Proceedings of the Second Congress of the I.S.R.M. Beograd, Jugoslavija. 98. Wissmann, W., 1968. Zur statischen Berechnung beliebig geformter Stollen und.
TECHNICAL UNIVERSITY OF CIVIL ENGINEERING BUCHAREST FACULTY OF RAILWAYS, ROADS AND BRIDGES
CONTRIBUTIONS TO THE CONCEPT AND STRUCTURAL ANALYSIS OF PRECAST CIRCULAR LININGS FOR SHIELD DRIVEN TUNNELS
by TEODOR IFTIMIE
Ph. D. Thesis Resume
Technical Commision: - Prof. Dr. Ing. Iorgu Nicula - Supervisor UTC Bucuresti - Prof. Dr. Ing. Dan Stematiu - UTC Bucuresti - Prof. Dr. Ing. Tadeus Schein - U T Timisoara. - Prof. Dr. Ing. Anghel Stanciu - U T "Gh. Asachi" Iasi. - Dr. Ing. Dragos Teodorescu - I.S.P.C.F. Bucuresti. - Dr. Ing. Alexandru Wintze - Metroul S.A. Bucuresti.
BUCHAREST 1996
CONTENTS SUMMARY 1. INTRODUCTION 1.1. The problem 1.2. The general procedure of tunnel design 1.2.1. The choice of the construction procedure 2. THE EVOLUTION OF CIRCULAR TUNNEL SHIELDS 2.1. Introduction 2.2. Short history. Functions. Classification. 2.3. Different shield types. Field of utilisation 2.3.1. Non-pressurised front shields 2.3.2. Compressed air shields 2.3.3. Slurry shields 2.3.4. Earth pressure balancing shields (E.P.B.S.) 2.4. The evolution of the usage of different types of shields 2.5. Future performance. 2.6. Conclusions 3. LININGS FOR SHIELD DRIVEN TUNNELS 3.1. General observations 3.2. Construction systems for circular linings 3.3. Radial joints between segments 3.3.1. Generalities 3.3.2. Plane joints 3.3.3. Hinge joints 3.3.3.1. The mechanical behaviour of joints 3.3.3.2. The stresses and strains in hinge joints with convex-concave faces 4. STRUCTURAL DESIGN MODELS FOR TUNNELS. 4.1. General 4.2. Parameters considered in tunnel analysis 4.2.1. Ground parameters 4.2.2. Work parameters 4.3. The structural models classification 4.3.1. The one-dimensional model 4.3.1.1. The determination of the loadings which act upon a tunnel lining A. Active loadings B. Passive loadings C. Loadings models for the design of a circular tunnel D. The establishment of the model of the soil loadings, taking in account the circular form of the cross-section 4.3.1.2. Structural analysis models considering the structure-rock mass interaction A. Shultze-Duddeck method B. The method of polygonal chain on elastic supports B.1. The polygonal beam on elastic medium continuum contact. B.2. Particularities of the precast reinforced concrete linings calculation 4.3.2. The continuum model 4.3.2.1. The Morgan - Muir-Wood - Curtis method 4.3.3 Comparative study of the main calculation methods
4.4 The analysis of the influence of various factors. Parametric studies. The safety problem 5. THE STRESSES AND STRAINS STATE STUDY IN THE GROUND SURROUNDING FOR SHIELD DRIVEN TUNNELS 5.1. Introduction. 5.2. The study of the movement around a shield driven tunnel. 5.2.1. The ground movement description. 5.2.2. The methods of the direct estimation for the settlement. 5.2.3. The study of the ground movement on reduced models. 5.2.4. The monitoring of the ground movement by in situ measurement 5.2.5. Conclusions. 6. EXPERIMENTAL RESEARCHES REGARDING THE PRECAST SEGMENTAL LININGS BEHAVIOUR FOR THE SHIELD DRIVEN TUNNELS 6.1 Introduction 6.2. Laboratory model tests. 6.3. Full size rig tests on linings. 6.3.1 Rig test of the railway tunnel segments 6.4. Researches regarding the behaviour, during the construction, of the annular structure of some tunnels. 6.4.1. Convergence measurements at Beia tunnel 6.4.2 In situ researches upon the annular structure from segmental lining with steel-concrete mixed bolted segments, at the highway tunnel of Elba from Hamburg. 6.4.3. In situ researches upon the annular structure from reinforced concrete segments with hinge joints at the Birnova railway tunnel. 6.4.4. In situ researches upon the annular structure from reinforced concrete segments with plane joints at the Gibei railway tunnel. 6.4.5. In situ researches upon the annular structure from reinforced concrete segments with plane bolted joints at the Bucharest metro. 6.5 Conclusion 7. THE THEORETICAL STUDY OF THE STRESSES AND STRAINS STATE INTO THE ROCK MASS-LINING ASSEMBLAGE 7.1. Introduction. 7.2. The finite elements method, general presentation. 7.3 The modelling of the underground work construction. 7.3.1. Principles. Objectives. Limits. 7.3.2. The structure- rock mass assemblage discretizing 7.3.3. The modelling of the tunnel construction sequences. 7.3.4. Constitutive models, behaviour lows. 7.3.5. Conclusions. 7.4. The simulation of the shield driven tunnel execution. Numerical examples 7.4.1. Introduction. 7.4.2. The study of the deep tunnel behaviour. The CAV program utilisation. 7.4.3. The study of the mean-deep tunnel behaviour. The CESAR program utilisation. 7.4.4. Conclusions. 8. SUMMARY AND FINAL CONCLUSIONS 9 REFERENCES 1. INTRODUCTION
1.1. The Challenge. Underground structures represent a special component in the general field of construction. Engineering works which place below the ground either traffic (railway tunnels, road tunnels, and metros) or services (hydro-electric tunnels, or tunnels for water supply or sewage) constitute the most important aspect of underground construction, and are of particular economic significance on both a national and even on a global scale. Rational and economic execution of these works is therefore very important, and must be the main aim of all those specialists engaged in the enterprise. This can best be achieved by the development and application of advanced tunnel-driving methods and techniques on the one hand, and on the other hand, by creating a support system, which is both economical and safe. The achievement of the second aim is the purpose of this present study. 1.2. The general procedure of tunnel design. In the study, analysis, design and detailing of an underground structure it is necessary to have the independent participation of the following disciplines: geology, geotechnics, construction technologies, design of the sustaining structural elements, and contractual law. Experts in each of these disciplines are responsible only for their own specific areas of expertise, the final design decisions being taken as a result of the integrated co-operation of all the disciplines for unified development of the project. Fig. 1.1. shows the main elements in the design process of a tunnel.
Fig. 1.1. Design process for tunnelling (ITA Working Group 1988).
1.2.1. The choice of the construction procedure The choice of the construction procedure implies: • selection of an excavation method: by hand, by drill-and-blast, mechanical excavators, shields, etc. • selection of a lining and support system: anchors, shotcrete, steel ribs, in-situ concrete, precast concrete, iron segments or steel liner plates; • selection of transportation systems for materials and for tunnel spoil, and the ventilation system. The main factors that influence the choice of the construction method and overall cost are, in order of importance: the nature of the ground, the environment and the tunnel geometry. The environment is an important criterion, especially in urban areas, where disturbance to the residents may be required to be kept to a minimum. Many methods have been proposed to establish a connection between ground characteristics and tunnelling method, the intention of these being to guide the designer in the concept phase in the choice of support for the tunnel. An appropriate classification system for the various geological materials is a first step towards reliable prediction of their behaviour, and there are several such classification systems. The main advantage of these methods is to oblige the geologist and the designer to review and quantify all the geotechnical parameters, which influence the choice of method for constructing the underground works. The selection of a particular method requires the exercise of experienced judgement to ensure the appropriateness of the proposed method of support. Amongst the various methods of tunnelling which have been developed over the years, two procedures currently dominate the world tunnel-building market. These are, for rock tunnels, NATM, and, for soft ground, the shield method. In this study, only the latter aspect is considered. 2. THE EVOLUTION OF TUNNELLING SHIELDS 2.1. Introduction The field of underground construction has developed rapidly during the 1970s, and even more rapidly in the 1980s. This phenomenon may be attributed to two main influences - in part, to the increased movement of people, vehicles, materials, etc., constrained by the lack of space in urban areas, which increases demand; and in part to the development of the technology of shields in general, and of pressurised shields in particular, which makes a tunnelled solution more feasible. Table 2.1 TUNELS LENGHT (km) 1970 - 1980 Railways Romania Hungary Czechoslovaky Italy England Germany
Metro
Sh.
AM.
Sh.
AM.
2.2
3.5
2.3
10
3 0.3 66 53 159
25 5.2 5.2 30 212
Shield 1980 - 1990
Road Total
Railways Sh.
0 0 0.5 0.5 8 53
AM.
Metro Sh.
Meth. Road
Total
(%)
0 0 6 6 7 62
105.4 20 82 82 79 493
66 90 85 90 30
AM.
18 10.5 7.8 60.5 28 0 20 6 4 72 6 4 72 91 46 26 424 185 246
Table 2.1. gives the lengths of tunnels built for roads, railways and in metro during the 1970s and 1980s, for Romania and other 5 European countries. For Romania, the execution methods are also shown, and it may be seen that the use of the shield method in Romania has increased from 12% in the 1970s up to 66.5% in the 1980s. In many countries, this figure has risen to very high levels (85-90%), demonstrating the applicability of this method both in soft ground and in rock. 2.2. Short history. Functions. Classifications. Shield technology already has a respectable pedigree. A review of the major developments in the evolution of shield tunnelling is included later in this study. When a tunnel is driven, it is essential to maintain the stability of the underground construction, and to protect the environment, especially the aboveground structures. A shield must be designed and analysed as a combination of individual and interactive functions, which must achieve two major objectives: to maintain the stability of the tunnel, and to advance the face. The latest generation of pressurised shields can provide the following functions: excavation, spoil removal, faces advance, immediate temporary support, erection of the permanent lining, and the injection and separation of slurry. Shields are generally classified as either open-faced or closed-face. The choice of which type of machine to use in a particular instance obviously depends on the nature of the ground to be traversed. Figure 2.6. presents a selection guide, with a classification of shield types. The choice of a machine depends not only on the ground conditions - both its mechanical and hydraulic properties - but also on the dimensions of the tunnel and its location - the geometric parameters. Thus the choice of whether to use an open- or closed-faced machine is influenced by the following: • the geotechnical parameters of the ground (cohesion, friction angle, hydrostatic pressure, etc.); • the geometrical parameters of the project (the depth of the cover and the tunnel diameter). This study presents the different types of shields, differentiating between those in which the face pressure is maintained, either with bentonite or by earth pressure. 2.4 The evolution of the use of various types of shields. In order to analyse the evolution of the use of various shield types, several sources have been used. This data has been processed to show the various types of shields used as percentage, as a number and also as executed length, and also to show the percentage of use of shields with diameters between 5-7m and bigger than 7m, used mainly for works of railway tunnels and metros. From these statistics, some interesting conclusions may be drawn: • The use of classical non-pressurised shields has remained fairly constant in most countries: 42% between 1967-1987 and 45% between 1984-1992, while in Japan its use decreased from 31% to 10% between the same periods. • The use of compressed-air shields has decreased from 26% to 12%; • The use of bentonite shields has decreased slowly in Japan, from 38% in the period 19671984, compared with 34% in the period 1984-1992, although it has shown a slight increase in the rest of the world, from 16% to 22% between the same periods;
TYPE OF TUNNELING MACHINE
Manual shields
SOFT GROUND Homogeneous Complex
Soft
Hard
Confined boulder
HARD GROUND (ROCK) Complex Homogeneous Soft Hard
Aplicable in combination with auxiliary method
Aplicable
Shields with excavator
Blind shields
Mechanical Shields Boom head shields EPB Shields (Without high density slurry) Slurry shields
EPB Shields (with high density slurry) EPBS / Slurry (with disc cutter) Shiel type TBM
TBM (Unshielded)
Fig. 2.6. MHI tunnelling equipment selection guide (Mitsubishi, 1993).
• The use of EPBS shields shows a significant increase in Japan (22%) but a more a modest 12% in the rest of the world; • The superiority of the pressurised shields over the classical shield becomes even more evident when the lengths of tunnel driven are analysed. The tunnel lengths driven by EPBS represent 46.7% compared with 24.6% by classical shields. A decisive contribution (33.4%) to this success is the 60.985 m French section of the Channel Tunnel; • The percentage of the use of large diameters 5-7 m and over 7m, is also significant, constituting respectively 48%, and 63% for the number of shields and 72% for the length of tunnel driven. The use of pressurised shields also dominates the large diameter segment at 29% and 53% (Japan) or 60% for length. All these data prove a single fact, that the pressurised shield has progressively conquered the world market, and amongst these, the EPBS type seems to represent the future. 2.5. Future performance. Predictions of future performance of machines as complex as the pressurised shield depends very much on analysing the effectiveness and the effect of different measures which may be incorporated. Future performance can be defined by different parameters, such as: the driving speed while digging, the mean rate of advance, the coefficient of working time use, or the coefficient of total time use. From these parameters, the mean rate of advance is the parameter which best characterises the overall performance of a worksite, contractor or machine. The mean rate of advance represents the ratio between the effective length of tunnel drilled by the machine and the total working time. Statistics obtained from different tunnels show different mean rates of advance dependant on the shield type and on the nature of the ground - for example, shields of the EPBS type can attain a typical figure of 200-300 m/month. The prediction of rates of advance is a difficult and delicate task, in which consideration must be given to such varied factors as: geological conditions, project concept, the Contractor’s site organisation and experience, the capacity and design of the machine, and the contractual relations between different parties. 3. TUNNEL LININGS 3.1. General observations The choice of lining is a major factor in any tunnel project. The lining ensures not only the stability of the excavated opening, the integrity of the tunnel structure itself and the safe utilisation of the tunnel in service, but it also affects the rate of tunnel advance. The lining is also the most important single element in the total cost of a tunnel project. Various studies and investigations show that the lining cost represents between 14-15% of the total cost of the tunnel; this cost is a function of various factors, such as the nature of the ground, the construction method, and the materials used. 3.2. Construction systems for circular linings. Construction systems used in the shield method: • system with a precast lining; • system with two linings: a precast primary lining followed by a secondary in-situ lining;
• System with in-situ concrete lining, cast using special formwork. The tendency is to use a single precast lining, which serves both as immediate and as final support, also ensuring water-tightness. The main requirements of a precast concrete segmental lining are: • the capacity to resist rapidly the soil and water pressure without dangerous deformation or seepage; • the capacity to resist handling stresses; • the capacity to resist ram forces produced during shield advance; • the capacity to resist water seepage through the segments and joints, and also to resist corrosion; • economy during construction and maintenance (the usage of some more resistant materials can be more economical than the usage of less resistant materials). In a simplified scheme, it is necessary to reduce the more significant factors that affect the design of a satisfactory tunnel lining system (Fig. 3.1.).
Fig. 3.1. The scheme of the design of a linings system. Some basic factors recommended in the design of a tunnel lining, would be: the anticipated axial force to be maintained, to anticipate the bending deformations of the lining and to take into account the possible instability of the lining under compressive loads. Figure 3.2. is a schematic diagram, showing the development of the precast lining, by illustration of some representative type.
