Buckling Behavior and Design of Steel Liner Encased in Deep Water Tunnel under External Hydrostatic Pressure 大深度地下水道管の外水圧による座屈挙動 および設計に関する研究 February, 2009 JIANHONG WANG 王 剣宏 Buckling Behavior and Design of Steel Liner Encased in Deep Water Tunnel under External Hydrostatic Pressure J. H. Wang Dissertation submitted to the Faculty of the Science and Engineering, Waseda University in fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering Professor A. Koizumi (Chair) Professor H. Seki Professor T. Yoda Professor O. Kiyomiya Defense Date: September 16, 2008 Tokyo, Japan Key words : deep underground, water tunnel, separated-type structure, external hydrostatic pressure, buckling behavior, free pipe, restrained pipe, plain pipe, stiffened pipe, jointed pipe, buckling design Copyright 2009 by J. H. Wang Buckling Behavior and Design of Steel Liner Encased in Deep Water Tunnel under External Hydrostatic Pressure J. H. Wang (ABSTRACT) The rapid population increase and the urbanization progress have become the inevitable trends in the next decades. On the other hand, with the Global Warming processing and increasing of water resource pollution in recent years, the water scarcity is becoming seriously, particularly in developing countries. To keep the sustainable and stable water supply, the water transfer infrastructure is required to enhance, particularly the development of new water pipeline systems in urban area. However, the long term infrastructure development has gotten the ground and shallow underground space congested in large city, the construction of water pipeline has to consider the use of deep underground space. In the current study, the large diameter water main line is focused and investigated in terms of its design and construction. Since the water tunnel is mainly adopted for water main line, its structure is required to investigate firstly considering the cost and security of water tunnel. At present, there exist two kind water tunnels, the independent-type and integrated-type, which are defined according to the liner installation methods. The independent-type tunnel denotes a water tunnel in which the water pipe is placed and supported by base or tunnel invert and remains the enough space for check passage. It is commonly adopted for a common utility tunnel or high security water supply lines. For such tunnel structure, the security of pipe can be ensured for its structural simplicity and acting load of only internal pressure, however the larger tunnel cross section than required cause high construction cost. On the other hand, the integrated-type structure is denoted the water tunnel whose linings and liners are integrated through filling annular gap with concrete or other backfilling materials. This type tunnel structure has been widely used for pressure water tunnel in both urban and mountain regions in practice. Although the water tunnel can be built with smaller cross section, the acting load conditions are made complex. The water pipe is not only required to resist internal pressure but also the external groundwater pressure and earth pressure, meanwhile, the tunnel lining has to resist the vibration and pressure of flowing water. Therefore the heavy water tunnel has to be designed taking into all the acting loads account. Actually, to provide economic and reliable water supply utilities, the water tunnel in terms of rationalization of structure and construction has been studied in recent years. The separated-type structure as a new water tunnel structure has been proposed by Technical Research Committee of Deep Water Pipeline Construction (TRCWPC), in which the water tunnel with minimum cross section decided by hydraulic design is built and the water pipe is installed without backfilling the gap between liner and lining. Using this type structure, the tunnel cross section can be reduced and the structural design can also be simplified. In 2002, the scheme of separated-type structure was investigated and the construction cost and time reduction and structural safety were identified by numerical analysis and assembly tests. In 2003, the longitudinal extension, seismic performance and other structural elements were investigated, and the construction procedures was re-examined by lab tests in terms of constructability. However, many evidences indicate that high hydrostatic pressure may be built up between linings and liner due to the leakage in linings or the end of steel liner during operation. This is particularly true for a deep water tunnel, thus the buckling of steel liner under high external pressure must be considered. Where, the steel liner looked as an encased pipe, its buckling behavior under external hydrostatic pressure is investigated and clarified firstly, and then the corresponding buckling design method for liner buckling is established. Historically, the buckling theories of encased pipe can be classified into two groups in principle: rotary symmetric theories (free buckling) and single-lobe buckling theories (non-symmetric, restrained buckling). In the former, that the liner should be permanently rotary symmetric during buckling process is considered because the circumference of liner is elastically reduced under external pressure and the gap between liner and host develops until buckling occurs. Whereas in the later, the only one lobe is produced at detached part of liner and other part pipe maintains attaching to the host, considering the practical liner will be restrained at the crown or bottom and the gap and the range of detached liner at the opposite site will be enlarged under loads (self-weight, water buoyancy and external pressure, etc.), finally the single-lobe buckling occurs in detached liner. Although the argument about which is the most reasonable and rational resolution still goes on until now, in practice engineering it is undoubted fact that the non-symmetric theories prevails. The reason can be considered that the integrated-type water tunnel structure has prevailingly been applied for many years and almost liner failed in the non-symmetric buckling. As for the liner of separated-type water tunnel structure in the current study, both rotary symmetric buckling and non-symmetric buckling are possible to occur because the different support conditions are considered for different liner installation method. The liner can be installed with the rubber