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Geotechnical considerations for offshore gravity type structures with emphasis on foundation stability.. Gaard, Thomas C. 1982-03-31

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GEOTECHNICAL CONSIDERATIONS FOR OFFSHORE GRAVITY TYPE STRUCTURES WITH EMPHASIS ON FOUNDATION STABILITY UNDER STORM WAVE LOADING by THOMAS C. GAARD S., The University of California, Davis, 197 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1982 O Thomas C. Gaard, 1982 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cl^\ ^^^.,1^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 32/ ^ J z. %  DE-6 (3/81) 11 ABSTRACT A thorough discussion of offshore gravity type structures presently being used, or considered for use in the near future by the oil industry, is presented, along with a brief summary of the major types of structures now used offshore. Factors affecting the stability of offshore gravity type structures are discussed, from the evaluation of a suitable site and the selection of soil parameters, through installation and short-term foundation safety. A case study of the Ekofisk tank is included to show how geotechnical concepts are applied offshore. A thorough description of wave loading on offshore gravity structures is presented, including a discussion on how the design storm is used in geotechnical analyses. Existing stability methods are reviewed. The merits and shortcomings of each method are discussed with respect to their application offshore. Procedures for analyzing the stability of offshore gravity type structures subjected to storm wave loading are developed based on the method of slices. Both Janbu's (1973) Generalized Procedure of Slices and Sarma's (1973) method are adapted for offshore analyses. The latter method is modified to perform pseudo-three-dimensional analyses. A computer program GRAVSTAB developed for this purpose is described and applied to several example problems. The versatility of the method of analysis is demonstrated and results are compared with existing methods. iii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS .' iii LIST OF TABLES vLIST OF FIGURES viACKNOWLEDGEMENTS x NOMENCLATURE xi CHAPTER 1 : INTRODUCTION 1 CHAPTER 2 : THE OFFSHORE GRAVITY TYPE STRUCTURE 11 2.1 General Characteristics 12.2 Platforms For General Offshore Development 14 2.2.1 Concrete Platforms 15 2.2.2 Steel Platforms 8 2.2.3 Hybrid Platforms 23 2.3 Platforms For Arctic Development 25 2.4 Deep Water Platforms And Other Structures 27 2.5 Sources Of New Platform Technology 30 CHAPTER 3 : DESIGN, CONSTRUCTION AND INSTALLATION 31 3.1 Preliminary Considerations 33.1.1 Sources Of Loading 31 3.1.1.1 Environmental Loads3.1.1.2 Operational Loads 32 3.1.2 Environmental. Design Parameters 33 3.1.3 Site Selection And Soil Investigations 34 3.1.4 Selection Of Soil Parameters For Design 42 3.2 Platform Design 46 iv 3.2.1 Hydrodynamic Analyses 46 3.2.2 Geotechnical Analyses 7 3.2.3 Structural Requirements And Analyses 55 3.3 Platform Construction 57 3.4 Platform Installation 9 3.5 Platform Instrumentation 63 CHAPTER 4. : THE EKOFISK TANK - A CASE STUDY 67 CHAPTER 5 : CHARACTERISTICS OF WAVE LOADING 93 5.1 Ocean Waves 95.1.1 The Wave Climate 93 5.1.2 Wave Theories 4 5.1.3 Results Of Linear Wave Theory 97 5.2 Characterizing The Wave System5.2.1 Obtaining The Design Storm 100 5.2.1.1 Statistical Description 105.2.1.2 Geotechnical Equivalent 101 5.2.2 Application Of The Design Storm 102 5.3 Wave Loads On The Foundation System 104 5.3.1 Wave Forces Acting On*The Structure 104 5.3.2 Wave Forces Acting On The Foundation 108 5.4 Effect Of Cyclic Loading On The Foundation System ...109 CHAPTER 6 : PROCEDURES FOR ANALYZING THE STABILITY OF OFFSHORE GRAVITY TYPE STRUCTURES 115 6.1 Fundamental Considerations 116.2 Modelling The Wave-Structure-Soil System 120 6.3 Loading Applied To The Foundation 126 6.4 Available Stability Methods 128 6.4.1 Classical Bearing Capacity Approach 128 V 6.4.2 Other Bearing Capacity Formulations 136 6.4.3 NGI Slip Surface Method 140 6.4.4 Method Of Slices 144 6.4.5 Finite Element Analyses 146.4.6 Model Tests 150 6.5 Summary 152 CHAPTER 7 : APPLICATION OF THE METHOD OF SLICES TO OFFSHORE GRAVITY STRUCTURE FOUNDATIONS 154 7.1 The Method Of Slices 156 7.2 Loading Applied To The Foundation 158 7.3 Treatment Of The Applied Horizontal Force 160 7.4 Modified Janbu Method 161 7.4.1 Assumptions 167.4.2 Derivation Of Equilibrium Equations .161 7.4.3 Working Formulas • 164 7.5 Modified Sarma Method 167 7.5.1 Assumptions 169 7.5.2 Derivation Of Equilibrium Equations 170 7.5.3 Working Formulas 173 CHAPTER 8 : EXAMPLES AND APPPLICATION OF ANALYSES 175 8.1 Description Of Computer Procedure 178.2 Example 1 - A Multi-layered Cohesive Deposit 177 8.3 Example 2 - A Cohesionless Deposit: Ekofisk Tank ....183 CHAPTER 9 : SUMMARY AND CONCLUSIONS 189 REFERENCES 194 LIST OF TABLES Table I - Comparison of Fixed Offshore Platforms 16 Table II - North Sea Concrete Gravity Platforms 19 Table III - Gravity Platforms in Other Parts of the World .. 20 Table IV - Environmental Design Criteria for Some Offshore Areas 35 Table V - Geotechnical Concerns for Offshore Gravity Type Platforms 48 Table VI - Example of the Accumulated Effect of a 100-year Storm 85 Table VII - Some Results of Linear Wave Theory 99 Table VIII - Comparison of Existing Stability Analyses 153 Table IX - Geometry and Loading Data for Example 1 177 Table X - Comparison of Computed Safety Factors for Example 1 179 Table XI - Coefficients for Estimating Undrained Strength from Triaxial Compression Data 182 Table XII - Effect of Shear Zone Representation on the Safety Factor 184 Table XIII - Geometry and Loading Data for Example 2 185 Table XIV - Effect of A-parameter on the Safety Factor 188 vi i LIST OF FIGURES Figure 1.1 - Steel Jacketed Platforms 2 Figure 1.2 - Mobile Platforms 2 Figure 1.3 - The Ekofisk Tank 5 Figure 2.1 - Components of an Offshore Gravity Type Platform 13 Figure 2.2 - North Sea Concrete Gravity Type Offshore Platforms 8 Figure 2.3 - Tecnomare Steel Gravity Type Offshore Platform 22 Figure 2.4 - Hybrid Gravity Type Offshore Platforms 24 Figure 2.5 - Arctic Platform Designs 27 Figure 2.6 - Proposed Deep-water Platforms 29 Figure 3.1 - Loads Acting on an Offshore Structure 32 Figure 3.2 - Plan of Survey Lines - Grid: Local Transverse , Mercator Spheroid 37 Figure 3.3 - Typical Soil Profile as Identified by Borehole, Cone Pentration Test and Gamma Ray Logging 43 Figure 3.4 - Comparison of Shear Strength Values from Sample Testing and from CPT 45 Figure 3.5 - Possible Failure Modes for an Offshore Gravity Structure Foundation 50 Figure 3.6 - Possible Modes of Sliding Failure 51 Figure 3.7 - Stability Diagram for a Raft Foundation 53 Figure 3.8 - Installation Sequence for a Gravity Platform .. 60 Figure 3.9 - Detail of CONDEEP Base Structure 61 Figure 3.10 - Maximum Dome Contact Pressures Observed During Installation of the "Beryl A" CONDEEP 64 Figure 4.1 - Detail of the Ekofisk Tank Bottom 69 Figure 4.2 - Loads on the Ekofisk Tank for the 100-Year Wave 71 vi i i Figure 4.3 - Design Storm Data for the Ekofisk Field 71 Figure 4.4 - Typical Geotechnical Profile from Ekofisk Field 72 Figure 4.5 - Shear Strength Data from Ekofisk 72 Figure 4.6 - Predicted Rocking Displacements for the Ekofisk Tank 76 Figure 4.7 - Load-Settlement Curve for Ekofisk Tank 7Figure 4.8 - Ekofisk Settlement Data Relating Submerged Platform Weight and Storm Wave Data in the Early Months After Installation 79 Figure 4.9 - Settlement Data for Ekofisk Tank During Early Storms 7Figure 4.10 - Location of Pressure Gauges and Piezometers Beneath Ekofisk Tank 82 Figure 4.11 - Pore Pressures Observed Under Ekofisk Tank During the First Major Storm 82 Figure 4.12 - Pore Pressure Rise per Cycle Observed in Undrained Simple Shear with Cyclic Loading for Samples Prepared with Relative Densities of 80% 85 Figure 4.13 - Theoretical Prediction of the Pore Water Pressure Distribution Beneath the Ekofisk Tank for Relative Densities of 77% and 85% .... 90 Figure 4.14 - Most Critical Failure Surface Found in Stability Analysis of Ekofisk Tank for Wave Loads Applied Under Undrained Conditions 92 Figure 5.1 - Regions of Validity for Various Wave Theories . 98 Figure 5.2 - Profile of an Airy Wave 99 Figure 5.3 - Forces Acting on the Foundation of an Offshore Gravity Structure 105 Figure 5.4 - Typical Design Storm Representation Used in Geotechnical Engineering 107 Figure 5.5 - Stress Path for a Foundation Element with Partial Drainage Subjected to Storm Wave Loading ill Figure 6.1 - Effective Stresses in Soil for Still Water Conditions (i.e. No Wave Loads) 118 Figure 6.2 - Definition Sketch of Effective Foundation 122 Figure 6.3 - Transformation of Loads, to Foundation Base ....123 Figure 6.4 - Theoretical Rupture Surface Geometry 129 Figure 6.5 - Comparison of Different Proposals for the Value of Nr 132 Figure 6.6 - Geometry of Rupture Surface Used for an Effective Stress Bearing Capacity Solution ....138 Figure 6.7 - Geometry of Sliding Body Used by NGI ..141 Figure 6.8 - Geometry of Bearing Failure Surface Used in the NGI Slip Surface Method 141 Figure 6.9 - Comparison of Two- and Three-Dimensional Distorted Finite Element Meshes for an Inclined and Eccentric Load 148 Figure 6.10 - Effect of Load Eccentricity on Effective Bearing Area as Evaluated Using the Finite Element 149 Figure 7.1 - Representation of Analysis by the Method of Slices 155 Figure 7.2 - Geometry and Forces on a (Janbu) Slice 162 Figure 7.3 - Curve Used for Evaluating the Safety Factor ...169 Figure 7.4 - Geometry and Forces on a (Sarma) Slice 171 Figure 7.5 - Typical (Sarma) Slice Showing Side Forces 171 Figure 8.1 - Shear Strength Profile for Example 1 178 Figure 8.2 - Critical Shear Surfaces for Example 1 as Evaluated by Different Stability Methods 180 Figure 8.3 - Zones of Shear on the Potential Failure Surface and Relevant Laboratory Tests 182 Figure 8.4 - Critical Shear Surface for Example 1 Found from Computer Program GRAVSTAB 184 Figure 8.5 - Distribution of Pore Water Pressures in Foundation Soil Used in Example 2 185 Figure 8.6 - Critical Shear Surface for Example 2 188 X ACKNOWLEDGEMENTS The author wishes to thank his advisor, Professor W.D. Liam Finn for his technical guidance and valuable suggestions to improve the presentation of this thesis. Dr. Yogi Vaid's comments throughout the text were also helpful in packaging the final product. He would also like to thank his advisor and Dr. M. de St. Q. Isaacson for stimulating his interest in many aspects of offshore engineering - a field which the author intends to pursue wholeheartedly. Thanks are also due Dr. P.M. Byrne for many valuable discussions regarding theoretical aspects of stability analyses among other things. The program routine in GRAVSTAB used for applying Sarma's method to offshore platforms is an extension of an earlier program STESL by K.W. Lee and W.D. Liam Finn for the analysis of the stability of underwater slopes. Funding for the computer studies was provided by the National Research Council under grant No.1498 to Professor Finn. This assistance was appreciated. Permission to reproduce many of the figures used in this thesis was kindly granted by numerous people. To all my friends in Vancouver who made my last two years worth more than an education (and bearable) you are not forgotten. A final thanks to Dr. Isaacson for giving me an interesting job at this university that does not require a suit to be worn (ever) or a shave more than twice a week. xi NOMENCLATURE B0 - equivalent foundation width B - effective foundation width L0 - equivalent foundation length A0 - platform base area D0 - skirt depth below mudline fl' - effective unit weight of soil PH - horizontal wave load on platform Pv - vertical platform load at seafloor APV - vertical wave load on platform M - moment at seafloor Ap, - wave pressure on seafloor at tail end of platform Ap2 - wave pressure on seafloor at nose end of platform PA - active soil force on nose of foundation Pp - passive soil force on tail of foundation Pw - water pressure force on tail of foundation Ps - shearing resistance on sides of foundation Pj - shearing resistance on soil-soil interfaces at sides VBT - vertical load at foundation base VB ~ VBT Per un^fc width HBT - horizontal load at foundation base HB ~ HBT Per unit width MfeT - moment applied at foundation base h, - moment arm for active or water pressure force h2 - moment arm for passive soil force xi i h3 - moment arm for shearing resistance on foundation sides e - eccentricity HET - horizontal force applied to effective area HE - HET per unit width HST - horizontal force applied to sliding surface Hs - HST per unit width Fs - maximum shear resistance available from sliding surface per unit width g - load inclination factor a - normal stress o" - effective normal stress tr, - major principal stress 0"3 - minor principal stress u - total pore water pressure us - static pore water pressure uc - pore water pressure due to cyclic effects Au - pore water pressure due to dynamic wave pressure z - depth below mudline A - pore water pressure parameter 0 - friction angle or mobilized friction angle c - cohesion or mobilized cohesion tan0 - frictional resistance or mobilized friction resistance c' - cohesion in terms of effective stress tan0' - frictional resistance in terms of effective stress F - factor of safety applied to strength parameters cu - undrained shear strength - shear strength T - shear stress xi i i Qo - ultimate bearing capacity q0 - ultimate bearing pressure q' - surcharge N - bearing capacity factor for friction N - bearing capacity factor for cohesion N - bearing capacity factor for surcharge s- - bearing capacity shape influence factors d- - bearing capacity depth influence factors i- - bearing capacity load inclination influence factors {, - slice number otj - angle made by top of i-th slice with horizontal ^t - angle made by base of i-th slice with horizontal b^ - width of i-th slice xt- - x-coordinate of midpoint of top of i-th slice yt| - y-coordinate of midpoint of top of i-th slice xb^ - x-coordinate of midpoint of base of i-th slice yb^ - y-coordinate of midpoint of base of i-th slice xg^ - x-coordinate of centroid of i-th slice yg^ - y-coordinate of centroid of i-th slice xs- - x-coordinate of point of application of side forces for i-th slice ysc ~ y-coordinate of point of application of side forces for i-th slice h^ - height of i-th slice - vertical offset of thrust forces for i-th slice Ah£ - distance between base and line of thrust for i-th slice FV{, - vertical force on top of i-th slice FT(. - total vertical load on top of i-th slice FH: - horizontal force on top of i-th slice xiv FN-, - normal force on top of i-th slice FTC - tangential force on top of i-th slice US^ - pore water force on base of i-th slice ssL - shear force on one side of i-th slice x 2 - normal force on base of i-th slice N; - effective normal force on base of i-th slice Si - shear force on base of i-th slice E'v - lateral thrust applied to i-th slice T'v - vertical shear force at x=xj Q'v - assumed vertical shear force at x=xj v; - vertical resultant on base of i-th slice H; - horizontal resultant on base of i-th slice wt - total saturated weight of i-th slice w;' - effective weight of i-th slice UH; - resultant water force at x=xj - pore water pressure at base of i-th slice - normal stress on base of i-th slice - shear stress on base of i-th slice - shear strength at base of i-th slice - available cohesion on base of i-th slice tan0 'j - available frictional resistance on base of i -•th slice = 1 - mobilized cohesion on base of i-th slice tan0'^ - mobilized frictional resistance on base of i -th slice r t. - factor of safety on interslice face for i-th slice K - acceleration coefficient as a fraction of gravity X - vertical shear force multiplier - K for a given factor of safety V CHAPTER 1 INTRODUCTION The increase in global energy consumption and the prevailing geopolitical climate in the world have had disastrous effects on the cost and availability of petroleum to most consumers. This, along with the western world's desire to be energy self-sufficient, has led to the development of energy resources which were previously considered to be uneconomical. In an effort to meet the goals of energy self-sufficiency and (indeed) availability, oil companies have in recent years been increasing their exploitation of the vast reserves of oil and gas that exist beneath the continental shelves of the world's oceans. As consultants to the oil companies, engineers are required to (1) provide the technical input necessary for the implementation of hydrocarbon recovery schemes, and (2) to develop reliable methods for the design, analysis, and installation of the necessary offshore structures. Offshore platforms have been in existence since the 1920s when oil was discovered at Lake Maracaibo, Venezuela. These structures, usually made of concrete and piled into the soft nearshore sediments, were crude by today's standards but are important in that they constituted the beginning of the offshore oil industry (Bjerrum, 1973). The first "deep-water" fixed platforms constructed were the steel jacket or template type structures, similar to those shown in figure 1.1 used in the Gulf of Mexico, of which some several hundred have been built B) SELF-FLOATER A) CONVENTIONAL TYPE Figure 1.1 - Steel jacketed offshore platforms (After McPhee and Reeves, 1975) 3 there since the 1960s. These platforms are also familiar sights in other parts of the world, namely: Lake Maracaibo, the Persian Gulf, the North Sea, the Java Sea, the Gulf of Guinea, offshore California, and to a lesser extent, other locations (Martin and Shaw, 1974). Semi-submersible and jackup type exploratory drilling rigs such as the ones shown in figure 1.2 are also in widespread use throughout the world. In 1969, when the Phillips Petroleum Company discovered the first commercial oil field (the Ekofisk field) in the northern North Sea, engineers were faced with some new and difficult problems when designing the necessary structures for the development of this field. Because of the extreme hostility of the northern North Sea (24 meter high waves at this location) and the lack of nearby harbors (the closest being nearly 320 kilometers away) the need arose for a production platform in close proximity to the drilling platforms (which were to be of the conventional jacket type) which could function as a storage facility in poor weather when tanker loading would-be impossible (Bjerrum, 1973). From this need came the first offshore gravity type production platform; this is the famed Ekofisk tank designed by the C. G. Doris Company of France. The tank is shown being towed from the Norwegian coast to its home in the northern North Sea in figure 1.3. Interest in gravity type production platforms has increased steadily since 1973 when the Ekofisk tank was installed, primarily because of the short installation time required for a gravity type structure (no piling necessary in the unpredictable North Sea) and the successful operation of the Ekofisk tank 4 Figure 1.3 - The Ekofisk tank (See following page) (Reproduced with permission of the Royal Institute of Naval Architects, London.) 6 since installation, including a good performance through a major storm (90% of the design storm) which occurred six months after it was installed (Marion, 1974). More than twenty other gravity platforms have been installed to date in the North Sea and other areas of the world (Waagaard, 1977). The Ekofisk tank was the first offshore gravity type platform but not the first gravity type structure used offshore. Gravity type light towers had been used extensively in Sweden for many years prior to 1973 in shallower, nearshore waters, usually 20 meters deep or less (Stubbs, 1975). The Royal Sovereign light tower in the English Channel is perhaps a more familiar example' of a pre-1973 gravity type structure. These are relatively small structures which required few new design concepts, and construction and installation techniques at the time that they were installed. The installation of the Ekofisk tank, however, was a milestone in engineering design and marked a new era for offshore gravity structures. This tank required many new design procedures, construction methods, and installation techniques that had to be developed specifically for these purposes (Bjerrum, 1973; Gerwick and Hognstad, 1973; Marion, 1974). In recent years, North American engineers have been playing an increasing role in the development of offshore gravity structure technology, although few are formally trained in the area. With increasing exploration and utilization of oil and gas resources off the North American coast, the need for geotechnical engineers with a good working knowledge of offshore engineering will undoubtedly increase on this continent. 7 The purposes of this thesis are threefold: (1) to introduce the geotechnical engineer to the field of offshore engineering, specifically, to familiarize him with the special problems associated with gravity structures, (2) to present an overview of existing stability methods applicable to offshore gravity type structures, and (3) to develop an alternative procedure for analyzing the stability of an offshore gravity structure subjected to storm wave loading. This thesis may be divided into two sections. Chapters 2-5 deal with the first consideration, a general background in offshore engineering. The second part of this thesis, Chapters 6-8, is concerned wholly with the topic of foundation stability under storm wave loading. The preliminary chapters serve a dual purpose. First, they serve to give the reader unfamiliar with offshore engineering a good working knowledge of offshore gravity structures. Secondly, they provide him with an appreciation of the offshore environment and the special design and installation requirements for gravity type structures. This is necessary so that the foundation analyses may be viewed in perspective. The aim of the section on foundation stability under storm wave loading is to present an overview of the stability methods presently available for performing such analyses and to demonstrate the need for and then develop a simple, practical alternative method for effective stress analyses. Chapter 2 serves as an introduction to offshore gravity type structures. The general characteristics of a gravity 8 structure are described and the major types of platforms are discussed in some detail. Chapter 3 is concerned with the design, construction, and installation requirements for these structures. First, the sources of loading in the offshore environment are outlined. Next, the site selection, the offshore site investigation and the selection of geotechnical soil parameters is discussed in depth. Platform design requirements (hydrodynamic, structural, and geotechnical), construction techniques and installation procedures are then delineated. A short section on platform instrumentation finishes off this chapter. Chapter 4 presents a geotechnical case study of the Ekofisk tank. This chapter serves the purposes of demonstrating how geotechnical analyses are applied offshore and how performance observations may be used as a check on design assumptions and predictions. Chapter 5, the final chapter in the first part of this thesis, deals with wave loading on gravity platforms. A brief discussion of the wave climate is given and the modelling of ocean waves by wave theories and statistical means is discussed. The geotechnical equivalent of the statistical design storm, that which is used for cyclic loading studies and to determine the maximum load, is given particular attention. Finally, the characteristics of wave loading on the foundation system are discussed as they pertain to foundation analyses. This chapter also serves as an introduction to the next section. Chapter 6 presents the quantitative aspects of wave loading on the foundation system. Methods of determining platform 9 stability under storm wave loading are then discussed; the merits and shortcomings of each method when applied to the offshore gravity structure are emphasized. It becomes clear that there are two fundamental analytical lines of approach to the problem: bearing capacity theory and the finite element method. A simple to use limit equilibrium method for total stress analyses of clay foundations called the NGI (Norwegian Geotechnical Institute) slip surface method is described in detail. In Chapter 7 an alternative method of analysis based on the method of slices is presented. This method is along the lines of the NGI method. It is a pseudo-three-dimensional effective stress method based on Sarma's (1973) method of slices. Sarma's method of slices is not well known among practicing engineers and hence Janbu's (1973) method of slices is also adapted to perform a gravity structure stability analysis, although only in two dimensions. This was done so that existing slope stability programs may be modified to perform some of the analyses and also to install faith in the use of Sarma's slice method. In Chapter 8 a computer program GRAVSTAB developed to perform these analyses is described and several example problems are worked. The application of the method to both a total stress analysis and an effective stress analysis is made. The versatility of the method is shown. This method is of great practical value for working these types of problems. The discussion of gravity structures presented herein, although primarily concerned with platforms, is generally applicable to all large gravity type offshore structures. The 10 analytical procedures discussed and developed in this thesis are applicable to any offshore structure with a monolithic gravity type base, whether it is a platform, light tower, flare structure or other facility.1 1A flare structure is used for burning off excess gases, primarily methane, produced along with oil from a well. 11 CHAPTER 2 THE OFFSHORE GRAVITY TYPE STRUCTURE The following discussion of offshore gravity type structures includes all the major types of gravity structures presently in use and those which are being seriously considered by the . oil industry for use throughout the world in the near future. 