Tunnel Support- Use of Lattice Girders in Sedimentary Rock

Tunnel SupportUse of Lattice Girders in Sedimentary Rock Einar Hrafn Hjálmarsson Faculty of Civil and Enviromental Engineering University of Iceland Tunnel support Einar Hrafn Hjálmarsson 30 ECTS thesis submitted in partial fulfilment of a Magister Scientiarum degree in Civil Engineering Supervisor Sigurður Erlingsson, University of Iceland Advisor Haukur Eiríksson, Hnit engineering Faculty Representative Gísli Eiríksson Faculty of Civil and Environmental Engineering School of Engineering and Natural Sciences University of Iceland Reykjavík, October 2011 Tunnel Support, use of lattice girders in sedimentary rock. Tunnel Support. 30 ECTS thesis submitted in partial fulfilment of a Magister Scientiarum degree in Civil Engineering. Copyright © 2011 Einar Hrafn Hjálmarsson All rights reserved Faculty of Civil and Environmental Engineering School of Engineering and natural Sciences University of Iceland VR-II, Hjarðarhaga 2-6 107, Reykjavík Iceland Telephone: +354 525 4000 Bibliographic information:: Hjálmarsson, E.H., 2011, Tunnel support, use of lattice girders in sedimentary rock, Master’s thesis, Faculty of Civil and Environmental Engineering, University of Iceland, 78 pages. Printing: Háskólaprent ehf. Reykjavík, Iceland, October 2011 v Abctract Icelandic geology is highly influenced by its location on the Mid Atlantic ridge. Changes in geological conditions are therefore frequent in Icelandic tunnels and flexible rock support methods are required to deal with constant variation of rock mass properties. Main focus of this thesis is tunnel excavation and rock support in sedimentary rock mass. Instability problems in thick sedimentary rock layers in the Óshlíð tunnel are discussed. Installed support is evaluated using finite element modelling of the rock mass and bearing capacity calculations of the tunnel lining. Special attention is given to the usage of lattice girders and how it was used as both temporary support to secure safer working environment at the tunnel face and as a part of the final rock support for the tunnel. Good estimation of rock mass properties to construct a reliable finite element model is important. Limited laboratory test data of the sedimentary rock mass turned out to be a problematic factor. A number of uni-axial compression tests were made on core samples from sedimentary layers in the tunnel but no tri-axial tests were executed. That led to very conservative approach in the estimation of the rock mass parameters. Result from the modelling indicates that the tunnel lining used in the Óshlíð tunnel is sufficient if average material parameters are used for the sedimentary rock mass. The rock support although seems to be slightly insufficient if lower limit (90% of test data above) is used for sediment strength. These results emphasise the importance of estimation of rock mass properties in preparation of tunnel constructions. vi vii Index Figures ................................................................................................................................. ix  Tables .................................................................................................................................. xii  Acknowledgement ............................................................................................................. xiii  1  Introduction ..................................................................................................................... 1  2  Rock mass properties...................................................................................................... 2  2.1  Icelandic rock mass ................................................................................................. 3  2.2  Characteristics of rock mass .................................................................................... 5  2.2.1  Generalized Hoek-Brown criterion ................................................................ 5  2.2.2  Mohr-Coulomb failure criterion .................................................................... 6  2.3  Stresses in rock mass ............................................................................................... 8  2.4  Discontinuties in the rockmass .............................................................................. 10  3  Types of rock support ................................................................................................... 11  3.1  Rock bolts .............................................................................................................. 11  3.1.1  General ......................................................................................................... 11  3.1.2  Types of rock bolts....................................................................................... 12  3.2  Shotcrete ................................................................................................................ 14  3.2.1  General ......................................................................................................... 14  3.2.2  Shotcrete methods ........................................................................................ 16  3.2.3  Shotcrete mix ............................................................................................... 16  3.2.4  Steel fiber ..................................................................................................... 17  3.2.5  Curing of shotcrete ....................................................................................... 17  3.3  Wire mesh .............................................................................................................. 18  3.4  Shotcrete ribs ......................................................................................................... 19  3.4.1  General ......................................................................................................... 19  3.4.2  Types of shotcrete rips ................................................................................. 20  3.4.3  Installation procedure (Norwegian standard) ............................................... 22  3.4.4  Usage............................................................................................................ 22  3.5  Lattice girders ........................................................................................................ 23  3.5.1  General ......................................................................................................... 23  3.5.2  Types of lattice girders ................................................................................. 24  3.5.3  Installation procedure ................................................................................... 25  3.6  Steel arches ............................................................................................................ 29  4  Estimation of required rock support .......................................................................... 30  4.1  Rock quality designation index (RQD) ................................................................. 30  4.2  The RMR-system .................................................................................................. 31  4.3  Rock tunneling quality Index, Q-system ............................................................... 33  4.3.1  Determination of the Q-value ...................................................................... 33  4.3.2  Required rock support for estimated Q-value .............................................. 34  4.4  Numerical analysis ................................................................................................ 36  viii 5  Deformation of rock mass ............................................................................................. 37  5.1  Theory .................................................................................................................... 37  5.2  Modeling deformation curve .................................................................................. 39  5.2.1  Axisymmetrical FEM model ....................................................................... 40  6  Case study: Use of lattice girders in Óshlíð tunnel. .................................................... 43  6.1  Introduction ............................................................................................................ 43  6.2  Rock mass properties in Óshlíð tunnel .................................................................. 45  6.3  Estimation of rock mass parameters ...................................................................... 46  6.4  Calculated capacity of installed support ................................................................ 51  6.5  FEM-modelling ...................................................................................................... 54  6.5.1  Modelling method ........................................................................................ 55  6.6  Result ..................................................................................................................... 57  7  Conclusions .................................................................................................................... 62  Bibliography ....................................................................................................................... 63  Appendix ............................................................................................................................. 65  A.1 Rock classification systems. ..................................................................................... 65  A.2 GSI and D value in Hoek-Brown criterion ............................................................... 70  A.3 Óshlíð tunnel: Geological mapping for st 16.300-16.450 ........................................ 72  A.4 Bearing capacity of shotcrete lining. ........................................................................ 73  ix Figures Figure 2.1 Simplified geological map of Iceland (Weisenberger, 2010). ............................. 3  Figure 2.2 Mixed face. Basalt layers with approximately 2m sediment interbed. ................ 4  Figure 2.3 Mohr-Coulomb and Hoek-Brown failure criterions (Hoek, 2000) ...................... 7  Figure 2.4 σ H vs depth at various locations in Iceland. .......................................................... 9  Figure 2.5 Ideology of most stress criterias ......................................................................... 10  Figure 3.1 Fixing of a single block and systematic bolting (Palmstöm et al, 2000) ........... 11  Figure 3.2 Most used rock bolts in Icelandic tunnels. Figures from (Statens Vegvesen, 2000) ............................................................................................... 13  Figure 3.3 Application of spiling bolts in a weakness zone. ............................................... 14  Figure 3.4 Shotcreting ......................................................................................................... 15  Figure 3.5 Deflection curves for concrete slabs with and without steel fibers (Palmstöm & Nilsen, 2000) .............................................................................. 17  Figure 3.6 Normal and measured curing of C30 shotcrete (Einarsson, 2010) .................... 18  Figure 3.7 Wire mesh to support local unstable area. ......................................................... 19  Figure 3.8 Radial rock bolting of a shotcrete rib (Vegagerðin, 2008) ................................ 20  Figure 3.9 Single layered shotcrete rib (Statens vegvesen, 2009) ....................................... 21  Figure 3.10 Double layered shotcrete rib (Statens vegvesen, 2009) ................................... 21  Figure 3.11 Installation of single layer shotcrete ribs in Stjördal tunnel, Norway (Guðmundsson, 2011). ..................................................................................... 23  Figure 3.12 Thee and four-chord lattice girders (Jorimann, 2010) ..................................... 24  Figure 3.13 Scaling of under breaks and loose rock blocks ................................................ 25  Figure 3.14 Initial layer of shotcrete ................................................................................... 26  Figure 3.15 Installation of lattice girder .............................................................................. 26  Figure 3.16 Girder half embedded in shotcrete ................................................................... 27  Figure 3.17 Installation of spiling bolts ............................................................................... 27  x Figure 3.18 Lattice girder fully embedded with shotcrete .................................................. 28  Figure 3.19 Steel arch with sliding gap (Hoek et al.,2008) ................................................. 29  Figure 4.1 Example of RQD-value estimation (Hoek, Practical Rock Engineering, 2000) ................................................................................................................. 30  Figure 4.2 Rock support according to the Q-system (Hoek, 2000). .................................... 34  Figure 4.3 Generated element mesh of finite element model. ............................................ 36  Figure 5.1 Deformation vector around advancing tunnel (Hoek, Practical Rock Engineering, 2000) ........................................................................................... 