf'c b + N 2 )ME

)

21N 2 + 30Mf'c b − 15Nl t f'c b l t f '2c b

Bending moment in the joint Design value compressive strength concrete

In most studies only the first two stages (opened joint and non-plastic closed joint) are being used. Plastic behaviour in the joints implies very large rotations and very large deflections, which is undesirable in normal day use and in general will not and may not take place. Consequently only the first and second stages will be applied in the model of this study as well. The equation for the spring stiffness of an opened joint shows that the stiffness is related to the rotation φ , bending moment M and normal force N. The software application ANSYS however only knows a non-linear spring with a custom relation between the rotation and bending moment, without the normal force. However no problems arise, for the normal ring force varies only little along the ring’s circumference and is relatively easy to predict in advance. For a ring under uniform radial pressure p (see Figure 12) the normal ring force is retrieved as follows. The vertical component of the pressure at an angle θ is given by: pv = p cos θ The force due to this vertical pressure at a section with an angular span of dθ holds: fv = pv Rdθ Where: R

16

=

External radius


Now the total vertical force at a quarter of the ring is given by: π/2

Fv = b ∫0

π/2

fv = b∫0

pv Rdθ = − bpR

Equilibrium of vertical forces at this single part holds: ΣN v = 0 ⇔ N = Fv Hence the normal ring force N is: N = − bpR

(1)

p θ

dθ R

N

N

12 | Uniform pressure leading to normal ring force tunnel

If the average radial pressure on the tunnel is used for pressure p in equation (1), a rather accurate approximation of the normal ring force in the Janßen joint returns. The resulting non-linear relation between the rotation and bending moment is assigned to all non-linear rotational springs at the locations of the longitudinal joints in the model. An example of this relation is shown in Figure 13. moment M [kNm] 700

Mmax

600 500 400

joint opened b = 1000 mm ℓt = 350 mm E'c = 33.500 MPa N = 3848 kN

300 200 joint closed

100

rotation φ [mrad]

0 0

20

40

60

80

13 | Relation between rotations and bending moments in a Janßen joint

Obviously the Janßen method only focuses on the transfer of bending moments in the joint. Normal forces and shear forces are transmitted in a far more straightforward way, namely by linear translation springs. Provided that the joint is subjected to a pure compressive normal force the joint will not be noticed at all, the normal force is simply transferred from one tunnel segment to the other. The joint itself does not elongate by the compressive force. A spring in tangential direction should therefore prevent such elongation. In other words: a spring stiffness of infinity should be assigned to the spring. The software application however requires an actual value. A parameter study showed that a value of 1010 kN/m results in minor insignificant values only; hence this value was selected. So why not use a beam element when an infinite stiff behaviour in the axial direction is required? The beam element has a property to transmit bending moments, shear forces and normal forces by definition from one end to the other. For this joint however only normal forces and shear forces (frictional forces) should be transferred, bending moments are taken care of by the Janßen joint. Hence a beam element is unsuitable.


17

The behaviour of frictional shear forces in the longitudinal joint is partly similar to the one of normal forces. If a shear force is directed from one concrete surface to another, it is assumed that no deformations occur until the ultimate capacity is reached and the connection fails at once. Then an instant infinite deflection takes place. In the model of the longitudinal joint this property is represented by a linear translation spring with a stiffness of 1010 kN/m in radial direction. After the framework analysis it should be verified whether the friction force capacity was exceeded or not. At the end three springs are necessary to model a longitudinal joint between two tunnel segments: a rotation spring with a non-linear spring stiffness according to Janßen’s method and linear translation springs in radial and tangential direction with high spring stiffnesses.

3.2.3

Ring joints

Contact and thereby exchange of forces in the ring joint occurs in the so-called bearing pads. In shield driven tunnels with the German thrust jack configuration dowels and sockets have been applied now and then. Originally these dowels were added to make positioning of the tunnel segments during assembly easy. Easy positioning requires some clearance between the dowel and socket for movement. In the initial situation prior to loading the dowel and socket are not in contact. Only at relatively large deformations of the tunnel ring contact might occur. Again this is something that’s undesired due to the large deformations and possible peak moments by the new ring interaction areas. No dowels and sockets are applied in the French thrust jack configuration (which will be applied on the tunnel for motorway A13/16 in the following section), thus this problem will not occur. Interaction in de ring joints is restricted to the bearing pads only, at the same position along the segmental length as where the thrust jack plates are located. In most Dutch shield driven tunnels so-called packing materials have been used at the contact area of the ring joint. For instance plywood planks are frequently used. The idea behind this phenomenon is that if the ring joint rotates the plywood plank will deform unevenly, but remains in contact with both concrete tunnel segments. Now the complete contact surface would still be available for the transfer of frictional forces in the ring joint. The thickness of plywood planks automatically defines the location of the ring contact areas. The Green Heart Tunnel showed that packing materials could easily be left out of the design as well. Its joints made use of pure concrete to concrete contact only. The bearing pads were defined by a few millimetres of additional concrete and behaved just as well. The force responsible for interaction between two adjoining rings is a friction force in radial direction. Just like the frictional shear forces in longitudinal joints, this transfer is modelled by use of a linear translation spring in radial direction again. The stiffness of this spring however does not equal infinity. Springs in ring joints represent both the stiffness of the concrete to concrete contact behaviour and the stiffness of the tunnel segments bordering the joint. If one returns to the original modelling of the concrete tunnel segments themselves the reason becomes clear: the beam elements representing the tunnel segments are located at the centre lines of the segments. Even though only half the segmental width is taken into account, the elements remain at the original centre lines. Hence at the cutting faces of the narrowed segments now (see Figure 14). Recollection of the skipped third dimension (tunnel axis) shows that there actually is a full segmental width in between the beam elements of both adjacent rings. Over that width deflections might occur by interaction forces in the ring joint. This will additionally define the total ring joint spring stiffness.

18


14 | Beam elements (and nodes) in t he centre line of a tunnel segment

The total spring stiffnes of the ring joint is determined as follows: 1 1 1 1 1 = + = + ⇔ krj = k rj ,segment k rj k rj , segment k rj ,contact k rj ,segment ∞ Where: krj = Total spring stiffness of the ring joint krj,segment = Spring stiffness by the segments krj,contact = Spring stiffness by the contact area (supposed to equal infinity (∞)) By considering the cross-sectional surface of the half tunnel segment in a vertical plane along the tunnel’s axis, a spring stiffness may be assigned to that particular part of the structure. It is assumed that the magnitude of the axial thrust jack force (still present in the ring joint contact areas) is of such extend that no rotation occurs in the joint. By forcing one of the sides upward a deflection u occurs due to a vertical force F. A linear bending moment distribution results, with its maximum values at both ends. Figure 15 displays this simple mechanism. The symmetric bending moment distribution allows reducing the problem to a beam clamped at one end and loaded by a force F at the other free end. The maximum bending moment simply holds: b M = F 2 The rotation of the loaded endpoint (in the ring joint in the original problem) is then given by: M( b / 2) Fb 2 θ= = 2EI 8EI Where: I = Moment of inertia in the radial-tangential plane

F

F u

u/2

b/2 b

15 | Determination of spring stiffness for ring joints


19

The deflection in the reduced problem is u/2. A rule of thumb provides: u 2 Fb 3 = 3 θ b / 2 = 2 24EI Then the stiffness of a spring representing this phenomenon is defined as: F 24EI = 3 k rj = u/2 b One ring interaction location is handled. Therefore the moment of inertia for the French thrust jack configuration is defined by half the segmental length in tangential direction. Hence: I = 121 l segm / 2 ⋅ h 3 Where: b h Di ℓ segm

= = = =

nsegm =

Full segmental width Lining thickness Internal tunnel diameter π( Di + h) Segmental length = n segm Number of segments per ring

The final spring stiffness for one ring connection holds: 24EI πEh 3 ( Di + h) krj = 3 = b n segm b3 For a tunnel diameter of this research project’s case study, values of approximately 106 to 107 kN/m are found depending on the lining thickness and concrete Young’s modulus. In literature a value of 108 kN/m is recommended [11]. Unfortunately no explanation has been given about the background of this value. A spring stiffness of 107 kN/m is close to both the calculated values and the value from literature. Therefore this compromising spring stiffness will be used in the model.

3.2.4

Soil interaction

The mass of the soil and the hydrostatic pressure by ground-water generate a load on the tunnel. The removal of soil at the location of the present tunnel causes this load to work on the rings in the tunnel lining. Nevertheless the soil itself has a certain stiffness as well, which therefore contributes to supporting its own loads. During the description of the case study tunnel for motorway A13/16 it came forward that this shield driven tunnel is primarily embedded in compacted sand. Since this study does not focus on a detailed design of the tunnel, but tries to find an answer on the question whether the lining thickness of shield driven tunnels may be reduced by applying ultra high strength concrete, a simplified soil continuum is used now. Subsequently the complete soil continuum is assumed to consist of packed sand only. This only affects the loads on the tunnel, not the supporting function as will become clear from the following sections. Loads from the soil

In a uniform soil continuum vertical pressure by the soil mass and hydrostatic water pressure develop gradually over the depth. Above the ground-water table only a grain pressure occurs, beneath the total pressure consists of the hydrostatic water pressure and effective grain pressure (see Figure 16).

20


x0 ρsd

xw

σv,wl

ρsw

xtop

hθ θ

σw( θ ) σv( θ ) σ'v( θ )

σw

σv

16 | Vertical stresses in uniform soil continuum along a tunnel

The vertical pressure at the ground-water table σv,wt is given by: σ v , wt = (x0 − x w )ρsd Where: x0 xw ρsd

= = =

(2)

Soil surface relative to mark NAP Water level relative to mark NAP Specific gravity of dry soil

The total soil pressure at an angle θ of the tunnel’s external surface is: σ v (θ ) = σ v , wt + x w − x top + h θ ρsw

(3)

Where: xtop = Top of tunnel relative to mark NAP ρsw = Specific gravity of saturated soil Parameter hθ is the vertical distance between the tunnel top and the point at an angle θ along the tunnel’s external circumference. It’s defined by: h θ = (1 − cos θ )D / 2 (4) Where: D

=

External diameter of the tunnel

The hydrostatic water pressure and the effective vertical grain pressure combined equal the total vertical pressure. Hence for the effective vertical grain pressure it holds: σ' v (θ) = σ v (θ) − σ w (θ) (5) Where σw(θ) is the hydrostatic water pressure at the point with an angle θ , reading: σ w (θ) = x w − x top + h θ ρ w

(6)

Substitution of equations (2), (3), (4) and (6) in (5) defines the effective vertical grain pressure as: σ'v = (x0 − x w )ρsd + x w − x top + (1 − cos θ)D / 2 (ρsw − ρw ) All vertical pressures simply result from the soil continuum in Figure 16. Water pressure is omni directional, so its horizontal pressure and vertical pressure are equal at any point. Soil pressures do


21

not know this property. This is visible by the fact that soil is stable under a certain gravity slope. The angle of this slope is the internal friction angle and is responsible for the fact that the horizontal effective grain pressure is defined by: σ' h (θ) = K 0 σ' v (θ) (7) Where neutral soil support coefficient K0 is smaller than 1 for soil and is related to the internal friction angle ϕ by: K 0 = 1 − sin ϕ Just like the total vertical pressure, the total horizontal pressure combines effective horizontal grain pressure and the hydrostatic water pressure: σ h (θ ) = σ' h (θ) + σ w (θ) (8) Using these equations it’s relatively easy to determine the total vertical soil pressure at the top of the tunnel. This pressure results from the soil and water masses on top of the tunnel. Implying that the vertical pressure at the bottom of the tunnel should equal the pressure at the top, for the tunnel hole does not introduce any additional mass. However a point at the same depth just next to the tunnel is loaded by the complete soil overburden and should therefore have a significant higher vertical soil pressure. The vertical pressures at the points next to and below the tunnel are connected by the horizontal pressure. This implies that an instable situation would occur. As a result equilibrium will be generated resulting in a total pressure below the tunnel somewhere in between the original pressures of both points. A commonly adapted solution is given in [5]. It’s assumed that the effective vertical grain pressure along the full tunnel circumference equals the effective vertical soil pressure by the original overburden at the centre point of the tunnel. The hydrostatic water pressure remains unchanged, resulting in a pressure difference between the top and bottom of the tunnel. This generates a floating pressure component, which has been observed in the construction of actual shield driven tunnel as well. Tunnel rings are defined in a cylindrical coordinate system because of their circular shape. Therefore the vertical and horizontal soil pressures have to be translated into radial and tangential ones (Figure 17). For this purpose conversion equations from Appendix A.2 are used. They read: sin 2 θ ⎤ ⎧σ v ⎫ ⎧σ r ⎫ ⎡ cos 2 θ = (9) ⎨ ⎬ ⎢ ⎥⎨ ⎬ ⎩σ t ⎭ ⎣− cos θ sin θ cos θ sin θ⎦ ⎩σ h ⎭ σv( θ )

σr( θ ) σt( θ )

σh( θ )

θ

θ

17 | Orientation of vertical and horizontal versus radial and tangential soil loads

Now the radial total ground pressure holds after substitution of equations (5), (7) and (8) in (9): σ r (θ) = σ' v (θ)(cos 2 θ + K 0 sin 2 θ) + σ w (θ) This also can be written as: ⎛ 1 + K 0 1 − K 0 ⎞ σ r (θ) = σ' v (θ)⎜ + cos 2θ ⎟ + σ w (θ) 2 ⎝ 2 ⎠

22


Now two new scalars are defined: ⎛ 1 + K 0 ⎞ ⎛ 1 − K 0 ⎞ C1 = ⎜ ⎟ and C 2 = ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ So at the end the total radial ground pressure is transformed to: σ r (θ) = σ' v (θ)(C 1 + C 2 cos 2θ) + σ w (θ) From equation (9) the tangential ground pressure is isolated as: σ t (θ ) = −(1 − K 0 )σ' v (θ) sin θ cos θ By the introduction of scalar C2 this may be written as: σ t (θ ) = −C 2 σ' v (θ) sin 2θ Loads from the soil are introduced by forces at nodes in the framework model. The (radial) pressure is therefore converted to the (radial) force: bl Fr (θ) = σr (θ) n 2 Where l n is the influence length along the lining’s circumference for one node. At an arbitrary node in the “span” of the tunnel segment this length is defined as the full length of a beam element, at end nodes (in the longitudinal joints) it’s defined as half the beam element length. Excavation of soil by the TBM will induce deformations in the soil. In sand these deformations will occur at once. The adhesive property of clayish materials or sand with clay included will spread the deformations over a longer period of time and affect the tunnel in its serviceability stage. Hence these induced deformations will result in loading of the tunnel in tangential direction. The complete loss of loads in tangential direction (in case of sand) or the presence of a part of the tangential loads is only introduced by multiplying the full tangential load by a reduction ratio αt. In case of sand this ratio is set to zero. If deformations remain it holds that αt > 0. For the Botlek Rail Tunnel, which has been constructed in soil with more-or-less similar conditions to the soil of this report’s case study, a value of αt = 0,25 has been applied [17]. Hence that particular value will be used here as well. Reduction of the ratio to αt = 0 causes a bending moment saving of no less than 12 %. This shows one of the advantages of shield driven tunnel construction in uncontaminated sand. Support by the soil

In structures stiffness attracts bending moments. The same applies to tunnel structures, where both the tunnel lining and soil have certain stiffness. Hence the loads from the soil are mutually supported. The interaction is introduced to the model by attaching the soil stiffness to the tunnel lining as linear translation springs. The spring stiffness per unit of surface of the tunnel’s exterior for a cylinder with ovalisation deformations (the transformation of the initial circular shape to an deformed oval shape by the soil loading) is defined by [5]: ksoil , r E (10) = oed A soil ,r 2R Where: Eoed = R = Asoil,r =

Oedometer stiffness of soil External radius of the tunnel Surface the spring is representing

The Oedometer stiffness depends on the soil properties only. Similar to the soil loads the soil springs are attached to the tunnel segment’s nodes. The stiffness in equation (10) was given per unit of surface, hence it has to be converted as follows:


23

bl n Eoed 4R Soil springs are positioned between the mentioned nodes of tunnel segments and additional restrained nodes with an equal angle, but larger radius. This clearly visualises that these springs are no part of the tunnel structure itself, but represent soil. c soil ,r =

3.2.5

Validation of the model

Before the described model will be applied frequently in the research on the feasibility of reduced lining thicknesses for shield driven tunnels, a verification of the results’ accuracy is performed. Results are compared to the ones from the graduation thesis by Slenders [17] on modelling of the Botlek Rail Tunnel. A key stone had been included in that particular model. A verification model without a key stone is generated using the application LDesign of Blom’s dissertation [5] and compared to the results of this study’s model as well. Validation with Botlek Rail Tunnel including key stone

From the report by Slenders [17] the following dimensions were retrieved, which have been applied in the model for the Botlek Rail Tunnel: Internal diameter = 8,65 m Lining thickness = 0,40 m Jack configuration = German (jacks at 0, ½ and 1 segmental length) Segments per ring = 7 and a key stone Beam elements per segment = 6 (is possible for the German configuration) Height contact area in longitudinal joint = 0,17 m Spring stiffnesses longitudinal joints = r: 28·109 kN/m, t: 28·1010 kN/m Spring stiffness ring joint = 20·107 kN/m Angle of first longitudinal joint = 12,86° The concrete material was defined by: Young’s modulus Lateral contraction

= 38.000 MPa = 0,2

The soil loading was represented by stresses on top, sides and bottom without any floating component. The soil properties were: Soil stresses top and bottom = 0,448 MPa Soil stresses sides = 0,411 MPa Oedometer stiffness = 38 MPa In Slenders’ report another equation was used to translate the soil Oedometer stiffness into a spring stiffness. That equation was commonly used before, but appears to be valid only for uniformly loaded (hence without ovalisation) tunnel rings [5]. The equation was: csoil , r Eoed = A R Hence this equation results in a double spring stiffness compared to equation (10). In the model about to be validated a fictitious double Oedometer stiffness of 76 MPa has been used to prevent uneven results. Resulting bending moments from both models corresponded very well. Figure 18 shows both distributions of the bending moment and their local maximum values. At the sides and bottom of the tunnel only small deviations up to 2,5 % occur. At the top however obvious changes of the bending moment distribution take place due to the missing key stone. The deviation is still reduced to approximately 4,2 % only.

24


100 Moment M [kNm] 50 0 0 3 3

A ngl e θ

0

-50

[ ° ] 3 0

-76,5 -79,9 -100 0 0 3

6 0

0 7 2

BRT-model Model for this study

0 9

87,2 84,9

0 2 1

2 4 0

-77,4 -76,5 0 1 5

2 1 0

180

18 | Moment distribution in validation BRT-model and model from this study

The absolute maximum values of both distributions are located at the right-hand side of the model and vary just 2,5 %. It’s therefore concluded that the generated model shows good results and that variations between models with and without the presence of a key stone are only very small. Therefore no key stone is included in this study. Validation with LDesign

An additional verification of the model from this study will be performed using a verification model without a key stone. The software application LDesign is a straightforward solution to obtain quick results for calculations on the ring behaviour of shield driven tunnels. The same dimensions and properties have been used as in the previous validation, only without a key stone now. The application appears to use the same conversion for the soil spring stiffness as Slenders did, hence a fictitious Oedometer stiffness of 76 MPa is introduced again. Once more only very small deviations of the resulting bending moments were retrieved. The general distribution of those bending moments is similar for both models as Figure 19 shows. Deviations of the absolute maximum values were restricted to just 2,1 %. A likely explanation for this deviation might be blamed on the fact that the LDesign model iterates the actual normal force in the longitudinal Janßen joints, though the model from this study uses an estimated value. Both validations in this section demonstrate the model corresponds well to the results of existing studies with variations of the maximum bending moments up to 2,5 % only. It is therefore assumed that the model’s results are correct.


25

100 Moment M [kNm] 50 0 0 3 3

A ngl e θ

0

-50

[ ° ] 3 0

-76,5 -78,2 -100 0 0 3

6 0

0 7 2

LDesign Model for this study

0 9

86,8 84,9

0 2 1

2 4 0

-78,2 -76,5 0 1 5

2 1 0

180

19 | Moment distribution in validation model LDesign and model from this study

3.3

Model for the case study

This report investigates the feasibility of a reduced lining thickness for shield driven tunnels by use of a case study for the future motorway A13/16 in the north of Rotterdam. In this section the dimensions and proportions of the concrete tunnel lining, the thrust jack configuration and number of segments per ring for that particular shield driven tunnel are determined.

3.3.1

Thrust jack configuration

The clarification of shield driven tunnels at the beginning of this report already revealed the differences between two common European thrust jack configurations. The German configuration (with thrust jacks over the longitudinal joints and at the middle of the tunnel segment) appeared to have some specific properties that lead to damage in the concrete s egments quite easily: Large tensile forces due to the introduction of thrust jack forces Sensitive to cracking by uneven placement of the segments Positioning of ring interaction locations in the ring joint results in large peak bending moments In the French configuration (thrust jacks at ¼ and ¾ of the segmental length) these problems have been averted. Hence in the tunnel of this study the French thrust jack configuration is applied.   

3.3.2

Number of tunnel segments per ring

The Green Heart Tunnel was the first shield driven tunnel in the Netherlands with nine tunnel segments per ring rather than the usual number of seven segments. As a result the tunnel segments have a considerably lower weight and therefore are easier to transport and place. The maximum bending moment also reduces in case more segments per ring are used. The additional longitudinal joints reduce the total lining’s stiffness relative to the soil. Implying that a larger share of the bending moments is supported by the soil.

26


This philosophy has been applied in the design of tunnels for the British Channel Tunnel Railway Link (CTRL) project connecting central London and the Channel Tunnel by a new high speed railway line. Some tunnels in this tender have been constructed from steel fibre reinforced concrete. It is known that this material is less capable of bearing high bending moments compared to concrete with reinforcement bars. Accordingly nine segments per ring were used in the lining with an internal radius of just 7,15 m [2]. The proclaimed effect on the reduction of the maximum bending moment if more segments are used has been studied on the lining of this study’s shield driven tunnel. As a starting-point the number of nine segments from the Green Heart Tunnel has been used. Compared to a ring with eleven segments the maximum bending moment reduced by 4,6 % only. One should ask oneself whether this small reduction of the bending moment in the embedded stage is sufficient to compensate for the additional assembly actions and construction time by the increased number of segments to be placed in each ring. In practise the number of segments is only defined by an optimisation for logistics. The capacity of the production process, the capacity of lorries transporting segments to the construction site and handling space in the tunnel boring machine are taken into account. Experience demonstrated that seven to nine segments per ring are optimal for logistics purposes. Therefore nine segments per ring will be used in the tunnel of the case study. At the end of the report this decision will be reflected.

3.3.3

Dimensions of the tunnel segments

Proportions and dimensions of the concrete tunnel segments have been adopted from the final design of the Green Heart Tunnel provisionally. The segmental width is set to 2 m. The contact height in the longitudinal joints is 0,40 m in the Green Heart Tunnel, its segmental thickness is 0,60 m. This ratio of 2:3 is applied in this study as well. No absolute value is given for both, because various lining thicknesses will be analysed in the following section and chapters. A parameter analysis shows that a reduction of the ratio between the joint contact height and lining thickness to 1:2 results in a bending moment cutback of 4,7 %. Nevertheless rotations in these longitudinal joints (giving an indication of the lining’s deformations) increase by a staggering 55 %. Obviously the original ratio is adopted for this study’s tunnel. The contact height of the bearing pads in the ring joints is set equal to the one of the longitudinal joints, hence a ratio 2:3 with the segmental thickness. The width of the pads is fixed at 1,50 m in the meantime. The bearing pads do not affect the lining’s ring behaviour, yet do play a key role in the introduction of thrust jack forces. From the concrete material properties only the Young’s modulus and Poisson ratio are required for the model. Lateral contraction, or the Poisson ratio, equals 0,2 for all considered concrete strength classes C35/45, C100/115 and C180/210. The Young’s moduli vary and hold for each material: EC35/45 = 33.500 MPa (NEN 6720) EC100/115 = 45.000 MPa [6] EC180/210 = 65.000 MPa (French interim recommendations [1]) These values hold for short-term uncracked concrete and therefore appear relatively conservative for calculations on the ultimate limit state for the tunnels serviceability phase (more long-term than short-term). Reduction of these values might be considered if calculations on the lining’s ring behaviour appear to result in the governing situation. Therefore at the end of this report the value of the Young’s modulus will be reflected once more.

3.3.4

Soil properties

Properties of sand packed at the location of the future tunnel for motorway A13/16 were present at the engineering office of Gemeentewerken Rotterdam already and hold:


27

Eoed = ρsd = ρsw =

50 MPa 18 kN/m³ 20 kN/m³

Figure 20 shows the complete framework model for this study’s shield driven tunnel. Both rings were assigned a different diameter in order to clarify the picture. Note that the diameters in the actual model are equal.

R

L 2

R

L 1

R L 2

L 1

R

R

L 1

2 L

R

R

L 2

1 L

R

R

2 L

1 L

Lx R

R

Longitudinal joint in ring x Ring interaction

R

L 1

2 L

R

R

L 2

1 L

R

R

L 1

L 2

R

L 2

R

L 1

R

20 | Complete framework model for the shield dr iven tunnel of this study

3.4

Relation with the lining thickness

Basically this chapter investigates whether very and ultra high strength concrete are able to cope with the bending moments by the tunnel’s ring behaviour in a smaller lining thickness. Therefore knowledge of the development of this bending moment at changing lining thicknesses is required. A simple rule of thumb tells that the lining thickness of a shield driven tunnel in ordinary strength concrete equals 1/20 of the tunnel diameter. For the tunnel of motorway A13/16 this implies 750 mm. Provided that the ratio of all Dutch shield driven tunnels finished so far is used, a ratio of 1/22 or lining thickness of 675 mm is sufficient. Calculations in this study focus on a wide range of lining thicknesses along the standard value. Results from lining thicknesses between 100 and 1.000 mm with steps of 100 mm are examined for all three applied concrete materials.

