Numerical Response of Concrete-Lined Tunnels Crossing Active Faults

AN ABSTRACT OF THE DISSERTATION OF

Stephanie Lange for the degree of Doctor of Philosophy in Civil Engineering presented on May 16, 2019.

Title: Numerical Response of Concrete-Lined Tunnels Crossing Active Faults

Abstract approved: Scott A. Ashford

A challenge for municipal authorities of growing urban areas is to provide larger and faster transportation and utility networks that are safe and resilient to significant disruptions after an earthquake event and other disasters. Urban regions, like the San Francisco Bay or the Los Angeles area, are situated in seismically active regions. In these areas, underground structures, such as water ducts or metro tunnels, must cross active faults to ensure population sustainability. It is commonly known that earthquakes with magnitudes greater than M6.0 can cause significant damage to tunnels in seismically active areas. In particular, large strains due to fault offsets and ground shaking can lead to severe damage in the tunnel lining, (e.g., concrete spalling), which can lead to potential closure and disruptions to the transportation network. Examining the behavior of concrete lined transportation tunnels built through active fault zones is critical to ensure resilient design and safe operations. Under- standing the response of tunnels crossing active faults will further inform rehabilitation and maintenance measures and support tunnel safety. A 2D model of a circular reinforced concrete tunnel crossing an active fault is developed within the finite element framework OpenSees. A parametric study with varying structural and ground properties is performed. The effects of earthquake magnitude, geology, fault zone width, and structural properties of the tunnel are studied and assessed to develop novel tunnel design strategies to accommodate large fault motions and to minimize tunnel service disruptions. The research is based on three main questions. (1) Can we give guidance when it is acceptable for a tunnel to cross an active fault? (2) What is the influence of faulting on the circular tunnel lining in the cross-section, but also along the longitudinal tunnel axis? and (3) What consequences and generalizations can be drawn to support serviceability of the tunnel after an event? Guidance to engineers on these research questions include a possibly reduced length of retrofit measures along the longitudinal tunnel axis where the tunnel crosses the active fault zone. This assessment is based on localized strains and stresses in the concrete lining. The flexural displacement capacity of the tunnel beam ranges between 0.2 % and 2 % of the inner tunnel diameter. A corresponding earthquake magnitude threshold for reaching peak compression concrete strain can be as low as M5.5 at active fault crossings. A generalization that can be drawn is the interaction between the tunnel diameter and the fault zone width, where a fault zone width of less than 1 to 2 times the tunnel diameter might be a larger concern for the tunnel lining due to abrupt shearing. The applicability of such research to existing tunnels is assessed for the Berke- ley Hills tunnels crossing the active Hayward Fault. The ultimate threshold displacements of the tunnel beam model are correlated to earthquake magnitudes. An evaluation with charts results in similar earthquake magnitude thresholds for concrete failure compared to a specific 2D numerical analysis. This shows that a simplified chart assessment that predicts an approximate threshold of an earthquake magnitude the concrete lining is able to withstand, might be applicable for early stage projects, e.g. during a feasibility assessment. c Copyright by Stephanie Lange May 16, 2019 All Rights Reserved Numerical Response of Concrete-Lined Tunnels Crossing Active Faults

Stephanie Lange

DISSERTATION

submitted to

Oregon State University

in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Presented May 16, 2019 Commencement June 2019 Doctor of Philosophy dissertation of Stephanie Lange presented on May 16, 2019.

APPROVED:

Major Professor, representing Civil Engineering

Head of the School of Civil and Construction Engineering

Dean of the Graduate School

I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request.

Stephanie Lange, Author ACKNOWLEDGEMENTS

This work would not have been possible without the support of many individuals during my time at Oregon State University. I am grateful for their support and experience. Firstly, I would like to express my sincere gratitude to my advisor Dr. Scott Ashford, who has been supportive of my career goals and who worked actively to provide me with his support, guidance, and mentorship. He has provided me with unique academia and industry connections. His guidance helped me in key aspects of my life and work, and improved my presentation skills. I am thankful to my co-advisor Dr. Ben Mason. He has provided thoughtful comments, suggestions, and relevant references for the successful application for the National Science Foundation Graduate Research Fellowship, which is greatly appreciated. I was fortunate to work with my co-advisor Dr. Michael Scott. My sincere thanks to him for his support in structural analysis and especially in men- toring me in OpenSees. I am grateful for him presenting the research poster at the 11th NCEE conference, I was not able to attend. I also thank him for his support of the tunnel course, for not only being the professor on record, but also for advising on teaching and for grading all student work. All my students were pleased to get a deserved A. I am grateful for Dr. Michael McRae for initiating the research subject and supporting this research with detailed information on the Berkeley Hills Tunnels. I would like to thank him for his support of my tunnel engineering career and for giving valuable advice. I am truly grateful to him. I would also like to express my sincere thanks to my dissertation committee members, Dr. David Trejo and Dr. Edmund Dever, for offering support, encouragement, and guidance. I gratefully acknowledge the suggestions and support from OSU professors Dr. Burkan Isgor for information on concrete, and Dr. John Nabelek, Dr. Eric Kirby, and Dr. Andrew Meigs for personal talks about seismicity at faults. In pursuit of my teaching goals, I am thankful for Dr. Shane Brown for insightful suggestions, encouragement, and patience. I thank Dr. Jason Weiss for his support of teaching a tunnel course at OSU. I gratefully appreciate the financial support received towards my Ph.D. from the National Science Foundation Graduate Research Fellowship Program. I am also grateful to the funding received by the Jerry Yamamuro Geotechnical Fellow- ship. I am thankful to Connie and Lee Kearney for supporting Dr. Scott Ashford, who, in return, was able to support me. The informational support of six different tunnel companies in the U.S. is gratefully acknowledged. I am grateful to all of those, with whom I have had the pleasure to work during this and other related projects. During early stages of my research, I am thankful to Don Ballantyne who supported my research with his extensive knowledge and wisdom. I thank Dr. Armin Stuedlein for his support of the GIGSO group and the personal opportunity to organize a 2 days graduate student symposium at OSU. I thank the Kearney Hall office staff for their continuous support. I would like to express my sincere appreciation to Prof. Bray for a short but important personal talk at UC Berkeley and Prof. Arduino for a personal talk about OpenSees soil elements, 2D and 3D modeling. I appreciate their honesty. I would especially like to thank my friends at OSU, (almost there Dr.) Maggie Exton, Dr. Adriana Debora Piemonti, and all of the Owen grad office past and present, for their advice, opinion, moral support, willingness to listen, outdoor adventures, and friendship. Nobody has been more important to me in the pursuit of this Ph.D. than the members of my family. I thank you all for your love, support, and encouragement. I would like to thank my parents, Barbel¨ and Herbert Lange, my sister Kerstin, and my aunt and uncle, Tina and Benno Voß, for believing in me that whatever I want to do is achievable. Most importantly, I am so thankful to my loving husband Alan. I can‘t thank you enough for motivating and inspiring me throughout this experience, for your companionship, love, patience, constant emotional support, humongous understanding, and for making daily oatmeal and carrot-juice to keep le lait maternel running. I could never express my gratitude in words for you. I would like to express my love to my wonderful daughter Luciana, who provided unending inspiration and passion. I thank the trees (and recycled paper products) for their self-sacrifice that allows not only this thesis to be printed but also all the books, manuscripts, journals, and thank you - cards I used over the past half a decade.

***** Stephanie Lange Oregon State University May 2019 CONTRIBUTION OF AUTHORS

Dr. Ben Mason assisted with earthquake engineering, formulating parameter ranges, the interpretation of data, and editing. Dr. Michael Scott was involved with the numerical modeling, the sensitivity analysis, the interpretation of data, and editing. Dr. Scott Ashford assisted with the overall research topic, research question ﬁnd- ing, interpretation of data, and the application of the research results to an existing tunnel. TABLE OF CONTENTS Page 1 INTRODUCTION 1

2 ANALYSIS OF CONCRETE-LINED TUNNELS CROSSING ACTIVE FAULTS - BACKGROUND AND MODEL SET UP 5 2.1 Concentration on concrete structure ...... 5 2.1.1 Abstract ...... 5 2.1.2 Introduction ...... 6 2.1.3 Background ...... 8 2.1.4 Basic features of numerical model ...... 10 2.1.5 Preliminary results ...... 18 2.1.6 Conclusions ...... 20 2.2 Concentration on fault and ground conditions ...... 22 2.2.1 Abstract ...... 22 2.2.2 Introduction ...... 23 2.2.3 Basic features of numerical model ...... 24 2.2.4 Preliminary results ...... 27 2.2.5 Conclusions ...... 28

3 PARAMETRIC STUDY OF CONCRETE-LINED TUNNELS CROSSING AC- TIVE FAULTS 29 3.1 Abstract ...... 29 3.2 Introduction ...... 30 3.3 Material and methods ...... 32 3.3.1 Finite element model ...... 32 3.3.2 Fault model ...... 35 3.4 Numerical experimentation ...... 38 3.4.1 Parametric study setup ...... 38 3.4.2 Veriﬁcation and convergence ...... 39 3.4.3 Input parameters ...... 42 3.4.4 Assessment strategy ...... 44 3.4.5 Results ...... 44 3.5 Discussion ...... 46 3.6 Conclusions ...... 55 TABLE OF CONTENTS (Continued) Page 4 SENSITIVITY ANALYSIS 58 4.1 Abstract ...... 58 4.2 Introduction ...... 58 4.3 Theory ...... 59 4.4 Results ...... 60 4.4.1 Concrete and steel strength ...... 60 4.4.2 Tunnel geometry ...... 64 4.5 Results with varying material properties ...... 67 4.6 Discussion ...... 69 4.7 Conclusions ...... 70

5 STATIC FAULT OFFSET ASSESSMENT OF THE CONCRETE LINING OF THE BERKELEY HILLS TUNNELS 72 5.1 Abstract ...... 72 5.2 Introduction ...... 73 5.3 State of research ...... 75 5.3.1 Active faults ...... 75 5.3.2 Case histories of damages to tunnels through active faulting . 77 5.4 Assessment strategies ...... 81 5.4.1 Assessment with limit states and correlations ...... 81 5.4.2 Assessment set-up with charts ...... 83 5.4.3 Assessment set-up for numerical analysis ...... 84 5.5 Example study Berkeley Hills Tunnels ...... 85 5.5.1 Overview ...... 85 5.5.2 Geology ...... 88 5.5.3 Tunnel design ...... 89 5.5.4 Tunnel construction ...... 90 5.5.5 Assessment results with charts ...... 93 5.5.6 Assessment results with numerical analysis ...... 94 5.6 Discussion ...... 96 5.7 Conclusions ...... 99 TABLE OF CONTENTS (Continued) Page 6 CONCLUSIONS 101

BIBLIOGRAPHY 106

APPENDICES 113

Appendix A - Industry input 114

Appendix B - Correlation of surface displacements to moment magnitude 116

Appendix C - Strain along the tunnel alignment 127 LIST OF FIGURES Figure Page 2.1 Schematics of 2D baseline model. Plan view of spring-beam model with cross-section and vertical view of tunnel crossing strike-slip fault. 12

2.2 Calibration plot of beam-spring models with varying number of springs, with maximum displacement under a single point load as calibration parameter. Chosen: 19 springs over a tunnel length of 200 m results in 10 m element length between two springs...... 15 2.3 Graph above: Schematic of observed strike-slip fault offset of the San Andreas Fault after 1906 earthquake with u = 5 m offset. Graph below: Simpliﬁcation of strike-slip fault for computation. Offset, u, is exaggerated...... 16 2.4 Schematic tunnel beam displacement with 20 m fault zone width and different fault offset values: 25.80 m for earthquake M8.0, 10 m for M7.4, 2.0 m for M7.0, 1.0 m for M6.8, 0.50 m for M6.4, and 0.07 m for M6.0 (medium strong rock properties along the tunnel alignment outside the fault zone)...... 19

2.5 Schematic tunnel beam displacement for medium strong rock properties along the tunnel outside the fault zone; with 2.0 m fault offset for varying fault zone widths of 10 m, 20 m, and 100 m...... 20

2.6 Schematic 2D tunnel beam model. (a) plan view with offset, (b) vertical view...... 26 2.7 Schematic tunnel beam displacements with 2 m fault offset and varying fault zone width: 1 m, 10 m, 20 m, and 100 m...... 27

3.1 2D model: (a) Plan view of tunnel beam with perpendicular offset input. Springs are modeled on both sides, but only shown on one side for graphical reasons. (b) Schematic representation of the tunnel cross-section: The tunnel is embedded in rock. The lining is divided into fiber sections. Road, structural fill, and vehicle are only shown schematically for a Di = 10 m tunnel...... 34 3.2 (a) Schematic strike-slip offset, (b) Schematic of the modeled simplification of the strike-slip fault offset in combination with the displaced tunnel beam. Offset, u, is exaggerated. x-axis scales differ. . . 37 LIST OF FIGURES (Continued) Figure Page 3.3 Graphical tunnel beam model with (a) narrow and (b) wide fault zone widths...... 37

3.4 Parametric study chart showing the general sequence of operations from left to right...... 38

3.5 Convergence of fibers of the tunnel cross-section for a 300 m long tunnel beam-spring model, with 1 m element lengths, loaded with a 100 MN point load, P, at the center. Bold marker symbolizes chosen model with 4 radial and 16 circumferential subdivisions resulting in 64 fiber sections with a numerical model displacement value of 13.68 mm. Maximal deflection after Hetenyi´ (1946) results in 13.37 mm...... 42 3.6 Discretization of element length for a 300 m long tunnel beam-spring model, loaded with a 100 MN point load, P, at the center. The numerical model displacement computes to 13.68 mm. Maximal deflection after Hetenyi´ (1946) results in 13.37 mm...... 43 3.7 Axial force - moment (N-M) interaction diagrams for baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a fault zone width of 30 m, and varying rock characteristics along the tunnel. Image insertion shows a stress-strain curve of concrete with various strain and stress levels...... 46

3.8 Displacements along the tunnel axis for the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width...... 47 3.9 Moments along the tunnel axis for the baseline model with 1/20- ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width...... 48 3.10 Shear forces along the tunnel axis for the baseline model with 1/20- ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width...... 49 LIST OF FIGURES (Continued) Figure Page 3.11 Strain along the tunnel axis for the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width. tm = Tunnel meter...... 50 3.12 Strain across the tunnel cross-section at tunnel meter (tm) -3 of the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, and ks = 100 MPa...... 50 3.13 Displacement curves for baseline model in extremely weak rock with varying fault zone widths...... 51 3.14 Strain levels of RC tunnel cross-section in a plot of geometric ratio versus threshold displacements for a 30 m wide fault zone with varying inner tunnel diameter and ground conditions along the tunnel alignment...... 53 3.15 Strain levels of RC tunnel cross-section in a plot of fault zone widths versus threshold displacements for a geometric ratio of 1/20 with varying inner tunnel diameter and ground conditions along the tunnel alignment...... 54

3.16 Strain levels of RC tunnel cross-section in a plot of geometric ratio versus threshold displacements for a 30 m wide fault zone with steel strength of 420 MPa on the left and 275 MPa on the right...... 56

4.1 Sensitivity analysis for steel strength, fy, for the baseline model. (a) 3D image of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the steel strength versus the displacement. (d) Change in displacement with respect to the steel strength along the tunnel axis...... 61

4.2 Sensitivity analysis for concrete strength, fc, for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the concrete strength versus the displacement. (d) Change in displacement with respect to the concrete strength along the tunnel axis...... 63 LIST OF FIGURES (Continued) Figure Page 4.3 Sensitivity analysis for lining thickness, h, for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the lining thickness versus the displacement. (d) Change in displacement with respect to the lining thickness along the tunnel axis...... 65 4.4 Sensitivity analysis for the outer tunnel diameter for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the diameter versus the displacement. (d) Change in displacement with respect to the diameter along the tunnel axis...... 66

5.1 Topographic map of the San Francisco Bay area, California, USA by U.S. Geological Survey (2017) with location of the Berkeley Hills Tunnels and two major faults, San Andreas and Hayward Fault. . . . 74 5.2 (a) Strike-slip fault with a vertical slip plane, (b) Varying geology of a fault zone with one vertical fault plane. PSZ: Principle Slip Zone. (after Sibson (2003)) ...... 76 5.3 Damages of tunnel No. 5 of the Southern Paciﬁc Railroad documented after 1952 Bakersﬁeld earthquake: (a) Collapsed section, looking west, about 53 m inside the east portal, (b) major damage to concrete tunnel lining due to faulting. (Steinbrugge and Moran, 1954) ...... 78

5.4 Damages of the Longxi road tunnels documented after the 2008 Wenchuan earthquake (M7.9), China: (a) Collapsed section at the fault crossing, (b) major damage to concrete tunnel lining due to about 1 meter fault offset. (McRae et al., 2014; Yu et al., 2016) ...... 79

5.5 Plan view of the retrofit section of the Claremont water tunnel before and after 2.6 m (8.5 feet) localized fault offset (Caulfield et al., 2005). HFZ: Hayward Fault Zone...... 80 LIST OF FIGURES (Continued) Figure Page 5.6 Computer model visualization. (a) Plan view of 2D beam-spring model with fault offset. Springs are shown on one side for clarity, but modeled on both sides. (b) Tunnel cross-section A-A with discretized fiber sections of the concrete lining. Vehicle and installation material are shown schematically...... 86 5.7 Crack pattern of tunnel lining due to fault creep compared to geology (Brown et al., 1981, p. 90) ...... 87 5.8 Summarized geology across Hayward Fault along the BHT...... 88

5.9 Cross-section of BART Berkeley Hills tunnels with the liner design Brown et al. (1981). All dimensions in mm but U.S. steel sizes. E.g., the #9 bar has a nominal diameter of 28.65 mm...... 91

5.10 Use of timber and steel struts during construction of the BHT (McRae et al., 2014) ...... 92

5.11 Chart excerpt from chapter 3 with BHT data, h = 0.457 m with varying ground condition. One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points...... 94

5.12 N-M-interaction diagram for the BHT cross-section with reinforced concrete, an diameter of 5.33 m, and a lining thickness of 0.457 m. . . 95 5.13 (a) Displacements, (b) strains, (c) shear forces, and (d) moments along the tunnel axis for the BHT with Di = 5.33 m and h = 0.457 m for Lzone = 30 m (fault zone width), with varying geology outside the fault zone. One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points...... 97

5.14 Displacement vs. lining thickness for the BHT with correlation to earthquake magnitudes. Results from Figure 5.11 are shown as estimated data points with a black c One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points.ircle...... 98 LIST OF TABLES Table Page 3.1 Correlated earthquake magnitudes to maximum surface displacements after Wells and Coppersmith (1994). One standard deviation of earthquake magnitude, 1σ = 0.34 unit points...... 55

4.1 Deﬁnition of parameter sets for sensitivity analyses ...... 68 4.2 Response sensitivity results for parameter sets ...... 69 LIST OF APPENDIX FIGURES Figure Page 1 Plot of maximum surface displacement vs. moment magnitude of all selected data points including the regression from Wells and Cop- persmith (1994) and the adjusted regression...... 120 2 Plot of average surface displacement vs. moment magnitude of all selected data points including the regression from Wells and Cop- persmith (1994) and the adjusted regression...... 121 3 Comparison of surface displacements from adjusted regression equation over surface displacement regression equation from Wells and Coppersmith (1994) to correlated moment magnitudes...... 122

4 (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.2 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 129

5 (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.24 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 130

6 (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.3 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 131

7 (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.4 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 132

8 (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.33 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 133

9 (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.4 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 134

10 (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.5 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 135 LIST OF APPENDIX FIGURES (Continued) Figure Page

11 (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.67 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 136

12 (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.5 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 137

13 (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.6 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 138

14 (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.75 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 139

15 (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 1.0 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter...... 140 LIST OF APPENDIX TABLES Table Page 1 Tunnel dimensions ...... 115

2 Maximum surface displacements correlated with moment magnitudes...... 119

3 Average surface displacements correlated with moment magnitudes. 120 4 Range of moment magnitudes for comparison of Wells and Copper- smith (1994) max. surface displacement regression equation with the adjusted regression equation...... 123 5 Range of moment magnitudes for comparison of Wells and Copper- smith (1994) average surface displacement regression equation with the adjusted regression equation...... 123 6 Strike-slip earthquake data from Wells and Coppersmith (1994) and Wesnousky (2008). Strike-slip type RL = right lateral, LL = left lateral. Displacements max., MD = maximum displacement, avg., AD = average displacement...... 124