plates set around the liner and at a regular longitudinal interval as that proposed by TRCWPC, in such case the support condition is named the uniformly point support (briefly, uniformly support) and the liner is considered as a free pipe. Also the liner can be just placed and supported by linings directly or sand media for easy installation, where the support condition is named the locally surface support (briefly, locally support) and the liner is considered as a restrained pipe. For free pipe, its buckling should consider a rotary symmetric buckling because the liner is difficult to attach to host and the hoop thrust developed in pipe wall prevails under external pressure. On the contrary, the buckling of locally supported liner should be considered as a non-symmetric buckling (single-lobe buckling), since a part of liner always attaching to the tunnel lining can be restrained. Moreover, as for the buckling of free pipe, since the plain pipe is not able to provide a safe and economic liner, the using stiffened pipe is the general consideration. The rational design of a stiffened pipe should be discussed in terms of its buckling resistance capacity. This requires the study of buckling behavior of stiffened pipe. On the other hand, taking into account the restraining effects, the buckling of restrained pipe, should be investigated in terms of buckling of plain pipe and stiffened pipe. Otherwise, the buckling of jointed pipe is also required to investigate considering the welding manufacture of liner in-situ. Accordingly, this study is to investigate the buckling of free pipe, restrained pipe and jointed pipe under external pressure, and discuss the corresponding buckling solutions for water tunnel liner design. In addition, the rational water tunnel structure and liner installation method are discussed and proposed. The dissertation is composed of six chapters including Chapter 1 Introduction, Chapter 2 Buckling of free pipe under external pressure, Chapter 3 Buckling of restrained pipe under external pressure, Chapter 4 Buckling of jointed pipe under external pressure, Chapter 5 Water tunnel liner design and Chapter 6 Summary and conclusions. The contents of individual chapter are described as followings: In Chapter 1, as the study background, the water crisis and urban development were introduced. The water tunnel structure was also discussed as well as related problems/measures. In addition, the water tunnel structure was discussed and the buckling of liner for separated-type water tunnel structure is considered based on the existing buckling theories. Finally, the purpose and composition of this study are described. In Chapter 2, the encased liner as a free pipe was investigated especially, the ring-stiffened pipe was discussed in terms of buckling behavior and theoretical analysis. The existing theories on buckling, nonlinear and imperfection were reviewed, and the numerical analysis method was examined firstly. The buckling behavior of ring-stiffened pipe involving the two buckling forms of general buckling and local buckling and the buckling form change with flexural rigidity of stiffener was confirmed known. In addition, that the buckling behavior is not only related with the stiffener but also the pipe was also identified. Based on the bucking behavior, the buckling theoretical equations corresponding to the buckling form were derived using the potential energy principle and Ritz method, and the applications of these equations were discussed in terms of the critical pressure estimation and buckling behavior simulation. As the solution for stiffened pipe buckling the two stage method was presented. The validation of two-stage method was identified by comparing with the results of numerical analysis and literature experiment. Furthermore, the existing theories were also examined and the limitation for application was identified. In Chapter 3, the liners encased in water tunnel as restrained pipe, its buckling behavior was investigated through numerical contact analysis and experimental approach. The prediction method for buckling location was discussed and presented, as well as the application method for Amstutz’s equations. The buckling of encased liner was investigated using the numerical contact analysis with respect to the practical and theoretical loading conditions, and the single-lobe buckling behavior and mechanism were clarified. The existing buckling theories were examined in terms of single-lobe buckling equations proposed by Amstutz and Jacobsen, and enhancement factor theory. Meanwhile the presented buckling location prediction and application methods for Amstutz’s equation were verified. Generally, the single-lobe buckling equations proposed by Amstutz and Jacobsen can be used for plain pipe while not for stiffened pipe were identified. The experimental investigation of single-lobe buckling failure was carried out, and a new analytical solution of critical pressure estimation according to stiffening conditions was presented based on the experimental results and single-lobe buckling mechanism. The verification was also conducted by comparing with numerical analysis results. In Chapter 4, the influence of transverse joints was investigated through the experimental approach and numerical analysis. From the experimental approach, that the buckling critical pressure and the buckling forms changes with flexural rigidity of joint were identified. The semi-analytical solution for buckling of jointed pipe was discussed and presented by taking into account the flexural rigidity of joint, and the validation was identified by the results of experiment and numerical analysis. In Chapter 5, based on the results obtained in former chapters, the design method for steel liner installed in deep water tunnel was given using a practical example. The corresponding design methods were discussed and the steel liner in terms of the plain pipe and stiffened pipe were designed for conventional integrated type and new separated type tunnel structures and two support conditions in later structure, respectively. Based on the designed results, the water tunnel structure and the support condition were discussed and the most rational water tunnel structure for deep water tunnel was proposed in terms of the safety and economics of tunnel lines. Finally, in Chapter6, the results of this research are summarized briefly, and the conclusions were drawn. 