2.1 General Characteristics A gravity type structure rests directly on the seabed and has no subsurface foundation other than shallow skirts and ribs which portrude through the upper sediments to transfer the horizontal component of the disturbing force to deeper, stronger subsoils. In areas where the surficial sediments have adequate strength to prevent sliding or the raft foundation is excavated, skirts or ribs may not be present (Huntemann et al, 1979). Skirts have the added features of providing scour protection from currents and wave induced water motions and containing grout which is used during installation. They are therefore usually necessary unless the foundation has been excavated. To prevent sliding at the base of the structure or a shear failure beneath the structure, a vertical force on the foundation must be maintained in some proportion to the maximum horizontal load. This is accomplished by using a structure of ample weight with respect to the horizontal forces expected -hence the name gravity structure. 12 There are generally three distinct parts of a large gravity platform: the base caisson, the towers, and the deck. A typical North Sea concrete gravity platform is shown in figure 2.1. Other gravity platforms, although somewhat different, have many of the same features as the one shown. The deck is used as a work area and often houses living quarters for the men who service the platform equipment. The specific equipment on the deck depends on the exact use of the platform. Facilities for fresh water storage and other platform requirements are often housed in the towers. The well risers may also be contained within the legs.2 The base caisson is used during installation as a buoyancy chamber; the caisson is made up of a number of cells (either outwardly apparent as with the platform shown in figure 2.1, or compartmented within the caisson as with the Ekofisk tank shown in figure 1.3) which are used to systematically ballast the structure. The skirts are driven into the foundation soil and the structure is firmly seated by increasing the ballast. The cells are used as both ballast tanks and oil storage facilities when the structure is operational. These platforms are generally massive structures, particularly those platforms designed for the North Sea. The largest of the North Sea giants, a Doris type structure placed in the U. K.'s Ninian field in 1978, weighed 600,000 tons 2Risers are pipes through which crude oil flows out from the well and up to the platform in to be processed, pumped, or stored. 13 Figure 2.1 - Components of an offshore gravity type platform (Adapted from Klitz, 1980) 14 (Steven, 1981a). This platform is taller than a 50 story building from the base to the deck (not including the deck equipment) and nearly as wide at the base. A Sea Tank type structure also placed in U. K. waters in 1978 holds the depth record for a gravity platform - 152 meters (Furnes, 1978). Other gravity platforms in the North Sea are nearly as large, and although the platforms in other offshore areas are appreciably smaller, they are still very large indeed. Clearly these are enormous structures with unusual design and construction requirements. 2.2 Platforms for General Offshore Development Steel jacketed structures and gravity platforms form the core of structures presently used in offshore hydrocarbon recovery.3 The jacketed structures are much more numerous. Generally, the steel jacketed structure, and the gravity structures discussed in the following subsections, will not be used in water depths greater than about 250 to 300 meters. In deeper water, other types of structures will be used. 3 A jacketed structure was successfully placed in 312 meters of water in the Gulf of Mexico in 1980 (Morrison, 1980a). The installation of this type of structure in water of that depth is not seen as a trend for the future. The use of this type of structure was economically justified since the priority was a large number of wells which this platform, with its large base area, was able to provide (Morrison, 1980b). Several other jacketed platforms in similar water depths are planned for use in the Santa Barbara channel. Alternative platform designs such as those discussed in section 2.4 are not fully developed yet. 15 2.2.1 Concrete Platforms The concrete (reinforced and prestressed) gravity type platform was designed to meet specific requirements for the development of the Ekofisk field in the North Sea. Steel jacketed platforms, the only fixed offshore platforms existing at that time, could not be modified to include the required amount of storage. This requirement for storage was the primary reason that the gravity type platform was developed and remains as an important factor when choosing between steel jacketed and gravity type structures for field development. It should be remembered that the gravity structure is an alternative to the steel jacketed platform, not a replacement for it. The two platform types are quite different and generally applicable to different design and production considerations. A brief comparison of the concrete gravity type platform and the steel jacketed platform, the two most common types of (large) fixed offshore structures, is presented in Table I. Concrete was the first material used for building gravity type platforms and remains the most common for a variety of reasons, some being: construction techniques require less skilled labor than steelwork, the availability of concrete is generally better than that of high grade structural steels, and concrete is more corrosion resistant and has a longer fatigue life than steel in the marine environment (Stubbs, 1975). The latter two reasons are very important since maintenance is expensive and repairs are difficult, if even possible, offshore (Billington, 1979). 16 Table I Comparison of Fixed Offshore Platforms STEEL JACKETED PLATFORM CONCRETE GRAVITY PLATFORM ADVANTAGES -Much industry experience -Generally cheaper for mild environments -Design less site-specific -More flexible to changes during fabrication -Good for areas with deep soft sediments ADVANTAGES -Requires little specialized labor -Greater production capacity -Easy to incorporate storage -Short installation time -Larger deck -Almost complete at tow-out for early production start -Longer fatigue life -More corrosion resistant DISADVANTAGES -Requires very skilled labor -Long, costly installations -Hard to provide storage -Relies on the availability of high-grade steel -Difficult to inspect for damage -Need more deep borings -Problems with driving large diameter piles -Shorter fatigue life -Less corrosion resistant PISADVANTAGES -Inflexible to design/const, changes-very site specific -Design more critical to specific water depth -Seabed must be relatively flat and level -Requires good bearing soils -Need good knowledge of shallow sediments Adapted from Bell (1974). 17 As of 1981, fourteen concrete gravity type platforms have been installed in the North Sea (Furnes, 1978; Steven, 1981b). These platforms are of four different types: the Doris, Andoc, CONDEEP, and Sea Tank designs. The Doris design is that of the Ekofisk tank (shown in figure 1.3) and looks somewhat different in general appearance than the other North Sea designs shown in figure 2.2. Design conditions, analysis techniques, and construction methods are, however, virtually the same for all these platforms. Each platform type was modified somewhat for the specific on-site design criteria: design wave height, water depth, and production requirements (storage capacity and deck installations). Hence, the size and shape of each platform is distinct. A summary of these platforms is given in Table II along with some of their important features. Three concrete gravity platforms have been built off the coast of Brazil (Franco, 1976) and one offshore Louisiana (Huntemann et al, 1979). These platforms are box-shaped and significantly smaller than the pedestal shaped North Sea giants. Four steel gravity type platforms off the Congo coast (Lalli, 1977), and a flare offshore Brazil (Burns and D'Amorim, 1977), are the only other large offshore gravity type structures in the world outside of the North Sea. A list of these platforms, also giving some of their important features, is given in Table III. Gravity platforms made of materials other than concrete are primarily special designs for particular applications. This is especially true of the all-steel gravity structure. A) CONDEEP DESIGN B) SEATANK DESIGN C) ANDOC DESIGN Fig. 2.2 North Sea concrete gravity type offshore platforms (Compiled from Sjoerdsma, 1975a) 00 19 Table II North Sea Concrete Gravity Platforms DESIGN FIELD/ COUNTRY WATER DEPTH DESIGN WAVE BASE WIDTH PURPOSE DATE Doris Ekofisk (Norway) 70 24.0 93 P-S 1973 CONDEEP Beryl A (U.K.) 120 29.5 100 D-P-S 1975 CONDEEP Brent B (U.K.) 142 30.5 100 D-P-S 1975 Doris Frigg CDP1 (U.K.) 96 29.0 101 D 1975 Sea Tank Frigg TP1 (U.K.) 104 29.0 72 P 1976 Doris Frigg MP2 (U.K.) 94 29.0 101 B 1976 CONDEEP Brent D (U.K.) 142 . 30.5 100 D-P-S 1976 Andoc Dunlin A (U.K./Hoi.) 152 30.5 104 D-P-S 1977 CONDEEP Statfjord A (Norway) 149 30.5 1 10 D-P-S 1 977 CONDEEP Frigg TCP2 (Norway) 1 04 29.0 100 T-B-P 1 977 Doris Ninian (U.K.) 139 31.2 140 D-P 1978 Sea Tank Brent C (U.K.) 142 30.5 100 D-P-S 1 978 Sea Tank Cormorant A (U.K.) 152 30.5 100 D-P-S 1978 CONDEEP Statfjord B (Norway) 144 30.5 152 D-P-S 1981 D=Drilling P=Production S=Storage T=Treatment B=Booster Note: All dimensions are in meters. Adapted from Furnes (1978). 20 Table III Gravity Platforms in Other Parts of the World DESIGNER/ CONSTRUCTION FIELD/ COUNTRY WATER DEPTH DESIGN WAVE BASE SIZE PURPOSE DATE Tecnomare (Steel) Loango (Congo) 89 9.4 3@ 1 8 1 D 1976 Tecnomare (Steel) Loango (Congo) 89 9.4 3@ 1 8 1 D 1976 Tecnomare (Steel) Loango (Congo) 89 9.4 3@1 8 1 D 1976 Tecnomare (Steel) Loango (Congo) 89 9.4 3§18' P 1977 Petrobas (Cone-Box) RGdeNorte (Brazil) 13 ? 46x53 D-P 1978 Petrobas (Cone-Box) RGdeNorte (Brazil) 13 ? 46x53 D-P 1978 Petrobas (Cone-Box) RGdeNorte (Brazil) 13 ? 46x53 D-P 1978 ARCO (Cone-Box) Louisiana (U.S.A.) 4 ? 23x34 D-P 1978 'This design has three base pads (tripod) Note: D=Drilling P=Production Note: All dimensions are in meters. Note: This list may be incomplete. 21 2.2.2 Steel Platforms Steel gravity platforms similar to the one shown in figure 2.3 were first installed off the Congo coast in 1976 (Lalli, 1977) and are presently being built for other locations, including one for the North Sea (Agostoni et al, 1980).* These platforms have some unique features which were developed to solve some special foundation problems. The steel gravity platform was developed for use in the Loango field off the Congo coast (Lalli, 1977),5 Seabed conditions there consist of a rocky uneven bottom (McPhee and Reeves, 1975) which is too hard for piles to be driven into and too unyieldingly uneven for the base slab of a concrete structure. The only type of structure that could be placed economically on the rocky seabed and provide the required amount of storage was a steel-based gravity platform on base pads. (Pile driving in these sediments would require all prebored holes, a lengthy and expensive operation.) The Tecnomare platform, with its tripod arrangement of legs, may be used in areas where the seabed is uneven or inclined by jacking up the legs to compensate for differences in topography (Offshore "The installation of this platform has been delayed due to numerous problems, including a strike at the construction yard. The platform is not expected to produce before 1983 or 1984 (Steven, 1981c). 5The platform was initially conceived for general offshore areas and later for consideration in the Sicilian channel; however, none of these platforms were ever built. The platform was fully developed for use off the Congo coast where it was first installed. 22 Figure 2.3 - Tecnomare steel gravity type offshore platform (After Lalli, 1975) 23 Europe, 1974). Since no highly skilled labor force was available in the Congo, the platforms had to be built in Europe and towed the 8500 kilometers to the Congo. Towing speed and stability requirements contributed to the design shape. A steel gravity platform was chosen for the Maureen field in the North Sea primarily because of reservoir considerations (Lalli, 1977). The Maureen field is a so-called marginal field, that is, one which has very limited potential. Since pipelines are not economically justified, storage is required. In the event that the field is not profitable, the structure may be removed and relocated to a comparable site with a minimum of structural damage; steel was chosen over concrete for this reason. Unfortunately, steel, gravity platforms suffer from many of the same setbacks as steel jacketed platforms, primarily steel cost and availability, and the need for a highly skilled labor force to build them. Hence, they will remain as an alternative to the concrete structure, not a replacement for it. 2.2.3 Hybrid Platforms The hybrid gravity platform (Hansen and Ingerslev, 1977; McPhee and Reeves, 1975) consists of a steel space frame mounted on a concrete raft. Two different platform designs are shown in figure 2.4. The hybrid is an attempt to combine the best features of both the steel jacketed and concrete gravity structures. It does offer some distinct advantages over its parents, but it also suffers from some of the same problems, namely: the need for a highly skilled labor force, the 24 Figure 2.4 - Hybrid gravity type offshore platforms (Compiled from Lalli, 1975, and McPhee and Reeves, 1975) 25 availability of high grade structural steel, and the requirement that the seabed be relatively flat and level with adequate bearing strength. The design utilizes a gravity base primarily to cut down installation time and cost and a space frame superstructure to attract smaller wave forces. Because the superstructure is lighter and the wave loads smaller than for the all concrete structure, the base may be decreased in size and the overall weight reduced by 65% to 75% (McPhee and Reeves, 1975). If bearing soils are weak, the base size may be increased to avoid overstressing the soil. Two construction methods add to the platform's flexibility (Hansen and Ingerslev, 1977). The raft may be towed to the site and installed, then the space frame may be connected and the deck mated, or the components may be built independently and joined upon completion at a protected nearshore location before tow-out. The first construction method allows for a gravity type platform to be used when draft restrictions are critical, that is, when no deep water construction site such as a fjord is located nearby; the raft and tower are floated out separately maintaining towing stability with much less draft. The second construction method may be employed when an early installation date is critical by taking advantage of the modular construction. No hybrid structures have yet been installed. 2.3 Platforms for Arctic Development Gravity type platforms have also been designed for use in the Arctic. Among these designs are the monocone (Stenning and 26 Schumann, 1979) and multiple-leg structure (Kliewer and Forbes, 1980), both of which are shown in figure 2.5. These structures were designed for shallow ice-infested waters of the Beaufort Sea. Horizontal ice loads are reduced by the nature of platform geometry. The surface piercing cylindrical leg(s) reduce the area exposed to thinner ice flows, while the conical5 sections below the surface are designed to fail the thicker ice sheets in flexure instead of compression, thereby greatly reducing the lateral loads on the structure. The multiple-leg structure is made of steel, which is more resistant to ice scraping and gouging than concrete. The monocone is constructed of reinforced concrete covered with steel armor to resist damage from moving ice flows. No platforms of either design have yet been installed. Artificial islands are also gravity type structures. Several types of artificial islands have been proposed for exploratory drilling structures in the Beaufort Sea, including the caisson retained structure (de Jong and Bruce, 1978) also shown in figure 2.5. One of these structures was recently completed by Dome Petroleum, Ltd. in the Canadian sector. These structures consist of eight steel wall sections attached via flexible joints that can move under ice loading to transfer the large horizontal load to the soil core - which is of sufficient size to resist shearing failure within the island. Although the structure's core is built of soil, not concrete or steel, it is fundamentally a gravity structure, since stability is achieved by providing a sufficient vertical force on the foundation (weight) to resist failure from horizontal loading. (a) Multiple Leg Gravity Type (b) Monocone (c) Caisson Retained Island Figure 2.5 - Arctic platform designs (Compiled from (a) Kliewer and Forbes, 1980, (b) Bercha and Stenning, 1979, and (c) de Jong and Bruce, 1978) 28 2.4 Deep Water Platforms and Other Structures For water deeper than about 300 meters, the cost of installing gravity platforms and steel jacketed structures increases very rapidly.6 In these waters, alternative recovery methods are therefore required to make hydrocarbon recovery economically attractive. A number of structures have been proposed for deep water use including the articulated column (Moinard, 1979), the guyed tower (Finn et al, 1979), and the tension leg platform (Falkner and Franks, 1978), all which are shown in figure 2.6. These structures are likely to be used in water depths of between 300 and 600 meters (Morrison, 1980b). In deeper water, subsea completion systems (Burkhardt and Michie, 1979) will probably afford the only economical solution. The articulated structures, guyed towers, and tension leg platforms are also applicable to areas with small reservoirs of limited potential as well as to purposes other than drilling or production platforms, such as: flare structures, light towers, and tanker loading terminals where traditional designs would be uneconomical. They are designed to be compliant, that is, they move with the disturbing force somewhat instead of trying to act rigidly and prohibit all motion. The forces acting on the The depth at which these structures become uneconomical is influenced by several factors, namely: the state of current technology, the availability of alternative recovery methods, the nature and severity of environmental loading, and the estimated volume of recoverable hydrocarbons. This depth was chosen based on current publications. The economical depth of monolithic gravity type platforms may be less than this - about 200 meters. 29 (b) Articulated Column (c) Tension-legged Platform (After McPhee and Reeves, 1975) Figure 2.6 - Proposed deep-water platforms 30 structure are thus reduced and the amount of materials required are therefore significantly less than for traditional designs. For these structures, the gravity type base may be used as a foundation system (as opposed to the alternative choice, piles) and they are therefore of interest here. The articulated column is presently being used for flare structures in the North Sea (Sjoerdsma, 1975b) and for tanker loading terminals in both the North Sea (Sjoerdsma, 1975b) and offshore Brazil (Burns and D'Amorim, 1977). However, no drilling or production platforms of this design have been built to date. The North Sea structures are in water depths of between 106 and 150 meters (Moinard, 1979). 2.5 Sources of New Platform Technology For updating this list, the reader is referred to several of many magazines concerned with offshore oil technology, specifically: Ocean Industry, Offshore, Offshore Engineer, The  Oil and Gas Journal, and the Journal of Petroleum Technology. Another good source of information are the Proceedings of the Offshore Technology Conference which is held annually in Houston, Texas. CHAPTER 3 DESIGN, CONSTRUCTION AND INSTALLATION 31 3.1 Preliminary Considerations Before a platform can be designed, some assessment of the sources of loading for the particular area must be made, and the necessary environmental and geotechnical design parameters chosen. 3.1.1 Sources of Loading The sources of loading in an offshore environment are numerous and of varying degrees of importance at different locations. . Generally, they may be broken into two categories: environmental loads and operational loads. Figure 3.1 shows the primary loads that may act on an offshore structure. 3.1.1.1 Environmental Loads Environmental loads are defined as loads caused by natural phenomena - those over which man has no control. Environmental loads acting on an offshore structure include (1) the forces caused by interaction between moving fluids and the structure, such as: wind, waves, currents, or flowing soil, (2) forces due to bodies such as ice impacting the structure, (3) stresses induced by thermal gradients, and (4) forces resulting from earthquake induced accelerations in the structure. Environmental loads transmitted to the foundation are generally inclined, eccentric, and either transient or cyclic in nature. 32 WIND SERVICE JVA\^WVW EARTHQUAKE Figure 3.1 - Loads acting on an offshore structure 33 3.1.1.2 Operational Loads Loads other than environmental are classified as service or operational loads and include those caused by moving equipment and machine vibrations on or within the structure, and those caused by interaction with support vessels, such as: mooring loads, helicopter landings, and possible collisions with either flying or floating vessels. These are usually minor loads necessary only for the design of the deck structure, with the notable exception being collision with a large surface vessel (tanker) under power. Additionally, there may be loads imposed on the structure and its foundation by fluctuations in oil storage quantity, density, or temperature. These loads may be significant and must be considered in design; both minimum and maximum values of weight fluctuations must be specified for foundation design. Minimum values affect stability (overturning) as do maximum values (overstressing). Eccentric loads may also be imposed on the foundation by varying distributions of deck equipment and oil in the storage tanks. The latter of these may be significant and must either be designed for or prevented. 3.1.2 Environmental Design Parameters After environmental conditions for the chosen area have been identified, design parameters must be chosen using appropriate and acceptable methods. This is usually outlined by the regulatory agency with jurisdiction in the case who will often have their own standards (Department of Energy (U.K.), 34 1974; Department of the Interior (U.S.A), 1979; Det Norske Veritas (Norway), 1977; Internationale de la Precontrainte (France), 1977). In some cases, requirements may also be set forth by the underwriter (e.g. Lloyd's Register of Shipping) or the owners, who sometimes use professional society recommendations such as those of the American Petroleum Institute (1978). A partial listing of environmental parameters for some of the world's offshore areas is given in Table IV. A quick glance at the table will show that offshore platforms are subjected to harsh environmental conditions. The results presented in the table are for specific locations within the offshore areas listed and may be more or less severe than those at other locations within the area. 3.1.3 Site Selection and Soil Investigations Gravity type structures require a fairly level seafloor free of large boulders and other obstructions that may damage the base of the structure when it is installed, unless the foundation may be prepared prior to installation. Foundation preparation is.limited to water less than about 70 meters deep -at least on a grand scale (Gerwick, 1974). Recent advances in underwater equipment and diving technology have probably extended this depth somewhat. Surface deposits must be somewhat uniform to prevent excessive differential settlement and, while necessarily possessing adequate strength for stability, they must not be so strong as to prevent the penetration of skirts during installation if skirts are adopted in the design. 35 Table IV Environmental Design Criteria For Some Offshore Areas AREA WAVE HEIGHT (m) CURRENT SPEED (m/s) TIDAL FLUC. (m) WIND SPEED (m/s) ICE THICK, (m) GROUND ACCEL. (g's) Baltimore Canyon (Ward et al,'77) 30.0 ? ? -Beaufort Sea (Kliewer+Forbes,'80) 12.5 ? 2.81 ? 5.0 ? Georges Banks (Ward et al,'77) 25.2 ? - ? Gulf of Alaska (Augustine et al,'78) (Bea and Akky,'79) 40.5 34.0 ? ? ? ? ? ? - ? 0.41 Gulf of Mexico (Haring+Heideman,'78) (Berman et al,'78) 22.8 26.5 ? 2'.7 2.22 ? ? ? - ? ? North Sea (Offshore Europe,'74) 30.5 ? ? 363 54" - ? Offshore Brazil (Burns+D'Amorim,'77) 16.0 1 .8 1.9 385 - ? Offshore Congo (Lalli,'77) 9.4 ? ? ? ? 'Includes a 0.3 meter lunar tide and a 2.5 meter storm surge 2Includes both lunar tide and storm surge 3One hour sustained speed "Gust (several seconds) 5One minute sustained speed 36 Another site requirement is that the bearing soils have adequate strength to support the structure throughout its operational life; this includes both stability under design loads and the effects of repetitive loading (waves, earthquakes, etc.) on the various subsurface deposits. Additionally, settlement of the structure due to elastic displacements, primary and secondary consolidation, and cyclic compaction, must be within tolerable limits. This is generally a site requirement since little modification may be done to the design of a gravity type structure with regards to settlement. These basic concerns are investigated using information from preliminary surveys, which is usually enough to determine the adequacy of the site. The selection of a site for an offshore platform is primarily dictated by oil reservoir considerations. There is always, however, some latitude that can be used to optimize foundation conditions (de Ruiter, 1976). The reservoir requirements may usually be met by installing the platform(s) within a fairly large area, on the order of several square kilometers or more. A general survey of the specified area is conducted to determine the most likely sites to place a platform(s). At this time, the type and number of platforms to be placed are often unknown. The survey lines and test holes for one such survey conducted in the North Sea are illustrated in figure 3.2. Geological investigations, both regional and site specific, are made of the proposed area. The regional survey is often 1 1 1 1 1—: lettooo LEGEND Soo mtfres » f S\J*.