Fig. 3.2. Schematic diagram showing the development of precast linings. 3.3. The joints between segments. 3.3.1. General The existence of joints between the segments is a main characteristic of precast linings. There are two main types of joints - those where the abutting faces are planar, and those where they are curved. The analysis of the mechanical behaviour of these hinge joints must take account of the effects upon the bending moments caused by eccentricities and resistance to rotation occurring at the joints. In addition, the state of stress and deformations around the contact zones must be analysed in order to establish the influences upon the neighbouring areas, and also upon the whole structure. The development of a realistic calculation model for a precast lining supposes also a correct modelling of the joints between the segments, whose influence is very important upon the sectional dimensions. The static action of the lining will be determined in a large measure by its rigidity, i.e. by the overall capacity to resist deformation, the combined effect of deformation of the segments and deformation at the joints. In the majority of the reinforced concrete precast linings, the deformation at the joints has a significant effect on the deformation of the segments. Thus, the magnitude and distribution of the internal forces depend to a great extent upon the distribution and construction of the joints. Consequently, the most important problem in the design of a reinforced concrete precast lining is the determination of the bearing capacity (strength and deformation) of the joints between the segments. 3.3.2. Plane-faced joints The plane-faced joint is must be the oldest type of joint used to the precast linings. The analysis of the joints with plane faces starts from the hypothesis that it is not possible in practice
to have a perfect contact between the two faces. In a static system, the contact zone between two segments can be approximated by a partial joint with a rotational rigidity Cd. This rigidity CD can be calculated by taking the simple view (Fig. 3.5.), fact that the joint cannot resist any other than compressive forces, whose distribution is considered to be parabolic. On this basis, the rotation angle (α) and the rotation rigidity (CD) may be determined.
Fig. 3.5. Plane joint - calculation elements. From the figure 3.5 we can deduce: e= 1 (3r+a); N= 2 σrb(a-r); 8 3 M=Ne= 1 σrb(a-r)(3r+a); 12 e m= = M = 1 ( 3r +1); a Na 8 a r = 8m −1 ; a 3 1 σr a 3N ∆ = α = = = 2 a −r a −r E ⎛ r⎞ 2ab⎜1 − ⎟ E ⎝ a⎠ 27 N ; (3.3) Fig. 3.7 Plane joint – angular deformations 32ab(1− 2m )2 E on the basis on which the rotation rigidity CD can be determined: 1 1 c Na Na c dM c + CD= ; (1-2m)²= ; m= + M=N 2 2 α 2 2 α dα α 1 Na d Na c − c (α 2 ) = ; dα 2 4 α3
8 a²b(1-2m)³E = CD0 K; (3.4) 27 Na (1 − 2 m) K=3.55 (1 − 2 m) 3 M=CD*α= (3.5) 4 The moments and axial forces in the joint can be calculated for different eccentricities using the above formulae, from which the interaction curve M-N may be drawn.
CD=
CD =
Fig.3.9. shows the interaction diagrams for eccentric compression in the current section and in a joint with plane faces with flattened angles, for a precast lining of 30 cm thickness. It can be seen that the bending bearing capacity of joint is always smaller than that the adjacent reinforced section, attaining maximum 60% in the case of a small eccentricities, in the limits of the central core.
In order to preserve this effect of joint, as far as possible in calculations, the degree of elastic clamping was defined conventionally as the ratio between the moment at the joint and the moment in the adjacent section. This can take values between 1, for perfect clamping and 0, for a fully hinged joint, but in practice would vary between 0,6 and 0 for the connection being studied.
Fig. 3.9. The interaction diagram M-N. It can be written as a rigidity matrix [ r ] of a bar (j-k) with elastic clamping supports multiplying the rigidity matrix elements for a doubly clamped bar with some coefficients function of η . EA l 12EI η l3 z1
0 0
[r] = −
6EI 2 −ηk ηj z1 l2
4EI 3η j l z1
0
0
12EI η l3 z1 6EI 2 +η j − ηk z1 l2
6EI 2 −ηk ηj z1 l2 2EI 3η jηk
−
EA l 0 0
−
l
EA l
z1
0 0
12EI η l3 z1
6EI 2 +ηk ηk z1 l2
4EI 3ηk l z1
where: η=ηj+ηk+ηjηk ; z1=4 - ηjηk ; For ηj = ηκ = 1 we obtain the rigidity matrix for doubly clamped bar and for ηj = 1, ηκ = 0 or ηj= 0 si ηκ = 1, we obtain the rigidity matrix for clamped-articulated bar and articulatedclamped bar. For ηj= ηk= 0 we obtain the rigidity matrix for double articulated bar.
Thus the bending moments in the joints for different situations of joint opening can be calculated. The calculation process is iterative; it starts from a known angular deformation, which is successively corrected until it agrees with the rotation obtained from calculation. This process may be extended to the cracked sections, approximating the clamping degree corresponding to the new rigidity. The effect of the rigidity reduction by cracking upon the magnitude of the moments in the lining can be simply calculated. The case of bolted joints can be treated similarly. 3.3. 3. Joints with curve faces. Hinge joints.
This type of joint ensures a more centred transmission of the compression annular forces than the type with plane faces. A joint with curved faces can be classified by reference to the following: 1. the curvature of surfaces in contact; 2. the permitted angle of rotation; 3. procedures of joint realisation; 4. the means of maintenance of the initial shape. This study presents most representative types of joints, many of them are patented Fig.3.10.
Fig.3.10. The types of hinge joints.
3.3.3.1. The mechanical behaviour of hinged joints.
Under the action of external loads, adjacent precast elements rotate, one against the other. In order to analyse the rotation phenomenon, a hinge joint is considered, with convexconcave faces (Fig. 3.12). On the basis of a study on the way that rotation, loads and stresses are transmitted, it has established that, for joints placed in different positions in cross section, the supporting line and, with this, the resultant compression stress has an orientation parallel to the tangent in the non-deformed position. Knowing the friction coefficient for concrete (0.5-0.8) and the T/N ratio (0.2 most frequently), and considering the rotations of both elements as being equal (αv=αk) that ratio rk/rv can be determined which ensures the stability against sliding for different angles of rotation α (table).
α 1° 2° 3° 4° 5°
T/N=0.2 rk/rv > 1.062 1.128 1.198 1.274 1.354
rv=0.2 rk > 0.21 0.23 0.24 0.25 0.27
Fig. 3.12. Geometrical relationships to rotation. 3.3.3.2. Stresses and strains in a hinged joint with convex-concave faces
The compressive annular force is transmitted from one segment to another by means of a surface smaller than that one of the cross-section of the segment. There is therefore a partial loading of the surface, while in the joint zone there is a state of spatial tension. Tests have confirmed the hypothesis that the concrete strength is higher under the action of some partial loadings of the surface. Different authors agree that the effective stresses in concrete can attain the compressive cube strength Rpr, and some of them hold that it may even exceed it by up to 80% (Leonhard and Reimann). The stresses and strains in the hinged joint can be analysed by three procedures: the application of an analytical solutions to the washer’s problem; the application of a simple geometrical procedure of approximation; or by F.E.M. Both types of joints have been analysed with the aid of FEM, both with plane faces and with circular faces, in two situations - centred, and with eccentricity. The program provides the graphical representation of the stresses (isobar) for max., σ2 max.
σx,σy, σ1
Thus the zones with compressive stress concentrations and those tensile stresses can be identified, and curves may be drawn to show the variation of different stresses in different directions.
Fig. 3.15. Stress distribution in the joint under centered loadings. a) σx stress; b) σy stress. Such a representation, for the stresses σx,σy is presented in the figure 3.15. On the basis of such an analysis, the reinforcement in the end zones of the joints can be established with more precision and efficiency. The analysis of the behaviour of the joints between the precast segments of the annular structure, demonstrates the important role played by the joints in ensuring adequate bearing capacity and hence the safety of the structure. 4. METHODS OF STRUCTURAL ANALYSIS FOR TUNNELS 4.1. General The statical calculation and the tunnel dimensioning problem has been always among the most discussed and difficult problem for the design engineers. The judgement, experience and intuition of the designer are more necessary than in other cases, in order to compensate for the absence of codified rules in this field. The construction of an underground work in a rock mass has to replace the pre-existing natural states of stress with a new system of stresses of the combined structure of rock mass, hole and lining. The lining of the tunnel is a strange body in a system (the rock mass), where the initial state of stress has been disturbed by the excavation of the hole and the lining limits the deformation of the ground towards the centre of the hole, thus impeding collapse. 4.2. Parameters to be considered in tunnel design.
The calculations for the design of a tunnel lining use an array of parameters which may be taken into consideration directly and quantitatively. Alternatively, a parameter may be taken into account indirectly and qualitatively after consideration of the possible effects that the introduction of that parameter might have on the performance of the proposed method of construction. Amongst these parameters are those of both the ground and of the lining. 4.3. Classification of calculation methods.
Fig.4.1. gives an indication of various structural design models used to represent the circular lining; these are shown as much as possible in sequence of evolution the general share corresponds to the two domains of the mechanics of deformable media, strength of materials and theory of elasticity. The one-dimensional model treats the lining as a curved bar, and the loads are determined separately using various theories, of which the most commonly used are those of Protodiakonov and Terzaghi. The two- or three-dimensional model determines the state of stress and the deformations in the ground/lining structure according to elastic theory, using analytical or numerical methods. There are also some empirical methods, based on experience, observation, and also measurement. The tunnel designer faced with choosing a design method from these will be somewhat bewildered. The analysis below attempts to answer the question: “Which method is most suitable?”
Fig. 4.1.
Fig. 4.2. Calculation theories for vertical loads 4.3.1. The one dimensional model
Underground structures are statically indeterminate, as follows: • internal indeterminacy of the structure (strange body in the system) which is treated as one-dimensional element (curved bar) in the same way as in conventional above-ground structures. • external indeterminacy represented by the lining-rock interaction. Static calculations for underground structures are more difficult than for above-ground structures because of this external indeterminacy. There are two different groups of methods for treating this indeterminacy: a. Methods which ignore the lining-rock interaction, and consider only the external loads, with the assumption that the lining is either infinitely rigid or freely deformable. These methods are met more in text books than in real life, their application being completely random.
b. Methods which take account of the rock-lining interaction. The methods from this group differ according to the way in which this interaction is treated, the way in which they consider the lining and the author's name. The interaction can be in the form of an arbitrary action (Hewett and Johanesson); or reactions corresponding to the Winkler's law (4,5,6); or reactions determined according to Elastic theory (7). The fundamental simplification introduced in the "one-dimensional model" methods is that the totality of the factors linked by the ground is represented by the combination of the active loads (vertical and horizontal with parameters γ,ϕ, c ) and passive loads (the bed coefficient K). The method of the polygonal beam on springs is the only method which can take account of most of the parameters of the tunnelling process, including the execution phases, the shape (circular or arbitrary) and the structural form (single or double lining, varying joint and segments configurations). 4.3.1.1. Determination of the loading which acts upon the lining of a tunnel
In the case of the one-dimensional model, the loads acting upon the lining are considered external and are divided into two categories: • active loads, whose magnitude is independent of annular deformation; • passive loads, comprising the reactions given by the surrounding ground, which oppose deformation of the lining and the intensity of which is related to this deformation. Establishment of a ground loading model for a circular cross-section.
Although in reality the loads acting upon the tunnel are perpendicular to the tunnel surface, which most often is curved, in the current calculations they have been split into their vertical and horizontal components, just to ease the work of the designer. Various mathematical models and formulae have been developed specific to this field (tunnels), to determine the vertical loads. The well-known model of sliding planes defined by Coulomb has been adopted to determine the horizontal loads. The distribution of active earth pressure acting on the sides of the tunnel is calculated using the formula established by Coulomb, for the hypothetical case of the rectangular section. ka =
sin 2 (θ + φ ) ⎡ sin(φ + δ ) sin(φ − β ) ⎤ sin θ sin(θ − δ ) ⎢1 + ⎥ sin(θ − δ ) sin(θ + β ) ⎦ ⎣
2
2
The case of the rectangular tunnel is not entirely relevant, since tunnels in practice have curved shapes, and nowadays are generally circular, to suit the use of shields. In the case of a circular section: sin 2 (θ + φ ) ka = sin θ (sin θ + sin φ ) 2 In order to apply this formula, we must approximate the circular shape by using a polygonal one. In this way the horizontal pressure on each side of the polygon can be calculated as a function of its inclination to the horizontal (θ). It can be observed that the plane of sliding is tangential to the circular lining at the point where the soil pressure is theoretically zero.
The ordinates of the pressure diagrams on each segment are computed, and the resultants of the radial pressure at the nodes Pat are calculated. These are resolved into their horizontal and vertical components, which can be introduced directly into the static calculation, in the form of horizontal and vertical distributed loads. The distributions thus obtained differ noticeably from the results obtained by previous calculation methods, both for vertical loads and particularly for horizontal loads, but they correspond better to engineering intuition (Fig. 4.7.).
Fig. 4.7. Proposed loading model for active earth pressure diagrams. In order to see the influence of the new distribution of the loadings upon the forces acting on the different sections, comparative studies have been carried out with the currently-used loadings, vertical uniformly-distributed and horizontal trapezoidal (Fig. 4.8.a), both vertical and horizontal uniformly-distributed (Fig. 4.8.b), Terzaghi model (Fig. 4.8.c) and the proposed model (Fig. 4.8.d).
Fig. 4.8. Comparison between different loading models.
The moments obtained with the proposed loading model are significantly smaller (40-60%) than those obtained using the classical models. Although the results seem encouraging, the introduction of the new loading model into current practice must be made carefully, and it must be checked by in-situ measurement. Due to a dearth of proper measurements of the pressures imposed by the ground on the lining of a tunnel, we have to accept comparison of theoretical results, with sample measurements (where they exist) taken from the literature, and these show good correlation. The final conclusion which may be drawn from this analysis is that, to determine the loads upon a tunnel, the shape of transversal section (circular, horseshoe, rectangular, etc.) must be taken into account. 4.3.1.2. Structural analysis models considering the structure/rock-mass interaction An inquiry instituted by the International Tunnelling Association in the working group "General Conceptions of Tunnels Design", to which twenty-six member countries have responded, revealed that the calculation methods most frequently used are those of Schulze/Duddeck and Muir Wood. If the inquiry had included the countries of the old socialist block, a third method would have been among those most frequently used - the method of the polygonal chain with elastic support. B. The method of the polygonal beam with elastic supports Also known also as the method of springs, this method has indeed a high degree of general applicability, permitting the introduction of some new hypotheses regarding: • the shape of the cross-section; • the orientation and distribution of the active and passive loads in cross-section and longitudinally; • the conditions of contact between ground and lining; • the condition of contact between individual lining elements. The cross section of a tunnel made up generally from curved elements can be approximated by a polygonal system made up of bar type elements, which may be either plane or spatial (Fig. 4.14).
Fig. 4.14. Structural models for poligonal beam on springs.
The composite action between the structure and the rock-mass along the continuous contact zone, produces displacements towards the exterior (the bed zone), and this is modelled by deformable supports (elastic springs) positioned at the nodes, capable of carrying only compression. If it is assumed that there is no friction between the lining and the ground, the elastic springs will be disposed radial (i.e. in the direction of the bisector of the angle between adjacent bars). If there is frictional contact, this will be distributed between pairs of springs at each node disposed either normally or tangentially or vertically and horizontally, producing a resultant whose direction is at an angle (less than the friction angle) to the normal to the arch.