28


In Chapter 2 the vertical alignment of the tunnel for motorway A13/16 was assumed to be nearly horizontal at a depth of -24 m + NAP. This depth projection is therefore applied in all calculations. The underground crossing of high speed railway line HSL-Zuid forced the tunnel down to this level. Ground level at this location is at -4,74 m + NAP, hence a soil overburden of 19,3 m is present. For a tunnel with the standard lining thickness of 675 mm, this particular overburden corresponds to 1,2 times the external tunnel diameter, in other words: a depth projection of 1,2D. A parameter analysis on the bending moment capacity of concrete tunnel linings showed that the normal ring force is of great influence on this capacity, especially on the one for high strength steel fibre reinforced concrete. The ratio between the bending moment capacity and the actual present bending moment defines the margin or safety of the lining to cope with the moment. Therefore a comparison of the safety levels for all three concrete strength classes might be influenced by the normal ring force, which directly results from the tunnel’s depth projection. The normal ring force is directly proportional to the total soil cover at the tunnel centre. In the conversion of soil properties to soil loads the water pressure and effective grain pressure are average at that specific depth. Depths of two and three times the standard cover at the centre are also included in the calculations. The most shallow depth projection where each individual ring with an arbitrary lining thickness is able to compensate the buoyancy force by use of its own weight and the weight of the soil above, is determined at an overburden on top of the tunnel of 11,1 m (also see Appendix A.3). The depth projections in all calculations are defined as the soil cover (or overburden) on top of the tunnel and have been stated in Table 3. Table 3 | Notation and values depth projection tunnel Soil cover top tunnel1)

Notation

19.2 m

11,1 m

0,7D

27,4 m

19,3 m

1,2D

54,8 m

46,7 m

2,9D

82,2 m

74,2 m

4,6D

Soil cover centre tunnel

1)

Assuming a lining thickness of 675 mm

3.4.1

Maximum bending moment

In the sections on the implementation of the framework model for embedded ring behaviour it has been mentioned already that stiffness attracts bending moments in hyperstatic structures. Therefore the ratio between the soil stiffness and tunnel lining stiffness determines the magnitude of bending moments in the concrete tunnel lining. The thicker the lining, the stiffer it is, the higher the bending moments it has to cope with. This relation however is not linear. So to say the total stock of bending moments from the soil loads is restricted, implying that the increment of the resulting bending moment starts to descend at very large lining thicknesses. It approaches some kind of asymptote. In Figure 21a the non-linear phenomena is visible for the tunnel with an internal diameter of 14,9 m and a soil coverage of 11,1 m (0,7D_. From the Young’s moduli for the considered concrete strength classes in section 3.3.3 it follows that a lining of ultra high strength concrete is 94 % stiffer at similar thicknesses compared to ordinary concrete (65.000 MPa versus 33.500 MPa for uncracked moduli). At very thin lining thicknesses no reduction by the mentioned asymptote is observed yet. At vary low thicknesses (approaching to 0 mm) resulting in an additional bending moment for C180/210 compared to ordinary concrete C35/45 of 94 % as well. However when the lining thickness increases it becomes clear that stiffer concrete will reach the asymptote at an earlier level. The relative surplus for concrete materials with a higher Young’s modulus descends if the lining thickness increases. Figure 21b visualises this trend by means of a diagram with this relative surplus for C180/210 and C100/115 compared to C35/45.


29

] m N k [ t 1000 n e m o m 800 g n i d n 600 e b m u 400 m i x a M 200

0 0

200

] % [ 5 4 / 580 3 C C180/210 o t e v60 i t a l C100/115 e r t n40 e C35/45 m o m s20 u l p r u Thickness h [mm] S 0 400 600 800 1000 0

C180/210

C100/115

Thickness h [mm] 200

a

400

600

800

1000

b

21 | Maximum bending moment by ring behaviour for depth of 0,7D. a) absolute values; b) surplus relative to maximum moments for ordinary concrete C35/45

The normal ring force is not influenced by the tunnel’s lining thickness. Or actually: it shouldn’t be. In the model however the depth projection is defined by a certain soil cover on top of the tunnel and the internal diameter is fixed. So at an increasing lining thickness, the external diameter grows, forcing the heart of the tunnel (where the loads are defined and hence the normal ring force is related to) down by the additional thickness. The resulting slight increase of the normal ring force is presented in Figure 22. -2200 -2300

0

200

400

600

800

1000 Thickness h [mm]

C35/45 C100/115

-2400 ] N-2500 k [ e c r o-2600 f g n i r l -2700 a m r o N

C180/210

C180/210 C35/45

22 | Normal ring force by ring behaviour for depth of 0,7D

Similar to the normal ring force (see Figure 23a), the maximum bending moment in the lining is more or less directly proportional to the soil overburden on top of the tunnel’s heart. The size of the soil loads (at average defined at that point) is responsible once more. For a standard lining thickness of 675 mm the maximum values have been given in Figure 23b, apparently the ratios between all three concrete strength classes don’t change. Therefore no additional deviations of the maximum bending moments occur for the three concrete strength classes an increasing depth. Except for the increasing absolute values of course, as Figure 24 demonstrates. An increasing maximum bending moment either by an increasing depth or increasing lining thickness does not necessarily imply a loss of safety provided that the ultimate resisting moment (or bending moment capacity) increases with a similar or higher rate. The next section will therefore deal with the development of the bending moment capacity for the considered concrete strength classes and depth projections.

30


0

-2000

-4000

-6000

0

-8000 -10000

500

] m19,2 [ e n 27,4 i l e r t n e c m o r 54,8 f h t p e d l e n n 82,2 u T

(0,7D) (1,2D)

(2,9D)

(4,6D)

1500

2000

2500

Maximum bending moment [kNm]

Normal ring force [kN] ] m19,2 [ e n 27,4 i l e r t n e c m o r 54,8 f h t p e d l e n n 82,2 u T

1000

(0,7D) (1,2D)

C180/210 C100/115

(2,9D) C35/45 (4,6D)

a

b

23 | Effect of tunnel depth projection with a lining thickness of 675 mm. a) normal ring force; b) tangential bending moment

] m N k [ t n4000 e m o m g3000 n i d n e b 2000 m u m i x a1000 M

C180/210 C100/115 C35/45

] m N k [ t n4000 e m o m g3000 n i d n e b 2000 m u m i x a1000 M

Thickness h [mm]

0 0

200

400

600

800

1000

a ] m N k [ t n4000 e m o m g3000 n i d n e b 2000 m u m i x a1000 M

C180/210 C100/115 C35/45

Thickness h [mm]

0 0

200

400

600

800

1000

b

C180/210

C100/115 C35/45

Thickness h [mm]

0 0

200

400

600

800

1000

c

24 | Maximum bending moment by ring behaviour for soil overburden of: a) 1,2D; b) 2,9D; c) 4,6D

3.4.2

Bending moment capacity of the lining

Calculation procedure of the ultimate resisting moment

For concrete elements with a rectangular cross-section, ordinary concrete and one layer of steel reinforcement bars, the ultimate resisting moment Mu is determined as follows according to Dutch building codes: ∅ ⎞ h ⎛ M u = − Nc 0 ,389x u + Ns ⎜ h − c − ⎟ + Nrep (11) 2 ⎠ 2 ⎝ Where: Nc = Compressive normal force in concrete


31

xu Ns h c ∅ Nrep

= = = = = =

Height of the compressive zone Tensile force in steel reinforcement bars Height of rectangular cross-section Concrete cover for reinforcement bars Diameter of the reinforcement bars Representative acting normal force on the cross-section

This section will demonstrate that the acting normal force has a positive effect (up to a certain depth.) on the lining’s bending moment capacity Therefore the representative normal force Nrep has been introduced, that does not include any safety factors. Minimum values resulting from calculations on the embedded ring behaviour have been applied. 0,389 xu xu h

Nc Mu

Nrep Ns c+Ø/2

25 | Calculation definition ultimate resisting moment

The equation is based on the mechanism shown in Figure 25. Where the tensile force in the steel rebars is defined as: Ns = A s fs Where: As fs

= =

Steel area of rebars Representative strength of reinforcement steel

From the equilibrium of horizontal forces it follows: ∑ Fh = 0 ⇔ Nc = Ns + Nrep The height of the compressive zone is determined by: Nc x u = 0 ,75f'c b Where: f'c b

= =

Design value of concrete compressive strength Width of the rectangular cross-section

Scalars 0,389 in the basic equation for Mu and 0,75 in the determination of xu hide the fact that this calculation procedure assumes that the concrete is not able to cope with any tensile forces and that its ultimate compressive strain is twice the compressive yield strain. Subsequently the procedure assumes the strain in the concrete cross-section at the position of the rebars is of such tensile extent that the maximum tensile strength from the stress-strain diagram (see Figure 26) is reached. For calculations on the ultimate resisting moment of tunnel segments in this study a more comprehensive calculation procedure is required. Several properties of the described simplified procedure disagree with the following requirements: Tensile stresses from the concrete should be introduced for the steel fibre reinforced concrete materials very and ultra high strength concrete. The ratio between the ultimate compressive strain and compressive yield strain of concrete should be free, since this ratio does not equal 1:2 for very and ultra high strength concrete. 



32


The use of multiple reinforcement layers should be possible: bending moments in the same tunnel segment may be both positive and negative over time; hence reinforcement is required at the top and bottom of the cross-section. As a result one layer is located in the concrete compressive zone, so the rebars are under pressure as well; resulting in the next requirement: Stresses in the reinforcement layers should depend on the actual (concrete) strain at its location, not just on the maximum tensile strength for reinforcement steel.





] a P450 M [ s s300 e r t S 150

Maximum tensile stress

Strain [‰] -30

-20

-10

0

10

20

30

-150 -300 -450

26 | Stress-strain diagram of reinforcement steel

Additionally to the enumeration the tensile behaviour of ultra high strength steel fibre reinforced concretes turns out to be rather complicated. Consequently another procedure is required to include all four requisites in the calculation of the ultimate resisting moment. The calculation will be altered to make the stress-strain relations of concrete and steel the base of the entire determination. Therefore the same principles from the simplified determination of the building code are applied again, meaning the equilibrium of horizontal forces and the equilibrium of bending moments: ns

∑ Fh = 0 ⇔ Nc + j∑=1 Fsj = Nrep ns

M sj − M N ∑ M = 0 ⇔ Mu = Mc + ∑ j=1 Where: Fsj Mc,sj or N ns

= Tensile steel force for reinforcement layer j = Contribution to the ultimate resisting moment by the concrete (c), steel reinforcement layer j (sj) or the normal ring force (N) = Number of steel reinforcement layers

Obtaining the ultimate resisting moment Mu from these equilibrium equations demands a vast number of identical calculations, which makes working out by hand a very uninteresting activity. For that reason a Visual Basic .NET2 application has been implemented to do the job. In Appendix B the calculation process of this application has been reproduced. Stress-strain diagram for very and ultra high strength steel fibre reinforced concrete

The French interim recommendations on structures in ultra high strength concrete [1] also contained a stress-strain relation or diagram for this special material. Two different effects by the steel fibres were presented: strain hardening (Figure 27a) and strain softening (27b). Their use will be discussed at a later stage.


33

) σ ( s f'c s e r t s

) σ ( s f'c s e r t s

E ε1%

εu

ε0,3

E strain ( ε )

εel ε‘c

ε1%

εu

ε0,3

strain ( ε )

εel

ε' u

ε‘c

f1% fel fc

ε 'u

f1% fc fel a

b

27 | Stress-strain relations of ultra high strength concrete: a) strain hardening; b) and strain softening

The parameters from the diagrams in Figure 27 are defined by: f εel = el E'c Where fel represents the tensile stress where cracking of concrete occurs for the first time, namely the end of the linear elastic behaviour of concrete in the tensional area. The Young’s modulus E is retrieved from product data by the material’s manufacturer or from the French recommendations. w σ( w 0 , 3 ) ε0 , 3 = 0 , 3 + εel and fc = lc K The stress fc corresponds to the stress at crack width w0,3 = 0,3 mm in the initial characteristic stresscrack width relation for the material (the stress-strain relation is originally derived from the stresscrack width relation). Parameter l c represents the characteristic length of the cross-section, for rectangular and T-shaped sections it holds: l c =

2 3

h , where h (in mm) is the height of the cross-

sectional area. Scalar K is an orientation coefficient for the steel fibres. Values between 1 and 1,75 may be used, depending on the use and relative thickness of the segment. σ( w1% ) w ε1% = 1% + ε el and f1% = K lc The stress σ1% represents the stress at crack width w1% = 0,01 H of the characteristic stress-crack width relation. Dimension H is the height of the bending test specimen, corresponding to 100 mm in the interim recommendations. εu =

lf

4l c

(12)

Where l f corresponds to the length of the steel fibres. f 'c E Where f'c is the compressive design strength. ε' u = 3 ‰ ε'c =

(13)

Calculations on the ultimate limit state of the structure demand an additional safety factor on strain εel and the tensile stresses fc and f1% according to the French recommendations. This partial safety factor for fibres holds: γ f = 1 ,3 The value of stress fc defines whether the concrete behaves with strain hardening or strain softening: only if fc > fel strain hardening will occur. The phenomenon of strain hardening is induced by the steel fibres. The effect on the total ultimate resisting moment however is only extremely small and may therefore be neglected in this calculation.

34


No recommendations on the material of very high strength concrete exist so far. The concrete strength class C100/115 in this study includes steel fibres as well; as a result it’s presumed that the French recommendations apply on this material as well [6]. The product BSI-Céracem by Eiffage is applied for ultra high strength steel fibre reinforced concrete C180/210 in this study. Its stress strain relation is determined by use of several parameters in the interim recommendations: f'c = 126 MPa fc = fel= 9,1 MPa [1] f1% = 7 MPa (value has been scaled from values used in [9]) E = 65.000 MPa = 20 mm (amount of fibres 200 kg/m³ or 2,5 vol-%) [19] lf For very high strength steel fibre reinforced concrete C100/115 a mixture described in [6] is used, which principal values read: f'c = 69 MPa fc = fel= 5 MPa f1% = 4 MPa (scaled from C180/210) E = 45.000 MPa ε'c = 1,75 ‰ (this value from the building code is larger than the one resulting from equation (13) and therefore assumed critical) lf = 40 mm (amount of fibres 125 kg/m³ or 1,6 vol-%) ] a P M -80 [ s s e r t S -60

] a P M-120 [ s s e r -100 t S

-80 -60

-40

-40 -20 -20 Strain [‰]

Strain [‰] 30

20

10

0

-10

15

10

b

5

0

-5

a

28 | Stress-strain diagrams steel fibre reinforced concretes a) C100/115; b) C180.210

Development ultimate restisting moment

The simplified calculation method for the ultimate resisting moment gives an idea of the transformation of this capacity when the lining thickness or normal ring force (related to the tunnel’s depth projection) changes. The equations again: ∅ ⎞ h ⎛ M u = − Nc 0 ,389x u + Ns ⎜ h − c − ⎟ + Nrep 2 ⎠ 2 ⎝ Ns = A s fs

∑ Fh = 0 ⇔ Nc = Ns + Nrep x u =

Nc 0 ,75f ' b b

Enlargement of the lining thickness results in an increase of the lever arms for reinforcing steel (h – c – ∅ / 2) and the contribution by the normal ring force (h / 2). This obviously results in a higher


35

bending moment capacity at thicker linings, even though the stress development in the concrete cross-section didn’t change yet. If the correct values for the steel stresses (reflecting on the steel force Ns) are adopted – which applies to the calculation in the application – the steel rebars might be located within the height xu of the concrete compressive zone. The reinforcement is under pressure now as well, revolving its initial positive contribution to the total bending moment capacity into a negative one. Then again the concrete compression force Nc is affected by the force in the reinforcement bars, which then reshapes the height of the compressive zone. The equation for compression force Nc shows that this force is lower in case of rebars under compression in stead of tension. This variation however is only very small compared to the acting normal ring force Nrep. As a result this phenomenon does not prevent the steel force from turning to tension if the lining thickness increases from very slender to thick. This is an additional positive effect on the bending moment capacity of the cross-section. Provided that no reinforcement bars are used, as is most desirable from the point of cost saving for very and ultra high strength steel fibre reinforced concrete, the contribution by the reinforcing steel to the total bending moment capacity is obviously omitted. Nevertheless these concrete materials are able to resist some tensile stresses in their own cross-section. But as a result of the stress-strain relation (see Figure 28) the value of these tensile stresses decreases if the distance from the concrete compression zone increases. Hence if the lever arm for the contribution to the ultimate resisting moment increases, the tensile stress actually decreases. On top of this the height of the tensile zone decreases if the lining thickness increases (because of equation (12) on page 34), causing a smaller positive contribution in stead of a larger one as in case of deep beams or linings. This negative property undermines the contribution of steel fibres to the bending moment capacity, which is thereby relatively inadequate. An advantage of very and ultra high strength concrete should now exist of the capability to cope with compressive normal forces in a narrower concrete compression zone compared to ordinary concrete. By this means the negative contribution to the ultimate resisting moment is restricted. At equal acting normal forces the height of the concrete compression zone remains almost unchanged (some distortion occurs by the length of the tension zone). The contribution to the bending moment capacity is thereby independent of the lining thickness. The most significant advantage of a small compression zone is subsequently reached if the contribution by rebars in ordinary concrete is relatively small because of a small lever arm, hence if the lining thickness is low. In that range the length of the tension zone is larger and therefore more effective as well. As soon as the lengths of both the concrete compression and tension zone are very small compared to the lining thickness, the lever arm of the normal ring force only causes the bending moment capacity to grow. Implying that at relatively low normal ring forces and high lining thicknesses the resulting ultimate resisting moment for ultra high strength concrete might be smaller than the one for ordinary concrete with reinforcement bars. For these rebars a steel quality of FeB435/500 and reinforcement percentage of 0,22 % (minimum requirement for concrete C35/45 according NEN 6720) on top and bottom of the cross-section are used, with a concrete cover of 35 mm and bars of∅12 mm. In Figure 29 the described phenomena are indeed observed for the most shallow depth projection. Remarkably the ultimate resisting moment for C100/115 surpasses the one of C180/210 from approximately 300 mm on. The higher compressive strength of the latter copes with the acting normal force in a smaller concrete compressive zone. The absolute height of the compression area defines the absolute height of the tensile region since they are interconnected in the stress-strain diagram. Consequently the contribution by the tensile force is inadequate at smaller lining thicknesses for C180/210, causing the bending moment capacity to approach the contribution by the normal ring force only at an earlier stage (dashed line in Figure 29).

36


] m N k [ t n e2000 m o m g1500 n i t s i s e r e1000 t a m i t l U 500

C35/45

C180/210

C100/115 C180/210

Contribution normal force Thickness h [mm]

0 0

200

400

600

800

1000

29 | Ultimate resisting moment at a depth of 0,7D

If the acting normal force Nrep (the minimum normal ring force) in the cross-section increases, the height of the concrete compression zone is primarily manipulated. By its enlargement the variations of the negative contributions for the considered concrete strength classes will rise. This is in favour of very and ultra high strength concrete. As soon as this favour exceeds the contribution ordinary concrete gets from its reinforcement bars, the bending moment capacity will be higher for ultra high strength concrete even at large lining thicknesses. A larger concrete compressive zone of course implies a larger height of the tensile zone in fibre reinforced concrete as well. Subsequently higher tensile stresses occur at relatively large distances from the compressive zone. These higher tensile stresses have a positive contribution to the comparison of higher strength concrete and ordinary concrete as well. The indisputable positive effect of the normal ring force or tunnel depth projection on the bending moment capacity of ultra high strength concrete is shown in Figure 30. The capacities of all considered concrete strength classes and depth projections are presented for a standard lining thickness of 675 mm. Concrete C180/210 commences with the smallest capacity of all at a depth of 11,1 m (0,7D) (normal ring force approximately 2.550 kN) because its tensile zone is inadequate already; at a depth projection of 74,2 m (4,6D) (normal ring force 11.100 kN) however it grew to own the largest capacity, exceeding the one of reinforced concrete C35/45 by no less than 120 %. Figure 31 displays the development of the bending moment capacity at increasing lining thicknesses for the three remaining depth projections. The fact that the increment of the compressive zone at high normal forces (depths) also causes the tensile zone to grow, is visible by the fact that the capacity of C100/115 exceeds the one of C180/210 at ever higher lining thicknesses. 0

500

1000

1500

2000

2500

3000

3500

Ultimate resisting moment [kNm] ] m19,2 [ e n 27,4 i l e r t n e c m o 54,8 r f h t p e d l e n n 82,2 u T

(0,7D) (1,2D)

(2,9D)

C100/115

C35/45 C180/210

(4,6D)

30 | Development ultimate resisting moment at various depth projections for lining thickness of 675 mm


37

] m N k [ t 5000 n e m o m4000 g n i t s3000 i s e r e t a2000 m i t l U1000

C35/45 C100/115 C180/210

] m N k [ t 5000 n e m o m4000 g n i t s3000 i s e r e t a2000 m i t l U1000

Thickness h [mm]

0 0

200

400

600

800

1000

C100/115

C180/210

Thickness h [mm]

0 0

200

a ] m N k [ t 5000 n e m o m4000 g n i t s3000 i s e r e t a2000 m i t l U1000

C35/45

400

600

800

1000

b

C180/210

C100/115 C35/45

Thickness h [mm]

0 0

200

400

600

800

1000

c

31 | Ultimate resising moment at depth projections of: a) 1,2D; b) 2,9D; c) 4,6D

The preceding discussion obviously demonstrates the main advantage of ultra high strength concrete. Not the material’s tensile strength, but its capability to cope with enormous normal forces in a very small cross-section is positive. Therefore all present bridges constructed from ultra high strength concrete have been prestressed to a very high extent. So actually it’s not the concrete itself, but the large prestress force that generates a high bending moment capacity capacity for thin beams.

3.4.3

Retrieving the required lining thickness

For all considered concrete strength classes the occurring bending moments and bending moment capacities are know for a wide range of lining thicknesses and depth projections. Division of the bending moment capacity by the occurring occurring moment returns a safety ratio which shows the amount of spare capacity for the tunnel segment’s cross-section. A required safety ratio is not included in Dutch building codes. Different from for instance bridges and buildings, no additional recommendations exist for the design of shield driven tunnels. In common structures like bridges loads are multiplied by a certain partial load factor. For shield driven tunnels however higher loads result in a higher normal ring force, which might positively influence the lining’s bending moment capacity. Hence a more favourable instead of less favourable situation is generated. As a result sometimes old codes were used, codes that included only one combined safety factor. This factor is applied on the reaction forces and bending moments, not on the loads. Application of the partial loading factor on the reaction forces and partial material factors on the cross-sectional capacities has a more-or-less similar result. For this study the latter principal is used in order to include all partial safety factors in the stress-strain diagram for ultra high strength concrete from the French interim recommendations. Partial safety factors for shield driven tunnels are slightly higher compared to ordinary structures to include the risks and doubts about calculation methods. In Appendix C an enumeration has been given for these factors according to Dutch and German building codes. Subsequently a partial load factor γ = = 1,5 is applied.

38


Determination of the resulting safety factors indicates a descending safety for all concrete strength classes if the lining thickness ascends. Hence the increment of the bending moment capacity is insufficient to keep up with the acting bending moment. Nevertheless the actual safety factor exceeds the required level for most considered thicknesses. x a m

x a m

M / 6 u M r o 5 t c a f y 4 t e f a S 3


C100/115 C180/210 C35/45

2

C100/115

C180/210

2

γ = 1,5

1

C35/45

γ = 1,5

1 Thickness h [mm]

0 0

200

400

600

800

Thickness h [mm]

0

1000

0

200

400

a

600

800

1000

b

x a m

x a m



C100/115

2

C35/45

C180/210 Thickness h [mm]

0 0

200

400

600 c

C100/115

2

γ = 1,5

1

C180/210

800

1000

γ = 1,5

1

C35/45

Thickness h [mm]

0 0

200

400

600

800

1000

d

32 | Development of safety factor Mu /Mmax at depth projections of: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D

At shallow depths it’s no surprise the safety factor of ultra high strength concrete C180/210 is lower than the one of ordinary concrete C35/45, particularly in thick linings. The reason of course is the small value of its bending moment capacity due to the relatively low acting normal force. Figure 32a indeed demonstrates this phenomenon. The other graphs in Figure 32 also show that the performance of high strength concretes improves compared to ordinary concrete at higher depths. At a soil cover of 74,2 m (4,6D) ordinary concrete C35/45 is unable to reach the required safety level. Up to a certain level this problem is relatively easy tackled by applying more steel bar reinforcement. As mentioned the trend of the safety development for the bending moment in embedded ring behaviour is downward at an increasing lining thickness. Hence the lining thickness where the required and occurring safety levels meet is an upper boundary condition for the lining thickness. In other words: a required lining thickness that apparently follows from other calculations may not exceed the requirement from this mechanism. For the considered amount of reinforcement in ordinary concrete, a lower boundary condition is present at high depths as well. The lining is unable to reach the required safety level. Values of the upper boundary conditions have been shown in Table 4 for the considered concrete strength classes and soil overburdens. Of course these upper boundary conditions do not imply that a solid tunnel (or a tunnel with an extremely large lining thickness) leads to failure by the embedded ring behaviour. As mentioned before the resulting bending moment in the lining will reach a maximum value at some point.


39

However at increasing lining thicknesses the bending moment capacity will keep on growing. Consequently at some point the bending moment safety level will ascend again and finally exceed the required safety level. At extremely high lining thicknesses a new lower boundary condition is created now. Table 4 | Upper boundary conditions on the allowable lining thicknesses in case of embedded ring behaviour Overburden

C35/45

C100/115

C180/210

11,1 m (0,7D)

n/a1)

910 mm

580 mm

19,3 m (1,2D)

1)

46,7 m (2,9D) 74,2 m (4,6D)

n/a

2)

595 mm

2)

1480 mm

610 mm

620 mm

590 mm

1020 mm 3)

(400 mm)

4)

(1.320 mm)

1)

No boundary condition present within the considered range of lining thicknesses, the safety level is higher than required at any point. 2) Value has been determined by extrapolation. 3) No upper boundary condition present, although a lower boundary condition occurs at 400 mm. 4) Required safety level is never reached by the considered cross-section. Extrapolation shows a thickness of at least 1.320 mm is required.