7 Strike-slip data from Wells and Coppersmith (1994) and Wesnousky (2008), continued from Table 6 ...... 125 8 Strike-slip earthquake data from Wesnousky (2008), continued from Table7 ...... 126 DEDICATED to my husband, our daughter, and my parents 1 INTRODUCTION

Urbanization of large cities with a dense population results in increased use of the underground for storage, shopping, and transportation. Many large cities, especially on the west coast of the United States, are built in seismically active regions (e.g., Seattle, San Francisco Bay area, and Los Angeles). As a result, it is important to consider the effects of active faulting on urban underground structures in large cities. Earthquake-induced large ground deformations, such as liquefaction, landslides, or fault displacements, can permanently damage underground structures (e.g., Hashash et al., 2001; Ha et al., 2008; ORourke et al., 2015). Seismic events, including the 1952 Bakersfield earthquake in California, with a moment magnitude (M) of 7.5, the 1999 Kocaeli (M7.4) and 1999 Duzce¨ (M7.2) earthquakes in Turkey, and the 2008 Wenchuan earthquake (M7.9) in China damaged several underground structures crossing active faults (Kontogianni and Stiros, 2003; Ulusay et al., 2002; McRae et al., 2014; Yu et al., 2016). General categories of seismic impacts on underground structures, such as tunnels, are (1) ground shaking; and (2) ground failure. The term ground failure usually refers to liquefaction, slope instability, and fault displacement, among others (Hashash et al., 2001). The main design considerations from U.S. guidelines for tunnels are based on wave propagation and ground motion. However, ground shaking has contributed to less damage compared to severe damage from large 2 permanent ground displacements due to ground failure. Prior tunneling research suggests designing underground structures to withstand small predicted fault displacements and to account for repairs of damaged structural elements after an earthquake event (Hung et al., 2009). Many researchers and practitioners use simplified beam-spring models, finite element (FE) methods, or finite difference (FD) methods to perform numerical modeling for tunnels (e.g., Anastasopoulos et al. (2008); Corigliano et al. (2011); Wang et al. (2012)). Although tunnel engineers have been working on earthquake-induced failures for decades, much of the knowledge is proprietary or limited to site-specific cases. To address the lack of a generalized interpretation framework for the influence of fault rupture on tunnel structures, the focus of this research is to evaluate the response of tunnels due to fault rupture displacements. The selection of the tunnel lining structure, material, and thickness all contribute to the response of the tunnel in an earthquake event with a considerable amount of fault offset. Many parts of a tunnel system are affected by seismic excitation, such as the portals, utility systems, and emergency exits. However, this research concentrates on the analysis of concrete-lined circular tunnels crossing active faults under fault offset. Here, failure of an underground structure is considered to be a structural failure. Ground failure is outside of the scope of this research. The present research investigates tunnel deformations and internal stresses and strains in the reinforced concrete (RC) lining based on a varying geological environment and tunnel geometry. The work is accomplished by numerical analyses and parametric studies using the finite element framework within the Open Sys- 3 tem for Earthquake Engineering Simulation (OpenSees) (McKenna et al., 2010). In- sights from the parametric studies may provide researchers and practitioners with important information. This dissertation will give considerations for the design of tunnels during early stage projects, like a feasibility assessment. The research is based on three main research questions: (1) Can we give guidance when it is acceptable for a tunnel to cross an active fault? (2) What is the influence of faulting on the circular tunnel lining in the cross-section, but also along the longitudinal tunnel axis? and (3) What consequences and generalizations can be drawn to support serviceability of the tunnel after an event? This dissertation is a collection of various manuscripts that address the research questions. The 2D model described in chapter 2 is comprised of two conference papers, which are published in their respective proceedings. The chapter is based on linear-elastic material behavior. The preliminary results and discussions during this chapter vary, depending on the ground conditions and the tunnel design characteristics. To inform modeling decisions, practical ranges of tunnel parameters were informed by half a dozen engineering design and construction companies in the U.S. The results from the inquiry are available in appendix A. Correlations from surface rupture displacements to earthquake magnitudes are based on Wells and Coppersmith (1994). Appendix B describes an adaption of the Wells & Cop- persmith formulation with additional earthquake data from Wesnousky (2008). Chapter 2 uses this adaption. Because of the large standard deviation of the Wells & Coppersmith regression, in to which the adapted regression falls, chapter 3 and onward use the well-known and established Wells & Coppersmith formulation 4 without adaption. Chapter 3 is an unpublished manuscript about the parametric study of the 2D model of a deep tunnel with non-linear behavior of concrete and steel materials. Further, it contains convergence and verification analyses, as well as an extended discussion of the results with multiple plots of displacements, correlating earthquake magnitudes with varying geometric ratios and fault zone widths. To determine the controlling tunnel beam element that reaches concrete failure first, an exhaustive evaluation of strains along the tunnel alignment was conducted and is placed in appendix C. The sensitivity analysis in chapter 4 identifies model sensitive variables. In detail, it examines material parameters from concrete and steel, as well as the tunnel geometry with lining thickness and inner tunnel diameter. Chapter 5 is an unpublished manuscript; it discusses an example application of the research to an existing tunnel - the Berkeley Hills tunnels. This chapter con- nects the results from the academic study to an engineering problem for an early stage assessment. The conclusions in chapter 6 summarize the major findings from each chapter, list limitations of this research, and connect to future work. 5

2 ANALYSIS OF CONCRETE-LINED TUNNELS CROSSING

ACTIVE FAULTS - BACKGROUND AND MODEL SET UP

Lange, S., Mason, H. B., Scott, M. H., and Ashford, S. A. (2018). Analysis of Concrete-Lined Tunnels Crossing Active Faults. North American Tunneling 2018 Proceedings, pg. 11-18. (The published conference paper has been reformatted from the original version for inclusion in the thesis.)

2.1 Concentration on concrete structure

2.1.1 Abstract

It is known that earthquakes with magnitudes greater than M6.0 can cause sig- niﬁcant damage to tunnels. In particular, large strains due to fault offsets lead to severe damage in the tunnel lining, (e.g., concrete spalling), which can lead to potential closure and disruptions to the transportation network. Examining the response of the soil-structure interaction between tunnels and active fault zones, where damaging earthquake ruptures with signiﬁcant offsets are possible, is critical to ensure resilient design and safe operation. Preliminary results of a calibrated 6

2D model of a circular reinforced concrete lined tunnel crossing an active fault show earthquake magnitude, fault geology, and structural properties of the tunnel are inﬂuential parameters. These results imply that novel tunnel design strategies are necessary.

2.1.2 Introduction

Urbanization of large cities with a dense population results in an increased use of the underground for storage, shopping, and transportation. Fast growing urban areas depend on transportation of individuals, which is one of the top priorities for city authorities. The construction of tunnels for individual traffic allows cities to grow and support their citizens lives. Many large urban areas on the west coast of the United States are built in seismically active regions (e.g., Los Angeles, San Francisco Bay area, Seattle), and as a result, it is important to consider the effects of active faulting on tunnel design. In general, seismic impacts on tunnels can be categorized into two groups: (1) ground shaking; and (2) ground failure. The term ground failure usually refers to liquefaction, slope instability, and fault displacement, among others (Hashash et al., 2001). The primary focus of this study is on the latter group, ground failure due to fault displacements. During fault rupture, large strains in the tunnel lining can cause spalling and potential closure. Accordingly, building tunnels through active fault zones that create damaging earthquakes is risky and is not considered during the planning phase of a tunnel project. In fact, conventional wisdom is to not construct tunnels 7 through fault zones, if at all possible, and to find alternative routes. If alternative routes are not available, especially in urban areas or in cases of vital infrastructure, prior research suggests to design the underground structure to withstand small predicted fault displacements and to account for repairs of damaged structural elements after an earthquake event (Hung et al., 2009; Hashash et al., 2001). Predicting the response of a tunnel crossing a seismically active fault zone is complex. Along with long-term maintenance, serviceability of a tunnel after an earthquake event is among the highest priorities for stakeholders, clients, and the public. The selection of the tunnel lining structure, material, and thickness all contribute to the response of the tunnel structure in an earthquake event with a considerable amount of fault offset. Many parts of a tunnel structure are affected by seismic excitation, such as the portals, water and ventilation systems, electronic systems, emergency exits, above ground buildings, and neighboring underground structures. However, this research will concentrate on the analysis of concrete- lined circular tunnels crossing active faults under large fault offset; specifically, the response of structural and geotechnical elements will be analyzed. The contribution to knowledge and understanding of tunnel engineering specifically under severe seismic and geological conditions such as active fault offsets will be accomplished by means of numerical analyses and parametric studies using the finite element framework OpenSees (McKenna et al., 2010) with appropriate constitutive models, boundary conditions, and initial conditions. Insights from the parametric studies may provide researchers and practitioners with important information on considerations for the design of tunnel projects. 8

2.1.3 Background

A thorough literature review on the subject of concrete-lined tunnels crossing active faults reveals that the topic is a relatively young research area with great future potential. Although tunnel engineers have been working on the topic for decades, much of the knowledge is still proprietary or confined to site-specific cases. In summary, the reviewed literature consisted of reconnaissance reports of damaged tunnels, numerical case studies of specific future tunnels, tunnels under construction, or in retrospect, and design approaches to accommodate active fault offsets. We will utilize the findings of the reconnaissance reports to inform our numerical modeling efforts. Notably, in the context of this paper, failure of an underground structure is considered to be a structural failure, cave-in, or otherwise unsafe for human use, requiring closure and repair or replacement. Kontogianni and Stiros (2003) compiled a list of tunnels affected by earthquakes with fault offsets over the previous 100 years. Their report shows underground structures such as railroad tunnels (mostly built during the nineteenth century) that cross active faults were damaged during earthquake rupture. They also discussed that road tunnels built after 1930 were damaged during earthquake rupture. In particular, the 1989 Loma Prieta earthquake (moment magnitude M6.9), the 1995 Kobe, Japan earthquake (M6.8), and the 1999 Duzce¨ (M7.2), Turkey earthquake damaged several underground structures crossing active faults. 9

Ulusay et al. (2002) reported on damage that occurred to several road and rail tunnels during the 1999 Kocaeli (M7.4) and Duzce¨ earthquakes in Turkey. Lit- tle damage occurred at the portals of the short TEM (Trans European Motorway) tunnels on the northern side of the Izmit Gulf in Turkey due to the 1999 Kocaeli earthquake. On the opposite side of the Gulf, the Bolu tunnels suffered active fault shear failure from the Duzce¨ earthquake while under construction, as well as ovaling due to the increase of stresses from additional dynamic loads during the earthquake event. Prentice and Ponti (1997) reported that the Wrights Tunnel beneath the Summit Ridge in the southern Santa Cruz Mountains, California was severely damaged with cave-ins and an offset of 1.5 m from the San Andreas Fault due to the 1906 San Francisco earthquake. Similar to the Bolu tunnels in Turkey, the damage on the Wrights tunnel was very localized. Desai et al. (1991) discuss design considerations for the Los Angeles water tunnels crossing active faults. The authors give special attention to ﬂexible joints between single tunnel elements to accommodate differential movements and to keep leakage at a minimum. Construction through fault zones is a difﬁcult task as seen with the Izmir Metro Tunnel in Turkey. Kun and Onargan (2013) discuss the excavation and construction of the metro tunnel through a fault zone, which raised problems for above-ground structures due to poor rock characteristics. The construction team used additional bolts, face nails, and jet grouting to strengthen the existing tunnel support system for safer above-ground structures. The construction through the geology of a fault zone is a critical point in the process of building 10 a tunnel, but not further discussed in this writing.

2.1.4 Basic features of numerical model

Numerical modeling of tunnel response is an important design tool, because the soil-structure interaction between the tunnel lining and the surrounding soil is complicated. Many researchers and practitioners use simplified beam-spring models, finite element (FE) methods, or finite difference (FD) methods to perform numerical modeling for tunnels (e.g., Keshvarian et al. (2004); Shahidi and Vafaeian (2005); Caulfield et al. (2005); Gregor et al. (2007); Anastasopoulos et al. (2008); Lan- zano et al. (2008); Corigliano et al. (2011); Wang et al. (2012); Luo and Yang (2013); Varnusfaderani et al. (2015)).

2.1.4.1 Modeling structure

Based on practical ranges of specific parameters, a baseline model was specified in this study. Figure 2.1 shows a schematic of the baseline model. Note that in Figure 2.1, the springs are shown only on one side of the tunnel to allow a cleaner image. During numerical modeling, the springs are modeled on both sides of the tunnel beam. The numerical model represents a simplified plan view of the tunnel interacting with the movement of a strike-slip fault rupture. 11

Baseline model characteristics:

• M7 earthquake with a fault offset of 2.0 m;

• Strike-slip fault with a dip angle of 90, which is characteristic for a vertical fault plane and an important characteristic of the San Andreas and Hayward Faults in the San Francisco Bay area;

• Rock properties for the surrounding ground of medium strengths (σ = 50 MPa), and rock properties of weak strengths to simulate the fault gouge (σ = 0.25 MPa);

• Homogeneous ground;

• Fault offset occurs on a single fault rupture plane;

• 90◦ crossing angle of the tunnel structure perpendicular to the fault plane;

• Circular tunnel cross-section in drained conditions;

• Ratio of tunnel lining thickness, h, to inner tunnel diameter, Di, of 1/20 , with

h = 0.30 m and Di = 6.0 m;

0 • Normal concrete strength of C25/30 with characteristic cylinder strength fc 0 of 25 MPa at 28 days (corresponding to fc of 4000 psi);

• Steel strength of 275 MPa (corresponding to U.S. steel grade 40);

• Tunnel length, L = 200 m, with fault offset in middle area. 12

Plan view: y Tunnel cross-section: D = 6.0 m Fault offset 2.0 m ratio 1/20 i h = 0.3 m 2.0 m x L = 200 m

Crossing angle 90°

Fault plane Tunnel x

Vertical view: z Fault plane with dip angle 90° to the horizontal axis

Tunnel

Figure 2.1: Schematics of 2D baseline model. Plan view of spring-beam model with cross-section and vertical view of tunnel crossing strike-slip fault.

The open-source, ﬁnite-element software framework OpenSees was chosen to simulate the tunnel response under imposed ground displacements. OpenSees is able to model simple 1D elements, like beams and springs, as well as 2D plane strain elements with a variety of constitutive models. Further, OpenSees is able to apply multiple point displacements simultaneously to simulate offsets at different locations of a tunnel crossing a fault zone. The scripting capabilities of OpenSees will be utilized to conduct parametric studies with varying elements including fault characteristics such as fault offset and fault zone width. To inform the modeling decisions, information about practical ranges of spe- ciﬁc tunnel parameters were given by half a dozen tunnel engineering design and 13 construction companies in the United States. The primary consensus of the parameters was:

• Lining thickness-to-inner diameter ratio ranging from 1/15 to 1/30, with 1/20 being the most found ratio;

0 • Concrete cylinder strengths fc between 25 MPa and 40 MPa (3500 psi to 6000 psi);

• Steel tensile strengths between 275 MPa and 500 MPa (grade 40 and 75);

• Initial support could be considered as structurally bearing, but is generally not considered due to quality issues with shotcrete during construction.

Together with the industry input, the structural model resulted in the following numerical elements:

• Force-based beam elements for the tunnel structure;

• Zero length elements for the soil springs;

• Circular ﬁber section for the tunnel cross section with inner and outer rein- forcements.

For the purpose of obtaining preliminary results, the concrete and steel materials are defined as elastic. The spring materials are defined as compression only with no tension. The numerical model was validated with known closed-form analytical solutions of simple beams, such as maximum moment and deflection of beam with 14 single point load, reaction forces and maximum moment of beam with prescribed vertical displacement of one end support, or maximum moment and deflection of beam on elastic ground after Hetenyi´ (1946). To further calibrate the numerical model, the radial and circumferential discretization of the concrete lining in the tunnel cross section was varied to obtain adequate results without the need for excessive computations. The same procedure was performed with the number of springs (cf. Figure 2.2), or distance between two beam nodes along the tunnel length. The distance between two beam nodes was set to 10 m after calibration, which coincides with a typical length of a cast-in-place tunnel lining segment.

2.1.4.2 Modeling fault

The ground offset is based on a strike-slip fault. Therefore, the numerical model is a horizontal view of the tunnel cutting through the strike-slip fault. Kasahara (1981) found that the displacement ﬁelds associated with a strike-slip fault can be viewed as a shear dislocation. Surface displacement measurements, such as from the San Andreas Fault after the 1906 San Francisco earthquake (M7.8), show that the strain accumulation away from the fault goes to zero at around 20 km (Figure 2.3). With general tunnel lengths in the order of hundreds of meters to some kilometers, it can be assumed that the side of the tunnel that is left of the fault zone stays in the original position and the side right of the fault zone will be dislocated according to the fault offset. This detail can be seen in the Figure 2.1 showing the schematics of the 2D baseline model. The fault offset is introduced 15

-0.15 Beam-spring model with 7 springs: -0.2

-0.25

-0.3

-0.35

-0.4

-0.45

-0.5 7 Springs

Maximum displacement at midpoint, d (mm) -0.55

-0.6 19 Springs -0.632 mm

-0.65 0 5 10 15 20 25 30 35 40 45 50 Number of springs

Figure 2.2: Calibration plot of beam-spring models with varying number of springs, with maximum displacement under a single point load as calibration parameter. Chosen: 19 springs over a tunnel length of 200 m results in 10 m element length between two springs. into the numerical model as nodal displacements of one side of the tunnel. Figure 2.3 shows the preceding modeling simplification. This simplification of offset in one direction instead of half the offset in each direction as observed in strike-slip faulting, puts the same relative displacement on the structure, but may influence the reaction of the surrounding ground, which 16

Observed strike-slip (m) fault offset

km 10 20 km

Strike-Slip Fault

Modeled strike-slip fault offset (m)

km 20 40 km

Figure 2.3: Graph above: Schematic of observed strike-slip fault offset of the San Andreas Fault after 1906 earthquake with u = 5 m offset. Graph below: Simplifica- tion of strike-slip fault for computation. Offset, u, is exaggerated. will be examined in the parametric studies. To understand what a specific fault offset displacement means in terms of the severity of an event, these displacements are correlated to earthquake magnitudes. Herein, we assume that the minimum moment magnitude required to cause concrete damage to tunnels is M6.0. The literature on correlations of various rupture parameters was analyzed to describe earthquake magnitudes with respect to specific fault offsets relatively close to the earths surface where tunnels are typi- cally located. The literature review resulted in a conservative adoption of the well- known correlations of Wells and Coppersmith (1994) with additional earthquake data from more recent events from Wesnousky (2008). The following formula- 17 tions were used to correlate the parameters, maximum and average fault offset and earthquake magnitude:

M = 6.712 + 0.912 · log(MD) M = 6.996 + 0.948 · log(AD) (2.1) where MD is the maximum surface displacement in meters (m), and AD is the average surface displacement in meters (m). Notably, to simplify the modeling of static fault rupture, the propagation of the rupture path from the bedrock through the overburden soil to the surface is not included in this study.

2.1.4.3 Modeling rock material

We consulted the tunneling professionals to determine the type of ground that tunnels through seismically active areas normally traverse and found that tunnels built in earthquake zones in the Paciﬁc Coast area are mainly in softer ground, such as sandy/gravelly soils, up to medium strong rock materials. The preceding knowledge formed the basis of the range stiffness of the rock surrounding the tunnel, which is modeled with springs. The springs are deﬁned as elastic no-tension materials. Since the model is a horizontal cut, it is assumed that there is no gap between the tunnel lining and the surrounding ground, as it might be above the crown of a tunnel. The following parameters used in the analyses are based on Barton (1978), Johnson and DeGraff (1988), and Marinos and Hoek (2001): 18

• Fault gouge with uniaxial compressive strength of 0.25 MPa and spring stiff-

ness of Eg between 0.2 - 2 GPa

• Clayey sandstone or shale with uniaxial compressive strength of 5 MPa and

Eg between 2 - 30 GPa

• Sandstone with uniaxial compressive strength of 50 MPa and Eg between 30 - 80 GPa

The uniaxial compressive strengths are given as an indicator for fault gouge material, low strength, and medium hard rocks. The foregoing values indicate three different levels of possible ranges of ground conditions along the Paciﬁc Coast of the United States and are not exact measurements.