Fig. 4.15 Types of springs Initially, arbitrary assumptions are made for the thickness of the bed zone and implicitly of that of separation (90-100 for the separation zone), and the system is solved by computer using the general methods of statics (the force or displacement method). After the first iteration, the contact zone structure may be reduced by elimination of any springs in tension, or it may be extended if high compression is found in the last support. The deformability of the elastic springs is determined on the basis of Winkler’s hypothesis by means of the bed coefficient K. This method allows the introduction into the calculations of some relatively complex hypotheses, such as: • the introduction of an initial deformation of given magnitude, representing a possible void between the ground and lining, considering the bed coefficient null or reduced; • the simulation of the elastic-plastic behaviour of the ground by changing the bed coefficient after a specified amount of deformation; • variation of bed coefficients from node to node, to allow for ground stratification (layering); • consideration of the joint type between the segments (hinge or plane joint); • the performance of a second order calculation. However, this model represents the simplest and the most efficient calculation-oriented model, which explicitly introduces the phenomenon of ground - lining interaction, and permits the solving of an almost unlimited variety of applications. B1. The polygonal beam in continuous contact with an elastic medium.
The solution with supports at nodes appeared in the period when calculations were performed manually. With the introduction of computers, it is now possible to attack the “continuous contact” problem. The difference lies in the fact that the reactions, instead of being applied at the nodes, are distributed along the bars and vary with node displacement. The calculation method is based on two fundamental hypotheses: • the reaction of the elastic medium is proportional to the vertical displacements of the bar (Winkler 's theory); and • the displacements of an arbitrary section of the bar may be completely determined, starting from the displacements at both ends.
The ground reaction may be found from the expression: ⎡ s2 s s 2 s3 s2 s s2 ⎤ q = - bk ⎢θ yi (1 − − θ yi s(1 − ) + ∆ zi (1 + 2 3 − 3 2 ) + ∆ zj ( 3 − 2 ) 2 ⎥ l l) l l l l l ⎦ ⎣ and allows the reactions and bending moments, due to the re-distributed load q, to be calculated by simple integration from the ends of each doubly-clamped element. Hence, it is possible to formulate the supplementary rigidity matrix for the doubly-clamped bar with continuous support, including the axial force, for a bar of unit width b=1:
⎡ 140 µ ⎢ l2 ⎢ ⎢ 0 ⎢ ⎢ 0 kl 3 ⎢ 420 ⎢ 70 µ ⎢ l2 Kb = ⎢ ⎢ 0 ⎢ ⎢⎣ 0
70 µ l2
0
0
156 l2 22 l
22 l 4
0
0
0
54 l2 13 − l
13 l
140 µ l2
−3
0
0 0
0 54 l2 13 l 0 156 l2 22 − l
⎤ 0 ⎥ 13 ⎥ − ⎥ l ⎥ −3 ⎥ ⎥ 0 ⎥ ⎥ 22 ⎥ − ⎥ l ⎥ 4 ⎥ ⎦
The total matrix of a double damping bar supported on elastic medium will be: K=Kel +Kb where : Kel = the rigidity magnitude of the unsupported bar; Kb = the supplementary rigidity matrix for elastic supporting. This rigidity matrix has been included in the computer program TIME, written for the plane structure by the author in 1989. The program module for the case of continuous support has been made validated by means of a simple example (the continuous beam on elastic supports with a central concentrated load) for which the exact solutions have been calculated using Winkler's functions. A comparison has been made for a precast lining of tunnel, computed using two different methods (Fig. 4.19): (a) with polygonal beams on springs and (b) with circular beam on continuous contact. Comparing the bending moment and axial force diagrams of both methods, the method that assumes continuous support gives a moment reduction of about 5% and a 2% increase of axial force, a more preferable solution; this may be due to the increased accuracy of the calculation. The calculation time is shorter for this method, and the input of data is easier.
Fig. 4.19 M and N diagrams for discrete contact (a) and continuous contact. (b). B2. Particularities of the reinforced concrete precast linings analysis.
It is normal practice to simplify the calculation by assuming that the lining is a continuous ring, and to ignore imperfections in casting. However, such a simplification does not accurately reflect the actual behaviour of the precast lining and is not therefore an efficient guide to choosing the most practical solution. A second simplification, equally important, is to consider the erected lining to be perfectly circular, and loaded in accordance with theory, ignoring deficiencies in grouting and erection eccentricities. Regardless of the type of joint, it is simply not possible in practice to construct a lining which conforms to the theoretical assumptions. The solution normally adopted is to perform the calculations based on the premise that indentations are already formed in the mating surfaces of the joints, i.e. that there is no point loading.
Fig. 4.20 Types of contacts between joints with plane faces. In chapter 3.3 above a detailed study was made of plane and curved faced joints between segments. For joints with curved faces, the displacement of the point of contact occasioned by rotation of the faces is generally small and has little influence on the state of stress in the segments. By comparison with joints with plane faces, however, the existence of the initial indentations in the curved faces, supported by the edges of the segment, has a significant effect on the static performance of the lining, even without the action of external loads. Thus, in a circular lining with initial indentations at the joints, the bending moments are due not only to the action of the non-uniform component of the load, but due also to the axial force.
The effect of the axial force across joints with initial indentations may be represented in the statically calculations by various auxiliary constructions, presented in this work (Fig. 4.21), from which the most appropriate to the real model, is that based on rotational rigidity as defined in chapter 3 above.
Fig. 4.21 Schematisation of joints opening. By performing iterative calculations on this model the eccentricity eo and the rotation rigidity Cd may be obtained. This forms the basis for the computer program called TIME. 4.3.3. Comparison of the principal methods of calculation.
The designer of a tunnel is usually faced with a difficult choice when it comes to deciding which calculation method to adopt. Having reviewed the alternatives -- a brief look at some, a more detailed look at others -- the following comparative study can facilitate the decision. In view of the popularity of precast linings, it is obvious that in such a comparative study of the different methods, the main factor to be considered is the accuracy and relevance of the selected model with respect to the peculiarities of the actual lining being proposed. A further factor in the choice of which methods should be considered in this study is the frequency of use of those methods. Four methods meet this criterion: • the Schultze/Duddeck (S+D) and Muir-Wood (M.W.)methods, reported, in an inquiry performed by ITA in 26 countries, as the most used; • the method of polygonal beams on springs (M.L.P.), mainly used in the countries of the old socialist chain, omitted from the ITA inquiry; • the FEM, a method generally accepted also for tunnel calculation which will be investigated more closely later. The principal features of these different models are that the Schultze/Duddeck method considers the structure and the supports to be continuous, while the polygonal beam method considers the structure with internal links at the joints and supported at the nodes. In both these methods, the active loads are considered as external and they are computed with one of the previously described methods. By contrast, the Muir-Wood method and the FEM consider the problem as a continuum model; the first method gives an analytical solution, while the second produces a numerical solution. The comparative study which follows assumes a shield-driven railway tunnel with the lining made up from five segments plus key. The calculation results are presented in Fig. 4.27. For uniformity of the loading hypotheses, a relaxation of the initial stresses has been assumed, giving a 60% reduction of the vertical load. However, the results obtained are not sufficiently different to eliminate the doubt and to give definitive confidence in one particular method. Nevertheless, all the methods give similar figures for the bending-moments, while the FEM and polygonal beam methods show agreement for the shape of the bending-moment diagrams.
Fig, 4.27. Comparison of the methods. In this comparison, the magnitude of the sectional forces is relatively unimportant when it comes to dimensioning the segments and designing appropriate reinforcement, whereas the shape and the sign of the bending-moment diagram are more critical. The existence of the joints between the segments of hinge is taken into consideration only by the polygonal beam method and by FEM with a corresponding discretization of the joint. The polygonal beam on springs method is perfectly suited for calculation by computer and able to reflect the behaviour of the peculiarities of this type of lining, occasioned by the type and rigidity of the joints, and by their number and location in the section. The use of FEM with its possibilities for elastic-plastic analysis and rheology can form a reference for the classical methods. The polygonal beam on springs method, as improved by conclusions from the FEM study, especially for loadings, can remain for the future a simple, efficient and fast tool for the analysis of precast linings. Efficient analysis can be ensured only by in-depth knowledge of the main factors which influence the behaviour of this type of lining. 4.4 The analysis of various factors influence. Parametric studies. The safety problem. 4.4.1. Introduction.
The aim of the tunnel engineer, as of any other engineer, is to achieve safe construction. There hardly exists another form of construction in which the safety problem is as complex as for tunnels. The establishment of a safety concept for tunnels is very difficult, impossible, even, due to the multitude, complexity and dispersion of the factors involved. It is essential that the safety aspects of each of the influencing parameters are examined. There are an almost unlimited number of possible combinations of partial safety factors which
have an influence on the overall safety. The results of the static calculations are also influenced by other factors and parameters, from which we put here the most important: • structure composition (number of segments, emplacements in the section, joint type); • initial deformation of the ring of segments before ground resistance is mobilised; • bed coefficient K; • deformations of plane joints with resistance to rotation; • angle of internal friction, and the coefficient of lateral pressure; • lining rigidity; • consideration of the tangential bed; • effect of non-linear geometry. This list, which is by no means exhaustive, shows examples of where the engineer can introduce partial safety factors for a number of different parameters. If, for example, for each individual hypothesis a safety factor is assumed, then the possibility of damage is reduced as the number of combined hypotheses increases. But, in practice, the engineer needs a guide, and the simpler such a guide is to use, the more useful it would be. The simplest way to reach this aim is by undertaking some parametric studies. By such a process, individual coefficients may be established for the influence exerted by each of the different parameters, and listed in order of the amount of influence exerted by each. 4.4.2. Analysis of the influence that the structural composition has upon the behaviour of the precast lining.
In the literature, there are accounts of numerous tests to establish, analyse and classify the factors which influence lining behaviour in general and precast linings in particular. For precast linings, the essential factors required for good structural analysis are: • number of segments; • type of joints between segments; • preservation of the initial circular shape. The influence of these factors upon the forces acting in the cross-section is reflected in the following structural analyses: 1. analysis of the influence due to the number of segments and their placement in the section; 2. analysis of the joint type and of the influence due to initial deformations, for two configurations of segments in the section; 3. analysis of the influence due to the rotational rigidity of plane joints, for two configurations. 1. The influence due to the number of segments and their placement in the cross-section has been analysed in three variants of the structure, with four, five and six segments respectively in the cross-section and with two possibilities of placement, either as a normal segment or as a key (Fig. 4.28.) The sectional stress (M and N) obtained by calculation (Fig.4.29) have been compared with those obtained for a continuous un-jointed cross-section, taken as the control. The variation of the maximum bending-moment, both in position and in magnitude, varies with the section structure. The maximum bending-moment decreases as the number of segments in the section increases, but it is also a function of the where the joints are located with respect to the vertical axis.
Fig.4.28 Types of performed structures.
Fig.4.29 Diagrams of the moments and axial forces.
The minimum value of the maximum bending-moment, is obtained in the variant 6B, being only 32% of the maximum moment of the control. The axial force increases also with increase number of segments, attaining the maximum, also for the variant 6B, of 29% of that in the control. The representation of the results obtained in the interaction diagrams M - N, (Fig. 4.30) allowing for safety coefficients, leads us to the following conclusions: • the configuration type B, seems to be more advantageous than type A from the safety point of view with respect to the values for M and N, giving a maximum for variant 6B; • it is observed that the application zone, with respect to the total interaction diagram, is reduced in the direction of the axial force, where considerable resources exist. From the analysis of the sectional forces M and N, it is clear that the variant 6B is the most advantageous. A definitive ordered list can be given only after the analysis of the state of deformations. In order to have a more clear image, the variation in maximum displacements is shown as a function of the variation of the bed coefficient K (Fig. 4.31). An increase of displacements is apparent when the bed coefficient K decreases, i.e., the ground is weaker.
Fig. 4.30 Interaction diagrams and safety coefficients The variant 6B is shown to be the most deformable, and for values of K under 1daN/cmc, the maximum displacements to the key become unacceptable. The variation in M with K, shows that as the bed coefficient K increases its influence upon the bending-moment decreases. This influence becomes less as the general rigidity of the lining decreases. It can be shown that, for the type 6B, the bending-moment variation as a function of K is practically nil. In conclusion, we can say that, when a tunnel structure (segments number and placements in section) is adopted, the nature of the ground must be considered first. Consequently, we can deduce that, for hard soils (K> 2daN/cmc), more elastic structures, such as types 5B, 6A and 6B, can be adopted, whereas for weak soils (K < 2 daN/cmc), the use of more rigid structures (types 5A, 4B) is recommended.
Fig. 4.31 The variation of M and ∆ diagram as a function of K.
2. Analysis of the type of joints and of the influence of initial deformations. The first analysis performed on three structural variants has supported the existence of the hinge joint type and requirement to preserve the circular form of the ring. In practice, the joints may also have plane faces, and the circular shape is not always maintained, sometimes with significant deformations being recorded even during the erection phase.
A comparative study of plane and hinged joints was indicated, and also of the influence which initial deformations might have upon the magnitude of the sectional stresses. This analysis was carried out on the variant of structure with 5 segments, in the A and B configurations, both for symmetrical and for asymmetrical deformation. On the basis of the results of the static calculations, presented in Fig.4.32, some observations may be made: • the shape of the bending-moment diagram and the magnitude of the bending-moments for a continuous ring are appreciably different from a ring made up of precast elements, regardless of the configuration or the joint type; • the use of plane joints instead of hinged joints led to an increase in bending-moments of up to 150%; • deformation, especially asymmetrical, from the circular shape in the initial stage, particularly those permitted by a lack of injection, caused an increase in bending-moments of 1.35 - 1.65 times.
Fig.4.32 The influence of the joint type (A,P), segment configuration (1,2), and of initial deformation (s.d., a. d.) upon the sectional forces M and N. In conclusion, the use of hinged joints is recommended, as also that the circular shape be preserved using an appropriate procedure (correct injection, interior arches to keep the shape, temporary screws). 3. Analysis of the influence due to the rotational rigidity of plane joints, for two configurations. In 3.3.2. the behaviour of the plane joint is analysed, the rotational rigidity Cd and the elastic clamping degree η being defined. In this manner, the value of the bending-moments in the joints can be calculated for different situations of joint opening, and also the influence that the rotational rigidity of the joints has upon the bending-moments. The analysis has been made on the five segments variant, with the two configurations A and B and centred plane joints, without primary openings.
For the type 5A lining, it may be observed (Fig. 4.33) that the variation in the degree of elastic clamping in the above joints, from 0 (hinge) to 1 (fully clamped), has no influence upon the value of the bending-moments in the current sections 1,3 or in the joint 2. This may be explained by the fact that the joints 2 are positioned in a region where the bending-moment is zero, so that the rotations of the two adjacent plane faces are sensibly nil, which corresponds to a rotational rigidity 0 and hence, zero bending-moment in the joints and no influence in the current sections. The 5B type of lining, with the closing segment at the key, has the adjacent joints subjected to large rotations, so having a strong influence on the bending-moment in the structure. Figure 4.34 a , presents the variation in bending-moment for two current sections 1 and 3 and two joints 2 and 4, as a function of the variation in the degree of elastic clamping of the joints adjacent to the key segment. It can be observed that the moments increase up to η > 0.3, in the
joints and also in the current sections, beyond which point the increase of η no longer has any influence on the bending-moment. However, analysing on the interaction diagram M - N (Fig.4.34.b), the domain which includes this example, it may be seen that, for joints, the possibility of η variation up to 0.1.is limited in practice, and it can thus be concluded that the rotational rigidity, taken into account in the calculation by the degree of elastic clamping, influences the bending-moment function and the location of the joints in the section. If the joints are located in zones of zero bending-moment, with big rotations, this will lead to bending-moments in the joints and to an increase in bendingmoments in the current sections. From this viewpoint, the 5B variant is less advantageous than the 5A variant, where plane joints are used. It can be observed that the maximum bending-moment in the key appears in the case of the hinged joint, and decreases as η increases. In the joint (2), the bending-moment given by the rotational rigidity is negative, so it has a favourable effect, discharging the section from the key. Consequently, it is recommended that a model with elastic clamping be used, and not a hinged model with, in order approximates reality more closely. Another analysis has been done to simulate a reduction in rigidity due to cracking and investigate the stresses in the lining, using also elastic clamping (Fig. 4.36). In addition, the influence of the soil parameters and the influence of lining rigidity upon the behaviour of the precast structure have been analysed.
