STRESSES AROUND UNDERGROUND OPENINGS CONTENTS

Transcription

1 STRESSES AROUND UNDERGROUND OPENINGS CONTENTS 6.1 Introduction 6. Stresses Around Underground Opening 6.3 Circular Hole in an Elasto-Plastic Infinite Medium Under Hdrostatic Loading 6.4 Plastic Behaviour Around Tunnels 6.5 Zone Of Influence 6.6 Excavation Shape And Boundar Stress Oval Shape 6.6. Ovaloidal Shape Openings Square Opening With Rounded Corner Excavations With Special Shapes 6.7 Stress Distribution Due To Development Of Fractured Zone 6.8 Tunneling In Stratified Rock And Block Rock 6.9 Rock Fractures And Scale/Size Effect 6.10 Numerical Modelling For Stresses Around Underground Opening 6.11 Effect Of Width To Height (W/H) Ratio 6.1 Tunneling In Weak Rock 6.13 Effects of Planes of Weakness on Stress Distribution 157

2 LECTURE INTRODUCTION Underground openings have a wide variet of applications like tunnels built for highwas and railroads, water suppl and sewage tunnels, underground power stations, storage caverns etc. With such a vast range of underground applications, it is necessar to understand the various aspects of underground openings and their stress and deformation characteristics. Rocks are initiall stressed and an opening created cause a changes in the initial stress. The design of an underground structure in rocks differ from other tpes of structural design in terms of the nature of loads acting on the sstem. The post excavation state of stress in the structure is the resultant of initial state of stresses and stresses induced b excavation. Hence the determination of the state of stress is necessar for an design analsis. The stud of stresses around underground openings gives an insight into the basic mechanisms like displacements and the stress fields and helps to provide suitable support for the underground opening. The major conditions around an opening can be classified as in-situ stresses due to the overburden rock, induced stresses due to the excavation for the opening and traffic loads not significant in the case of deep tunnels. Different tpe of tunnels and underground excavation include road tunnels, rail tunnels, rapid transit tunnels, water tunnels, sewage tunnels, hdroelectric tunnels, service and utilities tunnels, station buildings etc. These tunnels are in hard rock, weak rock, stiff soil, soft soil or mixed ground. Cutting or the excavation technique plas a great role in the overall design. Rock excavation can be made adopting an of the following, Drilling and blasting Using tunnel boring machines (TBM). Road headers Sequential excavation with small mechanical equipments 158

3 Figure 6.1: Different tpe of tunnels and underground excavations (Representation purpose onl) Ground water / pore pressure influence in tunnels is crucial and need to be considered during the design. Tunneling is an art and involves geolog, geotechnical engineer, structural engineer, mechanical engineer, electrical engineer and man other disciplines. Tunneling involves excavation (rock/soil) opening and depends upon excavation machine technolog. For the support sstem to sustain/protect the excavation one need rock/soil mechanics and support material technolog. 159

4 6. STRESSES AROUND UNDERGROUND OPENING These 3D problem can be reduced to a D if the stresses on the peripher of a long tunnel are studied b the sectional analsis. The problem is reduced to the stud of the stresses, on the peripher of an opening in a thin sheet of elastic or elasto-plastic medium. The mathematical difficulties in the theor of elasticit are reduced, if the boundaries of the bodies coincide with the co-ordinate sstem used. Thus the problems involving spheres or spherical/circular openings in an in-finite medium can be solved more easil if spherical coordinate sstem is used with the center of the co-ordinate sstem at the center of the sphere. Figure 6.: Sectional view of a long tunnel Figure 6.3: Elemental stress state in polar co-ordinate sstem 160

5 To transform the stresses from rectangular co-ordinate sstem to polar co-ordinate sstem the following transformation equations are used. sx + s sx s sr = cos θ + τx sin θ sx + s sx s = + cos θ τx sin θ sx s τr θ = τx cos θ sin θ The back formulations of horizontal and vertical stresses are given b, + sr sr sx = cos θ τx sin θ + sr sr s = + cos θ + τx sin θ sr τx = τr θ cos θ sin θ Figure 6.4: Stresses around circular tunnel in plain strain 161

6 Problem of a hole in an infinite plate is of special interest in the rock mechanics field because it corresponds to the problem of a long horizontal tunnel at depth in a uniform rock formation. Considering an infinite plate of thickness 't' with a circular hole of radius 'a' at the origin. If the applied stresses in the horizontal (x) and vertical (z) directions are σ x & σ z respectivel, the stress components are (G. Kirsch, 1898), 4 a a x z x z = ( s + s ) 1+ ( s s ) 1+ cos θ r r 4 1 a 1 3a 4a r = ( x + z) + ( x z) + 4 s s s 1 s s 1 cos θ r r r 4 1 3a a τr θ ( sx sz) = 1 + sin θ 4 r r a = radius of the circular opening θ = central angle with x-axis r = radial distance of the element from the center of the opening For σ x = 0, the maximum tangential stress is three times the applied stress the applied stress and occurs at the boundar on the X-axis, that is θ = 0 or π. θ = π/ or 3π/, the tangential stress at the boundar of the opening is equal to the applied stress but is of opposite in sign. Case 1: Hdrostatic case When, σ x = σ z = p (compressive), i.e. for the case of hdrostatic loading: a s r = p 1 r s θ p 1 a = + r τ rθ = 0 16

7 Figure 6.5: Circular opening in hdrostatic stress field If σ x = σ z, the maximum tangential stress occurs at the boundar of circular opening and is equal to two times the applied stress. The radial stress at the boundar is equal to the two times of applied stress and occurs on a plane at 45 o to the boundar. Thick wall clinder subjected external pressure This problem corresponds to the problem of a tunnel or shaft lining (with a internal and external radius 'a' and 'b' respectivel) in a rock formation having a hdrostatic stress field (P). Figure 6.6: Tunnel linings in plain strain with internal radious 'a' and external radious 'b' in hdrostatic stress field 163

8 This problem corresponds to the problem of a tunnel or shaft lining in a rock formation having a hdrostatic stress field. The radial, tangential and shear stresses corresponding to lining with internal radious 'a' and external radious 'b'. b P σ r = b a o a ( 1 ) r b P a σ o θ = ( 1 ) + b a r τ r θ = 0 If one takes a limit for b infinit, it can be seen that, the results are same as circular opening in hdrostatic stress field. 164