2.1.5 Preliminary results

Preliminary results in Figures 2.4 and 2.5 show threshold displacements of the tunnel structure at different fault zone lengths and with different amounts of fault offsets. Preliminary results show a direct influence between the amount of fault offset and the quantity of strain in the tunnel lining, which is anticipated. Further, the preliminary results show an influence of fault zone length beyond the fault zone in terms of strains in longitudinal direction of the tunnel alignment. With longer fault zones, this result diminishes; the strains in the tunnel elements beyond the 19 fault zone do not show a long influence for a longer fault zone, which means that the strains and therefore damages to the tunnel lining are very localized. This effect still has to be verified by comparison with short fault zones.

35 0.07 m Fault offset 30 0.5 m 1 m 25 2 m 10 m 20 25.8 m

5 Transverse displacement (m)

-5 0 20 40 60 80 100 120 140 160 180 200 Tunnel length (m) Figure 2.4: Schematic tunnel beam displacement with 20 m fault zone width and different fault offset values: 25.80 m for earthquake M8.0, 10 m for M7.4, 2.0 m for M7.0, 1.0 m for M6.8, 0.50 m for M6.4, and 0.07 m for M6.0 (medium strong rock properties along the tunnel alignment outside the fault zone).

These preliminary results are in an early stage of research and need further evaluation. Research is ongoing and further results are forthcoming. 20

2.5

2 10 m Fault Zone Width 20 m 1.5 100 m

0.5

Transverse displacement (m) -0.5 0 20 40 60 80 100 120 140 160 180 200 Tunnel length (m) Figure 2.5: Schematic tunnel beam displacement for medium strong rock properties along the tunnel outside the fault zone; with 2.0 m fault offset for varying fault zone widths of 10 m, 20 m, and 100 m.

2.1.6 Conclusions

This research focuses on the structural lining and soil-structure-interaction of a circular tunnel and aims to expand the knowledge on the subject. First steps to accomplish the main goal of understanding the behavior of concrete-lined circular tunnels crossing active faults were taken by varying fault offset and with it the earthquake magnitude impacting the tunnel lining. Behavioral insights from the parametric studies may provide researchers and practitioners with important information on considerations for the design of tunnel projects. Ulusay et al. (2002) described active fault damages in the existing Bolu tunnels and found that the damages were very localized and not dangerous to the long-term stability of tunnel structures. This could coincide with the preliminary results of a larger fault zone with more localized strain accumulation in the tunnel lining. A construction solution to accommodate fault offset with the knowledge of a 21 larger fault zone and localized damages was analyzed by Russo et al. (2002). These authors analyzed the design of the rehabilitated and ﬁnished Bolu tunnels, which uses the articulated design solution with seismic joints in the concrete lining to accommodate a fault offset of 50 cm over a length of 100 m. The case study of a previously damaged tunnel, such as the Bolu tunnels with an adaption of the ﬁnal lining is an example of executed earthquake engineering.

Beyond the scope of this research study, which is still ongoing, are considered parameters for improvements in future work.

• Varying the dip angle of fault from 0◦, 30◦, up to 90◦;

• Including different fault types: dip-slip normal, reverse, thrust faults;

• Simulating static and dynamic (creep) fault displacement for numerical modeling;

• Combine static and dynamic fault displacement and dynamic tunnel shaking due to earthquake waves;

• Consider the propagation of rupture path from the bedrock through the overburden soil to the surface interacting with the tunnel.

Investigation of this last point in connection with a future immersed tunnel had begun by Anastasopoulos et al. (2008), but further research is necessary to understand the underlying behavior of underground structures in soft ground. 22

Lange, S., Mason, H. B., Scott, M. H., and Ashford, S. A. (2018). Analysis of concrete-lined tunnels crossing active faults. Proceedings of the 11th National Con- ference in Earthquake Engineering, Earthquake Engineering Research Institute, Los Angeles, CA. 2018. (The published conference paper has been reformatted from the original version for inclusion in the thesis.)

2.2 Concentration on fault and ground conditions

2.2.1 Abstract

A challenge facing municipal authorities in growing urban areas is providing larger and faster transportation networks that are safe and resilient to significant disruptions after an earthquake event and other disasters. The construction of tunnels for individual traffic and mass transit through active fault zones became a necessity despite the risks. It is now known that earthquakes with magnitudes greater than M6.0 can create significant damage to tunnels in seismically active areas. In particular, large strains due to fault offsets and ground shaking lead to unacceptable stresses in the tunnel lining, which can cause subsequent spalling and potential tunnel closure. Examining the response of concrete lined tunnels built through active fault zones, where damaging, large-magnitude earthquake ruptures with significant fault offsets are possible, is critical to ensure resilient design and safe 23 operation. A parametric study with a 2D beam-spring model with varying ground and structural properties is performed with OpenSees. The effects of earthquake magnitude and geometry of the fault area are studied and assessed. Preliminary results imply a relatively localized area of damage to the tunnel structure.

2.2.2 Introduction

Many large urban areas on the west coast of the United States are built in seismically active regions (e.g., Los Angeles, San Francisco Bay area, Seattle), and as a result, it is important to consider the effects of active faulting on urban underground structures connecting large cities. General categories of seismic impacts on underground structures, such as tunnels, are (1) ground shaking; and (2) ground failure. The term ground failure usually refers to liquefaction, slope instability, and fault displacement, among others (Hashash et al., 2001). The primary focus of this study is on ground failure due to fault displacements. In the context of this paper, failure of an underground structure is considered to be a structural failure, cave-in, or otherwise unsafe for human use, requiring closure, repair or replacement. Prior tunneling research suggests designing underground structures to withstand small predicted fault displacements and to account for repairs of damaged structural elements after an earthquake event (Hung et al., 2009, FHWA). A literature review on the subject of active faults with crossing tunnels showed that engineers have been working on this specific problem for decades. But much of 24 the knowledge is still confined to site specific cases, such as a discussion about design considerations of Los Angeles water tunnels with specific flexible joints to accommodate differential movements (Desai et al., 1991) or reported damages that occurred to several road and rail tunnels during the 1999 Kocaeli and Duzce¨ earthquakes in Turkey (Ulusay et al., 2002). Notably, many researchers and practitioners use simplified beam-spring models, finite element (FE) methods, or finite difference (FD) methods to perform numerical modeling for tunnels (e.g., Anasta- sopoulos et al. (2008); Corigliano et al. (2011); Wang et al. (2012)). Our research concentrates on the analysis of active faults with large fault offsets being crossed by concrete-lined circular tunnels; specifically, the response of geotechnical and structural elements are analyzed. The work is accomplished by means of numerical analyses and parametric studies using the finite element framework OpenSees (McKenna et al., 2010). Insights from the parametric studies may provide researchers and practitioners with important information on considerations for the design of tunnel projects.

2.2.3 Basic features of numerical model

The predicted ground offset is based on strike-slip faults with magnitudes M6 to M8. Therefore, the numerical model is a plan view of the tunnel cutting through the strike-slip fault as seen in Figure 2.6. The fault offset is correlated to the earthquake magnitude to provide a general understanding of the impact. The literature 25 review resulted in a conservative adoption of the well-known correlations of Wells and Coppersmith (1994) with additional earthquake data from more recent events (Wesnousky, 2008). The following formulations were used.

M = 6.712 + 0.912 · log(MD) M = 6.996 + 0.948 · log(AD) (2.2) where MD is the maximum and AD is average surface displacement, respectively, both in meters. We consulted the tunneling profession to determine the type of ground that tunnels through seismically active areas normally traverse and found that tunnels built in earthquake zones near the Paciﬁc Coast of the United States are mainly in softer ground, such as sandy/gravely soils, up to medium strong rock materials. The following parameters of uniaxial compressive strengths and elastic modulus of rocks are given as an indicator for fault gouge material, low strength, and medium hard rocks:

• Fault gouge with uniaxial compressive strength of 0.25 MPa and spring stiff-

ness of Eg between 0.2 - 2 GPa

• Clayey sandstone or shale with uniaxial compressive strength of 5 MPa and

Eg between 2 - 30 GPa

• Sandstone with uniaxial compressive strength of 50 MPa and Eg between 30 - 80 GPa

The foregoing values indicate three different levels of possible ranges of ground conditions along the Paciﬁc Coast of the United States and are included in the para- 26 metric study. They are not exact measurements.

Plan view: y Tunnel cross-section: Displacement controlled D = 6.0 m fault offset, 2.0 m ratio 1/20 i h = 0.3 m 2.0 m x L = 200 m

Crossing angle 90°

Vertical Tunnel x (a) fault plane Vertical view: z Fault plane with dip angle 90° to the horizontal axis

Tunnel

(b) x Figure 2.6: Schematic 2D tunnel beam model. (a) plan view with offset, (b) vertical view.

Based on the practical engineering ranges of previously deﬁned parameters, a baseline model was speciﬁed in this study with the following characteristics. The baseline model is shown in Figure 2.6. The tunnel beam elements with linear- elastic concrete and steel properties, supported by Elastic No Tension (ENT) soil springs are subjected to a nodal displacement perpendicular to the tunnel axis representing a maximum offset possible until the capacity of material characteristics. The tunnel is assumed to be circular with homogeneous ground and in drained conditions, meaning no hydrostatic pressures are acting on the tunnel lining. 27

2.2.4 Preliminary results

The following graph in Figure 2.7 shows preliminary displacement results of the tunnel beam with a fault offset of 2.0 m and varying fault zone widths.

2.5

2 1 m Fault Zone Width 10 m 1.5 20 m 100 m 1

0.5

Threshold displacement (m) -0.5 0 20 40 60 80 100 120 140 160 180 200 Tunnel length (m) Figure 2.7: Schematic tunnel beam displacements with 2 m fault offset and varying fault zone width: 1 m, 10 m, 20 m, and 100 m.

A very narrow fault zone (1 to 10 m) might result in an abrupt but concentrated displacement regime. The preliminary results for wider fault zones (20 to 100 m) might give reason to assume very concentrated displacement regimes depending on the fault offset mechanism: offset on one end of the zone, as assumed here, or gradually over the whole zone. These preliminary results are in an early stage of research. Research is ongoing and further results are forthcoming. 28

2.2.5 Conclusions

The preliminary results show clearly an interdependency between the fault zone width and the length of tunnel that shows a concentrated displacement regime. In addition, the visible change in displacement could coincide with localized damage to the tunnel. Further research is necessary. Beyond the scope of this research study, which is still ongoing, are considered parameters for improvements for future work: (a) Varying the dip angle of fault from 0◦, 30◦, up to 90◦; (b) Including different fault types: dip-slip - normal, reverse, thrust faults; (c) Simulating static and dynamic (creep) fault displacement for numerical modeling; (d) Combining static and dynamic fault displacement and dynamic tunnel shaking due to earthquake waves; (e) Consider the propagation of rupture path from the bedrock through the overburden soil to the surface interacting with the tunnel. 29

3 PARAMETRIC STUDY OF CONCRETE-LINED TUNNELS

CROSSING ACTIVE FAULTS

3.1 Abstract

A challenge for municipal authorities of growing urban areas is to provide larger and faster transportation networks that are safe and resilient to significant disruptions after an earthquake event and other disasters. To allow cities to grow over the past several decades, the construction of tunnels for individual traffic and mass transit through active fault zones became a necessity despite the risk and special design and maintenance considerations, some of which may not have been under- stood at the time of construction. It is now known that earthquakes with magnitudes greater than M6.0 can create significant damage to tunnels in seismically active areas. In particular, large strains due to fault offsets and ground shaking lead to unacceptable stresses in the tunnel lining, which can cause subsequent spalling and potential tunnel closure. Examining the response of concrete lined tunnels built through active fault zones is critical to ensure resilient design and safe operation. A 2D model of a circular reinforced concrete tunnel crossing an active fault is developed within the finite element framework OpenSees. A parametric study with varying structural and ground properties is performed. The effects of earthquake magnitude, geology 30 and geometry of the fault area, and structural properties of the tunnel are studied and assessed to imply novel tunnel design strategies to accommodate large fault motions and to minimize tunnel service disruptions.

3.2 Introduction

Earthquake-induced large ground deformations, such as liquefaction, landslides, or fault displacements, can permanently damage underground structures (e.g., Hashash et al., 2001; Ha et al., 2008; ORourke et al., 2015). Recent events, including the 1989 Loma Prieta earthquake (moment magnitude (M) 6.9), the 1995 Kobe, Japan earthquake (M6.8), the 1999 Kocaeli (M7.4) and 1999 Duzce¨ (M7.2), Turkey earthquakes, and the 2008 Wenchuan earthquake (M7.9) in China damaged several underground structures crossing active faults (Kontogianni and Stiros, 2003; Ulusay et al., 2002; Yu et al., 2016). Although tunnel engineers have been working on earthquake-induced failures for decades, much of the knowledge is proprietary or limited to site-specific studies. Results of site-specific cases with fault displacements show a confined fault rupture with significant damage to underground structures - not only tunnels, but also pipelines and utility ducts - because the imposed offset introduces large strains into the lining of an underground structure. The main design considerations from U.S. guidelines for tunnels are based on wave propagation and ground motion. However, ground shaking has contributed to lower incidents and damage compared to severe damage from large permanent 31 ground displacements from ground failure, such as fault rupture. After the guidance of the Federal Highway Administration (FHWA) guideline by Hung et al. (2009), tunnels affected by permanent ground failure are to be analyzed similar to pipelines with a simplified beam-spring model. To address the lack of a generalized interpretation framework of fault rupture influence of tunnel structures, the focus of this research is to evaluate the behavior of tunnels due to fault rupture displacements with a parametric study. This research investigates the tunnel threshold deformations and internal stresses and strains in the reinforced concrete (RC) lining based on varying geological environment and tunnel geometry. Different rock strengths and changing fault zone widths surrounding the tunnel define the geological environment. The tunnel geometry varies with a geometric ratio of thickness of tunnel lining over inside diameter of the tunnel. Results of the parametric study for tunnel response to strike-slip fault rupture are shown graphically. The results are plotted in a graph showing either a change in geometric ratio or a change in fault zone width as the horizontal axis with respect to the tunnel beam displacement capacity on the vertical axis for two limit strain states - axial tension strain of steel and peak compression strain of concrete. Self-posed questions by the authors help analyze the results. (1) How does the tunnel lining move relative to the fault offset? (2) What is the magnitude and distribution of stresses in the tunnel lining due to the fault offset? (3) How much deformation could be applied and how does it correlate to earthquake magnitudes? (4) What is the influence of varying fault zone lengths? (5) What tunnel length is 32 affected by large stresses and strains, and how do they inform cracking? and (6) What is the influence of the tunnel surrounding rock?

3.3 Material and methods

3.3.1 Finite element model

Modeling options for tunneling are manifold. Most software have a user-friendly graphical interface to implement soil and tunnel elements. Finite element programs, like PLAXIS, FLAC, or RS2 (formerly Phase2) by Rocscience are robust programs and widely used for engineering globally. However, most software are for one specific tunnel project with a single set of parameters. FLAC is able to automate a parametric study with the help of a built-in scripting language FISH. Academia uses the open source finite element software framework OpenSees (McKenna et al., 2010) with advanced capabilities for modeling and analyzing nonlinear responses of systems and is specifically designed for parameter studies. The purpose of this research and the need for a script-able parametric study, displacement controlled fault offset, and the option of advanced modeling and program- ming, led to the selection of OpenSees for this analysis. The two dimensional (2D) model of the tunnel uses a force-based but displacement- controlled fault offset to perform the numerical experimentation. The 2D model, which is shown in Figure 3.1, consists of multiple nonlinear force-based beam- 33 column elements simulating a 300 m tunnel beam in the longitudinal direction. The tunnel beam cross-section is a tube-like reinforced concrete fiber section with 0 concrete C 25/30 ( fc = 3,500 psi) with a Young’s Modulus, E, of 26,700 MPa, a varying concrete lining thickness, h, between 0.20 m and 1.0 m, and a varying inner tunnel diameter, Di, between 6 m and 15 m. The smaller limit of the inner diameter corresponds, for instance, to a single track train tunnel, and the upper limit of Di = 15 m corresponds, for instance, to a multi-lane, double-deck road tunnel, such as the Alaskan Way Viaduct replacement tunnel in Seattle, WA (WSDOT, 2019). The fiber section consists of patches and layers, forming the concrete lining and the longitudinal reinforcement steel, respectively. The 300 m length can be seen as either a complete tunnel with portal structures or as a tunnel segment simulating only the fault region. Therefore, the tunnel beam is fixed in all three directions at the west end but only fixed in x-direction at the east end (cf. Figure 3.1a), which allows fault offset in y-direction. The tunnel length was varied and then set to 300 m, so that the fixed boundaries have no influence on the fault offset. The concrete longitudinal reinforcement consists of a maximum yield strength of 420 MPa (U.S. steel grade 60). This steel is herein assumed for newer tunnel con- structions. Because this research is also concerned with the performance of existing tunnels, results with steel strength 275 MPa (U.S. steel grade 40) will be shown. Circumferential reinforcement was neglected for modeling simplicity, because the main research concentration is the tunnel alignment. The reinforcement ratio is set to As/Ac = 1.5 % without exceeding the limit for standard concrete structures in 34

Plan view: Tunnel offset y, North Fault zone widths section 0.1m to 100m

Offset

L = 300 m x, East Major fault plane

A Crossing angle 90°

A Tunnel cross Displacement x, East section: D i and h controlled tunnel offset (a) Tunnel beam model in plan view

Tunnel cross-section A-A Concrete lining Lining thickness, h Outside diameter, D o Outer clear Tunnel cover 8 cm

Inner clear cover 5 cm

Inside diameter, D i

Rock Longitudinal rebars (Inner and outer layer) (b) Section A-A

Figure 3.1: 2D model: (a) Plan view of tunnel beam with perpendicular offset input. Springs are modeled on both sides, but only shown on one side for graphical reasons. (b) Schematic representation of the tunnel cross-section: The tunnel is embedded in rock. The lining is divided into fiber sections. Road, structural fill, and vehicle are only shown schematically for a Di = 10 m tunnel. 35 leaning on the U.S. building code ACI 318-14 (2014) for concrete buildings. The concrete clear cover after ACI 318-14 (2014, pg. 335) amounts to 8 cm (3 in) for the concrete permanently in contact with the ground and 5 cm (2 in) for inside the tunnel with some extend exposed to weather. Further, the model assumes a drained tunnel with no hydrostatic water pressure impacting the tunnel lining. Also, rock and fault zones are assumed homogeneous (i.e., no fault inhomogeneities are modeled). Elastic no-tension (ENT) soil springs are placed on both sides of the tunnel beam along the entire alignment to simulate the rock. More specifically, the springs are zero-length sections connected at the ends of each tunnel beam element. Equivalent spring values correspond to uniaxial compressive strengths for extremely weak rock (0.25 MPa), weak rock (5 MPa), and medium strong rock (50 MPa) (Barton (1978), ISRM, pg. 348). The spring value simplifies the ground and is multiplied by the tributary length along the tunnel elements, and times the outer diameter of the tunnel, to capture not only the length but also the height of the tunnel.

3.3.2 Fault model

The simulation of the fault offset is based on a strike-slip fault, but is applicable to normal faulting as well. For this study, the numerical model is a horizontal view of the tunnel and fault system. Kasahara (1981) found that the displacement fields associated with a strike-slip fault can be viewed as a shear dislocation. In strike-slip 36 faulting, some observations show a displacement along two fault planes resulting in a displacement in two opposite directions. Surface displacement measurements made after the MW7.9 1906 San Francisco earthquake show this displacement schematically in Figure 3.2a. It also shows that the strain accumulation away from the fault goes to zero at around 20 km. General tunnel lengths are usually on the order of hundreds of meters to several kilometers. The authors used modeling simplifications. It is assumed that one side of the tunnel stays in the original position and the other side of the tunnel is dislocated according to the fault offset, as shown in Figure 3.2b. This simplification is often used for physical experimentation on pipelines and tunnels where one side is moved to simulate a fault rupture, such as ORourke et al. (2015) or Abdoun et al. (2009). In addition, the authors assumed a fault zone with a fractured area and one major slip plane at the right end of that fractured area. Other faults might have fractured zones left and right of the slip plane or several slip planes scattered across the fault zone width (Sibson, 2003). A natural variation of the fault zone widths is part of this research. Figure 3.3 shows such variation. The fault offset correlates to earthquake magnitudes to provide a general understanding of the impact. Correlations developed by Wells and Coppersmith (1994) relate earthquake magnitude to maximum surface displacements from strike- slip faulting, M = 6.81 + 0.78 · log(MD) (3.1)

(1σ = 0.34 unit points) 37

(a) Large scale (b) Tunnel scale Fault zone width observations model Fault zone width Tunnel position after fault rupture Surface displacement after fault rupture u u

km 20 40 km m m

Strike-slip fault Initial tunnel position with 2 fault planes before fault rupture

Figure 3.2: (a) Schematic strike-slip offset, (b) Schematic of the modeled simpliﬁca- tion of the strike-slip fault offset in combination with the displaced tunnel beam. Offset, u, is exaggerated. x-axis scales differ.