Fig.4.34 The variation of M as function of η (a) and M - N diagrams (b) for the 5B lining type.
Fig.4.36 The M variation as a function of η, which simulates cracking at the key a) and the corresponding M - N diagrams b) for lining type 5A. These parametrical studies have demonstrated the influence that variation of the various factors has upon the magnitude of the sectional forces M,N or upon the displacements. On the basis of the results obtained, an order of priority can be proposed for consideration of the amount of influence exerted by each of the analysed factors. It is suggested that the order of these main factors be: the influence of the initial deformation (Ci=1.5), the influence of the type of joint (Ci=1.5), the influence of the section structure (Ci=1.3), and the influence of the ground characteristics (Ci=1.2). The coefficients for these influences so established, represent the possible deviation from the computed values, due to variations in the parameters. A combination of the several individual factors will lead to a total influence coefficient, which can be assimilated to a global safety coefficient. A value of 2.5 - 3.0 for this coefficient seems reasonable for underground structures, bearing in mind the complexity of this type of work. 5. THE STUDY OF THE STATE OF STRESSES AND STRAINS IN THE GROUND AROUND A SHIELD DRIVEN TUNNEL 1. Introduction The combined behaviour of the lining and rock mass in construction of a tunnel is very complex, due to the fact that the nature and characteristics of the ground are defined only approximately, and the method of construction and the way in which it is implemented can have a significant influence. The excavation of a tunnel produces changes in the state of stress in the ground around the tunnel, which determine the extent of any displacement of the periphery of the tunnel. This displacement is resisted by the structural support, which is loaded and deformed together with the ground, thus producing a passive reaction.
The associated tunnel design problem can be defined as follows: • the determination of the state of stresses and strains in the ground around the hole; • the determination of the state of stresses and strains into the sustaining structure; • the determination of the loads on the lining and of settlement at the surface. As we have seen above, conventional tunnel design methods used treat the above mentioned problems separately. The stresses and strains in the rock mass and in the lining, the loads on the lining and the settlement at the surface are computed by various methods, quite independently of each other. In reality, all the elements of the tunnel are interconnected and influence each other. A very close approximation to actual behaviour can be simulated using FEM. The role of the FEM is not to replace the conventional methods, but more to fill the existing gaps in our conception and to help in improving existing technology. Used properly, FEM can also clarify our understanding of the complex problems created in the ground when a tunnel is built. 5.2 The study of the ground displacement around a shield driven tunnel In underground structures, the notion of safety is different from that used for above-ground structures. So, for shallow-depth tunnels driven in soft ground and in urban areas, safety is linked more to tunnel stability than to considerations of allowable stresses. Such a tunnel is considered to be stable when the deformations have stopped increasing. Experience shows that displacement of the ground cannot be avoided no matter what method is adopted. and that these can propagate through the surrounding ground to reach at the surface. The extent and magnitude of such displacements depend upon the nature of the ground, the tunnel geometry, the technology and the method of application in the field. The designer of a tunnel must be able to predict, on the one hand, the maximum displacement likely to occur, and the origin of the loads on the lining, and on the other hand the permissible limits of the deformation which buildings in the neighbourhood of the tunnel are capable of sustaining.
The purpose of this chapter is to study ground displacement, using theoretical means, simplified models or in-situ measurements. 5.2.1 Description of ground displacements. In the literature, many authors have described the displacement produced around a tunnel in various conditions of ground and environment, and driven by various techniques. Peek (1969) is generally recognised as the pioneer of the analysis of ground movements around tunnel driven at shallow depth in soft ground. On the basis of an analysis of observation obtained from twentyfive completed tunnels, Peek made some recommendations for the design of tunnels and established empirical rules for estimating ground settlement at the surface. When the ground movement reaches the surface, a settlement trough is formed.
Most authors have represented the ground movement in two dimensions, longitudinal and transverse, and in three directions: vertical, horizontal and longitudinal, that is to say, the settlement trough may be measured in terms of the net settlement occurring at various locations, but this would ignore that the ground does not simply move down, it moves horizontally as well. The representation in the longitudinal sense permits visualisation of the progression of displacement as a function of the shield position which varies with time (Fig. 5.1).
Fig 5.1 The longitudinal representation of displacements function of the shield position The origin of the displacements can also be examined. The general profile of such a curve permits the definition of four zones of displacement: 1. ahead of the face; 2. at the shield; 3. in the zone where the lining is erected; 4. long term settlements. The representation of the movement in the transverse sense permits the visualisation of the zone of influence of the tunnel for the different stages of shield advance. Peek has proposed a representation of the ground surface settlements, which is very broad. (Fig.5.2)
Fig.5.2 The transversal representation of the movement 5.2.4 Monitoring ground movement by in situ measurements. The presentation of results of measurements for some shield-driven tunnels.
If in the past, the tunnels were constructed starting from observations previously accumulated in similar conditions, nowadays the use of in-situ measurements has become current practice for improved knowledge of the ground and for sizing the structural elements. Measurements can be divided into three groups, according to the point in the excavation cycle at which they are taken: • measurements before work starts, to determine the nature of the ground and the initial state of stress in the rock mass; • measurements during construction, to establish the new stresses and strains in the rock mass and in the lining and to verify the design hypotheses and calculation data; • measurements when the tunnel is in service, for monitoring safety.
We are concerned here mainly with the second group. The purposes for which these measurements are taken are: • the analysis of the behaviour of the ground traversed by the tunnel; • the monitoring of displacements and, sometimes, their prevention or limitation; • the checking of adequacy of the method of execution; • the checking of the design concepts and hypotheses adopted; • the adaptation of the working methods to suit the actual behaviour of the ground. Ground movements are generally monitoring by measuring the displacements, and the displacements usually measured are those that occur on the surface, for these are easy to be measure at reasonable cost and with good results. From the numerous examples present in the literature, some representatives are presented below. Hansmire and Cording (1975), in their work "Field measurements of ground displacements about a tunnel in soil", establish the principles of measuring displacements around a shielddriven tunnel. The research was made on a portion of the Washington Metro, comprising two circular tunnels of 6.4m diameter at a spacing of about 11m between centre-lines and some 12m below the surface. The ground traversed consists of sand, gravel and clay layers. The phreatic level originally at crown level was lowered during construction to the level of the invert. These Authors were the first to have traced the curves of isovalues for distortion and volume variation in the vicinity of the tunnel, and these curves have allowed the effect of compression of the vault to be visualised and understood. A large amount of data was obtained from reading each instrument, in correlation with shield advance. These data have been processed and plotted to show surface settlement, longitudinal vertical displacements, transversal horizontal displacements, and longitudinal horizontal displacements. For a better understanding of the behaviour of the ground around a tunnel when the state of stress-strain is modified, total deformations, component deformation and the volumetric deformation have been calculated from the measured horizontal and vertical displacements. The ground displacements have been measured along three axes: vertical, horizontal normal to the tunnel axis and horizontal parallel to the tunnel axis. The total displacement at a point has been computed as the result of the vertical and horizontal displacements in the plane normal to the tunnel axis. Those points at which the total displacement has the same value have been joined by curves, giving the paths of the isovalues for total displacements (Fig.5.25). Inspection of these curves has permitted the definition of some behaviour zones (Fig.5.29)
Fig.5.25 Isovalues of total displacements
The following zones are now hereby defined: • the expansion (extension) zone above the tunnel; • the compression zone laterally from the right legs; • the zone of maximum distortion. The expansion zone is located above the key, where an increase in volume of the soil occurs. The boundary of this zone can be established by tracing the isovalues of the volumetric specific deformation, as also can the boundary of the compressed zone be deduced from the sides of the tunnel. The zone of maximum distortion is defined by the tangents to the isovalue curves for maximum specific deformation.
Fig.5.29 The transversal representation of the different behaviour zones. Finally, the limit of the zone of decompressed soil can be approximately demarcated by considering that rupture surfaces are formed by shear along the zones with maximum tangential specific deformation. The shape so obtained perfectly suggests the well-known crushing vault. The work is, without doubt, the first documentation of the institution, treatment and interpretation of the measurements of displacements around a tunnel driven in soft ground. In her 1991thesis: "Creusement de galeries a faible profondeur a l'aide d'un tunnelier a pression de boue. Mesure in situ et etude theorique du champ de deplacements", Anne Pantet also obtains very interesting results from in-situ monitoring of ground movement. One chapter of this thesis is dedicated to the study of ground movements, carried out during the driving of some tunnels in soft ground with slurry shields. Measurements from three projects have been processed and interpreted: • Lille metro (D=9.0m) - precast segments; • Villejust railway tunnel (D=9.0m) - precast segments; • Lyon metro (D=5.8m) - extruded in-situ concrete. For each work, the surface settlements have been analysed, and also the vertical and horizontal displacements of the rock mass, in two directions, parallel to and normal to the tunnel axis. The manner of representing the development of settlement in longitudinal profile has been
improved by the dividing the total settlement value Smax into three terms, ascribed to different positions of the shield (Fig. 5.30): a. partial displacement in front of the tunnel; b. partial displacement beside the shield; c. partial displacement behind the shield. A comparison of the settlements in these three phases, as percentages of the total settlement, for the three tunnels, is given in the Table. A first conclusion is that, even if all the shields have the front closed and pressurised, an important percentage of the settlement (15 - 35%) occurs ahead of the face. The majority of settlement (55 - 67%), however, occurs after the shield has passed, the exception being the Lyon metro due to the use of extruded in-situ concrete. It is also apparent that settlement above the shield is the most frequent, which could seem unlikely due to the rigidity of the shield. The causes may be ascribed to the bead of the cutting edge, iconicity of the shield or steering. The analysis of displacements in the zone with deep cover (30 - 47m) at the Villejust tunnel is interesting. At the surface, the settlements are almost unobservable (max. 8mmm). The maximum settlement recorded above the key is between 60 and 125mm. Most of this displacement (80%) was developed behind the tail of the shield, thus demonstrating the partial effectiveness of injections. The plotted curves of isovalues for the vertical displacements (Fig. 5.34) displays: • a general symmetry of the vertical movements with respect to the vertical axis; • the focusing of a very big zone of displacements above the tunnel; • the zero isovalue line is below the horizontal axis. By processing the deformations in this way as shown in Fig.5.35, the following behaviour zones can be identified: • the expansion zone (tension)above the key; • the lateral zones of compression. By calculating the tangential deformations and the layout of their isovalues, the discharge vault can be drawn.
Fig.5.34 Isovalues of total vertical displacements.
6. EXPERIMENTAL RESEARCH INTO THE BEHAVIOUR OF PRECAST LININGS FOR SHIELD-DRIVEN TUNNELS 6.1 Introduction
If the notion of a tunnel comprises the combination of lining and ground, the lining is, of course, the main component. The lining ensures the stability of the hole, limits ground deformations, permits the safe use of the tunnel in service, and represents a principal component in the total cost of a tunnel (15 - 40%). All these are good reasons why the lining has been the most studied and researched element. Precast linings, because they are so widely used, have been in the first line of these researches. In order to analyse the behaviour of such linings, various methods and means have been developed. Three types of tests have been performed: • small-scale models laboratory tests; • full-size rig tests; • monitoring of the behaviour of extruded in-situ lining. Each test type has its own specific methods for determination the state of stresses and strains in the reinforced concrete structure such as photo elasticity (tests on model) and tensometry (tests on stand and in situ). For the tests on the stand or in-situ, levelling and measurements are done with extensometers, inclinometers, deflectometers, etc. The results of the tests must be processed and interpreted carefully by competent specialists. The tests on models demonstrate the usefulness of the method, for verifying the hypotheses upon which the calculations are based, and for comparing the actual stresses with those obtained by calculation, especially for problems for which satisfactory solutions cannot be found. Because of the relative simplicity of installation, relatively low cost, the relative ease of creating models and the ease with which changes to the model can be made or the parameters varied, these types of tests have been used successfully, and can form a first step of study. Many studies have been performed on models of circular shape, to determine the external pressure and the state of internal stresses and strains. However, in the majority of such tests, continuous rings have been used, a hypothesis which is not greatly relevant to this study, which is concerned with precast lining. Consequently, it was appropriate to select for this study a case which takes account of the existence of joints between segments. 6.3 Full size stand test on the linings. This type of test is closer to conditions on site, but they are rarely used due to the high cost. This work makes use of the tests carried out by Orlov (1961) on a fixed ring stand to study the static action of the linings of tunnels and wells of circular profile. The purpose of the stand test on the linings is to check the influence upon the static action of the following factors: the tension links in transverse and longitudinal joints, squatting of the ring, the elasticity of the supporting medium, variability in the dimensions of the segments, lack of erection precision, etc.
From these tests, it was possible to determine specific characteristics for each type of lining with regard to load-bearing capacity, and resistance both to crack formation and to deformation, the weak points of the different types of lining being ascertained by assessing the mode of failure.
6.3.1 Stand test on railway tunnel segments, (De=8.85m).
The stand test of the annular structure of a simple railway tunnel, driven by circular shield, had been a very costly and complicated operation. In order to check the load-bearing capacity of the individual elements of the annular structure of such a tunnel, a special steel stand was built, and loading schemes were developed to simulate the actions as realistically as possible. The structure of the railway tunnel cross section is presented in Fig, 6.8. The external lining comprises four current segments and a closing segment. The segments are of three types, depending upon their location in the ring. These types are: a key segment type K, two side segments type L and two invert segments type M. The transverse joints are partial hinges; the longitudinal joints are tongued-and-grooved. From observations made in the field, and also from the calculations for different loading combinations, the sectional stresses (M and N) had the same sign on the same segment, as follows: • the key segment type K is in tension at the intrados; • the side segments type L is in tension at the extrados; • the invert segments type M are in tension at the intrados. Fig. 6.8 Cross section of railway tunnel. 1 ground; 2 - injection; 3 - segment type K; 4 segment type L; 5 - segment type M; 6 closing segment type D; 7 - invert segment type F; 8 - banquette segment; 9- sealing;10 inner lining; 11 - Cables channel; 12 drainage channel; 14 - clearance. Having thus established the size and direction of the principle forces, a programme of testing could be tailored to appropriate for each of the segment type (K,L,M). However in view of the difficulty of creating uniformly distributed loadings similar to those deduced from the calculations or found to be exerted by the ground in practise, a loading scheme was adopted whereby point loads were applied at the key such similar sectional stresses were induced (Table A). These point loads were imposed by means of two 100t hydraulic jacks. In order to reproduce a support system similar to that obtaining in practice, reinforced concrete support blocks were used, fixed in place by steel cradling. During the testing process, the specific strains were measured along the intrados, extrados and at the centre of the cross-section. In addition, the horizontal and vertical displacements were measured, and also the width, depth and extent of any cracks which developed. The results of the tests, from which those for the K-type and L-type segments are presented in Fig.6.9, 6.10 and 6.11, form a database which may prove to be very useful in predicting the loading level during the execution, by measuring the cracks and opening up of any joints, and also give guidance in the estimation of reserve load-bearing capacity.