大深度地下水道管の外水圧による座屈挙動 および設計に関する研究 （ 概 要 ） 王 剣宏 近年，地球の気候変動にともない世界各地で水不足の問題が生じている．都市部では 長い期間にわたって各種のインフラ施設や地下構造物の建設が進められてきており，現 在ではその中浅深度の地下は相当に混雑している．水の供給を安定化するために，都市 部に新たな導水管路施設や水道管路施設を建設する場合には，大深度地下を使用せざる を得ない状況になってきている． 従来からの管路施設の構造は 2 つに大別される．まず，シールド工法や開削工法など によりトンネルを構築するが，その後，トンネルの内側に支持架台を設けて，その上に 鋼管やダクタイル管を載せて収容する構造（以降，独立構造と呼ぶ）と，トンネルの内 側に管を挿入し，両者の間に間詰めをして一体化する構造（以降，一体型構造と呼ぶ） である．独立構造の場合は，水道管には内水圧が作用するだけであり構造的には単純で 問題もほとんどないが，トンネル内に支持架台や管路を収容するため，大きなトンネル 断面が必要となり，とくに大深度に構築する場合には不経済となる．一方，一体型構造 の場合にはトンネル断面は小さくてすむが，管は内水圧だけでなくトンネルに作用する 土水圧の変動やトンネルの変形などの影響を受け，逆に，トンネルは管内の水流による 振動や水圧の変動，水温の影響などを直接受けることから，これらをあらかじめ考慮に 入れて設計しておかなければならず，両者ともその構造は重厚にならざるを得ない．ま た，間詰め材の硬化にともなう体積の収縮や間詰め材の充填の困難さによる管頂部での 空隙の発生なども避けられない． 大深度の高水圧下のトンネルに収容する管路に，その安全性と合理性を確保するため に，トンネルと内挿管との間に間詰めをしないで，それぞれ独立の構造とする分離型構 造が提案されている．これは前述した独立構造のものと基本的な考え方は同じであるが， トンネルの内径と内挿管の外径との差を可能な限り小さくしようとするものである． (財)「水道技術研究センター」と「大深度水道管路布設技術研究会（現水道管路シール ド技術研究会）」は，2002 年に共同で分離型水道管構造を提案し，それ以後，この構 造に対して，間詰めの充填層を必要としないため，トンネルの掘削断面の縮小が図れる こと，トンネルと内挿管とが独立であり，相互に干渉が生じないため，内挿管は内圧を 受けるだけの設計でよく経済性にすぐれていること，必要に応じて一次覆工と内挿する 鋼管を併行して施工することができ，工期の短縮やコストの縮減が図られることなどを， 机上の検討や各種の実験および解析を行って確認してきている．この分離型構造は設計 上も明確であり，トンネルは外部からの土水圧に対して，また，内挿管は内水圧に対し てそれぞれ設計を行えばよく，土水圧や内水圧が変動しても相互に影響を与えないとい う利点をもつ．とくに，内挿管は内水圧のみが作用するならば，薄肉管を使用すること が可能となる． しかし，水道管を収容するトンネルは必ずしも完全に防水されたものではなく，分離 型構造による水道管であっても，長期にわたる供用中には外水圧が作用する可能性があ る．トンネルと内挿管との間の狭いすき間を完全に漏水が満たせば管には外水圧が作用 し，それが大深度地下に建設される管であれば，とくに大きな外水圧が作用することに なる．そのような場合には内挿管の座屈が懸念される．このため，あらかじめ管の座屈 挙動を把握し，合理的な座屈設計法を十分に検討しておく必要がある．内挿管の座屈は その安全性を考えるうえで重要な問題でありながらいまだ解決できておらず，国内外で 水道管の座屈事故が発生している． このような座屈による事故を防ぐためには，厚肉の内挿管を使用せざるを得ない． トンネル中の鋼管の座屈に関しては，従来から，自由管の座屈理論（多波数座屈理論） と拘束管の座屈理論（シングルロブ座屈理論）との 2 つがある．前者は外水圧の作用に よって円周長が縮み，トンネルと管とのすき間が拡大するため，座屈は自由管と同じに 考えてよいとするものである．後者は管に作用する水圧により管が変形し，その一部分 がトンネルと接触することによって拘束される一方で，非接触部分は内側に曲げ変形し， いわゆるシングルロブを形成して座屈すると考えるものである．分離型水道管の場合に は，その支持状況によってどちらの現象も発生する可能性がある．水道管とトンネルの 間に等間隔にゴム板などを入れて水道管を支持する分離型水道管構造の場合には，両者 の間のすき間が保たれるから自由管として考えられる．一方，施工の容易性を考え，水 道管とトンネルの間のインバート部分にのみ砂やゴム板などを入れて支持する構造の 場合には，自重と静水圧による初期変形を考慮して拘束管としての座屈を考える必要が ある． 本研究は，以上のような背景から，分離型構造をもつ内挿管を想定して，その外水圧 による座屈挙動を検討したものであり，対象として補剛されていない内挿管と，断面方 向にリング状の補剛部材をもつ内挿管とを取り扱っている． 本論文は６章で構成されている．以下には各章ごとにその概要を示す． 第 1 章は序論であり，本研究の背景，導水管路や水道管路の構造，それらを大深度地 下に構築する場合の問題点，従来からの管の座屈理論のもとに分離型水道管の座屈検討 を行い，最後に研究目的および本論文の構成を提示している． 第 2 章は「自由管の外水圧による座屈」について述べた章である．まず，従来の研究 を中心に，これに考察を加え，自由管の座屈理論式，非線形座屈および初期不整の考え 方，座屈の数値解析方法などを示した．つぎに，数値解析法を用いてリング状の補剛材 をもつ自由管（以降，自由補剛管と呼ぶ）の座屈挙動を検討し，補剛材の剛性に応じて， 全体座屈と局部座屈の２つの座屈形態があることを示した．また，補剛材の曲げ剛性お よび補剛間隔，鋼管の径，厚さおよび長さが座屈荷重や座屈形態に与える影響を検討し た．最後に，自由補剛管の座屈は単純圧縮状態から急激に変形が生じ不安定状態となる 現象であり，座屈が発生するまでは水圧の作用方向が変わらないことから，従来の座屈 解析手法を適用することができると考え，鋼管には薄肉シェルの曲げ理論，補剛材には リング梁の理論を適用し，座屈形態に応じて Ritz の方法を用いて，実務に簡単に使え る自由補剛管の座屈理論式を導き，その具体的な座屈計算の方法，すなわち，「二段階 座屈計算法」と座屈挙動のシミュレーションの方法を提案した．それらの妥当性は数値 解析および既往の実験結果を用いて検証を行ったうえで，従来の理論式と比較して考察 を加えた． 第 3 章は「拘束管の外水圧による座屈」について述べた章である．この章では，まず， 従来の Amstutz や Jacobsen のシングルロブ座屈理論と座屈設計の実務によく使われて いる増大係数の理論，および FEM による接触解析を考慮した座屈解析方法について述べ た．また，実際の分離型水道管の浮力と自重とを考慮して，シングルロブ座屈が発生す る位置の予測方法を検討し，それと Amstutz の式の適用方法についての提案を行った． 一方，FEM による数値解析に，浮力と自重を考慮した接触解析法を適用し，補剛材がな い場合と補剛材がある場合の鋼管の座屈解析を行って，提案した座屈位置の予測方法と Amstutz 式の適用方法を検証した．その結果，補剛材がない鋼管の場合には，Amstutz および Jacobsen の理論式による理論値が FEM による解析値とほぼ一致するが，補剛材 がある場合にはこれらの式が適用できないことがわかった．そのため，簡単な実験を行 ってシングルロブ座屈の破壊メカニズムを検討し，それと Jacobsen の理論にもとづい て，補剛状況に応じたシングルロブ座屈の理論解析方法を提案し，その妥当性を数値計 算により検証した． 第 4 章は「継手がある鋼管の外水圧による座屈」について述べた章である．ここでは， 継手の継手剛性が異なる７つモデルの簡単な座屈実験を行い，継手の剛性が座屈荷重や 座屈形態に与える影響を調べ，継手の剛性を考慮した鋼管の座屈の理論的な解析方法を 提案して，それが実験結果や数値解析結果に符合することを確認した． 第 5 章は「水道管の設計法」についての提案とその設計例を示し，設計結果に検討を 加えた章である．まず，水道管の設計方法の手順を示し，従来の一体型構造の鋼管の場 合，分離型構造の鋼管で円周方向に均等に支持されている自由管の場合と，インバート のみで支持されている拘束管の場合とに分けて水道管の座屈設計例を具体的に示した． 設計結果をもとに，一体型構造の鋼管の場合には充填の欠陥による座屈への影響を検討 し，分離型構造の場合には，自由補剛管と拘束補剛管を対象に局部座屈設計法を具体的 に提示した．さらに，それらの設計の結果にもとづいて，水道管の最適と思われる構造につ いて考察を加え，提案を行った． 第 6 章は本研究を総括した「総括および結論」であり，本研究の主な内容をまとめ， 得られた知見と今後の課題を示した章である． Acknowledgments This work was spiritually and financially supported by many people, organizations, where I would like to thank them. I would especially like to show my appreciation for my supervisor, Professor A. Koizumi, for his unlimited encouragement, support, and friendship during my master and doctoral studies. In the past half and five years, professor A. Koizumi did not only teach me specialized knowledge, instruct my doctoral studies, but also carried about my living and helped me in all aspects. Special thanks are also given to Professor H. Seki, Professor T. Yoda, Professor O. Kiyomiya for their kindly reviewing the thesis and giving good advices. Particularly, I would like to thank Professor T. Yoda for his invaluable teachings and insights into buckling theory, and Columbia University Research Scientist Li for his English correction and improvement of this thesis. The colleagues at Koizumi laboratory have also provided me with valued assistance and treasured friendship during my residency there and deserve to be acknowledged. Particularly, I would also like to specifically thank fellow graduate students N. Mitsuda and Y. Amano of Waseda University for their valuable discussions and assistance with various technical details. I am extremely grateful to all of my friends and family who have supplied me with unlimited encouragement and friendship throughout my years of study. Though too numerous to name, they have all contributed to making me a better scientist and a better person. Finally, financial support from the Ushio Scholarship Foundation, the Japan Iron and Steel Federation, Waseda University and Atsumi International Scholarship Foundation is greatly acknowledged. J. H. Wang February, 2009 Table of Contents Chapter 1 Introduction 1.1 Background..................................................................................................................1 1.1.1 Worldwide Water Crisis ..........................................................................................1 1.1.2 Utilization of Urban Underground Space.................................................................2 1.2 Water Tunnel...............................................................................................................4 1.2.1 Linings .....................................................................................................................4 Unlined Tunnels................................................................................................................4 