ANALYTIC SOLUTIONS OF ELASTIC TUNNELINGPROBLEMS Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof.dr.ir..
Trang 1ANALYTIC SOLUTIONS OF ELASTIC TUNNELING
PROBLEMS
Trang 3ANALYTIC SOLUTIONS OF ELASTIC TUNNELING
PROBLEMS
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir J.T Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op 9 september 2002 om 10:30 uur
door Otto Erik STRACK Bachelor of Geological Engineering, University of Minnesota
geboren te ’s-Gravenhage
Trang 4Dit proefschrift is goedgekeurd door de promotor:
prof.dr.ir A Verruijt
Samenstelling promotie-commissie:
Rector Magnificus, voorzitter
Prof.dr.ir A Verruijt, Technische Universiteit Delft, promotor
Prof.dr.ir F.B.J Barends, Technische Universiteit Delft
Prof.dr.ir J.A Battjes, Technische Universiteit Delft
Prof.dr.ir R de Borst, Technische Universiteit Delft
Prof.dr H.G Meijer, Technische Universiteit Delft
Prof.dr.ir F Molenkamp, Technische Universiteit Delft
Prof.dr C Sagaseta, University of Cantabria, Spain
Published and distributed by: DUP Science
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Delft University Press
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ISBN 90-407-2327-3
Keywords: Elasticity, Tunnel, Complex Variables
Copyright © 2002 by O.E Strack
All rights reserved No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or me-chanical, including photocopying, recording or by any information storage and retrieval system, without written consent from the publisher: Delft University Press
Printed in The Netherlands
Trang 5To Sonja and Otto.
Trang 7Analytic Solutions of Elastic Tunneling Problems
The complex variable method for solving two dimensional linearly elastic prob-lems is used to obtain several fundamental analytical solutions of tunneling problems
The method is used to derive the general mathematical representation of problems involving resultant forces on holes in a half-plane Such problems are encountered in geomechanics during the excavation of tunnels When tunnels are excavated the removal of the weighted material inside the tunnel causes the ground under the tunnel to rebound This in turn causes a resultant force, referred to here as the buoyancy force, to act on the tunnel until the stresses return to a state of equilibrium The mathematical representation derived in this thesis shows that the displacements will usually be unbounded at infinity
in solutions of two-dimensional problems involving resultant forces acting on holes in a half-plane
The unbounded behavior of the displacements at infinity causes the deter-mination of the settlements due to tunnel excavations to have a certain degree
of arbitrariness A point must be chosen at which the vertical displacements are fixed, referred to in this thesis as the point of vertical constraint For the solutions in this thesis, it is assumed that the ground will exhibit a high stiffness for stresses which are below a predetermined level Plots of the stresses are used to determine the point nearest to the tunnel at which this predetermined level has been reached If the stresses continue to dissipate beyond this point, then this point can be used to fix the displacements As can be expected, the choice of this point has a large effect on the displacement field, but does not influence the stresses and strains
A general solution is derived for a single, circular tunnel in an elastic half-plane that includes a buoyancy force The boundary conditions in this solution are that the surface of the half-plane is stress-free and that the displacements are specified along the tunnel boundary This general solution is used in the rest of the thesis as the framework for considering three fundamental tunneling problems: the problem of a rigid, buoyant tunnel, the ground loss problem, and the ovalization problem
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Trang 8viii Abstract
The problem of a rigid, buoyant tunnel is encountered when a very stiff lining with negligible weight (in comparison to the weight of the excavated material) is placed inside a freshly excavated tunnel The problem is solved by assuming that constant displacements occur around the tunnel periphery This corresponds to the case in which there is a large amount of friction between the lining and the tunnel The solution shows that large amounts of heave can result from such excavations, but that the amount of heave is strongly dependant on the choice of the point of vertical constraint
The ground loss problem is encountered when the process of tunnel excava-tion or the placement of a tunnel lining causes the ground to converge towards the tunnel In the solution it is assumed that this convergence is uniform The solution shows that the settlement trough due to ground loss is wider than those usually encountered in practice In the solution the amount of ground loss along the surface is larger than the ground loss around the tunnel in most cases, and a simple formula is given for the maximum trough depth for tunnels whose depth