-/& vEUEL-i • BoftCHOuE TrtAtK. WITH TI" PCSITIOWS •* UNI rerJmwnoN TEST *J cXPLeCKTieWWCU. c-Figure 3.2 - Plan of survey lines (After Offshore'Soil - grid: local transverse Mechanics, 1976) 38 made using only presently available data. The history of the area is investigated, with particular attention paid to factors such as: the location of buried channels, deltaic clays, and tectonic movements. ' Environmental influences are assessed, including: regional and local rates of deposition, proximity to submarine canyons, and permafrost (Garrison and Bea, 1977). On-site studies are made using specially outfitted survey ships, some of which are over one hundred meters in length. The distribution patterns of minor seafloor features, angles of local slopes, and water depths in the proposed area are found from bathymetry. Geophysical seismic surveys are made at the same time as the bathymetrical studies and from the same ship. Acoustical profiling is done using electronic transducers which impart an acoustical pulse to the water by either a sound (Pinger), mechanical (Boomer), or spark (Sparker) disturbance. The resulting acoustical transmission, after travelling through the water, reaches the seabed and is reflected back by the various subsurface strata. The returning signals are picked up by hydrophone streamers or arrays (Offshore Europe, 1974). High resolution (Pinger or Boomer) surveys are used to gather information on the upper sediments, those less than about 30 meters deep, and low resolution (Sparker) surveys are conducted to determine the characteristics of deeper strata, although in somewhat less detail. Some seabed samples are necessary to provide specific geotechnical knowledge of the upper sediments - those most critical to the design of a gravity type structure. Shallow samples are taken with a gravity corer, vibratory sampler, or 39 other sampling device. Alternatively, cone penetrometers may be used to characterize the surficial sediments, provided some samples are taken for correlation purposes. At least one deep boring (100 to 150 meters) is required to provide information on the sediments within the range of interest for foundation design. The positions of boreholes and the locations of other tests should be known accurately relative to the future structure. This may be accomplished by deploying an array of transponders on the seafloor and locating all boreholes and other test locations with respect to the transponders. The electrical signals emitted by the transponders allow them to be readily located from the surface (McClelland, 1977). A preliminary site selection may be made based on data from the aforementioned tests. When a site has been chosen for further investigation, a carefully planned field program must be developed (Hitchings et al, 1976). The cost of offshore investigations is extremely high, on the order of tens of thousands of dollars per day for a large survey ship and crew (Braun, 1974); therefore, the program of testing must be thoroughly prepared by the geotechnical consultant before the ship is on-site. All geotechnical testing is performed by the consultant's geotechnical personnel and other operations are supervised by his inspectors (de Ruiter, 1976). Engineers on-board monitor the incoming data continuously to make on-the-spot decisions about the location and extent of in-situ tests. The base of the platform should fall within the area investigated in the final on-site survey. The structure once 40 on-site can be positioned only approximately over the proposed site since it will be constantly in motion under wind, wave, and current excitation. The expected accuracy in positioning must be established and will govern the size of the area to be explored. Error margins in positioning of 50 meters are typical (McClelland, 1977). The site investigation is conducted from a ship which is subject to constant motion, as is any drillstring or piece of equipment connected to it. Special equipment has been developed to try to compensate for this motion (Taylor, 1976), however, complete success has not and will not be achieved. Therefore, whenever possible, in-situ testing is done with equipment that rests directly on the seafloor and requires no rigid connection to the surface vessel, only flexible control cables and hydraulic lines. The site investigation, although well planned, does not follow a strict course. Information gained from early tests is used to determine the need for later tests. This should be kept in mind when reading the following text. The seabed topography is mapped in detail using side scan sonar, which is good to about 0.5 meters, and submersibles (de Ruiter, 1976). Obstacles such as boulders must be accurately located and their significance assessed. If they are too large, the site will not be suitable unless they can be removed. If not,-an alternative site may have to be chosen. A number of borings are made to varying depths. The number of tests depend on the uniformity of the soil profile. Generally three to five shallow boreholes (to 30-40 meters) and 41 at least one deep borehole (to 100-200 meters) are sunk (de Ruiter, 1976). Three corings has been suggested as an absolute minimum (George, 1976). Samples are usually taken at intervals of 1.0 to 1.5 meters over the first 15 meters, then less frequently (McClelland, 1977). The soils can only be identified with certainty where samples are taken since the drilling mud and cuttings exit at the seafloor (Low, 1975). Samples are usually extruded on-board, classified, and checked for quality. Some samples are selected for on-board tests, including routine classifications (Atterberg limits, grain size distribution, etc.) and unconfined compression tests, while others are prepared for the on-shore laboratories, where consolidation, triaxial, and other complex tests are carried out. Cone penetrometers are used extensively to check the uniformity of shallow deposits and to estimate the penetration resistance that will be encountered by the dowels and skirts during installation.7 They are also used for classification purposes and for estimating the undrained strength of clays and relative density of sands. The number of penetration tests depends on the uniformity of the soil profile with five to fifteen being typical numbers (de Ruiter, 1976). Additional acoustical profiling on a fine grid may be necessary if abrupt 'Dowels are cantilever rods which jut out of the base of the platform 5 m or so. They are used to stabilize the structure and prevent it from moving while it is being ballasted and the skirts are being imbedded. 42 changes in stratigraphy are detected (de Ruiter, 1976). A valuable qualitative picture is presented from cone penetration logs that may be used to assess the reliability of boring data. Down-the-hole penetrometers may be used in deeper boreholes to measure the density and shear strength of deeper sediments. Gamma ray logging can be done down the boreholes for a minor increase in cost. The gamma ray log provides a continuous picture of the borehole and is useful as a qualitative tool showing stratification and the presence of cohesive soils. These soils are marked by an increased gamma ray count. Other in-situ tests may be carried out in addition to those mentioned, including vane shear tests which are applicable in areas with soft clays (McClelland, 1977). 3.1.4 Selection of Soil Parameters for Design Design parameters are chosen based on data from both laboratory and in-situ tests. The borehole profiles are interpreted from cone penetration logs, gamma ray logs, and samples. Samples are used for identifying the deposits and for making site-specific penetrometer correlations. The in-situ tests provide a continuous picture of the profile and are important for identifying interfaces and small scale features, such as thin seams or lenses of varying material in larger seemingly uniform layers, since a continuous record of the drill mud or cuttings is not available. The results of one such borehole interpretation are shown in figure 3.3. The necessary design parameters are found by using established laboratory tests and in-situ methods properly 43 10 -H 15 •£ a. O 25 35 I—40-Soil profII* Fine to medium umd with diell froamentt (dente)-Small tilt fraction Few imotf graved Grey tilty cloy with teoms of fine land and tilty landdtiff fo very stiff) Fine fa medium land (dense) Seomi of silry clay Grey illty clay with teomt of lilt and tilty fine umd(very ttlff) Silly land layer? Silly land layer? Silty fine land with of tilty cloy(dente) Silly clayfoord) Flo.5 TYPICAL SOIL PROFILE AS IDENTIFIED »Y BOREHOLE,CONE PENETRATION TEST AND V RAY LOG Figure 3.3 - Typical soil profile as identified by borehole, cone pentration test and gamma ray logging (After George, 1976) 44 correlated to the offshore site. Oedometer tests are used to obtain consolidation data. The stress-strain characteristics of the foundation soils and the effective shear strength parameters are found from triaxial tests. Simple shear and direct shear testing may also be done. Relative density and modulus values are best determined from in-situ tests, such as the cone penetrometer with the aid of empirical correlation charts. The modulus value is particularly sensitive to sample disturbance. When choosing design parameters "one should always be aware of the limitations imposed by the conditions under which investigations at sea have to be carried out," (de Ruiter, 1976). Obtaining the shear strength parameters presents many difficulties, some being: - The ability to obtain samples of high quality - Inherent scatter in laboratory data - Deciding which type of shear tests are applicable to the problem (Rowe, 1975) - The natural variability of offshore deposits When choosing a profile, it must be remembered that an estimate which is too conservative may result in millions of dollars having to be spent to increase the platform size so that a reasonable factor of safety against sliding or bearing failure is obtained, while too liberal an estimate may result in the complete loss of a multi-million dollar investment and many lives. Results of both in-situ and laboratory shear tests for one particular site are presented in figure 3.4. The interpretation 45 CONE RESISTANCE fkPal— 0 500 WOO 1500 2000 2500 0 COHESION c (kPa)—• 40 60 80 100 CD < I UJ CO I 1-CL g I Figure 3.4 - Comparison of shear strength values from sample testing and from CPT (After de Ruiter, 1976) 46 of this profile is certainly subject to personal opinions and prejudices. A thorough knowledge of the types of tests and the conditions under which they were performed is essential when making such a decision. The cone penetrometer results provide a useful check on the laboratory data. 3.2 Platform Design The basic design considerations for a gravity platform are discussed briefly in this section. Detailed descriptions of design procedures are beyond the scope of this thesis. The reader is therefore directed to references if a deeper study is required. 3.2.1 Hydrodynamic Analyses Hydrodynamic analyses are performed primarily to provide the structural engineer with information on the magnitude and nature of wind, current, and wave loads. The pressure distributions caused by these loads on the structure, which will vary both spatially and temporally, are required for the design of the various components. The total forces acting on the structure, found from integrating the pressure distributions with respect to the spatial coordinates, are required for the design of the foundation. The nature of wave loading on both the structure and the foundation system is discussed in detail in Chapter 5. Wave loads are usually by far the most important fluid loads encountered. Many analytical procedures have been developed to calculate the forces due to waves interacting with 47 gravity type structures. In addition to theoretical analyses, model testing is often employed to provide a check on results (Garrison, 1977), since hydrodynamic theories cannot account for irregular shapes, interference effects, and other such problems that exist with real structures, except by approximate.numerical methods or through the use of empirical coefficients (which are usually based on laboratory tests on small scale models). Analytical methods are used to calculate the forces on individual members of the structure, since modelling these members individually is impractical. Scour potential is investigated using model tests since no acceptable analytical theories exist. Even model tests are not very reliable due to scale effects (Maidl and Schiller, 1979) and difficulties in modelling the soil. However, a valuable qualitative picture of the on-site scour potential may be drawn from these tests. Model tests are invaluable for providing information on towing resistance and motions, floating stability, damage stability, and submergence behavior when touching down (Offshore Europe, 1974). For these operations, model tests are relied upon heavily and are- always incorporated into the design procedure. 3.2.2 Geotechnical Analyses After preliminary surveys are completed and a detailed site investigation has been carried out for a possible gravity platform site, comprehensive geotechnical analyses begin. These analyses are summarized in Table V. 48 Table V Geotechnical Concerns For Offshore Gravity Type Platforms 1) INSTALLATION A) Penetration resistance of dowels and skirts B) Pore pressure dissipation and exit of confined water C) Bearing pressure on cells and slab D) Grouting procedures 2) CONTACT BETWEEN SEAFLOOR AND STRUCTURE A) Scour around or under structure B) Reduced area for bearing or sliding resistance 3) STABILITY UNDER PSEUDOSTATIC LOADS A) Sliding B) Bearing failure C) Overturning 4) SETTLEMENT A) Immediate elastic B) Primary consolidation C) Secondary consolidation D) Cumulative storm and/or earthquake effects 1) Strain softening in clays 2) Densification due shear stress reversals in sand 5) DISPLACEMENTS UNDER PSEUDOSTATIC LOADS A) Horizontal displacements B) Vertical displacements 6) EFFECTS OF CYCLIC LOADING A) Pore pressure rise B) Reduction in shear strength C) Decrease in stiffness D) Associated problems 1) Excessive horizontal displacements 2) Rocking 3) Liquefaction 7) DYNAMIC BEHAVIOUR A) Resonance B) Operational requirements 8) INSTRUMENTATION A) Installation B) Performance monitoring 49 The penetration resistance of dowels and skirts requires a good knowledge of the upper sediments, usually known from extensive cone penetrometer testing. This resistance, which will vary over the site, may be estimated using the standard bearing capacity equations of Meyerhof (1963) or Hansen (1970). The resistance to dowel or skirt driving is essentially the ultimate bearing capacity of the upper sediments. Penetrometer correlation charts may be used to estimate the bearing capacity factors. The local contact pressures on the base slab may be estimated by elasticity theory, provided that the detailed topography of the seafloor is known (Bjerrum, 1973). Generally, the slab is designed for a specified base pressure, since the topography and distribution of the upper sediments are not precisely known. Other installation problems will be discussed in a subsequent section on platform installation procedures. Good contact between the seafloor and the slab is necessary to prevent undermining of the foundation from water motions and to insure adequate area for foundation stability. This is achieved by grouting underneath the slab after the platform has been imbedded as far as possible. Proper grouting procedures and composition must be specified. There are a number of possible failure modes for an offshore gravity structure foundation, including: horizontal sliding, bearing failure, rocking, and liquefaction. These failure modes are shown in figure 3.5. The stability of the platform is investigated to insure that horizontal sliding or a deep-seated bearing failure does A) SLIDING B) BEARING CAPACITY Fig. 3.5 Possible failure modes for an offshore gravity structure foundation (Adapted from Hove and Foss, 1974) en o 51 not occur. There are a number of possible mechanisms for a horizontal sliding failure that must be considered in order to find the most critical one. These are shown in figure 3.6. Bearing failure is investigated using the bearing capacity formulas of Meyerhof (1963) or Hansen (1970), and for clay deposits, a simple limit equilibrium method called the (Norwegian Geotechnical Institute) slip surface method (Lauritzsen and Schjetne, 1976). The simple bearing capacity equations give a rough estimate of the bearing strength of foundation soils and are convenient and easy to use. They are not strictly applicable to layered deposits or the ocean wave or earthquake problems where loading conditions are more complicated than the formulae can account for. Therefore, finite element analyses are also performed to provide a more detailed investigation of the bearing stability. Overturning failure is not a problem if load eccentricity is not too high. This potential problem may be avoided by increasing the base size if necessary. A stability diagram is shown in figure 3.7 to show the basic relationship between vertical and horizontal loads as they relate to foundation stability. The magnitudes of the loads, of course, depend on the foundation size and the strength of the foundation soils. Calculations for elastic and consolidation settlements are similar to those which are done for any other structure. Of interest here is that the depth of influence, i.e. the size of the stress bulb, is substantially larger for these huge structures than for most projects on land, and therefore, the corresponding settlements are usually greater. In addition to 52 (ol PASSIVE WEDGE FAILURE (t» DEEP PASSIVE FAILURE (dl SLIDING FAILURE IN SHALLOW WEAK ZONE WITH WIDELY SPACED SKIRTS 1LIJ11J.IJJ1I.111. ' K ... . . • (.) SLIDING FAILURE IN SHALLOW WEAK ZONE AVOIDED WITH CLOSELY SPACED SKIRTS (f) SLIDING FAILURE IN DEEP WEAK ZONE Figure 3.6 - Possible modes of sliding failure (After Young et al, 1975) 53 Figure 3.7 - Stability diagram for a raft foundation (Adapted from Young et al, 1975) 0 54 these calculations, some assessment of the effect of cumulative storm or earthquake loading on settlement must be made. Laboratory tests are necessary to determine this influence and appropriate procedures must be used to estimate the amount (Andersen, 1976; Finn et al, 1977; Lee and Albaisa, 1974). Cyclic loading and its effects on the foundation soils are investigated in an approximate way. For wave loading, where the period is on the order of several seconds or more, pseudostatic analyses are performed for stability and displacement calculations. Cyclic effects are modelled by changing the soil properties to account for pore water pressure generation. For earthquake loading, full dynamic analyses are required with suitable effective stress computer programs. Displacements of the structure under pseudostatic wave loads are estimated using the finite element method. These displacements include vertical, horizontal, and rocking motions. The results are very sensitive to the values chosen for the soil parameters. The effect of cyclic loading may be incorporated here by estimating the pore water pressure rise and decreasing the soil stiffness and shear strength to account for this. Liquefaction potential is assessed from laboratory test data either directly with cyclic triaxial tests modified to allow for partial drainage (Lee and Focht, 1975a) or indirectly using analytical methods (Rahman et al, 1977). Preliminary studies based on simplified undrained analyses (Bjerrum, 1973) may be useful to assess the need for more advanced investigations, either laboratory or analytical. Most dynamic analyses are performed by the structural 55 engineer who requires soil parameters to model the stiffness and damping characteristics of the foundation soils. The evaluation of these soil parameters is the job of the soils engineer. Seismic considerations have not had a significant influence on gravity type structures designed to date. However, earthquake engineering for offshore gravity structures is receiving increasing attention as structures of this type are being considered for seismically active areas. A thorough discussion of this topic is beyond the scope of this thesis and the reader therefore is referred to several publications that deal with this subject, namely: Watt et al (1978) and Seines (1981). Two design codes may also be referenced: those of Det Norske Veritas (1977) and the American Petroleum Institute (1978). A final concern of the geotechnical engineer is the requirement for monitoring the installation and subsequent performance of the gravity platform. This is covered in detail in a later section. 3.2.3 Structural Requirements and Analyses The final structural analyses and design may proceed when the environmental loads have been defined and the soil investigations are complete. An important requirement of offshore structural design is that the structure be designed for construction, tow-out, and installation loads, in addition to the usual procedure of designing for the maximum forces expected during operational life. The three main components of the structure: the deck, 56 towers, and base caisson, have distinct design requirements. The deck, which is usually made of steel, must resist corrosion (which is higher in the splash zone than elsewhere) and fatigue failure throughout the platform's life, which is usually between 20 and 30 years. The critical points in the design are, however, not the deck but the towers and base slab (Sjoerdsma, 1975b). The towers must be designed to prevent implosion under the large hydrostatic forces due to the structure's deep draft and the additional wave induced pressure loading. The differential hydrostatic pressure acting at the base of the towers may be larger during the construction or tow-out phases, when the cells are not filled with ballast, than when the platform is on-site and operational. This must be investigated to determine the critical design load. The caisson will also have to resist implosion from large differential hydrostatic pressures if there are cellular compartments (as with the platforms shown in figure 2.2). It must also be strong enough to resist damage from foundation loads. These may be locally high due to contact pressure with objects such as boulders on the seabed or to the high resistance of dense sand pockets to deformation during installation. According to one code (Federation Internationale de la Precontrainte, 1977), the slab must be designed for 200 t/m2 at all locations to account for uncertainty in soil investigations and penetration resistance and for higher values if dense sand is found from soil investigations. This value is about an order of magnitude higher than the uniform bearing pressure calculated for the slab. 57 For the base slab, it is readily apparent that the critical design loads are those encountered upon installation. Other critical design conditions are not so obvious. For example, the steel gravity type platforms towed from Europe to the Congo were designed for greater wave loads expected during the tow than they would be subjected to once installed in the relatively calm waters offshore the Congo coast (Lalli, 1977). The dynamic analyses required for a large gravity type structure are numerous and involved. Since many of the loads acting on the structure (wind, waves, and earthquakes) contain components of many frequencies, spectral analyses are required for both resonance studies and fatigue calculations. In addition to the signal being random, the structure will probably not be symmetrical and different directions of loading will have to be investigated. For a more thorough presentation of the structural requirements for an offshore gravity structure, the reader is referred to several papers concerned wholly with this topic (Penzien, 1976; R0ren and Fames, 1976; Waagaard, 1977; Watt, 1979). 3.3 Platform Construction Of paramount importance in the construction of a large (concrete) gravity platform is the availability of suitable sites for dry docks, shallow water construction areas, and deep water construction sites. The base section of the platform is built in an excavated 58 dry dock. When the raft has been completed and the caisson walls raised to a predetermined height, the dry dock is flooded, the cofferdam removed, and the base section floated up and towed out to a shallow water site (usually a nearby or attached bay or fjord) where it is secured by mooring cables (Clausen, 1976). Compressed air may be used under the foundation to add buoyancy if there are problems in floating it out of the dry dock (Derrington, 1977) or to cut excavation costs (Werenskiold, 1977). At the shallow water construction site the base caisson is completed and the towers are erected. When the towers are completed, the structure is then towed out to a 'deep water construction area where it partially submerged by flooding ballast compartments in the base section and towers, and then moored. It is at this site that the deck is usually mated (Sjoerdsma, 1975b). The deck, which was built onshore, is loaded onto two barges (or old tankers). These barges are then towed out to where the platform is and positioned so that the deck is over the towers. The platform is then partially unballasted to raise the deck off the barges and onto the towers (Clausen, 1976). The structure, virtually complete, is now ready for tow-out. The transport of the structure between the various construction sites and then the tow-out to sea for installation must be carefully planned before construction begins. There must be adequate bottom clearance and room to maneuver the structure throughout all the towing routes. These considerations are the responsibility of a maritime consultant 59 who is well versed in these practices (Werenskiold, 1977). 