Shotcrete Lining................................................................................................................4 Un-reinforced Concrete Lining.........................................................................................5 Reinforced Concrete Lining..............................................................................................5 Lining Selection for Urban Water Tunnel.........................................................................5 1.2.2 Liners .......................................................................................................................6 Industrial Pipe...................................................................................................................6 Welded Pipes in Place.......................................................................................................6 1.2.3 Problems and Measures………................................................................................7 Leakage/Seepage...............................................................................................................7 Water-proof Measures.......................................................................................................7 Drainage Measures...........................................................................................................8 1.3 Water Tunnel Structure...............................................................................................9 1.3.1 Existing Water Tunnel Structure .............................................................................9 1.3.2 Problems of Traditional Water Tunnel Structure...................................................10 1.4 Water Tunnel in Current Study..................................................................................11 1.4.1 Deep Water Tunnel.................................................................................................11 1.4.2 Separated-type Water Tunnel Structure.................................................................12 1.4.3 Buckling Considerations.........................................................................................13 External Hydrostatic Pressure........................................................................................13 Buckling Types................................................................................................................13 Buckling Analysis............................................................................................................14 1.5 Purpose and Organization of Current Study..............................................................15 1.5.1 Purpose of Current Study.......................................................................................15 1.5.2 Organization of Dissertation..................................................................................15 References.......................................................................................................................17 Chapter 2 Buckling of Free Pipe under External Pressure 2.1 Introduction…………………………………………...............................................19 2.1.1 Plain Pipe vs. Stiffened Pipe..................................................................................19 2.1.2 Existing Buckling Theories Review.......................................................................21 Plain Pipe........................................................................................................................21 Stiffened Pipe..................................................................................................................22 Nonlinear and Imperfection Theory................................................................................24 2.1.3 Numerical Analysis Method and FEM Software...................................................26 2.2 Buckling Behavior of Ring-Stiffened Pipe ..............................................................29 2.2.1 Generalization on Buckling of Stiffened Pipe.......................................................29 2.2.2 Investigation through Numerical Analysis............................................................30 Effects of Stiffness and Spacing of Stiffeners..................................................................30 Effects of Pipe Geometries..............................................................................................31 Summary…......................................................................................................................32 2.2.3 Consideration and Subject......................................................................................33 2.3 Derivation of Theoretical Buckling Equations for Ring-Stiffened Pipe...................34 2.3.1 Introduction............................................................................................................34 2.3.2 Strains in Shell.......................................................................................................35 2.3.3 General Buckling Equation....................................................................................36 Strain Energy in Pipe......................................................................................................36 Strain Energy in Stiffener……………............................................................................37 Work done by External Pressure....................................................................................38 Total Potential Energy and Solution..............................................................................39 Critical Pressure and Buckling Wave.............................................................................41 2.3.4 Local Buckling Equation........................................................................................43 2.4 Application and Verification of Buckling Equations................................................44 2.4.1 Two-Stage Method................................................................................................44 2.4.2 Verification of two-stage method..........................................................................45 Verification by Experiment.............................................................................................45 Comparison between Two-stage