3.5

Conclusions

Resulting normal ring forces and bending moments in the tunnel ring enlarge linear proportional to the depth of the tunnel’s centre line. The bending moment significantly increases if the lining thickness ascends. Alterations of the tunnel ring’s design details, such as extension of the number of tunnel segments per ring, result in a very small reduction of the bending moment only. Steel fibres in very and ultra high strength concrete hardly contribute to the total bending moment capacity (or ultimate resisting moment). Thereby the bending moment capacity of very and ultra high strength steel fibre reinforced concrete strongly depends on the acting normal ring force. The thicker a lining of very or ultra high strength concrete, the smaller the contribution of steel fibres to the bending moment capacity becomes. As a result thin linings with a high acting normal ring force (hence at large depths) provide the best use of these materials’ capacities concerning normal forces and bending moment capacity when compared to reinforced ordinary concrete. In very shallow tunnels (overburden of 11,1 m (0,7D)) with relatively thick linings (over 400 mm) the bending moment capacity of reinforced ordinary concrete C35/45 grows bigger than the one of ultra high strength concrete C180/210. Lacking of steel reinforcement bars in the latter is responsible for this. The lining’s bending moment safety level – the safety margin to cope with the resulting bending moments – for ring behaviour of an embedded tunnel decreases if the lining thickness ascends. Consequently the thickness where the required safety level is hit is the maximum value that can be applied; hence an upper boundary condition of the required lining thickness. Reinforced ordinary concrete C35/45 has no upper boundary condition in the considered range of lining thicknesses. In very deep tunnels the high normal ring force reduces the bending moment capacity of reinforced ordinary concrete that is now unable to reach the required safety level. Therefore a lower boundary condition for embedded ring behaviour appears for this material at great depths. All boundary conditions on the required lining thickness by embedded ring behaviour have been displayed in Table 4 and Figure 33.

40


0

200

400

600

800



0,7D 1,2D C180/210

C100/115

2,9D

4,6D

C35/45

33 | Upper boundary conditions on the allowable lining thicknesses in case of embedded ring behaviour (lower boundary for C35/45)


41

42


Chapter 4

Grouting phase 4.1

Introduction

Tunnel rings are assembled within the shield of the tunnel boring machine (TBM). Therefore its diameter is larger than the tunnel’s external one. Behind the machine a tail void between the soil and tunnel is created during the excavation process. In order to prevent loss of soil support and possible settlements at ground level, the semi-liquid material grout is injected in the void. Grout – that consists of sand, water, potentially cement and several additives like bentonite or plasticizers – has some special properties that make it suitable for its specific task. The material is not actually liquid, but has a very low plastic yielding shear strength (viscosity) in the beginning. In other words: the material has a low resistance against flowing. If it flows (visco-plastic behaviour) it can easily be transported and injected into the void. Nevertheless if the yielding shear strength has not been exceeded yet, the material is able to resist some shear stresses, which is not the case for a liquid. If grout is subjected to a compressive pressure (ground and water pressure in this case) for some time, water will be squeezed out of the material and therefore the viscosity increases, thus the material stiffens. Over time it will turn into a stiff shell making interaction of the tunnel rings and the surrounding soil possible. As long as the grout didn’t harden yet the tunnel would like to float up in the semi-liquid material. The support of the tunnel then completely changes and is no longer comparable to the principal applied in Chapter 3 for a tunnel embedded in soil. Interaction between soil and lining is not acting in fresh grout. In Blom’s dissertation [5] came forth that the bending moment in the grouting phase, or uplift loading case, might play a key role in the determination of the required lining thickness. The resulting bending moment of this loading case appeared to barely depend on the lining’s thickness. However, because the bending moment capacity remains unchanged compared to the calculations in the previous chapter, the safety factor for this loading case will obviously ascend. In Blom’s study the required safety level for the Botlek Rail Tunnel was reached at a lining thickness of approximately 1/20 D. So if the lining thickness 1/20 D of a tunnel with ordinary reinforced concrete is determined by the fact that its ultimate resisting moment equals the design value of the bending moment of the grouting phase, this might help to retrieve the required lining thicknesses for another concrete strength class. Implying the other way round it should also hold that the bending moment capacity of another type of concrete with a certain lining thickness h equals the bending moment capacity of ordinary reinforced concrete with a lining thickness of 1/20 D. Now the required thickness h can be resolved. The analysis on the Botlek Rail Tunnel in Blom’s study has been performed for an overburden of approximately 1,4D. The question should be raised whether this hypothesis applies to deeper tunnels as well. Subsequently the effects on the ultimate resisting moment for the tunnel lining of motorway

43

A13/16 have been inquired at an increasing normal ring force, hence at an increasing depth. Although the tunnel may no longer be in full contact with the soil, the normal ring force should still make equilibrium with the ground pressure and therefore develops identical to the normal ring force in the embedded situation. For an unreinforced cross-section with concrete unable to cope with any tensile forces, the equilibrium of horizontal forces holds according to the simplified calculation method for the ultimate resisting moment on page 31: ∑ Fh = 0 ⇔ Nc = Nrep At an increasing acting normal force on the cross-section the height of the compressive zone will grow directly proportional to that force: Nrep Nc = xu = 0 ,75f ' b b 0 ,75f ' b b Its negative contribution to the bending moment capacity therefore grows with a parabolic proportional rate: 0 ,389N rep2 Mc = − Nc 0 ,389x u = − 0 ,75f' b b The positive contribution by the normal force itself obviously ascends with a directly proportional rate only: h M N = N rep 2 Hence the total ultimate resisting moment is a combination of these parabolic and linear equations: 0 ,389Nrep 2 h + Nrep M u = Mc + M N = − 0 ,75f' b b 2 At relatively low normal forces the linear part of this equation is predominant and the capacity will rise if the normal force increases. However at some point a turning point is reached and the capacity will drop due to the parabolic part. For this simple case that point is located at 8.785 kN for concrete C35/45. This is clearly visible in Figure 34 as well, where line (a) represents unreinforced concrete. A reinforcement layer at the bottom of the cross-section will elevate the ultimate resisting moment (as described in last chapter). If however the normal force reaches such high values that the tensile stress in the reinforcement layer drops or even turns into compression, the capacity will consequently decrease. At line (b) in Figure 34 this provokes ordinary concrete with a reinforcement percentage of 0,22 % to descend from approximately 7.900 kN. 0

250

500

750

0

1000

1250

1500

2000

Ultimate resisting moment [kNm] 0D

2000

0,7D 1D

1,2D 4000

(a)

(b)

2D

6000 2,9D 8000 ] N k [ 10000 e c r o f g12000 n i r l a 14000 m r o N

4D

(a) Unreinforced C35/45 (b) Reinforced C35/45

3D

4,2D 5D 6D

] m / m [ p o t m o r f h t p e d l e n n u t d e t a m i t s E

34 | Development ultimate resisting moment over the depth for 675 mm thick reinforced concrete C35/45

44


Nonetheless there are no reasons to assume that the development of the grout phase bending moment contains just such turning point. So a lining of reinforced concrete with a thickness of 1/20 D might not be governing for the grout moment at every tunnel depth. Hence the behaviour of the tunnel in the grouting phase should be investigated in more detail. For that the application LDesign is used, which contains a function for calculations on the uplift loading case of shield driven tunnels. LDesign was originally implemented for research on Blom’s dissertation [5]. Section 4.2 shortly describes the background of the model. Section 4.3 focuses on the relation between the safety factor for the grout bending moment and the lining thickness. Section 4.4 displays several conclusions regarding the behaviour of shield driven tunnels in the grouting phase and the required lining thickness for that particular phase.

4.2

Modelling of the grout phase

A framework model has been presented in [5] which is able to approximate grout pressures during the building phase of a shield driven tunnel. By these pressures resulting bending moments in the tunnel lining are retrieved. This section briefly explicates the basics of the model.

4.2.1

Background of the uplift loading case model

The shield driven tunnel and the injected fresh grout are surrounded by water pressure at all time. If the grout makes no contact with the tunnel lining the water pressure only loads the tunnel and it’s the tunnel’s own dead weight that should compensate the upward floating component of the hydrostatic pressure. However at usual lining thicknesses this won’t be the case. Nevertheless in actual tunnel construction a stable situation is reached at some point, the tunnel does not float up completely. So even in fresh grout some contact between the grout and lining must exist. In the introduction to this chapter it was mentioned that grout has some capability to cope with shear stresses. Only if this capacity is exceeded grout will flow. At that moment the so-called shear yield strength is reached. If the shield driven tunnel moves upward due to the floating component the grout shear stress will grow till it reaches its maximum value. By addition of its downward component to the tunnel’s dead weight it follows: DW + ∫ τ grout ,tan gential ,vertical = πDhbρ concrete + Dbτ yield Where: DW = Dead weight tunnel ∫ τgrout ,tan gential ,vertical = Vertical component of tangential frictional shear stress between tunnel and grout D = h = b = ρconcrete = τyield =

External diameter tunnel Thickness tunnel lining Width tunnel segment Specific gravity concrete Shear yield strength grout (0,0015 MPa [5])


45

Fictitious top support

Soil

Injected grout Dead weight tunnel

TBM Plastic shear yield stress grout

Grout pressure as load

35 | Forces and definitions in the uplift loading case [5]

Grout surrounds the tunnel; therefore it’s this material that transmits the water pressure to the tunnel. In addition the mentioned contact between tunnel lining and grout influences the pressure in the grout as well. The product of both, the actual grout pressure, now also includes the floating component and should consequently equal the downward directed forces. In Figure 35 this principal has been visualised. The definition now holds: ∫ σgrout ,radial ,vertical = DW + ∫ τgrout ,tan gential ,vertical Where:

∫ σgrout ,radial ,vertical =  

Vertical component of the radial grout pressure, consisting of:

Water pressure Lining-grout contact pressure

Hence the upward component of the total grout pressure equals the downward components by the tunnel’s dead weight and the tangential loading of the tunnel by the grout’s shear stresses. As a result for the vertical component of the grout pressure it holds: ∫ σ grout ,radial ,vertical = πDhbρ concrete + Dbτ yield Provided that the grout pressure develops as a hydrostatic pressure along the lining, it may be written as:

∫ σgrout ,radial ,vertical =

π

D 2 bρeq

4 Where ρeq is an equivalent specific gravity assigned to the grout. Accordingly it holds: πDhbρ concrete + Dbτ yield πhρ concrete + τ yield ρ eq = =4 π 2 Dπ D b 4 Assuming ordinary concrete C35/45 ( ρconcrete = 24 · 10 -6 N/mm³), a lining thickness of 675 mm and grout with a shear yield stress of 0,0015 MPa the equivalent specific gravity holds for the tunnel of motorway A13/16: −6 π ⋅ 675 ⋅ 24 ⋅ 10 + 0 ,0015 −6 3 3 ρeq = 4 = 4 ,1 ⋅ 10 N / mm = 4 ,1 kN / m (14.900 + 2 ⋅ 675)π This implies that vertical equilibrium occurs if the specific gravity of the floating component in grout is lowered from the specific gravity of water ρw = 10 kN/m³ to the equivalent specific gravity ρeq = 4,1 kN/m³. Now the tunnel’s dead weight and the grout shear stresses are able to compensate the remaining upward component of ρeq. According to Blom’s dissertation this reduction of the upward pressure has been observed in measures on site as well. Besides, finite element models where a tunnel embedded in a uniform soil continuum is loaded by the hydrostatic water pressure return the

46


same result. This indicates that the grout-lining contact pressure behaves like some kind of uniform bedding. Having a closer look on the grout pressures resulting from comprehensive finite element analyses shows that the grout pressure does not develop perfectly hydrostatic. A pressure peak arises on top of the tunnel. This behaviour will be explained by analysing the development and creation of the grout pressure: If grout has just been injected in the tail void along a certain tunnel ring, this ring is not yet moving upward. The considered ring has only recently left the tunnel boring machine and the next ring, that is still within the machine, prevents any vertical movement. Though as soon as the ring does move, the grout shear stress builds up. At the top of the tunnel grout is compressed between the moving tunnel and fixed soil. The shear yield strength has not been reached yet; hence grout is not able to flow away from the top. As a result the grout pressure above keeps increasing due to interlocking. Only if the shear yield strength is exceeded somewhere, grout at that particular location is able to flow from the top to the sides and bottom of the tail void. Because of the tunnel’s upward movement a relatively low grout pressure was generated in the void at the bottom of the tunnel. By flowing grout this phenomenon is partly compensated. Provided that the grout-lining contact pressure behaves like a uniform bedding, a local pressure increase is simulated by a local enlargement of the bedding’s stiffness, an additional fictitious bedding. In his study Blom presumed that this additional stiffness diminishes from the tunnel’s top to its sides in accordance with a quadratic cosine distribution. His study also shows that the magnitude of the pressure increment is related to the soil overburden, the soil’s specific gravity and the tunnel’s external radius. Consequently the stiffness of the fictitious top support is defined as: k fictitious = ξ cos 2 (θ) ⋅ k uniform Where: ξ=

h s ρs −1 R ρw

With: hs = kuniform = R = ρs = ρw = θ =

Overburden soil Stiffness uniform bedding External radius tunnel Specific gravity of soil Specific gravity of water Angle along tunnel circumference (0° at the top)

At the standard depth with an overburden of 19,3 m (1,2D) the ξ-ratio for the tunnel from this study holds: 19 ,3 20 ξ= − 1 = 3 ,75 14 ,9 / 2 + 0 ,675 10 If the stiffness of the uniform bedding for the grout-lining contact pressure is set to a very low value of for instance 1 MPa, the pressure is able to develop smoothly without significant bending moment reduction due to ring behaviour as described in the previous chapter. Now a value of 3,75 · cos²(0) + 1 = 4,75 MPa holds for the total bedding stiffness of the top support , which is the sum of the fictitious and uniform bedding (see Figure 36).


47

] a P M5 [ t r o p4 p u s s 3 u o i t i t 2 c i F 1

Angle [°]

0 0

45

90

135

180

225

270

315

360

36 | Fictitious top support

This calculation procedure (implemented in LDesign) has been applied on the Green Heart Tunnel in Blom’s study. The resulting total grout pressure, consisting of the water pressure and grout-lining contact pressure, was practically similar to the pressure resulting from finite element models. Bending moments in the tunnel lining result from this grout pressure in LDesign.

4.2.2

Complete and incomplete grouting

The uniform bedding along the tunnel represents a smooth reduction of the upward force. The reduction is induced by the grout-lining contact pressure. Nevertheless situations occur when this reduction is unable to get shape. For instance if free water is located between the tunnel and grout. A grout-lining contact pressure is prevented. If very liquid grout is used, a reduction is hardly noticed. On the other hand very stiff grout may result in a similar behaviour. If the tunnel starts to move upward in this case, the shear yield stress will be exceeded at a later stage or maybe not at all. Hence a grout flow from the top to the bottom of the tail void is prohibited. A gap between lining and grout may commence, which fills up with water. So again water prevents a reduction by the grout-lining contact pressure. If grout is injected incompletely along the tunnel’s circumference, no smooth reduction occurs as well. In all these situations the uniform bedding is omitted in the model. Hence the fictitious top support only remains. The situations are referred to as incomplete grouting. Also in case of complete grouting, subtraction of the water content from the grout is of main importance. Hardened grout will generate a stiff skeleton that supports the tunnel and hence makes cooperation of the soil and tunnel to bear the ground and water pressures possible. From that moment on ring behaviour of the embedded tunnel lining takes place. The porosity of sand subtracts the water from the grout relatively easy. In more dense or saturated soils like clay and peat a long period of time is required to squeeze out all water and therefore the grouting phase will stay active for a longer period as well.

4.3

Relation with the lining thickness

This section studies the requirements by bending moments from complete and incomplete grouting on the required lining thickness for motorway A13/16’s tunnel.

4.3.1


The magnitude of the bending moment in the tunnel lining is defined by the extent to which the upward floating pressure pushes the lining into the fictitious top support of the described model. The upward floating component is reduced by the tunnel’s dead weight and by the shear stress in the grout for both complete and incomplete grouting. In case of complete grouting a smooth development of the grout-lining contact pressure along the tunnel is possible, in case of incomplete grouting it isn’t. Therefore complete grouting has and incomplete grouting has no uniform bedding along the full tunnel perimeter. A uniform bedding causes the tunnel ring to be pushed more softly

48


into the fictitious top bedding; hence the resulting bending moment will be lower for complete grouting. The tunnel lining’s thickness has a major influence on its own dead weight, which reduces the upward floating pressure. So the thicker the lining, the less pressure is available to push the tunnel into the top support. For both ordinary concrete C35/45 and very high strength concrete C100/115 the specific gravity is approximately 24 kN/m³, for ultra high strength concrete C180/210 it is 28 kN/m³. As a result the reduction of the upward pressure is more effective for the latter. Which at the end results in a lower bending moment. For slender linings the lining stiffness is significantly lower than for thick linings. Therefore in the model slender linings will be influenced by the relation between the lining stiffness and bedding stiffness of the fictitious top support. This implies the comparison of concrete strength classes in case of thin linings might be influenced by their Young’s moduli as well. The combination of both described mechanisms on the grout bending moment is displayed in Figure 37 for a tunnel depth of 11,1 m (0,7D). The figure shows the resulting moments for complete and incomplete grouting. Up to a lining thickness of approximately 300 mm the stiffness ratio between the concrete tunnel lining and fictitious top support for the grout-lining contact pressure obviously influences the maximum bending moment. The type of concrete with the highest Young’s modulus, C180/210, returns the largest moments now. However from a thickness of approximately 600 mm it’s the dead weight of the tunnel lining that causes the bending moment to descend now with an almost linear pace. As a result the concrete materials with similar specific gravities, C35/45 and C100/115, return the same maximum bending moment, independent of their unequal Young’s moduli. ] m N k [ t n e 800 m o m g 600 n i d n e b t 400 u o r G 200

C100/115 Incomplete grouting

C35/45

C180/210

Complete grouting

C100/115

C180/210 C35/45 Thickness h [mm]

0 0

200

400

600

800

1000

37 | Bending moment by complete and incomplete grouting at a depth of 0,7D

Variations between resulting bending moments for complete and incomplete grouting are a result of the difference in bedding stiffnesses at the top and bottom for both situations. The differences are induced by the ξ-factor, which reads: h ρ ξ = s s −1 r ρw The stiffness ratio between the top and bottom support holds (ξ + 1) : 1 for complete grouting and holds ξ : 0 for incomplete grouting. If the soil cover on top of the tunnel increases, the ξ-factor increases as well. As a result the stiffness ratio for complete grouting approaches the one of incomplete grouting more and more. Hence the variations in the bending moments for both situations will decrease at an increasing depth. At the same time the stiffness ratio for incomplete grouting between the fictitious top support and the tunnel lining changes as well. The top bedding stiffness increases compared to the lining stiffness. The resulting bending moment for incomplete grouting therefore reduces at larger depths. The result is shown in Figure 38. The bending moment by complete grouting (line a) does approach the descending bending moment by incomplete grouting (line b) more and more as predicted.


49

0

200

400

600

800

0

1000

1200

1400

Grout bending moment [kNm] 0D

2000

0,7D 1D

1,2D 4000 (a)

(b)

(c)

2D

6000 2,9D 8000 ] N k [ 10000 e c r o f g12000 n i r l a 14000 m r o N

3D 4D

4,2D 5D

(a) Complete grouting (b) Incomplete grouting

6D

(c) Mu,1/20D / γ

] m / m [ p o t m o r f h t p e d l e n n u t d e t a m i t s E

38 | Development bending moment by complete and incomplete grouting over the depth for 675 mm C35/45

The introduction to this chapter discussed the hypothesis that the bending moment capacity of a lining in reinforced ordinary concrete with thickness 1/20 D might represent the bending moment by grouting. The exceptional development of the capacity at increasing depths however resulted in the conclusion that this might not apply to all tunnel depths. Therefore the bending moment capacity has been shown in Figure 38 once more. Now to verify the assumption (note that the capacity has been divided by the applied safety factor of 1,5 on resulting bending moments). It turns out that the bending moment capacity indeed doesn’t represent the grout moment at each depth. Especially at ] m N k [ t n e 800 m o m g 600 n i d n e b g 400 n i t u o r G 200

C100/115

C35/45

Incomplete grouting

C180/210

C100/115 C180/210 Complete grouting

C35/45

Thickness h [mm]

0 0

200

400

600

800

1000

a ] m N k [ t n e 800 m o m g 600 n i d n e b g 400 n i t u o r G 200

] m N k [ t n e 800 m o m g 600 n i d n e b g 400 n i t u o r G 200

C100/115 C180/210

Incomplete grouting

C35/45

Complete grouting

C35/45 C100/115 C180/210

Thickness h [mm]

0 0

200

400

600

800

1000

b

Incomplete grouting C100/115 C35/45 C180/210 Complete grouting

C35/45 C100/115 C180/210

Thickness h [mm]

0 0

200

400

600

800

1000

c

39 | Bending moment by complete and incomplete grouting at depths: a) 1,2D; b) 2,9D; c) 4,6D

50


larger depths the moment by grouting is overestimated. On the other hand the figure also shows that at a soil overburden of approximately 1D both moments are exactly similar. Depths of 0,5D to 1,5D are rather common for shield driven tunnels in the Netherlands. The hypothesis in [5] resulted from calculations on the Botlek Rail Tunnel with a soil cover of more or less 1,4D. It is therefore to be expected that the bending moment capacity and bending moment by grouting were this close in that particular tunnel as well. As a result a required lining thickness of approximately 1/20 D would be found. For the tunnel of motorway A13/16 the resulting bending moments by complete and incomplete grouting have been given in Figure 39. Only the remaining depth projections have been shown, refer to Figure 37 for the results of the most shallow projection.

4.3.2

Bending moment capacity

The normal ring force by the grout pressure from these calculations is no different from the one by calculations on the ring behaviour of the embedded lining. The function of this normal force is to make equilibrium with the pressures from the surrounding soil continuum; there is no difference in these loads. As a result the bending moment capacities of the lining do not change either, hence the developments from the previous chapter still apply. These distributions were shown in Figures 29 and 31 already.

4.3.3

Retrieving the required lining thickness

Compared to ring behaviour of an embedded tunnel lining the bending moments resulting from complete and incomplete grouting are relatively even at various lining thicknesses. Model analyses in the previous sections demonstrated that this holds for thicker linings especially. The bending moment capacity however does not alter compared to embedded ring behaviour. It therefore still shows a steady increase, as shown in the previous chapter. As a result the bending moment safety factor (ultimate resisting moment divided by the actual bending moment) now has an ascending development at increasing lining thicknesses. Only at the shallowest tunnel depths and very thin lining thicknesses a descending safety factor is observed for incomplete grouting. In Figure 40 the safety developments of all depth projections and concrete strength classes are given for complete and incomplete grouting. Obviously the safety level rises dramatically at higher depths. The resulting bending moment hardly changes, but the bending moment capacity shows a very significant increase, especially for the higher strength concrete materials C100/115 and C180/210. A required safety factor for the bending moment of γ = 1,5 has been introduced to the figure again (the same factor as used for ring behaviour on an embedded lining). Hence boundary conditions on the required lining thicknesses can be retrieved once more. The ascending development results in a lower boundary condition this time. So the actual lining thickness should be greater than or equal to the value found for grouting.


51

x a m

x a m


M / 6 u C180/210 M r o 5 t c a f y 4 C100/115 t e f a S 3 C35/45

C35/45

2

C100/115 C35/45

C100/115

1 C180/210 0 200

400

600

C100/115

γ = 1,5

1


C180/210

C180/210

2

γ = 1,5

0

C35/45

Thickness h [mm]

0

1000

0

200

400

600

a

800

1000

b x a m

x a m


M / 6 u M r o 5 t c a f y 4 t e Incomplete f a grouting S 3

Complete grouting

C100/115 C180/210

2

C180/210

Complete grouting

Incomplete grouting

2

γ = 1,5

1

1 C35/45 0 0

200

400

600


1000

γ = 1,5

5 1 / 1 0 0 C 1


0 0

200

400

600

c

800

1000

d

40 | Bending moment safety factor for complete and incomplete grouting at depths: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D

The resulting lower boundary conditions by complete and incomplete grouting for the required lining thickness have been listed in Table 5. Table 5 | Lower boundary conditions on the allowable lining thicknesses in case of complete and incomplete grouting C35/45 Depth 11,1 m (0,7D) 19,3 m (1,2D)

Complete 315 mm 370 mm

C100/115 Incompl. 725 mm 640 mm

Complete 1)

n/a

Incompl. 855 mm

1)

n/a

C180/210 Complete

Incompl.

1)

805 mm

1)

n/a

605 mm

n/a

635 mm

2)

46,7 m (2,9D)

425 mm

535 mm

180 mm

365 mm

155 mm

290 mm

74,2 m (4,6D)

525 mm

575 mm

230 mm

320 mm

125 mm

225 mm

1)

No lower boundary condition present within the considered range of lining thicknesses, safety level is exceeded at all values. 2) Estimated value

4.4

Conclusions

The bending moment capacity of a standard lining thickness of 1/20 D is not representative for the resulting bending moment by grouting at all considered depth projections in a tunnel lining of reinforced ordinary concrete C35/45. Especially at relatively large depths the bending moment is overestimated. Consequently slightly more slender lining thicknesses might be applied at these depths if grouting appears to be governing. The grout pressure in the tail void between tunnel and soil knows a local pressure increase on top of the tunnel. The bending moment by grouting is defined by the extent in which the tunnel is pushed into this pressure increment (described as fictitious top support) by the upward floating components

52


of the grout pressure (consisting of the hydrostatic water pressure and grout-lining contact pressure). The scale of the local pressure increase depends on the tunnel’s depth projection. Incomplete grouting results in significantly higher bending moments in shallow tunnels compared to complete grouting. However if the tunnel depth increases, the bending moment by complete grouting approaches the value of incomplete grouting more and more. The absolute value of the latter will somewhat descend at greater depths. The resulting bending moment is only slightly dependent on the lining’s thickness. The stiffness of the tunnel rings, such as the concrete Young’s modulus and the number of tunnel segments per ring, do not considerably influence the resulting bending moments. The weight of the tunnel ring however decreases the net upward floating force and therefore reduces the extent in which the lining is pushed in the fictitious top support. Especially at high lining thicknesses this results in a considerable reduction of the resulting bending moment for the heaviest of all considered concretes, ultra high strength concrete C180/210. Very and ultra high strength concrete require only very small lining thicknesses at high depths to cope with the bending moment by complete and incomplete grouting. At ascending lining thicknesses the bending moment safety level for both complete and incomplete grouting increases. The lining thickness at the required safety level is therefore a lower boundary condition of the final required thickness. Table 5 and Figure 41 show all boundary conditions. 0

200

400

600

800


] m19,2 [ e n 27,4 i l e r t n e c m o 54,8 r f h t p e d l e n n 82,2 u T

0,7D 1,2D

2,9D

C35/45 C100/115 C180/210

4,6D

41 | Lower boundary conditions on the allowable lining thicknesses in case of complete and incomplete grouting


53

54


Chapter 5

Additional structural mechanisms 5.1

Introduction

Shield driven tunnels are constructed within the protection of the shield of a tunnel boring machine (TBM). After assembly of a complete ring, the TBM moves forward by pushing its thrust jacks on the bearing pads of the newest assembled ring. The high thrust jack forces that are hereby introduced in the lining result in tensile bursting forces deep in the material and in high compression stresses under the thrust jack plate. In section 5.2 the relation between the lining’s thickness and the introduction of thrust jack forces is investigated. Moving forward of the tunnel boring machine also causes the one-but-last ring of the lining to leave the protection of the shield. Now the ring is loaded by the surrounding grout pressure and will start to deform. The latest ring however is still covered by the shield; hence no loading is present to initially deform it. As a result tunnel rings close to the TBM are subjected to torque in order to change the initial circular shape into an ovalised one by the loading with semi-liquid grout. This torsion phenomenon is also known as the trumpet effect and will be discussed in section 5.3.