37  Figure 5.2 Longitudial deformation profile based on elastic and empirical models for Rt =4,6m and Rp =7,45m. ................................................................................ 39  Figure 5.3 Axisymmetric model of the tunnel .................................................................... 40  Figure 5.4 Deformation vs decreasing inclusive e-modulus. .............................................. 41  Figure 5.5 Deformation curve for walls .............................................................................. 41  Figure 6.1 Overbreak due to weak sedimentary layer in Óshlíð tunnel. ............................. 43  Figure 6.2 Use of lattice girders and spiling bolts in Óshlíð tunnel. ................................... 44  Figure 6.3 Typical layered rock mass strata in the Vestfirðir peninsula (Guðmundsson et al., 2007). ............................................................................ 45  Figure 6.4 Normal distribution of UCS data ....................................................................... 47  Figure 6.5 UCS vs E i -module .............................................................................................. 47  Figure 6.6 Hoek and Brown failure envelop for lower limit strength parameters of sediment. ........................................................................................................... 48  Figure 6.7 Hoek and Brown failure envelop for mean strength parameters of sediment. ........................................................................................................... 48  Figure 6.8 Hoek and Brown failure envelope for basalt ..................................................... 50  Figure 6.9 Installed tunnel support at station 16.400 and 16.410. ...................................... 51  Figure 6.10 Cross section of lattice girder used in Óshlíð tunnel. Units are in mm. .......... 51  Figure 6.11 On left: Assembled lattice girders (Efla, 2011). On right: Final shotcrete lining with lattice girders (Pedersen, Kompen, & Kveen, 2010). .................... 51  Figure 6.12 UCS result of shotcrete in Óshlíð tunnel. ........................................................ 53  Figure 6.13 Case 1 and 2. Station 16.400. ........................................................................... 54  xi Figure 6.14 Case 3 and 4. Station 16.410. ........................................................................... 55  Figure 6.15 Maximum wall deformation and plastic radius. X indicates a shear failure in the rock mass and o indicates tension failure. ................................... 55  Figure 6.16 Deformation curve for case 1 ........................................................................... 56  Figure 6.17 Inclusion modulus vs deformation ................................................................... 56  Figure 6.18 Result for case 1. Lower limit strength of sediment 12m in thickness from tunnel invert. ............................................................................................ 57  Figure 6.19 Results for case 2. Mean strength of sediment 12m in thickness from tunnel invert. ..................................................................................................... 58  Figure 6.20 Results for case 3. Lower limit strength of sediment 12m in thickness from mid section. .............................................................................................. 58  Figure 6.21 Results for case 4. Mean strength of sediment 12m in thickness from mid section. .............................................................................................................. 59  Figure 6.22 Highest moments on the boarder of sediment and basalt in case 3. ................. 60  Figure 6.23 Result for case 4 with variable k value from 0,5 to 1,5. .................................. 61  xii Tables Table 3.1 Properties of commonly used rock bolts (based on Palmstöm & Nilsen, 2000) ................................................................................................................. 14  Table 3.2 Typical mix for C40 wet Shotcrete ..................................................................... 16  Table 4.1 Rock mass Rating system-RMR (enlarged in appendix A1) (Hoek, Practical Rock Engineering, 2000) ................................................................... 31  Table 4.2 Guidelines for rock support and excavation of 10 span tunnel according to RMR system. (Hoek, Practical Rock Engineering, 2000) ................................ 32  Table 4.3 ESR values (Hoek, Practical Rock Engineering, 2000) ...................................... 34  Table 4.4 Typical parameters for estimation of Q value in Iceland (Loftsson, 2009). ....... 35  Table 5.1 Rock mass parameters for axisymmetric FEM-model ........................................ 40  Table 6.1 UCS values of intact sedimentary rock samples in Óshlíð tunnel ...................... 46  Table 6.2 Rock mass properties of lower limit strength of sediment rock mass. ................ 49  Table 6.3 Rock mass properties of mean strength sediment rock mass. ............................. 49  Table 6.4 Min requirements for shotcrete in Óshlíð tunnel ................................................ 52  Table 6.5 Main model parameters ....................................................................................... 54  xiii Acknowledgement I would like to thank the following people for their advice and assistance during my work on this thesis. Dr. Sigurður Erlingsson for his guidance, helpful discussions and high interest in the project. Haukur Eiríksson for his guidance and helpful advice. Björn Harðarson for providing data and literature. Hlynur Guðmundsson for providing data and pictures. Gísli Eiríksson for providing test data from Óshlíð tunnel. Special thanks to my parents for all their support. xiv 1 1 Introduction First road tunnel in Iceland was excavated through Arnadalshamar in 1948. Since then 9 other road tunnels have been constructed along with numerous tunnels associated with construction of hydro power plants. Each tunnel has contributed to the knowledge of the Icelandic rock mass characteristics and usage of classification systems or other methods to estimate required rock support in the tunnels. Tunnel support methods in Iceland have mainly been derived from experience and conventions in the Norwegian tunnelling industry and the Norwegian standard of tunnelling has been the foundation of Icelandic tunnel design. Q-value is used in the Norwegian standard to classify rock mass into rock support categories. Norwegian rock mass is however completely different from the Icelandic rock mass and adjustments has therefore been needed to adopt the Q-system to Icelandic tunnelling conditions. One of the newest challenges in Icelandic tunnelling was excavation through thick sedimentary layers in Óshlíð tunnel (also referred to as Bolungarvík tunnel). Rock support method based on the usage of lattice girders was used for the first time in Icelandic tunnelling to deal with low strength sedimentary rock layers. Detailed description of the usage of lattice girders in sedimentary rock will be carried out in this report along with short overview of other main support methods used in Icelandic tunnelling. Pros and cons of all methods will be evaluated. Numerical analysis will be used to estimate deformation and stresses acting on such lining for real cases from Óshlíð tunnel. Bearing capacity of reinforced shotcrete lining with lattice girders will be calculated according to the Eurocode 2 standard. Foundation of high quality numerical analysis is the input parameters used to describe the rock mass behaviour around the underground opening. Main properties of rock mass characteristics will therefore be discussed and methods used to estimate rock mass properties of sedimentary layer in Óshlíð tunnel. Support method based on lattice girders has been used around the world for decades because of their flexibility and easy assembling ability. Since the knowledge of the usage of lattice girders in Iceland is very limited a detailed description of their application in Óshlíð tunnel is considered contributing to the knowledge base of the Icelandic tunnelling industry. 2 2 Rock mass properties Reliable design of an underground excavation requires reliable estimation of the rock mass surrounding the underground opening (Hoek, 2000). Rock mass is a complex matrix of intact rock and weakness zones and number of parameters influence its strength, deformability, permeability and stability behaviour. According to Kirkaldie (1988) at least 28 parameters could be used to describe rock mass behaviour (Kirkaldie, 1988). - 10 rock material properties - 10 discontinuities properties - 8 hydro geological properties To include all these parameters in estimation of rock mass is hard or even impossible. Limited number of the most representative parameters is therefore often used to describe the rock mass behaviour. Number of design and classification systems has been derived for this purpose and some of them are described in later chapter (Palmstöm & Nilsen, 2000). Compression and shear strength of rock along with properties of the discontinuities in the rock mass is usually the foundation of classification and design system in geotechnical engineering. Knowledge of the virgin stress field around the underground opening is also necessary to estimate deformations and stress concentrations around the opening The estimation of rock mass properties is usually the most challenging factor in geotechnical design. Quality of the structure design is never higher than the quality of the data used to estimate the rock mass parameters. Pre-geological investigation plays therefore an important role in the design process of every underground structure and should be comprehensive enough to make the design believable and reliable. Rock mass properties are highly dependent on its origin. Properties of igneous rock which have solidified from magma differ slightly dependent of the geological circumstance it’s formed in but rock mass of sedimentary type has completely different characteristics. Basic knowledge of geology is there for needed to choose a suitable design or classification systems. 3 2.1 Icelandic rock mass Iceland is located on the boundary between the Eurasian and North American tectonic plates called the Mid Atlantic ridge. The plates drift apart at the speed of 1-2 cm/yr and the void between the plates is constantly filled with igneous rock. The North Atlantic opened around 60 million years ago and has from that time formed the North Atlantic sea floor generated from the modern day plate boundary the Mid Atlantic ridge (Sigmundsson, 2006). Figure 2.1 Simplified geological map of Iceland (Weisenberger, 2010). The youngest rock is located near the volcanic active zones of Iceland and the oldest rock, 14-16 million years old, is located in the north west and east coast. Icelandic bedrock consists 80-90% of sub aerial basalt, 5-10% of acidic and igneous rocks and 5-10% of sedimentary interbeds. Typical Icelandic bedrock consists of relatively thin basalt layers with thin scoria layers on top and bottom of each layer and thin sedimentary interbeds. The strata usually dip slightly towards the volcanically active zone. Since the basalt layer are usually thin <10m a mixed face condition is very common in Icelandic tunnels and sometimes the dominating condition in tunnel projects (Harðarsson, 1991). Mix faced condition can be seen on Figure 2.2. 4 Figure 2.2 Mixed face. Basalt layers with approximately 2m sediment interbed. Other rock formations like dykes and breccias are also frequently crossed during excavation of tunnel. Sedimentary intebeds is also frequent in Icelandic rock mass but are usually thin layered, from few centimetres to 1 or 2 meters. Occasionally thicker sedimentary layers occur. Main focus of this report is tunnel excavation in sedimentary rock and properties of sedimentary rock will therefore be most discussed. Sedimentary rock can be divided in two main categories by its formation: 1. Where material particles has been transported to the place of deposition, known as clastic or detrital rock. Conglomerates, sandstones, siltstones, mudstones are of that kind. 2. By an aggregation of organic matter or chemically/biochemically. Limestone, chalk and coal are of that kind. Sedimentary rock is not interlocked like basalt rock but cemented together with an intergranual matrix by diagenesis (i.e. hardening of loose materials to rock) (Palmstöm & Nilsen, 2000). Sedimentary rocks in the Tertiary bedrock are mostly fine grained tuffaceous interbeds and some thicker conglomerates. The Pleistocene bedrock is usually more abundant of 5 sedimentary rock, mostly sandstones and conglomerates of glacial origin (Harðarsson, 1991). Thick sedimentary interbeds can cause some serious instability problems specially if joined with low strength, swelling potentials or sensitivity to water exposure. 2.2 Characteristics of rock mass Estimation of rock mass characteristics is required to design an underground excavation. Methods such as the generalized Hoek-Brown criterion and Mohr-Coulomb failure criterion can be used to describe the characteristic behaviour of rock mass like strength and deformations. Measured data from core samples are often used to estimate the properties of intact rock (no weakness planes) and from that point through empirical approach to estimate the behaviour of the overall characteristics of the rock mass surrounding an underground opening. Strength of intact rock sample is usually higher than the overall strength of the rock mass and method are therefore needed to convert measured data from core samples to the rock mass(Hoek, 2000). 