Method and Existing Buckling Equations..................47 2.4.3 Buckling Behavior Simulation...............................................................................50 Theoretical Analysis by Two-stage Method....................................................................50 Numerical Analysis by Finite Element Method..............................................................50 Verification of Buckling Behavior..................................................................................51 References.......................................................................................................................54 Chapter 3 Buckling of Restrained Pipe under External Pressure 3.1 Literature Reviews ...................................................................................................57 3.1.1 Single-Lobe Buckling Theory of Plain Liner.........................................................58 3.1.2 Single-Lobe Buckling Theory of Stiffened Liners.................................................60 3.1.3 Other Buckling Theories of Restrained Liner........................................................61 3.2 Contact Analysis Technology....................................................................................63 3.2.1 General Introduction...............................................................................................63 3.2.2 Contact Bodies Definition......................................................................................63 3.2.3 Analysis Procedure and Parameters.......................................................................64 Analysis Procedure.........................................................................................................64 Contact Tolerance...........................................................................................................65 Friction and Separation..................................................................................................66 3.3 Pre-Discussions on Buckling of Restrained Liner....................................................68 3.3.1 Prediction on Buckling Location...........................................................................68 3.3.2 Application of Existing Buckling Equations..........................................................70 3.3.3 Methodology and Objective ..................................................................................72 3.4 Numerical Analysis of Buckling of Encased Liner...................................................73 3.4.1 Numerical Analysis Modeling................................................................................73 Finite Element Analysis Modeling..................................................................................73 Contact Analysis Setting.................................................................................................76 3.4.2 Buckling of Plain Liner..........................................................................................78 Analysis Models and Cases.............................................................................................78 Numerical Results and Discussions................................................................................79 Summary..........................................................................................................................91 3.4.3 Buckling of Stiffened Liner...................................................................................92 Numerical Analysis and Results......................................................................................92 Numerical Results Discussion and Theoretical Solutions Examination.........................96 3.5 Experimental Investigations of Single-lobe Buckling Failure..................................98 3.5.1 Experiment Design.................................................................................................98 3.5.2 Test models and Experiment Setup......................................................................100 3.5.3 Experimental Results and Discussions.................................................................103 Failure Deflection and Loads........................................................................................103 Stains Distribution and Behavior..................................................................................106 3.6 New Solution for Single-Lobe Buckling of Stiffened Pipe.....................................117 3.6.1 Derivation of Single-lobe buckling Equations.....................................................117 3.6.2 New Analytical Solution......................................................................................120 Judgment of Stiffening Condition..................................................................................120 Rigidly Stiffened Pipe....................................................................................................120 Weakly Stiffened Pipe....................................................................................................122 3.6.3 Verification of New Solution................................................................................123 References......................................................................................................................125 Chapter 4 Buckling of Jointed Pipe under External Pressure 4.1 Introduction…………….........................................................................................127 4.2 Experimental Investigation......................................................................................128 4.2.1 Testing Models and Material................................................................................128 4.2.2 Experiment Setup..................................................................................................129 4.2.3 Experimental Results and Discussion...................................................................130 Buckling Modes and Critical Pressure..........................................................................130 Buckling Behavior Discussions.....................................................................................132 4.3 Numerical and Theoretical Investigation.................................................................138 4.3.1 Numerical