(a) Narrow fault zone width (b) Wide fault zone width Major fault plane Major fault plane Fault zone Fault zone

Tunnel beam

Figure 3.3: Graphical tunnel beam model with (a) narrow and (b) wide fault zone widths. where MD is the maximum surface displacement in meters (m) at one location along the surface fault rupture. Notably, to simplify the modeling of static fault rupture, the propagation of the rupture path from the bedrock through the overburden soil to the surface is not included. Bray et al. (1994) give further information on fault rupture propagation through overlying soil. The static analysis is displacement-controlled, meaning, to put a specific displacement onto the tunnel beam, simulating the fault offset, a specific amount of force needs to be applied at one side of the tunnel beam. The Newton solution algorithm with a convergence tolerance of 1e-10 with a maximum number of ten 38 iterations was chosen to generate a nonlinear approximation of the solution. The convergence test with a ’Norm Displacement Increment’ specifies a tolerance on the norm of the displacement increments at the current iteration. The load for the displacement-controlled offset, is applied in a multitude of steps to capture small differences. The tunnel beam uses three Gauss-Legendre integration points along the length of each element. Fiber sections are used across the area of the tunnel cross-section.

3.4 Numerical experimentation

3.4.1 Parametric study setup

Proof of Concept Input Parameters Computation Results / Output Interpretation

Transverse Earthquake Fault Zone Width Displacement Magnitude Convergence Parametric Study Geology (Spring Internal Forces + (V, M) Limit States Stiffness) Offset Input Verification Tunnel Geometric Stress / Strain (σ / ε) Graphs / Charts Ratio (h/D i)

Axial Force - Constitutive Be- Moment (NM) Industry havior (Strength) Interaction Application

Figure 3.4: Parametric study chart showing the general sequence of operations from left to right.

Figure 3.4 shows the general sequence of analysis, which starts with conver- 39 gence analyses of mesh size and beam length, and a proof of concept by verifying the numerical model with known closed-form analytical solutions. The parameter ranges are given for fault zone width, spring stiffness for varying rock strengths, and tunnel geometry with the lining-thickness-over-inner-diameter (h/Di) - geometric ratio. Constitutive behaviors, such as non-linear concrete and steel strengths, are part of the beam elements of the numerical model, they do not vary, and are implemented in the circular reinforced concrete ﬁber sections. Transverse displacement results correlate to earthquake magnitude interpretations with the Wells & Coppersmith formulation. Each step will be discussed in detail herein.

3.4.2 Veriﬁcation and convergence

The verification of the numerical beam model is done with a closed-form solution by Hetenyi´ (1946, pg. 63) of a fixed beam on elastic foundation. The problem to analyze is the following: a L = 300 m beam, totally fixed at both ends, is loaded with a point load of P = 100 MN at the midpoint of the beam. The relevant closed-form analytical equations for deflection, d, and bending moment, M, at the midpoint of the beam are given by Hetenyi´ (1946) as:

Pλ cosh(λL) + cos(λL) d = · (3.2) 2k sinh(λL) + sin(λL)

P cosh(λL) − cos(λL) M = · (3.3) 4λ sinh(λL) + sin(λL) 40 with r k λ = 4 (3.4) 4EI and 12Eg D k = · (3.5) i 2

where k is the subgrade modulus by Klar et al. (2005), i is the estimated distance from the center axis to the inﬂection point of the greenﬁeld settlement trough directly above the tunnel, assumed to be i = 10 m (Leca and New, 2007), and D is

the outside diameter of the tunnel. The Young’s modulus of the ground, Eg is cho-

sen for extremely weak rock to Eg = 100 MPa. The above parameters are specific for the analytical solution. EI is the flexural rigidity of the tunnel beam with E = 26700 MN/m2 for concrete C25\30. The numerical analysis uses linear-elastic material behavior, firstly to verify the results with the closed-form analytical solution, and secondly as a model basis for extending to a non-linear material behavior to lessen the possibility for the mesh being a significant source of error. The following parameters are used by both methods, the analytical and the numerical one: the

tunnel geometry is Di = 6 m and h = 30 cm, which results into an outside diameter of D = 6.6 m. For the numerical model, the spring stiffness of the ground for extremely weak rock is 100 MN/m3 times the tributary length along the tunnel beam and times the outside diameter D, and the Poisson’s ratio, ν for weak sandstone is 0.2 after Gercek (2007). The analytical solution by Hetenyi´ (1946) results in a maximum deﬂection at the midpoint of d = 13.37 mm. The analytical deﬂection compares well with the numerical displacement of 13.68 mm. 41

Figure 3.5 shows convergence as a function of fiber section size, where fiber section size is defined by radial times circumferential subdivisions. From Figure 3.5, 64 fibers were selected, four radial and sixteen circumferential subdivisions, to represent our tunnel cross-section. The subdivisions allow to evaluate strains and stresses at different positions in the cross-section. The convergence analysis for the longitudinal direction, which is shown in Figure 3.6, determines the convergent element length. The maximal deflection at the midpoint of the 300 m long beam with 0.5 m short elements is d = 13.68 mm. At 5 m element lengths, the deflection with the OpenSees model is 13.68 mm. the deflection curve in Figure 3.6 starts to descent after the element length of 5 m. These results lead to the convergence of the length of the beam elements of 5 m or less. The longest beam elements in the final numerical beam model are 5 m long, which coincides with other researchers, such as Anastasopoulos et al. (2007), then shorten to 1 m elements and shorter near the fault offset. A shorter element discretization in the vicinity of the fault offset allows (1) for smaller fault zone width variations and (2) for a smoother evaluation of displacements and internal strains and stresses. The moment, calculated using the formula by Hetenyi´ (1946) is M = 236 MNm. The OpenSees program gives a maximal moment with 5 m element length of M = 137 MNm. The case here presented with a single point load and a large tunnel diameter, results in values of the same order of magnitude. The moments by Hetenyi´ (1946) are larger compared to the numerical model. The displacement results are near identical for engineering purposes. It can be concluded that the numerical model acts similar to established closed-form analytical solutions. 42

14.3 Fiber sections visualization:

14.2

14.1

13.9 1 6 Varying radial subdivisions 1 through 6

13.8 13.68 mm Maximum Displacement, d (mm) 13.7 13.67 mm

13.6 0 20 40 60 80 100 120 140 160 180 200 Fibers Figure 3.5: Convergence of fibers of the tunnel cross-section for a 300 m long tunnel beam-spring model, with 1 m element lengths, loaded with a 100 MN point load, P, at the center. Bold marker symbolizes chosen model with 4 radial and 16 circumferential subdivisions resulting in 64 fiber sections with a numerical model displacement value of 13.68 mm. Maximal deflection after Hetenyi´ (1946) results in 13.37 mm.

Nonetheless, engineering judgment needs to be applied to interpret results from both methods.

3.4.3 Input parameters

The parametric study consists of 252 combinations and is conducted with four varying parameters:

1. Fault zone widths: Varies between 0.10 m and 100 m. 43

102

d = 13.68 mm 1 m 5 m element length

101 Beam-spring model visualization: 30 m Maximum Displacement, d (mm)

100 10-1 100 101 102 103 Element Length (m) Figure 3.6: Discretization of element length for a 300 m long tunnel beam-spring model, loaded with a 100 MN point load, P, at the center. The numerical model displacement computes to 13.68 mm. Maximal deﬂection after Hetenyi´ (1946) results in 13.37 mm.

2. Geology - Rock strength: The numerical model uses ENT soil springs with a

modulus of subgrade reaction, ks = (C· Eg)/R for a tunnel with a radius R,

a modulus of elasticity of the ground Eg, and a factor C ranging between 0.5

and 3.0 (Maidl et al., 2014, pg. 105). The chosen ks-values range between 100 MN/m3 for extremely weak rock as a lower bound, 1 000 MN/m3 for weak rock, up to 15 000 MN/m3 for medium strong rock as an upper bound, with values for Young’s modulus after Johnson and DeGraff (1988).

3. Tunnel geometric ratio, h/Di from 1/15 to 1/30: (a) Concrete lining thickness, h ranges between 0.20 m and 1.0 m, and (b) inner tunnel diameter val-

ues, Di, vary between 6 m, 10 m, and 15 m. Successful constructed circular 44

or near circular tunnels tend to have a relative narrow geometric ratio.

3.4.4 Assessment strategy

Results for internal forces, stresses and strains, and axial force-moment (N-M) interaction diagrams are interpreted using multiple limit states. Multiple limit states are deﬁned as follows: (a) service limit state (SLS) at the beginning of yielding of steel at 0.21% tension steel strain, and (b) ultimate limit state (ULS) at 0.3% of compression concrete strain. Reaching SLS is characterized by the steel taking on tension and resulting in cracking of concrete, either visible inside the tunnel or in- visible below the road or on the outside of the tunnel lining, which is in contact with the rock. After ULS is reached, it is assumed that the concrete is spalling and crushing and therefore in a state of failure.

3.4.5 Results

Due to an imposed offset, the tunnel in longitudinal direction experiences shear, but also normal forces and moments which vary across different geometric ratios, ground, and fault parameters. Therefore, the theoretical capacity of a tunnel cross-section at multiple limit states is ﬁrst evaluated with a baseline model with N-M-interaction diagrams. Concrete shear is not considered and outside of the scope of this research. After establishing the sequence of evaluation for the base- 45 line model, graphical visualizations combine results from the parametric study in the discussion, in section 3.5. The baseline model consists of a 300 m long tunnel model with a tunnel geometry of 1/20-ratio with h = 30 cm and Di = 6 m and and a 30 m wide fault zone. The geology of the fault is assumed with a spring stiffness of 100 MPa. The geology left and right of the fault is the only parameter we vary for the baseline evaluation with a spring stiffness of 100 MPa for extremely weak rock, 1 000 MPa for weak rock, and 15 000 MPa for medium strong rock. This baseline was chosen due to the fact that it resembles a standard single track train tunnel. The axial force - moment interaction diagrams for different strains of the baseline model are shown in Figure 3.7. The combination of axial force and moments for each strain state place along or inside the envelope. Displacement curves in Figure 3.8 along the tunnel axis illustrate clearly a one- sided fault offset of the baseline model. The displacements of the tunnel beam around the major fault plane are distributed across the fault plane but show a distinct shearing. The maximum displacement of the moved tunnel portion is 11.2 cm for the case with extremely weak rock and 3.3 cm for medium strong rock. Mo- ments along the tunnel axis in Figure 3.9 show peak values at the point of largest bending and Figure 3.10 shows maximum shear forces at the point of zero moment, which coincides with the major fault plane. These two plots show clearly the behavior of double curvature bending. Figure 3.11 illustrates the strains along the tunnel beam with the peak strain at tunnel meter (tm) -3. Figure 3.12 shows the strain distribution over the tunnel 46

105 Yielding of steel at = 0.21% (at ) -2 1 Peak compressive concrete strain is = -0.3% (at ) 2 Strains for extremely weak rock ( = 0.25 MPa) Strains for weak rock ( = 5 MPa) -1.5 Strains for medium strong rock ( = 50 MPa)

-1

Axial Force, N (kN) -0.5

0 0 0.5 1 1.5 2 2.5 Moment, M (kNm) 105 Figure 3.7: Axial force - moment (N-M) interaction diagrams for baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a fault zone width of 30 m, and varying rock characteristics along the tunnel. Image insertion shows a stress-strain curve of concrete with various strain and stress levels. cross-section for the point of largest strain of the baseline model in extremely weak rock (σ = 0.25 MPa). Compression is reached at the ”North” part of the tunnel and tension is taken up by the reinforcement of the ”South” part of the tunnel. ”North” and ”South” can also mean the crown (top) or the invert (bottom) of the tunnel, depending on the fault mechanism - either normal or strike-slip faulting.

3.5 Discussion

The moment and shear forces plots in Figures 3.9 and 3.10 illustrate peak values in a localized area. This might indicate a localized region of damage along the tunnel 47

0.2 Legend: 0.18 = 0.25 MPa) . . . Weak rock ( = 5 MPa) 0.16 - - - Medium strong rock ( = 50 MPa) Major fault plane 0.14 0.12 30 m fault zone width 0.1 11.2 cm 0.08 0.06 0.04 5.9 cm

Threshold displacement (m) 0.02 3.3 cm 0 -150-125-100 -75 -50 -25 0 25 50 75 100 125 Longitudinal tunnel axis (m) Figure 3.8: Displacements along the tunnel axis for the baseline model with 1/20- ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width. axis. Figure 3.13 shows localized displacement curves for the baseline model with varying fault zone widths. The maximum threshold displacement of the baseline model of 11.2 cm for extremely weak rock with a fault zone width of 30 m is 1.9 % of the inner tunnel diameter of 6 m. For a 1 m wide fault zone width, the displacement of 3.6 cm is only 0.6 % of the inner tunnel diameter of 6 m. The largest tunnel diameter in extremely weak rock results in a displacement of 16.6 cm, which is 1.1 % of the inner diameter. In summary, in extremely weak rock, the maximum displacements reached vary between 0.7 % and 2 % of inner tunnel diameter. In medium strong rock, the range of displacements varies from 0.2 % to 0.6 % of inner diameter. Larger values are for wide fault zone widths coupled with smaller 48

105 3.0 Legend: 2.5 = 0.25 MPa) . . . Weak rock ( = 5 MPa) 2.0 - - - Medium strong rock ( = 50 MPa) Major fault plane 1.5

1.0

0.5

0.0

Moment, M (kNm) -0.5 30 m fault zone width -1.0

-1.5

-150-125-100 -75 -50 -25 0 25 50 75 100 125 Longitudinal tunnel axis (m) Figure 3.9: Moments along the tunnel axis for the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width. diameters. Small values are for narrow fault zone widths coupled with large tunnel diameters. The small range of values goes over all geometric ratios. In all cases, the length of affected tunnel is about 30 m before and after the major fault plane. A 60 m band of high strains point to a relative localized damaged zone, that might need to be repaired after a severe event. This conclusion of the results was observed in China after the 2008 Wenchuan earthquake with a magnitude of M7.9, when the Longxi tunnel collapsed at an active fault crossing (Yu et al., 2016). Large-scale split-box and centrifuge tests of HDPE pipelines and similar fault rupture interactions done by ORourke et al. (2015), Abdoun et al. (2009), and others, result in double curvature bending and peak stresses close to the fault trace. These 49

104 15.0 Legend: = 0.25 MPa) 12.5 . . . Weak rock ( = 5 MPa) - - - Medium strong rock ( = 50 MPa) 10.0 Major fault plane

7.5

5.0

2.5

Shear force, V (kN) 0.0

-2.5 30 m fault zone width -5.0 -150-125-100 -75 -50 -25 0 25 50 75 100 125 Longitudinal tunnel axis (m) Figure 3.10: Shear forces along the tunnel axis for the baseline model with 1/20- ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width. results correspond qualitativey to the results of this study. An important observation to discuss, is the displacement of the tunnel beam for fault zone widths larger than 10 m for the baseline tunnel in extremely weak rock (Figure 3.13): The displacement curves do not vary for fault zone width of 10 m and larger. It is assumed, that the smaller tunnel diameter in combination with extremely weak fault gouge material gives the tunnel beam a larger possibility to deform. Graphical evaluations of strain levels at varying limit states of SLS and ULS are shown in Figures 3.14 and 3.15. Figure 3.14 presents comparisons of geometric ratio to threshold displacements for varying tunnel diameters and rock strengths. 50

0.024 Legend: = 0.25 MPa) 0.02 . . . Weak rock ( = 5 MPa) - - - Medium strong rock ( = 50 MPa) 0.016 Major fault plane (30 m fault zone width) 0.012

0.008

Concrete strain (-) Beam element at tm -3 0.004 Crushing of concrete at 0.3% Yielding of steel at 0.14% 0

~ 20 m -0.004 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 Longitudinal tunnel axis (m) Figure 3.11: Strain along the tunnel axis for the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, with varying geology outside the fault zone width. tm = Tunnel meter.

εcu = -0.3 %

Compression neutral axis

center line 6.6 m 6.0 m Tension

6.6 m εy = 1.2 %

Figure 3.12: Strain across the tunnel cross-section at tunnel meter (tm) -3 of the baseline model with 1/20-ratio of lining thickness, h = 0.30 m, to inner tunnel diameter, Di = 6 m, for a 30 m wide fault zone, and ks = 100 MPa.

The ﬁgure shows clearly, that larger geometric ratios, meaning larger RC lining thicknesses coupled with an increased outer tunnel diameter, yield larger possible displacements at the ultimate limit state. An increase of the lining thickness by 100 %, with constant reinforcement ratio of 1.5 % and a constant inner tunnel 51 diameter, thus constant tunnel clearance, results in an increase of the outer tunnel diameter and causes an increase of possible displacement of the tunnel beam by an average of 33 %. This trend is visible over varying tunnel diameters and rock conditions. Analyzing different fault zone widths within Figure 3.15, a fault zone width, for instance, of 10 m results in a nearly 30 % increase as well. All graphs show a positive jump in possible displacement at speciﬁc fault zone widths. These distinctive results suggest a substantial increase in possible displacements for fault zone widths larger than 1 to 2 times the inner tunnel diameter. A second observation from Figure 3.15 is the plateau of constant displacement for fault zone widths larger than 1 to 2 times the tunnel diameter. It suggests a large enough space is available along the tunnel alignment for the tunnel beam to deform into a double-

0.2 Legend: 0.18 __ Fault zone width of 20 m and larger . . Fault zone width of 10 m 0.16 - - Fault zone width of 1 m - . Fault zone width of 0.1 m Major fault plane 0.14 0.12 10 m fault zone width 20 m 0.1 30 m 11.2 cm 0.08 50 m 100 m 0.06 0.04

Threshold displacement (m) 0.02 3.2 cm 3.6 cm 0 -150-125-100 -75 -50 -25 0 25 50 75 100 125 Longitudinal tunnel axis (m) Figure 3.13: Displacement curves for baseline model in extremely weak rock with varying fault zone widths. 52 curvature bending due to the fault offset. Small displacements at shorter fault zone widths indicate a predominance of abrupt shearing, which is less favorable for a RC structure. A double curvature bending of the tunnel beam above yielding of steel results in cracking of concrete. If these cracks grow at the outside of the tunnel, which is in contact with the rock, it might result in new water ways damaging the concrete lining even further, resulting in reduced capacity. The double-curvature bending in combination with the direction of the fault offset might help to narrow down the affected area. For practical reasons, a membrane with possible injection open- ings between the membrane and concrete lining might help concentrate the water ways to a minimum. Also, engineers and public authorities should discuss project speciﬁcs, such as if a doubling in reinforced concrete lining, and with it doubling in concrete volume, is worth 33 % increase of possible displacements, considering a larger excavation diameter, increased material need and transport, increased disposal needs, etc. A correlation of tunnel displacements with earthquake magnitudes after Wells and Coppersmith (1994) result in a range of magnitudes between M5.5 for displacements of about 0.025 m up to around M6.5 for displacements of 0.2 m; see Table 3.1. The regression equation inhibits a relatively large standard deviation of 1σ = 0.34 unit points. It can be said, that for tunnels in medium strong rock a smaller earthquake of about M5.5 might result in concrete failure of the tunnel lining. On the other hand, a tunnel in extremely weak rock might be able to withstand a larger earthquake of up to around M6.5 before concrete failure occurs. Other tun- 53

Figure 3.14: Strain levels of RC tunnel cross-section in a plot of geometric ratio versus threshold displacements for a 30 m wide fault zone with varying inner tunnel diameter and ground conditions along the tunnel alignment. 54

Figure 3.15: Strain levels of RC tunnel cross-section in a plot of fault zone widths versus threshold displacements for a geometric ratio of 1/20 with varying inner tunnel diameter and ground conditions along the tunnel alignment. 55 nel equipment with less ﬂexibility like telecommunications or water/wastewater lines might be affected and need special care.