Table A
Fig.6.9 Test of segment type K.
Test of segment type L.
Fig.6.10 Diagrams of strains ε mm/m, for test segment type K.
Fig. 6.11 Curves showing strains ε mm/m, for test segment type K.
Foto 1,2 View of the steel stand
Foto 3 Hydraulic jacks
Foto 4 View of final cracks
6.4 Research into the behaviour of the annular structure of some tunnels during construction. Model tests and rig tests, although otherwise very useful, cannot completely imitate the complexity of the behaviour of segments in the field, and it is therefore to take some in situ measurements. Research into the stresses, strains and deformations surrounding shield driven tunnels presents many difficulties related to specific driving conditions. When the measuring equipment is installed actually in a tunnel under construction, factors may become apparent which are not previously foreseeable.
In order to establish what forces and bending-moments are acting in the lining, it is necessary to measure both deformations across the tunnel axis, and stresses acting both at the intrados and at the extrados. Obviously, a tunnel does not lend itself to the convenient taking of measurements at the extrados, and unconventional techniques have to be developed. Two types of measurements are used in the case of the tunnels made up from precast elements: measurements of the stresses and strains in the individual segments; and measurements of the deformed state of the lining as a whole. The measurement of strain within the segments by means of measuring devices buried within the concrete can provide more reliable data concerning the behaviour of the lining. There are two principle methods of measuring strain within the concrete: electronic methods (vibrating chord transducers) and mechanical methods (mechanical devices with large measuring basis). The stresses in the lining can thus be determined on the basis of measured deformations and strains. Interpretation of these measurements provides validation of the various models used to decide the dimensions of the lining, the basis for the structural design, and the values assumed for the different design parameters. This puts at the designer's disposal a useful database for adapting theoretical designs to suit field conditions. 6.4.1
Convergence measurements at the Beia 2 tunnel.
The Beia 2 tunnel is a simple railway tunnel driven by circular shield (D=9,0m) through horizontally stratified marl. The primary lining comprised five standard segments and one key. In this tunnel, measurements were taken in two stretches of the tunnel: Rings 518 - 532 and Rings 572 - 585. The measurements taken were: the chord lengths and the offset from the centre of the chord to the face of the standard segments, the opening of the joints and the widths of any cracks. The results of the measurements for two rings in this area are presented in Fig.6.12 and table 6.2. These measurements were interpreted by reference to the data obtained from the individual rig tests on the segments, and hence the load-bearing capacity of the annular structure could be estimated
Fig.6.12 Table 6.2
6.4.3 Investigation of the segmental lining with hinge joints of the Birnova railway tunnel.
The Birnova railway tunnel. was driven through marl, and lined with an annular structure of reinforced concrete segments, similar to that presented in Fig.6.8, with hinged joints as shown in Fig.6.18. In order to measure the deformations, mechanical devices with a large base measurement have been made (Fig. 6.19), which collect the deformations along a particular length of tunnel. The estimation of these deformations was made using both mechanical device and tenso-resistive apparata. Before being installed in the tunnel, the devices were calibrated, and an initial measurement was taken for the loaded state.
Fig. 6.18 Hinge joint.
Fig. 6.19 Mechanical device The experimental ring (199) is located towards the tunnel exit, under a soil cover of 9m. After erection of the ring in the tail of shield, a new measurement of the devices were taken, and the initial measurements were plotted. The next measurements were taken 24 hours after erection, by which time the ring had left the tail, and the last measurement was taken at 72 days, just before the in-situ secondary lining was cast. The data deduced from measurements has been presented in two ways: • diagrams of transversal strains at the measuring points (Fig. 6.21). • curves showing the variation of strains at 5cm from the intrados, at the mean axis, and at 5cm from the extrados (Fig.6.22);
Fig. 6.20 Mechanical devices location.
Fig. 6.21 Diagrams of transversal strains.
Foto 5 Mechanical device build in the reinforcement cage
Foto 6 Segments for experiment
Foto 7 First riding
Fig. 6.22 Strains in the segmental lining 72 days after installation. Only the final results at 72 days are shown, because those at 24 hours were too small to be significant. The sectional stresses M and N have been determined on the basis of the measured strains, and these have been compared with the calculated values in Fig. 6.23.
Fig. 6.23 Diagrams of the stresses M and N. a – analytical; b – measured. After analysis of the results of these measurements, some conclusions can be drawn: • the measurement, for the first time in a railway tunnel, of the strains from inside the tunnel by means of a mechanical device, has given very useful results, precise enough to improve the understanding of the behaviour of this type of lining;
• the symmetrical configuration of strains in all fibres has led to good shape of the ring and symmetrical loading; • the form and sign of strains and stresses on each segment is in concordance with results obtained by calculation; • the change of sign of the deformations at the external and internal fibre at the joint between the segments proves that the joint functions correctly as a hinge; • the bending moments established on the basis of the strain measurements have the same profile as those established by calculation, and their values differ by less than 20%. The axial forces are about 15% lower. The measurements taken at the Birnova tunnel confirm the predicted behaviour of a precast segmental lining with hinged joints. 6.4.4 Investigation of the segmental lining with plane joints of the Gibei railway tunnel.
The Gibei tunnel is the longest (L=2240m) shield-driven railway tunnel in Romania with a precast primary lining. The tunnel was driven through very mixed ground, made up mainly from clays and sands. The structure adopted for the primary lining of the experimental ring for the Gibei tunnel is the same structure as that at the Birnova tunnel described above, the only difference being the joint between segments, which is a plane type with tongue-and-groove (Fig.6.24). At the location chosen for the installation of the experimental ring, the primary lining was made up of a single type K key segment, the remainder of the segments being of type M.
Fig 6.24 Tongue-and-groove plane joint The ground cover over the tunnel is 70m, and comprises alternating layers of sandy clays with sands above. The measuring device used was similar to that used at Birnova, but with a smaller measuring base (500mm). Another difference with respect to Birnova is that the secondary insitu lining was also instrumented and monitored. The experimental research was carried out on Ring 420. The instrumentation of the precast segments was installed at the Precast Enterprise Brasov casting yard. The measurements were begun on 12 April 1983, three more readings being taken between then and 4 May 1983, when the secondary ring was cast. The initial measurements of the secondary lining were taken in June 1983, and taking of further measurements continued until October 1983. Thirteen measuring devices were installed in the primary lining; four measuring devices were installed in the secondary lining. The data gathered from these measurements are presented in the same manner as those from Birnova, with variation curves and diagrams of the transverse strains, to which are added diagrams showing the changes in deformation with time in the representative sections.
The following curves are shown: • curves showing the variation of strains at three points in the cross-section (intrados, extrados and mean axis) in the three steps of measuring for the primary lining (Fig.6.26, 6.27) and at the intrados for the secondary lining (Fig.6.29); • curves showing the variation with time of strains in the key segment and in the middle of the L segment (Fig.6.30); The sectional stresses M and N have been determined on the basis of the strains from the most frequently measured sections, for the three measuring steps. The analysis of the deformations measured at different intervals of time after installation, together with in-situ observations, has led to some conclusions which agree generally with the design hypotheses.
Fig.6.26 Strains in the primary lining
Fig. 6.27 Diagrams of transversal strains ε mm/m in external segmental lining
The experimental research performed at the Gibei tunnel on the behaviour of an annular structure composed of two linings, one precast with plane joints and the other monolithic, have proved useful and interesting, both for design and construction. A comparison with the research at the Birnova tunnel leads to the same conclusions as deduced theoretically. Thus, for the same segment configuration, the use of plane joints between segments, compared with hinged joints, leads to increased stresses concentrations at the edges of the segments, and the introduction of supplementary bending moments.
Fig.6.28 Strains at the intrados of the secondary lining 130 days after placing. Another important element affecting the final state of stress, which has a greater influence upon linings with plane joints than on hinged-jointed linings is the extent to which the circular shape of the lining is maintained during erection. The stress variation in the same section, from one loading stage to another, due to various factors (support during installation, unsymmetrical loading, etc.), suggests the adoption of envelope diagrams for calculation.
Fig.6.29 Transversal strains in the secondary lining. The study of the development over time of the stresses and strains in such a lining (Fig. 6.30) shows that, generally, significant increases in loading occur in the first few days after installation, these increases reducing with time. However, situations may arise where significant increases in loading develop with time (for example, in the Gibei tunnel), due to some phenomenon such as clay swelling, and such situations may justify the adoption of a secondary lining.
Fig. 6.30 Variation of transversal strains over time. 6.4.5 Investigation of the segmental lining with plane joints and bolts of the Bucharest metro. The Bucharest metro structure is composed from a single segmental lining with 5 curent segments and a key (Fig. 6. 32). The joints between segments are plane joints with curve bolts (Fig. 6.31). The experimental ring (733) is located on the first meto line, in Republica area. Instrumentation of the ring was made with mechanical devices built in the segments in concreting phase. The results of measurements are presented in Fig. 6.33-6.35.
Fig.6.31 Plane joint with curve bolts
Fig. 6.32 Bucharest Metro Segmental lining. Mechanical devices location. One of conclusions is that the bolted joints take bending moments just from first stage and enlarge the bending moment on segments.
Foto 8 Mechanical device
Foto 9, 10 Mechanical device build in the reinforcement cage
Foto 11 Segments for experiment
Fig. 6.33 Diagrams of transversal strains ε mm/m
Fig. 6.34 Strains in the segmental lining
Fig. 6.35 Diagrams of the stresses M and N. 6.5 Conclusions Analysis of the experimental research carried out on models, on test rigs, and on site, upon different annular structures made up of precast elements, allows some general conclusions to be drawn. 1. The existence of the joints between the segments has a decisive influence on the distribution of stresses and strains in the annular structure, generally producing a change in sign of the stress from one segment to another. 2. The type of joint (plane or hinged) influences both the distribution and the order of magnitude of the stresses, this fact being particularly noticeable in the zones adjacent to the joint. 3. The presence of bolts influences the behaviour of plane joints to a greater or lesser extent, as a function of the sense of the joint opening, and generally increases the bending moment in the lining. 4. The distribution of stresses and strains in the cross-section is generally non-linear, due to the curved shape of the segments and the loading (eccentric compression). 5. The number of joints and their location in the ring determine the sense and magnitude of the sectional stresses and of any possible damage. 6. In cases where deformation can take place (for example, where contact grouting is not carried out) or where loading is very high, the deformation of the precast elements may greatly
exceed those calculated, plastic zones appear, redistributions of stresses occur but the structure may still remain in equilibrium. 7. The bending moments determined experimentally agree well with those obtained by calculation, both as regards shape of the bending-moment diagram and as regards values, for the case where hinged joints are used. However, if the joint type departs from the hinge joint (plane joint with or without bolts), the bending moments obtained from measurements vary from those calculated, especially as regards shape. 8. The axial forces determined experimentally present lower values than those obtained by calculation, but nevertheless approximate to the Polygonal Beam Method. 9. The loading on the lining develops after the ring leaves the tail of the shield, over a short time period, until equilibrium is attained, after which supplementary loads rarely appear.
7. THE THEORETICAL STUDY OF THE STRESSES AND STRAINS STATE IN THE ROCK MASS-LINING ASSEMBLAGE. 7.1 Introduction Experimental investigations, especially those carried out in-situ, are important sources of information for designers of tunnels. But such sources are not generally available to the designer, being expensive and difficult to arrange. Also, truly comprehensive investigations, which follow both the ground deformations and those of the lining, and the pressures on the lining, are very rare indeed. In these conditions, the only tool available to the designer that can take account of the complex phenomena of the combination of lining and rock mass is theoretical analysis based on finite elements.
The most important of the methods of analysis of the behaviour of tunnels, have been treated in chapter 4. These methods still remain the basis of conceptual design and remain fundamental to such practice. However, there is a requirement for a better performing analysis and for a design tool based on sound engineering principles, capable of evaluating even the most complex situations without contradictory results. The possibilities offered by the FEM for such analysis are both tempting and encouraging, and have been found by many researchers to be an effective tool in the study of underground structures. 7.3 Modelling of underground construction. 7.3.1 Principles. Objectives. Limits Any system for modelling the behaviour of the rock mass/lining combination using FEM must represent as faithfully as possible, the following aspects: • the behaviour of the rock mass, evidenced by its capacity to resist the new state of stress, which depends upon the initial state of stress, the resistance parameters of the rock and the shape and size of the excavated space; • the interaction and contact phenomena between the rock mass and the lining, which depend upon: the bending-moment of the lining after installation, their flexibility, the construction phases and the nature and efficiency of injection behind the lining; • the tunnelling method and the construction sequence; • the effect of some specific loadings, such as: self-weight, filling injections, consolidation and water-tightness injection, and water pressure.
FEM permits a better representation of reality, and increases the variety of hypotheses which may be taken into account in calculation: • consideration of gravitational forces;
• • • •
non-linear , anisotropic and time-dependent behaviour; better definition of boundary conditions; the actual geometry of the rock mass and of the lining; consideration of the void behind the lining with evaluation of the shape and size of the hole and the closing mechanism, or the mechanism of filling by injection.
When the structural model is being devised, consideration must be given to: • whether the problem as a whole is to be defined, a function of the complexity and importance of the objective of the study; • whether the elements of the lining are to be modelled in one, in two or in three dimensions, a function of how the problem is to be approached; • selection of a model of the behaviour of the ground, a function of its nature (soil or rock) and the modelling of the discontinuities given by the stratification, slices etc; • how to model the interaction between lining and rock mass; • how to model the phenomenon of decompression and relaxation of the ground, together with simulation of the construction phases. The practical application of the FEM to the analysis of underground structures leads to important differences in dealing with the terrain type (soft ground or rock), which are presented in the table below. Medium continuity Ground behaviour models Relative rigidity lining-ground Effect of deformations
Soft ground Relative approximations Generally elastic-plastic Strong Important
Rock Discontinuities predominate Generally elastic Weak Often negligible
The main objectives of using FEM are: • the analysis of the behaviour of the ground and structures, especially for cases with complex geometry or complicated execution methods. It also represents a useful tool to assist in understanding the phenomena which produce disturbances. • the evaluation of the feasibility of a project and an appreciation of the necessity to reinforce some rock mass zones, and the most suitable method for such reinforcement; • the estimation of anticipated deformations, in order to appreciate their influence upon existing structures and to provide comparison elements for measurements in the field; • the general development of studies which permit the proposal of simplified procedures for calculation and dimensioning. Although these advantages are unarguable, FEM is nevertheless also subject to limits and difficulties in use to which the designers must pay attention. There are two alternative ways in which FEM can be used: • the use of simple models with few and easily measured parameters, which can lead in some cases to results far removed from reality; • the use of complicated models (with rigorous modelling of the initial state of stresses, rheological behaviour, etc.), with more accurate results but using a large number of parameters whose determination, at the scale of the whole rock mass, is simply not practicable. The tendency in practice, which permits the partial removal of the preceding doubt, is to make the analysis in several steps:
• in the first step, simple models are used and comparative parametrical studies are made, which permit identification of the predominant characteristics of the behaviour: • in the second step, the models and the characteristics are calibrated by means of experimental results obtained by in-situ tests; • finally, the use of more sophisticated models. Besides the general difficulties described above, local difficulties can also arise in: • establishing correct and representative discretisation and defining the initial conditions; • determining field parameters (the laws of behaviour); • modelling of the excavation phases. In order to understand and to appreciate the role of these difficulties, results and observations obtained from various investigations into the use of FEM in the field of tunnelling have been analysed. 7.3.2 Discretisation the structure-rock mass assemblage. The discretisation represents the first step of calculation with FEM, and can decisively influence the validity of the results. The discretisation must be made in such a way as to respect the following conditions: • to respect the shape of the underground structure; • to permit the evaluation of the initial effects on the rock mass; • to reproduce the geological conditions, identifying the limits of the different rock categories and the position and orientation of bedding planes and joints; • to permit the simulation of the execution sequences (excavation, erection of lining, injection) according to the proposed program; • to permit the modelling of the discontinuities of the lining; • to ensure a gradation of the element dimensions according to the stress gradients.