2.2.1 Generalized Hoek-Brown criterion Hoek and Brown proposed in the 1980’s a method to estimate the strength and properties of a jointed rock mass called Hoek-Brown failure criterion. The method is based on estimation of interlocking between rock blocks and shear conditions in the joints. This method was derived to be used to estimate strength of jointed rock mass where rock blocks are small relative to the excavation considered. The method has been modified over the years but the version introduced in this chapter is a modified version from 2002 (Hoek, Carranza, & al, 2002). The following series of equations represents the criteria: o′ 1 = o′ 3 +o cì _m b o′ 3 o cì +s] u Eq. 2-1 m b = m ì c [ uSI-100 28-14Ð ¸ Eq. 2-2 s = c uSI-100 9-3Ð Eq. 2-3 o = 1 2 + 1 6 (c -uSI¡15 -c -20¡3 ) Eq. 2-4 where o' 1 and o' 3 are the maximum and minimum effective principal stresses at failure, m b is the Hoek-Brown constant for the rock mass and m i is the Hoek-Brown constant for the intact rock samples, s and a are constants related to the rock mass characteristics and o cì is the uniaxial compressive strength of intact rock sample. GSI is the geological strength index introduced by Hoek in 1994 to simplify the conversion between the intact rock strength and the rock mass strength. 6 A disturbance factor D is used to take into account the disturbance from blasting and stress relaxations in the rock mass. To estimate the value of m i the equation 2-1 is used with S=1 and a=0,5 and becomes: o′ 1 = o′ 3 +o cì _m ì o′ 3 o cì +1] 0,5 Eq. 2-5 A series of triaxial test on core samples can therefore be used to determine the value of m i and o cì . Authors of the method recommends that series of at least five triaxial tests should be used (Hoek, Carranza, & al, 2002). Authors also recommend that the range of o' 3 should be equally distributed between zero and 0,5 time the intact compression strength. o' 3 can be set to zero in equation 2-5 to reveal the uniaxial compression strength of the rock mass and becomes: o c = o cì s u Eq. 2-6 The tensile strength of the rock mass can be found in a similar way by setting o' 1 equal to zero and the tensile strength becomes: o t = - so cì m b Eq. 2-7 Figure that can be used to estimate the value of GSI and D can be found in appendix A2. 2.2.2 Mohr-Coulomb failure criterion The Hoek-Brown failure criteria is well suited for jointed or heavily jointed igneous rock types like basalt. But for other rock types like sedimentary or metamorphic rock the Mohr- Coulomb failure criterion can be a better choice (Hoek, 2000). Estimation of shear strength can be made by the Mohr-Coulumb eqution: 7 ¡ ] = C +o ′ n tan(0) Eq. 2-8 where θ is the internal friction angle of the intact rock sample, C is the cohesion and o i n is the normal stress acting on the plain of failure (Erlingsson, 2009). The Mohr-Coulomb equation can also be written as: o 1 = 1 +sin (0) 1 -sin (0) o 3 + 2Ccos(0) 1 -sin (0) = C - +tan (ç)o 3 Eq. 2-9 Uniaxial compression strength and tensile strength can be derived from equation 2-9 by putting o 1 and o 3 to zero respectively, thus o 1 = 2Ccos(0) 1 -sin (θ) = o c Eq. 2-10 o 3 = 2Ccos(0) 1 +sin (0) = -o 1 Eq. 2-11 Figure 2.3 Mohr-Coulomb and Hoek-Brown failure criterions (Hoek, 2000) 8 2.3 Stresses in rock mass Stresses within the rock mass are of great interest when designing an underground excavation. Compressive stresses are by convention positive and tensile stresses have negative prefix. Both high and low stresses can cause instability problems in underground excavation. If stresses exceed the compression strength of the rock mass it yields and low stress condition can cause rock blocks to slide due to low normal stress in the joints. Stresses surrounding an underground opening are influenced mainly by the stress condition prior to the excavation (virgin stresses) and the geometry of the opening. Number of components influences the direction and magnitude of the virgin stress field. Main components influencing magnitude and direction of the stress field are (Palmstöm & Nilsen, 2000): Gravitational stresses are caused by gravitational force that pulls all material towards the centre of the earth. Gravitational stresses increase with depth and 0,027 MPa/m is a common number to calculate the vertical components of the gravitational stress field. Poisson ratio is often used to describe the relationship between the vertical and horizontal component of the gravitation stresses in elastic materials. Poisson ration does not however correspond very well with stress conditions in jointed rock mass. Topographic stresses are caused by uneven ground surface. Uneven surface causes stress concentrations and uneven stress field in the rock mass near the ground surface like in the bottom of deep valleys and mountain slopes. The horizontal component is the overall stress field can become the dominant factor near the surface in such cases. Tectonic stresses are mainly caused by plate tectonic or the continental drift. Major faults and folds are caused by tectonic stresses. Variation in tectonic stresses is high and that is one on the reasons why the horizontal component of virgin stresses are much more difficult to estimate than the vertical one. Residual stresses. During the earlier geological stages of the rock mass it locks in some internal stresses. Vetical component can generally be calculated by equation 2-12. o ¡ = Jcptb(m)xu,u27 HPo m , Eq. 2-12 The letter k is usually used to describe the ratio between horizontal and vertical stresses and the horizontal stress field becomes: o H = kxo ¡ Eq. 2-13 Origin and behaviours of virgin stress field can be very complex and hard to estimate. The vertical component of the stress field can easily be calculated with certain accuracy but the 9 horizontal components will always be a rough estimation unless some rock stress measurement has been performed. Stress measurements are time consuming and expensive but also very necessary to perform if a design of an underground excavation need relatively accurate estimation of the horizontal stress field. Number of stress measurements has been done in Iceland, mostly related to construction of hydroelectric power stations. Figure 2.4 shows overview of various stress measurements in Iceland. Figure 2.4 σ H vs depth at various locations in Iceland. Teigbjarg, Sandfell and Blanda (Haimson, 1981) Reyðarfjörður (Haimson & Rummel, 1981) Fárskrúðsfjörður tunnel (Dahle, 2005) Figure 2.4 shows that horizontal stresses at relatively shallow depth gives in most cases higher k value than measurements at greater depth. This is not surprising since both topographic and tectonical stresses are responsible for higher portion of horizontal stresses at shallow depth. Measurements of horizontal stress around the world also show that the value of k tends to be high at shallow depth and decreases at depth (Hoek, 2000). 0 100 200 300 400 500 600 700 800 0 5 10 15 20 25 30 d e p t h   [ m ] σ Hmax [Mpa] Reydarfjordur Teigsbjarg Sandfell Blanda Fárskrúðsfjörður tunnel σv (0,027xd) σH/σv = 0,5 σH/σv = 1,5 10 2.4 Discontinuties in the rock mass Discontinuities divide the rock mass into blocks of different scales. The blocks can be categorised by their scale as (Palmstöm & Nilsen, 2000): 1. First order fault block defined by larger weakness zones or faults. 2. Second order blocks formed by small weakness zones or seams. 3. Third order blocks formed by normal joints 4. Small discontinuities such ad bedding or schistosity partings. 5. Small fragments or grains in the rock. Discontinuities that form blocks that fall into category 5 are usually considered as part of the rock property and generally included in the strength characterisation of the rock. Shear strength of discontinuities is an important factor in estimation of characteristic of jointed rock mass. Numerous criterias and test methods have been derived to estimate shear strength of discontinuities (Hoek, 2000), but since the main focus in this report is cemented sedimentary rock, a detailed description of those criterias is outside the scope of this report. Figure 2.7 shows the main ideology of shear stress criterias in discontinuities. Figure 2.5 Ideology of most stress criterias 11 3 Types of rock support Rock support is added to improve stability of underground opening. Pre-geological investigation are used to reveal the main characteristics of the rock mass surrounding the opening and a design is made to deal with various geological conditions in the tunnel. Geological conditions often change very rapidly on the tunnel route and that require flexible support methods that can be quickly adjusted to the current circumstances (Palmstöm & Nilsen, 2000). Rock bolts and shotcrete are dominant support methods in Icelandic and Scandinavian tunnelling industry and has been for a long time. Shotcrete rips were first prescribed as an optional support method in Hvalfjörður subsea tunnel in 1996 but were not used until in Fárskrúðsfjörður tunnel in 2003 (Harðarson, 2011). Steel arches are not commonly used but have been used for example in Oddskarðs tunnel and hydro tunnel in the Kárahnjúkar project (Loftson, 2011) Main rock support methods will be described in this chapter and there pros and cons discussed. 3.1 Rock bolts 3.1.1 General Development of rock bolts began in the 1920s and has since then become the most dominant support method in underground construction (Luo, 1999) Rock bolting is a flexible method very commonly used for rock support. Rock bolts are frequently used as initial support at the tunnel face to obtain safe working conditions for the crew and they also form part of the final rock support (Palmstöm & Nilsen, 2000). After securing loose block at the excavation face by spot-bolting, a systematic bolting is often used. Then a previously defined pattern of rock bolts are installed based on the geological conditions. Geological mapping, Q-value or other methods are mainly used for the design of the required pattern to secure the rock mass. A systematic bolting can be installed at the end of excavation or during excavation of the tunnel. Figure 3.1 shows spot-bolting and systematic bolting. Figure 3.1 Fixing of a single block and systematic bolting (Palmstöm et al, 2000) 12 The following equation for bolts lengths has been suggested by Palmstöm (2000) for bolting of single loose block: IB ¡oo] = 1,4 +u,16Ðt _1 + u,1 Ðb ] Eq. 3-1 Ib wuII = 1,4 +u,u8(Ðt +u,Swt) _1 + u,1 Ðb ] Eq. 3-2 where Db = the block diameter (in meters) Wt = the tunnel wall height (in meters) Dt = the diameter or span of the tunnel (in meters) 3.1.2 Types of rock bolts Rock bolts are produced of various types and lengths. Rock bolts can be categorized as follows: - End anchored bolts. - Fully grouted bolts. - Combination bolts. - Swelling bolt - Self drilling bolts End anchored bolts can be used when fast activation of the bolts is needed. The anchor can be expansion shell, polyester grout or mortal (cement grout). End anchored bolts are often used for spot bolting to secure unstable blocks at the tunnel face. Fully grouted bolts are passive bolts and are not activated in the installation phase. Deformations in the rock mass will eventually activate the bolt after the grout has cured. Fully grouted rock bolts are often used for systematic rock bolting. Cement based grout is mostly used to fasten the bolt but polyester is also possible to use, especially for short bolts. Grout gives the bolt protection from corrosion but the bolts are also often galvanized or epoxy coded. Combination rock bolts are end anchored bolts that can be fully grouted afterwards. The most common use combination bolts in modern Scandinavian tunnels are CT-bolts. Polyethylene sleeve gives the CT-bolt additional protection from corrosion. CT-bolt are relatively stiff since it is both end anchored and fully grouted and are therefore very convenient for use in weak rock or heavily jointed rock. CT-bolts are however not 13 applicable in rock mass where high deformation can be expected since its stiffness can lead to failure in the rock bolt (Statens Vegvesen, 2000). Figure 3.2 Most used rock bolts in Icelandic tunnels. Figures from (Statens Vegvesen, 2000) Swelling rock bolts can be used as initial rock support but are not qualified as permanent rock support by the Icelandic road authorities. Swelling bolts have direct contact with the rock mass and therefore in great threat of corrosion. Installation procedure is however very fast and therefore favourable under special circumstances. Self drilling rock bolts are sometimes used in very weak or heavily jointed rock mass. Sometimes it’s very hard to keep the bore hole unblocked after drilling and therefore the installation of normal rock bolts are hard or impossible. That problem can be avoid with the self drilling rock bolts since it doesn’t require stable bore hole since it drills itself in. Self drilling rock bolts are however expensive because the drill bit at the end of the bolt cannot be retained (Statens Vegvesen, 2000). Rock bolts are sometimes installed ahead of excavation to support assumed unstable rock mass or rock mass with insufficient rock cover. Those rock bolts are called spiling bolts and are installed in a fan shaped pattern oriented 10-25°relativly to the tunnel axis. Spiling bolts are usually fully grouted with spacing of 30-80cm (Palmstöm & Nilsen, 2000). Figure 3.3 shows typical application of spiling bolts. 