5.2

Introduction of thrust jack forces

In the French and German thrust jack configurations the thrust jacks are grouped in pairs of two on a thrust jack plate. Axial forces are introduced in the tunnel segments via the plates on the bearing pads of the newest tunnel rings. These axial forces are supposed to be the only forces present, since the ring is still protected by the TBM’s shield and therefore not loaded by grout yet. Although the values of the introduced thrust jack forces are equal for both European configurations, they handle different systems to spread out the compression forces in the tunnel segments. Consequently variations in the tensile bursting stresses occur. In order to spread the thrust jack force over the segment’s tangential and radial directions, so-called compression trajectories have to bend away from the initially axial introduction direction. For the compression trajectories widen now and point in different directions (see Figure 42a), a tensile force in between them should take care of the equilibrium of horizontal forces. By help of the lattice analogy, where compression trajectories are schematised as compression struts, the total resulting tensile force can be determined (Figure 42b). In case of the French thrust jack configuration this system will occur just like described. By the applied positioning of thrust jacks on the segment, each jack plate is located at the exact centre line of its own half of a segment (two thrust jack plates per segment). Now the thrust jack force of each jack plate is able to spread over the width of its own segmental half and the tensile bursting stresses will appear as described. The high force introduced and the lateral contraction property of concrete result in deformations of the segmental shape close to the thrust jacks (see Figure 44a). As a result tensile stresses will appear in the deformed sections and cracks might occur. However if cracking does occur the deformation is no longer restricted and the tensile stresses will fade away.

55

t h g i e h e c n a b r u t s i d

compression

schematisation tension bursting force

compression trajectories

b

a

42 | Development of bursting stresses. a) Compression trajectories; b) Latice analogy

The German thrust jack configuration has no such decent spreading of compression trajectories over the full segment. The thrust jack plates in the middle of the segments do not result in any additional tensile stresses (hence no problems for concrete). The jack force introduced on the edge of the segment however, over the longitudinal joint, is unable to spread symmetrically to two sides and no compensating tensile bursting force can be created. Still the force will spread over the only side that’s present. As a consequence of the changed direction of the trajectories a completely different mechanism to ensure horizontal equilibrium of forces is introduced. Deep in the segment a compression strut links the forces of the jacks on the tension edge and in the middle. In between the actual introduction spalling force locations, so just under the segment’s front face, a tensile spalling force is created (see Figures 43 and 44b). This tensile force is a structural force needed for equilibrium, so cracking compression will not decrease its magnitude. Serious cracking with large 43 | Tensile force in latice analogy for German thrust jack configuration crack depths may therefore result in the German thrust jack configuration. Obviously the French configuration is more effective in spreading the introduced thrust jack forces from the TBM. This fact contributed to the decision made in Chapter 3 to apply this particular configuration in the tunnel of motorway A13/16. This section will discuss the resulting high compressive stresses under the thrust jack plates and the tensile bursting stresses deeper in the segments by the introduction of thrust jack forces for the French configuration. First of all the magnitude of this thrust jack force should be determined.

spalling stresses spalling stresses bursting stresses

bursting stresses compression

French

German

Tension Compression

a

56

b 44 | Stress distribution in tunnel segments due to introduction of thrust jack forces in the: a) French and b) German thrust jack configurations [5, 18]


5.2.1

Magnitude of the thrust jack force

For a long time total thrust jack forces for all jacks combined have been roughly estimated and scaled based on values for existing tunnels. In a study by De Rijke [12] however a model has been implemented to determine the forces based on soil properties and dimensions of the tunnel lining and tunnel boring machine. The so-called TBM Shield Equilibrium Model calculates the resulting thrust jack forces for each individual tunnel segment. For this a ring with only seven segments is assumed. A conversion to nine segments per ring will therefore be required in this study. Most properties for the soil have been discussed in previous chapters already, such as for the calculations on ring behaviour of an embedded lining. Additionally a friction coefficient between the soil and TBM shield is required (value of 0,45 for sand follows from tables in [12]), as well as socalled Ky,front and Ky,side coefficients, which both equal the average of the active and neutral soil support coefficients Ka and K0: 1 − sin ϕ ⎫ Ka = Ka + K0 ⎪ 1 + sin ϕ ⎬K y ,front = K y ,side = 2 K 0 = 1 − sin ϕ ⎪⎭ The internal friction angle ϕ = 30° was already mentioned in earlier calculations in this report. Hence it holds: 0 ,33 + 0 ,50 = 0 ,42 K y , front = K y ,side = 2 Dimensions of the tunnel boring machine for the shield driven tunnel of motorway A13/16 are not known yet. Therefore values from the machine of the Botlek Rail Tunnel from De Rijke’s study are used to make an estimation. Just like in the mentioned machine the external radius of the TBM is 250 mm larger than the one of the shield driven tunnel. The length of the excavation chamber is 2 m and the length of the shield is 9,9 m. De Rijke also proves that a reduction of the shield length results in a significant decrease of the actual thrust jack forces. For now however it is not guaranteed that such shortening can be achieved; hence for this study the conventional length of 9,9 m is assumed. For a description of the TBM Shield Equilibrium Model’s principal is referred to the study by De Rijke [12]. This report will show the final part of the calculation only. Here the principal horizontal force H jacks,total and the bending moment M jacks,total on the machine’s face for all thrust jack forces together, which is a result of the mentioned model, are converted to individual thrust jack forces. The resulting principal forces for the considered tunnel with a lining thickness of 675 mm and a depth projection of 19,3 m (1,2D) are: H jacks , total = −1 ,34 ⋅ 10 5 kN; M jacks,total = 2 ,02 ⋅ 10 4 kNm These reaction forces can be converted to a distributed load along the tunnel’s circumference. The values of this load on the tunnel’s top and bottom are given by: M jacks , total H jacks , total − q bottom = πR jacks 2 2πR jacks q top =

−H jacks , total πR jacks

− q bottom

Where R jacks is the radius of the jacks’ centre line, which equals the radius of the centre line for the tunnel segments. Consequently R jacks holds for the applied lining thickness: R jacks = 12 (Di + h ) = 12 (14 ,9 + 0 ,675) = 7 ,79 m This implies that the loads are: q bottom = 2 ,84 ⋅ 103 kN/m; q top = 2 ,63 ⋅ 103 kN/m


57

z qtop

I A H

B qmean y,q

G

C

F D E

qbottom

45 | TBM Shield Equilibrium Model with line load to determine thrust jack forces

The mean value of the distributed load along the tunnel’s circumference (qmean) is easy to determine, just like the variation between the top and bottom values (q∆): q mean = 12 (q top + q bottom = 2,73 ⋅ 103 kN/m q ∆ = q bottom − q top = 212 kN/m At each arbitrary angle θ in the lining’s circumference a cumulative force based on the distributed load can be defined. This cumulative force is given by: F( θ) = −R jacksq mean θ − 12 R jacksq ∆ cos θ If such a force is determined at each longitudinal joint of one individual tunnel segment, the resulting force by both thrust jack plates on that particular segment is: Fsegment = F2 − F1 Where: F1 = −R jacks q mean θ 1 − 12 R jacks q ∆ cos θ 1 F2 = −R jacks q mean θ 2 − 12 R jacks q ∆ cos θ 2 The axial force on one group of thrust jacks (one thrust jack plate) in this segment is now simply defined as: Fgroup = 12 Fsegment Letters have been appointed to all tunnel segments in Figure 45. Table 6 displays the determined values and calculation principal of the actual thrust jack forces for the tunnel segments based on the described method. Table 6 | Calculation of axial thrust jack forces Segment

58

θ1 [rad]

θ2 [rad]

F1 [kN]

F2 [kN]

Fgroup [kN]

A (top)

1,75

2,44

-36.970

-51.326

-7.178

B

2,44

3,14

-51.326

-65.978

-7.326

C

3,14

3,84

-65.978

-81.016

-7.519


D

3,84

4,54

-81.016

-96.350

-7.667

E (bottom)

4,54

5,24

-96.350

-111.751

-7.700

F

5,24

5,93

-111.751

-126.958

-7.604

G

5,93

6,63

-126.958

-141.803

-7.423

H

6,63

7,33

-141.803

-156.286

-7.241

I (top)

7,33

8,03

-156.286

-170.575

-7.415

According to the TBM Shield Equilibrium Model the maximum thrust jack force is 7.700 kN for this tunnel configuration at a depth of 19,3 m (1,2D). Including a safety factor γ jack = 2 it now holds [4]: F jack = γ jackFgroup = 2 ⋅ −7.700 = −15.400 kN The safety factor used is relatively high compared to the factor which has been applied on the bending moment in prior calculations. The main reason is the uncertainty about original thrust jack force estimates. However since the model by De Rijke is no longer an estimation, but a calculation based on soil and TBM properties, a lower safety factor may be suggested. Nevertheless a safety factor of 2 is still common now and will therefore be used in this study. If the tunnel lining thickness increases, the external diameter of the TBM increases as well. Consequently the maximum thrust jack force grows (see Figure 46a). For the tunnel embedded in soil (Chapter 3) the normal force developed directly proportional to the depth of its centre. For the thrust jack force there is nothing different, as Figure 46b shows.

-13500

0

200

400

600

800

0

1000

-10000

-20000

-30000

Thrust jack force [kN]

thickness h [mm] ] m19,2 [ e n 27,4 i l e r t n e c m o r 54,8 f h t p e d l e n n 82,2 u T

-14000 -14500 ] N-15000 k [ e c r -15500 o f k c a j t -16000 s u r h T

-40000

(0,7D) (1,2D)

(2,9D)

(4,6D)

a

b

46 | Development thrust jack forces over: a) Lining thicknesses for depth of 1,2D; b) Tunnel depth

5.2.2

Compressive stresses beneath the thrust jack plates

As a result of the axial thrust jack forces high compressive stresses occur in the concrete under the jack plates. These stresses are given by: F jack F jack σ c , jack = = A jack a l a b Where a l and a b are the length and width of the thrust jack plate. In Chapter 3 it has been mentioned that the estimated dimensions of the thrust jack plates are based on dimensions used in the Green Heart Tunnel. The width of the plate therefore is 1.500 mm and its height is ⅔ of the lining thickness. Hence it holds: a l = 1.500 mm; a b = 23 h In the determination of the thrust jack forces a tunnel lining thickness of 675 mm and a soil cover of 19,3 m (1,2D) were given as an example. By reapplying this model the stress under the thrust jack plate is given by:


59

σ c , jack =

F jack a l a b

=

− 15 ,4 ⋅ 10 6 N = 22 ,8 MPa (14) 1.500 ⋅ 23 h mm 2

The maximum allowed compressive stress under the jack plates is known in the Dutch code NEN 6720 as f’co. Hence the following requirement should be met at any time: σc , jack ≤ f'co According to the code the maximum allowed compressive stress is given by: f' co = f ' c

l b

a l a b

(15)

Where f’c is the design value of the compressive strength for concrete and l and b are the length and width of the segmental cross-section over which the thrust jack force will spread in the end. The latter dimensions are defined by the minimum values of: ⎧a l + 2sl ⎫ ⎧a b + 2s b ⎫ ⎪ a +d ⎪ ⎪ a +d ⎪ ⎪ l ⎪ ⎪ b ⎪ = and b min l = min ⎨ ⎨ ⎬ ⎬ ⎪ 5a l ⎪ ⎪ 5a b ⎪ ⎪⎩ 5 b ≥ a l ⎪⎭ ⎪⎩ 5l ≥ a b ⎪⎭ Dimension d is the height of the so-called disturbance area and s l and s b are the differences between the length and width respectively of the thrust jack plate and half the area available for spreading (see Figure 47). sb ab

al

sl

d

a b + 2s b

a l + 2s l A-A'

A-A'

47 | Definitions schematised stress distribution under a thrust jack plate

For the tunnel with a lining thickness of 675 mm the dimension a l + 2s l , that equals half the segmental length, and a b + 2s b is given by: a l + 2sl = l segm / 2 =

π(Di + h ) π(14.900 + 675) = = 2.718 mm 2n segm 18

a b + 2s b = h Tunnel segments transmit the axial thrust jack force to older rings via bearing pads in the ring joint. These pads are situated right behind the thrust jack plates. So only half of the segmental dimension in axial direction is available for spreading the force. The other half is used to narrow the spread force again in order to lead it to the bearing pads (see Figure 47). Therefore the disturbance length d is half the segmental width: 2.000 = 1.000 mm d= 2 Determination of all remaining parameters is possible now:

60


h − 23 h 1 2.718 − 1.500 sl = = 609 mm; s b = =6h 2 2 ⎧ a l + 2s l = 2.718 mm ⎫ ⎪ a + d = 2.500 mm ⎪ ⎪ l ⎪ l = min ⎨ ⎬ = 2.500 mm ⎪ 5a l = 7.500 mm ⎪ ⎪⎩5 b(≥ a l ) = 3.375 mm ⎪⎭ a b + 2s b = h ⎧ ⎫ ⎪a + d = 2 h + 1.000 mm ⎪ h if h ≤ 3.000 mm ⎫ h ≤ 3.000 mm ⎪ b ⎪ ⎧ 3 =     → b = h b = min ⎨ ⎬ ⎨ ⎬  1 2 5 a 3 h h 1 . 000 mm if h 3 . 000 mm = + > 3 3 b ⎭ ⎪ ⎪ ⎩ ⎪⎩ 5l(≥ a b ) = 12.500 mm ⎪⎭ Accordingly parameter b equals the lining thickness h for all thicknesses considered. The governing value for parameter l however varies within the considered range. The following definition applies: ⎧ a l + 2s l ⎫ if h ≤ 300 mm ⎫ ⎪ a + d ⎪ ⎧a l ⎪ l ⎪ ⎪ ⎪ if 300 < h < 500 mm ⎬ l = min ⎨ ⎬ = ⎨5h ⎪ 5a l ⎪ ⎪a + d if h ≥ 500 mm ⎪ ⎭ ⎪⎩5 b(≥ a l )⎪⎭ ⎩ l The maximum value of the permitted compressive stress is now defined by substitution of all parameters in equation (15): ⎧f 'c 1 ,5 ⎫ if h ≤ 300 mm ⎪ ⎪ h l⋅h ⎪ ⎪ f 'co = f 'c if 300 < h < 500 mm ⎬ (16) == ⎨f 'c 2 1.500 ⋅ 3 h 10 2 ⎪ ⎪ ⎪⎩f 'c 2 ,5 if h ≥ 500 mm ⎪⎭ For a lining thickness of 675 mm the permitted values for the three concrete strength classes are: f 'co , C 35 / 45 = f'c ,C 35 / 45 2 ,5 = 0 ,6 ⋅ 45 ⋅ 1 ,58 = 43 MPa f 'co , C100 / 115 = 0 ,6 ⋅ 115 ⋅ 1 ,58 = 109 MPa f 'co , C180 / 210 = 0 ,6 ⋅ 210 ⋅ 1 ,58 = 199 MPa In Figure 48 the development of these values along an ascending lining thickness has been included. From a comparison of the permitted stresses and the actual compressive stress by the introduction of thrust jack forces in equation (14) (22,8 MPa), it appears that all concrete strength classes are able to cope with the compressive stresses under the thrust jack plate in case of a lining thickness of 675 mm and a depth projection of 19,3 m (1,2D). Equation (14) also shows that the resulting compressive stress for constant thrust jack forces is related to the inverse lining thickness and will therefore decrease with an asymptotic extent at increasing thicknesses. However the real thrust jack force of thick linings is slightly larger than the one of slender linings (Figure 46a). Nevertheless the global development of the resulting concrete compressive stress is hardly assaulted by this variation, as Figure 48 proves. The locations where the lines of the actual and permitted values meet demonstrate that the minimum required lining thicknesses for the concrete strength classes are: h ≥ 390 mm for C35/45 h ≥ 175 mm for C100/115 h ≥ 100 mm for C180/210


61

0

0

200

400

600

800

1000 h [mm]

-20

f'co,C35/45

-40 -60 -80 -100 ] a P-120 M [ -140

f'co,C100/115

k c a j , c

-160 σ s s-180 e r t S

f'co,C180/210

48 | Compressive stress under thrust jack plates for a depth of 1,2D

If the depth projection of the tunnel increases, the thrust jack force grows as well (see Figure 46b). Consequently the concrete compressive stress under the thrust jack plate ascends accordingly. No changes occur in the permitted compressive strength, irrespective of the depth. The occurring stresses for all four considered tunnel depths and the permitted stresses for all three materials have been combined in Figure 49a. Again minimum required lining thicknesses can be isolated, which have been included in Figure 49b. Obviously concretes with a higher compressive strength require significantly thinner tunnel linings to cope with the compressive stresses introduced by thrust jack forces. Especially in very deep tunnels the advantage of C100/115 and C180/210 is clearly visible. At the end this may turn out to be a very positive property of very and ultra high strength concrete, since these materials manifested splendid behaviour for deep tunnels in calculations on the uplift loading case in the previous chapter as well. 0 -50 -100

0

200

k c a j , c

-300

600

800

1000

0,7D f'co,C100/115 1,2D

0

200

400

600

800

h [mm]

f'co,C35/45

deeper

-150 -200 ] a P-250 M [

400

2,9D

f'co,C180/210

4,2D

σ

s-350 s e r t S

a

1000

thickness h [mm] ] m19,2 [ e n 27,4 i l e r t n e c C 1 m 8 o 54,8 0 r f /2 h 1 t 0 p e d l e n n 82,2 4,6D u T

C35/45

C100/115

b

49 | Introduction of thrust jack forces over the depth. a) Compressive stress under thrust jack plate; b) Required lining thicknesses

5.2.3

Tensile bursting stresses

The introduction to this chapter already described that bursting tensile stresses are generated by spreading of the thrust jack forces (compression trajectories) in tunnel segments. The extent of the spreading and therefore the magnitude of the tensile stresses strongly depends on the variations between the dimensions of the introduction and final spreading surfaces. This relation has been described by Iyengar in a diagram (see Figure 51). In this diagram the development of tensile bursting stresses (as a fraction of the fully spread compressive stress) over the depth of the segment is shown for various interrelations of the introduction width (β) and the spreading width (a). Spreading of the introduced force occurs in one direction only, namely in the x-y plane of the figure. Nevertheless dimensions of the thrust jack plates are smaller than the dimensions of the tunnel segment in both radial and tangential direction; hence tensile bursting stresses in both directions should be considered.

62


βr =

β t = 1.500

2 3

h

0 0 0 . 1 =

a r = h

1 t

a

a r = h A-A'

A-A' radial

tangential

50 | Spreading of compression trajectories in tangential and radial direction in tunnel segments

In the radial direction of the tunnel segments, the introduction width is restricted to the height of the thrust jack plates, the spreading width is the thickness of the concrete segments. Hence: β r = 23 h ⎫ β r 2 = ⎬⇒ ar = h ⎭ ar 3 In Iyengar’s diagram a ratio between the maximum radial tensile bursting stress σcr and the spread compressive stress σcm is found, reading: σ cr = 0 ,15 σ cm σcx σcm

0,5

β a =0

y

σcm

=

F jack ab

σcx

F jack

0,1

0,4

a

β

x

β

σcy

0,2

β a =0,3

0,3

a

b

0,4 0,5 0,2 0,17

0,15

0,6

0,67

0,7 0,1

0,8 0,9 x a

0 0

0,25

0,5

0,75

1,0

51 | Diagram of Iyengar for tensile bursting stresses due to introduction of thrust jack forces

In order to determine the final spread compressive stress σcm both the spreading height and width are required. The spreading height (in radial direction) has been retrieved already and equals the lining thickness, or segmental thickness. Determination of the spreading length (in tangential


63

direction) however is more complicated. Iyengar’s definition of dimensions as presented in Figure 51 assumes that the introduced compression force is completely spread over the full cross-section after a distance that equals the spreading width a. During the calculations on the compressive force below the thrust jack plates it came forward that only half of the segmental width (hence 1.000 mm, also see Figure 50) is available as a spreading depth. The tangential width of the jack plate however is 1.500 mm (βt in Figure 50) and is therefore larger than the spreading depth (at1 in the figure) at all time. This is impossible according to the normal definition. As a result an alternative method should be used to retrieve the spreading distance in tangential direction. Maximum spreading in Iyengar’s diagram occurs if β = 0. Over the full spreading depth a, the trajectories should spread over a distance of a/2 to either side. So maximum spreading of the introduced thrust jack force on the sides of the jack plates holds a/2 as well. In this expression a equals at1 from Figure 50 in this case. Consequently the maximum spreading length in tangential direction of the thrust jack force is: a t = βt + a t1 = 1.500 + 1.000 = 2.500 mm This dimension is smaller than half the segmental length (½ π(Di + h)/9 = 2.720 mm for h = 675 mm) and is therefore thought to be critial for the spreading length. The complete spreading area is known now, so the compressive stress after full spreading may be retrieved by use of the definition from Figure 51: F jack F jack F jack σcm = = = ab a ra t 2.500h Subsequently the ratio between the tangential tensile bursting stresses and the spread compressive stress is found as well. According to the diagram it is given by: β t = 1.500 ⎫ β t = 0 ,6 ⎬⇒ a t = 2.500⎭ a t Tensile bursting stresses in radial and tangential direction are now determined by: 0 ,15F jack F jack σ σ cr = cr σ cm = = σ cm 2.500h 16 ,7 ⋅ 10 3 h 0 ,17 F jack F jack σ σ ct = ct σ cm = = σ cm 2.500h 14 ,7 ⋅ 10 3 h The thrust jack force F jack has a positive value under compression, for it points in the direction defined by Figure 51. Similar to the development of the compressive stress under the thrust jack plates, the tensile bursting forces due the jack force introduction are related to the inverse lining thickness as well. Figure 52 shows both resulting stresses for an overburden of 19,3 m (1,2D). ] a P M [ t c / r c

fc,C180/210

8

σ

s e s s6 e r t s g n i t 4 s r u B

σct

σcr

fc,C100/115

fc,C35/45

2

h [mm]

0 0

200

400

600

800

1000

52 | Radial and tangential tensile bursting stresses due to the introduction of thrust jack forces

64


In case no additional steel bar reinforcement is included, the following requirement should be met: σ cr ≤ fc and σ ct ≤ fc Where fc is the design value of the tensile strength for concrete. These values hold 1,65 MPa (NEN 6720), 5 MPa and 9,1 MPa for concrete materials C35/45, C100/115 and C180/210 respectively. For the applied relation between the height and width of the thrust jack plate on one side and the dimensions of the tunnel segment on the other side, tensile bursting stresses in the tangential direction are governing. Figure 52 thereby returns the following minimum required lining thicknesses for the tunnel in order to resist tensile stresses by the introduction of thrust jack forces at a depth of 19,3 m (1,2D): h ≥ 635 mm for C35/45 h ≥ 195 mm for C100/115 h ≥ 105 mm for C180/210 Apparently the lining thicknesses required by the tensile stresses are higher than the ones dictated by compression under the thrust jack plates (also see Figure 48). In case of reinforced ordinary concrete C35/45 however the possibility exists to include rebars to deal with the tensile stresses. Hence for this material the lining thickness required by the resulting compression stresses is governing. Again the actual thrust jack force is the only input parameter of the calculations that depends on the tunnel’s depth projection. In Figure 53a the resulting tangential tensile bursting stresses are presented. The required lining thicknesses of the considered concrete materials are shown in Table 7 and Figure 53b. ] a P M [ t c 25 σ 4,2D s s e r t 20 s 2,9D g n i t 15 s r u B 10 1,2D

0

400

600

800

1000

thickness h [mm]

fc,C180/210

0,7D

fc,C100/115

5 fc,C35/45

h [mm]

0 200

200

400

600

800

1000

] m19,2 [ e n 27,4 i l e r t n e c C m 1 8 o r 54,8 0 f / 2 h t 1 p 0 e d l e n n 82,2 4,6D u T

C35/45

C100/115

a

b

53 | Introduction of thrust jack forces over the depth. a) Maximum tangential tensile bursting stresses; b) Required lining thickness per concrete strength class by bursting stresses

Table 7 | Lower boundary conditions on the allowed lining thicknesses by the introduction of thrust jack forces C35/45 Depth

Compr.

C100/115 Tension

Compr.

C180/210

Tension

Compr.