Correlated earthquake Displacement - Fault offset, m magnitude, M 0.025 0.035 0.050 0.075 0.100 0.125 0.150 0.175 0.200 M with maximum displacement 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.2 6.3

Table 3.1: Correlated earthquake magnitudes to maximum surface displacements after Wells and Coppersmith (1994). One standard deviation of earthquake magnitude, 1σ = 0.34 unit points.

A reduction in steel strength from 420 MPa to 275 MPa (from US grade 60 to 40) results in an earlier yielding of steel and an average reduction of ULS displacements by 5 %. Figure 3.16 compares both baseline models with higher and lower steel strengths. The sensitivity analysis in chapter 3 shows a minor inﬂuence of the steel strength.

3.6 Conclusions

To address the lack of a generalized interpretation framework, a parametric study of a 2D beam-spring model of a tunnel crossing an active fault was evaluated with varying ground and tunnel characteristics. Results show a clear localization of damages. Maximum threshold displacements of the tunnel beam due to fault offset until the reinforced concrete is in failure, varies between 0.2 % and 2 % of the inner tunnel diameter. A doubling of the tunnel lining thickness results in 33 % increase of possible displacements. A variation in fault zone width shows that the tunnel lining is at higher risk 56

Figure 3.16: Strain levels of RC tunnel cross-section in a plot of geometric ratio versus threshold displacements for a 30 m wide fault zone with steel strength of 420 MPa on the left and 275 MPa on the right. of damage under a smaller earthquake if the fault zone width is less than 1 to 2 times the tunnel diameter. Short fault zone width with distinct fault planes result in sharp shearing of the tunnel and lower capacity. A magnitude M6.0 and greater can damage underground structures in seismically active areas. Directly at the fault, an earthquake with a magnitude around M5.5 coupled with unfavorable conditions, such as medium strong rock or a short fault zone width and a small tunnel diameter, might damage the underground structure until concrete failure. On the other hand, a fault offset equivalent to an earthquake of about M6.0 can be withstood by large tunnels in softer ground. Since this research is designed to serve early stage projects for evaluating the tunnel structure due to a fault rupture, a detailed analysis during a later stage of the project with specific project properties is indispensable. However, it is impor- 57 tant to understand the implications and applicability of this research through an industry application for an existing tunnel crossing an active fault. Limitations of the presented work are the failure criterion, which is defined by reaching the peak compression concrete strain at ULS with bending of the tunnel beam and no concrete shear; also steel is defined only in tension. Additional limitations are the use of only static fault rupture, no ground shaking nor other external forces. Further research needs to be done to investigate these limitations, e.g. by differing modeling approaches, such as shell elements for the tunnel beam, or a consideration of flexure-shear-interaction of the reinforced concrete with the modified compression-field theory (MCFT) (Bentz et al., 2006). Gaps between concrete lining and surrounding rock after an event, which might show undesired water ways, may be of importance. A variation of concrete strength with and without post-stressing would further this research in terms of displacement capacity. An- other aspect of future work is the seismic wave amplification, that might happen in mountains where tunnels are often build through. 58

4 SENSITIVITY ANALYSIS

4.1 Abstract

It is important to know the effect of variability of variables of a numerical model. A sensitivity analysis is a preferred tool to assess the effect of uncertain variables in a numerical model. Finite difference and direct differentiation methods are two approaches used in this chapter to assess the sensitivity of the tunnel response to modelling parameters. The ﬁnite difference method serves as a control to the direct differentiation method, which is incorporated in the numerical ﬁnite element framework OpenSees. Results show that material properties such as concrete and steel strengths, within the ranges assessed, have a smaller impact on the response sensitivity than geometric properties, like the tunnel diameter or the concrete lining thickness. The use of the parametric study results in combination with the sensitivity results gives a better assessment of the response of tunnels crossing active faults, and shows which parameters control the response sensitivity of the numerical model.

4.2 Introduction

Sensitivity analysis is a means to assist decision making (Eschenbach and McK- eague, 1989), but also to assess the uncertainty of computational engineering mod- 59 els. It helps to identify sensitive and important variables to give information on reliability and robustness of the system despite different parameter values (Scott and Haukaas, 2008). For this research, the response sensitivity from material properties and tunnel dimensions are important variables and will be analyzed further. The basis for the sensitivity analysis is the ﬁnite element model described in chapter 3.

4.3 Theory

To perform a sensitivity analysis, two approaches are available: (1) the finite difference method (FDM) and (2) the direct differentiation method (DDM) (Scott and Haukaas, 2008). With the finite difference method, one parameter is changed at a time while all other parameters stay constant at their nominal value. The FDM approximates the derivative δx x − x ≈ 2 1 (4.1) δy ∆y where x1 is the response for the model with respect to y1, x2 is the response for the model with respect to y2, ∆y = y2 − y1. The change from y1 to y2 has to be small enough to see a calculable difference but not too large to achieve a secant rather than tangent perturbation (Munoz-Rojas˜ et al., 2004). With the direct differentiation method, controlling equations within the finite element code of a software program are directly differentiated and then implemented back into to finite element code. This approach requires initial effort but results in a higher accuracy and lower computational cost. (Scott and Haukaas, 60

2008; Guo and Jeffers, 2014) The DDM is implemented in OpenSees and computes the response sensitivity at each load step after the response has converged. The fault offset is computed with a displacement controlled analysis. The graphical output of the DDM analysis shows the computed sensitivity corresponding to the fault offset displacement. To identify the impact of one variable at a time on the outcome at the same load level, the baseline model data will be used: Di = 6 m, h = 0.3 m, Lzone = 30 m, the rock strength is set to the extremely weak uniaxial compressive strength of σ = 0.25 MPa with an approximate spring stiffness of 100 MN/m3 times the outer diameter of the tunnel times the tributary length along the tunnel axis.

4.4 Results

4.4.1 Concrete and steel strength

A change in material strengths is a plausible adjustment during early stages of a project. A varying concrete and steel strength is useful to examine existing structures with lower strength materials as well. The graphical results in Figure 4.1 originate from the DDM analysis using OpenSees. The major fault plane is at tunnel meter 0. The maximum displacement of 0.112 m of the baseline model due to fault offset in the subplot (b) in Figure 4.1 is based on a steel strength of 420 MPa. The sensitivity analysis uses this value as its nominal value. The subplots (a), (c), and (d) visualize the DDM analysis results for the steel strength. Figure 4.1d will 61 be used for evaluation. The peak values surround the major fault plane.

(a) (b) 0.15

0.2 0.1

0 0.05

0 -0.2 5 -150 0 -50 -5 050 10 -0.05 150 -5 Threshold displacement (m) -150-100-50 0 50 100150 Threshold displacement (m) Longitudinal tunnel axis (m) Tunnel axis (m) ( u/ fy) fy (m) (c) 10-5 (d) 0.15 2

0.1 1

0.05 0 fy) fy (m)

u/ -1

0 (

-0.05 -2

Threshold displacement (m) -2 0 2 4 -150-100 -50 0 50 100 150 ( u/ fy) fy (m) 10 -5 Longitudinal tunnel axis (m)

Figure 4.1: Sensitivity analysis for steel strength, fy, for the baseline model. (a) 3D image of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the steel strength versus the displacement. (d) Change in displacement with respect to the steel strength along the tunnel axis. 62

The FDM analysis uses the computed ultimate limit state displacement u420MPa

= 0.1121758 m from the baseline model with a steel strength of fy,420MPa = 420 MPa as the nominal value. A second displacement with an increased steel strength of fy,420.1MPa = 420.1 MPa computes to u420.1MPa = 0.1121755 m. The change in displacement calculates to

(u420.1MPa) − (u420MPa) ∆u ≈ ( ) · fy = −0.0013 m ( fy,420.1MPa) − ( fy,420MPa)

These results with respect to a small change in steel strength are small and support the performed DDM analysis in Figure 4.1. The nominal value of the concrete strength is 25 MPa. Figure 4.2 shows the response sensitivity with respect to the concrete strength. The subplot 4.2d visualizes the response sensitivity. 63

(a) (b) 0.15

0.2 0.1

0 0.05

0 -0.2 5 -150 0 -50 -5 050 10 -0.05 150 -5 Threshold displacement (m) -150-100-50 0 50 100150 Threshold displacement (m) Longitudinal tunnel axis (m) Tunnel axis (m) ( u/ fc) fc (m) (c) 10-5 (d) 0.15 2

0.1 1

0.05 0 fc) fc (m)

u/ -1

0 (

-0.05 -2

Threshold displacement (m) -2 0 2 4 -150-100 -50 0 50 100 150 ( u/ fc) fc (m) 10 -5 Longitudinal tunnel axis (m)

Figure 4.2: Sensitivity analysis for concrete strength, fc, for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the concrete strength versus the displacement. (d) Change in displacement with respect to the concrete strength along the tunnel axis. 64

4.4.2 Tunnel geometry

A sensitivity analyses for the tunnel lining thickness, h, and the inner tunnel diameter, Di, result in the Figures 4.3 and 4.4. Both analyses show negative trend values on each bottom right sub-ﬁgure. Chapter 2 on the parametric study discusses the variation of both parameters. A detailed look at the results of the parametric study leads to the same trend results of the sensitivity analysis.

• With a tunnel situated in the lowest rock strength, crossing a 30 m wide fault

zone, with 6 m inner tunnel diameter, and a geometric ratio of h/Di = 1/15, thus a lining thickness of h = 6 m/15 = 0.40 m, the displacement due to fault offset computes to 0.1196 m.

• A 10 m diameter tunnel with a geometric ratio of 1/15 - a lining thickness of h = 0.33 m, results in a displacement of 0.1119 m. 65

(a) (b) 0.15

0.2 0.1

0 0.05

0 -0.2 0.01 -150 0 -50 050 -0.05 150 -0.01 Threshold displacement (m) -150-100-50 0 50 100 150 Threshold displacement (m) Longitudinal tunnel axis (m) Tunnel axis (m) ( u/ h) h (m) (c) (d) 0.15 0.01

0.1 0.005

0.05 0 h) h (m) u/

0 ( -0.005

-0.05 -0.01

Threshold displacement (m) -0.01 0 0.01 -150-100 -50 0 50 100 150 ( u/ h) h (m) Longitudinal tunnel axis (m)

Figure 4.3: Sensitivity analysis for lining thickness, h, for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the lining thickness versus the displacement. (d) Change in displacement with respect to the lining thickness along the tunnel axis. 66

(a) (b) 0.15

0.2 0.1

0 0.05

0 -0.2 0.05 -150 0 -50 0 50 -0.05 150 -0.05 Threshold displacement (m) -150-100 -50 0 50 100 150 Threshold displacement (m) Longitudinal tunnel axis (m) Tunnel axis (m) ( u/ D) D (m) (c) (d) 0.15 0.04

0.1 0.02

0.05 0 D) D (m) u/

0 ( -0.02

-0.05 -0.04

Threshold displacement (m) -0.05 0 0.05 -150 -100 -50 0 50 100 150 ( u/ D) D (m) Longitudinal tunnel axis (m)

Figure 4.4: Sensitivity analysis for the outer tunnel diameter for the baseline model. (a) 3D graph of the response sensitivity. (b) Displacement curve along the tunnel alignment. (c) Change in displacement with respect to the diameter versus the displacement. (d) Change in displacement with respect to the diameter along the tunnel axis. 67

4.5 Results with varying material properties

In addition to the sensitivity analyses of the baseline model parameters for steel 0 ( fy = 420 MPa) and concrete ( fc = 25 MPa) strengths, analyses with a wide range 0 of material properties from fc = 20 MPa to 55 MPa and fy = 275 MPa to 550 MPa were performed. Table 4.1 describes the deﬁnition of each parameter set. Table 4.2 0 shows the sensitivity response of each parameter set for the variables fc, fy, D, and h. The values in Table 4.2 for concrete and steel are lower by two to three magnitudes compared to the values for the tunnel geometry, D and h. 68 (m) 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 h (m) 6 6 6 6 6 6 6 6 6 6 6 6 D (MPa) s 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 E 60 60 60 60 60 60 60 60 40 60 75 80 (US grade) y f 420 420 420 420 420 420 420 420 275 420 520 550 (MPa) y f (MPa) 24900 26700 28300 29900 31400 32800 34300 35700 24900 24900 24900 24900 c E (psi) 0 3000 3500 4000 5000 6000 6500 7500 8000 3000 3000 3000 3000 c f -20 -25 -30 -35 -40 -45 -50 -55 -25 -25 -25 -25 (MPa) 0 c Table 4.1: Deﬁnition of parameter sets for sensitivity analyses f Name fc20MPa fc25MPa fc30MPa fc35MPa fc40MPa fc45MPa fc50MPa fc55MPa fy275MPa fy420MPa fy520MPa fy550MPa 1 2 3 4 5 6 7 8 9 10 11 12 Row 69

Table 4.2: Response sensitivity results for parameter sets

0 0 Row Name δu/δ fc · fc (m) δu/δ fy · fy (m) δu/δD · D (m) δu/δh · h (m) 1 fc20MPa -5.43E-05 -8.49E-05 -7.68E-02 -5.06E-02 2 fc25MPa -1.06E-05 -1.77E-05 -3.18E-02 -9.01E-03 3 fc30MPa -7.63E-06 -1.42E-05 -2.99E-02 -6.09E-03 4 fc35MPa -8.78E-06 -1.41E-05 -3.26E-02 -7.07E-03 5 fc40MPa -9.75E-06 -1.40E-05 -3.51E-02 -7.91E-03 6 fc45MPa -1.05E-05 -1.36E-05 -3.73E-02 -8.59E-03 7 fc50MPa -1.17E-05 -1.35E-05 -4.00E-02 -9.58E-03 8 fc55MPa -1.30E-05 -1.33E-05 -4.28E-02 -1.07E-02 9 fy275MPa -1.10E-05 -1.08E-05 -3.03E-02 -9.46E-03 10 fy420MPa -1.06E-05 -1.77E-05 -3.18E-02 -9.01E-03 11 fy520MPa -8.05E-06 -1.83E-05 -3.01E-02 -6.54E-03 12 fy550MPa -7.03E-06 -1.69E-05 -2.94E-02 -5.56E-03

4.6 Discussion

Sensitivity analysis shows that a change in steel strength has a small influence on the response sensitivity within the ranges evaluated here. Due to a decrease in steel strength the threshold displacement of the response sensitivity increases. The results of the sensitivity analysis show that the change in concrete strength has a small influence on the response sensitivity of the numerical model. Thus, initial stages of concrete creep have a minor impact. However, the long-term behavior might have an impact, but needs more research. The long-term behavior of concrete creep is outside the current scope of this research. The sensitivity analysis for geometric parameters of the tunnel shows a larger influence on response sensitivity than the material properties. Increasing the single parameter of the lining thickness results in an increased stiffness which follows a decrease in displacements. An increase in inner tunnel diameter results in a dis- 70 placement decrease as well, with the understanding of keeping the concrete area of the cross-section the same. A detailed analysis of the parametric study results shows a very similar decrease in displacement with increasing tunnel diameter. E.g. an outer tunnel diameter of D = 6.8 m with h = 0.40 m increases to D = 10.67 m with h = 0.33 m, where the cross-sectional area stays similar, results in a threshold displacement decrease from 11.96 mm to 11.19 mm. However, the combination of both increasing inner diameter and lining thickness causes the displacement to increase. Larger diameters and thicker linings make the tunnel beam stiffer and able to resist more load. The sensitivity analysis showed that the same load level was reached with less displacements compared to smaller, thinner tunnels. But due to the stiffness increase of the tunnel lining, more load can be applied and in the end, this results in a larger displacement for the whole tunnel system.

4.7 Conclusions

The sensitivity analysis identifies variables that affect the tunnel response. The response sensitivities from a change in steel or concrete strength are less considerable than a change in tunnel geometry. The tunnel geometry has a much larger impact on the response sensitivity of the model than material properties. The sensitivity analysis is based on a fixed displacement, with an internally fixed load limit, and varies single parameters, such as the yield strength of the steel. For a numerical model with a reduced steel strength to reach the same fixed 71 displacement, less load needs to be applied. In other words, more displacement is possible. The parametric study, on the other hand, assumes a fixed limit state, such as the SLS, and the displacements as an independent variable. The sensitivity analysis results of an increasing displacement with decreased steel strength relates well to Figure 3.16 of the parametric study in chapter 3. Figure 3.16 shows that a numerical model with lesser steel strength results in less displacements to reach the fixed SLS compared to a higher steel strength. 72

5 STATIC FAULT OFFSET ASSESSMENT OF THE CONCRETE

LINING OF THE BERKELEY HILLS TUNNELS

5.1 Abstract

A growing population in urban areas need safe and resilient public transportation systems. Urban locations, like the San Francisco Bay or Los Angeles areas, are in addition situated in seismically active regions. Underground structures, such as water ducts or metro tunnels, must cross active faults to ensure population sustainability. Understanding the response of tunnels crossing active faults will further inform rehabilitation and maintenance measures and support tunnel safety. This report assesses displacements from the impact of static fault offset on concrete linings of existing tunnels. This assessment is for early stage projects, such as during a feasibility assessment. The Berkeley Hills tunnels crossing the active Hayward Fault was studied. The resulted displacements due to fault rupture are correlated to earthquake magnitudes to provide a general understanding of the impact. An assessment with published charts results in similar earthquake magnitude thresholds for concrete compression failure compared to a 2D numerical analysis. This shows that a simpliﬁed chart assessment, to predict an approximate threshold of an earthquake magnitude the concrete lining is able to withstand, is applicable for early stage projects. 73

5.2 Introduction

With a continuous population increase of urban areas, municipal authorities see the need to adapt the public network. Specifically, individual and public transportation systems in seismically active regions need to be brought up to code or expanded in size and number. Over decades, researchers and engineers, such as Hashash et al. (2001), Wang et al. (2012), Anastasopoulos et al. (2008), analyzed underground structures, like metro tunnels, with seismic wave impact. New urban development makes it necessary to consider tunnels crossing active fault zones. Common sense and technical guidance suggests to avoid crossing active faults (Hung et al., 2009), because active fault rupture creates intense bending and shearing of the underground structure. However, public water and transit tunnels in California cross active faults, such as the Berkeley Hills tunnels (BHT) that cross the active Hayward Fault. Figure 5.1 shows the 1968 constructed twin tunnels (Brown et al., 1981) of the BHT in relation to the San Francisco Bay area and two major faults, the San Andreas Fault and the Hayward Fault. New investigations show a 72 % probability of an earthquake magnitude 6.7 or higher occurring by 2043 in the San Francisco Bay Area (Detweiler and Wein, 2018). The main deliverable of the present study is to assess the reinforced concrete lining of an existing tunnel structure crossing an active fault to inform early stages of a retrofit project or the design of retrofitting measures. This study will assess the threshold displacements in the concrete lining due to fault offset of the BHT in two ways, (1) by use of charts, and (2) by numerical analysis, to infer capabil- 74

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Figure 5.1: Topographic map of the San Francisco Bay area, California, USA by U.S. Geological Survey (2017) with location of the Berkeley Hills Tunnels and two major faults, San Andreas and Hayward Fault. 75 ities. Graphical tools will help engineers to interpret ﬂexibility boundaries and earthquake thresholds of the concrete lining. This report is an application of a parametric study of 252 combinations of varying parameters - the inner tunnel diameter, lining thickness, fault zone width, and ground conditions. The parametric study is described in more detail in chapter 3.