14 Figure 3.3 Application of spiling bolts in a weakness zone. Table 3.1 Properties of commonly used rock bolts (based on Palmstöm & Nilsen, 2000) Type of rock  bolt  Diameter  [mm]  Yield strength  [kN]  Failure load  [kN]  Elongation at  failure  Elongation for 3  m bolt [mm]  Standard  bolt length  [m]  Round steel bar  20  60‐70  100  8%  240  0.8‐6.0  Deformed bar  20  120  150  3%  90  0.8‐6.0     25  220  250  1%  30  0.8‐6.0  CT‐bolt  20  120  150  3%  90  1.5‐6.0     22  200  250  2%  60  1.5‐6.0  Hollow bolt  27  100  130  8%  240  2.0‐6.0  3.2 Shotcrete 3.2.1 General Shotcrete is a widely used method for tunneling support. This type of rock support is obtained by spraying concrete on the rock surface. Shotcrete for rock support has been used for several decades and has become increasingly popular because of its favorable properties together with high capacity and flexibility (Palmstöm & Nilsen, 2000). 15 Figure 3.4 Shotcreting The main advantages of shotcrete as rock support are (based on Palmstöm & Nilsen, 2000): - Ready to use on short notice (batching plant on site) - Framework is not needed - Applicable in uneven excavation profile - Easy to combine with other support methods - Flexible deformation properties. The main disadvantages are: - Low tension strength (can be improved by mixing fibers or reinforce with casting steel) - Can collapse when applied on swelling rock types - Hard to apply on rock types with low cohesion (some types of sedimentary rock) - Hard to apply on rock with flowing water. Shotcrete can be used in multiple variations as rock support: 16 1. Ordinary shotcrete spayed in layers of up to about 60-100 mm thickness. 2. Mesh reinforced shotcrete. This is produced by first spraying a layer of concrete before installing a mesh with typically 5-6 mm diameter steel bars. Then a second and sometimes more layers are applied to cover the mesh entirely. 3. Fiber reinforced shotcrete. Steel or plastic fibers are mixed with the wet concrete. 4. In combination with steel ribs, lattice girders or steel beams. Al these mentioned methods are typically used with combination of rock bolt to fasten loose block or to fasten the wire mesh or steel sets (Palmstöm & Nilsen, 2000). 3.2.2 Shotcrete methods Wet mix shotcreting is now the dominant shotcrete method in modern construction of tunnels and has almost replaced the previous dry mix method. The difference between these two methods are that the wet-mix method uses fully mixed concrete and blows it out of the nozzle of the shotcrete robot using compressed air. The mixing water is however not added to the dry-mix until in the nozzle in the dry-mix method. That gives the operator the change to adjust the dosage of water during shotcreting. The wet mix method is however more convenient for ordinary tunnel shotcreting since it has less rebound, higher capacity and gives better working conditions. 3.2.3 Shotcrete mix When producing concrete one uses a various types of recipes based on what properties of the concrete one wishes to get. Table 3.1 shows a typical mix for concrete used for shotcrete support in tunnelling in Norway (similar in Iceland). Table 3.2 Typical mix for C40 wet Shotcrete Portland cement 470 kg Micro-silica 8% Aggregates 0-8mm 1670 kg Superplasticizer (BNS) 5 kg Plasticizer (lignosulphonate) 3,5 kg Steel fibers 50 kg Accelerator (modified silicate) 5% The amount of micro-silica and accelerator are calculated as percentage of cement. Micro- silica in the mix improves the strength properties of the shotcrete and makes it easier to distribute the steel fibers in the fresh shotcrete. It also reduces permeability and improves frost resistance (Palmstöm & Nilsen, 2000). 17 3.2.4 Steel fiber Unreinforced shotcrete is a brittle material that experiences cracks and displacement under relatively low tensile stress compared to its compression strength. Steel fibers have been used since the late 1950’s to improve the concretes tensile properties. By adding 1 vol % of steel fibers into a shotcrete mix can increase the load capacity of a C50 shotcrete slab by 85% (Palmstöm & Nilsen, 2000). The usage of steel fibers has made it possible to use a combination of rock bolts and fiber reinforced shotcrete instead of full cast lining. A full cast lining is a very time consuming process and expensive so replacing that with rock bolts and fiber shotcrete is very economically favourable. Figure 3.5 Deflection curves for concrete slabs with and without steel fibers (Palmstöm & Nilsen, 2000) Plastic (polypropylene) fibers can also be used in increase the tensile strength of concrete and are often used for surface shotcrete for example as fire-protection for PE-water membrane. 3.2.5 Curing of shotcrete To increase the curing rate of shotcrete it is necessary to use so called accelerator. It´s mixed with the shotcrete in the nozzle and reacts with cement in the shotcrete. The dosage is usually 4-8% of cement weight in the mixture. Usage of accelerator gives shotcrete initial curing to some degree so it doesn’t fall of the rock surface. The influence of the accelerator on curing and measurements of compression strength of core specimens of shotcrete from Óshlíð tunnel can be seen on figure 3.6. After 15 hours the compression strength has reach 20 MPa. Shotcrete with no accelerator will reach this value after 4-5 days (Einarsson, 2010). 18 Figure 3.6 Normal and measured curing of C30 shotcrete (Einarsson, 2010) When estimating the bearing capacity of shotcrete it’s necessary to have information on the curing rate of the shotcrete. Bearing capacity of a shotcrete lining is time dependant but the stresses form the tunnel wall is mainly dependent on the distance from the tunnel face. Right timing to activate a tunnel lining is therefore critical and is discussed further in relations with ground reaction curves in Chapter 5. 3.3 Wire mesh Wire mesh is commonly used with combination of rock bolts and shotcrete. Use of fiber reinforced shotcrete has however reduced the need for wire mesh since installation of wire mesh is time consuming and is therefore only used where fiber reinforced shotcrete is not considered to be sufficient (Palmstöm & Nilsen, 2000). Wire mesh is also commonly used in combination with shotcrete ribs and lattice girders to distribute forces from the rock mass to the bearing elements. 0,1 1 10 100 0,1 1 10 100 U C S   [ M P a ]   Curing time [hours] 6 % accelerator 4% accelerator No accelerator Measured Óshlíð tunnel 6% Measured Óshlíð tunnel 5% 19 Figure 3.7 Wire mesh to support local unstable area. Wire mesh is also used where low cohesion between shotcrete and rock surface is causing fallouts of shotcrete. These circumstances often occur during shotcreting of very wet, heavily jointed or sedimentary rock surfaces. Local shear failures due to pull weight of the shotcrete is also common in rock types with low cohesion strength and are often supported with wire mesh before shotcreting. Main advantages: - Increases shear strength and ductility of a shotcrete lining. - Reduced fallout of shotcrete during shotcreting. Main disadvantages: - Installation is time consuming and therefore replaced by steel fiber if possible. 3.4 Shotcrete ribs 3.4.1 General Then the normal support methods like shotcrete, rock bolt and wire mesh are not considered sufficient to secure long term stability in the rock mass, one must consider alternative heavier rock support methods. A full concrete lining in weak tunnel rock is a time consuming and expensive support method. Therefore a reinforced shotcrete ribs are a very favourable alternative to be used where it’s applicable (Palmstöm & Nilsen, 2000). 20 Shotcrete rips are built by fastening rebar steel along the tunnel profile with rock bolts and then covered with shotcrete. Diameter of the rebars is usually 20mm (Norwegian standard) and number of rebars and spacing between rips is adjusted to the local condition and required bearing capacity. Figure 3.8 Radial rock bolting of a shotcrete rib (Vegagerðin, 2008) 3.4.2 Types of shotcrete rips Shotcrete rips can be either single or double layered as shown on Figure 3.7 and 3.8. Single layered shotcrete ribs are constructed by tying the rebars to a fastening rod fixed at the end of the rock bolts. Main disadvantage with single layered rips are their low moment capacity. Rebars on top and bottom of a double layered rip however provide the rip with much larger moment capacity. 21 Figure 3.9 Single layered shotcrete rib (Statens vegvesen, 2009) Figure 3.10 Double layered shotcrete rib (Statens vegvesen, 2009) 22 3.4.3 Installation procedure (Norwegian standard) Before installing the first (or only) layer of rebars, the rock surface must be smoothened with shotcrete. This layer is often 100-200mm thick fiber reinforced shotcrete (Palmstöm & Nilsen, 2000). After smothering of rock surface, the following procedure takes place: - Rock bolts installed with 1-1,5 m separation based on rock class (Norwegian standard) - Fixing bars are fastened on all rock bolts. - Reinforcement bars (20mm) are fastened (welded or tied) to the fixing bars. - The rib is shotcreted so that minimal shotcrete cover is 50mm (75mm for subsea tunnels) 40mm spacing from reinforcement bars to the smoothening shotcrete is required before shotcreting the rip. Min 110mm spacing between parallel rebars is required. Point 2-4 in the procedure is repeated to build a double layer shotcrete rip (Pedersen et al, 2010). 3.4.4 Usage Shotcrete rip can be considered as a beam that experience load from the deforming rock mass. Because of the circular formation of the rip the load from the rock mass is transferred to axial stress in the rib. The rib can however also experience moment where the point load or uneven loads from the rock mass occur. Steel reinforcement is supposed to grant the rip with enough moment capacity to withstand those moments. Shotcrete rips are also commonly used where overburden of the tunnel is low. Where stresses and deformation are low can the ribs be installed relatively far from the face if the short time stability is sufficient to provide the workers with safe working conditions at the tunnel face. Therefore they can be installing in greater number to gain productivity. Figure 3.11 shows installed shotcrete ribs in the Stjördal tunnel where overburden is only 2-4m (Guðmundsson, 2011). 23 Figure 3.11 Installation of single layer shotcrete ribs in Stjördal tunnel, Norway (Guðmundsson, 2011). Main advantages of shotcrete rips are: - Material is on site or available with short notice. - Low preparation time. - Adapt to irregular profile. - High moment capacity if double layered Main disadvantages are: - Needs to be rock bolted in the whole contour. - Difficult to use as a support for spiling bolt if profile is irregular. - Low moment capacity if single layered. 3.5 Lattice girders 3.5.1 General Lattice girders have been used for tunnel support since the late 1970s. In recent decade a move from heavy rolled steel arches to lighter, more manageable lattice girders has taken place (Komselis et al, 2005). Steel arches and lattice girders have basically the same function. They can be used as an element of temporary support lining or a part of the permanent lining. Lattice girders were for example used as part of the permanent lining in the underground transport system in Bavaria, Munich in Germany with great success (Baumann & Betzle, 1984). 24 Lattice girders are very similar to normal shotcrete ribs, the main thing distinguishing these two support methods are the installation method. The final product is a bar reinforced shotcrete lining. Main advantages of lattice girders are (based on Komselis et al, 2005): - Simple and fast installation - Solid support for spiling bolts - Temporary support for shotcrete until in gains sufficient strength to support itself. - Immediate support in the area of the tunnel face. - High moment capacity Main disadvantages are: - Needs to be ordered in time or constructed on site (min 1-2 weeks in Iceland). - Leads frequently to increased usage of shotcrete 3.5.2 Types of lattice girders Two types are most commonly used, three-chord and four-chord. Three-chord lattice has triangular section with a larger bar (25-40mm) at the apex and two smaller diameter bars at the base corners. Sinusoidal bars (10-12mm) separates the apex and the main bars. Four- chord lattice has four equally sized bars (20-40mm) at the corners of rectangular section. Sinusoidal side bars and cross bars (16mm) separates the main bars. Figure 3.12 Thee and four-chord lattice girders (Jorimann, 2010) Design diameters stated in this chapter are the most common used bar diameters in lattice girder production. Other bar sizes is also possible (Komselis et al, 2005). 25 3.5.3 Installation procedure Installation procedure of lattice girders is highly related to the geological circumstances and the purpose on their installation. It can be used as a temporary rock support to secure a safer working environment for the workers at the face or considered as part of the final rock support lining. Following description of installation procedure is considered applicable for installation of lattice girders where sedimentary rock is partly or the only rock type in the profile and where the rock mass stresses are not preventing that final support lining is at least partly installed very close to the tunnel face. The installation procedure is in ten steps and every step will be described. 