3.4 Platform Installation The platform, being partially submerged for stability, is towed out to location by an array of tugboats. This is an extremely delicate operation that must be very well planned and coordinated by the maritime consultant. The positioning and submerging of the structure is also his responsibility. Weather forecasts are used to choose a sailing time and are constantly monitored and updated to insure calm seas for the tow-out (Werenskiold, 1977). The structure once on-site can be placed only approximately on location. Because of the high inertia of such a large structure, even when moving very slowly, there will no doubt be some finite motions at the moment of touchdown, especially if there is a current present (Watt, 1976). The sea must be relatively calm at the time of installation to avoid excessive motions of the structure that may damage the caisson and its appendages. The installation sequence is shown in figure 3.8. The structure is systematically ballasted once on-site to stay level while sinking. The rate of submergence is carefully monitored so that the structure does not impact the seafloor heavily and damage the bottom slab, skirts or ribs. To aid in placing the structure and minimizing damage to both the structural and soil components of the foundation, steel dowels which portrude several meters below the skirts are provided. The dowels penetrate the seafloor under the weight of the platform as it sinks and provide resistance to horizontal motion 60 (c) Skirt driving W) Grouting Figure 3.8 - Installation sequence for a gravity platform (Adapted from Watt, 1976) 61 which could break off the skirts or ribs or gouge out the foundation soils, impairing stability under storm conditions. The base detail of a CONDEEP type platform installed in the North Sea is shown in figure 3.9. As the structure is further submerged, the skirts and ribs penetrate the foundation soil. To keep the structure vertical during skirt penetration into the seabed, which in general will be irregular due to a sloping seafloor and varying soil conditions at the site, large moments may be applied to the foundation by ballasting appropriate cells thereby driving the skirts deeper (Clausen, 1976). Care must be taken to allow ample time for water entrapped within the skirt compartments to flow out from underneath the slab. If the platform is lowered too fast, high current velocities may result and cause channels to be eroded underneath the structure that may lead to more erosion and threaten the stability of the platform (Gerwick, 1974) . To insure good contact between the base slab of the structure and the foundation soils, the space between them is usually grouted utilizing a built-in piping system in the base provided for this purpose (Callis et al, 1979). Grouting usually begins after a few points on the base have touched down. The structure must be submerged slowly to allow excess grout to flow out from underneath the structure without damaging the foundation soils or .overstressing the skirts (Watt, 1976). Submersibles may be used to monitor the success of grouting operations (Callis et al, 1979). The base of the structure is usually instrumented so that 62 k 50jn j Figure 3.9 - Detail of CONDEEP base structure (After Clausen, 1976) 63 decisions can be made during installation about the amount of penetration possible. If excessive pressure is exerted on any of the foundation components from either pushing objects such as boulders into the seafloor or increased driving resistance from say a lense of dense sand, the submergence may be halted and grouting to fill the interskirt spaces may commence. For one CONDEEP structure very high pressures were experienced on the base of one cell during submergence - probably from the high deformation resistance of a lense of dense sand that was undetected during the soil investigations (Clausen, 1976). The decision to stop driving the platform to prevent structural damage to the slab was made based on information from instruments built into the caisson. The data available from the instrument interpretations is shown in figure 3.10. After grouting is completed, some form of scour protection may be placed depending upon local soil conditions and expected water particle velocities near the structure. Gravel mats connected to the structure and rolled out after installation is completed have been used (Offshore Europe, 1974). 3.5 Platform Instrumentation Platforms are generally well instrumented to (1) aid in installation, and (2) to provide information on the performance of the structure during its operational life. Although the cost of instrumenting the structure is high, the money saved in construction costs is more than offset by this (McClelland, 1977) since materials and labor are reduced by not having to increase dimensions to account for uncertainties in installation 64 « 200 Design Maximum Allowable Value 2? (A S 100 0) E 50 I Expected Maximum Values 10 15 20 Dome Number Figure 3.10 Maximum dome contact pressures observed during installation of the "Beryl A" CONDEEP (After Clausen, 1976) 65 loads (soil reactions). Instrumentation for measuring platform response provides data for future design on pore pressure rise, lateral displacements, etc. after the platform is installed. The following instrumentation has been used to monitor the installation of platforms now on site (DiBagio et al, 1976): - Wavedata by means of a buoy anchored near the platform - Bottom clearance by means of echo-sounders installed under the base of the caisson - Draft by ,means of pressure transducers mounted near the base - Ballast water level in cells and towers by means of pressure transducers within these compartments - Bending moments and axial forces in dowels from strain gauges - Water pressure in skirt compartments beneath the caisson during penetration and contact grouting by means of differential pressure transducers - Verticality from a biaxial inclinometer - Base contact pressures using earth pressure transducers mounted flush on the slab - Strain in reinforcing steel in base slabs and cell walls by means of strain guages in the reinforcement - Short term settlement by means of pressure measurements in a closed hydraulic system Other instrumentation has been used to monitor the performance of these platforms (DiBagio et al, 1976): - A complete system for oceanographical/meteorological measurements (wave, tide, current, wind and temperature data) - Base contact pressures by means of earth pressure transducers mounted flush on the slab - Structural strain at the base of the towers, giving the moments from wave action transferred to the foundation, from strain gauges - Linear accelerations and displacements at the base, at mid-66 height of the towers, and at deck level - Angular accelerations and displacements at the base and deck levels - Long-term horizontal and vertical displacements by means of a flexible telescopic casing installed under the caisson - Pore pressures in the foundation soil by means of piezometers installed beneath the platform A computer operated digital data acquisition system is used to process data as it is received with the computation of basic statistical data being processed on-line and stored on a magnetic tape (Clausen et al, 1975). 67 CHAPTER 4 THE EKOFISK TANK - A CASE STUDY The Ekofisk tank has been the subject of numerous papers (Bjerrum, 1973; Braun, 1974; Clausen et al, 1975; Duncan, 1972; Gerwick and Hognstad, 1973; Lee, 1976; Lee and Focht, 1975a; Lee and Focht, 1975b; Marion, 1974). Being the first large offshore gravity structure installed, it naturally received a lot of attention in the engineering community. The Ekofisk tank is familiar to almost everyone involved in offshore platform design and construction and has a relatively large body of literature associated with it. With these points in mind, a discussion of the Ekofisk tank would appear to be useful as a means of presenting geotechnical concepts and the application of theories in the offshore environment. A geotechnical case study of the tank is presented herein. A general description of the Ekofisk tank may be found in several sources (Gerwick and Hognstad, 1973; Marion, 1974; Offshore Europe, 1974). The details of the design, construction, and installation of this platform are discussed in depth in these papers and will only be highlighted here. The tank was built near Stavanger, Norway, then towed over 400 kilometers from the Norwegian coast to the Ekofisk field in the middle of the North Sea where it was placed on June 30, 1973. After being positioned, the structure was ballasted with water to imbed it in the foundation soils. Positioning errors were 10 meters off target and 3°50' out of orientation (Marion, 68 1974). The base of the structure, shown in detail in figure 4.1, is covered with 5 cm high corrugated steel plates and has 40 cm high skirts along the periphery and 40 cm high ribs underneath the central structure; these were provided to obtain full contact with the seafloor soil. After skirt driving was completed, nylon mats attached just above the skirts were rolled out by divers and rocks dumped on them to provide protection against scour (Gerwick and Hognstad, 1973). Additional sand ballast was added after placement to achieve a maximum negative buoyancy for the tank. The submerged weight of the tank after this ballasting was 190,000 metric tons (Clausen et al, 1975). The final cost of the platform including installation was in excess of $28 million (Offshore Europe, 1974). 8 The structure is nearly circular in plan, resembling a square with rounded corners, with an approximate diameter of 93 meters. It is 90 meters high and rests on the seabed in 70 meters of water. One million barrels of crude oil may be stored in the central reservoir which is roughly 45 meters square in plan and 70 meters high; this reservoir is composed of nine lobes for maximum structural strength (Gerwick and Hognstad, 1973) each with walls nearly one meter thick at the base (Offshore Europe, 1974). Surrounding the reservoir is a perforated breakwater designed to reduce wave loads on the tank, which extends from about 12 meters above the water surface to "This figure is in 1973 U.S. dollars. 69 Figure 4.1 - Detail of the Ekofisk tank bottom (After Clausen et al, 1975) 70 the base slab where it is rigidly attached. The heavily post-tensioned base slab is 6 meters thick and extends beneath the entire structure forming a huge solid raft foundation covering an area of 7360 m2 (Offshore Europe, 1974). The buoyant weight of the tank is now about 190,000 metric tons and in static water exerts an average pressure of about 25.8 t/m2 on the foundation soils. Under wave loading, there will be a fluctuating component of the vertical stress which is on the order of 5% of the static pressure (H0eg, 1976). This fluctuating vertical load is in phase with the horizontal force and moment (Schjetne, .1976). For the Ekofisk tank, the magnitude of the fluctuating vertical force is about 10,000 metric tons for the design wave. Therefore, when analyzing the foundation for design wave conditions, a vertical force of 200,000 metric tons acting on the foundation is used; this corresponds to a uniform vertical pressure of about 27.2 t/m2. For the 100-year design wave, a horizontal force of about 78,600 metric tons will act on the tank (Bjerrum, 1973). Since this resultant force will act above the seafloor, a moment will be applied to the foundation. The magnitude of this moment is approximately 2,800,000 ton-meters (Clausen et al, 1975). The tank is shown schematically in figure 4.2 with the loads acting on it corresponding to the 100-year wave. Some design storm data is shown in figure 4.3. Foundation conditions at the Ekofisk field are typical of the North Sea: alternating layers of dense sands and heavily overconsolidated clays. A typical geotechnical profile from the Ekofisk field is shown in figure 4.4. The upper 26 meters are 71 ft = 78,6001 ^36m L ^93m-Ja^lOpOOt Pv =19Q000t «X " - S W.L, S70m Figure 4.2 - Loads on the Ekofisk tank for the 100-year wave 5000 WAVES 15 MRS DURATION * & 400 20 40 60 60 FT I I L_ 0 5 <0 IS 20 25 M. WAVE HEIGHT. H Figure 4.3 - Design storm data for the Ekofisk field (After Lee and Focht, 1975a) 72 « -Mm ,j 50 m 100 m 150 ml-• —1 20 m 7C m ^^^^^^^^^^^^ Sand Cljy Sind Figure 4.4 - ^j«^otechnical)Profile fro. EkoHsk Mela -Om Sea floor jc-70m UNDRAINED SHEAR STRENGTH (t/m') 11 '° Si" IS" uj uj o in SO 60 Fin* land Stiff sandy clay Fint und Hard clay * i • < • 0 • • > • • UU triaxial • Unconfintd compression test o Pocket penetrometer Figure 4.5 - Shear strength data from Ekofisk lArter Clausen et al, 1975) 73 comprised of extremely uniform fine sand with a thin clay seam at about 18 meters below the seafloor; the upper few meters have a relative density on the order of 100% (Bjerrum, 1973). This high density is most probably due to the effect of countless waves that have passed overhead since the sand was deposited (Bjerrum, 1973). Shear stresses are induced in the soil from passing waves because of the varying pressure distribution they impose on the seafloor (Henkel, 1970). These stresses cycle back and forth and may compact the sand if they are sufficiently large. This is termed "preshearing" (Bjerrum, 1973) and has been demonstrated to be an important source of soil compaction in the laboratory (Lee and Focht, 1975a). The stiff clay beneath the sand has an undrained shear strength of about 40 t/m2 (Bjerrum, 1973). The clay seam at 18 meters below the mudline is substantially weaker. Some shear strength data is presented in figure 4.5. Foundation studies for the platform were performed independently by McClelland Engineers, Ltd. and the Norwegian Geotechnical Institute. The Norwegian Geotechnical Institute represented Det Norske Veritas, the agency responsible for approving the platform safety for the Norwegian government, and was responsible for checking the foundation safety independently of McClelland's findings. There was a great concern about safety since an oil spill of possibly one million barrels would be disastrous (Duncan, 1972). Final approval of the tank required that it could not be used for oil storage for several months after installation; thus, if a failure occurred during this period, no spill could take place (Clausen et al, 1975). 74 Preliminary studies carried out at the Norwegian Geotechnical Institute (NGI) were reported by Duncan (1972). Finite element modelling of the foundation, taking into account the nonlinear behaviour of the soils, was done to predict the elastic settlement of the structure and the displacements expected under storm wave loading. Elastic settlement was estimated to be about 30 cm for the tank due to its own weight (Duncan, 1972). Assuming that the load-settlement curve for elastic settlement is linear, this would imply that the structure would move up and down about 1.5 cm when subjected to the fluctuating vertical force of the design wave. The horizontal displacements of the tank were estimated to be about 15 cm back and forth for the design wave (Bjerrum, 1973). Concurrent with these linear displacements are rocking motions which result from the cyclic moment. NGI estimates showed that subject to the design wave, one side of the base slab would move down 30 cm while the opposite side would move up about 45 cm. Superimposed on the elastic settlement due to the platform weight, this would mean that one side would lift about 15 cm off the sand (Duncan, 1972) resulting in a possibly unstable situation. Because the platform would almost undoubtedly be subjected to numerous storms before the 100-year design storm would hit, the tank was expected to settle from the preshearing effect and the sand under the tank to densify and become stiffer reducing the motions (Duncan, 1972). This settlement and accompanying increase in stiffness meant that rocking motions expected for the 100-year wave could be modified 75 from the original estimates. New 'calculations indicated that the base slab would move up about 15 cm on one side and down 15 cm on the other (Bjerrum, 1973). This implied that the base slab would not be lifted off the soil, and. at all times would exert a positive pressure on the foundation (Bjerrum, 1973). Results of these studies are shown in figure 4.6. The NGI finite element studies of displacements included the effect of the pore water pressure change at the seafloor due to the passage of waves overhead (Bjerrum, 1973). They did not, however, include the effects of cyclic loading on the foundation soils. Rahman et al (1977) have shown that pore water pressure ratios of about 20% and 8% will occur under the edges and center of the tank, respectively, when subjected to the 100-year storm. Their analysis is for a relative density of 85%, which is less than the relative density in-situ, and includes the effects of partial drainage in the sand. Although their results are not strictly correct, they do show that excess pore water pressures will develop under the tank during design storm loads. This has been confirmed from observations on-site using piezometers and pressure gauges installed underneath the structure (Clausen et al, 1975). Expected rocking displacements would then be greater than reported by Bjerrum (1973) for the NGI analyses since the increase in pore water pressure under the tank would decrease the stiffness of the upper sand layer. Settlement observatons have been reported by Foss (1974) and Clausen et al (1975). A load-settlement curve is shown in figure 4.7 for the installation phase of the tank. After touchdown, the platform was ballasted to seat it firmly on the 76 Figure 4.6 - Predicted rocking displacements for the Ekofisk tank (After Duncan, 1972) igure 4.7 - Load-settlement curve for Ekofisk tank (After Clausen et al, 1975) 77 foundation soils. During this time the seabed was being deformed from both elastic compression and plastic displacements. The plastic displacements corresponded to the penetration of short concrete skirts and to the flattening of the seafloor beneath the structure; since the bearing capacity of undulations and mounds would be exceeded as the platform seated, these features were destroyed and the platform settled. This "bedding settlement" should correspond to the skirt depth for a flat seafloor, indicating full skirt penetration and base contact with the seafloor. The load-settlement curve became nearly linear when the submerged weight reached 50,000 tons. The penetration of the 40 cm high skirts into the seafloor at this time was about 35 cm. It is important that the load-settlement curve became linear. This indicated that (essentially) full contact between the base of the structure and the seafloor was achieved (Clausen et al, 1975). If the linear portion of the load-settlement curve is extrapolated back to zero submerged weight and forward to 190,000 tons, the elastic settlement of the tank due to its own weight may be established. This results in an elastic settlement of about 10 cm, substantially less than the 20 cm predicted by NGI.9 The discrepancy may be due to the use of stiffness parameters in the finite element analyses that were 9One should note that Duncan's (1972) reported estimate of 30 cm for elastic settlement of the tank was made before final discrepancies in the value of soil parameters were cleared up (Lee and Focht, 1975a). This estimate was later changed to 20 cm (Braun, 1974). 78 not really representative of the "undisturbed" soil; stiffness parameters were probably chosen much too conservatively as a consequence of the difficulty and uncertainty associated with offshore soil testing and sampling - particularly at that time -a decade ago. Settlement continued to occur after the platform was installed. The time history of settlement for the first seven months after installation is shown in figure 4.8. Most of the ballast was added to the tank in the first few days after it was placed. The load-settlement curve previously discussed was developed for this time interval. Settlement in the early days of July, 1973, Was due to the elastic response from increased submerged weight of the platform as it was ballasted. Settlement in the following months may be attributed to a variety of factors, namely: increased submerged weight of the platform (from more ballasting), consolidation in the clay, and wave action on the tank. The amount of settlement due to the increased ballast load can easily be estimated from extrapolating the load-settlement curve in figure 4.7. The overall elastic settlement would be about 10 cm and would not increase after the tank was fully ballasted. At the end of ballasting, settlement was observed to be about 13 cm (see figure 4.8). Hence, the consolidation and wave induced settlement was on the order of 3 cm up until the middle of October (when ballasting was terminated). During this time the sea was relatively calm as seen from the wave data in the figure. Note that the wave heights shown are the significant wave heights (a statistical parameter) not the 79 Figure 4.8 - Ekofisk settlement data relating submerged platfo weight and storm wave data in the early months after installation (After Clausen et al, 1975) MQ0OO r itqpoo 5 | tOQOOO S KtOOO *1* 1974 Figure 4.9 - Settlement data for Ekofisk tank during early storms (After Clausen et al, 1975) 80 maximum wave heights; these values must be increased by approximately 80% to find the maximum wave heights (Sarpkaya and Isaacson); the exact increase depends on statistical data which is not available. In November, the platform was subjected to several major storms. The first storm hit on 6 November and a settlement of about 2 cm occurred during the next few days (Clausen et al, 1975). On 19 November, the major storm of the year occurred. When this storm hit, the platform instrumentation was out of service and the wave data was estimated from a nearby weather ship, the "Famita" (Foss, 1974). Estimates put the maximum wave height at about 22 meters, or about 90% of the 100-year design wave - truly a significant storm. The platform settled about 5 cm during the period of 16 November to 20 November (Foss, 1974). Total settlement during November was about 7 cm, of which most probably occurred in the sand. The dense sand consolidated under the action of repeated shear stress reversals (Clausen et al, 1975). A detailed record of the November settlements is shown in figure 4.9. After the storm of 19 November subsided, no detectable additional settlement of the platform occurred for the next two months. From mid-December 1973 to July 1974, the platform was observed to settle another 1-3 cm (Clausen et al, 1975). This was most likely due to consolidation in the clay. Total settlement one year after installation was approximately 24 cm. This was within the 20-40 cm range predicted by NGI (Clausen et al, 1975). They estimated that the initial settlement would be 20 cm, and that another 15 cm would occur over the life of the 81 structure from storm effects (Braun, 1974). Unfortunately, more recent data is not available to extend the settlement-time curve shown in figure 4.8. Reported differential settlement of the platform after installation was 13 cm from the northeast (high) to the southwest (Clausen et al, 1975). (This, however, may not all be settlement since the seafloor was uneven and perhaps slightly sloping.) Although seemingly large, this corresponds to the platform being off vertical by only about one-twentieth of a degree. In the twelve months after installation, additional differential settlements of about 2 cm in the east-west and 6 cm in the north-south directions occurred (Clausen et al, 1975). Of interest, perhaps, is how the settlement data was obtained. Sightings were made on a nearby jacketed platform founded on deep piles. This platform had been placed more than a year before the Ekofisk tank and was not expected to settle noticeably during the period under consideration (Foss, 1974). The development of excess pore water pressure under the Ekofisk tank has been the subject of several studies. Fortunately, some pore water pressure data from Ekofisk is available to compare with theoretical predictions. The platform base was instrumented with seven pressure gauges and the underlying soil with twelve piezometers. A description of the installation of piezometers beneath the tank is given by Clausen et al (1975). The arrangement of these devices is shown in figure 4.10. Data from the storm of 6 November, the first major storm to hit the platform, is shown in figure 4.11. Several important 82 ®: n rtftrt to gauge no. Figure 4.10 -K^ati?nn.ff^rfSSUre 9au9es and piezometers beneath Ekofisk tank (After Clausen et al, 1975) i 5 K O < 10 ui in m x a. ui a 20 25 75 SO •5 go \ 95 o 0)(T) ®_ Hydrostatic (for wattr dtpth • 67.5 m)-\_ LEGEND: _K 4 th Nov. 1322- uBD _ H 6 th Nov. nSS-uiS. ~ I < {*)•• n rtftrt to gaugt no. -X— ®\ H 85 TO 75 W 85 go TORE WATER PRESSURE 11/m'l —U D © 0) ,® rtftrt to 1 gaugt no. o o < in < ui z X *— a. Ui o "5 0 2 ( 6 8 PORE WATER PRESSURE INCREASE It/m') DURING 6. NOV STORM Figure 4.11 - Pore pressures observed under Ekofisk tank during the first ma-jor storm (After Clausen et al, 1975) 83 observations may be made. First of all, the pore water pressures increased during the storm at all test locations. Secondly, the maximum pore pressures developed in the sand occurred not at the platform base, but some distance below it. Thirdly, pore water pressure in the clay seam was substantially higher than in the sand, perhaps indicating that partial drainage occurred in the sand. And finally, the pore water pressures that developed in the sand beneath the clay seam were significantly less than those developed in the sand not far above it. Typical pore water pressure ratios were on the order of 3% to 7% in the sand. Unfortunately, the instruments were not working for the storm of 19 November, which was nearly as large as the design storm. Considerable consolidation had occurred in the sand from previous storms by this time and may have affected the pore pressure response considerably. No other pore water pressure data has been made available, and thus no particular conclusions about the effects of preshearing on pore water pressure response may be made here. Early theoretical studies of pore water pressure generation were reported by Bjerrum (1973). He assumed that the sand could not drain at all over the course of the storm and therefore data from undrained laboratory shear tests was directly