Tension

11,1 m (0,7D)

315

460

125

140

n/a

n/a1)

19,3 m (1,2D)

390

635

46,7 m (2,9D) 74,2 m (4,6D)

680 2)

1030

1)

175

195

100

100

2)

315

380

175

200

2)

410

575

260

305

1260 1880

1)

No boundary condition within the considered range of lining thicknesses, safety level exceeds the required level at all time. 2) Values have been determined by extrapolation


65

5.3

Torsion in tunnel segments by the trumpet effect

At the back of the tunnel boring machine a tunnel ring leaves the protection of the TBM-shield as the machine excavates and moves forward. Thereby that particular ring is loaded by the pressures of fresh injected grout and starts to deform. Rings which are still protected by the TBM shield are not loaded yet. If it is assumed that these rings in the machine behold their circular shape, the segments in the ring leaving the TBM will twist in order to adapt to the ovalised shape by the grout loading (see Figure 54). Because of the vertical narrowing of the tunnel ring, this mechanism is also known as the trumpet effect. Torsion in the twisted segments is a consequence of forced deformations. It is therefore not a direct failure mechanism; it only leads to possible cracking hence damage. This type of cracks however has been observed quite frequently during the construction of existing shield driven tunnels. A comparison of the considered concrete strength classes is therefore interesting, in order to find out whether this type of damage is possible to occur in very and ultra high strength concrete as well.

Lt

Grout loading

TBM shield A

Protected segment

A

Deformed segments

Change of shape at top of the ring (side view) B

Initial circular shape

Grout loading

TBM shield B

Ovalised shape

Protected segment

Segments subjected to torsion

Deformed segments

Change of shape at the side of the ring (top view)

54 | Torsion in tunnel segments by deformation due to grout loading

5.3.1

Torsion capacity of the lining

In a horizontally ovalised tunnel (“lying egg”) the maximum diameter shortening by the ovalisation deformations is observed in vertical direction (diameter between top and bottom of the tunnel), the maximum diameter enlargement occurs in horizontal direction (als see Figure 54). As a result the torsion problem can be restricted to a distance between locations with maximum (top) and minimum (at an angle of π/4 from the top) deformations. The torsional length Lt is therefore given by: π( r + 21 h) Lt = 4 Where r is the inner radius and h is the lining thickness. The torsional distance in axial direction determines the amount of damage to a big extent. The exact distance is not known however. Depending on the interaction of rings in this direction possibly one or perhaps multiple segmental widths are required to generate the deformation. The torsion width is therefore indicated as Bt for now. θ't

u2

Bt

55 | Rotation of tunnel segment in vertical plane

66


In the vertical plane of the tunnel’s longitudinal direction (axial-radial plane) the tunnel segments subjected to torsion will rotate over an angle of θ’t (see Figure 55). This angle is defined as: u θ't = 2 Bt Herein u2 is the maximum ovalisation deformation (half the diameter shortening) caused by the grout loading case. Per unit of length along the tunnel circumference the torsion angle is given by: θ' 4u 2 θt = t = L t bπ( r + 21 h) The torsional moment is generally defined as: T = GI t θ t With G being the shear modulus of concrete and It the torsional moment of inertia: ⎛ E h ⎞ G= ; I t = 13 Bth 3 ⎜⎜ 1 − 0 ,6 ⎟⎟ with h < Bt 2(1 + ν ) Bt ⎠ ⎝ Where E is the Young’s modulus and ν is the Poisson ratio of concrete. The torsional moment creates a linear shear stress in the concrete tunnel segments. Its maximum value at the top and bottom of the segments is given by: T (17) τt = Wt Where Wt is the elastic torsional modulus: 2 1 3 Bt h Wt = again with h < Bt h 1 + 0 ,6 Bt If it’s assumed that the shear stress τt is the only stress present (hence stresses by the thrust jack forces are omitted for now), the shear stress is a principal tensile stress and therefore should not exceed the tensile strength fshr: τ t < fshr Substitution of the parameters in equation (17) gives the torsional shear stress: E h ⎞ 4u 2 3 ⎛ 1 ⎜ ⎟ 3 B t h ⎜ 1 − 0 ,6 ⎟ 2(1 + ν) Bt ⎠ bπ(r + 21 h) ⎛ GI t θt 2Ehu2 h 2 ⎞ 2Ehu2 ⎝ ⎜ ⎟ τt = = = − = 1 0 , 36 2 2 1 Wt Bt π(r + 21 h)(1 + ν) ⎜⎝ Bt ⎠⎟ Bt π( r + 12 h)(1 + ν) 3 Bt h 144 244 3 h ≈1 1 + 0 ,6 Bt Hence it also holds: τt < fshr

2Ehu2 Bt πfshr ( r + 21 h)(1 + ν) < fshr ⇔ u 2 < ⇒ Bt π(r + 12 h)(1 + ν ) 2Eh

(18)

According to the Dutch building code NEN 6720 the mean tensile strength f cm is the boundary condition for tensile stresses for calculations on cracks. Hence: fshr = fcm = 1 ,4fcrep = 1 ,4(1 ,4fc ) = 2fc (19) The maximum permissible deformation to prevent cracking by torsion appears to depend on several concrete properties is presented in equations (18) and (19) (note that the Poisson ratios are similar for all considered concrete strength classes):


67

fc E Consequently this results in the following relation for the three types of concrete: u 2 ~ 1,65/33.500 = 4,9 ⋅ 10-5 for C35/45 u2 ~

u 2 ~ 5,0/45.000 = 11 ⋅ 10-5 for C35/45 u 2 ~ 9,1/65.000 = 14 ⋅ 10 -5 for C35/45 Accordingly C180/210 is able to resist deformations with a magnitude of almost triple the ones for C35/45 before cracking occurs. The bending moment by grouting was found not to depend on the concrete stiffness in the previous chapter. As a result the higher Young’s moduli for very and ultra high strength concrete ensure smaller deformations at similar lining thicknesses and bending moments than for ordinary concrete. Hence the capacity to prevent cracking is higher and the actual deformations are lower, an advantage of very and ultra high strength concrete. The permitted value of deformation u2 is nearly proportional to the inverse of the lining thickness according equation (18). If the torsion width Bt is assumed to be one segmental width (2 m) [5], the development of the permitted value is as shown in Figure 56. ] m m [ 2 u n80 o i t c e l f 60 e d m u m40 i x a M

C180/210

C100/115

20 C35/45 h [mm]

0 0

200

400

600

800

1000

56 | Maximum allowed deformation u2 due to torsion only in tunnel segments

Influence of thrust jack forces

In this chapter has been discussed before that thrust jacks from the TBM introduce large forces into the concrete tunnel segments. This force is transmitted to older rings by bearing pads in the ring joints. Therefore the axial force will be present in the twisted tunnel segments as well. Such a multiaxial stress pattern is described by Mohr’s circle (Figure 57). This circle translates the multi-axial stresses in the principal compressive stress σ1 and principal tensile stress σ2. From the circle these principal stresses are defined by: σ 1 ,2 =

1 2

(σ

xx

+ σ yy ) ±

1 4

(σ

xx

− σ yy )2 + τ tj 2

Stresses σxx and σyy represent the maximum tensile bursting stress and the compression stress respectively due to the introduction of thrust jack forces. If it is assumed that torsion may occur to tunnel segments within the TBM occasionally, no normal ring force is present. Hence stresses by the thrust jack forces and torsion are the only stresses in Mohr’s circle. Then parameter τtj is the shear force by torsion, now defined by: τ tj = σ xy = σ yx

68


σyx

F jack direction principal stress σ2

T

y σyy

( σyy,σyx )

direction principal stress σ1

σ2

σ1

σxx,σyy

σyx σxy

x

( σxx,σxy )

σxx

σxy

b

a

57 | Multi-axial stresses in twisted tunnel segments. a) Definition of stresses; b) Mohr’s cirle

The principal tensile stress by the multi-axial stresses is restricted by boundary fshr again: σ1 =

1 2

(σ

xx

+ σ yy ) +

1 4

(σ

xx

− σ yy )2 + τ tj 2 ≤ fshr

If fshr equals fcm , the maximum allowed value for shear force τtj can by isolated by: (fcm − 12 (σxx + σyy ))2 ≥ 14 (σxx − σ yy )2 + τtj2

⇒ fcm 2 +

1 4

(σ

2 xx

+ σ yy 2 + 2σ xxσ yy ) − (σ xx + σ yy )fcm ≥

1 2

(σ

2 xx

+ σ yy 2 − 2σ xxσ yy ) + τtj 2 2

τtj ≤ fcm 2 − fcm (σ xx + σ yy ) + σ xxσ yy

Values for stresses σxx and σyy result from the calculations on tensile bursting stresses in the previous section. At a depth projection of 19,3 m (1,2D) the admissible value for shear force by grouting τtj now develops as presented in Figure 58. Obviously the allowed values for very and ultra high strength concrete benefit from the introduced compressive forces in thin linings in particular. As a consequence even higher deformations are allowed. If the initial concrete tensile strength fshr is replaced by τtj these deformations are described by equation (18) once more: Bt πτtj (r + 12 h)(1 + ν ) u2 < 2Eh If concrete is unable to cope with the combined compressive and tensional stresses by the introduction of thrust jack forces, the permissible value of τtj reduces to zero. That’s what occurs at a lining thickness of approximately 300 mm for ordinary concrete C35/45 in Figure 58. Evidently the tunnel segments are unable to resist any deformation u2 now. At deeper tunnel projections this phenomenon will appear for the other concrete strength classes as well. In Figure 59 the permitted displacements have been visualised for all considered depths. Higher compressive and tensile bursting stresses at higher depths result in a decrease of the local peaks at slender lining thicknesses.


69

] m m [ m30 c f d n 25 a

τtj,C180/210

j t

τ

s e 20 s s e r t S15

fcm,C180/210

τtj,C100/115 fcm,C100/115

10 5

τtj,C35/45

fcm,C35/45

h [mm]

0 0

200

400

600

800

1000

58 | Allowed shear stress τtj at a depth of 1,2D

] m m [

] m m [ 2 u100 n o i t a 80 m r o f e d 60 d e t t i m r 40 e P 20

2

u100 n o i t a 80 m r o f e d 60 d e t t i m r 40 e P 20

C180/210

C100/115 C35/45

0 200

C100/115 C35/45

h [mm] 0

C180/210

400

600

800

h [mm]

0

1000

0

200

400

a

2

C180/210

C100/115 C35/45

h [mm]

0 0

200

400

600

800

1000

b ] m m [ 2 u100 n o i t a m 80 r o f e d 60 d e t t i m r 40 e P 20

] m m [ u100 n o i t a 80 m r o f e d 60 d e t t i m 40 r e P 20

600

800

1000

C180/210

C100/115 C35/45

0 0

200

400

c

600

800

h [mm] 1000

d

59 | Development of allowed deflection u2 at depths: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D

5.3.2

Torsion displacements in the uplift loading case

Bending moments in complete and incomplete grouting have different developments, as described in the previous chapter. Deformations resulting by the lining’s ovalisation will follow the development of bending moments to a high extent. Resulting bending moments by incomplete grouting slightly decreased at higher tunnel depths. On the other hand bending moments by complete grouting increased at higher depths, but were very low at shallow projections. Bending moments hardly depended on the concrete’s properties. Therefore, variations of the deformations for all considered concrete strength classes only depend on the concrete Young’s modulus. Figure 60 indeed shows such a development of the ovalisation deformations for a lining thickness of 675 mm. At this particular lining thickness ordinary concrete C35/45 is incapable of resisting the

70


resulting stresses by the introduction of thrust jack forces at the deepest tunnel projection. Hence no deformation has been given there. 0

5

10

15

20

25 Deformation u2 [mm]


0,7D C35/45

1,2D

C35/45 Incomplete grouting

2,9D 0 1 2 / 0 8 1 C

4,6D

Complete grouting C100/115

60 | Development of actual deflections due to ovalisation of tunnel in the uplift loading case for a lining thickness of 675 mm

A reduction of the lining thickness significantly reduces the lining’s total stiffness (EI) against deformations. Consequently the deformation for slender lining thicknesses increases dramatically. The small reduction of the bending moment by incomplete grouting that has been observed at very slender linings is of hardly any help. Figure 61 shows the principal. ] m m [ 2 u250 n o i t a 200 m r o f e d150 l a u t c A100

C100/115 C180/210

C35/45

C100/115 C35/45 C180/210

50 Thickness h [mm]

0 0

200

400

600

800

1000

61 | Deflections by complete and incomplete grouting at overburden of 19,3 m (1,2D)

The dimension available for torsion in the tunnel’s axial direction (Bt) is set to one segmental width again. A safety factor for the resistance against cracking by torsion in the trumpet effect is defined as the permissible value of deformation u2 divided by the actual occurring deformations due to the uplift loading case. Figure 62 contains the development of this safety factor for incomplete grouting at a depth of 19,3 m (1,2D). Apparently the more flexible behaviour of thin linings (resulting in high permissible deformations as presented in Figure 59b) is unable to compensate the larger increment of the occurring deformations at these thicknesses (Figure 61). Consequently very large lining thicknesses are required to prevent cracking by torsion. This does not hold for ordinary concrete C35/45 only, which is unable to prevent cracking without applying additional rebars in the considered range of thicknesses anyway. However it does hold for the steel fibre reinforced concretes very and ultra high strength concrete. At greater tunnel depths the occurring ovalisation deformations by incomplete grouting slightly descend. At thick linings however the permitted value hardly changes (except for C35/45 at the deepest location). Consequently the minimum required lining thickness for torsion cracks by incomplete grouting descends at an increasing depth projection. Table 8 and Figure 64 show that development.


71

n o i s r o t r 4 o t c a f y t e 3 f a S

C180/210

2 C100/115

γ = 1

1


0 0

200

400

600

800

1000

62 | Safety factor for incomplete grouting against cracking by torsion for an overburden of 1,2D

Table 8 | Lower boundary conditions on the allowed lining thicknesses to prevent cracks by the trumpet effect C35/45 Depth

5.4

C100/115

C180/210

Complete

Incompl.

Complete

Incompl.

Complete

Incompl.

11,1 m (0,7D)

760

1255

435

780

255

565

19,3 m (1,2D)

890

1230

550

770

365

555

46,7 m (2,9D)

980

1115

583

700

375

510

74,2 m (4,6D)

1110

1170

560

655

295

460

Conclusions

Introduction of thrust jack forces

The thrust jack force from the tunnel boring machine increases directly proportional to the tunnel’s depth. An ascending lining thickness results in a slight enlargement of the tunnel’s diameter; this causes an increment of the resulting thrust jack force linear to the additional diameter, which is only relatively small compared to the increment of the lining thickness. High compressive concrete stresses occur under the thrust jack plates by the introduction of the jack forces. In case of the French thrust jack configuration tensile bursting stresses deeper in the segment below the jack plates are the only structural tensile stresses resulting from the introduction. Provided that the height of the thrust jack plates is linked to the lining thickness, the thrust jack force only slightly increases if the jack plate’s height grows. The resulting stresses however descend at a more extensive rate; hence the safety level increases at an ascending lining thickness. The required lining thickness in order to cope with the tensile bursting stresses in tangential direction is governing above the required thicknesses by the compressive stresses. In reinforced ordinary concrete it is assumed that reinforcement bars will be utilised to cope with the tensile forces, as a result the compressive stresses are governing for this material. Required thicknesses for very and ultra high strength concrete are significantly lower than for reinforced ordinary concrete. From a depth projection of approximately 50 m (3D) a larger lining thickness is required for C35/45 than the thickness of 1/20 D from the commonly applied rule of thumb. All required lining thicknesses by the introduction of thrust jack forces have been shown in Figure 63.

72


0

200

] m19,2 [ e n 27,4 i l e r t n e c m o r 54,8 2,9D f h t p e d l e n n 82,2 4,6D u T

400

600

800

_ compressive stress + tensile bursting stress

Thickness h [mm]

+

C35/45 unreinforced

_

_

+

C180/210

C35/45

+

_

1000

C100/115

63 | Lower boundary conditions on the allowed lining thicknesses by the introduction of thrust jack forces

Torsion by the trumpet effect

If tunnel segments leave the protection of the tunnel boring machine’s shield, they’re loaded by the grout pressure. As a result the initial circular shape will ovalise, causing torsion in these tunnel segments. Stresses by torsion result in principal tensile stresses. Stresses by the introduction of thrust jack forces positively affect the permissible torsinal shear stress up to a certain depth. The ovalisation deformations are defined by the tunnel’s diameter alteration during complete or incomplete grouting. These deformations decrease at a growing tunnel depth. If thinner linings are applied the resulting deformation will increase dramatically. Therefore the safety against cracking by torsion increases at an ascending lining thickness. Required lining thicknesses to prevent cracking are very high for all considered concrete strength classes. If the standard lining thickness of 1/20 D is applied for ordinary concrete cracking will occur if no additional reinforcement is used to prevent this. Figure 64 gives all boundary conditions for the required lining thickness in order to prevent cracking by the torsional trumpet effect in case of both complete and incomplete grouting. 0


200

400

600

800

1000


(ic) Incomplete grouting (c) Complete grouting 0,7D 1,2D

(c)

(ic)

2,9D (c) (c) 4,6D

C180/210

(ic)

(ic) C100/115

C35/45

64 | Lower boundary conditions on the allowed lining thicknesses to prevent cracks by the trumpet effect


73

74


Chapter 6

Evaluation lining thickness reduction 6.1

Introduction

Relations between several failure mechanisms and the required lining thickness for shield driven tunnels were investigated in the previous chapters. Each individual mechanism resulted in a boundary condition on the required lining thickness in order to cope with the forces or bending moments from that mechanism. Four different depth projections for the potential tunnel of future motorway A13/16 in Rotterdam were considered. This chapter will bring all boundary conditions on the lining thickness together, in order to find a minimum required thickness that meets the requirements by all mechanisms. Section 6.2 will deal with this. Consequently will be visible what, considered a certain depth projection, the governing mechanism for the tunnel is. For long a discussion is going on about this question of governing behaviour in the design of shield driven tunnels. Perhaps this report may give some help to come to a conclusion in that discussion. More interesting however is to find a way to reduce the influence of the governing mechanism(s) and so to reduce the lining thickness even more. Section 6.3 will shortly discuss some alterations of the considered tunnel which might lead to this goal, focussed on the high strength steel fibre reinforced concretes.

6.2

Importance of the construction phase

In this report requirements on the lining thickness for the shield driven tunnel of motorway A13/16 were given by the bending moment for ring behaviour of an embedded tunnel lining, the bending moment for complete and incomplete grouting, the tensile and compressive stresses by the introduction of thrust jack forces and the principal tensile stress by torsion in the segments during the grouting phase. Figures 65 to 67 show the collection of boundary conditions on the lining thickness for the concrete strength classes C35/45, C100/115 and C180/210 respectively. If the design of a shield driven tunnel’s lining thickness is based on the strength related boundary conditions, it is restricted by the conditions for incomplete grouting, the introduction of thrust jack forces and the embedded ring behaviour for all concrete strength classes considered. A so-called ULS area is presented in the diagrams, restricted by these boundary conditions. In ordinary concrete reinforcement bars have to be applied to take care of the tensile stresses by the introduction of thrust jack forces. The bending moment by incomplete grouting is governing at shallow tunnel depths. At deeper projections the lining thickness is dictated by the introduction of thrust jack forces. For ordinary concrete a lower boundary results from calculations on the structure’s embedded ring behaviour as well at very large depths. However, cracks by torsion are still likely to occur within these ULS areas. In ordinary concrete C35/45 cracks will show up at lining thicknesses below approximately 1.100 mm. Very high strength concrete C100/115 cracks at thicknesses lower than 700-800 mm. Finally C180/210 will crack below approximately 500 mm. The implications of governing behaviour by the mentioned mechanisms will be discussed in this section.

75

0

200

400

600

800

1000


j a c j a c k s t e n k s s . c o g m l . r o p r p . m u

19,2 0,7D ] m27,4 1,2D [ e n i l e r t n e c m54,8 2,9D o r f h t p e d l e n n 82,2 4,6D u T

t c o m p l.

l . p m o c n i n o i r s t o

o c n i t u r o g

r in g b e ha v i o ur

t o r s i o n c o m p l.

65 | Boundary conditions on lining thickness for C35/45 0

200

400

600

800

1000


19,2 ] m27,4 [ e n i l e r t n e c m54,8 o r f h t p e d l e n n 82,2 u T

0,7D 1,2D

j a c k s j a c t e n k s s .

c o m p r .

2,9D

l . p m c o i n t u r o g

t o r s i o n c o m p l.

g r o u t c o m p l.

4,6D

l. p m o c in n io r s to r v i o u e h a b g r in

66 | Boundary conditions on lining thickness for C100/115 0

200

400

600

800

1000


19,2

j a c k ja s t c e k n s s c .

] m27,4 [ e n i l e o r t m n p e r . c m54,8 2,9D l. o r p f m h t o c p e t u d l ro e g n n 82,2 4,6D u T

t o r s i o n c o m p l .

. p l m o i n c u t o g r l .

p m o c i n n io r s t o

r u o i v a h e b g ni r

67 | Boundary conditions on required lining thickness for C180/210

An economic design of a certain structure is commonly described as that the structure’s construction phase may not be governing in stead of its serviceability phase. The serviceability phase is mainly included by the bending moment in ring behaviour of the embedded lining. This bending moment however results in an upper boundary condition of the required lining thickness. In other words: only in case of large lining thicknesses this element of the serviceability phase will be governing. A better description of the most economic design of a shield driven tunnel should therefore focus on the smallest possible lining thickness in the construction phase.

76


6.2.1

Torsion by trumpet effect governing

Torsion by the trumpet effect in the tunnel segments during the construction phase leads to very large required lining thicknesses according to the presented diagrams. The required lining thicknesses for reinforced ordinary concrete C35/45 never becomes as low as the value of 1/20 D (750 mm) given by the commonly applied rule of thumb for the lining thickness of shield driven tunnels. Therefore it appears that cracks by the trumpet effect will appear in the tunnel segments if the thickness of 1/20 D is still applied. The use of reinforcement bars in the expected cracking zones may significantly reduce the development of large crack widths. On the other hand torsion by the trumpet effect only occurs in an extremely small period of time compared to the tunnel’s full life span. Only if the considered ring is pushed out of the tunnel boring machine this type of torsion occurs. The part out of the TBM ovalises by the pressures from the grout, although deformation on the part within the machine is prevented by the newer ring which is not loaded at all. As the excavation process moves on the newer ring will ovalise as well, that implies that the deformation differences and therefore torsion fade away. Subsequently the cracks formed will be compressed and might completely close. Torsion is a result of a forced deformation of the considered ring. Cracks resulting from such mechanism are damage, but do not lead to a direct loss of the tunnel’s structural safety and thereby to potential failure. Therefore the question should be raised whether or not cracks by the trumpet effect are acceptable. The diagrams in Figures 65 to 67 that show by means of the ULS area that this leads to a significant reduction of the required lining thickness and cost saving. Of course one should remember that damage caused by cracks may lead to a reduction of the tunnel’s durability over time. Some relatively simple adjustments of the tunnel’s construction process might lead to a reduction of crack formation by torsion. For instance if more rings are located within the TBM shield the ovalisation deformations are spread over multiple rings and consequently cracks are reduced. Nevertheless this implies an enlargement of the shield’s length that results in higher thrust jack forces according to section 5.2.1. For ultra high strength concrete an increase of this force is no problem at all, since only at very deep projections the thrust jack force is governing. A large grow of the thrust jack force might lead to governing behaviour of this force for ordinary concrete at shallower depths. In practice so-called adjusters are occasionally used as well. These steel trusses are placed within a new ring to prevent it from deforming up to a certain level. Consequently the ovalisation deformations are gradually introduced to the tunnel segments in multiple rings. If the full deformation is present the adjusters are removed. Finally an optimised grouting regime can be used to prevent ovalisation of the tunnel. For that purpose grout pressures have to be varied along the tunnel’s perimeter. If the tunnel intents to deform as a “lying egg” higher grout pressures are applied on the sides in order to force it into its original circular shape again.

6.2.2

Grouting phase and introduction of thrust jack forces governing

Provided that torsional cracks are prevented or else are considered as an acceptable case of damage, other failure mechanisms will be governing for the minimum lining thickness. In case of shallow tunnels this holds for the bending moment by incomplete grouting for all considered concrete strength classes. For deeper tunnel projections the mechanism for introduction of thrust jack forces will lead to the required thickness. For reinforced conventional concrete it is assumed that reinforcement bars will take care of the tensile stresses introduced by the thrust jack forces. Hence the compressive stresses under the thrust jack plate are governing. For steel fibre reinforced concrete the tensile strength from the concrete itself should cope with the tensile stresses; hence now the line in the diagrams for tension by the thrust jack forces is governing in deep tunnels. At very large depths the normal ring force reaches such high values that ordinary concrete with the considered amount of steel reinforcement is unable to generate a bending moment capacity that


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reaches the required safety level. Subsequently at great depths the ring behaviour of an embedded lining in ordinary concrete results in a lower boundary condition for the lining thickness. 0

200

400

600

800

1000



0,7D 1,2D

C35/45 2,9D C100/115 C180/210 4,6D

68 | Effect of t he combined boundary conditions on the required lining thicknesses

Figure 68 shows the combined ULS areas and minimum required lining thicknesses for all concrete strength classes. From the diagram it obviously follows that it is now impossible to agree on one lining thickness only that defines the required governing thickness for all arbitrary depth projections. For ordinary concrete C35/45 however the range is relatively small up to an overburden of 50 m (3D). Here the required thickness varies from 1/21 D to 1/26 D, all very close to the value from the rule of thumb 1/20 D. At large depths very and ultra high strength steel fibre reinforced concretes C100/115 and C180/210 only require very slender linings. According to incomplete grouting and the introduction of thrust jack forces for instance, concrete C180/210 only requires a lining of 255 mm (1/58 D) at a depth of 61 m (4D). At the most shallow projection however no less than an additional 215 % of concrete lining is required, resulting in a thickness of 805 mm, or 1/19 D. The application of shield driven tunnels in ultra high strength concrete at great depths only seems essential. However in order to reach these depths the tunnel should lead through more shallow ground as well. So how should a lining thickness of approximately 255 mm fit in there? Or is the creation of very slender tunnel structures from ultra high strength steel fibre reinforced concrete virtually impossible? These questions will be dealt with later on in this chapter.