1) Proper scaling of loose rock blocks is always the first step in every sequence of installation of rock support. It minimizes the risk of fall down of shotcrete and makes the working environment safer at the tunnel face. Scaling of under breaks in the profile must also take place at this stage before shotcreting. Scaling in sedimentary rock with very low cohesion strength can though be risky. Few cubic meter of lost shotcrete is then better than risking large overbreaks by scaling to harshly with an excavator.      Figure 3.13 Scaling of under breaks and loose rock blocks 26 2) After scaling the tunnel walls and roof must be secured by installation of shotcrete. Installation of the initial layer of shotcrete in sedimentary rock can be very time and material consuming if the rock is weak. The weight of the shotcrete is sometimes causing too much shear stress in the rock and causing both shotcrete and rock to fall down. The shotcrete operator should though be able to build up the shotcrete shell by starting at the floor or harder rock and slowly cover the unsupported area.        Figure 3.14 Initial layer of shotcrete 3) In this case(as usually) the lattice girders is the first structural member to be installed since the initial shotcrete layer is not considered a structural member but a safety measure and is not involved in bearing capacity calculations since it’s thickness varies. The lattice girder is assembled on the tunnel floor and raised by an excavator or by the drill rig. To make sure that the girder is at the right position it must be check by a laser guidance system or a surveyor.    Figure 3.15 Installation of lattice girder   4) The girders must be fastened into place. All girders must at least be fastened as close to the floor as possible. This is done to prevent that the end of the girder slides from the wall when forces from the deforming rock mass starts to act on the 27 girder. Extra rock bolt can be added if hard stable rock is somewhere in the profile. 5) Wire mesh is usually installed behind the lattice girder. It both adds tensional strength to the shotcrete between the girders and also help with transferring the load to the main bearing units which in this case is the bar reinforced concrete beam build up by lattice girder and shotcrete. Rebars must be installed to overlap the junctions of the girder to unify the moment and shear capacity of the girder. 6) Shotcrete is now used to cover the wire mesh and lattice girder. In this round the girder is only half embedded in shotcrete to prevent that the girder is overloaded with uncured shotcrete that is not able to support itself.       Figure 3.16 Girder half embedded in shotcrete   7) The girder is now used as a cantilever for spiling bolts used to support the rock for next blasting round. The holes for spiling bolts are drilled through the lattice girder and the bolts installed. By installing a tight series of spiling bolt the rock ahead has been supported and chances of overbreaks reduced.    Figure 3.17 Installation of spiling bolts 28 8) The next tunnel round is now excavated. 9) After repeating stage 1-5 above in order to install the next lattice girder the previously installed girder is connected to the new one with side bars and covered with shotcrete. A continuous supporting lining is achieved by connecting all girders with sidebars.     Figure 3.18 Lattice girder fully embedded with shotcrete 10) Installation of the previous lattice girder is now finished and stage 6-10 can now be repeated for the current lattice girder. As previously stated is this working procedure of lattice girders adjusted for use in conditions where sedimentary rock layers are causing instability at the tunnel face. In other geological condition the work procedure could be slightly different and must always be adapted to the local conditions. Similar work procedure was successfully used in the construction of Óshlíð tunnel. 29 3.6 Steel arches Steel arches and lattice girders are very similar construction elements in tunnels. They can work as a temporary support or as a part of final support lining. Arches are often used as a temporary support in multiple drift tunnelling since it easy to disassemble (U.S. Department of transportation, 2011). Steel arched are rarely used in Iceland but has been used in Kárahnjúkar hydro project and Oddskarð tunnel. Based on (U.S. Department of transportation, 2011) and (Hoek et al., 2008): Main advantages of steel arches are: - Immediate support after installation - Excellent support for spiling bolts - Easy to disassemble and for forming temporary shotcrete wall in multiple drift excavation - High deformability if granted with sliding joint. See Figure 3.19 main disadvantages are: - Heavy and expensive - Poor bonding with shotcrete Figure 3.19 Steel arch with sliding gap (Hoek et al.,2008) 30 4 Estimation of required rock support Estimations of required rock support is usually based on observation, experience and personal judgment of those involved. Engineers generally uses support guidelines or methods to back up their estimation of required rock support. Three types of methods are mostly used for this purpose (Palmstöm & Nilsen, 2000): - Analytical methods, involving analysis of stress distributions and deformations using methods like numerical analysis, analogue simulation or physical modeling. - Observational methods, like the New Australian Tunnelling Method which uses measurements of movements in the rock mass during excavation. Observations are off course also used to check if the chosen installed rock support was the right way to go or not and adjustment made if required. - Empirical methods, often illustrated in table or graphs that connect classification of rock mass to curtain rock support. Number of empirical methods has been derived such as the RMR system end the Q-system. General overview of the most commonly used empirical methods will be given in this chapter and the use of the Final Element as an analytical method will be used in case studies later on in this report. 4.1 Rock quality designation index (RQD) Deere et al. developed the RQD system in 1967 to estimate the quality of rock mass from drill cores. The RQD value is defined as the summarized length of all core pieces longer than 100mm divided by the total length of drilled core (Hoek, 2000). Figure 4.1 Example of RQD-value estimation (Hoek, 2000) Nowadays the RQD system is mostly used as a sub-system in the RMR and Q-system. 31 4.2 The RMR-system The RMR-system was introduced by Bieniawski in 1973. The tunnel route is divided into section where the geological properties are the same or similar and the tunnel support is assigned to those sections according to Table 4.1 (Palmstöm & Nilsen, 2000). Six geotechnical parameters are used to classify the rock mass (Erlingsson, 2009): 1. Uniaxial compression strength 2. RQD-value 3. Spacing of discontinuities 4. Conditions of discontinuities 5. Ground water conditions 6. Orientation of discontinuities Table 4.1 Rock mass Rating system-RMR (enlarged in appendix A1) (Hoek, 2000) 32 Results from the RMR classification are given in number from 0-100. Table 4.2 shows how guidelines have been derived for rock support in 10m span tunnel for estimated RMR value. Table 4.2 Guidelines for rock support and excavation of 10 span tunnel according to RMR system. (Hoek, 2000) 33 4.3 Rock tunneling quality Index, Q-system 4.3.1 Determination of the Q-value The Q-system has been the dominated rock mass classification system since 1980 in Iceland. The system was developed by Barton et al. in 1974 based on tunnelling experience mainly in Norway and Sweden (Palmstöm & Nilsen, 2000). In this classification system are the joints and other discontinuities measured or estimated from core samples or visible rock mass surface and thereafter the Q-value estimated for the rock mass. (Loftsson, 2009) The Q-value is a number that varies on logarithmic scale from 0,001-1000 and is defined as: µ = RµÐ [ n x [ ¡ [ u x [ w SFR Eq. 4-1 where: RQD is the Rock Quality Designation [ n is the joint set number [ ¡ is the joint roughness number [ u is the joint alteration [ w is the joint water reduction The Q-value can be considered as a function of tree parameters; where each parameter represents different aspects of general rock mass strength (Erlingsson, 2009): RçÐ ] n is and estimation of block sizes. ] r ] c is an estimation of shear strength in joints. ] w SPR is an estimation of the active stresses in the rock mass. Table to determine the value of individual parameter of the Q-system can be found in the Appendix A1. The Q-system has been used in Iceland to estimate required rock support in tunnelling since 1980. Determination of individual parameters has been adjusted to Icelandic rock mass based on experience in various tunnel projects in Iceland (Loftsson, 2009). The Icelandic road Administration published a report in 2009 (Loftsson, 2009), where this experience is used to establish some guidelines to estimation of individual parameters of the Q-values for different types of Icelandic rock mass. An overview of the guidelines for determination of different rock types for Icelandic rock mass is given in Table 4.4 34 4.3.2 Required rock support for estimated Q-value Barton et al. introduced in 1974 an addition parameter, Excavation Support Ration (ESR) to take into account a factor of safety correlated to the importance of tunnel construction. Table 4.3 gives Bartons suggested ESR values for different types of tunnel constructions. Table 4.3 ESR values (Hoek, 2000) Figure 4.2 shows estimated support needs for given Q-value and span of tunnel (Hoek, 2000). Support giuedlines from the Norwaigan standard for estimated Q-value is in the Appendix A1. Figure 4.2 Rock support according to the Q-system (Hoek, 2000). 35 Table 4.4 Typical parameters for estimation of Q value in Iceland (Loftsson, 2009). Rocktype RQD‐ value  J n  value  Basalt Usually 60-85 for low or medium jointed. 30-60 for heavily jointed rock mass Usually 9 but 12 for flaky olivine basalt. Scoria For hard well cemented, 50-80 but 30-50 for low cemented rock mass Usually no joint systems. 9-12 is used for well cemented and 15- 20 for weak cemented. Sediment Values of 40-60 are common but 10-30 if layered. If UCS <3Mpa then 10. 9 in most cases but 6 if not jointed. 15-20 if very weak UCS< 3 Mpa Breccia Usually immeasurable so lowest value of 10 is typically used. 15-20 is normal value. But use 20 if an excavator is able to tear it down. Dykes Usually 50-70 for for low or medium jointed. 20- 40 for heavily jointed rock mass Usually 12-15. Use 15 if very jointed. J r value J a value Basalt Usually 2-3. Usually 2-3. 2 if fillings are very thin and 3 if filling involves clay particles. Up to 4-8 if depending on thickness and amount of clay. Scoria Usually 2-3. But 2 or lower if the rock mass is shifted. Usually 2-3 and similar to the surrounding basalt. Sediment Usually 3-4 since joints are usually smooth. Use 3 for sandstone and conglomerate rock. Usually 3-4. 2 can be appropriated for hard sandstone. Breccia Usually 0,5-1 since joint are usually smooth. Usually > 4. Up to 8-12 if filling are preventing interlocking of rock blocks. Dykes Usually 2-3 if joints are irregular. Usually 2-3. If filling include clay then 4. J w SRF Basalt 1 if the tunnel is dry < 5l/min. 0,66 if water pressure is < 2,5 bar and 0,33 if water pressure is > 2,5 bars. Usually 60-85 for low or medium jointed. 30-60 for heavily jointed rock mass Scoria 1 if the tunnel is dry < 5l/min. 0,66 if water pressure is < 2,5 bar and 0,33 if water pressure is > 2,5 bars. For hard well cemented, 50-80 but 30-50 for low cemented rock mass Sediment 1 if the tunnel is dry < 5l/min. 0,66 if water pressure is < 2,5 bar and 0,33 if water pressure is > 2,5 bars. Values of 40-60 are common but 10-30 if layered. If UCS <3Mpa then 10. Breccia Usually < 1 since water flow is often connected with breccia. 0,33 if water pressure is 2,5-10 bars. Usually immeasurable so lowest value of 10 is typically used. Dykes Usually 0,66 if dry or 1 if low water flow. 0,33 if water pressure is higher than 2,5 bars. Usually 50-70 for low or medium jointed. 20-40 for heavily jointed rock mass 36 4.4 Numerical analysis Numerical modelling of rock mass is done by dividing the rock mass into large number of individual elements and use computer to calculate how they interact for given stress and stiffness conditions. The finite elements method used in this report is in the category of continuous models which means that the rock mass is considered to be continuous rock mass and only a limited number of discontinuities (joints, faults, etc.) may be used. To construct a finite element model of rock mass one must define geological properties of the rock mass and the virgin stress field. Element mesh of the rock mass is then generated to divide the rock mass into individual elements and boundary conditions determined. Finally the magnitude of stresses and deformations are calculated for every nodal point in the mesh for given rock mass and boundary conditions (Palmstöm & Nilsen, 2000). The computer program Phase2 (version 7) is used in this report to construct finite element model of rock mass surrounding tunnel in sedimentary rock mass. The FEM is used to calculate the deformations occurring during excavation and stresses that reacts on installed rock support. Phase2 is able to automatically generate the element mesh. An example of generated mash can be seen on Figure 4.3. Notice how the elements get smaller as it closes on the excavation profile. Figure 4.3 Generated element mesh of finite element model. 