applicable. The dimensions of the structure are such that full drainage cannot take place during the storm; the amount of drainage, of course, depends on the permeability of the soil and length of the drainage path. Hence, his assumption of no drainage taking place was not unfounded. The amount of pore water pressure rise for a single cycle in undrained shear was determined from 84 laboratory tests, the data being shown in figure 4.12. By representing the design storm by the number of waves in specified height bands (i.e. a histogram), the number of shear stress cycles at a particular amplitude may be found since the wave.forces are known. Knowing the number of cycles applied at each shear stress amplitude, the pore water pressure developed under undrained conditions may be estimated by summing up the contribution of all cycles. This is demonstrated in Table VI. Bjerrum (1973) found for a relative density of about 90%, a pore water pressure ratio of about 31% would be developed beneath the platform, assuming undrained conditions. This analysis, being very simple and conservative, is good for demonstrating whether further analyses are required. It should be noted that no consideration of the distribution of stresses beneath the platform was considered in this analysis. The results of more advanced analyses are reported by Lee (1976) and Lee and Focht (1975a). There was some uncertainty with regards to the in-situ relative density (Lee and Focht, 1975a). Preliminary site investigations at the Ekofisk field made by McClelland Engineers, Ltd. suggested that the sand was medium dense to dense with a relative density of about 80%. Early studies indicated that the sand under the tank might liquefy under the cyclic storm loads. For this reason, an extensive program of cyclic testing was carried out on samples taken from Ekofisk, and further in-situ tests were performed to better define the relative density of the sand for correlation with laboratory test results. The preliminary tests performed to assess the liquefaction 85 UvwrBrassl*vct: T /or' H' vi Figure 4.12 - Pore water pressure rise per cycle observed in undrained simple shear with cyclic loading for samples prepared with relative densities of 80% (After Bjerrum, 1973) Table VI Example of the Accumulated Effect of a 100-year Storm (After Bjerrum, 1973) Height of waves: m Number of waves, N O TO 4-8 48S 007 0-006 2-9 8-12 471 012 0013 61 12-16 282 017 0030 8-5 16-20 121 0-22 0 065 7-9 20-24 32 0-26 0150 4-8 24-26 3 0-30 0-300 0-9 Total 1394 311 86 potential were standard undrained cyclic triaxial tests used for earthquake studies. Data for samples compacted at three relative densities, 63%, 77%, and 100%, showed that liquefaction (defined here as when the ratio of the excess pore water pressure to the effective confining pressure is equal to unity) would take place in the tests with relative densities of 63% and 77% when subjected to design storm cyclic shear stresses (Lee and Focht, 1975a). Since the tests used for assessing earthquake liquefaction potential are not really applicable to the ocean wave problem where preshearing will densify the sand before design loads occur and partial drainage will take place, additional tests were performed to reassess the liquefaction potential taking- these factors into account. In this set of tests, samples at different relative densities were sheared in undrained cyclic triaxial tests at low stress levels and then allowed to reconsolidate, simulating the effects of preshearing. To investigate the beneficial effects of partial drainage, a laboratory test procedure was developed to model this. This laboratory procedure is outlined by Lee and Focht (1975a). First, the permeability of the sand was established and the time period for 10% consolidation to occur beneath the tank was evaluated based on plane (and radial) flow conditions. This time period was estimated to be 500 seconds (125 seconds for radial flow) and was converted to an equivalent number of waves for the 10% consolidation time period, equalling about 50 (12.5 for radial flow). The samples were then tested undrained for this number of cycles. The pore water pressure rise was noted, then the back pressure was increased to 90% of this amount and 87 the drainage line opened to allow the sample to consolidate by 10% of its excess pore water pressure. The drainage line was then closed and the sample was tested undrained for another 50 (12.5) cycles. Testing continued in a similar fashion until the samples either liquefied or reached equilibrium. From this type of testing, it was found that a sample compacted to 77% relative density would not liquefy. Shortly after the second stage of laboratory tests had been completed, data from additional cone penetrometer testing at the site became available. This data indicated that the sand was extremely dense with a relative density of nearly 100% (Lee and Focht, 1975a). Additionally, the permeability of the sand was determined to be much lower than what had been found previously. A series of new tests were performed on the sand compacted to 100% relative density and tested under conditions of undrained shear. Tests were performed on samples that were both unconsolidated and consolidated to simulate the effects of preshearing. From these test results, it was concluded that the sand possessed adequate resistance to liquefaction, with the preshearing of samples adding additional cyclic strength (Lee and Focht, 1975a). The problem of pore water pressure generation beneath the tank was investigated after installation by Rahman et al (1977) who formulated the problem mathematically. They represented the soil by finite elements, with linear.stress-strain behaviour, and considered the distribution of stresses within the soil mass from both the weight of the tank and the applied wave loads. Their method is formulated as follows: The zone of directional 88 randomness of the waves is assumed to be sufficiently wide so that loading on any plane passing through the vertical axis of the platform is essentially the same as all the others when time averaged. Hence, the problem can be approximated as being axisymmetric with respect to loading, and therefore, pore water pressure generation and dissipation. The equation for radial and vertical consolidation is then formulated to include a pore water pressure generation term whose parameters are defined by data from undrained cyclic shear tests. The rise in pore water pressure measured in undrained cyclic triaxial tests is found for different cyclic shear stress levels and curves of number of cycles versus pore water pressure are obtained. The coefficients of these curves are used in the pore water pressure generation function. The time history of loading is approximated by a histogram and the loads are applied incrementally to the platform. The storm is applied by time stepping as follows: a given number of cycles at a certain stress level (corresponding to an equivalent number of waves of a given height) are applied (through the use of the pore water pressure generation function) and the resulting pore water pressures are then allowed to drain for an amount of time corresponding to the number of waves. The procedure is continued until the storm is over, that is, when all the waves have been represented by the time stepping procedure. Results of their studies showed that allowing for partial drainage is extremely important for predicting the correct pore water pressures developed underneath the Ekofisk platform. They found that if the foundation sand had a relative density of 77%, 89 liquefaction would not occur; in fact, maximum pore water pressure ratios would be less than about 30% beneath the entire foundation. A Bjerrum (1973) type of analysis at this relative density would indicate that the sand would have liquefied under the tank (Rahman et al, 1977). Rahman et al's (1977) type of analysis can provide information on the distribution of pore water pressures beneath the tank. The other methods cannot. Some results of their studies are shown in figure 4.13. It is of interest to note that maximum pore water pressures are developed under the edges of the platform, not beneath the center. This will affect all stress analyses, and is of particular significance when predicting rocking motions. Stability analyses were carried out for the tank to insure safety under storm wave loading. For lack of better methods, the bearing capacity equations of Hansen (1970) were used. Several problems were encountered when trying to apply this well-known bearing capacity formula to the Ekofisk tank (Bjerrum, 1973). First of all, the bearing capacity factors used in the equation were determined semi-empirically for model footings of a very small size. When extrapolating these results to the Ekofisk tank with a base dimension of about 93 meters, considerable scale effects were induced. The value of the bearing capacity factor Nr was decreased to take this into account (Bjerrum, 1973). The reduction of Nr with footing size may actually be attributed to a decrease in the friction angle with an increase in the mean principal stress. Secondly, the inclined load factor proposed by Hansen (1970) had never been used on a foundation with such a high ratio of horizontal to 90 1 1 I r 0,-TT* -i 4 i r> lb* Himry of CajkMstwit liom Timt - hra Dr*85% », »kf •!0'9C*n/MC Figure 4.13 - Theoretical prediction of the pore water pressure distribution beneath the Ekofisk tank for relative densities of 77% and 85% (After Rahman et al 1977) 91 vertical force (about 38%). A thorough review of model test results led to the conclusion that the inclination factor of Hansen (1970) was acceptable for the high ratio of horizontal to vertical load (Bjerrum, 1973). This factor reduced the bearing capacity to one-fifth of its value for vertical loading only. Finally, the bearing capacity of the tank would be influenced by drainage conditions. Since the wave force would go from zero to a maximum value in one-quarter of a wave length (about 4 seconds), virtually no drainage could occur. This problem of (essentially) undrained bearing capacity had never been investigated before, since complete drainage is usually assumed for foundations on cohesionless soil (Bjerrum, 1973). To model this, the undrained friction angle found from triaxial tests, increased from 34° to 36* to account for assumed plane strain conditions, was used for stability calculations. A plasticity solution was carried out to determine the most critical failure surface for the design loads (Bjerrum, 1973). This analysis was quite complicated since the pore water pressure distribution affected the effective stresses which determined the rupture surface. A lengthy and difficult iteration procedure was required to find the rupture surface. The result of this work is shown in figure 4.14. Unfortunately, no factor of safety was reported. 92 Figure 4.14 93 CHAPTER 5 CHARACTERISTICS OF WAVE LOADING 5.1 Ocean Waves Ocean waves are generally the most important environmental phenomenon that ocean engineers must deal with when designing structures for the offshore environment. Although earthquakes or ice loading may apply the largest horizontal forces on a structure in some areas, wave loading will nonetheless be an important consideration and must be investigated. Waves in the ocean come in a variety of forms, including: wind waves, ship-generated waves, tsunamis, and tides. In the open ocean where the water is sufficiently deep to prevent significant tsunami shoaling and tides are not restricted by narrow passages, wind generated waves will be the most important of these forms with regards to offshore structure design. These will be the only ocean waves considered in this thesis. 5.1.1 The Wave Climate .The generation of wind waves is a complex phenomenon where energy from the blowing wind is transferred to water particles at the air-sea interface by pressure gradients and frictional forces which subsequently set the water into motion (Kinsman, 1965). The amount of energy that can be put into a wave system depends on the duration, intensity and direction of the wind, the fetch (the sea distance over which the wind blows), the frictional resistance of both the seafloor and air-sea 94 interface, and internal energy dissipation. Wind waves may be classified as being either sea or swell. The former are still under the influence of the generating wind, while the latter travel across the ocean surface virtually unaffected by the wind. Empirical charts have been developed to estimate some characteristics of these waves from meteorological data (e.g. Shore Protection Manual, 1977). Waves in the ocean are very complex and do not conform to precise mathematical modelling. The sea is characterized by numerous waveforms of varying shape, length, height, speed, and direction, all superimposed on each other in an everchanging arrangement. For this reason, the sea is modelled statistically using spectra to take these factors into account. These spectra are approximations at best, and do not actually define a particular "sea-state" at any time. Very limited data is extrapolated to obtain the statistical properties of the wave system. For engineering purposes, it is useful to describe the ocean surface by a train of uniform waves of specific height and period travelling in water of constant depth. This is the most simplistic model of ocean waves and is often adequate for design purposes. Numerous theories have been developed for this situation. 5.1.2 Wave Theories All the analytical wave theories make some of the same basic assumptions (McCormick, 1973). They differ in the way in which the governing equations and boundary conditions are 95 mathematically formulated. Common to all are the assumptions that the water is incompressible and that flow is irrotational (no shear stresses at the air-sea interface or at the seafloor). From potential flow theory, this implies that a velocity potential must exist and satisfy the Laplace equation. This equation is an expression of continuity for irrotational flow and requires a number of boundary conditions to solve it. These are as follows: (1) the bottom is impermeable, nondeformable, and horizontal - a no flow boundary (seabed boundary condition), (2) the pressure at the air-sea interface is constant (dynamic free surface boundary condition), and (3) the flow at the air-sea interface is in accordance with the geometry and motion of the free surface (kinematic free surface boundary condition). Additionally, since the velocity potential should be cyclic in nature it is assumed to be periodic with both spatial and temporal variation. Analytical wave theories vary in complexity and accuracy depending on how they approximate the boundary conditions. The simplest theory for ocean waves is the linear theory presented by Airy (1845). He assumed that the periodicity was sinusoidal and that the free surface boundary conditions could be linearized. With these assumptions, the solution of Laplace's equation subjected to the four boundary conditions results in the velocity potential having only one term, which depends on the wave period and height, the static water depth, and the depth of a reference point below the static water level. It is sinusoidal and periodic in the direction of propagation with time. From the velocity potential, other equations may be 96 derived for water particle accelerations, velocities and displacements, wave induced pressure on the seafloor, etc. The computed surface waves are known as Airy waves. Other wave theories commonly used are the higher order Stoke's (1880) theories, particularly the second and fifth. The free surface boundary conditions in these theories are estimated to higher orders by a perturbation process. The resulting velocity potential has the same number of terms as the order of the theory, and is a series approximation. The individual terms are sinusoidal; however, the waveform, being comprised of different sinusoidal forms superimposed on each other, is not. These waves are characterized by steeper crests and shallower troughs than linear (sinusoidal) waves. For shallow water, where the bottom significantly affects the travelling surface gravity wave, the waveform may be approximated by the Jacobian elliptical cosine (cn) function (Korteweg and De Vries, 1895) to any order desired. These are the cnoidal wave theories. They compare well with wave tank tests in shallow water, but are complicated and difficult to use (Shore Protection Manual, 1977). Numerical wave theories have also been developed. Dean's (1965) theory, which is the best known, is based on stream functions instead of velocity potentials and requires the use of a computer to solve the equations for any given set of wave parameters. Its use is limited in engineering applications because the method, due to its complexity, cannot be used in most wave force theories. The regions of validity for the best known wave theories 97 are shown in figure 5.1. Clearly no one theory can be regarded as being the best for all applications. 5.1.3 Results of Linear Wave Theory Linear wave theory, besides being the simplest to use, is more reliable than the other analytical theories over a greater range of conditions. It does not suffer from numerical instability as most of the other theories do when applied in regions beyond their (calculated) range of validity (Sarpkaya and Isaacson, 1981). For these reasons, it is the most widely used wave theory by practicing engineers. Additionally, most wave force theories assume that the waves may be represented by linear theory, although the wave length used in the resulting wave force equations may be computed using another wave theory, usually Stoke's fifth order theory. Linear theory is used extensively in spectral wave force calculations (Bea and Lai, 1978). The profile of a linear wave is shown in figure 5.2, and some results of linear wave theory are presented in Table VII. Note that only the wave height, water depth, and either wave length or period are needed to define a linear wave. This is also the case for other wave theories. The wave length and period are related by the dispersion relation, which is derived from the velocity potential. 5.2 Characterizing the Wave System Since wind waves are random in nature, they are best described statistically. Approximations may then be made to 98 005• | i i 1 1 r 0.00005' o.OOl 0.002 0.005 0.01 0.02 005 0.1 0.2 d Figure 5.1 Regions of validity for various wave theories (After Sarpkaya and Isaacson, 1981) 99 Wove speed,c L 5 B z • d d k = 2JT-/L 6 - kx-a)t Wove period,T = L/c Surface elevation shown ot t = 0 Figure 5.2 - Profile of an Airy Wave (After Isaacson, 1980) Table VII Some Results of Linear Wave Theory (After Sarpkaya and Isaacson, 1981) Velocity potential irH cosh (ks) A *= — sin 6 kTsinh(kd) EH cosh (ks) . m — • sin e 2u> cosh (kd) Dispersion relation c2 = -y- = f - tanh (kd) k2 k Surface elevation H T| • — COS e Horizontal particle displacement H cosh (ks) . tB • sin 6 ' 2sinh(kd) Vertical particle displacement H sinh (ks) t • i COS 6 * 2sinh(kd) Horizontal particle velocity nH cosh (ks) u • cos 6 T sinh (kd) Vertical particle velocity irH sinh (ks) . w * — . . „ sin 6 T sinh (kd) Horizontal particle acceleration 8u 2ir2H cosh (ks) — c —. sin 8 8t T2 sinhftd) Vertical particle acceleration aw 2w2H sinh (ks) — * x cos S at T2 »inh(kd) Pressure 1 „ cosh (ks) p •= -pgz + -pgH—___cose 2 cosh (kd) Group velocity - 1 Ii 2kd 1 00 2 [ iinh (2kd)JC Average energy density E^ipgH2 100 characterize the wave system in simpler terms for the purposes of foundation design. 5.2.1 Obtaining the Design Storm The design storm is usually found by extrapolating data from wave records. This data is often rather sparse and must be representative of storm wave conditions to use the statistical methods developed for defining the design storm. 5.2.1.1 Statistical Description The distribution of wave heights for a particular sea-state may be characterized by a Rayleigh distribution, assuming that the free surface is Gaussian for a specific recording interval, usually 6 hours. The assumption that the free surface variation for a particular sea-state may be represented by a Gaussian distribution corresponds well with observations. Data from the recording interval is assumed to be described by a 10 minute sample which is representative of the 6 hour recording interval. To describe the variation of sea-states over the long-term (years), it is convenient to represent each recording interval by one statistical parameter, the significant wave height, denoted Hs. This is defined as the average height of the one-third highest waves in the wave record, i.e. the recording interval. For any recording interval the significant wave height may be computed without much difficulty . (usually by a digital computer). A probability distribution may be fitted to the significant wave heights from numerous records to estimate the significant 101 wave height for some remote event (e.g. the design storm). This is usually done using the extreme value statistics of Gumbel (1958). The probability of a rare event occurring may be found for a specified recurrence interval (e.g. 100 years). Thus, the significant wave height for the design storm may be estimated from wave records. The distribution of wave heights within the design storm may be found using short-term statistics - the Rayleigh distribution. The whole procedure may be repeated to find the distribution of wave periods for the design storm. The duration of a storm may be days, however, for practical purposes some time limit must be chosen. A design storm of twelve hours is often used (Isaacson, 1981). The storm is assumed to buildup, peak, and decay during this time. The duration of the design storm will affect the wave statistics. 5.2.1.2 Geotechnical Equivalent For geotechnical purposes, this type of representation is not very useful in practice. Therefore, for a specified design storm, it is useful to transform the statistical distributions of wave heights and periods into a histogram relating wave heights to frequency of occurrence (i.e. number of waves of some height) and a curve defining the wave height - wave period relationship. This, in geotechnical literature, is known as the "design storm". Because the distribution of wave heights during a storm is represented by a Rayleigh distribution, the number of waves in any particular band of heights will be known. This is easily transformed into a histogram. The histogram could have as many 102 bands as there are waves in the storm. This, needless to say, would be impractical. Generally five to fifteen divisions is acceptable, depending on the type of analysis to be performed and the accuracy desired. Six (Bjerrum, 1973) to sixteen (Lee and Focht, 1975a) divisions have been used for pore water pressure generation studies. The duration of the design storm is also of interest, since the amount of pore water pressure dissipation occurring in granular deposits will be sensitive to this. Bjerrum (1973) suggests that the storm may be assumed to buildup over six to nine hours, maintain full-storm conditions for three to nine hours, then subside in another six to nine hours. He used the worst six hours of the design storm to analyze the pore water pressure buildup under the Ekofisk tank. These 6 hours of storm contained 1394 waves. Lee and Focht (1975a) used a group of 5000 waves to characterize a thirteen hour storm for the Ekofisk tank. Such a large group of waves appears to be excessive since the smaller waves will have little effect on pore water pressure generation. This is confirmed by Rahman et al (1977) who used a six hour storm to analyze the same problem. They found that equilibrium pore water pressure ratios of a few percent at most were quickly achieved and maintained at the lower cyclic stress ratios produced from the numerous smaller waves. 5.2.2 Application of the Design Storm Using the actual time history of the storm is impractical. Therefore, the geotechnical design storm approximation may be used. Of primary interest here is when to apply the maximum 103 wave during the design storm to find the most critical condition for stability. Bjerrum (1973) assumed that it is conservative to apply the maximum wave at the end of the design storm when pore water pressures would be the highest. Based on an undrained analysis of the sand during the storm, the pore water pressures would indeed be the highest at the end of the storm. His reasoning with regard to maximum pore water pressures corresponding to the critical time to apply the maximum wave is sound, however, the end of the design storm is not necessarily the most critical with respect to stability. For cohesionless soils, some drainage will take place during the storm and the maximum pore water pressures under the foundation will probably occur at the height of the storm or just thereafter. Rahman et al (1977) assumed that the storm is characterized by smaller waves increasing in height to a maximum (the design wave), then decreasing in a similar fashion, as Bjerrum did (1973) and they applied the waves to the foundation system with this order in mind - representing the time history of loading in an approximate way. They found that for the Ekofisk tank (founded on fine sand), the maximum pore water pressures would occur just after the peak of the storm. For this type of analysis, the critical application of the maximum wave would be just after the peak of the storm. Intuitively, this seems correct for foundations on sand. For foundations on clay, where no substantial drainage can take place during the storm, the usual procedure is to apply the design wave to the structure at the end of the storm (Schjetne, 1976). This approach is conservative, but not unduly so, at least for stiff clays 104 (Andersen et al, 1976). 5.3 Wave Loads on the Foundation System Wave loads on the foundation system consist of the forces exerted on the structure and transferred to the soil by the raft and the pressure on the exposed seabed due to travelling surface gravity waves. Both must be considered when designing the foundation. Because the period of wind waves is on the order of two to twenty seconds, forces on the foundation may be considered to act pseudostatically for stress analysis. The effects of cyclic loading on the soil should be modelled appropriately. The loads acting on the foundation of a gravity structure subjected to wave action and the resulting soil reactions are shown in figure 5.3. 