6.2.3

Ring behaviour of embedded tunnel

Calculations on the ring behaviour of the embedded tunnel lining showed a descending safety ratio at increasing lining thicknesses (except for ordinary concrete C35/45 at very large depths). As a result the boundary condition of the lining thickness by this mechanism is the requirement on the maximum allowed thickness. The bending moment by incomplete grouting for ultra high strength concrete C180/210 however requires such high lining thicknesses at shallow depths, that these thicknesses exceed the value for embedded ring behaviour. This actually implies that no tunnel from this material can be constructed at these shallow depths if incomplete grouting takes place. Interpolation within the diagram from Figure 67 demonstrates the minimum required depth is 22m (1,4D) now. For a lining of very high strength concrete C100/115 the boundary conditions by incomplete grouting and the introduction of thrust jack forces on one side and embedded ring behaviour on the other side nearly meet at the most shallow and most deep tunnel projections. As a result this material may be applied in the considered depths only if no changes are made. At more shallow depths a situation as described for C180/210 occurs, where the bending moment by grouting requires too large a thickness, at deeper projections the introduction of thrust jack forces requires a too high lining thickness. In Chapter 3 it has been stated that a relatively high concrete Young’s modulus would be applied in the calculations on embedded ring behaviour, the modulus for short-term uncracked concrete. If this

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failure mechanism turned out to be governing for the tunnel’s structure, the value of the modulus could be reconsidered. Hence the lower value of cracked concrete might be used for instance; resulting in a reduction of the actual bending moments and therefore in an increment of the lining’s bending moment safety level. The calculations on the embedded ring behaviour returned an upper boundary condition of the lining thickness. So as a consequence of the lower Young’s modulus for concrete higher lining thicknesses may be applied according to this failure mechanism. However the application of thicker linings does not lead to a more economic design. Therefore it doesn’t seem to make sense to reduce the resulting bending moments by ring behaviour in order to retrieve an enlargement of the possible range of tunnel depths and a reduction of the lining’s thickness. This implies other adaptations of the lining’s design in order to obtain a reduction of the ring behaviour’s bending moment, as for instance the extension of the number of tunnel segments per ring, are pointless according to the calculations in this report. Subsequently the lower boundary conditions of the required lining thickness should be shifted to the left in the diagrams. This therefore applies to the bending moment by incomplete grouting and the introduction of thrust jack forces. The goal may be achieved in three ways: by an enlargement of the lining’s capacities, by a reduction of the reaction forces or by a combination of both. The next section will demonstrate some possible solutions.

6.3

Improvement of behaviour steel fibre reinforced concrete in tunnels

The contribution of steel fibres in very and ultra high strength steel fibre reinforced concrete to the lining’s bending moment capacity is very small. The normal ring force is the main reason for the large capacity compared to reinforced ordinary concrete at great depths. This has been concluded in Chapter 3 already. Now however this phenomenon is of major importance to the large variations between the required lining thicknesses at very shallow depths and at high tunnel depths for the concretes C100/115 and C180/210. As mentioned before the difference between the minimum lining thickness at great depth and the lining thickness at the most shallow depth holds 215 % for ultra high strength concrete. As a result several improvements of the lining’s bending moment capacity at shallow depths and reduction of the resulting forces are desired. In this section both will be taken into consideration, in particular focussing on the steel fibre reinforced very and ultra high strength concretes. At first an increment of the bending moment capacity is considered in section 6.3.1, next reductions of the resulting forces and bending moments in the grouting phase and due to the introduction of thrust jack forces are discussed in sections 6.3.2 and 6.3.3 respectively.

6.3.1

Addition of steel bar reinforcement

The fact that reinforcement bars in a lining of ordinary concrete C35/45 are a great advantage for the development of its bending moment capacity compared to very and ultra high strength steel fibre reinforced concrete, has been extensively discussed in Chapter 3 already. As soon as the tunnel depth increases, the normal ring forces rise and the capacities of C100/115 and C180/210 will finally exceed the one of reinforced concrete C35/45. At shallow depths however the bending moment capacity of the steel fibre reinforced concrete materials is insufficient to obtain a significant reduction of the lining thickness. Application of reinforcement bars in the cross-sections of these materials as well, might be a legitimate option to increase the capacity and tackle the problem. In order to test this proposition an example will be discussed where additional reinforcement is applied to the original steel fibre reinforced concrete C180/210. Regardless of the lining thickness 14 rebars per full segmental width, with a diameter of 20 mm (14 ∅20), are used at the bottom as well as at the top of the cross-section. Due to the dense concrete matrix a concrete cover of only 10 mm is sufficient [8].


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The discussion on the calculation method of the bending moment capacity showed the contribution by the rebars is most effective at a relatively thick lining in case of ordinary concrete. Only at large lining thicknesses the tensile force in the steel bars reaches its maximum level. Besides, the lever arm for its contribution is defined by the lining thickness as well, resulting in a larger bending moment capacity. In case of ultra high strength concrete another additional fact occurs. By now it’s known that the maximum tensile strain from the stress-strain diagram for ultra high strength concrete is easily exceeded in thick linings and shallow tunnel depths due to the small concrete compressive zone. The same applies to the steel rebars. If the strain at the location of the reinforcement bars exceeds the ultimate strain for steel because of a very small concrete compressive zone, the complete steel force by this reinforcement layer would disappear in the bending moment capacity calculation. A lower maximum compressive strain results in a larger height of the concrete compressive zone and therefore returns the steel force to the cross-section. Consequently the bending moment capacity still shows an increase at an ascending thickness. Figure 69a clearly shows this phenomenon. In Figure 69b the increment in terms of percentage of the ultimate resisting moment due to the reinforcement bars is displayed. ] m N k [ t n2000 e m o m g1500 n i t s i s e r 1000 e t a m i t l U 500

1,2D reinforced 0,7D reinforced

1,2D unreinforced 0,7D unreinforced Thickness h [mm]

0 0

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] % [ 5 4 / 580 3 C o t e v60 i t a l e r t n40 e m o m s20 u l p r u S 0

0,7D

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Thickness h [mm] 0

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b

69 | Utimate resisting moment with and without additional rebars: a) absolute values; b) surplus by rebars

Even though Figure 69b shows that the effect of the reinforcement bars is not at its optimum at very thin lining thicknesses, the actual increment is sufficient to result in a significant lower required lining thickness by the bending moments in the grouting phase. The lining thickness required by embedded ring behaviour is influenced by the increased bending moment capacity as well. The required safety level is now met at each considered lining thickness and depth in case ultra high strength concrete with reinforcement bars is applied. Hence the original upper boundary condition is no longer visible and the applicable ULS area is extended significantly. In Figure 70 both the original unreinforced and reinforced situations are presented. Higher amounts of reinforcement have been included in the figure as well. Obviously huge savings of the required lining thickness can be realised by the addition of steel rebars. In case 56 bars per full segmental width are used, a lining thickness of only 265 mm or 1/56 D is sufficient. At larger depth projections less reinforcement is needed in order to obtain the same lining thickness. As a consequence similar tunnel segments can be produced with a reinforcement percentage depending on the projected depth of each individual segment.

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0

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0 0 0 reinforcement: Ø 2 Ø 2 Ø 2 5 6 4 2 2 8

19,2 j a c ] k s 27,4 m [ t e e n n s i l . e r t n e c m54,8 2,9D o r f h t p e d l e n n 82,2 4,6D u T

2 0 Ø 1 4

f . i n e r u n

r u d oi e v c a r h of e n b i e g r n ni u r

70 | Influence of reinforcement bars on the required lining thicknesses for C180/210

The improvement of the lining’s bending moment capacity by the application of reinforcement bars nearly reduced the required lining thickness to the original minimum value of 255 mm from the previous section. At the most shallow projection however very high amounts of reinforcement are required by the bending moment for incomplete grouting. A reduction of the actual resulting forces by this mechanism might obtain an even better and more considerable result. The following section gives an example of that solution.

6.3.2

Reduction of uplift force in grouting phase by ballast

The maximum bending moment in a tunnel lining in freshly injected grout is determined by the extent in which the lining is pressed into the fictitious top support by the upward floating component of the grout pressure (combined hydrostatic water pressure and grout-lining contact pressure). The fictitious top support imitates a local pressure increase at the top of the tunnel by interlocking of grout in the tail void. Both complete and incomplete grouting were considered in the calculations. Complete grouting requires, especially at shallow tunnel depths, a significant lower bending moment capacity than incomplete grouting. The latter however may occur for various reasons. Considering the present grout injection techniques it is assumed that incomplete grouting can not be prevented easily and therefore will take place. A reduction of the resulting bending moment by grouting should be achieved by a reduction of the top support’s stiffness or by a reduction of the total upward floating force. The most easy solution of both is to reduce the upward floating force. It is not the gross floating force (roughly by the hydrostatic development of the water pressure), but the net floating force that can be altered. The weight of the tunnel and (to a limited extent) the shear stresses in the grout, act against the upward floating force and reduce its final net value. An enlargement of the total tunnel weight therefore prevents the lining to be pushed into the top support by a high upward floating force. Additional mass in the tunnel segments themselves is inappropriate; this adds up to the costs for transport of these prefabricated segments. Moreover the additional weight is required in the grouting phase (construction phase) only, not during the serviceability phase. To restrict floating of the tunnel during the grouting phase mass of a temporary kind is thus required. Straightforward ballast in the tunnel’s interior, for instance sand fill, seems most suitable and most simple to use. At the same time as the grout is injected sand is now dumped at the bottom of the tunnel. If the grout has hardened to such degree that the additional mass is no longer needed, sand can be moved from the back of the fill to the front of the tunnel boring machine to serve as ballast for the newest rings once more. During construction of shield driven tunnels some fill is created by sand or a part of the future internal structure already (see Figure 71). The resulting levelled surface provides a carriage way for dump trucks to transport the excavated soil from the TBM out of the tunnel. By not only defining the


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height of this fill on pure functional grounds, but on temporary structural grounds as well, a reduction of the required lining thickness may be achieved.

71 | Levelled tunnel interior during construction Green Heart Tunnel

The software application LDesign, that was used to perform calculations on the grout uplift loading case, has no function to add additional mass to the tunnel. The final net uplift force however is reduced by the weight of the tunnel lining. Therefore the specific gravity of the concrete lining (the specific gravity for C180/210 holds ρC180/210 = 28 kN/m³) is increased in new calculations to imitate a sand fill (specific gravity of ρsand = 18 kN/m³) at similar lining thicknesses. By iteration a specific gravity for the concrete lining is searched for that meets the required bending moment safety level of 1,5 at a specific lining thickness. In other words: for each lining thickness the lining’s specific mass is altered up to the moment that the resulting maximum bending moment equals the bending moment capacity divided by the required safety level of 1,5. The additional weight of the tunnel lining is now easily converted to a required height of the sand fill from the top of the circular tunnel tube. For the shield driven tunnel of motorway A13/16 several required sand heights have been determined for ultra high strength concrete with and without additional steel bar reinforcement (14 ∅20). The most shallow tunnel depth projections only have been considered. The required lining thickness should be reduced to the minimum value from the original calculations as much as possible. Therefore only values close to 255 mm have been applied, hence 200, 300 and 400 mm. The maximum required sand fill has a height of 5,8 m for these cases (most shallow depth, 200 mm without rebars). Figure 72 displays the required sand fills for all situations. In case a lining thickness of 300 mm steel fibre reinforced concrete C180/210 with the additional reinforcement bars from the previous section is used, temporary sand heights of 4,0 and 2,7 m are necessary at the most shallow 11,1 m (0,7D) and standard 19,3 m (1,2D) tunnel depth projections respectively. 200 mm

300 mm

] m [ l l i f d n Reinforced a Unreinforced S 14 Ø20

0,7D

400 mm



6

6

6

4

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1,2D

0,7D

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0,7D

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1,2D

0,7D

1,2D

0,7D

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72 | Required sandfill to resist bending moment by incomplete grouting for C180/210

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A proper combination of additional rebars in the tunnel lining in order to increase the bending moment capacity and a (temporary) sand fill in order to reduce the upward floating forces in the grouting phase, is able to contribute to the reduction of the required lining thickness of steel fibre reinforced ultra high strength concrete C180/210 in shallow tunnels. Now the governing situation of the construction phase is reduced for this type of shield driven tunnels. The application of steel reinforcement bars in the tunnel segments cause an enlargement of the safety level in the tunnel’s serviceability phase as well.

6.3.3

Reduction of thrust jack force

The diagrams in section 6.2 proved that the introduction of thrust jack forces leads to governing required lining thicknesses at great depths. For reinforced ordinary concrete C35/45 this occurs from a soil cover of approximately 38 m (2,3D) already. From this point the required lining thickness grows nearly linear to the depth’s increment. For steel fibre reinforced high strength concretes the tensile bursting stresses in the segment are responsible for this. In case of reinforced ordinary concrete it is assumed that these tensile forces are carried by the reinforcement completely, as a result the compressive stress under the thrust jack plates is governing for this material. An enlargement of the jack plate’s surface reduces both the resulting tensile and compressive stresses. The previous chapter proved stresses in tangential direction are governing for the applied plate dimensions; hence the plate’s width should be increased firstly. It is important however to remember that the force that is induced in the plates by two thrust jacks from the tunnel boring machine, must be able to uniformly spread over the full width and height of the plates in order to introduce a constant pressure in the concrete. If the width of the plate increases, this becomes more and more difficult. Nevertheless suppression of the symptoms of thrust jack force introduction is not the only way to solve the problem. In the study by De Rijke [12], which has been used to retrieve the actual trust jack forces in this report, also contains some advises to reduce these forces. Measures vary from a reduction of friction by the cutting wheel and the TBM’s shield to reduction of the required axial length of this shield. For more details on possible reductions of this failure mechanism is referred to De Rijke’s report. A reduction of the thrust jack force results in a more or less linear proportional reduction of the required lining thickness.

6.4

Conclusions

Torsion by the trumpet effect during the construction phase leads to damage by cracks. If the rule of thumb for a lining thickness of 1/20 D is used, cracks will appear in reinforced ordinary concrete C35/45. Even if damage by torsion is neglected the construction phase of the shield driven tunnel is very important for the determination of the required lining thickness. Incomplete grouting in shallow tunnels and the introduction of thrust jack forces in deep tunnels now dictate the required thickness. The rule of thumb 1/20 D leads to a safe tunnel lining in reinforced ordinary concrete up to a depth of approximately 50 m (3D). No simple rule of thumb is present for the required lining thickness of steel fibre reinforced very or ultra high strength concrete at each arbitrary tunnel depth projection. The required thickness of tunnel linings in plane ultra high strength steel fibre reinforced concrete is lowest in deep tunnels. Reinforced ordinary concrete behaves best at a depth projection of 2D. The best application of plane very high strength concrete is in between both previous materials. In order to reach high depths for tunnels in very and ultra high strength concrete shallow depths should be passed as well. Reinforcement bars should be added to these steel fibre reinforced materials to make construction of shallow tunnels possible as well. Tunnel linings with a thickness from only 1/56 D are possible now.


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In all shallow tunnels the required lining thickness by incomplete grouting is very large. This effect can be repressed by adding additional temporary weight to the tunnel (for instance by sand fill inside the tunnel) in order to reduce the total uplift force from the water and grout pressures.

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Chapter 7

Conclusions and recommendations Conclusions

The construction phase of shield driven tunnels is of major importance for the determination of a required lining thickness. For shallow tunnels the bending moment by incomplete grouting dictates the tunnel’s strength and therefore the required thickness. In deep tunnels concrete stresses by the introduction of thrust jack forces are governing. In all considered concrete strength classes cracks are likely to occur due to torsion in tunnel segments leaving the tunnel boring machine (known as the trumpet effect) if the lining thicknesses dictated by the tunnel’s strength are applied. Also in case the thickness of a lining from reinforced ordinary concrete is determined by the common rule of thumb 1/20 D, torsion cracks occur. Therefore the tunnel’s construction phase can not be neglected in the design process. Ring behaviour of a tunnel embedded in soil during the serviceability phase does not result in a governing required lining thickness. The safety level for the bending moment in this failure mechanism decreases if the lining thickness increases. As a result the requirement found for the lining thickness is an upper boundary condition. Hence the applied thickness may not exceed this relatively high value. Consequently alterations of the tunnel’s design in order to reduce the bending moment in embedded ring behaviour only do not result in a lower required lining thickness. Measures to decrease the governing behaviour of the construction phase however, do result in thinner linings. Steel fibres in very and ultra high strength concrete hardly contribute to the lining’s bending moment capacity. If the lining thickness increases this minor contribution decreases even more. Consequently slender tunnel linings in ultra high strength concrete with steel fibre reinforcement only are most favourable at very high tunnel depths, where high normal ring forces are present. At a depth of approximately 62 m (5D) concrete C180/210 only requires a lining thickness of 255 mm (1/58 D). At very shallow depths however thicknesses of over triple this minimum value are necessary. The same happens to very high strength concrete C100/115 that requires a lining thickness of only 375 mm at a depth of 46 m (2,9D). As a result no simple rule of thumb can be set for the required lining thickness of a shield driven tunnel in steel fibre reinforced very or ultra high strength concrete at each arbitrary depth. The rule of thumb 1/20 D for the lining thickness of reinforced ordinary concrete proved to be on the safe side up to a tunnel depth projection of approximately 3D for the considered tunnel. Tunnel linings in very or ultra high strength concrete can be constructed only if reinforcement bars are added. The wide scattering of required lining thicknesses over the depth of a tunnel in plain steel fibre reinforced concrete, makes it impossible to construct tunnels of these materials in both more shallow and deep grounds. Reinforcement bars increase the lining’s bending moment capacity to a high extent and therefore reduce the governing behaviour of incomplete grouting. In the considered tunnel with an internal diameter of 14,9 m a lining thickness of only 265 mm (1/56 D) has been

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presented for a heavily reinforced tunnel segment in ultra high strength concrete at each depth projection. Temporary measures can be taken to reduce the governing behaviour of incomplete grouting during the construction phase of a shield driven tunnel. Reduction of the net uplift floating force by use of additional mass in the tunnel has a positive effect on the required lining thickness. Consequently tunnel linings in ultra high strength steel fibre reinforced concrete with additional reinforcement bars are able to reach thicknesses below 250 mm or 1/60 D. Recommendations

More (temporary) measures might be possible to reduce governing behaviour and damage due to mechanisms in the construction phase of shield driven tunnels. This holds especially for torsion in the tunnel segments that leads to cracking in all considered concrete materials and most lining thicknesses. Research on how to prevent this topic can result in a significant decrease of damage in tunnel linings during the construction phase. Studies on modelling of the behaviour of shield driven tunnels in fresh grout should focus on solutions to prevent governing behaviour of this mechanism. Especially in shallow tunnel depths prevention of high resulting bending moments by grouting is needed in order to reduce the required lining thickness. This study showed that the minimum required lining thickness for very and ultra high strength concrete very much depends on the tunnel’s depth projection. Only a very large tunnel diameter has been investigated up to now. Likewise research on the applicability of these concrete materials in shield driven tunnels with smaller diameters is needed. Comparison of both studies might show that the application of shield driven tunnels in ultra high strength concrete is perhaps more favourable for large or small tunnels. Only strength related mechanisms have been studied in this report. Requirements on the deflections have been omitted. Calculations on torsion in the grouting phase however showed that very large deflections occur during incomplete grouting of very slender tunnel linings. Adaptations of the lining’s cross-section for slender ultra high strength concrete in order to generate stiffer behaviour might therefore be needed to meet the requirements. Hence more detailed research on quality constraints, like deflections, durability and fire safety, should be performed before a shield driven tunnel in ultra high strength concrete is constructed.

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References 1.

Acker, P., C. Baloche et al., Ultra High Performance Fibre-Reinforced Concretes, Interim Recommendations. Bagneux Cedex (France): Setra and AFGC, January 2002

2.

Bambridge, C., Focusing on fibres: CTRL experience. In: Tunnels & Tunnelling International, March 2006

3.

Blom, C.B.M., A.P.M. Plagmeijer, Boren van tunnels met niet-ronde vormen . In: Cement № 6 2003, pages 74-80 (Dutch)

4.

Blom, C.B.M., Concrete linings fors hield driven tunnels , lecture note. Delft: Delft University of Technology, March, 2006

5.

Blom, C.B.M., Design philosophy of concrete linings for tunnels in soft soil , dissertation. Delft: Delft University Press, December 2002

6.

Bruijn, H.J. de, HS², Literatuur- en voorstudie: hogere sterkte beton nader belicht . Utrecht: Delft University of Technology and Holland Railconsult, December 2005 (Dutch)

7.

Burgers, R., Non-linear FEM modelling of steel fibre reinforced concrete for the analysis of tunnel elements in the thrust jack phase , thesis. Delft: Delft University of Technology, September 2006

8.

CRC description. Hjallerup (Denmark): CRC Technology Aps (download at www.crc-tech.com)

9.

Hollander, J. den, Technical feasiblity study of a UHPC tied arch bridge , thesis. Delft: Delft University of Technology and Ingenieursbureau Gemeentewerken Rotterdam, May 2006

10. Kaptijn, N., Zeerhogesterktebeton, Toepassingen , handout. Utrecht: Rijkswaterstaat Bouwdienst DIO, 2002 (Dutch) 11. Pruijssers, A.F. et al., Toetsingsrichtlijn voor het ontwerp van boortunnels voor weg- en railinfrastructuur L500. Gouda: Centrum Ondergronds Bouwen, September 2000 (Dutch) 12. Rijke, Q.C. de, Innovation of stress and damage reduction in bored tunnels during construction based on a shield equilibrium model , thesis. Utrecht: Delft University of Technology and Holland Railconsult, February 2006 13. Rijkswaterstaat, SATO Deel 5 Tunneldetails. Utrecht: Bouwdienst Rijkswaterstaat (Dutch) 14. Rijkswaterstaat, Startnotitie Rijksweg A13/16 Rotterdam . Rotterdam: Ministerie van Verkeer en Waterstaat, Directoraat-Generaal Rijkswaterstaat Zuid-Holland, November 2005 (Dutch) 15. Rijkswaterstaat, Technische haalbaarheidsstudie tunnelverbinding A6/A9. Utrecht: Ministerie van Verkeer en Waterstaat, Directoraat-Generaal Rijkswaterstaat, September 2002 (Dutch) 16. Schumacher, P., Rotation Capacity of Self-Compacting Steel Fiber Reinforced Concrete , dissertation. Delft: Delft University of Technology, November 2006 17. Slenders, B.M.A., Modellering van boortunnels , thesis. Utrecht: Delft University of Technology and Projectorganisatie HSL-Zuid, January 2002 (Dutch) 18. Waal, R.G.A. de, Steel fibre reinforced tunnel segments . Delft: Delft University Press, January 2000 19. Walraven, J.C., Ultra-hogesterktebeton: een material in ontwikkeling . In: Cement № 5 2006, pages 5761 (Dutch)

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Table of figures Figures 1 | Volume of concrete increases quadratic proportional to tunnel diameter 2 2 | Visualisation of this report’s structure 4 3 | Dimensions and elements of the tunnel lining 6 4 | Thrust jack configurations: a) German method; b) French method 7 5 | Uneven support of tunnel segments for: a) German configuration; b) French configuration 7 6 | Concrete materials: significant increase of density and homogeneity for BSI 8 7 | Pedestrian Bridge of Peace in Seonyu, South Korea 9 8 | Surroundings of motorway A13/16 in the north of Rotterdam 10 9 | Fitting two directions in one tunnel tube. a) Two tunnels with large useless free areas; b) One tunnel with less free space 11 10 | Cross-section of tunnel for motorway A13/16 with personal cars only (scale 1:150) 12 11 | Schematic representation alignment of the tunnel for motorway A13/16 12 12 | Uniform pressure leading to normal ring force tunnel 17 13 | Relation between rotations and bending moments in a Janßen joint 17 14 | Beam elements (and nodes) in the centre line of a tunnel segment 19 15 | Determination of spring stiffness for ring joints 19 16 | Vertical stresses in uniform soil continuum along a tunnel 21 17 | Orientation of vertical and horizontal versus radial and tangential soil loads 22 18 | Moment distribution in validation BRT-model and model from this study 25 19 | Moment distribution in validation model LDesign and model from this study 26 20 | Complete framework model for the shield driven tunnel of this study 28 21 | Maximum bending moment by ring behaviour for depth of 0,7D. a) absolute values; b) surplus relative to maximum moments for ordinary concrete C35/45 30 22 | Normal ring force by ring behaviour for depth of 0,7D 30 23 | Effect of tunnel depth projection with a lining thickness of 675 mm. a) normal ring force; b) tangential bending moment 31 24 | Maximum bending moment by ring behaviour for soil overburden of: a) 1,2D; b) 2,9D; c) 4,6D 31 25 | Calculation definition ultimate resisting moment 32 26 | Stress-strain diagram of reinforcement steel 33 27 | Stress-strain relations of ultra high strength concrete: a) strain hardening; b) and strain softening 34 28 | Stress-strain diagrams steel fibre reinforced concretes a) C100/115; b) C180.210 35 29 | Ultimate resisting moment at a depth of 0,7D 37 30 | Development ultimate resisting moment at various depth projections for lining thickness of 675 mm 37 31 | Ultimate resising moment at depth projections of: a) 1,2D; b) 2,9D; c) 4,6D 38 32 | Development of safety factor Mu/Mmax at depth projections of: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D 39 33 | Upper boundary conditions on the allowable lining thicknesses in case of embedded ring behaviour (lower boundary for C35/45) 41 34 | Development ultimate resisting moment over the depth for 675 mm thick reinforced concrete C35/45 44 35 | Forces and definitions in the uplift loading case [5] 46