37 5 Deformation of rock mass Stability in advancing tunnel in weak rock is strongly related to the deformation occurring after excavation. Deformations of the rock mass starts about one and half diameter of the tunnel in front of the tunnel face and has reached 25-33% of the final deformation already at the tunnel face. In weak rock mass these deformations might already result in instability at the tunnel face depending on the rock mass strength and the in situ stress conditions (Hoek, 2000). Figure 5.1 Deformation vector around advancing tunnel (Hoek, Practical Rock Engineering, 2000) In order to determine the appropriated timing of the installation of specific tunnel support one must have knowledge of the rock mass deformation behaviour and displacement capacity of the support (Hoek et al., 2008). Creation of longitudinal deformation profiles and ground reaction curves are convenient tools to gain knowledge of the rock mass behaviour. Longitudinal deformation profiles can be measured in situ or created by using analytical or numerical methods such as the finite element method (FEM). Use of analytical approach is however only possible for very simple shape of excavation (circular) and isotropic elastic or elastic-plastic rock mass. To model more complex shaped excavation profile and rock mass with plastic behaviour one must use axisymmetric or three dimensional finite element models. Empirical best fit analysis can though be used to create deformation profiles if maximum displacement and plastic radius has been measured or modelled. 5.1 Theory Panet derived in 1995 a relationship base on elastic analysis for short term longitudinal deformation profile for known maximum deformation u max (Hoek et al., 2008): 38 u ¡ u mux = 1 4 + S 4 _1 -_ S S +4J t ] 2 _ Eq. 5-1 where d t =X/R t , X is the distance from face and R t is the runnel radius. Alternative expressions have been derived for elastic analysis for deformation profiles suggesting that the deformation doesn’t form a continuous curve in front and behind the advancing tunnel face. Unlu and Gercek suggested in 2003: u ¡ u mux = u 0 u mux + A u (1 -c B c d t ) ¡or X < u Eq. 5-2 u ¡ u mux = u 0 u mux + A b (1 -_ B b A b +J t ] 2 ) ¡or X > u Eq. 5-3 where u 0 is the deformation at the tunnel face and A a , A b , B a and B b are functions of the Poisson ratio: u 0 u mux = u,22u +u,19 Eq. 5-4 A u = -u,22u -u,19 B u = u,7Su +u,81 Eq. 5-5 A b = -u,22u +u,81 B b = u,S9u +u,6S Eq. 5-6 where υ is the Poisson’s ratio. Numerous empirical best fit solutions based on plastic modelling and empirical best fit to actual measured closure date have been suggested. Based on measured data, Chern in 1998 suggested the following expression (Hoek, er al, 2008): u ¡ u mux = _1 +c [ -d t 1,1 ¸ _ -1,7 Eq. 5-7 The shape of the deformation curve is however directly connected to the radius of the plastic zone R p . Therefore Hoek et al. (2008) have suggested the following relationship for short term longitudinal deformation curve: u 0 u mux = 1 S c -0,15P r Eq. 5-8 where P r is given as: P ¡ = R p R t _ Eq. 5-9 39 u ¡ u mux = u 0 u mux c d t ¡or X < u Eq. 5-10 u ¡ u mux = 1 -_1 - u 0 u mux ] c - 3d t 2P r ¡or X > u Eq. 5-11 Relationship for short term longitudinal deformation curve derived by Hoek et al.(2008) will be used as part of the case study in Chapter 6. Figure 5.2 shows comparison of elastic and empirical models mentioned above. Figure 5.2 Longitudial deformation profile based on elastic and empirical models for Rt =4,6m and Rp =7,45m. 5.2 Modeling deformation curve An axisymmetric FEM analysis can used to model the response of an elastic-plastic rock mass around non-circular cross section. The process can best be explained by an example. In this chapter, a axisymmetric model and best fit empirical analysis is used to construct a deformation curve for a given rock mass and field stress circumstances. Cross section and rock mass properties in Table 5.1 from Óshlíð tunnel are used in this example. Rock mass properties are analyzed data from Mannvit consulting engineering office from core samples taken during construction of the tunnel (Mannvit, 2009). 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 ‐20 ‐10 0 10 20 30 40 U r / U m a x Distance from face [m] Hoek Chern Elastic model R t  =4,6m  R p  =7,45m  40 Table 5.1 Rock mass parameters for axisymmetric FEM-model Rock‐ type  Stiffness   Tensile  strength   Friction  angle   Cohesion  UCS  Hoek Br.‐ parameter Hoek Br.‐ parameter  Hoek Br.‐ parameter E m   T 0   θ  c   σ ci   m b   s  a     [MPa]  [MPa]  [°]  [MPa]  [MPa]          Basalt  12270  80  4,09  0,00855  0,503  Sediment  2266  ‐0,79  56  1,66              The field stress is considered to be 16,2 MPa in vertical direction and 6,48 MPa in horizontal direction. That corresponds to 600m over burden and k value of 0,4. This example should be considered as hypothetical and will not be used in the case study in Chapter 6. 5.2.1 Axisymmetrical FEM model For this example a tunnel of 9,2 m width is considered to be crossing a 12m thick sedimentary rock embedded in a basalt rock mass. Location of the sedimentary layer is considered to be in the invert of the tunnel cross section. Figure 5.3 Axisymmetric model of the tunnel As the tunnel advances the support from the tunnel face is reduced. To simulate this effect an elastic material is placed inside the excavation profile and its stiffness reduced in stages. At stage 1 the E-modulus of the inclusive rock mass is the same as the surrounding rock mass and zero at the final stage to simulate a fully excavated tunnel far from the tunnel face. Deformations for a single point can therefore be collected for each modulus of the inclusion material. Figure 5.4 shows the results of the deformations in the walls in the example. 41 Figure 5.4 Deformation vs decreasing inclusive e-modulus. According to the axisymmetric model (see Figure 5.4) the maximum average mid wall displacement is 74mm for unsupported tunnel far from the tunnel face. The maximum plastic radius can be estimated at the final stage of the model as 7,45m. Deformation curve as a function of distance from the tunnel face can now be created by using the empirical relationship suggested by Hoek (see Chapert 5.1) since both maximum deformations and plastic radius have been estimated in the model. The curve is given in Figure 5.5. Figure 5.5 Deformation curve for walls The deformation curve also shows how much of the total displacement has already taken 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 0 0,02 0,04 0,06 0,08 0,1 I n c l u s i v e   E ‐ m o d u l e   [ M P a ] Wall deformation [m] Left wall Right wall Average 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 ‐10 0 10 20 30 W a l l   d e f o r m a t i o n   [ m ] Distance from face [m] Deformation curve Installing of support  Installing of support 1,5m from face 42 place before installation of rock support. In this case the rock support is considered to be installed 1,5 m from the face. Figure 5.5 shows that at that time, the deformations in the rock mass have reached 33mm or 45% of its total deformation. Figure 5.4 shown that deformation of 33mm has occurred when the inclusion modulus has been decreased to 100MPa. Support from the tunnel face at 1,5m distance is therefore considered equal to inclusion modulus of 100MPa. This information could be used to make an FEM-model that would allow undisturbed deformation up to 33mm and make the installed support 1,5m from the tunnel face react with all deformation after that. The method of generating relaxation to a certain inclusion modulus can be used in FEM- modelling to replace the function of load split. The load split function divides the deformations so that the tunnel support will only react with deformations that are considered to take place after activation of the rock support. 43 6 Case study: Use of lattice girders in Óshlíð tunnel. 6.1 Introduction Óshlíð tunnel is located in the Vestfirðir peninsula at the north west corner of Iceland and connects two small towns, Hnífsdalur and Bolungarvík. The tunnel is a 9,2m span road tunnel with 53,75 m 2 cross section and a length of 5,2 km. Óshlíð tunnel were constructed in 2008-2010 and are categorized as low traffic tunnel according to the Norwegian road tunnel standard with annual average daily traffic (AADT) around 780. The tunnel was excavated by traditional drill and blast method. This case study involves a FEM-design of a shotcrete lining through sedimentary rock in the Óshlíð tunnel. Pre-geological investigation indicated number of sedimentary interbeds along the tunnel route. Original design of the tunnel support suggested that those sedimentary interbeds would be dealt with by conventional rock support like rock bolts, shotcrete, wire mesh and shotcrete ribs with spiling bolts according to convention in Icelandic tunnel construction. During the construction of the tunnel some instability problems were experienced in the tunnel crown when excavated through sedimentary interbeds that in some cases lead to collapse of the tunnel crown right after blasting (Figure 6.1). An alternative working method for tunnel support was therefore initiated to deal with these circumstances. That support method involved installation of lattice girders and spiling bolts. Figure 6.1 Overbreak due to weak sedimentary layer in Óshlíð tunnel. 44 Since the use of lattice girders was not expected during the design of the tunnel, the tunnel lining involving lattice girders was designed on site with participation of engineers from both contractor and the supervision and approved by the tunnel designer. Figure 6.2 Use of lattice girders and spiling bolts in Óshlíð tunnel. The FEM program, Phase 2, is used to estimate the forces that react on the tunnel lining under the current geological circumstances. The design code Eurocode 2 is further used to estimate the bearing capacity of the lining. Only limited investigations of the rock mass properties were available. Number of uni- axial compression tests (UCS) were made to estimated the intact rock strength of both sedimentary interbeds and basalt. No tri-axial tests results were available to estimate the geological parameters in the Hoek-Brown or Mohr-Coloumb failure criterion. To investigate how much this limitation of available data will infect the design both average and lower limit strength parameters were used. At station 16.360 (TM 2.095 from Hnífsdalur side) a sedimentary layer began to arise from the bottom of the tunnel. 10m thick sedimentary interbed had been logged in core hole at st. 16.100 and the current layer was considered to be the same one. Exploratory holes indicated that the layer could be 12-15m thick and according to the inclination of the strata it would be in the profile for at least 80-100m. Excavation revealed though that tectonic fault had shifted the strata in station 16.390 and excavation in sediment layer turned out to be around 50m. Geological circumstances around station 16.400 to 16410 is used in this case study since UCS and cohesion test data are available for that sedimentary layer. Geological mapping can be found in the Appendix A3 of the area discussed. 45 6.2 Rock mass properties in Óshlíð tunnel Óshlíð tunnel are located in the Vestfirðir peninsula at the north west corner of Iceland. The rock mass of the tunnel route is part of the oldest rock mass in Iceland. The rock mass strata is formed by highly alterated basalt layers and numerous sedimentary layers, some with very low compression strength. The rock mass strata dips 5-7°to south west and since the tunnel route is almost parallel to the dip direction it goes through around 500m of the rock mass strata. Most basalt layers are around 3-8m thick but goes up to 20m in thickness. (Guðmundsson et al., 2007) Pre-geological investigation of the tunnel route indicated that the tunnel route would cross numerous sedimentary layers with thickness from 1-5m. Much thicker sedimentary layers was though encountered during excavation of tunnel and lead to use of heavier rock support than previously anticipated. Figure 6.3 shows typical layered rock mass strata in the Vestfirðir peninsula. Figure 6.3 Typical layered rock mass strata in the Vestfirðir peninsula (Guðmundsson et al., 2007). 46 6.3 Estimation of rock mass parameters The reliability of a FEM-design is never higher than the reliability of the input parameters (Palmstöm & Nilsen, 2000). Therefore an effort is made here to estimate required rock mass properties to be able to construct a reliable FEM-model of the rock mass. Hoek and Brown failure criterion will be used to mimic both the basaltic and sedimentary rock. The choose of Hoek and Brown criterion instead of Mohr-Coulumb failure criterion is based on that no tri-axial data are available and therefore hard to create a reliable Mohr-Coulumb failure envelope. Table 6.1 shows UCS (Uni-axial compression strength) data measured on intact rock samples in Óshlíð tunnel at station 16.350 (Mannvit, 2009). Table 6.1 UCS values of intact sedimentary rock samples in Óshlíð tunnel Depth [m] UCS[MPa] E[MPa]  3,8  18,3  2630  4,85  14,6  2320  5,15  15,2  3160  5,75  10,9  2440  6,35  4,7  1015  6,45  10,9  2250  6,75  4,7  1240  7,45  17,5  3725  7,55  11,6  2850  7,65  10,0  2350  8,4  10,9  3015  8,95  5,7  940  9,2  7,0  1480  9,4  5,4  1475  10,35  14,9  2520  10,45  12,6  2840  Mean  10,9 2266  St dev  4,5    UCS values for the sediment layer tested in this borehole are highly distributed. Value of standard deviation of the data is 42 % of the mean value. Statistical analysis is therefore required to justify a single value for using as a UCS in the FEM model. By using statistical analysis it is possible to determine the percentage of values in the data set that exceeds a certain value (see Figure 6.4). 