analysis................................................................................................138 4.3.2 Flexural Rigidity of Joint......................................................................................139 4.3.3 Semi-Analytical Solution......................................................................................140 4.3.4 Verification of Numerical and Theoretical Solutions...........................................141 4.4 Summary..................................................................................................................144 References......................................................................................................................144 Chapter 5 Water Tunnel Liner Design 5.1 Design Conditions and Procedure...........................................................................145 5.2 Steel Liner Design for Internal Pressure..................................................................148 5.2.1 Steel Liner Thickness Estimation.........................................................................148 5.2.2 Expansion Deformation........................................................................................148 5.3 Steel Liner Design for External Pressure................................................................149 5.3.1 Steel Liner of Integrated Structure.......................................................................149 5.3.2 Steel Liner of Separated Structure........................................................................151 Uniformly Supported Liner............................................................................................151 Locally Supported Liner................................................................................................153 5.4 Summary and Discussions.......................................................................................156 References.....................................................................................................................158 Chapter 6 Summary and Conclusions 6.1 Summary...………………………...........................................................................159 6.2 Conclusions…………………..................................................................................161 List of Figures Fig.1.1 Estimated distribution of water scarcity in 2025…...............................................1 Fig.1.2 Schematics of water tunnel structure....................................................................9 Fig.1.3 Buckling collapse of steel liner of conventional water tunnel structure.............10 Fig.1.4 Schematics of deep water tunnel in current study...............................................11 Fig.1.5 Liner support conditions in separated-type water tunnel………………………12 Fig.1.6 Buckling collapse of steel liner of separated water tunnel structure...................14 Fig.2.1 Typical stiffeners.................................................................................................20 Fig.2.2 Wall cross-section of stiffened pipe.............................................................21 Fig.2.3 Buckling forms of stiffened pipe.........................................................................23 Fig.2.4 Equilibrium paths for perfect and imperfect shells.............................................26 Fig. 2.5 Local buckling and general buckling..................................................................29 Fig.2.6. Effects of stiffness and spacing on buckling of uniformly stiffened pipe..........30 Fig.2.7 Effects of pipe dimensions on buckling of uniformly stiffened pipe..................31 Fig.2.8 Buckling behavior of uniformly Stiffened pipe (spacing S1>S2)........................32 Fig.2.9 Variations of an infinite small element and flexural deformation.......................35 Fig. 2.10 Schematic of model and loading......................................................................36 Fig.2.11 Work mechanism of infinite small element during buckling............................38 Fig.2.12 Comparison of numbers of buckling waves......................................................48 Fig.2.13 Comparison with Kendrick theory....................................................................48 Fig.2.14 Comparison with Bryant theory........................................................................48 Fig.2.15 Comparison with Tokugawa theory..................................................................49 Fig.2.16 Comparison with Timoshenko theory...............................................................49 Fig.2.17 Buckling behavior (Model1) ............................................................................52 Fig.2.18 Buckling behavior (Model2,9) .........................................................................52 Fig.2.19 Buckling behavior (Model3, 10) ......................................................................52 Fig.2.20 Buckling behavior (Model4) ............................................................................52 Fig.2.21 Buckling behavior (Model5) ............................................................................52 Fig.2.22 Buckling behavior (Model6) ............................................................................52 Fig.2.23 Buckling behavior (Model7,8 ).........................................................................52 Fig.2.24 Buckling deformation........................................................................................52 Fig.3.1 Schematic of single-lobe buckling......................................................................57 Fig.3.2 Analytical model of single-lobe buckling...........................................................59 Fig.3.3 Various gap-corrected hydrostatic buckling data compared with Boot-Glock theory for perfectly close fitting, perfectly circular liners.................................62 Fig.3.4 Initial state of encased pipe.................................................................................68 Fig.3.5 Potential equilibrium state of encased pipe before buckling..............................69 Fig.3.6 Buckled shape of restrained pipe .......................................................................69 Fig.3.7 Schematic profile of contact analysis modeling.................................................73 Fig.3.8 Elastic-perfectly- plastic stress-strain behavior..................................................74 