The discretisation process is very important for the analysis of ground displacements and stresses in the lining. Many studies have been carried out for attaining the first aim, but for the second aim there is a paucity of bibliographic references, even though the principle concern regarding underground structures is to ensure the safe bearing capacity of the lining. For tunnels in rocks, the rock mass is the main support element, and the lining is secondary; in soft ground, however, the roles are reversed, for the lining after erection becomes the main support element for the redistribution of forces. In conclusion, in this field, the analysis of stresses in the lining is an even more important objective than determining surface settlements. The realistic determination of the stresses in the lining is crucially dependent on the discretisation of the surrounding rock mass and on the boundary conditions imposed. The use of a continuous model (Fig.7.1.a) leads to unsatisfactory results especially for shallow depth tunnels. Ahrens and others (1982) record the use of another type of partially continuous model for analysis by FEM of a tunnel in soft ground (Fig.7.1.b and 7.2). The results show that the partially continuous model is closer to practical reality. The other important factor for obtaining realistic results is the correct discretising of the lining. The lining can be discretised in one- or two-dimensional elements, a function of the degree of detailing required. Unfortunately, in the examples studied, the most frequently used discretisation, the continuous two-dimensional one, does not take into account, or does not adequately model, the discontinuities (the joints between segments) which are, in fact, essential elements in the correct definition of the state of stress in the lining.
7.3.3. Modelling the execution phases. The modelling of the phases of the construction of a tunnel is a difficult problem, due to the three-dimensional phenomena which are developed, and due to the succession of the phases which causes interference between them. The basis of modelling the phases is again whether the analysis is two- or three-dimensional. Various studies and measurements made during tunnelling have proved that there is a three-dimensional effect present in the zone around the excavation face, where a three-dimensional analysis is recommended. At two to three diameters behind the face, the state of stress becomes stabilised longitudinally and a two-dimensional analysis is appropriate. The two-dimensional analysis can be extended also along the face zone with sufficient accuracy in the following situations: • when the ground is good, and the predominant ground displacements occur after the shield has passed, in order simply to close the tail void; • when the ground is self-supporting even after the shield has passed, and the contact between the ground and the lining is made by void injection or by expanding the lining.
Three-dimensional modelling of a shield-driven tunnel suffers also from many problems associated with pressure on the face, with the effect of the shield shoving forces and the shield weight, and with the closing of the void or its filling by injection. The model must allow for: the application of the stresses in the face, the distribution of the stresses around the periphery of the hole, the sequential application of the weight of the shield, of the shoving forces and the addition of the weight of the lining. The two-dimensional model is very attractive and widely used, as it can be seen in the literature. Three types of two-dimensional models are used (Fig. 7.10): • the transversal model; • the longitudinal model; • the axi-symmetrical model. The most usual is the transverse model, based on the hypothesis of deformations in plan. The use of the two-dimensional model, with its obvious advantages, is conditioned by the necessity to take account of the face phenomena, an operation which can be achieved by various calculation artifices. These artifices follow the two principal aspects of creating a tunnel: the excavation phase and erection of the lining, which represent the two extreme situations, a lined tunnel and an unlined tunnel. Other important artifices refer to the simulation of face advance and the closure of the tail void. 7.3.4. Constitutive models, laws of behaviour. Assumptions made regarding how the ground will behave must be in accordance with the forces which appear in the rock mass after the underground construction is complete. From this viewpoint the tunnel is extremely complex, the paths of the stresses around the excavated void being various and complicated. The determination of field parameters is made by carrying out classical tests, which produce states of stress. These, however, may not be in accordance with what is met in the field. The choice as to the manner of behaviour is a very significant problem, but it is also a very difficult one, which depends on the experience and the intuition of the designer. Nowadays, there are many models which simulate the field behaviour; the user can choose from the simplest model, with linear elastic behaviour, to the most complicated, incorporating rheological behaviour.
The problem is to choose the best model to be used in the numerical analysis of a tunnel, knowing the difference existing between the complexity level of the model and the accuracy of the parameters being used. There are, in fact, some conclusions regarding the efficiency of utilisation of the various models. The first conclusion is that any type of behaviour model can be used successfully for the determination of the settlements at the surface, if the field parameters are correctly chosen. Another conclusion is that the utilisation of a three-dimensional analysis leads to better results that a two-dimensional analysis model of behaviour. Finally, a very important element in the success of the analysis is the ability to simulate of the execution phases, which can impose a two-dimensional analysis with respect to a three-dimensional one, or a simpler model instead a more complex one. 7.4. Simulation of the execution of a shield-driven tunnel. Numerical examples. 7.4.1 Introduction. From an analysis of the experimental and theoretical studies made on some shield driven tunnels, some conclusions have been formulated which serve to improve the modelling and the development of a concept to tackle analysis of the construction process for a shield-driven tunnel using FEM. These conclusions have been used as guidelines in the numerical examples which follow, for the simulation of the execution of a shield driven tunnel. The example chosen for study represents a shield-driven railway tunnel, corresponding to a simplified representation of real tunnels, such as those at Gibei and Birnova.
Thus, two variants have been studied, corresponding to the two situations: a a deep tunnel, the limit of the influence zone being considered as about three diameters concentrically out of from the excavated hole. The analysis was done using the CAV program; b a medium depth tunnel, considering a zone extending the full height up to the surface and to 50m on either side of the tunnel. This analysis was done using the CESAR program. 7.4.2 Study of the behaviour of a deep tunnel, using the CAV program.
The CAV program uses a two-dimensional model, and assumes plane deformation. The elements used are: iso-parametrical, rectangular, and one-dimensional and of contact. The ground is assumed to behave both elastically and elastic-plastically. Simulation of the excavation phases is made by means of a decompression algorithm, which starts from the forces equivalent to the initial stresses, and reproduces any change in the state of the rock mass, the state of stress determined in an excavation phase becoming the initial state of stress for the following phase. The three-dimensional effect in the face zone and the effect of the face advance are simulated by using a relaxation coefficient. The excavation and installation or concreting of the lining is simulated by activating and de-activating elements from specific zones. A much finer discretisation is used around the joints between segments to model their effect. Two numerical analysis were carried out, one assuming elastic behaviour and the other elastic-plastic. The hypotheses suitable for a deep tunnel were used, to allow comparison with classical methods of calculation based on theory of elasticity . The results show good correlation between these methods, regarding the distribution of horizontal and vertical stresses in the case of an unlined hole (Fig.7.25). The calculation was repeated, considering the effect of face advance and the influence of the installation of the precast lining. The results obtained from this first numerical study of a deep tunnel, compared with the results obtained by in-situ research upon works with similar characteristics, presented in previous chapters, has permitted the statement of some useful observations for the improvement of the modelling of such work.
One achievement of this first study is the successful modelling of the hinged joint between segments.
Fig.7.33 Diagrams of σθ stresses.
Fig. 6.34 M and N diagrams 7.4.3 Study of the behaviour of a medium-depth tunnel. The use of CESAR program. This second study was performed on the same type of tunnel, corresponding to the Gibei tunnel described above, but considered now as of medium depth and analysed up to the surface.
The simulation was done using the CESAR computer program. This program is based on FEM and is particularly well-adapting to solving problems of tunnels and underground structures. The possibilities of this program are multiple: two- and three-dimensional analysis, various models of elasto-plastic behaviour, various contact element types, a facility to simulate face advance in a two-dimensional analysis, graphical pre- or post-processing, etc. In view of the possibilities offered by this program, several studies may be considered for monitoring the influence upon the stresses and deformations in the rock mass and in the lining, of different factors, such as: • the construction phases and the effect of the face advance; • elastic or elasto-plastic ground behaviour; • the existence of joints between segments, their type, number and disposition; • the closing of the tail void between the lining and the excavated contour; • long term modification of the mechanical characteristics of the ground.
Ground
Es µ γ Geotechnica l model c φ Concrete Es µ γ Model
Fig. 7.35 Discretisation of soil
A.
50 MPa 0.35 0.02 MN/mc Mohr Coulomb 0.1 Mpa 20° 33000 MPa 0.2 0.025 MN/mc Linear elastic
Fig. 7.36 Discretisation of hinge joint
B. Fig. 7.36 Discretisation of the lining
In the study which was actually carried out, three variants of the lining were analysed: A a precast lining of five segments and a key, as used in the Channel tunnel; B similar to A, but with the key in the invert, as used in railway tunnels in Romania; C a continuous lining, applicable to the shield method only in the case of an extruded concrete lining, but often used in models presented in the literature. In order to improve the calculated results, with respect to the experimental ones, a new system of modelling was used, incorporating relaxation coefficients. Three relaxation hypotheses have been tried: a uniform relaxation, λ=constant for variant "C'; b non-uniform relaxation, λ=0 in the invert and λ=0.3 for the rest of the section, in the first phase, and λ=0.2-0.7 on the entire perimeter in the second phase, for variants "A" and "B"; c non-uniform relaxation, λ=0.6 in the arch, a load of 0.2MPa at the invert and λ=0.3 elsewhere, in the first phase, and λ=0.7 uniformly in the second phase, for variants "B" and "C". d non-uniform relaxation λ=0 in the invert and λ=0.3 elsewhere in the first phase, λ=0.5 in the arch in the second phase and λ=0.2 on the entire perimeter in the third phase, for variant "A". The reasoning behind this modelisation starts from the following observations: 1. The lack of correlation between displacements revealed by experiment to be large at the key and close to zero at the invert, and those obtained by calculation with λ constant on the contour, which are equal at key (downwards) and at invert (upwards). 2. Between the face and the tunnel lining is the shield, of rigid construction, with a considerable weight (300-800t), comparable to that of the ground it displaces. The invert supports this load over a reduced area, thus impeding or reducing the decompression of the ground. There is therefore a difference between the decompression which occurs at the invert and at the crown of the tunnel. 3. The excavated contour is of necessity greater than the extrados of the lining, to allow for the tail skin, and for steering tolerance. The gap between them varies in width, from a maximum at the key to near zero at the invert, where the lining tends to drop down to rest on the invert after being ejected from the tail. It is not possible in practice entirely to prevent this by injection, and there is consequently a bigger displacement at the key than there is at the invert. Variant "A". The configuration of the segments in section and the discretizing of this type of lining, are shown in Fig. 7.36. Two cases have been considered: • elastic-linear ground behaviour (variant A1); • elasto-plastic ground behaviour with Mohr- Coulomb rupture criterion (variant A2). To allow for the effect of the advancing face, the calculation has been done in two steps: • step I - corresponding to the un-supported hole before the lining is installed; • step II - corresponding to the installed lining well behind the face. Variant A1. The λ coefficient distribution on the section is non-uniform, being zero at the invert and 0.3 for the rest of section (the b hypothesis of relaxation). The isovalue curves of total displacement shown in Fig.7.39 give good correlation as regards shape and values with those obtained experimentaly from the Villejust tunnel in the area where H=50m.
The same figure also shows the curves for displacements of the vertical axis of the tunnel, and along the horizontal at the ground surface, and the displacement of the excavated contour towards the interior of the hole. It can be seen that the displacements at the ground surface are small (max. 15mm vertically), and the width of the settlement trough exceeds the limits of the discretized domain, which is well known and has no influence on the rest of the results. From an analysis of the total displacements, it was found that the zone with significant displacements is that above the key, extending to a height of one diameter, and this agrees with observations from the experimental research.
Fig. 7.39 Isovalue curves of total displacements - Variant A1. The state of stress in the ground is shown by the curves of the isovalues of the stresses σx and σy, for the second step (Fig. 7.40). On the same figure, are also shown the variation curves of the vertical and horizontal stress σy and σx, in the initial and final stage. It can be seen that the zone of disturbance and concentration of stress is relatively reduced, about three diameters around the hole. Another very important problem was the study of the state of stresses in the lining. On the basis of the stresses calculated at various sections, the program automatically determines the sectional stresses M and N and draws the diagrams (Fig. 7.41). These diagrams have been compared with those calculated by the polygonal chain method. From the bendingmoment diagrams, it can be seen that the shape and the sign of the diagram on each segment differs, and there are also important differences regarding the values. The diagrams of axial forces also differ, both in profile and especially in values. Nmax. determined with FEM is about three times greater. Because the axial force found by experiment at the Birnova and Gibei tunnels is also smaller than that computed by the polygonal chain
method, it is necessary to improve the modelling used with FEM, in order to represent reality more closely. The divergency of the bending-moment diagrams of the two methods is due to the relaxation hypotheses adopted in this variant for FEM.
Fig. 7.41 M and N diagrams, variant A1. To improve the results and achieve better correlation between the two methods, the dhypothesis of deconfining has been used in variant A2, which corresponds better to the manner of progressive loading of the lining (Fig.7.42).
Fig.7.42 M and N diagrams, variant A1.1 Ground pressure will act continuously on the deformed structure, and this pressure can be considered uniform on the excavated profile (phase 3).
This loading hypothesis corresponds better to reality and leads to a bending-moment diagram closer to that for the polygonal chain method (PCM), and to smaller axial forces, coming closer to those observed in experiment Variant A2 studies the same type of lining and goes through the same steps as A1, the only difference being the assumption of elasto-plastic ground behaviour, based on the Mohr-Coulomb rupture criterion. The same parameters were used as in variant A1, and thus, the isovalue curves of total displacements have shapes and values very close to those of variant A1. The use of Mohr -Coulomb criterion for the ground, combined with the decompression artifice based on the coefficient λ, presents some difficulties, due to the tendency of exaggerated extension of the plasticization in the first phase, due to the lack of support and the action of the forces acting on the excavated perimeter. In reality, in this phase the deformation phenomenon, and the subsequent plasticisation, are limited by the presence of the shield, and depend upon rupture in the face and upon the magnitude of the void around the shield. The insignificant increase of displacements with respect to variant A1 indicates that the assumption of elasto-plastic behaviour has little effect. The sectional stresses M and N in the lining are almost unchanged with respect to variant A1, thus proving the insignificant role of plasticisation upon the stresses in the lining. Variant B. This variant corresponds to the type of structure much used in Romanian railway tunnels. This type of lining has a wide degree of utilisation, and has been the subject of a multitude of theoretical analyses. In addition, it has been the subject of in-situ tests at the Birnova and Gibei tunnels. As a result, several studies have been done, with various hypotheses, for the purpose of calibrating numerical models with experimental results. The study B1.1 differs from variant A1 only by the inverse configuration of the segments in section, so the stresses and deformations in the ground differ but little, due to the small discretization changes, and hence these are not shown here. The sectional stresses M and N have been determined and are shown (Fig.7.45) in order to make a comparison with variant A1 and with the results obtained with the PCM or by in-situ tests. The bending-moments agree well with those obtained with PCM and even to the experimental ones, but the axial forces are about three times greater. The study B1.2 is an attempt to improve on the results obtained in variants A1 and B1.1, both regarding the magnitude of displacements and of bending-moments, by trying a new way of relaxation (c). In the first step, a progressive variation of the λ coefficient on the section contour was considered -- 0.6 on the key segment and 0.3 in the rest -- and in addition the loading (0.2MPa) on the invert imposed by the weight of the shield was taken into consideration. In the second step, the effect of the tunnel roof closing in on the lining was simulated with a loading (0.2MPa) on the key segment. The isovalue curves of the total displacements for the two steps (Fig.7.46) show an encouraging approximation to the shape and values obtained experimentally at the Villejust tunnel. The ground displacement resulting between the initially excavated profile and the displaced one satisfactorily simulates the closing of the gap between ground and lining which would occur in the case of imperfections in the injection process.