90% of UCS measurement are exceeding 5,2 MPa and that value will be used as input parameter in the FEM-model. The mean value of 10,9 MPa could also be used but in that case the rock mass would in 50% of cases be less favourable than the parameter used in the design and would be hard to justify. It will however be used here as a comparison. Figure 6.5 shows the correlation between E-modulus and UCS. 47 Figure 6.4 Normal distribution of UCS data Figure 6.5 UCS vs E i -module Measurements of cohesion strength in this same borehole indicated mean cohesion strength of c= 0,79MPa and c 10% = 0,37 MPa. The stiffness modulus E i10% should be around 1366 MPa according to Figure 6.5. Other UCS measurements of sedimentary rock in Óshlíð tunnel indicates UCS values from 1,5-9,8 MPa (Pálsson, 2009). That supports the conservative approach of using UCS 10% . 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 P r o b a b i l i t y D e n s i t y UCS [MPa] Density 10% 90% 0 500 1000 1500 2000 2500 3000 3500 4000 4,0 6,0 8,0 10,0 12,0 14,0 16,0 18,0 20,0 S t i f f n e s s E [ M P a ] UCS [MPa] 48 Figure 6.6 Hoek and Brown failure envelop for lower limit strength parameters of sediment. Figure 6.6 shows the Hoek-Brown failure envelope (red) for the lower limit strength of the sediment layer. Figure 6.7 shows the Hoek-Brown failure envelope if mean values are used for UCS strength and corresponding E i -modulus. Figure 6.7 Hoek and Brown failure envelop for mean strength parameters of sediment. 49 The blue line on Figure 6.6 and 6.7 shows approximated Mohr-Coulomb failure envelope to the Hoek and Brown envelope. Table 6.2 Rock mass properties of lower limit strength of sediment rock mass. Selected rock properties  Hoek Brown parameters Mohr‐Coulomb fit  Rock Mass Parameters  UCS  5,2  MPa  mb  4,972 cohesion  1,239 MPa  tensile strength  ‐ 0,589 MPa Intact E‐modulus  1366  MPa  s  0,5629 fric. angle  28,12 °  UCS  3,606 MPa GSI  95  a  0,5    E‐modulus  1266 MPa mi  6              D  0,1                             Table 6.3 Rock mass properties of mean strength sediment rock mass. Selected rock properties  Hoek Brown parameters Mohr‐Coulomb fit  Rock Mass Parameters  UCS  10,9  MPa  mb  4,972 cohesion  2,11 MPa  tensile strength ‐1,234 MPa Intact E‐modulus  2262  MPa  s  0,5629 fric. angle  33,05 °  UCS  8,177 MPa GSI  95  a  0,5    E‐modulus  2096,5 MPa mi  6              D  0,1                             First column of Table 6.2 and 6.3 shows the selected parameters. GSI value is selected relatively high since this specific type of sedimentary rock is well cemented and doesn’t have any joint systems. m i value of 6 is recommended value (Hoek, 2000) for sedimentary rock with silty or clayey texture if no tri-axial values are available to derive a complete Hoek-Brown failure envelope. D value of 0,1 is chosen since damage from blasting is considered minor. Choose of higher D value does however not reduce the rock mass strength considerably. The vertical component of the virgin stress field can be calculated from overburden (in this case 400m) according to Eq. 2-12. As previously stated is the horizontal component of the stress field harder to estimate if no stress measurements have been performed. Considering that the most problematic factor in the excavation of the Óshlíð tunnel was instability in the tunnel crown and previous stress measurements in Iceland, a k value of 0,5 is considered appropriate. The virgin stress field around the excavation is therefore estimated: o v = 1u,9 HPo onJ o H = S,4S HPo Rock mass parameters of the basalt rock are shown on Figure 6.8. The rock mass parameters for the basalt layer are chosen relatively conservative. Modelling shows however that increasing the rock mass strength of the basalt has minimal affects on the modelling results and the sediment rock parameters are the dominant factor due to its very low strength. 50 Figure 6.8 Hoek and Brown failure envelope for basalt 51 6.4 Calculated capacity of installed support Installed tunnel support in all cases that will be analysed (see further in Table 6.5) are lattice girders c/c 1500mm, 150mm minimum fiber reinforced shotcrete cover between girders and 6mm wire mesh behind the girders. Figure 6.9 shows cross section of the installed tunnel lining and Figure 6.10 shows cross section of the lattice girder. Figure 6.9 Installed tunnel support at station 16.400 and 16.410. Figure 6.10 Cross section of lattice girder used in Óshlíð tunnel. Units are in mm. Figure 6.11 On left: Assembled lattice girders (Efla, 2011). On right: Final shotcrete lining with lattice girders (Pedersen, Kompen, & Kveen, 2010). 52 Tunnel lining in rock with non-hydrostatic stress condition will always experience both axial stresses and moments. M-N (Moment vs Axial force) envelope is therefore useful to estimate if the tunnel lining is able to tolerate applied loading from the rock mass. M-N envelope can be derived by calculating the ultimate bearing capacity of the lining for given strain condition (Eiríksson, 2010). Eight strain conditions will be used to derive the M-N envelope: 1) Pure axial force 2) Zero strain in tensional bars (positive moment) 3) Equilibrium. Max strain in shotcrete and yielding in rebars at the same time (positive moment) 4) Pure moment (positive moment) 5) Pure tension 6) Pure moment (negative moment) 7) Equilibrium. Max strain in shotcrete and yielding in rebars at the same time (negative moment) 8) Zero strain in tensional bars (negative moment) Same strain conditions are used to derive M-N envelopes for structural members in buildings (columns and beams) that experience both axial forces and moments. Calculated bearing capacity of the lining in Figure 6.9 can be seen in Appendix A4. Phase2 can be used to derive M-N envelopes. The method used by the program is however purely elastic and M-N envelope is constructed for each structural member. This is not realistic for reinforced shotcrete lining due to bonding between the shotcrete and the reinforcement. That method is however more realistic for lining of shotcrete and steel beams since bonding strength between shotcrete and steel beams is very low. Characteristic strength of the shotcrete used in Óshlíð tunnel was C30/37 and other requirements can be seen in Table 6.4. Table 6.4 Min requirements for shotcrete in Óshlíð tunnel UCS category     C30/37    Chloride contends category  C1 0,20 Material category  3 Max grain size  8 mm  Min cement content  420 kg/m 3   Max v/c  0,45 Steel fiber content  40 kg/m 3   Energy absorption     E700    53 UCS-testing on core samples displayed in Figure 6.12 shows that 99% of core samples had higher UCS strength than 30MPa. Figure 6.12 UCS result of shotcrete in Óshlíð tunnel. Material partial factor for concrete γ c is given as 1,5 in Eurocode 2. Design strength of shotcrete in Óshlíð tunnel should therefore be (Eurocode 2, 2002): ¡ cd = ¡ ck y c = SuHPo 1,S = 2uHPo Eq. 6-1 Eurocode 2 however notes that the partial factor γ c can be lowered if reliable measurements can be used to confirm the real strength of the concrete. Partial factor γ s for steel is 1,15. 0 1 2 3 4 5 6 7 8 9 10 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 N u m b e r   o f   t e s t s UCS‐strength of shotcrete [MPa] 54 6.5 FEM-modelling Four cases will be modelled that represents the geological circumstances at stations 16.400 and 16.410. Geological mapping of the area can be found in Appendix 4. Estimated rock mass parameters from Chapter 6.3 are here used to construct FEM-models that are considered to have similar characteristic as the rock mass in those locations in Óshlíð tunnel. Table 6.5 shows the main model parameters for each case. Table 6.5 Main model parameters Case 1  Case 2  Location        st 16.400  Location       st 16.400  Sediment layer  12m  Sediment layer  12m  Position of sediment  From invert up  Position of sediment  From invert up  Rock mass parameter  Low strength (Table 6.2)  Rock mass parameter  Mean strength (Table 6.3)  Virgin stress field     400 m overburden, K=0,5  Virgin stress field     400 m overburden, K=0,5 Case 3  Case 4  Location        st 16.410  Location       st 16.410  Sediment layer  12m  Sediment layer  12m  Position of sediment  From middle section up  Position of sediment  From middle section up  Rock mass parameter  Low strength (Table 6.2)  Rock mass parameter  Mean strength (Table 6.3)  Virgin stress field     400 m overburden, K=0,5  Virgin stress field     400 m overburden, K=0,5 Rock mass parameters for the basalt are given in Figure 6.8 Figure 6.13 and 6.14 shows position of the sedimentary layer (brown) in the tunnel profile each case. Figure 6.13 Case 1 and 2. Station 16.400. 55 Figure 6.14 Case 3 and 4. Station 16.410. 6.5.1 Modelling method This chapter shows how axisymmetrical modelling of deformation curves are used to construct reliable FEM-model of the geological situations of each case. All cases are modelled in the same way and modelling process of case 1 is therefore only shown in this chapter, Case 1. St: 16.400, 12m Sedimentary layer from tunnel invert with lower limit strength. Figure 6.15 Maximum wall deformation and plastic radius. X indicates a shear failure in the rock mass and o indicates tension failure. The analysis shows that maximum wall deformation for unsupported tunnel far from the tunnel face is 40mm (80mm convergence) and maximum plastic radius is 7,26m. Figure 56 6.16 shows a deformation curve derived from the information according to equation 5-8 to 5-11. Figure 6.16 Deformation curve for case 1 Figure 6.16 shows that 18mm deformation has already occurred 1,5m from the tunnel face. The rest of the deformations will react with the tunnel lining. Figure 6.17 Inclusion modulus vs deformation 0 0,01 0,02 0,03 0,04 0,05 0,06 ‐10 ‐5 0 5 10 15 20 25 30 d e f o r m a t i o n   [ m ] Distance from face [m] Support activated 1,5m  from tunnel face 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0 200 400 600 800 1000 1200 1400 W a l l   d e f o r m a t i o n   [ m ] Inclusion modulus [MPa] 57 Figure 6.17 created with same method as described in Chapter 5.2, shows that the tunnel face is still supporting the tunnel profile 1,5m from the face with similar effect as if material with stiffness of 500 MPa where placed inside the excavation. Tunnel relaxation is therefore generated until the inclusion modulus is 500 MPa. It corresponds to 1,5m from the tunnel face. All deformation after that will build up stresses in the tunnel lining. All four cases are modelled by the same method but different inclusion modulus according to their corresponding axisymmetric deformation model. 6.6 Result M-N envelopes for both characteristic strength f ck and reduced design strength f cd according to Eurocode 2 are displayed on the following figures in this chapter. Result of axial forces and moments acting of the tunnel lining are scattered on the M-N envelopes of the lining on Figures 6.18-6.21 for each case. Figure 6.18 Result for case 1. Lower limit strength of sediment 12m in thickness from tunnel invert. ‐1000 0 1000 2000 3000 4000 5000 6000 ‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 A x i a l   f o r c e   [ k N / m ] Moment [kNm/m] Characteristic strength Eurocode Modeled stresses 58 Figure 6.19 Results for case 2. Mean strength of sediment 12m in thickness from tunnel invert. Figure 6.20 Results for case 3. Lower limit strength of sediment 12m in thickness from mid section. ‐1000 0 1000 2000 3000 4000 5000 6000 ‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 A x i a l   f o r c e   [ k N / m ] Moment [kNm/m] Characteristic strength Eurocode Modeled stresses ‐1000 0 1000 2000 3000 4000 5000 6000 ‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 A x i a l   f o r c e   [ k N / m ] Moment [kNm/m] Characteristic strength Eurocode Modeled stresses 59 Figure 6.21 Results for case 4. Mean strength of sediment 12m in thickness from mid section. The results clearly establish how important reliable estimations on rock mass properties are for the finite element design of the shotcrete lining. The lack of tri-axial test data to estimate properties of the rock mass leads to more conservative approach in selection of rock mass parameters and might therefore lead to design of heavier support than necessary. Case 1 and 3 shows values outside or on the M-N failure envelope when conservative values are used for rock mass parameters. All values are however inside the failure envelope of the lining if the mean strength of the rock mass is used. Increased diameter of the rebars in the lattice girder would grant the lining with enough moment capacity to include all value inside the failure envelope in cases 1 and 3. Case 3 is the most challenging case. Figure 6.22 shows that the highest moments occur on the border of basalt and sediment. This is probably due to squeezing effect on the boarder since the stiffness of the sedimentary layer is one order of magnitude lower then for the basalt. ‐1000 0 1000 2000 3000 4000 5000 ‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 A x i a l   f o r c e   [ k N / m ] Moment [kNm/m] Characteristic strength Eurocode Modeled stresses 60 Figure 6.22 Highest moments on the boarder of sediment and basalt in case 3. To investigate how changes in k value (ratio σ H / σ V ) influence the results was case 4 repeated for k value 1,0 and 1,5. Result for case 4 with variable value of k can been seen on Figure 6.23. Hydrostatic stress condition is constructed in the model if k value is changed to 1,0. Figure 6.23 shows that it results in lower moments compared to non-hydrostatic stress conditions. The tunnel lining will however fail if k value of 1,5 is selected since it results in very high moment in the tunnel lining along with very increased axial force. As previously stated is the k value estimated to be around 0,5. The changes that the k value is higher than 1,0 is unlikely considering the tunnel depth and that instability problems were in most cases isolated to the tunnel crown. 