89

36 | Fictitious top support 48 37 | Bending moment by complete and incomplete grouting at a depth of 0,7D 49 38 | Development bending moment by complete and incomplete grouting over the depth for 675 mm C35/45 50 39 | Bending moment by complete and incomplete grouting at depths: a) 1,2D; b) 2,9D; c) 4,6D 50 40 | Bending moment safety factor for complete and incomplete grouting at depths: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D 52 41 | Lower boundary conditions on the allowable lining thicknesses in case of complete and incomplete grouting 53 42 | Development of bursting stresses. a) Compression trajectories; b) Latice analogy 56 43 | Tensile force in latice analogy for German thrust jack configuration 56 44 | Stress distribution in tunnel segments due to introduction of thrust jack forces in the: a) French and b) German thrust jack configurations [5, 18] 56 45 | TBM Shield Equilibrium Model with line load to determine thrust jack forces 58 46 | Development thrust jack forces over: a) Lining thicknesses for depth of 1,2D; b) Tunnel depth 59 47 | Definitions schematised stress distribution under a thrust jack plate 60 48 | Compressive stress under thrust jack plates for a depth of 1,2D 62 49 | Introduction of thrust jack forces over the depth. a) Compressive stress under thrust jack plate; b) Required lining thicknesses 62 50 | Spreading of compression trajectories in tangential and radial direction in tunnel segments 63 51 | Diagram of Iyengar for tensile bursting stresses due to introduction of thrust jack forces 63 52 | Radial and tangential tensile bursting stresses due to the introduction of thrust jack forces 64 53 | Introduction of thrust jack forces over the depth. a) Maximum tangential tensile bursting stresses; b) Required lining thickness per concrete strength class by bursting stresses 65 54 | Torsion in tunnel segments by deformation due to grout loading 66 55 | Rotation of tunnel segment in vertical plane 66 56 | Maximum allowed deformation u2 due to torsion only in tunnel segments 68 57 | Multi-axial stresses in twisted tunnel segments. a) Definition of stresses; b) Mohr’s cirle 69 58 | Allowed shear stress τtj at a depth of 1,2D 70 59 | Development of allowed deflection u2 at depths: a) 0,7D; b) 1,2D; c) 2,9D; d) 4,6D 70 60 | Development of actual deflections due to ovalisation of tunnel in the uplift loading case for a lining thickness of 675 mm 71 61 | Deflections by complete and incomplete grouting at overburden of 19,3 m (1,2D) 71 62 | Safety factor for incomplete grouting against cracking by torsion for an overburden of 1,2D 72 63 | Lower boundary conditions on the allowed lining thicknesses by the introduction of thrust jack forces 73 64 | Lower boundary conditions on the allowed lining thicknesses to prevent cracks by the trumpet effect 73 65 | Boundary conditions on lining thickness for C35/45 76 66 | Boundary conditions on lining thickness for C100/115 76 67 | Boundary conditions on required lining thickness for C180/210 76 68 | Effect of the combined boundary conditions on the required lining thicknesses 78 69 | Utimate resisting moment with and without additional rebars: a) absolute values; b) surplus by rebars 80 70 | Influence of reinforcement bars on the required lining thicknesses for C180/210 81 71 | Levelled tunnel interior during construction Green Heart Tunnel 82 72 | Required sandfill to resist bending moment by incomplete grouting for C180/210 82 73 | Tangential cross-section of the longitudinal joint (left) and an overview of the stresses in the reduced beam problem of Janßen 98 74 | Stresses in the Janßen joint in case the compressive yield strain has been exceeded (left) and the stress-strain relation for rotation in the joint (right) 100

90


75 | Cartesian coordinate system (x-y-z) and cylindrical coordinates system (r-θ -z) 105 76 | Distances in a Cartesian x-y and r-t-(θ) coordinate system for a point (ux , uy) 105 77 | Distributed loads on the element of the tunnel structure 106 78 | Required depth projection tunnel to prevent floating 108 79 | Forces and lever arms in the calculation 109 80 | Stress-strain diagram of concrete with tensile behaviour 110 81 | Stress-strain diagram reinforcement steel 111 82 | Determination equation by nodes of a linear element 111 83 | Adapted stress-strain diagrams in case material factors are excluded from the strength calculations 123

Tables Table 1 | Lining thickness over diameter ratio for Dutch shield driven tunnels 5 Table 2 | Classification of concrete strength classes [19] 8 Table 3 | Notation and values depth projection tunnel 29 Table 4 | Upper boundary conditions on the allowable lining thicknesses in case of embedded ring behaviour 40 Table 5 | Lower boundary conditions on the allowable lining thicknesses in case of complete and incomplete grouting 52 Table 6 | Calculation of axial thrust jack forces 58 Table 7 | Lower boundary conditions on the allowed lining thicknesses by the introduction of thrust jack forces 65 Table 8 | Lower boundary conditions on the allowed lining thicknesses to prevent cracks by the trumpet effect 72 Table 9 | Equations for the longitudinal Janßen joint 103 Table 10 | Partial safety factors for strength (resistance) according NEN 6720 and DIN 1045 neu 119 Table 11 | Partial loading factors NEN 6720 (safety class 3) and DIN 1045 neu 120


91

92


Appendices

Table of contents Appendix A Derivations A.1 Janßen joint A.2 Transformation of coordinate systems A.3 Uplift of embedded tunnel

95 97 105 107

Appendix B Ultimate resisting moment for steel fibre reinforced concrete

109

Appendix C Safety factors in shield driven tunnels

117

93

94


Appendix A

Derivations A.1 Janßen joint A.2 Transformation of coordinate systems A.3 Uplift of embedded tunnel


97 105 107

95

96


A.1 Janßen joint The longitudinal joint is an inhomogeneous part of the tunnel structure and therefore needs extra attention in the modelling process of ring behaviour for a shield driven tunnel. The joint transfers a bending moment and a normal force by contact; it’s unable to transfer tensile forces and opens in case of a relatively high bending moment. Modelling of a contact area usually means longer calculation times. If a solution is present to simplify this problem, it would be favourable for the framework analysis. A simplified solution has been presented by Janßen. The contact problem was reduced to the problem of a beam, which is unable to cope with any tensile stresses. The width and height in the tangential cross-section of the beam equals the contact height of the longitudinal joint. Opening of the joint is symbolised by formation of a “crack” in the complete tensile zone of this beam. The formation of this crack depends on the applied normal force and bending moment at that particular joint. The stiffness of the beam, also called Janßen joint, now represents the stiffness of the longitudinal joint. By applying this stiffness (being a function of the normal force and bending moment) to a non-linear rotational joint it’s possible to simplify the contact problem in a longitudinal joint in a framework analysis. This appendix discusses the derivation of Janßen’s method and computes some useful attributes of the Janßen joint in advance.

Derivation rotational stiffness of a Janßen joint The lining of a shield driven tunnel is loaded by a bending moment and a normal force. This holds for the longitudinal joint as well. As discussed before the stiffness of the Janßen joint depends on both the normal force and the bending moment. In order to compute the stiffness first the stresses in the joint are being examined in figure 73. The parameters used in this figure represent: lt = Height and width of the contact area M N R uM xu σM σN σ φ

= = = = = = = = =

Bending moment Normal ring force Reaction force Displacement at edge of the Janßen beam by the bending moment Height of the contact area still subjected to a compressive stress Stress by bending moment Stress by normal force σM + σN Rotation in the Janßen joint


97

N M lt

lt

φ uM lt

σN

– +

σM

– R

xu

–

σ

73 | Tangential cross-section of the longitudinal joint (left) and an overview of the stresses in the reduced beam problem of Janßen

Depending on the ratio between the bending moment and the normal force the joint will be opened or closed. Both situations are discussed separately. This will also be done for the opened situation when concrete behaves plastic.

Rotational stiffness of a closed longitudinal joint The rotational stiffness of a spring is defined by the equation: M cr = φ

(1)

From this equation the bending moment M is known. However the rotation φ still needs to be derived. By having a look at figure 73 this rotation is given by: 2u M φ= (2) lt

In order to determine the displacement uM , first of all the stress by the bending moment shown is needed: 6M σM = 2 (3) bl t Where: b = Width of the tunnel segment For the strain resulting from this stress it holds: σ εM = M Ec

(4)

Where: Ec = Young’s modulus of concrete Next the deflection uM is easy to determine: uM = εMl t

(5)

Substitution of equations (3), (4) and (5) into (2) gives the rotation by σM: 12M φ= E c bl 2t

(6)

Now substitution of equation (6) into equation (1) results in the constant rotational stiffness of a closed Janßen joint depending on the stiffness and lay-out of the contact area: E bl 2 cr = c t (7) 12

98


As soon as the maximum stress by the bending moment σM surpasses the constant stress by the normal force σN , the joint opens. Therefore equation (7) only holds if the following criterion is met: σM ≤ σN (8) The stress σN is given by: N σN = bl t

(9)

By substituting equation (3) and (9) into (8) it holds: Nl t M≤ (10) 6 In order to retrieve the rotation when opening of the Janßen joint first occurs, equation (10) is substituted into equation (6). Now the criterion for using equation (7) as the rotational stiffness of the Janßen joint may be written as: 2N φ≤ Ebl t As long as this criterion is met, the longitudinal joint is closed.

Rotational stiffness of an opened longitudinal joint In case the joint opens, a different approach is needed in order to determine the rotational stiffness. This determination starts with the fact that equilibrium of forces should be present: ΣN = 0 ⇔ N = R (11) The reaction force R is given by: σ bx u (12) R= 2 By combining both equations (11) and (12) it holds for the height of the contact area still under pressure: 2N xu = (13) σ b Equilibrium of bending moments is required as well: x ⎞ ⎛ l ΣM = 0 ⇔ M − N⎜ t − u ⎟ = 0 3 ⎠ ⎝ 2 By substitution of xu (equation (13)) in equation (14), it follows: −4 N σ= ⎛ 2M ⎞ − 1⎟⎟ 3 bl t ⎜⎜ ⎝ Nl t ⎠ The strain ε , the displacement u and the rotation φ in the joint are given by: σ u ε= , u = εl t and φ = Ec xu

(14)

(15)

(16), (17) and (18)

After substitution of equations (13), (15), (16) and (17) into equation (18), the rotation φ is given by: 8N φ= (19) 2 ⎛ 2M ⎞ − 1⎟⎟ 9 bl t E c ⎜⎜ N l ⎝ t ⎠ Hence the equation for the non-linear rotational stiffness of the Janßen joint in case of opening holds:

⎛ 2M ⎞ − 1⎟⎟ 9 bl t E c M⎜⎜ N l M ⎝ t ⎠ = cr = φ 8N

2

Rotational stiffness opened joint with plastic concrete stresses


99

At an increasing rotation φ of the longitudinal joint, a point will be reached where the maximum strain in the joint reaches the compressive yield strain of concrete. From this point the layout of stresses in the joint will be different from the diagram in figure 73. Therefore a new overview will be given: N M lt

lt

f'c φ

u lt

R1 x1

R2 x2 –

Ec ε'c

ε'cu

σ

74 | Stresses in the Janßen joint in case the compressive yield strain has been exceeded (left) and the stress-strain relation for rotation in the joint (right)

New parameters in these figures are: x1 = Height of the compressive zone up to the compressive yield strain x2 = Height of compressive zone from the yield strain to the ultimate strain R1 = Reaction force from the compressive zone at part x1 R2 = Reaction force from the compressive zone at part x2 f'c = Design compressive strangth of concrete ε'c = Compressive yield strain ε'cu = Ultimate compressive strain The compressive yield strain ε'c is defined by: f' ε' c = c Ec In this case the rotation φ is given by the equation: ε' l f' l φ= c t = c t x1 E c x1 Or for the first part of the compressive zone x1 it holds: f' l x1 = c t Ec φ

(20)

Now the reaction force R1 will be computed by (after substitution of equation (20)): f ' c x 1 b R 1 = 2 The other way round holds for x1: 2R 1 x 1 = f ' c b At all time equilibrium of forces should be present, resulting in: ΣN = 0 ⇔ R 2 = N − R 1

(21)

(22)

(23)

The second part of the compressive zone x2 is defined by the following equation after including equation (23):

100


x2 =

R2 N − R1 = f' c b f ' c b

(24)

Equilibrium of bending moment is defined by: x l ⎛ x ⎞ ΣM = 0 ⇔ M − N t + R 1 ⎜ 1 + x 2 ⎟ + R 2 2 = 0 2 2 ⎝ 3 ⎠ After substitution of equations (22), (23) and (24) equation (25) can be rewritten as: ⎛ 2R 1 lt (N − R 1 ) b ⎞⎟ (N − R 1 )2 b ⎜ + =0 M − N + R1⎜ ⎟+ 2 f' c 2 f' c ⎝ 3f' c b ⎠

(25)

(26)

Now by using the substitution of equation (20) in (21) in the new equation (26), the rotation φ can be isolated as the only unknown. Resulting from Maple 6 substitutions, the rotation is given by the comprehensive equations: φ 1 , 2

(6N ± =

)

21N 2 + 30Mf' c b − 15Nl t f' c b l t f' c2 b 6(− 2Mf' c b + l t Nf' c b + N )E c 2

(27)

By using the final value of the non-plastic opened joint rotation as a boundary condition, the correct equation of both can be determined. Hence the Janßen rotational stiffness (equation (1)) for the plastic situation of the joint is known: 6(− 2Mf' c b + l t Nf' c b + N 2 )ME c M c r 1 , 2 = = φ 1 , 2 6N ± 21N 2 + 30Mf ' b − 15Nl f' b l f ' 2 b

(

c

t

c

)

t

c

Derivation of forces in a Janßen joint In this study some checks have been executed on the reactions in the longitudinal joint. The maximum strain should not exceed the ultimate compressive strain of the concrete. Calculations on the bending moment in the joint also showed that it converges to a maximum value. This value will be computed in this appendix as well.

Upsetting strain The upsetting strain, or ultimate compressive strain, is a material related property. This study therefore investigates the effects of this property on the maximum strain actually occurring in the joint. The upsetting strain εmax will be given as a function of the rotation φ. In case of a closed joint equation (2) will be substituted into equation (5): φl t = εMl t 2 Where εM represents the strain by the bending moment in the joint. The strain due to the normal force is given by: σ N εN = N = E' c E' c bl t Now the total strain εmax in a closed joint holds: φ N ε max = ε M + ε N = + 2 E' c bl t For the calculation of the maximum strain in case of an opened joint firstly equation (18) will be substituted into equation 17: φx φx u = εl t ⇔ ε = u lt

Substitution of equation (13) in this equation gives:


101

ε=

2φN bl t σ

(28)

Now equation (15) is introduced to equation (28) and ε is replaced by εmax , resulting in the final equation for the maximum strain in an opened joint: − 3φ ⎛ 2M ⎞ ⎜ ε max = − 1⎟ 2 ⎜⎝ Nl t ⎠⎟ Next if concrete plasticity occurs in the opened Janßen joint, the maximum strain is retrieved from figure 74 and holds: ε' (x + x 2 ) f' c (x 1 + x 2 ) ε max = c 1 = (29) x1 E c x1 Substitution of equations (20), (24) and (21) in (29) results in the following equation for the maximum strain if plasticity occurs: f ' l b + 2NE c φ ε max = c t 2E c f' c l t b

Maximum bending moment in opened and plastic longitudinal joints The non-linear expression for the rotation of an opened Janßen joint (equation (19)) contains a part which might cause an invalid division by zero. The rotation diverts to infinity as soon as it reaches a certain maximum bending moment, Mmax of the opened non-plastic Janßen joint. This bending moment is found by isolating the part between brackets in equation (19). The maximum bending moment is derived by setting this equation equal to zero: 2M −1 = 0 Nl t By reordering this equation and replacing M by Mmax , we find: M max = N

lt

2

In the non-linear equation for the rotation of the Janßen joint in case of concrete plasticity (equation (27)) too, a possibility for an invalid division by zero exists. Hence this expression knows a maximum bending moment as well. This maximum is derived by the equilibrium of: −2Mf ' c b + l t Nf' c b + N 2 = 0 Reordering and again replacement of M by Mmax now results in the maximum bending moment for a plastic Janßen joint: lt N2 M max = N + 2 2f' c b

Summary of equations representing the Janßen joint By assuming the rotation in the Janßen longitudinal joint increases and the normal force N is known, three different stages are given: 2N 1. Closed joint, φ ≤ Ebl t 2N and ε ≤ ε' c Ebl t

2.

Opened joint, φ >

3.

Concrete behaving plastic in joint, ε' c < ε ≤ ε' cu

The equations for the rotational stiffness and reactions in the Janßen joint for all three situations are summarised in Table 9.

102


Table 9 | Equations for the longitudinal Janßen joint Closed Rotation

Rotational stiffness

Maximum strain


φ=

12M bl 2tE'c

cr =

bl 2tE'c 12

εmax =

N φ + 2 E'c bl t n/a


Opened φ=

Plastic behaviour

8N

⎛ 2M ⎞ 9bl tE'c ⎜⎜ − 1⎟⎟ ⎝ Nl t ⎠

⎛ 2M ⎞ 9bl tE'c M⎜⎜ − 1⎟⎟ ⎝ Nl t ⎠ cr = 8N εmax =

⎞ −3φ ⎛ ⎜ 2M − 1⎟ ⎜ 2 ⎝ Nl t ⎠⎟

Mmax = N

lt

2

⎛ 6N ± 21N2 + 30Mf' b − 15Nl f' b ⎞l f'2 b ⎜ ⎟ t c c t c ⎠ φ1,2 = ⎝ 2 6(− 2Mf'c b + l tNf'c b + N )E'c

2

2

cr 1,2 =

6(−2Mf'c b + l tNf'c b + N2 )ME'c ⎛ 6N ± 21N2 + 30Mf' b − 15Nl f' b ⎞l f'2 b ⎜ ⎟ t c c t c ⎝ ⎠ εmax =

f'c l tb + 2NE'c φ 2Ec f'c l tb

Mmax = N

lt

2

+

N2 2f'c b

103

104


A.2 Transformation of coordinate systems A cylindrical coordinate system (r-θ -z) is best used to model the tube of a shield driven tunnel. The loads on the tunnel however are given by a Cartesian coordinate system (x-y-z). Equations to transform one system into the other will be described in this appendix. Besides the conversion of Cartesian soil stresses into cylindrical soil forces on the nodes of the tunnel will be discussed as well. First of all both systems have been shown in figure 75. x r θ

z

z

y

75 | Cartesian coordinate system x-y-z ( ) and cylindrical coordinates system r( θ -z )

Both the Cartesian and cylindrical coordinate systems share the same axis z. Therefore this appendix only focuses on the transformation of the 2-dimensional Cartesian x-y and circular r-θ systems. Stresses and forces act on a surface. The angle θ of this surface in the Cartesian system x-y is fixed. Implying forces perpendicular to the axis r might occur as well. Therefore an axis t will be added to the circular r-θ system. Now a new coordinate system r-t is created, showing the Cartesian x-y system rotated over an angle of θ . x

r ur

ux

(ux, uy ) ur2

θ

ur1 y

uy θ

ut1 ut2

ut t

76 | Distances in a Cartesian x-y and r-t-(θ) coordinate system for a point (ux, uy )

The dimensions from figure 76 are given by: u r 1 = u x cos θ u r 2 = u y sin θ u r = u r1 + u r 2 u t 1 = u y cos θ u t 2 = u x sin θ u t = u t1 + u t 2


105

These equations will be combined in one matrix equation for the transformation of the Cartesian x-y coordinate system into the r-t-(θ) coordinate coordinate system: ⎧u r ⎫ ⎡ cos θ sin θ ⎤ ⎧u x ⎫ (1) ⎨ ⎬=⎢ ⎥⎨ ⎬ ⎩u t ⎭ ⎣− sin θ cos θ⎦ ⎩u y ⎭

Transformation of Cartesian x-y distributed loads to cylindrical r-t forces As mentioned before loads by the soil will be given (see later on in this appendix) in a Cartesian x-y system. The vertical loads are positioned at the x-axis, the horizontal loads at the y-axis. The forces on the nodes of the tunnel lining will be given in a cylindrical r-t-(θ) system. system. The radial and tangential forces are located at the r- and t-axis respectively and will be retrieved from distributed loads at these axes. x

Fv

r

θ

cos θ

y

σv t

σr

σh

σt sin θ 1

Fh

θ

77 | Distributed loads on the element of the tunnel structure

From figure 77 the following relation holds for the forces Fv and Fh in the Cartesian system: 0 ⎤ ⎧σ v ⎫ ⎧Fv ⎫ ⎡cos θ ⎨ ⎬=⎢ ⎨ ⎬ sin θ⎥⎦ ⎩σ h ⎭ ⎩Fh ⎭ ⎣ 0

(2)

By substituting equation (2) into equation (1) (u ,r , u , t, ux and uy have been replaced by σr , , σ , t, σv and σh respectively) the relation between the horizontal and vertical distributed loads and the radial and tangential distributed loads is known (although σr and σt have been visualised as concentrated forces in figure 77, their influence width of 1 makes them equal to a distributed load): sin 2 θ ⎤ ⎧σ v ⎫ ⎧σ r ⎫ ⎡ cos 2 θ ⎨ ⎬=⎢ ⎥⎨ ⎬ ⎩σ t ⎭ ⎣− cos θ sin θ cos θ sin θ⎦ ⎩σ h ⎭

106


A.3 Uplift of embedded tunnel A tunnel embedded in soil should be in vertical equilibrium at all time. Hence the upward floating force by the hydrostatic water pressure should be compensated by the mass of the tunnel and mass mass = 0,9 is applied on the masses of the soil on top of this tunnel. For safety reasons a safety factor of γ mass of both tunnel and soil. By use of Figure 16 on page 21 of the actual report the vertical distance between the water table and an arbitrary point at the tunnel’s circumference with an angle θ holds: h θ , w = x w − x t + R(1 − cos θ) Where: xw xt R

= = =

Water table relative to mark NAP Top tunnel relative to mark NAP External radius tunnel

Hence the water pressure at this location holds: σ w ( θ) = h θ , wρ w Where: ρw

=

Specific gravity water

The vertical water force on the tunnel lining at an angle θ is then given by: F = −Rσ w (θ) cos θ So the total bottom force by the hydrostatic water pressure is then defined by: 3 π 2

Fw , up = ∫ Fdθ = Rρw ( 2x w − 2x t ) + R 2ρw ( 2 + 21 π) 1 π 2

For the cross-sectional surface of the soil on top of the tunnel it holds: π A soil = 2R (x 0 − x t + R ) − R 2 2 Where: x0 = Ground level relative to mark NAP Consequently the force (mass) of this soil continuum is: Fsoil = ρ soil A soil Where: ρsoil =

Specific gravity soil

Provided that the weight of the tunnel is omitted, it should hold: Fwater = γ mass Fsoil

(1)

The weight of the tunnel reads: ⎛ 2 ⎛ D i ⎞ 2 ⎞ Ftunnel = ρ concrete π⎜ R − ⎜ ⎟ ⎟⎟ ⎜ 2 ⎝ ⎠ ⎠ ⎝ Where: ρconcrete = Specific gravity concrete


107

Di

=

Internal diameter tunnel

Now should hold: Fwater = γ mass (Fsoil + Ftunnel )

(2)

The software application Maple 6 has been utilised to solve equations (1) and (2). As a result the values of xt as presented in Figure 78 were retrieved for the soil and tunnel properties from this study. 0

200

400

600

800

(NAP) 0 ] -2,5 m [ t x -5 P A N -7,5 k r a m o -10 t e v i t -12,5 a l e r l e -15 n n u t f -17,5 o p o T


Ground level (-4,74 m + NAP)

l u n n e h t t u g g i e w e a d n g d i n d u u l c n I n E x x cluding dead w eight t t t unnel

78 | Required depth projection tunnel to prevent floating

Each individual considered lining thickness should be prevented against floating. Subsequently the value of a lining thickness of 100 mm has been applied. This one requires the deepest depth projection (if the tunnel’s dead weight is included). Obviously thicker linings are heavier and therefore require a more shallow minimum depth projection. The minimum depth projection is set to: -15,85 m + NAP, hence a soil cover of 11,1 m (0,7D).

108


Appendix B

Ultimate resisting moment for steel fibre reinforced concrete Introduction This appendix presents the calculation principal for the ultimate resisting moment of a reinforced rectangular cross-section of concrete with any arbitrary stress-strain relation. The calculation has been implemented in a Visual Basic .NET2 application.

stress

strain

Ns1

xu

Nc,compr

Nd Nc,tens

Ns2

79 | Forces and lever arms in the calculation

Both equilibrium of horizontal forces and equilibrium of bending moments should exist at all time: ns

∑ Fh = 0 ⇔ N c + ∑ Fsj = N d

(1)

j=1

ns

∑ M = 0 ⇔ M u = M c + ∑ M sj − M N

(2)

j=1

Where: Nc Fsj ns Nd Mu M

= = = = = =

Normal force in concrete Steel force in reinforcement layer j Number of reinforcement layers Acting normal force Ultimate resisting moment Contribution to Mu by element defined by its subscript

The following subscripts will be used throughout the calculation: c = Concrete sj = Steel reinforcement layer j N = As a result of the acting normal force Nd The normal force in concrete is principally in compression (negative), hence the value of Mc is negative as well. Consequently the maximum value of Mu is reached if the contribution by concrete is minimised. The concrete strain in the upper fibre of the cross-section is therefore set to the ultimate


109

compressive strain of the material ε'cu. The concrete normal force Nc is then carried in the smallest height of the concrete compressive zone possible. The stress-strain diagram of concrete is spread over the height of the cross-section in order to determine its resulting bending moment contribution Mc. The equilibrium of forces from equation (1) is used to find the concrete strain at the bottom of the cross-section first.