Fig. 7.45 M and N diagrams for variant B1.1.
Fig. 7. 48 M and N diagrams for variant B1.2. The displacement at the ground surface is about 4cm vertically, and extends transversely beyond the limits of the chosen domain. The zone with significant displacements is also that above the key, extending more than one diameter vertically. The sectional stresses M and N in the lining (Fig.7.48) show significant increases with respect to the variant B1.1. The bending-moment in the key segment increases by about 50%, and Nmax increases by about 15%.
The main conclusion of these first two studies is that the axial forces are much more than those obtained by in-situ tests. This is due to the large percentage of loading (λ=0.7) imposed on the lining in phase 2. This loading results in a reasonable bending-moment, but an unacceptably high axial force. The modelling is thus in need of improvement, and this has been achieved by two related procedures: • a decrease of decompression (λ=0.5-0.2) from the phase 2, corresponding to the lining installation, in order to reduce the axial force; • the delimitation of a zone of highly decompressed ground above the key, in accordance with evidence from all the theoretical and experimental studies, which indicate a reduction of the elasticity modulus by a factor of between five and ten, in phase 2. Fig.7.49, shows several studies or modelling hypotheses of phase 2, based on the procedures proposed hereabove.
Fig. 7.49 Loading hypotheses for variant B, These studies can be split into three: 1 Studies which follow the influence of the elasticity modulus variation in the crown delimited over the key, (2,3,4 Fig. 7.50. For λ=0.5 the elasticity modulus has been varied from 50MPa up to 10 and 5mpa.
Fig. 7.50 M and N diagrams of B variant and 2,3 and 4 hypotheses.
The first conclusion given by the λ decrease to 0.5, is the obtaining of an axial force within reasonable limits (1.4 -2.25 MN). The second conclusion is that the bending-moments increase with the decrease of E from the zone delimited over the key. So, for E decreased by 5 times M increases 1.58 times, and for E decreased by 10 times, M increases 1.79 times. Comparison with the results obtained at the Gibei tunnel, on the basis of insitu measurements, indicates the necessity for a further reduction of λ, which will be performed in the next study. 2. Studies which follow the coefficient λ variation influence (3,5,6 Fig.7.51) For E=10MPa in the crown above the key, λ is varied from 0.5 and 0.3. A decrease of the moments (1.24 - 1.64) and of axial forces (1.21 - 1.55) occurs with the λdecrease of 1.25 - 1.66 times, an almost linear relationship. The sectional stresses M and N approach even more closely to those obtained by in-situ measurements.
Fig.7.51 M and N diagrams, variant B, hypotheses 3,5,6. 3. Studies which follow the influence of the laws of behaviour of the ground. In studies 7 and 8 two laws of ground behaviour have been analysed : elastic-linear and elasto-plastic with the Mohr-Coulomb criterion. It must be pointed out that the use of an elasto-plastic law of behaviour for soft ground would lead to an extension of the plasticisation on the entire domain, in the first phase considering the excavated profile to be un-lined and loaded with decompression forces. In reality the plasticisation phenomenon is limited, as also is that of the deformation, by the presence of the shield in the first phase and of the lining in the second phase. Consequently, the law of behaviour has a reduced influence upon the sectional stresses (1.27 times for M and 1.36 for N) Fig 7.52.
Fig. 7.52 M and N diagrams, variant B, hypotheses 7 and 8. Variant "C". This variant, corresponding to a continuous lining, serves more as a comparison for the other, precast, variants, and bearing in mind that this modelling type has been and is still utilised. The only difference between this variant and variant B1.2 is a different lining structure, and the stresses and strains in the groud are very similar. Thus it is only necessary to consider the stresses M and N in the lining (Fig.7.53). The bending-moment diagram differs completely from those of precast linings and it has values much greater than variant B1.2 which has the same loading conditions. The N diagram coincides with that of variant B1.2. The immediate conclusion is that of the necessity to have joints and to consider the configuration of segments.
Fig.7.53 M and N diagram, variant C.
7.4.4. CONCLUSIONS
The numerical simulations which have been performed have led to some positive encouraging results, but they also reveal some inadequacies in the modelling of the finite elements, although it is possible to eliminate these. Two main objectives, which also define safety concepts for underground structures, have been monitored in this study: ground displacements and sectional stresses in the lining. The estimation of ground movements, which occur mainly as a result of closure of the tail void, and the study of the propagation of these movements, particularly up to the surface, are two of the fundamental aspects of the concept of shallow and medium depth tunnels. The theoretical ground movements have been estimated according to various hypotheses, and factors have been incorporated to try to take into account such phenomena as the initial conditions, the boundary conditions, the height of the ground cover over the tunnel, the extent of the discretized domain, the ground behaviour, the structure of the ground, and the effect of an advancing face and of the closure of the tail void. The first study considered a number of hypotheses which proved to be unrealistic, the theoretical results being very different from those obtained in the experiment with which they have been compared. The initial state of stresses was considered homogeneous in the discretized domain, which was assumed to extend only to three times the tunnel diameter around the tunnel; advance of the face was simulated using a relaxation coefficient applied uniformly around the excavated profile. Comparison of the results so obtained with experimental or rig studies followed, and was concerned mainly with the qualitative aspects of the representation of the various phenomena. Thus for estimating displacements, the qualitative aspects considered in relation to the in-situ tests were: • the dissipation of the displacements from tunnel to the surface; • the asymmetry of the development of displacements with regard to the horizontal diameter, being greatest at the key and a minimum under the invert, due to gravitation effects and to the presence of the shield in the zone between the face and the lining. The displacements calculated in the first study were symmetrical with respect to the horizontal diameter due to the use of a constant relaxation coefficient, and this was confirmed by the results of the second study. In order to eliminate this deficiency both from the analysis of the in-situ tests, and from the theoretical prediction of displacements, a non-uniform simulation was required, to allow for relaxation of the excavated profile during the advance of face. This was achieved by applying the λ coefficient variation to the profile, with a larger value at the crown and a smaller one on the invert, in the first phase. This approach is supported by reanalysis of the experimental results, from which it is evident that the loading imposed by the shield at the invert impedes deformation of this zone and maintains the states of stress much closer to the initial one, with respect to the upper part of the tunnel, where deformation and relaxation are unimpeded.
These differences in the processes of relaxation between crown and invert are maintained and even possibly accentuated up to and even beyond the time that the lining is installed and the shield advances, due to imperfections in the injection, the subsidence of the segmental ring, the formation of the non-uniform void and the influence of the shield on the invert. The use of this artifice, in the second study, has led to a hypothesis of the process of deformation around the tunnel which, qualitatively, is in accordance with the empirical reality found in small-scale models or in actual tunnels. The graphs of total displacements shows the formation, above the key, of a bulb of decompressed ground with significant deformations, corresponding to the crushing vault mentioned in classical studies and similar to that indicated by empirical research. Although the results obtained by using this artifice seem satisfactory as regards displacements, there is still some doubt with respect to simulating the loads imposed on the lining and implicitly the stresses in the lining. Thus the simulation of both the decompression and the formation of the bulb above the key do not properly represent the influence upon neither the lining nor the actually observed process of ground decompression and progressive detachment above the shield. In the cases considered, the tail void extends a maximum 6 cm above the key, which automatically implies a settlement of about 6cm. When the shield advances and the lining begins to come under load, the loosened ground above the arch loads the lining almost instantaneously, either directly or by means of the (more or less satisfactorily) injected material, The phenomenon of decompression, loosening and detaching of the ground accelerates in this zone, in which maximum displacements at the key are recorded, and the limits of the crushing arch are developed, which produce the main loading upon the lining. This progressive process of deformation and loading is not captured in the numerical simulation, due to limitations in the programs based on MEF, amongst which the most significant appears to be the consideration of the continuity throughout this period of successive detachment of material. Another limit is the unsatisfactory manner in which the various ground behaviour laws describe this phenomenon in general, and also the modelling of the volumetric variations which take place due to the tension which develops in the material. In fact, it must note that the main parameters which govern the behaviour of the ground are determined by triaxial compression tests, whereas the ground above the lining is actually subjected to tension. It is therefore apparent that a mode of behaviour of the ground must be established that will give a correct representation of the deformations, regardless of the state of stresses. The experimentation of an elastic-plastic behaviour law with a Mohr-Coulomb rupture criterion failed to provide relevant results in this respect. This is due to the limitation of the plasticisation phenomenon, due to the existence of the shield in the first phase, and of the lining in the second phase. However, the displacements and moments in the lining showed insignificant increases. Further parametrical studies in the framework of this same law and the further experimentation may lead to development of more comprehensive observations. From the analysis of the evolution of the displacements from the tunnel to the surface, determined in the variants of the second study, it appears necessary to extend the discretised
domain up to the surface for medium-depth tunnels, i.e. where the ground cover is up to ten times the tunnel diameter. As has been shown above, for tunnels in soft ground, safety is governed by the lining whose bearing role is predominant. Special attention has therefore been paid to the analysis of stresses in the lining. Two types of structure, continuous (extruded) and precast, have been analysed, and two configurations of the segments, according to several different processes of relaxation. The results obtained by numerical calculation (the M,N sectional stresses) have been compared with those obtained with the polygonal chain method or by in-situ research. Qualitatively, the results obtained with FEM are satisfactory, but quantitatively, important differences in the axial forces have been observed in the first studies. These deficiencies have been eliminated and the modelling has been improved in accordance with the in-situ measurements results, by reduction of the decompression coefficient and the introduction of a decompressed ground bulb with reduced E, in phase 2. By variation of E and λ parameters, the results obtained by calculation give good correlation with empirical results. On the basis of the results of the numerous studies made, it is possible to formulate some recommendations for modelling a shield-driven tunnel. The first phase of calculation, for the zone between the face and the lining, on which the shield lies, but which is considered to be un-supported, must simulate not only the process of deformation and decompression up to closure of the tail void, but also of that behind the lining. In order to simulate this process correctly, it is necessary to adopt a decompression coefficient which varies around the profile, with a value of zero at the invert and a maximum at the key. It should be noted that this phase does not refer to the working face with the classical λ=0.3, but corresponds to the period up until the lining starts to take up the load, by which time most of the deformation has taken place. The important feature of this step is the displacements which must correspond to those found in reality. The second phase of calculation, corresponding to the development of load in the lining, must represent this loading process as correctly as possible. In order to achieve this, two procedures can be utilised: 1 Applying the loading in two stages, of which the first corresponds to the loading on the key zone given by the crushing arch already formed on the shield, and represented by a decompression on this zone (λ=0.2); and the second stage corresponds to the mobilisation of pressure on the whole profile and this is represented by a uniform decompression on the profile (λ=0.3). 2 The introduction of the decompressed bulb of ground above the key, with E reduced with regard to the rest of the domain; this can be coupled with an elastic-plastic law of behaviour and which has as a result an increase in bending-moments even for a smaller decompression coefficient (λ=0.3-0.2), and also smaller axial forces, close to those found in practice. In this phase, the important results are the sectional stresses (M and N).
Following these recommendations, we have the certainty of obtaining of results (displacements at the ground surface and sectional stresses in the lining) which give good agreement with those obtained by in-situ measurements.
8. FINAL CONCLUSIONS. ; The use of precast structures in underground construction is closely connected to the circular shield method, whose advantages have led to its adoption in the most diverse of fields and in many different domains, from railway tunnels to metros, and from urban sewerage to hydroelectric undertakings. This method, and implicitly the precast lining, has conquered the entire world, representing the major part of the total volume of underground structures, which justifies the numerous studies and research carried out in this field of research, to which it is hoped that this present paper may make a modest contribution. ; The tackling of a subject in this field of study has as a pre-requisite a good knowledge of the technology and primarily of the machines used, the shields, their functions and performance. The last decades have marked an important evolution of the shield, from relatively simple machines to fully automated, but very costly, pieces of equipment. The correlation of a rich vein of statistical material has permitted the formulation of some conclusions referred to the use of various types of shields, classical and pressurised, of the more frequently used diameters and those with high performance. ; For shield-driven tunnels in soft ground, the lining represents the main element of stability and strength, but also the most important single element in the total cost. The major part of this paper is dedicated to the analysis of this essential structural element. The main elements which influence the design of a lining system (the segments configuration, material type, loading conditions etc.) are categorised and then detailed. ; The existence of the joints between the segments is the main characteristic of precast linings. The development of a structural model for a precast lining, as close as possible to reality, requires correct modelling of the joints between the segments, whose influence upon the magnitude of the sectional stresses is extremely significant. The mechanical behaviour of joints with plane faces, rigid to bending, and of hinged joints with convex-concave faces has been analysed in detail. For the joints with plane faces a relationship for the rotation angle and rotation rigidity has been deduced, with which the moments and the axial forces in the joints can be determined, for different eccentricities, having in view the shapes of the interaction curves MN.