5.3 State of research

5.3.1 Active faults

In a seismological sense, faults are planes of weaknesses that grow and develop by the cumulative action of earthquakes. Faulting is a sudden and dynamic rupture. When an earthquake occurs, it sends vibrations through the ground, which are strongest close to the fault plane and continue to the ground surface. The source of the vibration, or ground motion, is distributed along a section of the fault. And the intensity of the shaking is related to earthquake magnitudes. Researchers and engineers distinguish between three major classes of faults: normal, reverse, and strike-slip. A combination of two classes, e.g. normal and strike-slip, is often called oblique-slip. While normal and reverse faults show dip-slip displacements, a strike-slip fault displaces with a horizontal lateral motion on an inclined or near vertical plane (Scholz, 2019). Figure 5.2 illustrates a strike-slip fault with one principle slip zone (PSZ), but a fault zone can be composed of multiple slip zones and varying locations. The total 76

(a) (b) Figure 5.2: (a) Strike-slip fault with a vertical slip plane, (b) Varying geology of a fault zone with one vertical fault plane. PSZ: Principle Slip Zone. (after Sibson (2003))

slip might be distributed along several planes or the majority of the slip might occur at one PSZ, which is the case at the Hayward fault. Burford et al. (1978) found that slow slippage, sometimes called fault creep, is common to active faults. The fault gouge consist of weak material, often a hetero- geneous mixture of clay particles, which have lower friction and variably weath- ered rock matter (Scholz, 2019). The present study will concentrate on static fault offset with no fault creep. Brown et al. (1981) gives further information on creep impact by the Hayward Fault. Some of the active faults in the San Francisco Bay area in California are the San Andreas, Calaveras, and Hayward Faults (Detweiler and Wein, 2018). One 77 of the highest urbanized region in the United States is at the Hayward Fault with about 2.4 million people living there (Detweiler and Wein, 2018). The most recent damaging earthquake was in 1868. Earthquakes with a magnitude 6.3 and larger during the past 1900 years occurred in an interval between 97 and 183 years, on average 150 years. Detweiler and Wein (2018) say that a very powerful earthquake at the Hayward Fault can occur at any time, and is expected to severely damage residence and business buildings, private and public infrastructures and utilities. The probability that the earthquake occurs at the Hayward Fault is the highest with 33 %. A fault offset of up to 2 m might occur during a large earthquake of M6.8 to M7.0 including aftershocks.

5.3.2 Case histories of damages to tunnels through active faulting

Investigations of damages to tunnels have been reported by several researchers, such as Dowding and Rozan (1978), who created a list of 71 tunnels that experienced damage during an earthquake. Owen and Scholl (1980) found in connection with underground structure, that thick tunnel linings suffer more damage than thin linings, that are more ﬂexible. Sharma and Judd (1991) expanded previous data sets to 192 affected tunnels with information on ground behavior from 85 earthquakes. The following three examples are a review of case histories of existing tunnels that were subject to fault offset. BART Berkeley Hills Tunnels Seismic Retrofit Feasibility Study

Figure 3-1. Major Damage to Concrete Tunnel Lining in (a) Southern Pacific Railroad Tunnels No. 4, and (b) No. 5 (Steinbrugge and Morgan, 1954)

(a) BART Berkeley Hills Tunnels Seismic Retrofit Feasibility Study

(a) (b) Figure 3-3. Collapse Zone within Southern Pacific Railroad Tunnel No. 5 in the Tehachapi(b) Mountains following the 1952 Bakersfield Earthquake (Steinbrugge and Morgan, 1954) Figure 5.3: Damages of tunnel No. 5 of the Southern Paciﬁc Railroad documented after 1952 Bakersﬁeld earthquake: (a) Collapsed section, looking west, about 53 m inside the east portal, (b) major damage to concrete tunnel lining due to faulting. (Steinbrugge and Moran, 1954) Jacobs Associates and G&E Engineering Systems -25- Rev. No. 4 / January 6, 2014

5.3.2.1 Southern Paciﬁc Railroad Tunnel, Kern County, California, USA

The Bakersfield earthquake in 1952 ruptured with a moment magnitude (M) of 7.5 and the epicenter was near Bakersfield, CA. The Southern Pacific Rail crosses several faults while meandering between east and west. Multiple tunnels are part of the route just west of Bealville, CA which were damaged. Tunnel No. 5 expe- Jacobs Associatesrienced and G&E Engineering a lateral Systems displacement-27- of aboutRev. No. 4 1.5/ January m 6, 2014 (see Figure 5.3). (Steinbrugge and Moran, 1954) BART Berkeley Hills Tunnels Seismic Retrofit Feasibility Study

Figure 3-4. Damage to the Crystal Springs Outlet Tunnel following the 1906 San Francisco Earthquake (Lawson, 1908)

(a) (b) Figure 3-5. Collapsed Section of the Longxi Tunnel, China Following the 2008 Wenchaun Earthquake (Li, 2012) Figure 5.4: Damages of the Longxi road tunnels documented after the 2008 Wenchuan earthquake (M7.9), China: (a) Collapsed section at the fault crossing, (b) Jacobs Associates and G&E Engineering Systems -28- Rev. No. 4 / January 6, 2014 major damage to concrete tunnel lining due to about 1 meter fault offset. (McRae et al., 2014; Yu et al., 2016)

5.3.2.2 Longxi Road Tunnel, Sichuan, China

The Longxi twin tunnels are road tunnels crossing an active fault about 70 km west of Chengdu City, and only 18 km east of the epicenter of the 2008 Wenchuan earthquake. The fault zone is about 10 m wide and consists of highly fractured rocks, mostly sandstone and mudstone. The 2008 Wenchuan earthquake in Sichuan produced an earthquake with M7.9. The tunnel suffered heavy damages (see Figure 5.4), including a collapse at the fault crossing with major damages over 50 m tunnel length. (Lin et al., 2009; Yu et al., 2016) RETROFIT OF THE CLAREMONT TUNNEL 1135 80

Figure 6. Plan view of oversized tunnel section showing configuration of structural carrier pipe Figure 5.5: Planbefore view and after of 2.6 the m (8.5 retrofitfeet) of discrete section fault offset (HFZ: of Hayward the ClaremontFault Zone) water tunnel before and after 2.6 m (8.5 feet) localized fault offset (Caulfield et al., 2005). HFZ: Hay- depend strongly on the quality of geologic characterization carried out in the bypass ward Fault Zone.tunnel during construction. The first side drift will serve as a valuable pilot tunnel for such characterization. The potential for local blockage of the tunnel resulting from collapse of the tunnel crown has been addressed by incorporating an internal structural carrier pipe within 5.3.2.3 Claremontthe oversized water tunnel section. Tunnel, The carrier Alameda pipe is constructed County, of rolled steelCalifornia, with an USA internal diameter of 1.8 m (6 feet) and wall thickness of 76 mm (3 inches). The internal diameter was established based on minimum post-earthquake flow requirements of 492,000 cubic meters per day (130mgd) (assuming that complete blockage occurs around the carrier pipe), and the wall thickness was designed to accommodate ground The Claremontcollapse water loads. tunnel The pipe crosses is lightly restrained the Haywardby yieldable guide Fault posts at neareither end, the so BHT. It experi- that it is essentially free to rotate and shift as fault displacement occurs. As shown in Figures 5 and 6, the gap between the pipe walls and final tunnel lining is of sufficient ences creep. 35dimension years to after preclude construction the tunnel shell from was impinging completed on the carrier inpipe,1929 although (Caulfield et al., movement of the tunnel shell will cause the pipe to slide and/or rotate into the post- earthquake position. The ability for the pipe to move in a largely unrestrained manner 2005), the concretemeans linerthat the pipe in is the effectively area isolated of from the seismically Hayward induced Faultloading conditions. was cracked and the The discrete fault displacement depicted in Figure 6 is an idealization that results in conservative dimensions of the oversized tunnel section. While the primary fault invert was buckledoffset may (Brown occur across et a al., single 1981). well defined A plane, bypass it will more tunnel likely be covering distributed the Hayward across many planes over a distance of several meters. A distributed faulting condition Fault Zone washas finished the potential in to damage 2006 a with larger area an ofover-excavated the tunnel lining and increase profile the potential and an additional free-lying steel pipe to allow water to flow even after a fault offset, see Figure 5.5 (McRae et al., 2014). 81

5.4 Assessment strategies

A conventional tunnel liner design requires the analysis of the cross-section with ground or rock loads and other external pressures, like water. Tunnels, in general, behave better than above-ground structures during seismic shaking (Hashash et al., 2001), because they are embedded in one medium, the ground, and not divided into air and ground with varying resistance, like skyscrapers. However, fault ruptures produce a change in loading on the existing tunnel. The amount and orientation of the loading depend partially on the tunnel alignment relative to the fault movement, which causes compression or extension in addition to shear. Additional effects impacting tunnels crossing an active fault are the sense of fault movement - dip-slip or strike-slip, the amount of displacement, the fault zone width over which displacement occurs, the ground conditions left and right of the fault zone, and the dimensions of the underground opening. For this study, the authors are looking at fault movements, which can be found at strike-slip faults, but also at normal faults. These transversal fault movements affect the tunnel lining of underground structures, which is the long-term main support of the underground opening. This report studies circular reinforced concrete linings of tunnels.

5.4.1 Assessment with limit states and correlations

An assessment is the act of judging or deciding the value, quality, or importance of something. To identify and quantify the behavior of reinforced concrete, several 82 limit states are addressed:

1. SLS: Service Limit State, which corresponds to the start of yielding of steel, and

2. ULS: Ultimate Limit State, which symbolizes concrete failure at concrete peak compressive strain.

Both limit states are expressed in strains. When ﬁrst cracks appear, this is a sign that the steel is now taking on tension the concrete cannot take anymore and therefore the steel starts to yield. A lower steel strength is often found in existing tunnels that were built in the mid-20th century. With steel strength of fs = 275 MPa (U.S. grade 40) and the initial Young’s modulus of steel of E = 200 000 MPa, yielding of steel calculates to 275 MPa / 200 000 MPa = 0.14 % strain. The maximum compressive strain of a concrete at 28 days is called the concrete peak compressive strain. It is set to 0.3 % after the building code of the American Concrete Insitude (ACI) 318-14 (chapter 10.2.3). Important note to the reader: For the purpose of this study, no factors of safety are regarded. In addition to limit states of the reinforced concrete, an equivalent earthquake magnitude threshold is taken into consideration to provide a general understanding of the impact. With a correlation from Wells & Coppersmith (1994), it is possible to estimate the earthquake magnitude based on fault offset displacements of the tunnel lining. M = 6.81 + 0.78 · log(MD) (5.1)

(One standard deviation: 1σ = 0.34 unit points) 83 where M is earthquake magnitude and MD the maximum surface displacement due to strike-slip faulting. Correlations between earthquake magnitude and other rupture parameters, such as normal fault displacements, can be found in Wells and Coppersmith (1994). Two different paths of assessments are looked at: (1) with charts and (2) through numerical analysis. Both methods are described below. Comparison and evaluation with an example application follow in chapter 5.5, results.

5.4.2 Assessment set-up with charts

The ﬁndings of chapter 3, the parametric study, are pictured in evaluation charts to assess threshold displacement capacities of tunnel beams crossing active faults. These graphs show two limit states and correlate to earthquake magnitudes from strike-slip faulting. A set of these evaluation charts simplify a preliminary assessment of a tunnel. The following parameters need to be deﬁned:

• Equivalent inner tunnel diameter, Di,

• Equivalent concrete lining thickness, h,

0 • Concrete strength, fc,

• Fault geometry, speciﬁcally the width of the fault zone, Lzone, and

• Rock conditions surrounding the tunnel outside of the fault zone.

The charts assume a vertical fault plane and a perpendicular tunnel crossing of 84 the fault. Inﬂuence by other parameters, like concrete strength, was determined with a sensitivity analysis in chapter 4. The results of the sensitivity analysis show that the change in concrete strength has a small inﬂuence on the response sensitivity of the numerical model. Thus, initial stages of concrete creep have a minor impact. However, the long-term behavior might have an impact, but needs more research. The long-term behavior of concrete creep is outside the current scope of this research. The results of the sensitivity analysis show further, that the tunnel geometry parameters have a much larger impact on the response sensitivity of the numerical model than material properties within the limits of this study.

5.4.3 Assessment set-up for numerical analysis

The second method to assess a tunnel is by numerical analysis. The analysis uses the open source finite element software framework OpenSees (McKenna et al., 2010). The numerical model consists of simplified 2D beam elements with soil springs. Displacement controlled forces reflect the fault offset input. The tunnel beam section, to assess the BHT, is 300 m long and stretches over the fault zone width. The beam is fixed in all three directions at the west end and only fixed in two directions (axial and moment) at the east end. The y-direction is kept free to allow offset input. The beams are discretized in 5 m element lengths with elastic-no-tension soil springs at each element end. The tunnel cross-section contains of discrete fibers as circular patches and layers creating a reinforced concrete 85 tube with non-linear material behavior. Longitudinal reinforcement strengthens the tunnel lining with an outer and inner layer. Figure 5.6 visualizes the numerical model. Output in form of axial force-moment (N-M) interaction diagram, limit state displacements, strains, shear forces, and moments along the tunnel alignment will show the results and be used for assessment.

5.5 Example study Berkeley Hills Tunnels

5.5.1 Overview

The Berkeley Hills tunnels are 5 km (3.1 miles) long twin tunnels, meaning two parallel tunnels of similar geometry, owned by the Bay Area Rapid Transit (BART). They pass through the Berkeley Hills formation between stations Rockridge and Orinda east of Berkeley, CA and transport around 8200 passengers during one rush hour (BART, 2016). About 300 m from the west portal at Rockridge station, the twin tunnels cross the Hayward Fault. The Hayward fault zone along the tunnel alignment has a major principle slip zone of 30 m. The Hayward Fault is a highly active right lateral strike-slip fault where large earthquakes with surface ruptures occur. In addition to earthquakes, the Hayward Fault experiences creep which results in concrete liner cracking of both tunnels (cf. Figure 5.7). (Brown et al., 1981) 86

Plan view: 30 m fault zone width y, North Major fault plane Tunnel beam with D i and h Tunnel offset x, East section Offset

L = 300 m

Crossing angle 90° Displacement controlled A Tunnel tunnel offset

A x, East (a) Tunnel Cross Section A-A Outside diameter Lining thickness, h Concrete lining Tunnel Outer clear cover 8 cm

Inner clear cover 5 cm Inside diameter, D i

Rock Longitudinal rebars (b) (Inner and outer layer) Figure 5.6: Computer model visualization. (a) Plan view of 2D beam-spring model with fault offset. Springs are shown on one side for clarity, but modeled on both sides. (b) Tunnel cross-section A-A with discretized ﬁber sections of the concrete lining. Vehicle and installation material are shown schematically. 87

Figure 5.7: Crack pattern of tunnel lining due to fault creep compared to geology (Brown et al., 1981, p. 90) 88

1 - KJfm: Franciscan complex mélange (Cretaceous, late Jurassic) -- Sheared black argillite, graywacke, and minor green tuff.

2 - Jsv: Keratophyre and quartz keratophyre (late Jurassic) -- Highly altered intermediate and silicic volcanic and hypabyssal rocks.

3 - Ku: Undivided great valley complex rocks (Cretaceous) -- Sandstone, siltstone, shale, and minor conglomerate. Claremont Tunnel 4 - Tsm: Unnamed glauconitic mudstone (Miocene) -- Brown mudstone is interbedded with sandy mudstone. 7 9 7 5 - Tcc: Claremont chert (late to middle Miocene) -- Hayward Fault Laminated and bedded chert, minor brown shale, and 8 white sandstone. 7

6 - Orinda Formation (late Miocene) -- Distinctly to 6 indistinctly bedded, nonmarine, pebble to boulder Berkeley5 Hills Tunnels conglomerate, conglomeratic sandstone, coarse- to 4 medium-grained lithic sandstone, and green and red 3 siltstone and mudstone. 2 7 - Tmb: Moraga Formation (late Miocene) -- Basalt and 1 andesite flows, minor rhyolite tuff.

8 - Tst: Siesta Formation (late Miocene) -- Nonmarine siltstone, claystone, sandstone, and minor limestone.

9 - Tms: Interflow sedimentary rocks.

16.5° Digital Cartography by R.W. Graymer APPROXIMATE MEAN and R.E. Stamski, 1999-2000 DECLINATION, 1980 1 .5 0 1 2 3 4 5 Kilometer

Edited by Jan Zigler 1 .5 0 1 2 3 Mile U.S. Geological Survey Digital Raster TRUE NORTH MAP LOCATION Graphic (DRG) MAGNETIC NORTH

Figure 5.8: Summarized geology across Hayward Fault along the BHT.

5.5.2 Geology

The geology of the Berkeley Hills area is very complex and is indicative of a highly seismically active region (Graymer, 2000). Figure 5.8 shows the geology across the Hayward Fault zone along the BHT. Starting from west to east, the geology across the fault consists of (1) KJfm - Fanciscan complex melange:´ sedimentary rocks, (2) Jsv - Keratophyre and quartz keratophyre: volcanic rocks, and (3) Ku - Undivided great valley complex rocks: sedimentary rocks. The general rock strength can be interpreted as weak to medium strong rocks after ISRM (1978). The spring stiffness is a function of the modulus of subgrade 89 reaction, the tributary tunnel length, and tunnel diameter. The rock strength of the fault gouge is comparable to the uniaxial compressive strength, σ, of 0.25 MPa and that of the surrounding rock is between 5 MPa and 50 MPa. The corresponding modulus of subgrade reaction are 100 MN/m3 for the fault gouge and between 1000 MN/m3 and 15000 MN/m3 for the surrounding rock. The sedimentary rocks contain sheared and slickensided fragments left and right of the large fault zone, see Figure 5.7. Therefore, both material properties of the ground are assumed and analyzed. The complex fault zone of the Hayward Fault has a total width along the tunnel alignment of up to 300 m and includes multiple slip zones. The major slip zone, with the dominant offset value, has a width of 30 m.

5.5.3 Tunnel design

The original tunnel design considered fault rupture. With an assumption of a distributed displacement of 0.61 m horizontally and 0.305 m vertically along 100 m of tunnel across the Hayward Fault zone, the design included a possible oblique tunnel displacement of 1.3 m without the need for a larger tunnel clearance. The train tracks were designed to sit on wooden ties to allow easy realignment after a permanent ground offset. Creep was not considered during the design of the BHT. High lateral and vertical pressures exist on the tunnel lining in the Hayward Fault zone. Due to high rock loads, a near circular proﬁle was chosen during the design phase, a horseshoe shape was constructed in the end (see Figure 5.9). The twin tunnels have an equivalent inside diameter of Di = 5.33 m with a design liner thickness 90 of h = 0.457 m. Brown et al. (1981) and McRae et al. (2014) performed numerical analyses to assess pressure cell reading and to study seismic retroﬁt alternatives, respectively. Both reports use a thicker tunnel lining of 0.75 m to 0.8 m with no referenced resources. However, the required lining thickness by design will be used for further analysis. The reinforcement ratio for special tunnel sections calculates to 3.1 %; the standard reinforcement ratio design is around 1.5 %. Steel has a strength of fs = 275 MPa (U.S. grade 40) and the concrete type is a C 20/25 with a 0 compressive strength of fc = 20 MPa (3000 psi). (Brown et al., 1981)

5.5.4 Tunnel construction

The construction crew encountered difﬁculties during construction crossing the highly squeezing Hayward Fault zone. Timber and steel struts had to be placed in the invert (bottom) to stop the movements, which caused the side posts to be pushed towards each other, meaning into the tunnel. Figure 5.10 shows the timber and steel struts placement during construction. Construction of both tunnels were completed in 1968 after only three years. Further, they found water inﬂow (about 40 liters per second at any one face) during construction. To deal with the water, and reduce the hydrostatic pressure, the tunnel was constructed with a drainage system (Brown et al., 1981). The concrete liner cracking pattern in Figure 5.7 was cataloged in 1979, 11 years after completion, with vertical and normal cracks compared to the tunnel axis. These cracks were found over a length of 585 m crossing the Hayward Fault. Es- 91

Figure 5.9: Cross-section of BART Berkeley Hills tunnels with the liner design Brown et al. (1981). All dimensions in mm but U.S. steel sizes. E.g., the #9 bar has a nominal diameter of 28.65 mm. 92

BART Berkeley Hills Tunnels Seismic Retrofit Feasibility Study

Figure 2-3. Initial Ground Support Consisting of Steel Ribs and Invert Struts Used in the BART Figure 5.10: Use ofBerkeley timber andHills Tunnels steel struts within during the Hayward construction Fault Zone of the BHT (McRae et al., 2014) 2.1.2 Existing Conditions

Ongoing creep within the Hayward Fault Zone has resulted in offset, cracking, and some spalling of the tunnel lining over approximately 600 feet of tunnel. A detailed description of the condition of the tunnel lining within the Hayward Fault Zone is presented by Brown et al. (1981) and WLA (2006). Typical tunnel damage is shown in Figures 2-4 and 2-5. Using survey data compiled between 1985 and 1999, it can be determined that the tunnel has been subjected to ongoing creep in the reach of tunnel between Stations 1200+30 and 1201+30 at a rate of approximately 0.14 in./yr (3.5 mm/year, WLA, 2006). Using this average creep rate, it can be determined that the tunnel has been subjected to approximately 6 inches of offset since tunnel construction was completed. This offset has resulted in minimal remaining clearance to the train dynamic envelope beyond that required by the PUC between Stations 1200+30 and 1201+30.