5.3.1 Wave Forces Acting on the Structure Wave loads acting on the structure are found using formulas derived from potential flow theory with empirical coefficients. There are two basic methods for finding wave forces on structures: the design wave method and the spectral analysis method. In the spectral method, forces are defined statistically, whereas in the design wave method, forces are treated deterministically. When waves propagate past a structure, forces are exerted on it from both frictional and inertial effects caused by the moving fluid. The former component is highly nonlinear while the latter is not (Morison, 1950). If the lateral dimension of Figure 5.3 Forces acting on the foundation of an offshore gravity structure 106 a structure is significant compared to the wave length (20% or more), the water particle motions and waveform are greatly disturbed by the presence of the body as the wave passes; this must be taken into account when predicting the wave loads acting on the body (MacCamy and Fuchs, 1954). Diffraction theory was developed for this purpose (MacCamy and Fuchs, 1954) and may now be applied to large volume structures of arbitrary shape such as gravity platforms (Garrison, 1979; Hogben et al, 1977). Linear diffraction theory (developed for Airy waves) is presently used for both deterministic and probabilistic wave force calculations for gravity platforms (Isaacson, 1980). In the diffraction regime, the drag component is small and may be neglected, leaving only the inertial component. The inertial force on a vertical surface piercing cylinder computed from linear diffraction theory is represented by a single term for a given wave and varies sinusoidally in time (MacCamy and Fuchs, 1954). For the geotechnical engineer, the significance of this is that the overall wave forces on a gravity structure will vary nearly sinusoidally in time. Although the wave forces will differ for different waves in the storm, individual wave forces may be defined completely by a magnitude, oscillatory period, phase angle, and frequency of occurrence. For geotechnical purposes, the phase angle is unimportant except when finding the design pressure on the seabed corresponding to the maximum wave. The design storm may then be transformed from a wave height— frequency of occurrence histogram, for the given wave height— wave period curve, to one of force--frequency of occurrence. This representation is shown in figure 5.4. The variation of O 5 10 15 20 25 30 0 12 3 4 5 6 Wave Height (m) Time (hrs) (c) Horizontal force--wave parameter relationship (d) Time history of wave forces Figure 5.4 - Typical design storm representation used In geotechnical engineering o -j 108 cyclic "shear stresses during the storm may be be found in the same way once the horizontal forces are defined. 5.3.2 Wave Forces Acting on the Foundation For any given wave, the resultant vertical and horizontal forces may be computed from diffraction theory. The wave forces acting on the foundation are the same as the wave forces acting on the structure, however, a moment must be applied to the foundation to account for the resultant horizontal force acting some height above the seabed. The forces acting on that part of the seabed not under the raft are due to the weight of the overlying body of water and the influence of passing waves. The pressure at any location on the seafloor is composed of a steady and a fluctuating component. The steady component is uniform over the seafloor (assuming that the water depth does not change) and is nothing more than the normal hydrostatic pressure. The fluctuating component is the dynamic pressure due to particle accelerations in the wave and at the seafloor is very nearly equal to the hydrostatic pressure due to the weight of a column of water displaced from the static water level as the wave passes. Any appropriate wave theory may be used to find the seabed pressure distribution; the fluctuating component will be of nearly the same form as the free surface. Linear theory is commonly used and the resulting seafloor (cyclic) pressure distribution is sinusoidal. The steady component is uniform everywhere, and therefore is of no significance since it affects neither the effective 109 stresses or the stress gradients in the soil. The fluctuating component is of interest for two reasons, namely: (1) it does not act uniformly over the seafloor at any given time and must therefore be considered as an external load, and (2) it induces stress gradients which produce cyclic shear stresses in the soil. When finding the pressure distribution corresponding to the design wave, the phasing of the wave forces must be considered. The seafloor pressures near the platform will be less than the pressure amplitude. The maximum forces acting on a platform usually occur when the nodal points of the waveform are near the platform's vertical axis, i.e. when the waveform passes through the still water level near the platform's vertical axis. For long (large) waves, this means that the pressure curve has a node somewhere over the raft and that the maximum pressures on the seabed due to the wave will be some distance from the edge of the platform (approximately one-quarter of a wavelength away from the vertical axis of the platform). 5.4 Effect of Cyclic Loading on the Foundation System The effects of cyclic loading on the soil must be taken into account for all stress analyses. Such effects influence the safety of the platform with respect to failure (sliding, bearing, rocking, liquefaction or otherwise), as well as the platform motions during storm loading and long-term effects such as settlement. Using a storm histogram similar to the one shown in figure 5.4(d) to represent the time history of loading, the 110 stress path for an element beneath a sand foundation may look like that which is shown (unidirectionally) in figure 5.5. This representation is idealized for the purposes of illustration. The figure may be interpreted as follows. The first histogram band represents some number of cycles at one stress amplitude. This amplitude is the distance from point "a" to the effective stress axis. The resulting residual pore water pressure decreases the effective normal stress. Hence, movement from point "a" to "b". The next set of stress cycles are at a magnitude represented by the distance from point "c" to the horizontal axis. Pore water pressure reduces the effective normal stress to point "d". The stress path shown in the figure does not show the full path, i.e. the return to the horizontal axis (zero shear stress) for each set of cycles is not shown. This was left out to clearly illustrate the effects of cyclic loading on a foundation element. Similarly, the path continues as the cyclic stress amplitude increases to a maximum corresponding to the design wave, then decreases. Note that excess pore water pressures exist in the foundation element throughout the storm. Much data is available for sand tested under undrained conditions. This is due to the interest in earthquake induced liquefaction which has attracted scores of researchers. Partially drained sand behaviour has not been well studied, and little information is available on the subject. Partial drainage for cohesionless soils has not been directly modelled in laboratory shear tests. Instead, modified undrained cyclic triaxial .tests are used (Lee and Focht, 1 975a). 111 EFFECTIVE NORMAL STRESS, U' Figure 5.5 1 12 Pore water pressure generation in undrained shear depends on the characteristics of the sand, the magnitude of the static shear stress in the soil before testing, and the magnitude and time history of the applied cyclic shear stresses. The amount of pore water pressure generation in undrained shear may be estimated from laboratory tests that appropriately model the ocean wave loading problem (Lee and Focht, 1975b). Estimation of the pore water pressures developed under an offshore gravity type structure may be made using (1) laboratory tests such as those described by Lee and Focht (1975a) which model partial drainage and preshearing using modified cyclic triaxial tests, or (2) from numerical methods that model the soil by finite elements and solve the equations of radial and vertical consolidation (Rahman et al, 1977). Undrained analyses in sand (Bjerrum, 1973) are too conservative and are not appropriate for advanced studies. Cyclic loading of clay has recently been a subject of intensive study (Andersen et al, 1976; van Eekelen and Potts, 1978) as some of the gravity platforms in the North Sea are underlain by substantial clay deposits. The results of an extensive study on clay behaviour under cyclic loading were reported by Andersen et al (1976), and demonstrated several important concepts. First of all, shear strain may be used instead of pore pressure development as a parameter for describing response to cyclic loading. Cyclic shear strains are uniquely related to the effective stress and independent of the overconsolidation ratio or number of cycles. Secondly, the 113 effective strength parameters c' and tan0' are virtually unaffected by cyclic loading, but the undrained strength cw is. Thirdly, the undrained strength is a function of cyclic shear strain and the number of stress cycles applied. And finally, the higher the overconsolidation ratio for a given cyclic stress ratio, the fewer number of cycles are necessary to bring the sample to failure. Andersen (1976) developed a method based on the accumulation of cyclic shear strains to predict failure from excessive displacements. It has been found for insensitive clays that if the applied shear stress level is below some critical value, a state of non-failure equilibrium will be reached where the stress-strain curves follow closed hysteresis loops with no further increase in pore water pressure (Sangrey et al, 1969). If this critical value is exceeded, each loading cycle will cause a cumulative increase in pore water pressure and displacements which ultimately results in a shear failure. This failure will occur at a reduced undrained strength which is about two-thirds of the value for static loading. This critical cyclic shear stress level should be determined experimentally for each cohesive deposit (Bjerrum, 1973). Consolidation history is important when assessing the effects of cyclic loading on clay. For normally consolidated and slightly overconsolidated deposits, drainage after cyclic loading implies consolidation and an increase in the un.dirained strength. For heavily overconsolidated clays, swelling may occur after loading is terminated, with a corresponding reduction in strength (Schjetne, 1976). The effect of this on 114 platform safety is as follows: For foundations on highly overconsolidated deposits, the safety of the platform will decrease with subsequent cyclic loading. The safety of platforms on normally consolidated or slightly overconsolidated soil will increase in time. However, these deposits are usually unsuited for a gravity platform since displacements may be excessive under storm wave loading. 115 CHAPTER 6 PROCEDURES FOR ANALYZING THE STABILITY OF  OFFSHORE GRAVITY TYPE STRUCTURES 6.1 Fundamental Considerations The purpose of a stability analysis is to assess the margin of safety against an ultimate foundation failure. This margin of safety may be expressed in one of two ways: by a load safety factor or a material factor. The load safety factor is defined as the ratio of the load required to cause an ultimate failure to the design load, when the design strength of the soil is used. The material safety factor is the amount by which the strength parameters must be reduced to bring the soil to a state of limiting equilibrium under the design loads. The degree of strength mobilization in the soil is often how this latter result is expressed. The two safety factors will in general be di f ferent. Onshore, a safety factor of about three is commonly used to take into account uncertainties associated with the values of strength parameters, ground water conditions, loading conditions, and the reliability of analytical methods. Offshore, a much lower safety factor is used and a considerable amount of effort is spent in trying to better define the problem than is commonly done for most onshore projects. The use of more rigorous analyses is justified economically, since the degree of uncertainty will be less, and therefore, the factor of safety may be reduced. For this reason, bearing capacity theory 116 is generally used only for a preliminary estimate of the adequacy of a site for a gravity type structure. Limit equilibrium methods based on slip surfaces and the finite element method are usually employed. Evaluation of the safety factor requires three things: an estimation of the shear strength of the soil, estimation of the shear stresses in the soil, and the postulation of a failure mechanism. The shear strength of the soil may be defined by the Mohr-Coulomb failure criteria, which is Tr. = c + a tan0 (6.1) where c and tan0 are the shear strength parameters and C is the normal stress on the shear surface. In terms of effective stress, this equation is Tf= c'+ a'tan0' (6.2) where c' and tan0' are the effective shear strength parameters and 6" is the effective normal stress, defined as cr = <y - u (6.3) where u is the resultant pore water pressure. One should remember that each half wave cycle the direction of loading reverses. There are, therefore, both shear stress reversals and pulsating vertical stresses in the soil. Both will affect the magnitude of the pore water pressure at any time during the storm. It is convenient to discuss the pore water pressure in the soil in terms of its various components. The total pore water pressure, u, at any location is given by u = us + ue+ Au (6.4) where us is the hydrostatic component, uc is the residual pore 117 water pressure due to cyclic loading at any time during the storm, and Au is that part due to the changes in the principal stresses. The hydrostatic pore water pressure in the soil is simply us = ^d + Ywz (6.5a) where )(w is the unit weight of seawater, d is the still water depth, and z is the depth below the mudline. Note that this term includes the pressure due to the static body of water above the mudline. The effective stresses in the soil are, however, not influenced by the presence of an overlying body of water. This is illustrated in figure 6.1. This may be taken as us = V„z ' (6.5b) if the weight of the overlying body of water is omitted from all calculations. The component of pore water pressure in the soil due to changes in the principal stresses may be determined from (Skempton, 1954) Au = B[AO"3 + A (AC, - AC3 ) ] (6.6) where cr, is the major principal stress, o~3 is the minor principal stress, and A and B are dimensionless pore pressure parameters measured in the laboratory. This component will be a maximum or minimum (for dilation) for the design wave. The B-parameter may be taken as unity for offshore foundation analyses. For an elastic isotropic soil, the A-parameter is equal to one-third. This implies that the rise in pore water pressure is equal to the total stress increment. For other values of this parameter, the pore water pressure rise is not equal to the total hydrostatic stress increment. 118 d unnnij * I / Z / 1 (i) (2) Us u « l«(d • D0) 6' - q • ).Do * - Xwd + Hz u « *w(d • x) cr' - Y'z (2) Figure 6.1 - Effective stresses in soil for still water conditions (i.e. no wave loads) 119 Pore water pressures within the soil mass due to cyclic loading may be estimated using (1) data from cyclic shear tests with a numerical model such as Rahman et al's (1977) to account for consolidation, (2) data from modified cyclic triaxial tests such as those described by Lee and Focht (1975a) correlated with the in-situ stresses, or (3) from experience gained through observations of platforms instrumented for performance. A combination of these methods may be used. In addition to the pore water pressures generated as a consequence of cycling loads on the structure, excess pore water pressures will also be developed in the soil not under (or influenced by the presence of) the raft. Stress gradients are induced in the seabed from the pressure variations caused by passing waves (Henkel, 1970). If the waves are large enough and loading is sustained, liquefaction may occur (Finn and Lee, 1978). A stability analysis for a clay foundation may often be performed using a total stress analysis. The pore water pressure rise in a foundation element will be nearly equal to the total spherical stress increment for overconsolidated clay. The A-parameter is close to one-third for these clays (Skempton and Bjerrum, 1957). An effective stress analysis may also be performed. This method presents two difficulties, namely: the pore water pressures must be estimated and the effective shear strength parameters have to be evaluated. These parameters must be found from laboratory shear tests which require good quality samples. These are often unattainable. A total stress analysis may be performed without estimates of the pore water pressures 120 or laboratory shear test data. The undrained strength determined from in-situ tests such as the cone penetrometer or vane may be used directly, reduced appropriately for the estimated effect of cyclic loading (i.e. strain softening). This has been the case for most North Sea gravity structures founded on clay (Schjetne, 1976). It should be noted that for cohesive soils the pore water-pressures developed within the soil mass due to the weight of the platform may not have dissipated substantially by the time that a major storm hits the field. In this case, an effective stress analysis is required and loads less than the design wave loads may be used. A total stress analysis for the design storm loading, which may be assumed to hit the platform at a later date, may also be performed. Some sort of risk analysis will be required. For the stability analysis of a foundation on cohesionless soil an effective stress analysis is performed. This requires estimates of the pore water pressures within the soil mass. Any appropriate method may be used to estimate these pore water pressures. The friction angle must be determined from laboratory shear tests. Cone penetration resistance can help to establish the relative density which is needed for interpreting the tests. 6.2 Modelling the Wave-Structure-Soil System Since gravity platforms never approximate strip footings and rarely rest on the seabed without some sort of subsurface foundation, i.e. skirts and ribs, some assumptions regarding 121 geometry must be made so that analytical techniques may be used. The platform is usually modelled by an "equivalent rectangular foundation", that is, one with the same area as the actual platform base. The "effective foundation" is usually assumed to be at skirt-tip level (Lauritzsen and Schjetne, 1976). This assumption is good if failure does not extend up into the skirt compartments. The skirts should be spaced closely enough together to prevent failures of the kinds shown in figures 3.6(a), 3.6(b), and 3.6(d). A definition sketch of the effective foundation is shown in figure 6.2(a) for a two-dimensional representation and in 6.2(b) for a three-dimensional one. When modelling the foundation system, it is important to include all the forces which act on both the structure and the seabed adjacent to it. The points of application of the resultant forces, or the distribution of pressures, must be either known or assumed. The loads acting at the foundation base are required for a stability analysis. The wave loads, which are specified at the seafloor, must be transmitted to the foundation base by including the forces acting between the seafloor and the skirt tips in the resultant load. Figure 6.3(a) shows the forces acting on the foundation system, and figure 6.3(b) shows the resultant loads which are used in a stability analysis. These forces are defined qualitatively in the following paragraphs. The vertical load applied to the effective foundation, VBT, will consist of the buoyant weight of the platform Pv , the fluctuating vertical force due to the wave APV (which will be 122 (b) Three-dimensional representation Figure 6.2 - Definition sketch of effective foundation 123 APlX) nrrrr 1" AP(X) W»V« I I I I I l * R (a) Loads acting on the platform APOO rrrrrr r AP(X) IffNt/l 1 T 1 ' ' 1 » OIMBT  (b) Loads transferred to the foundation base Figure 6.3 - Transformation of loads to foundation base 124 downward for design conditions), and the added load from soil contained within the skirt compartments. Vertical shear forces acting on the periphery of the imbedded foundation may usually be ignored. The distribution of vertical stress beneath the raft will be nonuniform since loading is inclined and eccentric. This is usually taken into account by considering only that area of " the foundation base which is symmetrical with respect to the resultant vertical load. The resultant vertical load is then applied centrally on this "effective area" (Hansen, 1961). The effectve area is represented by BL0 in figure 6.2(b). It may be determined once the load eccentricity at the base is established. This eccentricity is initially unknown since the moment at the base of the structure depends on the soil forces acting between the mudline and skirt-tip level; these forces must be determined from an iteration procedure. However, the moment at base level may be approximated by choosing reasonable values for the soil forces. The effective area may then be established. The effective area will usually not be influenced significantly by the soil forces, particularly for shallow foundations. For one-dimensional eccentricity, that is, when loading is parallel to one side of the equivalent rectangular base, only one dimension of the equivalent foundation is reduced. This is normally the case for gravity structures since they are usually of similar width and length (i.e. they are approximately radially symmetric). Meyerhof's (1953) "effective width" principle is then used. The horizontal force which acts on the effective foundation, HRT, is somewhat more difficult to assess. This 125 force is equal to the resultant of the horizontal wave load PH and the horizontal components of all the forces acting between the mudline and skirt-tip level: the active soil force PA on the tail end of the foundation, the passive soil force Pp on the nose of the foundation, and shear forces Ps on the sides of the imbedded base. These forces must be estimated. Generally, the active and passive soil forces are assumed to act only horizontally. For structures with significant penetration into the seafloor, this assumption may not be reasonable. When the active soil force is negative, as for clay foundations, a tension crack is assumed to exist at the tail end of the platform. A water force, Pw, due to the dynamic wave pressure (which is not in equilibrium with the soil pore water pressures) acts in any crack. The shear forces acting on the sides of the foundation, Ps, reduce the horizontal force applied to the foundation base. This resistance is not included for a plane strain analysis. The distribution of horizontal force over the foundation base is assumed differently in various stability theories. The dynamic wave pressure acting on the seabed, Ap(x), varies roughly sinusoidally; the estimated variation depends on the wave theory used to calculate it. For the purpose of a gravity structure stability analysis, it is often adequate to apply the pressure uniformly on either end of the raft, taking magnitude and phasing into consideration, since the variation of this pressure over a short distance is usually minimal. The dynamic wave pressure will affect the magnitudes of soil forces acting on the structure above skirt-tip level. 126 The effects of cyclic loading on the foundation soils must be adequately taken into account in accordance with sections 5.4 and 6.1. The effect of consolidation history on clay and preshearing in sand, must be considered when choosing strength parameters appropriate for design storm analyses. 6.3 Loading Applied to the Foundation The vertical load at the effective foundation level, V61, is equal to the sum of the vertical load at the seafloor and the submerged weight of soil within the skirt compartments. This may be written as VBT= Pv + APV + (A0D0)i" (6.7) where Pv is the vertical platform load in the absence of environmental loading (the buoyant weight of the platform, which is not constant over the life of the structure), APV is the vertical load due to environmental loading, A0 is the area of the equivalent base, D0 is the depth of the effective foundation, and H' is the effective unit weight of soil. The horizontal load acting on the foundation base, HBT, is the resultant of the applied environmental load PH and all the horizontal forces acting on the platform between the seafloor and skirt-tip level. This may be expressed as H*T= PH + (P* or Pj - Pp (6.8) where the active soil force is defined by PA = [ (0.5rDo2+Ap,Do)tan2 (45'-0/2)-2cDotan(45'-0/2) ]L0 (6.9) Here, Apf is the dynamic wave pressure acting on the seabed at the tail end of the platform, 0 is the mobilized friction angle, 127 c is the mobilized cohesion, and L0 is the equivalent platform length. If PA is negative, a water pressure force Pw replaces it, given by Pw= Ap,D0L0 (6.10) The passive soil force is defined as Pp= [(0.5o'Do2+Ap2Do)tan2(45o+0/2)+2cDotan(45°+0/2)]L0 (6.11) where Ap2 is the dynamic wave pressure acting on the seabed near the nose of the platform. Note that Ap2 is a negative quantity. The preceding equations may be used to define plane strain loading. This is done by dividing each equation by the equivalent platform length L0 to find the force per unit length. For a three-dimensional analysis, the aforementioned equations may also be used, with one exception: shearing resistance on the sides of the foundation must be included. In this case, Equation (6.8) may be written as HBT = P« + (PA or Pw) - Pf - Ps (6.12) where the shearing resistance on the sides of the foundation Ps is defined by Ps = 2D0B0(c + O.5tf'Dotan0) (6.13) The moment applied at the foundation base, MB1, is the resultant of the moment at the seafloor M, and the moments due to all the forces acting between the seafloor and the foundation base. This may be expressed as MfeT= M + PHD0 + (PA or Pw)hi - Pfh2 - Psh3 (6.14) where h,, h2, and h3 are moment arms for the appropriate forces. These may be found from earth pressure theory. The effective width B may be found once the eccentricity is known. This may be written as 128 B = B0(1 ~ 2e) (6.15) where B0 is the equivalent foundation width and the eccentri city e is given by e = (MB7Afc-r)/B0 (6.16) 6.4 Available Stability Methods Presently, there are a number of analytical or numerical methods which may be used to assess the foundation stability of a gravity platform. These are: the bearing capacity methods, the NGI slip surface method, and the finite element method. Centrifuge tests may also be performed to investigate foundation stability. These methods are discussed in depth in the following sections. The problems encountered when applying them to offshore gravity structures are emphasized. In the following chapters, an alternative procedure based on the method of slices is presented. 