is more than twice the tunnel radius The tunnel is also shown to experience a considerable amount of settlement due to ground loss
The ovalization problem is encountered when the initial stresses in the ground are uneven in the horizontal and vertical directions In the solution it is assumed that the unevenness of the stresses causes the tunnel to deform from a circle into an oval The solution shows that the settlement trough is much narrower than for the ground loss problem A simple formula is given for the maximum trough depth for tunnels whose depth is more than twice the tunnel radius The three fundamental solutions discussed above are combined to illustrate their possible use as tools for the prediction of settlement troughs The inclusion
of buoyancy effects in the solution is shown to have a possibly considerable narrowing effect on settlement troughs Neglect of the buoyancy effect might well be a reason why settlement troughs calculated in theory are often wider than those measured in practice
O.E Strack, Analytic Solutions of Elastic Tunneling Problems Ph.D Thesis,
Delft University of Technology Delft University Press, 2002
Trang 9Part I Introduction and Theory
Chapter 3 Multiple Holes in a Half-Plane 7
§ 3.1 Complex Potentials for a Finite Plane with Holes 7
§ 3.2 Complex Potentials for a Half-Plane with Holes 8
§ 3.3 Boundary Equations for a Half-Plane with Holes 14
§ 3.4 Chapter Summary 16
Chapter 4 A Deforming Circular Tunnel 18 § 4.1 The Complex Potentials and Geometry of the Problem 19
§ 4.2 Solution for a Deforming Hole in a Half-Plane 21
§ 4.3 Chapter Summary 29
Part II Applications and Conclusion Chapter 5 Buoyancy of a Rigid Tunnel 33 § 5.1 Solution of the Problem 34
§ 5.2 Validation of the Solution 36
§ 5.3 Comparison to Yu’s Solution 37
§ 5.4 Stresses on and around the Tunnel 40
§ 5.5 Displacements along the Surface 41
§ 5.6 Comparison to a Finite Element Calculation 44
§ 5.7 Chapter Summary 46
Chapter 6 The Ground Loss Problem 48 § 6.1 Solution of the Problem 48
§ 6.2 Validation of the Solution 50
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Trang 10x Contents
§ 6.3 Comparison to Sagaseta’s Solution 52
§ 6.4 Stresses around the Tunnel 53
§ 6.5 Stresses on the Tunnel 54
§ 6.6 Displacements along the Surface 56
§ 6.7 Comparison to a Finite Element Calculation 58
§ 6.8 Chapter Summary 60
Chapter 7 Ovalization of a Circular Tunnel 61 § 7.1 Solution of the Problem 62
§ 7.2 Validation of the Solution 63
§ 7.3 Comparison to Verruijt and Booker’s Solution 65
§ 7.4 Stresses around the Tunnel 67
§ 7.5 Stresses on the Tunnel 68
§ 7.6 Displacements along the Surface 70
§ 7.7 Chapter Summary 72
Chapter 8 Prediction of Settlement Troughs 73 § 8.1 Peck’s Surface Settlement Formula 73
§ 8.2 Superposition of Ground loss and Ovalization 74
§ 8.3 Buoyancy Effects 76
§ 8.4 Chapter Summary 77
Chapter 9 Conclusion 79 Appendix A Multi-Valued Complex Potentials 81 §A.1 Derivation from the Stress Equations 81
§A.2 Evaluation of the Coefficients in the Potentials 85
§A.3 Extension to more than one Hole 88
Appendix B Implementation in a Computer Program 89 § B.1 Dimensionless Complex Potentials 89
§ B.2 Dimensionless Coordinates 91
§ B.3 Evaluation of the Stresses and Displacements 91
§ B.4 Comparison of Numerical and Analytical Solutions 93
Appendix C The Ovalization Boundary Condition 95 § C.1 Boundary Condition Along the Tunnel 95
§ C.2 Transformation of the Boundary Conditions 96
§ C.3 Fourier Expansion in the Transformed Plane 97
Trang 11The research carried out for this thesis was supported by Delft University of Technology and by Bouwdienst Rijkswaterstaat I am grateful for their support and for the opportunity to perform this study
My profound thanks are extended to Professor Arnold Verruijt who mentored
me throughout my Ph.D study His advising style was perfectly matched to
my approach to learning Over the past four years I was continuously inspired
by his approach to problems and teaching I am very grateful to have had the opportunity to work with him I would also like to thank the members of the geotechnical lab who have made my time in Delft a rewarding experience Of particular enjoyment for me were our daily coffee breaks in which we would discuss both the mundane and profound issues of the day I am especially grate-ful to André Koelewijn who corrected the Dutch translation of the propositions and abstract of this thesis
This work certainly would never have been completed, or perhaps even be-gun, without the strong support and encouragement of my wife Sonja Her questions from a layman’s perspective often required me to explain difficult concepts in an innovative way, many times allowing me to see the problem from another angle Lastly, I thank my son Otto, who breathed new life into
my research
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Trang 12xii Acknowledgements