On the basis of these curves, the degree of elastic clamping of such a joints can be determined, and this is used in the rigidity matrix of bars with elastically clamped supports, implemented in a computer program. Thus the bending-moment in the joints can be calculated for different situations of opening of the joints, and hence the influence on the rest of the section may be ascertained. The calculation can also be applied to bolted joints or cracked sections. For hinged joints, the mechanical behaviour has been studied, deducing relationships for sliding stability, and also the state of stresses and strains in the joint zone has been studied using MEF, which serves to suggest better reinforcement of the ends of the segments. ; The establishment of a structural model for a tunnel is a very complex problem, due to the multitude of parameters and the complexity of the system behaviour. For a better understanding of the place, role and evolution of the structural methods for tunnels, an original of categorisation of these methods has been devised. The topic generally falls within the ambit of
deformable media mechanics and of strength of materials (the one-dimensional method) and the elasticity theory (the continuum model). The methods belonging to the one-dimensional model are the classical, conventional ones, known generally after their authors and which compute the sectional stresses in the lining and the loads on the lining, using different un-connected methods. After discussing how the loads on a tunnel are derived, and describing different loading models for circular tunnels, this Paper presents a method to establish the soil loadings, taking into account the circular shape of the cross-section, in accordance with the results of the pressure measurements actually recorded in various tunnels. The new loading model is compared with the classical ones, by the determination of the sectional stresses (M and N), which are found to be remarkably lower. Although the results are encouraging, the introduction of the new model of loading into current design practice must be checked against in-situ measurements taken during the progress of the works. ; From the methods of the one-dimensional model, the most well-known and used are those which consider the ground - structure interaction. From this group, the most frequently used are the Schulze - Duddeck method and the method of the polygonal chain on springs. Both methods have been examined and analysed herein, in order to establish some comparisons. The conclusions are clear on the part of the polygonal chain method, which represents the most simple and efficient calculation model, which introduces explicitly the phenomenon of ground lining interaction, permits the solving of an almost unlimited variety of applications, and is supported by a considerable database. In this Paper, attempts have been made to improve this method by: a. Replacing the discrete contacts between the structure and ground with continuous contact. To this end, a rigidity matrix has been established for a doubly-clamped bar with continuous contact, which has been introduced into the computer program mentioned. A comparison has been made for a tunnel structure using both methods, which indicated a reduction in bending-moments of about 5% and a 2% increase in the axial force, i.e. more favourable conditions, for the variant with continuous contact; also the data introduction is easier. b. taking into consideration the action of the joints, by the introduction of bars with elastic clamping, whose stiffness matrix is based on the degree of elastic clamping, defined by the rotational rigidity. This model can allow for the changes due to each execution phase, by iterative calculation, recalculating in each step the eccentricity, the rotational rigidity and the degree of elastic clamping. The numerical examples and the conclusions after using this model are presented in the parametrical studies. ; The last chapter of the first part of this Paper analyses by parametrical studies the influence of various factors upon the principal quantities monitored -- sectional stresses and displacements -- and tries to define a safety concept. The establishment of a safety concept for tunnels is very difficult, due to the multitude, the complexity and the dispersion of the various influencing factors. The method adopted in this paper to overcome these difficulties is to carry out parametrical studies, on the basis of which individual influence coefficients are established for the various parameters, which can be ordered as a function of the amount of influence that each exerts, it being possible finally to establish a global coefficient which can be regarded as a safety coefficient. The parametrical studies have been performed for the most important factors and parameters, as follows:
a. The structural arrangement (the segments number and their placing in the section).
Three variants have been analysed, with four, five and six hinged segments in the section, and two possibilities of placing, with a segment or with hinged joint at key, and these have been compared with a continuous structure. For the analysed structure type, (railway tunnel, Di=8.25m) influence coefficients for M and N have been established, and even safety coefficients on the M-N interaction diagram. This analysis indicates that the most advantageous variant is the most elastic one, but the final image can be obtained only after analysis of the maximum displacements, which are a function of the bed coefficient. It should be noted that the deformation is large because the structure is more elastic. On the basis of this analysis, it can be seen that a relationship exists between the lining rigidity and the rigidity of the ground, orientative value limits being established also for the analysed structure. b. The type of joint and the initial deformation. This analysis was made on a variant made up of five segments, with two configurations of the segments in section, two joint types (hinged and with plane faces), and two deforming situations, symmetrical and unsymmetrical. The results of the static analysis demonstrate the influence of the two factors upon the magnitude of the sectional stresses. The adoption of plane joints can increase the bending-moments to one-and-a-half times those obtaining in hinged joints, and the deformation from the initial phase of the structure display increases of 1.35-1.65 times. These increases can be serious arguments for the recommendation to adopt the hinged joint and the necessity to keep the circular shape by various procedures (injection, former rings, temporary bolts). c. The rotational rigidity of the plane joints. On the basis of the rotational rigidity and of the degree of elastic clamping, defined in section 3 above, and using the model of a joint with eccentricity and elastic clamping, the influence of the eccentricity and rotational rigidity upon the sectional stresses in the joints has been analysed in theory, and measured in the field, for the variant of five segments and two configurations of placing. Useful conclusions may be drawn from modelling structures with plane joints: • the rotational rigidity, which may be taken into account in calculation by the degree of elastic clamping, influences the bending-moments, as a function of the positioning of the joints in the section; • the possibility of variations in the degree of elastic clamping is limited to at most 0.1, due to the reduced zone of action, generated by the small axial forces; • the modelling of an initial eccentricity is recommended, with elastic clamping and not with hinged joint, which give bigger moments. Also, rigidity is reduced by cracking, and this has been simulated as a process of redistribution of stress until a new state of equilibrium is reached. d. The soil parameters. The influences of the angle of internal friction, of the active thrust coefficient and of the tangential bed have been analysed. The results obtained seem to counteract preconceived ideas regarding the importance of these factors upon the magnitude of sectional stress, proving a more reduced influence than the factors previously analysed.
Finally the principal analysed factors have been sorted in order of importance, as a function of the magnitude of the individual influence: • the initial strain influence (Ci=1.5); • the joint type influence (Ci=1.5); • the section formation influence (Ci=1.3); • the ground characteristics influence (Ci=1.2). By combining several of these factors, a total influence coefficient is obtained, which can be regarded as a global safety coefficient. A value of 2.5 - 3.0 is proposed as reasonable for underground structures, in view of the complexity of such works. ; The second part of the paper is occupied by the study of the state of stresses and strains in the ground - lining assemblage of the shield driven tunnels. The problems affecting the design of tunnels may be defined as follows: • determination of the stresses and strains in the ground around the hole; • determination of the stresses and strains in the supporting structure; • determination of the loads on the lining and of the settlements at the surface.
The conventional methods, analysed in the first part, treat these problems as separate entities, without any connection between them. In fact, all the components of a tunnel are interconnected and influence each other. The only way in which the complex phenomena of the ground - lining assemblage may be treated as a unified whole is by empirical means, especially in-situ research and the finite elements method. Empirical research is usually oriented towards the two main problems, study of the ground and study of the lining. ; The study of the state of stresses and strains around the hole has been carried out entirely on the basis of documentary evidence, since such research has not been performed in Romania. An extensive study has been made of a large body of documentation from the specialised literature, containing theoretical studies, tests on models, measurements in trial tunnels and in tunnels under construction. This study has considered various working hypothesis, test models, field conditions and tunnel-driving machines, and as a result various ideas have been developed and general conclusions drawn, referring to the movement of ground around shield-driven tunnels.
The origins and the causes of the ground movements and their propagation longitudinally and transversally have been defined and classified, in general and in particular, as a function of the ground cover over the tunnel, i.e. of the ratio H/D. Three possible deformation mechanisms have been identified: • H/D < 2.5 (surface tunnels), where the settlement extends up to the surface, the volume of the displaced soil being limited by two shearing surfaces; • 2.5 < H/D < 5 (medium-depth tunnels), where the deforming mechanism is characterized by the formation of a strongly decompressed bulb of soil, limited by a rupture surface, known under the name of crushing arch; • H/D > 5 (deep tunnels), where the deforming mechanism is close to the theoretical models, with the development of a plastic zone around the hole. The established limits for the ratio H/D are orientative, and they depend greatly upon the nature of the ground. It is, in any case, obvious that the shield is the major factor which conditions the weak or strong development of settlement. The main elements which influence the behaviour and the performances of a shield are: maintaining the equilibrium of the face pressure and the avoidance of the soil losses; ease of steering; and the execution of continuous and uniform injection behind the tail.
In view of the number of the parameters involved in the development and evolution of deformations and displacements which cannot be precisely determined, the theoretical models of calculation must be calibrated by means of information obtained after such empirical studies. ; The estimation of ground movement is only one aspect in the conception of a tunnel. Equal attention must be paid to the study of the state of stresses and strains in the tunnel lining, which is undoubtedly the principal factor in stopping deformations and in the stability of the assemblage. In this Paper, the whole variety of possible tests has been analysed: on scale models in the laboratory, on full size rigs and in-situ. For the tests on scale models and full size rig tests, examples from the literature have been presented, highlighting the structural behaviour and the role of the joints.
For the primary linings used in railway tunnels in Romania, a program of measurements was devised and performed in concert with the Department of Reinforced Concrete at ICB. This program takes into account the construction and technical peculiarities of the different structures studied: • the rig test on the primary lining segments; • convergence measurements at the Beia tunnel 2; • in-situ research upon the primary lining of precast segments with hinged joint at the Birnova tunnel; • in-situ research upon the primary lining of precast segments with tongue-and-groove plane joints, and upon the secondary lining of monolithic concrete at the Gibei tunnel. Tests were performed using a custom-built rig on various types of segment, categorised by their manner of behaviour in the field, and these provide a very useful database for assessing the level of loading, the reserve bearing capacity and the risk of failure. Besides the in-situ measurements from the Birnova and Gibei tunnels, in which the author was personally involved, the paper also presents in-situ measurements made at the Bucharest metro, with plane bolted joints, and at the road tunnel under the River Elbe in Hamburg, where steel/concrete segments with plane bolted joints were used; the study thus encompasses the whole variety of structures and joints between segments. On the basis of these empirical studies, some general conclusions can be formulated, from which the most important ones are: • the existence of the joints between segments has a decisive influence on the distribution of stresses and strains; • the joint type (hinged or plane, with or without bolts) influences both the distribution and the order of magnitude of the stresses; • the number of joints and their placing on the lining profile determine the sense and magnitude of the sectional stresses; • the data obtained from experimental research, especially the sectional stresses and the deformations, are very useful elements to calibrate and improve the calculation models. ; Empirical research, especially in-situ, is a special source of information for designers of tunnels, but is not usually available to the designer, being expensive and difficult to arrange. In these conditions, the only tool accessible to the designer and able to allow for the complex phenomena of the lining/ground assemblage, is numerical analysis based on finite elements.
In this Paper, a general presentation of FEM has been made in relation to the modelling of underground work under construction, with specific principles, objectives and limits. After analysing the specific steps and elements of such a modelling, discretisation of the structure/ground assemblage, modelling the construction sequences, advancing of the face and
closure of the tail void, and selection of a theory to describe the behaviour of the ground, some very useful conclusions have been stated in order to improve: 1 The modelling of construction sequence in soft ground with the shield method, using FEM, leads to unsatisfactory results, due to: • deficiencies of the continuous model for the simulation of this problem; • inadequate modelling of the discontinuities of the structure (the joints between segments); • deficiencies in the simulation of construction sequences, of the face advance and of the closure of the tail void in two-dimensional modelling; • insufficient empirical data to calibrate the models. 2 The use of complicated rheological theory is not justified at this stage, in the conditions in which determination of the soil parameters is approximate, leading to irrelevant results; 3 A two-dimensional model should be used in preference to a three-dimensional one, due to a more reduced working volume, easier interpretation of the results and greater success of the artifices adopted. ; The conclusions from the theoretical and empirical studies made upon some shield- driven tunnels presented in this Paper, have served to improve the modelling and to clarify the concepts involved in tackling the analysis of the construction process of a shield-driven tunnel, using FEM. The checking of some hypotheses and the testing and validation of some new artifices, has been done by numerical examples. These examples have been made on a shield-driven railway tunnel, corresponding to the actual data of the Gibei tunnel, using two computer programs for finite elements, customised for underground structures.
a. The first example studies the behaviour of a tunnel considered deeply, using the CAV program, which uses a two-dimensional model and a ‘deformation in plan’ hypothesis, and the simulation of the excavation and face advance is made by artifices. In this example, a special discretisation was tried in the joint zone, in order to allow for the effect of a hinged joint. The results obtained with this first numerical study, compared with the results of in-situ research upon similar works, has permitted the statement of these observations useful for the improvement of the modelling of such works : • the computed displacements are symmetrical with respect to the horizontal diameter, in comparison with the measured displacements which are large at the key and zero in the invert; • the computed bending-moments in the segments are much smaller than those measured, but they correspond in sign, due to the functioning of the joint, which is a first success. b) The second study has been performed on the same tunnel type, considered this time to be of medium depth and analysed up to the surface, using the CESAR program. The possibilities offered by this program have permitted several studies to be tackled, to clarify the influence of various factors upon the state of stresses and strains in the ground and lining. Three types of transverse section have been analysed: • precast segmental lining and two configurations of five segments in section with the key at the crown (type A) • as above, but with the key in the invert (type B); • continuous lining (type C).
To improve the calculated results, with respect to these obtained empirically, a simulation was tried of non-uniform relaxation of the excavated profile during the process of advancing the face in the first phase. This proposal is supported by practical reality, for it was found that the load imposed by the shield on the invert, which impedes the deformation in this zone and maintains the state of stresses closer to the initial state with respect to the upper part where deformation and relaxation can take place. The use of this artifice has led to a particular pattern of deformation around the tunnel which accords with empirical results. It could also be shown by calculation that a bulb of decompressed ground is formed above the key, with significant deformations, corresponding to the crushing arch of the classical studies and similar to that suggested by empirical research. This artifice of variable relaxation is the most important contribution to the improvement of the two-dimensional model, which makes this instrument really competitive. The final part of the study is dedicated to the elimination of the last doubts, related to the discrepancies between those stresses in the lining determined by calculation and those found empirically. This last step is, in fact, calibration of the model with the results of empirical research, achieved by varying the main calculation parameters λ and E. Also in this step, the phenomenon of the bulb of decompressed soil has been successfully introduced, with E being reduced with respect to the surrounding ground, and the assumption of elastic-plastic ground behaviour, with Mohr-Coulomb failure criterion. The introduction of the bulb of decompressed soil has given excellent results, and represents a contribution to improvement in two-dimensional modelling. However, the use of an elasticplastic behaviour law gave irrelevant results, the plasticisation phenomenon being limited by the presence of the shield in the first phase and of the lining in the second phase. The final results of this step mark a success, with the calculated sectional stresses approaching those obtained experimentally. On the basis of the encouraging results of numerous studies, guidelines have been developed for the modelling of shield-driven tunnelling, which define the concept presented in the conclusions in the chapter 7. By following these guidelines, accurate predictions can be made of ground displacements and stresses in the lining which will be close to those revealed by in-situ measurements. ♦ This Paper meets some objective requirements from the field of underground construction, where there is a dearth of theoretical studies investigating these aspects to such extent and depth. The work carried out in the preparation of this Paper has involved much experience in the fields of design, of research and of monitoring of the construction process, and subsequently also of analysis, assimilation and processing required to produce this document. ♦ This Paper tackles a large variety of problems specific to the design of shield-driven tunnels, from the new technology of pressurised shields to the behaviour of the lining /ground assemblage and to the conceptualisation and structural analysis of the supporting structure, and has lead to significant potential for improvement and amelioration of some of these problems. ♦ The thorough study of the behaviour of the joints between segments, by analytical or numerical means, has led to the establishment of some new calculation elements, which allow more realistic structural analysis and dimensioning.
♦ The systematisation of the methods of structural calculation and the examples done by the most frequently used methods, put at the designer's disposal both an overall view and guidance towards the most appropriate method. ♦ The improvements brought to the polygonal chain method, integrated in a computer program and coupled with the suggested method of determining loads, give new life to this method and transform it into a competitive tool for the structural analysis of precast underground structures. ♦ The parametrical studies carried out during the preparation of this Paper have permitted the definition and classification of the principal factors which influence the behaviour of precast structures, and the establishment of some influence coefficients and of global safety coefficients, giving the basis of a possible safety concept. ♦ The experimental research on rig tests and in-situ, never before carried out upon the precast structure of railway tunnels, has proved a fruitful experience and has led to the creation of a very useful database for furthering the understanding of the behaviour of such structures and for the calibration and improvement of the calculation models. ♦ The introduction of some very efficient artifices, calibrated by in-situ measurement, has considerably increased the reliability and efficiency of numerical modelling of the shield method by FEM, which is now available to the tunnel designer to use in accordance with the guidelines given herein. ♦ The conclusions and results obtained in this work have a clear practical applicability, serving to provide guidance for any designer involved in the structural analysis of precast linings.
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