6.4.1 Classical Bearing Capacity Approach The stability of shallow foundations is often investigated using bearing capacity theory. Computation of the ultimate load Q0, or ultimate bearing pressure q0, is based on a simplified model of an infinitely long rigid strip footing of width B0 resting in a homogeneous deposit of effective unit weight cohesion c, and friction angle 0, at a depth D0. The footing is loaded with a central vertical load Q which is assumed to produce a uniform pressure q. The adjacent soil is loaded with a uniform surcharge q'. This representation is shown in figure 6.4. 129 Qo=q0B0 Figure 6.4 - Theoretical rupture surface geometry 130 The bearing capacity problem is represented mathematically by a rather cumbersome set of partial differential equations. A closed form analytical solution has not yet been found, although for special cases of this problem, solutions are available (e.g. Prandtl, 1921). The most widely recognized solution is that of Terzaghi (1943). He proposed that the ultimate bearing capacity be evaluated from q0 = -„'BNY+ cNc+ q'N. (6.17) 2 * * where the N-values are known as bearing capacity factors. These coefficients arise from the plasticity solution. Since Nt and N% are calculated for one rupture surface and Ny for another, this equation is an approximation (Hansen, 1970). The location of the theoretical rupture surface is different for each combination of c, 0, and q'. The equation is, however, conservative and errors are generally less than 20% (Lundgren and Mortensen, 1953). The Terzaghi (1943) solution is generally accepted, but the numerical values of the bearing capacity factors to be used in the equation are not. The bearing capacity factors arise from the plasticity solution and depend only on the friction angle -for a particular shape of the assumed rupture surface. It is this dependence on the shape of the assumed rupture surface that gives rise to the many different interpretations of these factors. The variations in the Nc- and N^-values for different solutions may be on the order of a factor of two, and the 131 differences in the N^-values may be even more. This discrepancy is of particular interest for gravity structures founded on cohesionless soil where the frictional (first) term in Equation (6.17) is predominant. Values of Ny as a function of the friction angle have been compiled by Andersen (1972) from the published results of several authors. His findings are shown graphically in figure 6.5. It is clear that there is considerable variability in the value of Ny for a given value of the friction angle depending upon whose results are used. Equation (6.17) is an approximate theoretical solution for a; very idealized foundation. Real foundations are never infinite in length and loading is often inclined Or eccentric or both. For gravity structures, loading is always inclined and eccentric. Eccentricities of 10% and inclined load factors of 0.2 to 0.4 are common.10 Since an analytical solution is not possible except for the simplest of cases, empirical factors have been employed to improve results. To extend Terzaghi's (1943) solution to include the effects of the shearing resistance of the soil above the foundation base, inclined loading, and different foundation shapes, Equation (6.17) is rewritten as (Hansen, 1961): q0 = — V BNy sy dr iy + cNeScd^i,. + q'N^d^i^ (6.18) where the s-, d-, and i-parameters are empirical coefficients 10The inclined load factor, denoted by &, is the ratio of the horizontal to vertical load. 132 Figure 6.5 - Comparison of different proposals for the value of Ny (After Andersen, 1972) 133 which represent the effects of foundation shape, imbedment depth, and load inclination, respectively. These empirical coefficients were found using plate loading tests by varying one parameter (set) at a time and corrrelating the results to Equation (6.18). Curves were fitted to the data to determine approximate analytical expressions. Equation (6.18) is the general formula for the bearing capacity of a rigid horizontal footing resting in a homogeneous horizontal deposit and is the basis of the two most widely used bearing capacity theories, those of Meyerhof (1963) and Hansen (1970). The two theories differ in their estimation of these coefficients and the N-values. Hansen's (1970) formulation is the method preferred offshore. This is due to several factors. The most obvious one is that gravity structure technology developed in Europe where engineers used Hansen's (1961,1970) theory for bearing capacity problems. A thorough study of this method, for application to cohesionless deposits, was made prior to installation of the Ekofisk tank (Bjerrum, 1973). The conclusion drawn from this study was that Hansen's (1970) method will provide acceptable results, even for large load inclinations, when used for the purposes it was developed for. (i.e. for total stress analyses of homogeneous deposits.) Eccentric loading is not treated by using empirical coefficients. The effective area approach proposed by Meyerhof (1953) is used instead. A concentric load is applied to the footing on a reduced area. The dimensions of this centrally loaded "effective area" are then used in the general bearing 134 capacity equation, Equation (6.18). Overall moment equilibrium is satisfied when using this technique. The horizontal force is also assumed to act only over the effective area. This method of treating eccentric loads is found to be conservative (Hansen, 1970). For total stress analyses of clay foundations, substantial sliding resistance may be mobilized on that part of the foundation base outside of the effective area. This resistance will reduce the horizontal force acting on the effective area; that force used in the bearing capacity calculation. Since the horizontal force is critical in reducing the ultimate bearing capacity, consideration of any amount of resistance that may be mobilized outside of the effective area is extremely important if undue conservatism is to be avoided. Bearing capacity theory may be modified to take this into account (Lauritzsen and Schjetne, 1976). The horizontal load taken on the foundation base outside of the effective area is subtracted from the total load. This force may be found using the procedure outlined in the following section. The bearing capacity solutions mentioned above assumed that the soil was homogeneous, with constant values of the cohesion, friction angle, and effective unit weight. This is, of course, a great simplification, since real soil deposits are never homogeneous. Offshore deposits in particular tend to be nonhomogeneous. They are usually layered and often consist of clay interbedded with sand, or vice versa. This is due to the nature of the depositional environment. Meyerhof (1963) proposed that when the aforementioned soil properties vary 135 within the deposit, average values should be used. This is reasonable if the variations are small. Since bearing capacity theory is based on the assumption that critical shear zones develop within the soil mass, average values within the potential failure body should be used instead of average values along the potential failure surface (Lauritzsen and Schjetne, 1976). The evaluation of suitable soil parameters to use in the analysis is quite subjective since the location of the rupture surface is often unknown. This problem becomes more critical as load inclination increases and the foundation soils become less homogeneous. For layered foundations, further assumptions regarding the location of the shear zones may be made. This problem was addressed by Terzaghi and Peck (1948) who developed a crude but useful procedure for treating this type of problem. More recent solutions have been presented by Button (1953) for a two-layer cohesive deposit, by Meyerhof (1974) for the case of a sand layer over clay, and by Davis and Booker (1973) for the case where the undrained strength increases linearly with depth. The case where a soft layer is sandwiched between two stronger materials was investigated by Yamaguchi and Terashi (1971). Other approximate techniques for dealing with multiple layer systems have been presented by .Brown and Meyerhof (1969) and Reddy and Srinivasan (1967), among others. These solutions are extensions of Terzaghi's (1943) theory. The extension of bearing capacity theory to layered foundations is quite subjective and approximate.. Fully drained conditions are usually assumed for a bearing 136 capacity analysis of sand. This is not the case for a large offshore gravity structure on this type of deposit which is subjected to storm wave loading. To be at all useful, the bearing capacity formula should take into account the pore water pressures developed in the soil. 6.4.2 Other Bearing Capacity Formulations The first bearing capacity method formulated to consider wave induced pore water pressures was that developed by Hansen (Bjerrum, 1973). This method was used to assess the stability of the Ekofisk tank and was reported in detail at a later date by Hansen (1976). The theory was developed based on the assumuptions that the soil is homogeneous and that a rotational type of failure about a point "o" below the foundation base will occur (see figure 4.14). A rigid-plastic failure mechanism was assumed and the the pore water pressures induced by the dilatancy of the sand were included (Hansen, 1976). In general, the solution to this problem is very difficult to obtain because the rupture surface, which is found based on the effective stress distribution, must first be computed from an estimated pore water pressure distribution. The computed displacement field then gives a corrected distribution of pore water pressures. This leads to an extremely complicated iterative solution. Residual pore water pressures due to cyclic loading are not considered. This method is not easy to use and has no advantages over other effective stress bearing capacity solutions (e.g. the following method). Janbu et al (1976) developed a two-dimensional bearing 137 capacity solution for a weightless soil with zero pore water pressure. They extended this solution to include soil weight and excess pore water pressures in an approximate way; the force equilibrium equations derived for the Generalized Procedure of Slices (GPS) method (Janbu, 1973) were numerically integrated; the integration was performed over a bearing capacity rupture surface similar to the one shown in figure 6.6. The stress distribution on the assumed rupture surface is found by employing the GPS method. Both the vertical and horizontal loads are assumed to act on the effective foundation area. This treatment is adequate for cohesionless soils. Their result is expressed by a modified bearing capacity equation. This is T= r(o;+ a - uw)tan0 (6.19) o~v+ a = BNy + (q'+ a)N^- ukNw (6.20) where T is the average horizontal shear along the base, r is the relative degree of horizontal shear mobilization, tan0 is the mobilized frictional resistance, o~v is the average vertical soil reaction over the base, a is the attraction (c/tan0), ufc is the average pore water pressure along the base, and NH is a dimensionless bearing capacity factor. An iterative solution is required to determine the safety factor since the soil forces are expressed in terms of the degree of strength mobilization. Curves were developed for the bearing capacity factors which may be used to speed up the solution procedure. A pore water pressure distribution in the soil corresponding to the maximum wave is assumed based on changes in 138 Figure 6.6 - Geometry of rupture surface used for an effective stress bearing capacity solution 139 the principal stresses over one loading cycle. The maximum pore water pressure at any location is found from Equation (6.6). Cumulative pore water pressures from cyclic loading are incorporated into the analysis by using a simple pore water pressure generation model. This is given by where n is the number of bands in the design storm histogram, m is a dimensionless pore pressure parameter obtained from cyclic load tests, and N is the number of cycles at any stress level. Since the concern is with relatively small pore water pressure ratios, not liquefaction, this type of pore water pressure model is used. A pore water pressure model such as Seed et al's (1976) arcsine formula is unnecessary. The pore water pressures on the assumed failure surface must be found by using an iteration technique, since the stresses and pore pressure parameters depend on the degree of strength mobilization in the soil. Another bearing capacity formulation was developed by Murff and Miller (1977) to analyze the foundation stability of a gravity platform. They approximated the set of partial differential equations derived for classical bearing capacity plasticity solution. This set of partial differential equations is solved numerically and allows more complex boundary conditions to be specified. Hence, factors such as inclined loading and irregular base geometry may be dealt with directly, instead of by using empirical coefficients as other bearing capacity theories do. Results are comparable to classical Afo", - o-3) (6.21) 140 theory and are somewhat conservative. Since the solution is found numerically, soil properties may vary with depth. The shape of the failure surface is necessarily defined mathematically to solve the equations, and therefore, it is constrained to a functional representation such as a logarithmic spiral. This constraint on shape limits the usefulness of this method for layered foundations. 6.4.3 NGI Slip Surface Method A slip surface method was developed at the Norwegian Geotechnical Institute to investigate the stability of offshore gravity platforms founded on clay. Details of this method have been reported by Lauritzsen and Schjetne (1976) and Schjetne (1976). An alternative approach to the bearing capacity formulation was desired that was simple to use, reliable, and applicable to offshore gravity structures founded on layered deposits. The NGI slip surface method offers some distinct advantages over bearing capacity theory, namely: complex loading can be accomodated somewhat more easily, the horizontal force is applied on both the effective area and non-effective area so that undue conservatism is avoided, and layered foundations may be analyzed directly since the soil properties may vary along the potential failure surface. This method is based on an assumed failure mechanism with the geometrical model of the "sliding body" shown in figure 6.7. The body has a constant cross section over the platform length and is cut off by vertical planes at the sides. Figure 6.7 - Geometry of sliding body used by NGI Figure 6.8 _ Geometry of bearing failure surface used in the NGI slip surface method (After Lauritzsen and Schjetne, 1976) 142 The surface of the sliding body is broken up into four sections as shown in figure 6.8: an active section "ab", a flat section "be", an inclined section "cd", and a passive section "de". The inclined section "cd" is directly beneath the effective area. The resistance to sliding for each of these sections is evaluated under force equilibrium conditions. The factor of safety is found from overall horizontal force equilibrium. Only the magnitudes of forces are considered, not the distribution, since moment equilibrium is not applied to the sliding body. Inherent in this method, is the assumption that the shearing resistance at the soil-soil interfaces on the side areas will reduce the horizontal force acting on the base. This resistance is assumed to act horizontally. Hence, Equation (6.12) may be rewritten to include this resistance: HB, = PH + <PA or Pw) - Pp - P& - Pt (6.22) where Px is defined as P3 = 0.4(2cAs) (6.23Here, c is the average cohesion and As is one side area (shown in figure 6.8 as "cefd"). The coefficient is used to account for the fact that the postulated failure mechanism is not the correct one. If the soil did indeed fail, there would be significant differences between the assumed vertical plane surfaces on the foundation sides and the actual failure geometry. A plane strain analysis is essentially being modified to do a pseudo-three-dimensional one. The 0.4 value was chosen to make the factor of safety agree with Hansen's (1970) 143 formulation for a homogeneous deposit, which was modified to reduce the horizontal force acting on the effective area. The horizontal force taken along the flat or sliding section "be" of the foundation base is found from H„ = c(B0-B)L0 (6.24) This is a mobilized force with the factor (B0-B)L0 being nothing more than the area of the sliding surface. The horizontal force applied to the effective area may then be found from HE, = HftT - HST (6.25) The NGI slip surface method is relatively simple to use. To find the critical slip surface, the angle °c is incremented in steps and the minimum factor of safety is established. ' The analysis may be done by hand, although use of a small computer program will speed up the analysis considerably, especially when there are a number of layers. For layered foundations, the NGI slip surface method usually predicts a lower safety factor than the bearing capacity formulas. This safety factor should be used in preference to the bearing capacity result. The geometry of the slip surface is fixed, only the angle changes; the passive zone is always at 45° with respect to the horizontal. The slip surface is constrained to the shape shown in figure 6.8. The slip surface used in calculations really has a sharp corner where the passive wedge starts. This is not shown in the figure. For layered foundations, average soil properties along the slip surface are used. A thin seam of weak material will therefore be represented only by a slight decrease in the average cohesion computed on the potential failure surface. This will have a minimal effect on the computed factor 144 of safety and discretion must be used when interpreting results. The possibility of a deep sliding type of failure, such as that shown in figure 3.6(f), occurring, cannot be properly assessed. The NGI slip surface method may only be used for total stress analyses of clay foundations. A direct extension of this method to an effective stress analysis is not possible. Since the distribution of shear and normal stresses along a potential failure surface is not considered within any of the four shear zones, the inclusion of stress dependent frictional resistance and pore water pressures would be exceedingly crude. The method of slices could be used for extending the NGI slip surface approach to treat these types of problems. 6.4.4 Method of Slices This technique has been mentioned for offshore gravity structure stability analyses (Eide, 1974; H0eg, 1976; Lauritzsen and Schjetne, 1976; Young et al, 1975), although no compre hensive treatment has yet been reported. None of the available slice methods are directly applicable to offshore gravity structures in their present forms. This technique will be adapted so that it may be used for offshore stability analyses in the following chapter. 6.4.5 Finite Element Analyses The finite element method may be used to assess foundation stability (Broughton, 1975; Prev0st et al, 1981a; Vaughan et al, 1976; Zienkiewicz et al, 1979), although it is used primarily for displacement calculations. This is a powerful technique 145 that can easily deal with complex loading and variable soil properties. A stress-strain model is used, which is more realistic than the rigid, perfectly plastic representation used in the bearing capacity and limit equilibrium methods to better model true soil behaviour. The stress-strain models used in finite element analyses vary widely. The finite element method is basically an extension of matrix structural analysis techniques which solve the equations of equilibrium for a set of structural members. The soil is discretized into "elements" and the force-displacement equations are written for the set of soil elements. The nonlinear, anisotropic, elastoplastic, path-dependent stress-strain properties of the soil may be modelled in finite element analyses by using appropriate constitutive relations (Prev0st et al, 1981a). Cyclic loading is treated by using a quasi-static approach. The results of finite element analyses are very dependent on the assumed constitutive relations used as input. Because soil stiffness parameters are required to perform the analyses, finite element studies are generally done only after the detailed site investigation has been carried out. A high degree of uncertainty is always associated with the in-situ stiffness parameters measured for offshore deposits. In the finite element method, the applied loads are incremented to stepwise approximate the stress-strain curve. If loading is carried out far enough, some soil elements will reach stress levels high enough to "fail", that is, they can no longer support an increased load. Since they are confined by other elements that have not reached a critical failure stress level, 146 large displacements of the failed elements cannot occur; failure is localized. A progressive failure will occur as more elements fail under an increasing load. Eventually, no more load can be added without excessive displacements (and load transfer) within much of the soil mass; this corresponds to a total failure. The ultimate bearing capacity, or applied vertical load at failure, is well defined for dense sands and insensitive clays, but for loose sands and sensitive clays it is not (Vesic, 1975). Fortunately, offshore gravity structures are usually founded on the former type of deposits where finite element analyses can generally distinguish a total bearing failure. Excessive displacements may occur rather suddenly upon application of the critical load increment. The ratio of the failure load to the design load will define the load safety factor since the design soil strength is used in the analysis. Considerable experience is required to interpret results from finite element analyses. Prev0st et al (1981a) performed an extensive series of finite element analyses. They compared their results with centrifuge test data for a model footing on plastic silt (Prev0st et al, 1981b) where the load was increased monotonically to failure. Both two- and three-dimensional analyses were performed; the former assumed plane strain conditions, while the latter modelled the foundation as a circular footing with three-dimensional constitutive relations. The three-dimensional analysis was found to adequately predict displacements at the failure state and loads observed in the model test. The two-dimensional results were found to be consistent with the experimental data. The (exaggerated) 147 distorted meshes for both two- and three-dimensional analyses at similar load inclinations and eccentricities are shown in figure 6.9. Note that the displacement patterns are similar. Prev0st et al (1981a) concluded that although two-dimensional finite element studies cannot "provide exact quantitative information about the behaviour of the soil-structure system, they would still provide useful answers regarding relative magnitudes of loads and displacements." This would imply that the plane strain assumption for foundation analysis may be adequate in many cases. The effect of load eccentricity was studied using the two-dimensional model. Some results are shown in figure 6.10. It is clearly evident that the effective bearing area reduces with increasing eccentricity. In fact, the effective bearing area appears to be very nearly equal to the effective area defined by Equation (6.15) for plane strain loading. Two-dimensional finite element methods cannot be adapted to perform a pseudo-three-dimensional analysis like, the NGI slip surface method. That is, shearing resistance at the soil-soil interfaces on the sides of the potential failure body cannot be included in the element equilibrium equations. A complete three-dimensional analysis can model this, but is exceedingly expensive to perform. Since many soil parameters are stress dependent, iteration techniques must be used to achieve stress compatability. This requirement for many iterations with a large set of simultaneous equations means that use of a computer with a large memory is mandatory. It also leads to the high cost of running these 148 (a) Two-dimensional — It s y wilF _ ' ' ' / I I \ \ S _ . (b) Three-dimensional Figure 6.9 Comparison of two- and three-dimensional distorted finite element meshes for an inclined and eccentric load (After Prevost et al, 1981a) 149 Figure 6.10 Effect of load eccentricity on effective bearing area as evaluated using the finite element method (After Prevost et al, 1981a) 150 types of computer programs. For a multimillion dollar platform, this may be of little significance. However, for smaller structures, computer costs can be an important consideration. Many engineers are unwilling to base their decisions solely on finite element analyses. This may be due to a distrust of the constitutive relations used to model soil behaviour or the numerical techniques employed in solving the equations. A high degree of uncertainty associated with the estimation of stiffness parameters is also an important factor. 