Jacobs Associates and G&E Engineering Systems -13- Rev. No. 4 / January 6, 2014 93 pecially across a 30 m length crossing the main slip zone in Figure 5.7, a comparison between the crack pattern and the geology illustrates the relationship between ground and liner. The fault along the tunnel shows maximum shear displacement at the major fault plane. The crack pattern also indicates an extension of the tunnel in the axial direction due to the strike-slip movement of the fault. But a detailed analysis of the crack pattern due to fault creep is outside of the scope of this research. Also it is to note, that there has not been a large enough earthquake to damage tunnels crossing the Hayward Fault at the Berkeley Hills, including the BHT but also the Claremont water tunnel, which received a seismic retroﬁt in 2006.

5.5.5 Assessment results with charts

The charts for the assessment are chosen from chapter 3. Both charts are for a tunnel system with a 6 m inner tunnel diameter, Di, a 30 m wide fault zone width,

Lzone, varying lining thicknesses, h, and a comparison between weak rock and medium strong rock. To visualize the BHT results, the charts from chapter 3 are adapted with a smaller displacement range and larger lining thicknesses. Input parameters of the BHT are the equivalent inner diameter of 5.33 m and a lining thickness of 0.457 m. The spring value of the rock strength is weak to medium strong. Taking the graphical tool, the BHT should be placed below the thresholds due to a smaller diameter. The charts in Figure 5.11 visualize the estimated data points of the BHT. 94

(a) Weak rock (b) Medium strong rock ( = 5 MPa) ( = 50 MPa)

ULS ULS

SLS SLS Threshold displ. (m) Threshold displ. (m) Earthquake magnitude Lining thickness, h (m) Earthquake magnitude Lining thickness, h (m)

Figure 5.11: Chart excerpt from chapter 3 with BHT data, h = 0.457 m with varying ground condition. One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points.

The maximum displacement until the steel yields for weak rock is just below 0.03 m and about 0.065 m until crushing of concrete starts. An earthquake with a magnitude of around 6.0 might fail the concrete lining. For medium strong rock, the results are lower. The displacement until yielding of steel is about 0.02 m and until crushing of concrete about 0.04 m. The graph for medium strong rock is not clear on the earthquake magnitudes. Using equation 5.1 with both displacement estimates, the concrete lining might start to fail at an earthquake magnitude of around M5.5.

5.5.6 Assessment results with numerical analysis

Fault offset introduces shearing into the tunnel beam, which results in shear forces, V, but also axial forces, N, and bending moments, M along the tunnel alignment. The N-M-interaction diagram in Figure 5.12 visualizes the state of internal forces 95

Yielding of steel at = 0.14% (at ) 105 1 -2.5 Peak compressive concrete strain is = -0.3% (at ) 2 N-M data for weak rock ( = 5 MPa) N-M data for medium strong rock ( = 50 MPa) -2

-1.5

-1 Axial Force, N (kN)

-0.5

0 0 0.5 1 1.5 2 2.5 3 Moment, M (kNm) 105 Figure 5.12: N-M-interaction diagram for the BHT cross-section with reinforced concrete, an diameter of 5.33 m, and a lining thickness of 0.457 m. of the loaded structure compared to its flexural capacity. Concrete shear is outside of the scope of this research and not considered. The graphs show data at yielding of steel and at peak compression concrete strain for the two rock characteristics of weak and medium strong rock. Figure 5.12 is specific for Di = 5.33 m, h = 0.457 m, and Lzone = 30 m. The displacement in Figure 5.13a shows double curvature bending due to a right-lateral strike-slip fault offset. The maximum displacement to reach peak 96 compression concrete strain computes to 6.2 cm for weak rock. Subfigure (b) visualizes the concrete strains along the tunnel alignment. The peak occurs near the major fault plane. The shear force and moment lines in subfigures (c) and (d), respectively, illustrate as well a sharp change at the major fault plane. Figure 5.14 shows a threshold displacement versus lining thickness chart with the limit states for the BHT for weak and medium strong rock based on the results from Figure 5.13. With equation 5.1, the threshold of earthquake magnitudes for weak rock lies at around M6.0, the threshold for medium strong rock at around M5.5.

5.6 Discussion

Both results from the chart assessment and the numerical analysis yield the same earthquake magnitude thresholds. Placing the estimated data points lower on the reference chart in Figure 5.11 resulted in very similar displacements compared to the numerical analysis. For a preliminary investigation, using the charts conservatively, places the tunnel in similar ranges of threshold displacements and earthquake magnitude ranges compared to a simpliﬁed numerical analysis. The crack pattern recorded in 1979 shows a cluster of cracks over a length of 30 m - about 20 m before and 10 m after the fault plane. These cracks formed due to continuous creep over the previous 11 years. The strain pattern along the tunnel beam from the numerical model in Figure 5.13b shows a length of 15 m that has reached yielding of steel due to a static fault rupture. Both results compare positive in terms of conﬁned cracking across the major fault plane. But further research 97

10-3 18 Legend: -0.04 Legend: 16 ____ Weak rock = 5 MPa 14 -0.02 - - Medium strong rock, = 50 MPa 30 m fault zone width Major fault plane 12 Major fault plane 0 10 8 0.02 3.6 cm 6 4 Crushing of concrete at 0.3% 0.04 Concrete strain (-) 2 6.2 cm Yielding of steel at 0.14% 0

Threshold displacement (m) 0.06 30 m fault zone width -2 ~ 15 m 0.08 -4 -150-125-100 -75 -50 -25 0 25 50 75 100 125 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 (a) Longitudinal tunnel axis (m) (b) Longitudinal tunnel axis (m) 104 105 17.5 4 Legend: Legend: 15.0 - - Medium strong rock 3 - - Medium strong rock 12.5 Major fault plane Major fault plane 10.0 2 7.5 1 5.0 0 2.5

0.0 Moment, M (kNm)

Shear force, V (kN) -1 -2.5 -2 -5.0 30 m fault zone width 30 m fault zone width -7.5 -3 -150-125-100 -75 -50 -25 0 25 50 75 100 125 -150-125-100 -75 -50 -25 0 25 50 75 100 125 (c) Longitudinal tunnel axis (m) (d) Longitudinal tunnel axis (m) Figure 5.13: (a) Displacements, (b) strains, (c) shear forces, and (d) moments along the tunnel axis for the BHT with Di = 5.33 m and h = 0.457 m for Lzone = 30 m (fault zone width), with varying geology outside the fault zone. One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points. 98

(a) Weak rock (b) Medium strong rock ( = 5 MPa) ( = 50 MPa)

ULS ULS

SLS SLS Earthquake magnitude Earthquake magnitude

Threshold displacement (m) Lining thickness, h (m) Threshold displacement (m) Lining thickness, h (m)

Figure 5.14: Displacement vs. lining thickness for the BHT with correlation to earthquake magnitudes. Results from Figure 5.11 are shown as estimated data points with a black c One standard deviation of earthquake magnitude, 1σ, is 0.34 unit points.ircle. needs to compare in detail the crack patterns due to creep to that of fault rupture. Researchers might be able to infer the length of affected or collapsed tunnel after an earthquake. Case histories showed tunnel linings dislocated to more than their thickness. Offsets of 1 m and more might lead to tunnel collapse. Even at smaller displacements, once the peak compressive strength of concrete is reached, the concrete strength decreases, resulting in a faster accumulation of displacements with less forces/pressure. After an event, fundamental surveys of the existing condition and following rehabilitation measures are necessary to ensure tunnel safety and service. Retrofit strategies might include hinges as part of the concrete lining or over-excavation as an articulate and flexible system to spread fault rupture over a longer distance. Post-stressing of the tunnel segment crossing the active fault will increase the axial force acting on the concrete lining, but also increases the mo- 99 ment capacity and therefore the rotational abilities. But the effectiveness of post- stressing is a question of existing ground conditions and the estimated earthquake intensity, among others. Additional research is helpful to fully understand the response of concrete- lined tunnels crossing active faults. Among other subjects, research might incor- porate the influence of a twin tunnel close-by. Further, heterogeneity of faulting, oblique faulting in horizontal and vertical directions, influences the response of underground structures and might prove important. A possible re-alignment of tunnels during a rehabilitation measure with the knowledge of fault activity on or near the most recent active fault traces (Taylor and Cluff, 1977) might be examined in future studies. The question of how fluids influence the fault rupture at tunnel depth is an issue for future research to explore.

5.7 Conclusions

This report is an investigation of the response of the concrete lining of the Berke- ley Hills tunnels crossing the active Hayward fault. It is an example application of an already published parametric study to assess displacement thresholds due to earthquake faulting, where reaching the peak compressive concrete strain is defined as failure of the concrete lining. Two assessment strategies were looked at, (1) with charts and (2) with numerical analysis. For the present report, four parameters were used as input variables - tunnel dimensions of inner diameter and lining thickness, fault zone width, and rock strength. The assessment is meant for 100 an early stage project, such as a feasibility study. The assessment uses charts conservatively, however, places the limit state displacement capacities of the tunnel lining in similar ranges compared to a simplified 2D numerical analysis. To create the numerical model, detailed information is necessary, but are highly uncertain during early stage projects. The careful chart assessment shows an alternative. Case histories showed that underground structures are vulnerable to seismic impact, especially at active fault crossings. The displacements of the case studies are between 1 m and 1.5 m and led to tunnel collapse. A 2 m fault offset due to the pending earthquake in the Bay area with M6.7 or greater, might result in severe tunnel damage. The structure cannot be build to withstand the forces from such a large earthquake. But research shows that earthquakes of around M5.5 can be accommodated by the design tunnel lining of the Berkeley Hills tunnels. How time, ground water, and fault creep influence the concrete strength and lining thickness of the existing tunnels is an open question and should be analyzed in conjunction with detailed surveys. Retrofit strategies, besides an articulate design and over-excavation, might include replacement of a tunnel section across the fault and enhanced with post- stressing to increase curvature capacity. Future work, investigating the influence of twin tunnels or other neighboring structures might prove important. Further, the heterogeneity of faulting affects the response of underground structures and might be explored in future studies. 101

6 CONCLUSIONS

Examining the response of concrete lined tunnels built through active fault zones is critical to ensure resilient design and safe operation. To address the lack of a generalized interpretation framework, a parametric study of a 2D beam-spring model of a tunnel crossing an active fault was evaluated. Case histories showed that underground structures are vulnerable to seismic impact, especially at active fault crossings. Damage localizations were monitored for the Bolu tunnels that suffered active fault shear failure due to the 1999 M7.2 Duzce¨ earthquake in Turkey, but also for the Longxi tunnel collapse due to an active fault offset of 1 m during the 2008 M7.9 Wenchuan earthquake in China. The research is based on three main research questions. (1) Can we give guidance when it is okay to cross an active fault? (2) What is the influence of faulting on the tunnel lining in cross-section and longitudinal direction? and (3) What consequences and generalizations can be drawn to support serviceability? Uniaxial stress states are considered for this study. Failure of an underground structure is considered to be a structural failure. The failure criterion is defined by reaching the peak compression concrete strain at the ultimate limit state with bending of the tunnel beam. Concrete shear is outside of the scope of this research. Steel is defined in axial tension. Further limitations of this research is the use of static fault rupture not coupled with ground shaking nor other external forces. 102

Answering research question (1): Earthquakes with a magnitude M6.0 and greater can damage underground structures in seismically active regions. Directly at the fault, an earthquake with a magnitude of about M5.5 coupled with unfavorable conditions, such as medium strong rock or a short fault zone width and a small tunnel diameter, might damage the underground structure until concrete failure. On the other hand, an earthquake up to around M6.5 can be withstood by large tunnels in weak rock conditions until concrete failure occurs. Maximum limit state displacements of the tunnel beam, until the reinforced concrete is in failure, varies between 0.2 % and 2 % of the inner tunnel diameter - from 0.03 m up to 0.18 m. A doubling of the tunnel lining thickness results in an increase of possible displacements by 33 %. A similar result shows the increase of inner tunnel diameter. From an increase from 6 m to 10 m (+ 67 %) derives a 25 % increase in displacements. However, the response sensitivity of the numerical model from a change in steel or concrete strength is less considerable than a change in tunnel geometry.

Answering research question (2): The tunnel lining cross-section shows concentrated compression either at the crown or the invert, depending on the location along the tunnel longitudinal axis; about 2/3 of the cross-section is in tension due to the fault impact. Further results of the numerical model with varying ground and tunnel characteristics show a localization of damages around the major fault plane along the tunnel longitudinal 103 axis. The computed localization of higher stresses averages to around 60 m tunnel length - about 30 m before and after the major fault plane.

Answering research question (3): Results show an interdependence between the fault zone width and the by damage affected length of tunnel. Fault zone widths of 30 m and longer do not show any difference in results. Shorter fault zone width result in smaller threshold displacements but similar affected length of tunnel. A variation in fault zone width shows a less risky behavior for the tunnel lining if the fault zone width is 1 to 2 times the tunnel diameter. Shorter fault zone widths show an abrupt shearing of the tunnel structure and result in lesser threshold displacements.

In summary, guidance to engineers in regard to the research questions include a possibly reduced length of retrofit measures across an active fault zone based on localized strains and stresses in the concrete lining. The flexural displacement capacity of the tunnel beam ranges between 0.2 % and 2 % of the inner tunnel diameter. A corresponding earthquake magnitude threshold for reaching peak compression concrete strain can lie as low as M5.5 at active fault crossings. A generalization that can be drawn is the interaction between the tunnel diameter and the fault zone width. A fault zone width of less than 1 to 2 times the tunnel diameter might be a larger concern due to abrupt shearing. Retrofit strategies, besides an articulate design and over-excavation, might include replacement of a tunnel section across the fault and enhanced with post-stressing to increase curvature ca- 104 pacity. To account for repairs of damaged structural elements after an earthquake event, the localization of damages should be taken into account. Structural hinges before and after the fault might help concentrate lining damage.

However, it is important to understand the implications and applicability of this research through an industry application for an existing tunnel crossing an active fault. The investigation of the response of the concrete lining of the Berkeley Hills tunnels crossing the active Hayward Fault is an example application. Two assessment strategies were looked at, (1) with charts and (2) with numerical analysis. The assessment with charts was done conservatively, however, places the limit state displacement capacities of the tunnel lining in similar ranges compared to a simpliﬁed 2D numerical analysis. To create the numerical model, detailed information is necessary, but are highly uncertain during early stage projects. The chart assessment shows an alternative to a preliminary numerical model. Case studies with damaged and collapsed tunnel linings due to active fault offsets experienced offsets between 1 m and 1.5 m. The pending earthquake in the Bay area with M6.7 or greater, might result in a 2 m fault offset and can result in tunnel collapse. The structure cannot be build to withstand the forces from such a large earthquake. But research shows fault offsets from earthquakes of around M5.5 can be accommodated by the design tunnel lining of the Berkeley Hills tunnels. How time, ground water, and fault creep inﬂuence the concrete strength and lining thickness of the existing tunnels is an open question and should be analyzed in conjunction with detailed surveys. 105

This assessment is designed for the use in early stage projects, such as a feasibility study, and by no means eliminates detailed surveys and analyses in support of the success of an underground structure project. Further research needs to be done to address limitations of this work, e.g. by differing modeling approaches, such as shell elements for the tunnel beam, or a consideration of flexure-shear-interaction of the reinforced concrete with the modified compression-field theory (MCFT) (Bentz et al., 2006). In addition, beyond the scope of this research study are: (a) Varying the angle of the fault crossing; (b) including different fault types; (c) simulating static and dynamic (creep) fault displacement for numerical modeling; (d) combining static and dynamic fault displacement and dynamic tunnel shaking due to earthquake waves; and (e) considering the heterogeneity of faulting which affects the response of underground structures and might be important. (f) Another aspect of future work is the seismic wave amplification, that might happen in mountains where tunnels are often build though. Further research in sight of engineering applications, needs to be done to investigate gaps between concrete lining and surrounding rock after an event, which might show undesired water ways even further away from the fault plane. Investigating the influence of twin tunnels or other neighboring structures during earthquakes might prove important as well. 106

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APPENDICES 114

Appendix A - Industry input

To stay close to practitioners, questions of practical ranges of parameters con- cerning earthquake magnitudes, tunnel lining structure, and concrete and steel strengths were asked to half a dozen leading tunnel engineering design and construction companies in the U.S. The main results are summarized below:

• Earthquake magnitudes: large earthquakes are considered to start at around M6.0.

• Tunnels build in earthquake zones in the Paciﬁc Northwest are mainly in soft ground, such as sandy/gravely soils, up to medium strong rock material.

• Ground materials encountered were in the line of claystone, siltstone, sandstone, glacial till.

• In general, the dimensions of a circular tunnel can be seen as the ratio of the

thickness of the tunnel lining over the inner diameter of the tunnel, h/Di.

• The thickness of segmental lining lies in general between 0.15 m and 0.75 m (DAUB, 2013).

• The range of the geometric ratio lies between 1/15 (smaller diameters) and 1/30 (larger diameters), with 1/20 being the most found ratio, see Table 1; 115

• The initial support system (shotcrete and bolts, or segmental lining) can be considered to share the load between the initial support and the ﬁnal lining of up to about 50 % of the ground loads, but is generally not done due to quality issues with the shotcrete during construction.

0 0 • Concrete strengths range between C25/30 ( fc = 4000 psi) and C40/50 ( fc = 6000 psi), and steel strengths range between 275 MPa and 500 MPa (U.S. grade 40 and 75).

• There are two general approaches to tunnel design with large ground loads or in this case large ground deformations: either the structure withstands the loads or it yields to it. In the case of fault offsets, it is not advised to build thicker tunnel lining since it attracts even more forces and will fail faster.

Table 1: Tunnel dimensions

Tunnel Tunnel lining thickness, h (m) Diameter Di (m) 0.2 0.25 0.3 0.35 0.4 0.5 0.6 0.75 1.0 5 1/25 1/20 1/17 1/14 1/12 1/10 1/8 1/7 1/5 6 1/30 1/24 1/20 1/17 1/15 1/12 1/10 1/8 1/6 8 1/40 1/32 1/27 1/23 1/20 1/16 1/13 1/11 1/8 10 1/50 1/40 1/33 1/29 1/25 1/20 1/17 1/13 1/10 12 1/60 1/48 1/40 1/34 1/30 1/24 1/20 1/16 1/12 15 1/75 1/60 1/50 1/43 1/38 1/30 1/25 1/20 1/15 18 1/90 1/72 1/60 1/51 1/45 1/36 1/30 1/24 1/18 20 1/100 1/80 1/67 1/57 1/50 1/40 1/34 1/27 1/20 116

Appendix B - Correlation of surface displacements to moment

magnitude

The behavior of concrete-lined circular tunnels crossing active faults is studied by applying a static fault offset or displacement to the tunnel structure. The offset value is a varying input parameter depending on the size of the earthquake. Since earthquake magnitude ranges are better known by researchers and the public alike, the offset displacement values are compared to earthquake magnitudes to obtain a better and more immediate understanding of the size of the earthquake needed to obtain such an offset.