Fig.3.9 Loading conditions (Loading cases) ..................................................................74 Fig.3.10 Loading procedure of water pressure in numerical analysis.............................75 Fig. 3.11 Pressure-displacement behavior with different gap ratio ( Model3 )...............80 Fig.3.12 a) Radial deformation of failure mode under LC1 (Model3)...........................82 Fig.3.12 b) Radial deformation of failure mode under LC2 (Model3) ..........................84 Fig.3.13 Comparison of theoretical and numerical results (Model1, R/t=200) .............86 Fig.3.14 Comparison of theoretical and numerical results (Model2, R/t=125) .............87 Fig.3.15 Comparison of theoretical and numerical results (Model3, R/t=75) ...............87 Fig.3.16 Comparison of theoretical and numerical results (Model4, R/t=50) ...............87 Fig.3.17 Comparison of theoretical and numerical results (Model5, R/t=35.7) ............88 Fig.3.18 Enhancement factor change with respective to the R/t and Δ/R.......................88 Fig.3.19 Local and general buckling modes ( S=1.5m, t=5mm, tr=30mm ) ..................93 Fig.3.20 Behavior of critical pressure to gap ratio ( S=0.5m ) .......................................94 Fig.3.21 Behavior of critical pressure to gap ratio ( S=1.0m ) .......................................94 Fig.3.22 Behavior of critical pressure to gap ratio ( S=1.5m ) .......................................94 Fig.3.23 Comparison of theoretical and numerical critical pressures ( t=5mm) ............95 Fig.3.24 Comparison of theoretical and numerical critical pressures ( t=10mm ) .........95 Fig.3.25 Comparison of theoretical and numerical critical pressures ( t=20mm ) .........95 Fig.3.26 Schematic of experiment design.......................................................................99 Fig.3.27 Experiment setup.............................................................................................101 Fig.3.28 Schematic of failure deflection........................................................................101 Fig.3.29 Measurement arrangement..............................................................................102 Fig.3.30 Loading and failure condition (Model 4) .......................................................103 Fig.3.31 Behavior of testing load vs. radial displacement at crown.............................104 Fig.3.32 Failure deformation mode………………………………..............................105 Fig.3.33 Stains distribution and behavior with testing loads (Model 1) ......................107 Fig.3.34 Stains distribution and behavior with testing loads (Model2) .......................108 Fig.3.35 Stains distribution and behavior with testing loads (Model3) .......................109 Fig.3.36 Stains distribution and behavior with testing loads (Model4) .......................110 Fig.3.37 Stains distribution and behavior with testing loads (Model5) .......................111 Fig.3.38 Stains distribution and behavior with testing loads (Model6) .......................112 Fig.3.39 Stains distribution and behavior with testing loads (Model7) .......................113 Fig.3.40 Average strain-testing load (Model 1) ...........................................................114 Fig.3.41 Average strain-testing load (Model 2) ...........................................................114 Fig.3.42 Average strain-testing load (Model 3) ...........................................................114 Fig.3.43 Average strain-testing load (Model 4) ...........................................................114 Fig.3.44 Average strain-testing load (Model 5) ...........................................................114 Fig.3.45 Average strain-testing load (Model 6) ...........................................................115 Fig. 3.46 Average strain-testing load (Model 7) ..........................................................115 Fig.3.47 Definitions in single-lobe buckling equations derivation...............................117 Fig.3.48 Comparison between analytical and numerical critical pressure (t=5mm).....123 Fig.3.49 Comparison between theoretical and numerical critical pressure (t=10m)....123 Fig.3.50 Comparison between theoretical and numerical critical pressure (t=20m)....124 Fig.4.1 Schematic view of experimental setup ............................................................129 Fig.4.2 Experimental setup………...............................................................................130 Fig.4.3 Typical buckling modes....................................................................................131 Fig.4.4 Failure mode of No.1a, b, c..............................................................................131 Fig.4.5 Buckling failure mode of No.2a, b, c...............................................................131 Fig.4.6 Buckling failure mode of No.3a.......................................................................131 Fig.4.7 External pressure vs. strain behavior (No.1a) .................................................133 Fig.4.8 External pressure vs. strain behavior (No.2a) .................................................133 Fig.4.9 External pressure vs. strain behavior (No.3a) .................................................133 Fig.4.10 External pressure vs. strain behavior (No.1c) ...............................................133 Fig.4.11 External pressure vs. strain behavior (No.2c) ...............................................134 Fig.4.12 External pressure vs. strain behavior (No.1b) ...............................................134 Fig.4.13 External pressure vs. strain behavior (No.2b) ...............................................134 Fig.4.14 Radial displacements (No.1a).........................................................................135 Fig.4.15 Radial displacements (No.1c) ........................................................................135 Fig.4.16 Comparison of critical pressures for different mode......................................135 Fig.4.17 Numerical analysis model…..........................................................................138 Fig.4.18 Buckling deformation change with η (model1)..............................................142 Fig.4.19 Estimation of effective flexural rigidity factor of joint η .............................142 Fig.4.20 Comparison of theoretical, numerical, and