Maximum concrete strain The force resulting from a line segment in a multiple linear stress development (such as presented in Figure 80) is determined first. s s e r t S

1

(1)

0

(2)

Strain

6 (3)

(6)

2

5 (5) 4 (4) 3

80 | Stress-strain diagram of concrete with tensile behaviour

The average stress in segment i is given by: εi

∫ f ( ε)

σav ,i =

εi −1

ε i − ε i −1

, then it holds for the normal force: N i = bh i σav ,i

The integral, that is the surface under the line segment, is simply determined for this linear development: εi f +f ∫ε f(ε) = i 2 i−1 (ε i − ε i−1 ) i −1 Of course this principal can be applied on all individual line segments in the stress-strain diagram. The total integral, from now on assigned as η , follows from the sum of the preceding individual integrals. Hence the integral in the upper fibre with strain ε0 up to point i with strain εi is: εi

εi

ε0

i −1

∫ f(ε) = ηi = ε∫ f(ε) + ηi−1

If this integral is determined for each node of the stress-strain diagram, the normal force in the concrete is known if the strain at arbitrary node i is the strain in the cross-section’s bottom fibre. This normal force then reads (by use of prior equations for Ni and σav,i): ηi N ci = hb εi − ε0 The total normal force at the cross-section is known by performing this calculation on each node in the concrete stress-strain diagram and by adding the steel forces due to the concrete strain at their individual location to it. The strain in reinforcement layer j simply reads: x sj (εi − ε0 ) ε sj = h The strain that is connected to this strain follow from the multiple linear stress-strain relation for reinforcement steel (see Figure 81):

110


⎧0 ⎪f / γ ⎪⎪ s ms fsj = ⎨fs ε sj /(ε sy γ ms ) ⎪− f / γ ⎪ s ms ⎪⎩0

if ε > ε su if ε sy ≤ ε ≤ ε su (0) if ε sy > ε > −ε sy (1) if - ε sy ≥ ε ≥ −ε su (2) if ε < −ε su

The steel force Fsj for reinforcement layer j is: π Fsj = σsjA sj met A sj = n sj ∅ sj 2 4 s s 1 e r t S

(1)

0

(2) 2

Strain

(3)

4

(4)

3

81 | Stress-strain diagram reinforcement steel

A complete set of resulting total normal forces in the cross-section on each critical point is gained if this calculation procedure is repeated at every node of concrete’s stress-strain relation and on the nodes of the steel relation for each individual reinforcement layer. Comparison of the forces in this set to the actual acting normal force Nd makes it possible to find the line segment in the concrete diagram where the actual strain in the bottom fibre is at. If, for instance, the steel force disappears (if εsu is exceeded) a non linearity is found. Now the concrete strain closest to neutral (= 0) is used. Hence for concrete and each individual reinforcement layer a small linear zone of strains (the line segment in its stress-strain diagram) is known. These zones are zc for concrete and zsj for reinforcement layer j. First of all: a straight line between two arbitrary nodes is defined by the following equation (see Figure 82): f (ε) = β(ε − εi ) + fi Where: β =

fi+1 − fi ε i+1 − ε i

σ

i+1

fi+1 ( i ) fi

i

ε εi

εi+1

82 | Determination equation by nodes of a linear element

The total resulting normal force is: ns ns η(ε) N tot = N c (ε) + ∑ Fsj (ε) = hb + ∑ α j (ε − ε zsj+1 ) + fzsj A sj ε − ε0 j=1 j=1


111

With: α j =

fsj ( ε zsj+1 ) − fsj ( ε z sj )

⇒ N tot

ε zsj+1 − ε zsj

A sj and η( ε) =

f (ε) + fzc 2

(ε − ε ) + η zc

zc

⎛ β ⎞ = ⎜ (ε − ε zc ) + fzc ⎟(ε − ε z c ) + η zc ⎝ 2 ⎠

⎛ β ( ⎞ ⎜ ε − ε zc ) + fzc ⎟(ε − ε z c ) + ηz c ns 2 ⎠ = ⎝ hb + ∑ α j ε − ε zsj + fz sj A sj ε − ε0 j=1

(

)

β ε(ε − ε zc ) + γ (ε − ε zc ) + η zc ns = 2 hb + ∑ α jε − α jε zsj + fzsj A sj ε − ε0 j=1 β Where: γ = − ε zc + fzc 2

By setting Ntot equal to Nd it follows: ns α ε − ε0 β β j ε(ε − ε 0 ) N d = ε 2 − εε zc + γε − γε zc + η zc − A(ε − ε 0 ) + ∑ hb 2 2 j=1 hb ns

α j ε zc − fzsj A sj

j=1

hb

Where: A = ∑

This can be written as: aε 2 − bε − c = 0 ns α ε N N ε β ns α j β j 0 With: a = + ∑ , b = γ − ε z c − A − d − ∑ , c = η zc − γε z c + Aε 0 + d 0 2 j=1 hb 2 hb j=1 hb hb Consequently the solution for the strain at the bottom fibre ε is: − b ± b 2 − 4ac 2a Only one of these two strains is positioned within all boundary conditions (boundary nodes of the zones in the stress-strain diagrams of concrete and steel). That value is therefore the maximum strain in the cross-section. ε 1 , 2 =

If the values of the scalars β and α j equal zero (0), parameter a will turn out to be zero as well. Consequently no valid solution is found by the equation for ε1,2. A simpler calculation principal can be used now. Namely the strain that is looked for is positioned at horizontal areas in all stress-strain relations. The zones (line segments) in these diagrams are still known. This implies that the forces in the reinforcement steel are known. Hence: ns

N c = N d − ∑ fzsj A sj j=1

The concrete force is generally defined as: f( ε) + fzc (ε − ε zc ) + ηzc fz (ε − ε zc ) + η zc η( ε) 2 Nc = hb = hb = c hb ε − ε0 ε − ε0 ε − ε0

if

f( ε) = fzc

, which is the case in an

horizontal segment of the stress-strain diagram. The maximum strain is now determined by:: N c (ε − ε 0 ) N ⎛ N ⎞ = fzc (ε − ε zc ) + η zc ⇒ ⎜ c − fzc ⎟ε = c ε 0 − fzc ε zc + ηzc hb hb ⎝ hb ⎠ Concluding: Nc ε 0 − fzc ε zc + ηzc hb ε= Nc − fzc hb

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The ultimate resisting moment results from the equilibrium of bending moments in equation (2) and generally holds: ns

M u = M c + ∑ M sj − M N j =1

For that purpose the stress-strain relation between strains ε0 and ε is spread over the entire height of the cross-section. The strain in each node is now converted to a lever arm relative to the upper fibre by: ε −ε xi = h i 0 ε − ε0 The bending moment Mc is constructed of the bending moment of all individual segments in the stress-strain relation combined: nc

M c = ∑ M ci i =1

These individual bending moments are determined by multiplying the surface under the stress-strain section by the lever arm of that particular section. Hence: the integral of the stress-strain diagram multiplied by the vertical distance x: xi + 1

xi + 1

xi

xi

M ci = b ∫ f( x)x dx = b

∫

x

f − β xi xi 2 ⎤ i+1 ⎡β βxi x + (fi - βxi x i )x dx = b⎢ xi x 3 + i x ⎥ 2 ⎣ 3 ⎦ xi 2

f − βxi xi ⎛ β (xi+12 − xi 2 ) ⎞⎟ b ⇒ Mci = ⎜ xi (xi+13 − xi 3 ) + i 2 ⎝ 3 ⎠ f −f Where: βxi = i+1 i x i+1 − xi For the bending moment by the reinforcement layers it holds: ns

ns

j= 1

j= 1

M s = ∑ M si = ∑ f(ε sj )A sj x sj Where the determination of steel strain εsj has been mentioned before and reads: x sj (ε − ε 0 ) ε sj = h The bending moment by the acting normal force is: h M N = N d 2 Consequently the ultimate resisting moment Mu is given by: ns f − β xi x i ⎛ β (x i +1 2 − x i 2 ) ⎞⎟ b + ∑ f(ε sj )A sj x sj + N d h M u = ⎜ xi (x i + 1 3 − x i 3 ) + i 2 2 ⎝ 3 ⎠ j=1

Validation with calculation principal Dutch building code NEN 6720 For this validation the following rectangular cross-section is considered: Height h = 500 mm Width b = 1.000 mm Concrete cover c = 35 mm Diameter rebars ∅s = 12 mm Partial load factor γ = 1 Acting normal force N = -2.000kN (druk) Reinforcement percentage ω = 0,21%, thus steel area As = 1.050 mm² Concrete strength class = C35/45


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With design strength f'c = 27 MPa and design tensile strength fc = 1,65 MPa Steel = FeB435/500 with design strength fs = 435 MPa The steel force is determined by: N s = A s fs = 1.050 ⋅ 435 = 456.750 N From the equilibrium of horizontal forces it follows: ∑ Fh = 0 ⇔ N c = N s + N = 456.750 + 2.000 ⋅ 10 3 = 2.456.750 N

The height of the concrete compressive zone reads: Nc 2.456.750 = = 121 ,3 mm xu = 0 ,75f' b b 0 ,75 ⋅ 27 ⋅ 1.000 Accordingly it holds for the ultimate resisting moment: Mu = Mc + Ms + MN Where: M c = −0 ,38N c x u = −0 ,3889 ⋅ 2.456.750 ⋅ 121 ,3 = −115 ,9 ⋅ 10 6 Nmm = −115 ,9 kNm M s = N s (h − c − ∅ s / 2 ) = 456.750 ⋅ (500 − 35 − 6 ) = 209 ,6 ⋅ 10 6 Nmm = 209 ,6 kNm MN = N

h 500 = 2.000 ⋅ 10 3 = 500 ⋅ 10 6 Nmm = 500 kNm 2 2

Hence follows: M u = −113 ,2 + 209 ,6 + 500 = 593 ,7 kNm The application, wherein the described calculation principal has been implemented, returns the following value: M u = 593 ,7 kNm The individual contributions are: M c = −115 ,9 kNm; M s = 209 ,6 kNm; M N = 500 kNm Consequently no variations are found between the values resulting from the custom calculation prinipal and the Dutch code.

Validation with method by Den Hollander Voor de validatie zijn de volgende data gebruikt: Height h = 500 mm Width b = 1.000 mm No rebars are applied Acting normal force N = -2.000 kN Concrete strength class = C180/210 with a stress-strain diagram as described in this report For this validation an Excel-sheet from the graduation thesis of Den Hollander [9] is used. That sheet searches for an equilibrium of horizontal forces by iteration. In order to retrieve a valid result from that method prestressing steel has to be added to Den Hollander’s calculation. A steel area of 1 mm² has been positioned at a distance from the upper fibre of 0 mm in order to exclude the steel force from the ultimate resisting moment. The model generated a bending moment capacity of: M u = 801 kNm

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The Visual Basic application from this study returned the following: M u = 803 ,9 kNm A difference of only +0,36 % is observed. Both in case of reinforced concrete and steel fibre reinforced concrete the application implemented returns very accurate ultimate resisting moments. Consequently it is assumed that this method can be applied in this study.


115

116


Appendix C

Safety factors in shield driven tunnels Building codes prescribe calculation methods and safety ratios for most standard structures. Several additional regulations deal with special requirements for various specific structures like bridges, or sheet piles. The structure of a shield driven tunnel shows several differences with normal structures like bridges or buildings, for instance its vital interaction with the surrounding soil. However no additional regulations exist for this type of structure. This chapter therefore discusses the policy from Dutch and German building codes from the bottom to present a safety philosophy for shield driven tunnels in the end. Safety factors in building codes exist to include the probability of several risk aspects in the calculation. In modern building codes these safety factors are split up in partial safety factors for the strength or resistance of the structure and for the stress on the structure by loading. Via a simple check it’s possible to check whether a structure is able to resist unforeseen risks as well. The check is given by the following basic equation: Rd ≥1 (30) Sd Where Rd represents the resistance of the structure, including partial safety factors for the strength, and Sd represents the stress to the structure, including partial safety factors for loading. The partial safety factors will be discussed in separate sections in this appendix. This appendix describes the policy in the Dutch and German codes for partial safety factors on the resistance side for concrete. The partial safety factors on the stress – or loading – side of the structure, including the influence of soil, will be dealt with as well. Next a combination of those partial safety factors and a safety philosophy that is applicable for the unique structure of shield driven tunnels are presented.

Resistance of the structure The risk of the strength (or resistance) of a material as built being less than the strength as foreseen in the design, is introduced by so-called material factors in calculations according to the Dutch concrete code NEN 6720 and the German building code DIN 1045 neu. These factors reduce the strength of a material in the input of a calculation. Implying the material factors are introduced before the calculation of the strength is performed. The use of material factors from the building codes for concrete is limited by a maximum strength class of concrete. The NEN is valid for strength classes up to C53/65 (ordinary concrete) and additional recommendations exist for values up to C90/105 (high strength concrete); the DIN can be used up to C90/105 as well. For ultra high strength concrete (C135/150 to C180/210) only French


117

recommendations [1] exist. Results from these recommendations will be used for very high strength concrete (C90/105 to C135/150) as well. Regulations for ordinary concrete and reinforcement steel will be discussed. This appendix shows the recommendations for very high strength and ultra high strength concrete.

Ordinary concrete Building codes know three levels of compressive strengths of concrete: 1. Characteristic compressive strength (f'ck) 2. Representative value compressive strength (f'crep) 3. Design value compressive strength (f'c) The final value, the design value, is used for strength calculations. In the Dutch NEN 6720 its value is defined by: f' crep f' c = (31) γ mc Parameter γ mc represents the material factor for concrete, with a value of 1,2. In case of accidental loading situation (gas explosion, collision and impact load) the material factor may be reduced to 1,0. For the representative compressive strength holds: f' crep = 0 ,72f' ck (32) The representative compressive strength is the long-term uniaxial compressive strength of concrete. The short-term uniaxial strength is retrieved by multiplying the 28-days cube strength of concrete by 0,85, from now on referred to as αu. The long-term strength follows from the product of the shortterm strength and another 0,85, referred to as αt. The product of both scalars is 0,72 and has been included in equation (32). Please note that the Eurocode no longer uses the 28-days cube strength for f'ck , it has been replaced by the 28-days uniaxial compressive strength from cylindrical test specimens. The original German code DIN 1045 used one overall safety factor to include all risks of a structure at once. Code DIN 1045 neu was adopted to implement the partial safety factors from the future Eurocode in a German code before the actual introduction of the Eurocode. Therefore the DIN 1045 neu makes use of the uniaxial compressive strength f'ck as well: f ' crep = α term f ' ck = 0 ,85f' ck (33) The value of material factor γ mc from the DIN neu has a different value from the NEN: for in situ structures the value holds 1,5; for prefabricated elements from a constantly supervised production process the value is reduced to 1,35. In case of extraordinary loading situations a value of 1,3 is sufficient. For reinforced ordinary concrete the tensile strength is negligible compared to tensile forces in steel reinforcement bars, therefore no tensile strength will be taken into account. Material factors for steel reinforcement bars are equal in both the Dutch and German codes. The design value for the steel strength fs is given by the equation: f fs = sk (34) γ ms Where fsk represents the characteristic steel strength. The value for the material factor for steel γ ms is given by 1,15 in both codes. It may be reduced to 1,0 in case of extraordinary loading situation in the German code and in case of gas explosions, collisions and impact loads in the Dutch code. Table 10 gives an overview of all partial safety factors related to the strength of the material for both concrete and steel. For a comparison of partial safety factors for concrete the ratio between the

118


characteristic and representative compressive strengths has been split up in the two α-s (αu and αt) that have been discussed after equation (32) already. This is essential to show the difference between the cube strength from the NEN and the cylindrical strength from the DIN. Table 10 | Partial safety factors for strength (resistance) according NEN 6720 and DIN 1045 neu Material

NEN 6702

DIN 1045 neu

1,2 (1,0) 1)

1,35 2) (1,3) 3)

Multiaxial to uniaxial αu

0,85

-

Short-term to long-term αt

0,85

Concrete Material factor γ mc

Total partial safety factor γ c = γ mc / ( αu α )t

1,67 (1,39)

0,85 1)

1,59 (1,53) 3)

Steel Material factor γ ms

1,15 (1,0) 1)

1,15 (1,0) 3)

1)

Reduced material factor for a gas explosion, collision or an impact load Factor for prefabricated elements, for in situ elements holds γ mc = 1,5 3) Reduced material factor for an extraordinary loading situation 2)

Very high strength and ultra high strength concrete It’s been mentioned before that the French recommendations [1] are the only present regulations for design calculations on ultra high strength concrete. To come to a conclusion for the adaptation of the strengths of this type of concrete in comparison to ordinary concrete the Dutch CUR Recommendation 97 and the Germen code DIN 1045 neu, both valid for concrete strength classes up to C90/105, will be included in the text below as well. Tensile strengths will be discussed separately in here, for they have been neglected in the text concerning ordinary concrete. Compressive stresses

In CUR Recommendation 97 a higher level of safety is required for concrete strength classes from C53/65. The additional safety is requested because of the more brittle behaviour of high strength concrete and has been included in an increment of the combined ratio (αu αt). This ratio should be multiplied by the following scalar α'c: 785 − f ' ck α' c = (35) 720 In the German DIN 1045 neu the safety should be increased from C55/67 as well. The Germans multiply the material factor of concrete by the scalar γ 'c: 1 γ ' c = ≥1 (36) f ' ck 1 ,1 − 500 Although both methods appear very different, this difference is actually relatively small. Inversion of the German increment and conversion of the included characteristic compressive strength from multiaxial to uniaxial in the Dutch equation, gives: f' f' 1 1 = 1 ,1 − ck and DIN: = 1 ,1 − ck ≥ 1 NEN: (37) and (38) γ ' c γ ' c 612 500 The CUR and the DIN both deal with reinforced concrete without the addition of steel fibres. Steel fibres however ensure a more ductile behaviour of concrete, resulting in less brittleness. According to [6] the recommended extra safety is superfluous for high strength, very high strength and ultra high strength steel fibre reinforced concretes. Implying the same safety strategy from as before can be applied for these ranges of concrete strength classes if steel fibres have been added. Tensile stresses


119

The increasing influence of tensile behaviour in very high strength and ultra high strength concrete mainly results from the steel fibres in the material. Among others the orientation of these fibres, which can be directed during the casting process, influences the magnitude of the tensile stress1. This fact can beneficially influence the resistance of the material; however if the orientation as casted changes from the orientation as designed, serious loss of resistance might occur. To cope with all unforeseen fibre related decrements of the structure’s strength, an additional partial safety factor is introduced in [1]. This factor adapts all tensile parameters influenced by the steel fibres from the stress-strain diagram in Figure 27 on page 34 of the actual report. The extra factor, γ f, should be added in ultimate limit state (ULS) calculations only. In serviceability limit state (SLS) related calculations, like deflection, the extra safety is redundant. The values of the partial safety factor for the influence of steel fibres γ f hold [1]: γ f = 1,3 for standard loading situations γ f = 1,5 for accidental loading situations The new equations for the influenced parameters from the stress-strain diagram for ultra high strength concrete in figure 27 are: σ( w 0 , 3 ) w 0 , 3 f ε 0 , 3 = + el and fc = (39) and (40) γ f E el γ f K lc w f σ( w 1% ) (41) and (42) ε 1% = 1% + el and f1% = γ f E el γ f K lc

Stress on the structure The risk of the stress on the structure being more severe, implying higher loads, then used in design calculations is introduced by so-called loading factors. These partial safety factors have been described in NEN 6702 and DIN 1045 neu. Loading on a shield driven tunnel is mainly introduced by the surrounding soil. Therefore this section will deal with this material especially and will try to combine its effects with the effects of loading factors.

Loading factors from the codes Loads in both the NEN and DIN are subdivided in three different categories: 1. Static loads (dead weight and other permanent loads) 2. Live loads (mobile loads and varying loads) 3. Prestress force Loading on a structure can be favourable and unfavourable relative to the overall loading situation. Table 11 shows all loading factors for the Dutch and German codes. In the Dutch code all loading factors can be reduced to the value of 1,0 in case of a gas explosion, collision or an impact load. Table 11 | Partial loading factors NEN 6720 (safety class 3) and DIN 1045 neu Static load γ g Loading situation

NEN

1)

DIN

Live load γ q 2)

NEN 3)

Favourable

0,9 (1,0)

1,0

0 (1,0)

Unfavourable

1,35 (1,0)

1,35

1,5 (1,0)

3)

3)

Prestress force γ p

DIN

NEN

DIN

0

1,0

1,0

1,5

1,0

1,0

3)

1)

NEN 6720; 2) DIN 1045 neu; 3) Reduced factors for gas explosion, collision or impact load

Markovic, I., High-Performance Hybrid-Fibre Concrete: Development and Utilisation , dissertation. Delft: Delft University Press, January 2006 1

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Soil Soil both supports and loads a tunnel structure. This interaction causes the tunnel to change its shape to an oval, as discussed in the previous chapter. This oval can be formed by an increment of the diameter in horizontal direction, a so-called lying oval, or by an increasing diameter in vertical direction, a standing oval. In both cases the main loading and supporting occur in different regions along the ring’s circumference. It might be necessary to appoint regions with either a loading or a supporting function. In future changes of the ground’s surface level or additional loads on the ground’s surface (for instance nearby structures) might occur. Finding the combination of loading and supporting regions that results in the most sever loading case of the tunnel lining is nearly impossible. Forces and moments in the lining resulting from calculations on the tunnel’s ring behaviour, very much depend on loads and support from the surrounding soil, which depend on its density and stiffness. Introduction of loading factors to the loads by the soil does not necessarily imply a more severe loading situation of the tunnel structure. Increasing loads normally result in an increasing ring force. A higher ring force itself can impose a higher moment capacity of the segments and can limit the rotations in longitudinal joints. Up to a certain degree of loading both given results influence the overall ring behaviour of the lining in a beneficial way. Risks related to the soil and therefore related to the loading and support of the tunnel should be introduced in a different way than codes prescribe for ordinary tunnels. The next section therefore discusses a overall safety philosophy for shield driven tunnels.

Safety philosophy shield driven tunnels In the design of shield driven tunnels loads on the structure mainly depend on the depth of the tunnel and type of soil surrounding it. In design calculations of ordinary structures this load would be multiplied by the partial loading factor before the calculation is performed. In tunnel design however an increasing load does not necessarily mean an increasing risk. Increasing loads imply increasing normal ring forces. Before it was mentioned this can beneficially influence the ring behaviour of the lining. Therefore a different approach to introduce the partial safety factors in design calculations on shield driven tunnels is required. Apply safety factors after calculation

To tackle the given problem safety ratios are normally applied after the calculation has been made. Implying that the stress on the structure (Sd) – the reaction force – is multiplied by safety factors ( γ ) and compared to its resistance (Rd) – the ultimate capacity of the structure for that particular type of reaction force. The rule from equation (43) is now given by: Rd ≥1 (43) γ S d Combining partial safety factors

Now it’s decided that safety factors will be used at the end of a calculation, two options for the way to apply these factors still remain: 1. Apply material factors in ultimate capacity calculations and introduce loading factors in the final check. Now the requirement from equation (43) can be rewritten as: R d ( γ mc , γ ms ) ≥ 1 (44) γ b S d


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Where γ b is the loading factor that represents the safety for interaction between the concrete tunnel lining and the surrounding soil. 2. Combine material factors and loading factor to introduce one overall safety factor in the final check only. The requirement now holds: Rd R = d ≥ 1 (45) γ m γ b S d γ S d Parameter γ m is the combined material factor for the materials in the lining and γ now represents the overall safety factor. Both approaches know a combined loading factor that should contain several risk aspects. Combined loading factor for approaches 1 and 2

The combined loading factor γ b represents a safety margin for interaction between the tunnel lining and the surrounding soil. Therefore it does not include a loading factor only, but deals with various uncertainties. For the Statentunnel in the light rail project RandstadRail in Rotterdam, the IGWR included the following features in the loading factor2: Nature of loading (dead weight, live load at ground surface, fire) Accuracy of the applied calculation method Segmented shield driven tunnels are relatively new types of structure Influences of placing tunnel segments and grouting is not fully understood yet Experience from other tunnelling projects The value for the partial loading factor that should cover this enumeration was set to γ b = 1,5.     

Combined material factor for approach 2

The IGWR decided to use the second approach (equation (45)) to combine partial safety factors from NEN-codes. For the combined material factor γ m a value of 1,15 was used. From Table 10 follows this factor holds for reinforcement steel instead of concrete. To prevent brittle fracture of a structure from reinforced concrete, reinforcement steel should yield first before a structure collapses. This implies the reinforcement bars have to fail, therefore their material factor has been introduced. The overall safety factor for the Statentunnel in Rotterdam therefore holds: γ = γ m · γ b = 1,7. In case of very high strength or ultra high strength steel fibre reinforced concrete tunnel segments additional steel reinforcement might be excluded. Now the tensile region of the concrete itself should fail. In case of the second approach a combined material factor of 1,2 should be used according NEN 6720 and 1,5 should be used for the DIN 1045 neu. Changes in the stress-strain diagram for approach 2

Material factors are introduced in the final check from equation (45). Hence material factors are excluded in the ultimate capacity calculations for all reaction forces. This implies a serious adaptation of the stress-strain diagram of concrete and reinforcement steel, for the stresses in all diagrams have been reduced by these material factors. Changes in the stresses might influence the size of the strains as well. Approaches to implement the deformed stress-strain diagram are: A. No changes to the strains are made. This method has been shown as approach A in figure 83. The basic shape of the diagram (ratio between ε'c and ε'u) shows no differences with the original diagram. However the slope of the linear elastic section, the modulus of elasticity of the material, increases.

Taffijn, E., J. Gerritsen, H. Pachen, RandstadRail DO Ringberekeningen gesegmenteerde tunnel Noorderkanaal – SFD. Rotterdam: Ingenieursbureau Gemeentewerken Rotterdam, May 2002 (Dutch) 2

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B. Elastic behaviour stays unharmed; compressive yield stress and tensile stress at point of cracking do change. Approach B in figure 83 shows the same linear elastic behaviour of the original and new diagram. A new value has to be assigned to the compressive yield stress. The figure shows the simple adaptation. Exactly the same change will be used for the tensile stress where cracking occurs for the first time (see figure 27 on page 34), this automatically adapts the values of the other points in the diagram to the new situation (except for the ultimate tensile strain). ) ' ( s f' s crep e r t s . f'c r p m o c

) ' σ ( s s f'crep e r t s . f'c r p m o c

approach A

original

⎛ γ f' ⎞ arctan⎜⎜ mc c ⎟⎟ ⎝ ε'c ⎠ ε'c

compr. strain ( ε')

ε'u

approach B

original

⎛ f' ⎞ arctan⎜⎜ c ⎟⎟ ⎝ ε'c ⎠

compr. strain ( ε')

ε' ε'c1 = c γ mc

ε'u

83 | Adapted stress-strain diagrams in case material factors are excluded from t he strength calculations

This study uses approach B to adapt the stress-strain diagram in case an overall safety factor is used (approach 2). Preservation of the linear elastic behaviour is thought to be more important than retaining the ratio between the compressive yield stress and the ultimate compressive stress. Former code for approach 2

It’s been mentioned before that the German code DIN 1045 neu was introduced in order to be able to use the partial safety factors, material factors and loading factors, even before the future Eurocode will be introduced. The former code DIN 1045 only used on overall safety factor. Its version from 1988 has been used in German tunnelling design up to January 1st 1996. The DIN 1045 subdivides two overall loading factors: γ = 2,1 in case the structure fails on compression γ = 1,75 in case the structure fails on bending Please note the difference based on failure of concrete (compression) or reinforcement steel (bending). In the present DIN 1045 neu this difference is established by the different material factors of both materials.


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TW Groeneweg - Shield Driven Tunnels in UHSC

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