Past research shows a high correlation of various rupture parameters such as surface displacement or rupture length to moment magnitude (e.g., Wells and Cop- persmith, 1994; Wesnousky, 2008). A fault offset from a strong earthquake is a displacement of the two continental plates, which manifests in a surface displacement. The question of why to use surface displacements as an input parameter on an underground tunnel structure arises here, which can be answered with the ﬁnd- ings of Wells and Coppersmith (1994) of the study of the relationship of surface to subsurface displacements. Their comparison of average subsurface displacement to maximum surface and to average surface displacements respectively show no variation as a function of earthquake magnitude. As a result, the distribution of 117 the data shows that the maximum surface displacement is more than the average subsurface displacement, which in return is more than the average surface displacement. Furthermore, the slip at seismogenic depth on the fault plane due to large earthquakes with magnitudes larger than M6 is visible as surface displacements. The seismogenic depth is the nucleation zone of earthquakes and where the slip on the fault plane occurs. Moreover, a geologic subsurface displacement is deﬁned as the displacement at the seismogenic depth of that region. The seismogenic depth varies widely between regions, but is at about 5 to 15 km below the surface (Scholz, 2002).

On grounds of the study of Wells and Coppersmith (1994) and the deﬁnition of subsurface in the geologic sense, it can be inferred to use the surface displacement as input parameters for fault offsets on underground tunnel structures.

Many researchers such as Wells and Coppersmith (1994), Wesnousky (2008), Hanks and Bakun (2008), Stirling et al. (2013) have developed several correlations between multiple rupture parameters and earthquake magnitudes to create scaling relationships for seismic-hazard analyses. Especially Stirling et al. (2013) listed a large compilation and comparison of magnitude-area and magnitude-length scaling relationships from 13 different sources. But Wells and Coppersmith (1994) modeled the relationship between moment magnitude and surface displacement among other relationships for various faulting regimes which will be the basis of the adapted regression model for this research. 118

Wells and Coppersmith (1994) used data from 244 historical earthquakes from the 1857 Ms8.3 Fort Tejon, CA, earthquake to the 1993 M6 Eureka Valley, CA earthquake to develop the empirical regression models. Note that Ms is the surface magnitude. Fourty-three events occurred on strike-slip faults and are the basis for the following regression equation for maximum surface displacement:

M = a + b · log(MD) (1) where M stands for moment magnitude, MD is maximum surface displacement in meters (m), a and b are equation coefﬁcients with a = 6.81 and b = 0.78. In addition to the maximum displacement, net displacement measurements along the entire surface rupture of the fault line are used to determine an average displacement per event. For the regression equation of average displacement and moment magnitude twenty-nine historical earthquake events were used. The regression equation has the same basis as for the maximum displacement, only with varying coefﬁcients a and b with a = 7.04 and b = 0.89.

To include recent large strike-slip earthquakes into the strike-slip regression models of Wells and Coppersmith, data from the well cited Wesnousky (2008) paper are added to the list, which can be viewed in Table 6 at the end of this chapter. The adjusted regression equations with the basis of the well trusted Wells and Copper- 119

smith (1994) regression model result into the following:

M = 6.712 + 0.912 · log(MD) (2)

M = 6.996 + 0.948 · log(AD) (3) where M stands for moment magnitude, MD is maximum surface displacement in meters (m), and AD is average surface displacement in meters (m). Figure 1 and Figure 2 show the data points from Wells and Coppersmith (1994) and Wesnousky (2008) as well as the regression lines from Wells and Coppersmith (1994) and the adjusted regression line, which takes all shown data into account.

The Table 2 and Table 3 list a range of displacements with correlated earthquake magnitudes to compare the Wells and Coppersmith (1994) regression models with the adjusted models. It can be shown that the two models give similar results for both the maximum and average surface displacements with less than M0.2 difference.

Table 2: Maximum surface displacements correlated with moment magnitudes.

Maximum Displacement - Fault offset, m Correlated Magnitude 0.1 0.25 0.5 1.0 1.5 2.0 5.0 10.0 20.0 Wells and Coppersmith (1994) 6.0 6.3 6.6 6.8 6.9 7.0 7.4 7.6 7.8 Adjusted regression 5.8 6.2 6.4 6.7 6.9 7.0 7.3 7.6 7.9

The conservativeness of the adjusted regression model can be compared to the 120

Wells and Coppersmith (1994) calculated maximum surface displacements as seen in Figure 3. The reference line at MDall = 1 symbolizes the same computed maxi- MDWC mum displacements with both regression equations.

8.5 (a) M = 6.81 + 0.78*log(MD)

(b) M = 6.712 + 0.912*log(MD) ( + 1 ) 8

7.5 ( - 1 )

Moment Magnitude, M 6.5

(a)

6 Data Wells and Coppersmith (1994) (b) Data Wesnousky (2008) Regression Wells and Coppersmith (1994) (a) Adjusted regression for all data (b) 5.5 10-1 100 101 102 Maximum Surface Displacement, MD (m)

Figure 1: Plot of maximum surface displacement vs. moment magnitude of all selected data points including the regression from Wells and Coppersmith (1994) and the adjusted regression.

Table 3: Average surface displacements correlated with moment magnitudes.

Average Displacement - Fault offset, m Correlated Magnitude 0.1 0.25 0.5 1.0 1.5 2.0 5.0 10.0 20.0 Wells and Coppersmith (1994) 6.15 6.5 6.8 7.0 7.2 7.3 7.7 7.9 8.2 Adjusted regression 6.05 6.4 6.7 7.0 7.2 7.3 7.7 7.9 8.2 121

The evaluation of the new model shows larger maximum surface displacements for earthquakes M7.4 and lower. Only for very large earthquakes with M7.4 and higher, the adjusted model results in slightly larger maximum surface displacements. For the average surface displacement, the point crossing the equal line is at M7.7. The similar results in combination with the use of more recent earthquake data suggests the use of the adjusted regression models from equations (2) and (3)

8.5 (c) M = 7.04 + 0.89*log(AD)

(d) M = 6.996 + 0.948*log(AD) ( + 1 ) 8

( - 1 ) 7.5

(d)

6 Data Wells and Coppersmith (1994) Data Wesnousky (2008) Regression Wells and Coppersmith (1994) (c) Adjusted regression for all data (d) 5.5 10-1 100 101 102 Average Surface Displacement, AD (m)

Figure 2: Plot of average surface displacement vs. moment magnitude of all selected data points including the regression from Wells and Coppersmith (1994) and the adjusted regression. 122 to be applied for this research to correlate the input parameter of fault offset to earthquake magnitude. The values used to compute the comparison seen in Fig- ure 3 are listed in Tables 4 and 5.

Both adjusted equations are applied to the conference papers in chapter 2. However, the in depth analysis of the parametric study in chapter 3, showed, that the standard deviations in the Wells & Coppersmith article for the regression of dis-

3 Ratio of Maximum Surface Displacements, MD Ratio of Average Surface Displacements, AD

2.5

(MD)

1.5 (AD)

0.5

7.4 7.7 0 5 5.5 6 6.5 7 7.5 8 8.5 9

Figure 3: Comparison of surface displacements from adjusted regression equation over surface displacement regression equation from Wells and Coppersmith (1994) to correlated moment magnitudes. 123 placement and moment magnitude for strike-slip faults is 0.29 magnitude points for MD and 0.28 magnitude points for AD. The adjusted regression including data from Wesnousky (2008) falls inside one standard deviation. As a result, the well- known and established Wells & Coppersmith correlation is the basis for the earthquake magnitude estimation in chapters 3 and 5.

Table 4: Range of moment magnitudes for comparison of Wells and Coppersmith (1994) max. surface displacement regression equation with the adjusted regression equation.

Moment Magnitude, M Maximum Displacements, MD in m 5.5 6.0 6.5 7.0 7.5 8.0 8.5 Adjusted regression, MDall 0.05 0.17 0.59 2.1 7.3 25.8 91.3 Wells and Coppersmith (1994), MDWC 0.02 0.09 0.40 1.8 7.7 33.5 147 MDall 2.24 1.81 1.46 1.18 0.95 0.77 0.62 MDWC

Table 5: Range of moment magnitudes for comparison of Wells and Coppersmith (1994) average surface displacement regression equation with the adjusted regression equation.

Moment Magnitude, M Average Displacements, AD in m 5.5 6 6.5 7 7.5 8 8.5 Adjusted regression, ADall 0.02 0.07 0.25 0.9 3.3 12.0 43.7 Wells and Coppersmith (1994), ADWC 0.03 0.09 0.30 1.0 3.4 11.5 38.6 ADall 1.42 1.31 1.21 1.12 1.03 0.96 0.88 ADWC 124 3.30 7.25 7.38 1.50 8.00 2.10 6.54 0.15 1.63 0.18 2.30 2.10 1.30 2.60 Avg., AD 6.10 2.00 5.90 4.35 9.40 0.20 0.40 1.30 2.60 0.38 5.20 2.70 0.60 3.60 3.40 10.00 14.60 12.00 Displacement (m) Max., MD 27 77 39 10 78 87 10 2.7 790 850 365 113 180 310 19.7 23.5 1200 1800 dyne-cm) ment (10e26 Seismic Mo- 7.90 8.02 7.92 6.83 6.92 7.67 7.22 8.14 6.25 6.88 7.03 7.34 6.63 7.23 7.26 6.63 7.47 7.63 Magni- Moment tude, Mw LL LL LL LL LL LL RL RL RL RL RL RL RL RL RL RL RL RL Slip Type Strike Date (m/d/yr) 04/18/1906 12/16/1920 08/10/1931 12/21/1932 05/19/1940 11/18/1951 03/18/1953 12/04/1957 06/28/1966 08/19/1966 01/05/1967 07/22/1967 04/19/1968 08/31/1968 01/04/1970 05/22/1971 02/06/1973 02/04/1976 Varto Kansu Bingol Mogod Tonghai Motagua Parkﬁeld Damxung Gobi-Altai Canakkale Earthquake San Francisco Kehetuohai-E Dasht-e-Bayaz Imperial Valley Mudurna Valley Cedar Mountain Borrego Mountain Luhuo/Xianshui He Iran China China China China China Turkey Turkey Turkey Turkey Location USA, CA USA, CA USA, CA USA, CA USA, NV Mongolia Mongolia Guatemala 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Nr. Strike-slip Values from Wells and Coppersmith (1994) Table 6: Strike-slip earthquake dataslip from type Wells RL and = Coppersmith rightAD (1994) lateral, = and average LL Wesnousky displacement. (2008). = left Strike- lateral. Displacements max., MD = maximum displacement, avg., 125 4.7 3.1 1.1 4.2 1.6 2.5 0.6 2.05 0.18 0.10 0.54 0.70 0.60 2.95 1.66 Avg., AD 3.50 0.80 1.50 0.12 0.20 0.92 1.50 1.10 6.20 6.00 9.10 7.90 3.50 7.40 3.30 1.90 4.40 1.50 Displacement (m) Max., MD 3 79 20 76 2.6 9.2 1.6 1.8 0.3 460 114 7.12 10.1 1.11 54.7 11.3 52.5 28.7 dyne-cm) ment (10e26 Seismic Mo- 7.23 6.53 6.64 6.00 6.20 6.61 7.13 6.83 7.74 7.34 7.90 7.30 6.70 7.70 6.90 6.80 7.60 6.30 Magni- Moment tude, Mw LL LL LL LL LL LL LL RL RL RL RL RL RL RL RL RL RL RL Slip Type Strike Date (m/d/yr) 11/24/1976 10/15/1979 01/23/1981 10/27/1985 11/24/1987 11/24/1987 11/06/1988 11/06/1988 07/16/1990 06/28/1992 01/09/1857 10/28/1891 11/02/1930 12/25/1939 05/19/1940 12/20/1942 11/26/1943 09/10/1943 Tosya Daofu Luzon Tottori Landers Kita-Izu Caldiran Erzincan El Centro Neo-Dani Earthquake Constantine San Andreas Erbaa-Niksar Elmore Ranch Imperial Valley Gengma, Yunnan Superstition Hills Lancang-Gengma Japan Japan Japan China China China Turkey Turkey Turkey Turkey Algeria Location USA, CA USA, CA USA, CA USA, CA USA, CA USA, CA Philippines 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Nr. Strike-slip Values from Wells and Coppersmith (1994) Strike-slip Values from Wesnousky (2008) Table 7: Strike-slipTable data 6 from Wells and Coppersmith (1994) and Wesnousky (2008), continued from 126 2.1 0.9 0.3 3.5 2.3 1.1 2.1 3.3 2.4 3.6 0.13 0.41 0.13 1.56 Avg., AD 3.50 2.00 0.40 0.78 0.50 0.50 6.20 6.70 5.10 5.20 5.00 8.70 8.30 8.90 Displacement (m) Max., MD 1.6 8.1 4.9 2.5 3.5 13.3 0.16 0.48 0.43 0.22 27.4 62.5 46.8 51.6 dyne-cm) ment (10e26 Seismic Mo- 7.35 6.70 6.10 6.40 6.40 6.20 7.60 7.20 7.10 6.90 7.00 7.80 7.80 7.70 Magni- Moment tude, Mw SS LL LL LL RL RL RL RL RL RL RL RL RL RL Slip Type Strike Date (m/d/yr) 02/01/1944 07/22/1967 04/08/1968 10/15/1979 07/29/1981 11/23/1987 07/16/1990 06/28/1992 08/17/1999 10/16/1999 11/12/1999 11/14/2001 11/14/2001 11/03/2002 Sirch Izmit Duzce Luzon Denali Kunlun Landers Mundurnu Earthquake Hector Mine Gerede-Bolu Borrego Mtn. Kunlun (spot) Imperial Valley Superstition Hills Iran China China Table 8: Strike-slip earthquake data from Wesnousky (2008), continued from Table 7 Turkey Turkey Turkey Turkey Location USA, CA USA, CA USA, CA USA, CA USA, CA Philippines USA, Alaska 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Nr. Strike-slip Values from Wesnousky (2008) 127

Appendix C - Strain along the tunnel alignment

The analysis of the response of the concrete-lined tunnels crossing an active fault uses steel tension and concrete compression strains. The fault offset displacement results in a double-curvature bending of the tunnel beam. Since there are two curves - one with compression at the crown of the tunnel (top) and tension at the invert (bottom), and the second one opposite of the ﬁrst - there is the need to analyze all beam elements with respect to the strain. Each beam element has an identifying number. Because this work concentrates on the tunnel behavior in its longitudinal direction, axial strains and curvatures along the tunnel alignment are recorded for each of the 252 parameter combinations. With the axial strain and curvature information for each beam, the maximum compression and tension strains along the tunnel alignment calculate to

D + 2 · h e = e ± ( i · κ) (4) a 2

where e is the maximum or minimum strain of the tunnel cross-section at one point along the longitudinal tunnel axis, ea is the axial strain of the tunnel cross- section, κ is the curvature of the tunnel cross-section at the same point, and Di and h are the inner tunnel diameter and the concrete lining thickness, respectively. The zero-strain point is at the height of the horizontal centerline of the cross-section. 128

Therefore, the maximum strain is either at the top or at the bottom depending on the direction of the bending. Because of concrete cracking, the maximum strains are tension strains. The beam elements that show the largest tension strain, are used for the detailed analyses in chapters 3 and 5 of this thesis. The detailed analyses look at steel and concrete strains of each parameter combination at two different levels - service limit state (SLS), and ultimate limit state (ULS). The SLS is deﬁned with the strain at which yielding of steel starts and the ULS is deﬁned with the concrete peak compressive strain when the concrete is in failure. The graphs show the strains along a deliberately chosen section of the tunnel between tunnel meter -25 and +25. Each controlling tunnel beam is marked with the tunnel meter. Fault zone widths of 100 m were omitted due to the fact that responses with Lzone = 100 m are the same responses as those with Lzone = 50 m. These graphs are underlying information to inform the research and not part of the published or to be published articles and papers. 129

0.03 0.03 Legend: Legend: __ Extremely weak rock __ Extremely weak rock Major fault plane Major fault plane 0.025 . . . Weak rock 0.025 . . . Weak rock - - - Medium strong rock - - - Medium strong rock Beam element, tm = -0.1

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

0.01 Beam element, tm = 3 0.01 Beam element, tm = 3

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

0.01 Beam element, tm = 3 0.01 Beam element, tm = 3

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 4: (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.2 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 130

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

0.01 Beam element, tm = 3 0.01 Beam element, tm = 3

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

0.01 Beam element, tm = 3 0.01 Beam element, tm = 3

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 5: (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.24 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 131

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

Beam element, tm = 3 Beam element, tm = 3 0.01 0.01

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 0.015 Beam element, tm = -3 Beam element, tm = -3

Beam element, tm = 3 Beam element, tm = 3 0.01 0.01

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 6: (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.3 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 132

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 0.015

Beam element, tm = -3 Beam element, tm = 3 Beam element, tm = -4 Beam element, tm = 3 0.01 0.01

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 0.015

Beam element, tm = -4 Beam element, tm = 3 Beam element, tm = -4 Beam element, tm = 3 0.01 0.01

0.005 Beam element, tm = 2 0.005 Beam element, tm = 2

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 7: (a) to (f): Strains along tunnel alignment for DI = 6 m, h = 0.4 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 133

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 7 Beam element, tm = 7 0.01 0.01 Beam element, tm = 2 Beam element, tm = 2

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 7 Beam element, tm = 7 0.01 0.01 Beam element, tm = 2 Beam element, tm = 2

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 8: (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.33 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 134

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 7 Beam element, tm = 7 0.01 0.01 Beam element, tm = 2 Beam element, tm = 2

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 7 Beam element, tm = 7 0.01 0.01 Beam element, tm = 2 Beam element, tm = 2

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 9: (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.4 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 135

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 7 Beam element, tm = 2 Beam element, tm = 7 0.01 0.01 Beam element, tm = 2

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 Beam element, tm = 3 0.015 Beam element, tm = 3

Beam element, tm = 2 Beam element, tm = 7 Beam element, tm = 2 Beam element, tm = 7 0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 10: (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.5 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 136

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

0.015 0.015 Beam element, tm = 2 Beam element, tm = -9 Beam element, tm = 2 Beam element, tm = -6 Beam element, tm = 4 Beam element, tm = 8 0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

0.015 Beam element, tm = 2 0.015 Beam element, tm = 4 Beam element, tm = 2 Beam element, tm = -6 Beam element, tm = 4 Beam element, tm = 8 0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 11: (a) to (f): Strains along tunnel alignment for DI = 10 m, h = 0.67 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 137

0.02 0.02 Beam element, tm = -0.1

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

Beam element, tm = 2 0.015 0.015 Beam element, tm = 4

Beam elements, tm = -9 Beam element, tm = 2 Beam element, tm = -7

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

Beam element, tm = 2 Beam element, tm = 2 0.015 Beam element, tm = 4 0.015 Beam element, tm = 4

Beam element, tm = -7 Beam element, tm = 9 0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 12: (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.5 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 138

0.03 0.03 Legend: Beam element, tm = -0.1 Legend: __ Extremely weak rock __ Extremely weak rock Major fault plane Major fault plane 0.025 . . . Weak rock 0.025 . . . Weak rock - - - Medium strong rock - - - Medium strong rock

0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

Beam element, tm = 2 0.015 0.015 Beam element, tm = 4

Beam elements, tm = -9 Beam element, tm = 2 Beam element, tm = -7

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02

Beam element, tm = 2 Beam element, tm = 2 0.015 Beam element, tm = 4 0.015 Beam element, tm = 4

Beam element, tm = -7 Beam element, tm = 9 0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 13: (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.6 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 139

Beam element, tm = -0.1 0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

Beam element, tm = 2 0.015 0.015 Beam element, tm = 4 Beam elements, tm = -9 Beam element, tm = 2 Beam element, tm = -8

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02 Beam element, tm = 2 Beam element, tm = 2

0.015 0.015 Beam element, tm = 4 Beam element, tm = 4 Beam element, tm = -8 Beam element, tm = -8

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 14: (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 0.75 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter. 140

Beam element, tm = -0.1 0.02 0.02

0.015 0.015 Beam element, tm = -1

0.01 0.01

0.005 0.005 bottom top Tension 0 0 Compression

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b)

0.02 0.02

Beam element, tm = 2

0.015 0.015

Beam elements, tm = -9 Beam element, tm = -8 Beam element, tm = 5

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d)

0.02 0.02 Beam element, tm = 2 Beam element, tm = 2

0.015 0.015

Beam element, tm = -9 Beam element, tm = 5 Beam element, tm = -9 Beam element, tm = 5

0.01 0.01

0.005 0.005

0 0

-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (e) (f)

Figure 15: (a) to (f): Strains along tunnel alignment for DI = 15 m, h = 1.0 m, Lzone varies from (a) to (f), rock strength varies on each subplot. tm = tunnel meter.