experimental results....................143 Fig.5.1 Flow chart for water tunnel design...................................................................146 Fig.5.2 Schematic view of water tunnel conditions......................................................146 Fig.5.3 Schematic profile of tunnel structure................................................................147 Fig.5.4 Design graph for estimating steel liner thickness.............................................149 Fig.5.5 Schematic view and notions of backfill defect region……………...................150 Fig.5.6 Relation between critical pressure and defect region (t=15mm) .....................150 Fig.5.7 Relation between critical pressure and thickness of steel liner.........................154 Fig.5.8 Relation between critical pressure and stiffener height (t10mm,S1.5m) .........155 List of Tables Table 2.1 Analysis models.............................................................................................30 Table 2.2 Material properties......................................................................................30 Table 2.3 Test models...................................................................................................45 Table 2.4 Aluminum (Al-6061) properties................................................................45 Table 2.5 Comparison between analytical and experimental results.....................46 Table 3.1 Analysis models (plain pipe) ..........................................................................78 Table 3.2 Gap cases (plain pipe)......................................................................................78 Table 3.3 Application range of Amstutz’s equation in critical pressure.........................78 Table 3.4 Critical pressure comparison of close-fitted pipe............................................88 Table 3.5 Analysis models (stiffened pipe) ....................................................................92 Table 3.6 Gap cases (stiffened pipe) ...............................................................................92 Table 3.7 Test models....................................................................................................100 Table 3.8 Material properties.........................................................................................100 Table 3.9 a) Collapse load of model as a part of pipe (in edge load).............................104 Table 3.9 b) Collapse load of model as an individual arch (in total load).....................104 Table 4.1 Test models....................................................................................................128 Table 4.2 Material properties.........................................................................................128 Table 4.3 Critical pressures and buckling modes..........................................................130 Table 5.1 Design results summary.................................................................................157 Nomenclature The liner is also expressed as pipe/cylindrical shell, and the lining as cavity/ host/ lining / RC segment. In addition, the buckling of free pipe and restrained pipe are also called the free buckling, multi-lobe buckling, rotary symmetric buckling and restrained buckling, single-lobe buckling, non-symmetric buckling, etc., respectively. The other symbols are noted as followings: Shell/ liner and Stiffener Geometry t : thickness of shell plat L : length of pipe/ liner D0 : outer diameter D : mean diameter R : mean radius of pipe/liner hr : height of stiffener tr : thickness of stiffener S : spacing of stiffener Be : effective width of stiffener Ar : cross-sectional area of stiffener Ir : second moment of inertia of stiffener X, Y, Z : global Cartesian coordinates x, y, z :cross-sectional cylindrical coordinates θ : angular coordinate of cross-section (θ = y/R) Material Properties fy : yield stress E : Young’s modulus V : poisson’s ratio Buckling Theory of Free Steel Liner Π : total potential energy of systems V : strain energy of cylindrical shell(with variations Vp1 Vp2) Vp1 : axial strain energy of cylindrical shell Vp2 : flexural strain energy of cylindrical shell Vs : strain energy of stiffener U : work done on shell by outside forces dS : increment of arc length u, v, w : displacements in cylindrical coordinates χ x0 , χ y0 : curvature change of middle surface εx0, εy0 , εz0 , γxy0 , γxz0 , γyz0 : strains at middle surface εx, εy , εz , γxy , γxz , γyz : strains at arbitrary surface σx, σy, σz, τxy, τxz, τyz, : stresses at arbitrary surface ε1, ε2, ε3 ,γ12, γ13, γ23 : nonlinear strains in principal material directions Ny : axial force resultants Zb : Baterdf parameter p : acting external pressure Pcr : critical external buckling pressure PcrG, PcrL : critical pressure of general and local buckling PL : limit external pressure of elastic buckling σcr : critical stress m, n : axial wave and circumferential wave numbers A, B, C: constant factors of displacement u, v and w α, Κ : non-dimensional variations for calculation buckling waves Buckling Theory of Restrained Steel Liner Δ : gap between steel liner and cavity R’: inner radius of tunnel lining t : liner plate thickness σN : axial stress α : one-half angle subtended to center of cylindrical shell by buckled lobe β : one-half angle subtended by new mean radius through half waves of buckled lobe Pcr : critical external buckling pressure F : cross-sectional area of stiffener and pipe shell between the stiffeners h : distance from neutral axis of stiffener to the outer edge of the stiffener J : second moment of inertia of stiffener and contributing width of the shell K : enhancement factor Fb : buoyancy of unit length liner W : self-weight of unit length liner ρ : new mean radius through the half waves of buckled lobe ls : circumferential length of single-lobe wave I : effective second moment of inertia of the stiffener Buckling theory of jointed steel liner kθ :flexural rigidity of joint η: effective flexural rigidity factor of a joint kθ* : ratio of flexural rigidity of joint to that of segment Design Study Variables Hs : over burden Hw : ground water table Pi : design internal pressure Po : design external pressure Fs : safety factor for buckling design Di:: necessary inner diameter (I.D.) σa: allowance stress
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