61 Figure 6.23 Result for case 4 with variable k value from 0,5 to 1,5. Q-value for area discussed was estimated 0,1. According to the Norwegian standard (see table in Appendix 1) is the required rock support as follows: - Spiling bolts with 200-300mm spacing. - Double shotcrete ribs with 1,5 -2m spacing - 150-250mm C35 fiber reinforced shotcrete between ribs. - 3-6m rock bolts systematically driven through the shotcrete ribs. The standard also notes that the shotcrete ribs can be replaced by lattice girders. These requirements correspond quite well to the installed rock support in the Óshlíð tunnel. ‐2000 ‐1000 0 1000 2000 3000 4000 5000 6000 ‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 A x i a l   f o r c e   [ k N / m ] Moment [kNm/m] Characteristic strength Eurocode k 0.5 k 1 k 1.5 62 7 Conclusions Author of this report had the privilege of working on site during the construction of the Óshlíð tunnel and followed the installations procedure of lattice girders and other rock support mentioned. Following conclusion are based on experience on site of Óshlíð tunnel and investigations in this thesis: Lattice girders are pre fabricated outside the tunnel and the time spent on building the beam is therefore shorter at the face compared to the shotcrete ribs. Lattice girder are therefore safer choose if the tunnel face is considered unstable. Shotcrete ribs are however more convenient to use if the short time stability of the working area is sufficient since it can better be adapted to irregular profile and therefore does not lead to high consumption of shotcrete when covered. The finite element modelling is quite helpful tool to estimate the force acting on tunnel lining and should be reliable if required rock mass and stress data are available. The core replacement method used here to derive a stiffness-deformation relationship worked well and gave realistic results. Lack of tri-axial data turned out to be a problematic factor and lead to very conservative approach in the estimation of the rock mass parameters. Pre-geological investigations should therefore include tri-axial testing on the rock samples at least on the rock mass that is considered to be the most problematic during excavation. Modelling of case 1 to 4 revealed that the highest moments occurred on the boundary of sediment and basalt rock as expected. Weakest point for these moments is where the squeezing effect is working perpendicular on the tunnel profile, in this case in the same height as the radius point of the tunnel profile. Changing the k-value from 0,5 to 1,5 tuned out to have quite dramatic influence on the results in case 4. Stress measurements are almost inevitable if FEM-design is to be used as a design tool for tunnels or other underground openings. Especially if the construction is planned at low depth. Bearing capacity calculations revealed that increased moments acting on the tunnel lining should be dealt with by increased diameter of the lattice girder chords or decreased spacing between girders. Tensile strength of shotcrete is low, even though mixed with steel fiber, that the most effective way to deal with higher moments in the lining is to increase the amount of reinforcement per tunnel metre. Modelling of four cases was carried out. Once the M-N envelopes of the lining and rock mass parameters had been estimated this turned out to be a relatively fast process. FEM- modelling could therefore be used on tunnel site to estimate required tunnel support in special cases if M-N envelopes have been calculated for different types of linings in forehand and parameters for various rock types estimated. Calculation of M-N envelopes was however very time consuming and very sensitive for even smallest errors. M-N envelopes should therefore not be calculated on tunnel site under time pressure and thoroughly reviewed. 63 Bibliography Baumann, T., & Betzle, M. (1984). Investigation of the performance of lattice girders in Tunneling. Munich Germany: Philipp Holzmann AG. Dahle, H. (2005). In situ rock stress in layered icelandic volcanic rocks-Fáskrúðsfjörður tunnel. Trondheim Norway: SINTEF. Efla. (2011). Óshlíðargöng-framkvæmdaskýrsla. Reykjavík: Vegagerðin. Einarsson, E. (2010). Óshlíðargöng- steypuframleiðsla (in icelandic). Reykjavík: BM Vallá. Eiríksson, H. (2010). Lecture notes- Concrete structures 2. Reykjavík: Háskóli Íslands. Erlingsson, S. (2009). Lecture notes in Engineering Rock Machanics. Reykjavík, Iceland: University of Iceland. Eurocode 2. (2002). Eurocode 2: Design of concrete structures. Brussels: European committee for Standardiziation. Guðmundsson, Á., Kaiser, S. Ó., & Ward, T. (2007). Óshlíðargöng, Aðstæður til gangagerðar milli Hnífsdals og Bolungarvíkur (in icelandic). Reykjavík: Jarðfræðistofan ehf. Guðmundsson, H. (25. 3 2011). e-mail refference. Iceland. Haimson, B. C. (1981). Hydrofracturing Stress Measurements-Hole FV-1, Teigsbjarg. Wisconsin U.S.A: Orkustofnun. Haimson, B. C., & Rummel, F. (1981). Hydrofracturing stress measurements in the IRDP drillhole at Reydarfjordur, Iceland. Wisconsin: Journal of Geophysical Research. Harðarson, B. A. (20. 9 2011). e-mail refference. Reykjavík , Iceland. Harðarsson, B. A. (1991). Tunneling Quality of Icelandic Rock. Reykjavík: JTS Geotechnical Services Ltd. Hoek, E. (2000). Practical Rock Engineering. Vancouver, Canada: Evert Hoek Consulting Engineer Inc. Hoek, E., Carranza, C., & al, E. (2002). Hoek-Brown Failure Criterion - 2002 Edition. Vancouver. Hoek, E., Carranza-Torres, C., & Diederichs, M. (2008). Integration of geotechnical and structural design in tunneling. Vancouver, Canada: Evert Hoek Consulting Engineer Inc. Jorimann. (2010). JB-Gittertrager. Walenstadt: Jorimann Stahl-Tunnelbau. Kirkaldie, L. (1988). Rock classification systems for engineering purposes. Philadelphia: ASTM. 64 Komselis, C., Blayney, N., & Hindle, D. (2005). The use of lattice girders in the construction of tunnels. London, UK: London mining and Mineral Consultants LTD. Loftson, M. (20. 9 2011). e-mail refference. Reykjavík, Iceland. Loftsson, M. (2009). Kortlagning á jarðfræði og styrkingum í jarðgöngum (in icelandic). Reykjavík: Vegagerðin. Luo, J. (1999). A new rock bolt design criterion and knowlage-based expert system for stratified roof. Virginia: Virginia faculty of the Virginia Polytechnic Institute. Mannvit. (2009). Bolungarvíkurgöng - setbergslög- Samantekt um athuganir (in icelandic). Reykjavík: Vegagerðin. Palmstöm, A., & Nilsen, B. (2000). Engineering Geology and rock Engineering. Oslo, Norwey: NBG. Pálsson, F. (2009). Enginnering geology of Oshlid tunnel, Iceland. London: Imperial collage of London. Pedersen, K., Kompen, R., & Kveen, A. (2010). Arbeider foran stuff og stabilitetssikring i vegtunneler. Oslo, Norway: Statens Vegvesen. Sigmundsson, D. F. (2006). Iceland geodynamics: crustal deformation of divergent plate tectonics. London: Springer/Praxis. Statens Vegvesen. (2000). Fjellbolting. Oslo, Norway: Statens vegvesen. Statens vegvesen. (2009). Prinsipp bergsikring i tunnel- E6 Trondheim Stjördal. Oslo: Statens vegvesen. U.S. Department of transportation. (2011). Technical manual for design and construction of road tunnels. Washington DC: Federal Highway Administration. Vegagerðin. (2008). Óshlíðargöng 4. hefti Byggingarvirki - Uppdrættir (in icelandic). Reykjavík: Vegagerðin. Weisenberger, D. T. (2010). Iceland. Sótt 26. 08 2011 frá DR. Tobias Weisenberger: http://www.tobias-weisenberger.de/6Iceland.html 65 Appendix A.1 Rock classification systems. RMR-system (Hoek, 2000) 66 Recommended parameter for Q-value 67 68 (Hoek, 2000) 69 Recommended rock support in Norwegian standard for certain Q-value (Pedersen, Kompen, & Kveen, 2010) 70 A.2 GSI and D value in Hoek-Brown criterion Estimation of GSI value in the Hoek-Brown failure criterion. (Hoek, 2000) 71 Estimation of D- value in the Hoek-Browm failure criterion. (Hoek, 2000) 72 A.3 Óshlíð tunnel: Geological mapping for st 16.300-16.450 (Ósafl, 2010) 73 A.4 Bearing capacity of shotcrete lining. Definitions of dimensions Definitions of material properties Only in tension Calculation of center of mass. b 1500mm := t 150mm := h g ¸ ( ¸ 195mm := b1 780mm := b g ¸ ( ¸ 180mm := b2 330mm := l 0 ¸ ( ¸ 2000mm := t2 95mm := h t t2 + := d1 16mm := h 0.225m = d2 20mm := dg h g ¸ ( ¸ d1 2 ÷ d2 2 ÷ := u 30deg := co 30mm := dg 0.177m = f cd ¸ ( ¸ 20MPa := f yd ¸ ( ¸ 435MPa := o 0.85 := A s1 ¸ ( ¸ d1 2 | \ | | . 2 t · 2 · := A s1 ¸ ( ¸ 4.021 10 4 ÷ × m 2 = A s2 ¸ ( ¸ d2 2 | \ | | . 2 t · := A s2 ¸ ( ¸ 3.142 10 4 ÷ × m 2 = A c ¸ ( ¸ b t · 0.5 t2 · b2 · + A s1 ¸ ( ¸ ÷ A s2 ¸ ( ¸ ÷ := A c ¸ ( ¸ 0.176m 2 = b eff ¸ ( ¸ 2 0.2 b b2 ÷ ( ) 2 · l 0 ¸ ( ¸ 0.1 · + ¸ ( ( ¸ · b2 + := b eff ¸ ( ¸ 1.044m = A s11 ¸ ( ¸ d1 2 | \ | | . 2 t · 2 · t 6mm 2 | \ | | . 2 b eff ¸ ( ¸ 150mm · + := A s11 ¸ ( ¸ 5.989 10 4 ÷ × m 2 = h cm ¸ ( ¸ b eff ¸ ( ¸ t · 0.5 · t · 0.5 b2 · t2 · t t2 3 + | \ | | . · + b eff ¸ ( ¸ t · b2 t2 · 2 + := h cm ¸ ( ¸ 0.069m = 74 Pure axial force Pure tension Equilibrium stage (positive moment) If 0 then adjust equation of compression zone. N d ¸ ( ¸ o f cd ¸ ( ¸ · A c ¸ ( ¸ · f yd ¸ ( ¸ A s1 ¸ ( ¸ A s2 ¸ ( ¸ + ( ) · + := N d ¸ ( ¸ 3.306 10 3 × kN · = N tog ¸ ( ¸ f yd ¸ ( ¸ A s11 ¸ ( ¸ A s2 ¸ ( ¸ + ( ) · := N tog ¸ ( ¸ 397.187kN · = c cu ¸ ( ¸ 0.0035 := c syd ¸ ( ¸ 0.00217 := d h g ¸ ( ¸ d2 2 ÷ := d 0.185m = x bal ¸ ( ¸ c cu ¸ ( ¸ c cu ¸ ( ¸ c syd ¸ ( ¸ + d · := x bal ¸ ( ¸ 0.114m = 0.8 x bal ¸ ( ¸ · ( ) t < 1 = F c ¸ ( ¸ o f cd ¸ ( ¸ · 0.8 · x bal ¸ ( ¸ · b eff ¸ ( ¸ · := F c ¸ ( ¸ 1.621 10 6 × N = F s1 ¸ ( ¸ f yd ¸ ( ¸ A s1 ¸ ( ¸ · := F s1 ¸ ( ¸ 1.749 10 5 × N = F s2 ¸ ( ¸ f yd ¸ ( ¸ A s2 ¸ ( ¸ · := F s2 ¸ ( ¸ 1.367 10 5 × N = N ud ¸ ( ¸ F c ¸ ( ¸ F s1 ¸ ( ¸ + F s2 ¸ ( ¸ ÷ := N ud ¸ ( ¸ 1.66 10 6 × N = M ud ¸ ( ¸ F c ¸ ( ¸ h cm ¸ ( ¸ 0.4 x bal ¸ ( ¸ · ÷ ( ) · F s1 ¸ ( ¸ h cm ¸ ( ¸ d1 2 ÷ | \ | | . · + F s2 ¸ ( ¸ d h cm ¸ ( ¸ ÷ ( ) · + := M ud ¸ ( ¸ 63.948mkN · = 75 Pure positive moment Pure negative moment Equilibrium stage (negative moment) x f yd ¸ ( ¸ A s2 ¸ ( ¸ · o f cd ¸ ( ¸ · 0.8 · b eff ¸ ( ¸ · := x 9.625 10 3 ÷ × m · = M u ¸ ( ¸ f yd ¸ ( ¸ A s2 ¸ ( ¸ · d 0.4 x · ÷ ( ) · := M u ¸ ( ¸ 24.756mkN · = d 1 ¸ ( ¸ h d1 2 ÷ := d 1 ¸ ( ¸ 0.217m = h c ¸ ( ¸ A s11 ¸ ( ¸ f yd ¸ ( ¸ · tan u ( ) · ( ) o f cd ¸ ( ¸ · := h c ¸ ( ¸ 0.094m = M u2 ¸ ( ¸ f yd ¸ ( ¸ A s11 ¸ ( ¸ · d 1 ¸ ( ¸ 2 h c ¸ ( ¸ · 3 ÷ | \ | | . · := M u2 ¸ ( ¸ 40.197mkN · = x bal2 ¸ ( ¸ c cu ¸ ( ¸ c cu ¸ ( ¸ c syd ¸ ( ¸ + d 1 ¸ ( ¸ · := x bal2 ¸ ( ¸ 0.134m = b eff ¸ ( ¸ 1.044m = h c2 ¸ ( ¸ 0.8 x bal2 ¸ ( ¸ · := h c2 ¸ ( ¸ 0.107m = F c2 ¸ ( ¸ o f cd ¸ ( ¸ · h c2 ¸ ( ¸ ( ) 2 tan u ( ) · := F c2 ¸ ( ¸ 3.381 10 5 × N = F s11 ¸ ( ¸ f yd ¸ ( ¸ A s11 ¸ ( ¸ · := F s1 ¸ ( ¸ 1.749 10 5 × N = F s2 ¸ ( ¸ f yd ¸ ( ¸ A s2 ¸ ( ¸ · := F s2 ¸ ( ¸ 1.367 10 5 × N = N ud2 ¸ ( ¸ F c2 ¸ ( ¸ F s2 ¸ ( ¸ + F s11 ¸ ( ¸ ÷ := N ud2 ¸ ( ¸ 2.143 10 5 × N = M ud2 ¸ ( ¸ F c2 ¸ ( ¸ h h cm ¸ ( ¸ ÷ 2 h c2 ¸ ( ¸ · 3 ÷ | \ | | . · F s2 ¸ ( ¸ d h cm ¸ ( ¸ ÷ ( ) · + F s11 ¸ ( ¸ h cm ¸ ( ¸ d1 2 ÷ | \ | | . · + := M ud2 ¸ ( ¸ 60.387mkN · = 76 Zero strain in tensional bars (Positive moment) center mass of compression zone h c3 ¸ ( ¸ d 0.8 · := h c3 ¸ ( ¸ 0.148m = A c3 ¸ ( ¸ t b eff ¸ ( ¸ · 0.5 b2 · t2 · h h c3 ¸ ( ¸ ÷ ( ) 2 tan u ( ) ÷ ¸ ( ( ( ¸ + := A c3 ¸ ( ¸ 0.121m 2 = h cm2 ¸ ( ¸ b eff ¸ ( ¸ 0.5 · t 2 · 2 h h c3 ¸ ( ¸ ÷ ( ) tan u ( ) b2 + 2 ¸ ( ( ( ¸ t 0.5 h c3 ¸ ( ¸ t ÷ ( ) · + ¸ ( ¸ · h c3 ¸ ( ¸ t ÷ ( ) · + b eff ¸ ( ¸ t · 2 h h c3 ¸ ( ¸ ÷ ( ) tan u ( ) b2 + 2 ¸ ( ( ( ¸ h c3 ¸ ( ¸ t ÷ ( ) · + := h cm2 ¸ ( ¸ 0.06m = F c3 ¸ ( ¸ o f cd ¸ ( ¸ · A c3 ¸ ( ¸ · := F c3 ¸ ( ¸ 2.057 10 6 × N = N ud3 ¸ ( ¸ F c3 ¸ ( ¸ F s1 ¸ ( ¸ + := N ud3 ¸ ( ¸ 2.232 10 6 × N = M ud3 ¸ ( ¸ F c3 ¸ ( ¸ d h cm2 ¸ ( ¸ ÷ ( ) · F s1 ¸ ( ¸ d d1 2 ÷ | \ | | . · + N ud3 ¸ ( ¸ d h cm ¸ ( ¸ ÷ ( ) · ÷ := M ud3 ¸ ( ¸ 28.218mkN · = 77 Zero strain in tensional bars (negative moment) Height of compression area. Center mass of compression zone. h c4 ¸ ( ¸ d 1 ¸ ( ¸ 0.8 · := h c4 ¸ ( ¸ 0.174m = A c4 ¸ ( ¸ h c4 ¸ ( ¸ t2 ÷ ( ) b eff ¸ ( ¸ · t2 b2 · 0.5 · + := A c4 ¸ ( ¸ 0.078m 2 = F c4 ¸ ( ¸ o f cd ¸ ( ¸ · A c4 ¸ ( ¸ · := N ud4 ¸ ( ¸ F c4 ¸ ( ¸ ( ) F s2 ¸ ( ¸ + := N ud4 ¸ ( ¸ 1.456 10 6 × N = h cm3 ¸ ( ¸ h c4 ¸ ( ¸ t2 ÷ ( ) b eff ¸ ( ¸ · h h c4 ¸ ( ¸ ÷ h c4 ¸ ( ¸ t2 ÷ 2 + | \ | | . · t2 b2 · 0.5 · t t2 3 + | \ | | . · + h c4 ¸ ( ¸ t2 ÷ ( ) b eff ¸ ( ¸ · t2 b2 · 0.5 · + ¸ ( ( ( ( ¸ := h cm3 ¸ ( ¸ 0.099m = M ud4 ¸ ( ¸ F c4 ¸ ( ¸ h cm3 ¸ ( ¸ 0.5 d1 · ÷ ( ) · F s2 ¸ ( ¸ d 1 ¸ ( ¸ co ÷ d2 2 ÷ | \ | | . · + N ud4 ¸ ( ¸ h cm ¸ ( ¸ 0.5 d1 · ÷ ( ) · ÷ := M ud4 ¸ ( ¸ 55.171mkN · = 78 Moment 0 M ud3 ¸ ( ¸ M ud ¸ ( ¸ M u ¸ ( ¸ 0 M u2 ¸ ( ¸ ÷ M ud2 ¸ ( ¸ ÷ M ud4 ¸ ( ¸ ÷ 0 | \ | | | | | | | | | | | | | | . := Moment 0 28.218 63.948 24.756 0 40.197 ÷ 60.387 ÷ 55.171 ÷ 0 | \ | | | | | | | | | | | | . mkN · = Nforce N d ¸ ( ¸ N ud3 ¸ ( ¸ N ud ¸ ( ¸ 0 N tog ¸ ( ¸ ÷ 0 N ud2 ¸ ( ¸ N ud4 ¸ ( ¸ N d ¸ ( ¸ | \ | | | | | | | | | | | | | | | . := Nforce 3.306 10 3 × 2.232 10 3 × 1.66 10 3 × 0 397.187 ÷ 0 214.258 1.456 10 3 × 3.306 10 3 × | \ | | | | | | | | | | | | | | . kN · = 1 ÷ 10 5 × 5 ÷ 10 4 × 0 5 10 4 × 1 10 5 × 1 ÷ 10 6 × 2.5 10 5 × 1.5 10 6 × 2.8 10 6 × 4 10 6 × Nforce Moment