6.4.6 Model Tests Model testing for stability problems is done using centrifuge tests (Andersen et al, 1979; Heijnen, 1981; Prev0st et al, 1981b; Rowe, 1975; Rowe et al, 1976). A small model foundation is placed on a carefully constructed soil profile in a bucket which is then mounted on an arm connected to a central shaft and spun. The deadweight bearing pressure is derived from the resulting centrifugal forces. A horizontal load may be applied to the model by means of a jack or cyclic loads may be imposed on the model by using a displacement-controlled servo-hydraulic actuator (Andersen et al, 1979). For tests with cyclic loading, water can be put on the soil surface to apply a back pressure; this may be necessary to prevent cavitation which will not occur in the field under high hydrostatic pressures. Irregular platform geometry is easily accounted for in centrifuge tests since a structural model of any shape may be made. Centrifuge tests do not suffer from one of the major problems that other model tests do: the inability to simulate 151 high stresses resulting from gravitational loads in the prototype. These tests are an attempt to predict foundation behaviour without much of the subjectiveness of numerical techniques. They do require field and laboratory test data to define the in-situ soil properties. Variable soil properties are modelled by building a soil profile with different layers using soil from the test site to control factors such as particle size and fabric. Clays are remolded and consolidated to the specified overconsolidation ratio. The more nonuniform the soil profile is in-situ, the harder it is to model. Generally, a few layers at most are used. Not only do the soil profile and loading history have to be representative of the prototype, so do the pore water pressures. For cohesionless soils where substantial drainage may take place in the model, in-situ consolidation is modelled by scaling the time factor. This is usually done by using pore fluids much more viscous than water. If similarity requirements between the model and prototype are satisfied, then the various factors influencing the test do not have to be distinguished separately (Heijnen, 1981). For example, the stresses within the soil mass do not have to be determined, since they are not used in setting up the test. In numerical studies, stresses are computed using parameters obtained from laboratory shear tests. The stresses existing in the centrifuge model soil are similar to those in the prototype soil if the soil profiles and loading are the same. Centrifuge tests, like finite element studies, find the load safety factor and provide information on displacement and failure modes. 152 Centrifuge tests have been used to investigate the foundation stability of existing platforms (Rowe, 1975) and for theoretical studies (Andersen et al, 1979; Prev0st et al, 1981b), but they have not yet been used in design. This will likely change in the future as testing procedures improve (Heijnen, 1981). The major drawback of - centrifuge testing is that it is time consuming, expensive, and can be conducted at a limited number of facilities. 6.5 Summary A summary of the existing stability methods applicable to offshore gravity structures is given in Table VIII. It is quite evident that there are two distinct classes of analyses: the relatively simple bearing capacity formulations and the NGI slip surface method, and the more sophisticated analyses which consider more realistic stress-strain behaviour. There is presently no analytical alternative to the crude bearing capacity approach or the NGI slip surface method except the finite element method. A simple effective stress method is needed which can adequately treat both layered foundations and complex loading. In the following chapters, such a technique, based on the method of slices, is presented. Table VIII - Comparison of Existing Stability Methods METHOD ADVANTAGES DISADVANTAGES .. AR1NG CAPACITY THEORY (CLASSICAL) -SIMPLE TO USE -SUITABLE FOR HAND CALCULATIONS -EASY TO PERFORM PARAMETER STUDIES -DOES NOT CONSIDER STRESS-STRAIN BEHAVIOUR -BASED ON FIXED GEOMETRY OF RUPTURE SURFACE -CANNOT TREAT COMPLEX LOADING CONDITIONS -LIMITED TO TOTAL STRESS ANALYSES -NOT GOOD FOR LAYERED FOUNDATIONS -SUBJECTIVITY OF BEARING CAPACITY FACTORS NGI SLIP SURFACE METHOD -RELATIVELY EASY TO USE -SUITABLE FOR HAND CALCULATIONS -APPLICABLE TO LAYERED FOUNDATIONS -EASY TO PERFORM PARAMETER STUDIES -DOES NOT CONSIDER STRESS-STRAIN BEHAVIOUR -BASED ON FIXED GEOMETRY OF FAILURE SURFACE -ONLY FOR TOTAL STRESS ANALYSES OF CLAY -DOES NOT CONSIDER DISTRIBUTION OF LOADS -PROBLEMS WITH THIN SEAMS FINITE ELEMENT METHOD (TWO-DIMENSIONAL) -CONSIDERS STRESS-STRAIN BEHAVIOUR -CAN ACCOMODATE IRREGULAR GEOMETRY -APPLICABLE TO LAYERED FOUNDATIONS -CAN STUDY SOIL-STRUCTURE INTERACTION -POSSIBLE TO STUDY PROGRESSIVE FAILURE -PROVIDES INFORMATION ON FAILURE MODES -CANNOT STUDY THREE-DIMENSIONAL EFFECTS -REQUIRES MANY SOIL PARAMETERS AS INPUT -MUCH DATA PREPARATION REQUIRED -EXPENSIVE ANALYSES -REQUIRES THE USE OF A LARGE COMPUTER CENTRIFUGE MODEL TESTING -MODELS STRESS-STRAIN BEHAVIOUR -CAN ACCOMODATE IRREGULAR GEOMETRY -APPLICABLE TO LAYERED FOUNDATIONS -TRUE THREE-DIMENSIONAL ANALYSIS -PROVIDES INFORMATION ON FAILURE MODES -EXTENSIVE PREPARATION REQUIRED -REQUIRES SPECIALLY TRAINED PERSONNEL -REQUIRES SOIL FROM FIELD SITE -EXPENSIVE ANALYSES -LIMITED TO FACILITIES WITH CENTRIFUGES Ul 150 types of computer programs. For a multimillion dollar platform, this may be of little significance. However, for smaller structures, computer costs can be an important consideration. Many engineers are unwilling to base their decisions solely on finite element analyses. This may be due to a distrust of the constitutive relations used to model soil behaviour or the numerical techniques employed in solving the equations. A high degree of uncertainty associated with the estimation of stiffness parameters is also an important factor. 6.4.6 Model Tests Model testing for stability problems is done using centrifuge tests (Andersen et al, 1979; Heijnen, 1981; Prev0st et al, 1981b; Rowe, 1975; Rowe et al, 1976). A small model foundation is placed on a carefully constructed soil profile in a bucket which is then mounted on an arm connected to a central shaft and spun. The deadweight bearing pressure is derived from the resulting centrifugal forces. A horizontal load may be applied to the model by means of a jack or cyclic loads may be imposed on the model by using a displacement-controlled servo-hydraulic actuator (Andersen et al, 1979). For tests with cyclic loading, water can be put on the soil surface to apply a back pressure; this may be necessary to prevent cavitation which will not occur in the field under high hydrostatic pressures. Irregular platform geometry is easily accounted for in centrifuge tests since a structural model of any shape may be made. Centrifuge tests do not suffer from one of the major problems that other model tests do: the inability to simulate 151 high stresses resulting from gravitational loads in the prototype. These tests are an attempt to predict foundation behaviour without much of the subjectiveness of numerical techniques. They do require field and laboratory test data to define the in-situ soil properties. Variable soil properties are modelled by building a soil profile with different layers using soil from the test site to control factors such as particle size and fabric. Clays are remolded and consolidated to the specified overconsolidation ratio. The more nonuniform the soil profile is in-situ, the harder it is to model. Generally, a few layers at most are used. Not only do the soil profile and loading history have to be representative of the prototype, so do the pore water pressures. For cohesionless soils where substantial drainage may take place in the model, in-situ consolidation is modelled by scaling the time factor. This is usually done by using pore fluids much more viscous than water. If similarity requirements between the model and prototype are satisfied, then the various factors influencing the test do not have to be distinguished separately (Heijnen, 1981). For example, the stresses within the soil mass do not have to be determined, since they are not used in setting up the test. In numerical studies, stresses are computed using parameters obtained from laboratory shear tests. The stresses existing in the centrifuge model soil are similar to those in the prototype soil if the soil profiles and loading are the same. Centrifuge tests, like finite element studies, find the load safety factor and provide information on displacement and failure modes. 152 Centrifuge tests have been used to investigate the foundation stability of existing platforms (Rowe, 1975) and for theoretical studies (Andersen et al, 1979; Prev0st et al, 1981b), but they have not yet been used in design. This will likely change in the future as testing procedures improve (Heijnen, 1981). The major drawback of - centrifuge testing is that it is time consuming, expensive, and can be conducted at a limited number of facilities. 6.5 Summary A summary of the existing stability methods applicable to offshore gravity structures is given in Table VIII. It is quite evident that there are two distinct classes of analyses: the relatively simple bearing capacity formulations and the NGI slip surface method, and the more sophisticated analyses which consider more realistic stress-strain behaviour. There is presently no analytical alternative to the crude bearing capacity approach or the NGI slip surface method except the finite element method. A simple effective stress method is needed which can adequately treat both layered foundations and complex loading. In the following chapters, such a technique, based on the method of slices, is presented. Table VIII - Comparison of Existing Stability Methods METHOD ADVANTAGES DISADVANTAGES ....AR1NG CAPACITY THEORY (CLASSICAL) -SIMPLE TO USE -SUITABLE FOR HAND CALCULATIONS -EASY TO PERFORM PARAMETER STUDIES -DOES NOT CONSIDER STRESS-STRAIN BEHAVIOUR -BASED ON FIXED GEOMETRY OF RUPTURE SURFACE -CANNOT TREAT COMPLEX LOADING CONDITIONS -LIMITED TO TOTAL STRESS ANALYSES -NOT GOOD FOR LAYERED FOUNDATIONS -SUBJECTIVITY OF BEARING CAPACITY FACTORS NGI SLIP SURFACE METHOD -RELATIVELY EASY TO USE -SUITABLE FOR HAND CALCULATIONS -APPLICABLE TO LAYERED FOUNDATIONS -EASY TO PERFORM PARAMETER STUDIES -DOES NOT CONSIDER STRESS-STRAIN BEHAVIOUR -BASED ON FIXED GEOMETRY OF FAILURE SURFACE -ONLY FOR TOTAL STRESS ANALYSES OF CLAY -DDES NOT CONSIDER DISTRIBUTION OF LOADS -PROBLEMS WITH THIN SEAMS FINITE ELEMENT METHOD (TWO-DIMENSIONAL) -CONSIDERS STRESS-STRAIN BEHAVIOUR -CAN ACCOMODATE IRREGULAR GEOMETRY -APPLICABLE TO LAYERED FOUNDATIONS -CAN STUDY SOIL-STRUCTURE INTERACTION -POSSIBLE TO STUDY PROGRESSIVE FAILURE -PROVIDES INFORMATION ON FAILURE MODES -CANNOT STUDY THREE-DIMENSIONAL EFFECTS -REQUIRES MANY SOIL PARAMETERS AS INPUT -MUCH DATA PREPARATION REQUIRED -EXPENSIVE ANALYSES -REQUIRES THE USE OF A LARGE COMPUTER CENTRIFUGE MODEL TESTING -MODELS STRESS-STRAIN BEHAVIOUR -CAN ACCOMODATE IRREGULAR GEOMETRY -APPLICABLE TO LAYERED FOUNDATIONS -TRUE THREE-DIMENSIONAL ANALYSIS -PROVIDES INFORMATION ON FAILURE MODES -EXTENSIVE PREPARATION REQUIRED -REQUIRES SPECIALLY TRAINED PERSONNEL -REQUIRES SOIL FROM FIELD SITE -EXPENSIVE ANALYSES -LIMITED TO FACILITIES WITH CENTRIFUGES 154 CHAPTER 7 APPLICATION OF THE METHOD OF SLICES TO  OFFSHORE GRAVITY STRUCTURE FOUNDATIONS In this chapter the method of slices equations are modified so that they may be used to analyze the foundation stability of an offshore gravity structure subjected to storm wave loading. A pseudo-three-dimensional technique not unlike that used in the NGI slip surface method is included in the analysis. A pore water pressure model based on changes in the principal stresses is used to account for wave induced pore water pressures. Use of this model with the method of slices is described in the following chapter. The method developed herein is also useful for analyzing clay foundations for a deep sliding type of failure. In these cases bearing capacity theory and the NGI slip surface method are inapplicable. The method derived from Sarma's (1973) method of slices is not difficult to understand or complex to use and is of significant practical value. A typical representation of the problem by the method of slices is shown in figure 7.1. The failure model used is similar to the one used by NGI, that which was shown in, figure 6.7. The only difference is that the shear surface may take on a different shape (i.e. it is not constrained to shape shown in figure 6.8 - straight lines defining the inclined and passive sections.) It is convenient to label the two parts on the structure base; the flat section "ab" will be referred to as the sliding section, while the portion "be" will be termed the 154 CHAPTER 7 APPLICATION OF THE METHOD OF SLICES TO  OFFSHORE GRAVITY STRUCTURE FOUNDATIONS In this chapter the method of slices equations are modified so that they may be used to analyze the foundation stability of an offshore gravity structure subjected to storm wave loading. A pseudo-three-dimensional technique not unlike that used in the NGI slip surface method is included in the analysis. A pore water pressure model based on changes in the principal stresses is used to account for wave induced pore water pressures. Use of this model with the method of slices is described in the following chapter. The method developed herein is also useful for analyzing clay foundations for a deep sliding type of failure. In these cases bearing capacity theory and the NGI slip surface method are inapplicable. The method derived from Sarma's (1973) method of slices is not difficult to understand or complex to use and is of significant practical value. A typical representation of the problem by the method of slices is shown in figure 7.1. The failure model used is similar to the one used by NGI, that which was shown in figure 6.7. The only difference is that the shear surface may take on a different shape (i.e. it is not constrained to shape shown in figure 6.8 - straight lines defining the inclined and passive sections.) It is convenient to label the two parts on the structure base; the flat section "ab" will be referred to as the sliding section, while the portion "be" will be termed the Figure7.1 Representation of stability analysis by the method of slices 01 01 156 effective area. Two different procedures are developed. The first method is an adaptation of Janbu's (1973) Generalized Procedure of Slices. This method was chosen because it is applicable to slip surfaces of arbitrary shape and because it is familiar to most foundation engineers. Adaptation of this method to a pseudo-three-dimensional is not easily made due to the way in which the equations are derived. An alternative method based on Sarma's (1973) formulation is presented. This method is easily adapted to perform a pseudo-three-dimensional analysis. Sarma's (1973) method of slices is not well known among practicing engineers. At a first glance, the method may appear to be an awkward approach to the stability problem. However, a more thorough study will show that the method is in fact quite logical and extremely versatile. The slice equations are derived in general form, that is, they are independent of the assumptions regarding external loading. When specific problems are analyzed, further assumptions are made, regarding for instance: the load distribution on the foundation base, the pore water pressures in the soil, the distribution of soil parameters with both depth and horizontal position, and the shape of a potential failure surface. These factors will vary with the type of problem to be analyzed, and hence, versatility is maintained. 7.1 The Method of Slices The method of slices is a limit equilibrium analysis which treats the soil as a rigid plastic material. The degree of 157 safety against an ultimate foundation failure is expressed as F = T|/T (7.1) where F = the factor of safety, the shear strength along some shear surface, and ^ = the shear stress along the same shear surface. The purpose of a stability analysis is to find the minimum value of F corresponding to the most critical stability condition. The determination of the factor of safety against an ultimate foundation failure requires estimates of (1) the shear strength of the soil along the most critical shear surface, (2) the shear stress along this surface, and (3) the location of this surface. The following sections will concern themselves with finding the shear stresses on a shear surface and deriving expressions for the factor of safety. Since the shear strength and shear stress will vary along the shear surface, the soil is broken up into slices. The shear strength may vary from slice to slice as will the shear stress on the base of each slice. If the factor of safety is assumed to be constant along the entire shear surface, then Equation (7.1) will be a weighted average for the factor of safety, or F = - l^(a; ) (7.2) n L-i where the a-t's are weighting parameters that depend on slice geometry and loading and n is the number of slices. The individual ratios of shear strength to shear stress may be examined to see if the ratio anywhere exceeds unity. This is not valid since the shear stress cannot exceed the available strength. Local overstressing within the potential failure body 158 may be examined by comparing the shear strengths to shear stresses along the slice interfaces. Equations will be developed to check if the failure condition is violated within the potential failure body. The shear strength may be defined by the Mohr-Coulomb failure criteria, which in terms of effective stress is (Equation 6.2) *tj= c'+ C7'tan0' (7.3) Equation (7.1) essentially defines a state of limit equilibrium. This equation may be rearranged to define the average shear stress in terms of the shear strength. This is simply *t = ^(1/F) • (7.4) where 1/F is the degree of strength mobilized in the soil. This is constant for any shear surface. Introducing Equation (7.3) into (7.4) yields the mobilized or equilibrium shear stress on the shear surface for each slice. This is ~CL = C'+ 6Jtan0; (7.5) where c^= c'/F, and (7.6atan0i= tan0'-/F (7.6b) These are the mobilized strength parameters. 7.2 Loading Applied to the Foundation The vertical load at the seafloor is increased by the effective weight of soil within the skirt compartments when applied at the foundation base. This may be written as • Vtvr= Pv + APv + (BoDoLo)*' (7.7) The vertical load applied to the foundation base per unit width 159 as used in the method of slices is simply v* = V^/Lo (7.8) The horizontal load acting on the foundation base is the resultant of the applied environmental load PH and all the soil forces acting on the platform between the seafloor and skirt-tip level. This may be expressed as (Equation 6.12) HM= PH + (PA or Pw) - Pf - Ps (7.9) where PA , Pw, Pf , and Pt are defined by equations (6.9), (6.10), (6.11) and (6.13), respectively. These equations are summarized here for reference purposes. PA= [ (O.5^'Do2+Ap1Do)tan2(45'-0/2)-2cDotan(45,-0/2) ]L0 (7.10) Pw= Ap^oLo (7.11PP= [(0.5*'Do2+Ap2Do)tan2(45°+0/2)+2cDotan(45'+0/2)]L0 (7.12) Ps= 2D0B0(c + O.5-i-'Dotan0) (7.13The horizontal force per unit width applied to the foundation base is Hft = HaT/L0 (7.14) The moment applied at the foundation base is the resultant of the moment at the seafloor and the moments due to all the forces acting between the seafloor and the foundation base. This may be expressed as (Equation 6.14) MaT= M + PHD0 + (PA or Pw)h, - Pph2 - Psh3 (7.15) It is useful to use the concept of effective area when treating the distribution of the horizontal force on the foundation base. Since there is only single eccentricity on the equivalent foundation base, the effective width is all that is required. This is defined as (Equation 6.15) B = B0(1 - 2e) (7.16) 160 with the eccentricity being given by (Equation 6.16) e = (7.17) Bo 7.3 Treatment of the Applied Horizontal Force The horizontal force applied at the foundation base may be separated into two parts: that which acts on the sliding surface "ab" and that which acts over the effective area "be". This may be written simply as HBT= HST+ HET (7.18) where HS1 is the force taken by the sliding surface "ab", and Hnis the force taken by the effective area "be". Equation (7.18) may be written as HB = Hs + HE (7.19) per unit width. The horizontal force taken by the sliding surface per unit width is given by Hs = \(c + o"tan0)dx (7.20) The horizontal force taken by the effective area per unit width may be found by inserting Equation (7.20) into (7.19) and rearranging. This yields f h HE = Hft - \(c + O"tan0)dx (7.21) The horizontal force taken by any slice may be found from FHL= He(bt/B) (7.22where bt is the width of a slice. Combining equations (7.21) and (7.22) yields FH-t = [HB-\(c + o"'tan0)dx] (b:/B) (7.23) 161 7.4 Modified Janbu Method The application of Janbu's (1973) Generalized Procedure of Slices (GPS) method to the analysis of offshore gravity structure foundations requires some modifications. Basically, what is required is the inclusion of horizontal forces acting on the tops of the slices (everywhere in the equations) and the determination of the magnitudes of these forces. They are initially unknown since the force taken outside the effective area on the sliding surface "ab" is expressed in terms of the degree of strength mobilized in the soil. An iteration procedure must be used to establish the horizontal force applied to the effective area, and hence to the slices. The derivation presented below follows that given by Janbu (1973) as closely as is possible so that a comparison between the two sets of equations may be made and so that existing slope stability programs can be easily modified. 7.4.1 Assumptions Janbu's (1973) Generalized Procedure of Slices is founded on the following assumptions: A - Plane strain conditions apply. B - The position of the line of thrust for the normal interslice forces is assumed to be known. C - The normal force on the base is assumed to act where the the total resultant vertical force intersects the base. 7.4.2 Derivation of Equilibrium Equations The following equations include all the forces acting on the slice shown in figure 7.2. The geometrical variables and FV, * S.W.L. FH, Figure 7.2 - Geometry and forces on a (Janbu) slice 163 forces for any slice are defined in the figure. These equations were developed for the loading defined in sections 7.2 and 7.3 and for the model shown in figure 7.1. The equations for the vertical and horizontal equilibrium of a slice are FT-fc + AT- = N-cosoi + SLsinc,; (7.24) AEi - FHi = N^sinc; - S-cosc;, (7.25where FTL = FVt + W, (7.26) AT; = T(t+1)-T(i) (7.27AE-, = E(L + 1 )-E(L)„ (7.28) Moment equilibrium about the assumed point of application of N-(mid-base) yields E-Ay; " AE-Ah; + T; b; + FH-.h; = 6 (7.29) Note that T-'bi-and E^-Ay,; are couples and that terms of second order have been neglected. This equation may be rearranged to find the interslice shear force T-, namely, T- = -E'tan6c + Ah i (AE - /b ^ ) - h-^FH^b-J (7.30) where tan6; = Ayc/bt (7.31For overall vertical equilibrium, the total vertical resultant on the shear surface must be equal to the weight of the body plus the boundary loads applied to the soil mass. If • v-t is the total vertical resultant on the base of a slice, then E[Vt] = Z[FT;] (7.32) From equation (7.24) note that V-L = NjCOSo-t + Sjsina; = FTZ + AT; (7.33) Introducing this equation into (7.32) yields l[ATr] = 0 (7.34Overall horizontal equilibrium requires that the total 164 horizontal resultant on the shear surface be in equilibrium with the horizontal boundary forces. If is the total horizontal resultant on the base of a slice, then E[H-j = He (7.35) From equation (7.25) it may be written that H; = N-sinc; - S^OSc^ = AE z - FH-t (7.36) Introducing this equation into (7.35) yields I[AE-J = 0 (7.37noting from Equation (7.22) that E[FHr] = He 7.4.3 Working Formulas The complete set of basic equations which must be satisfied for each slice is Ti = c; + (crL-u-t )tan0,; (7.38) o-t = PPC + TTi - "Cjtanoi (7.39AE = FH; + (PP; +TT- Jb-^tano; - T; b: ( 1 +tan2 at ) (7.40) T; = -E-ttan6! + Ahc (dEc/dbc ) - h;(dFH£/dbj) (7.41) where is the shear stress, c; the cohesion, 6"; the normal stress, Ui the pore water pressure, and tan0j the frictional resistance on the base of a slice. Additionally, PP-,, = FT-t /b-t , and (7.42) TTt = AT; /b; (7.43Note that Equation (7.38) defines the state of limit equilibrium, that (7.39) is the equation for vertical equilibrium of a slice, and that (7.40) is one equation for both vertical and horizontal slice equilibrium. Moment equilibrium for a slice of infinitesimal width is defined by Equation (7.41 ). 165 The requirement for overall horizontal equilibrium, from Equations (7.19) and (7.37) may be written as I[AE;;] = H6 - Hs - Ht (7.44) This may also be written as E[AEr] = Hs + E[FHj] - He (7.45) Inserting Equation (7.40) into (7.45) and rearranging, yields l[FHl + (PPi +TTL )b;tancx ] - E[*t;bf ( 1+tan2ar) ] = HB- Hs- E[FH- ] (7.46) The maximum horizontal resistance available from the sliding surface per unit width may be expressed as Fs = \(c' + cr'tan0')dx = (HS)F (7.47) Introducing = ~C$:/F and (7.47) into (7.46) and solving for F yields E[ tr.b;( 1+tan2a;) ] + F. F = - (7.48) E[ (PP-t+TT-)brtano-t ] + HB Introducing Equation (7.39) into (7.38) gives a general expression for the shear strength, which is T4.= c'- + (PPr+TTc -u£-vX-tanor)tan0V (7.49) Introducing = ^;/F into the above expression and solving for T^; yields c ',- + (PP- +TT- -u-L) tan0 '-T4.= -^ i ' ' (7.50) 1 + (1/F)tan0'; tanoc The average factor of safety for the general case is found by using Equation (7.48) with T4;. defined by Equation (7.50). For simplicity, several abbreviated terms, are used to define ' the factor of safety. For each slice the following abbreviations will be used: 166 BZ = (PPj+TTC)bitana- (7.51) AX = T$:b.( 1+tan2o-t) (7.52By inserting Equations (7.51) and (7.52) into (7.48), the formula for the average factor of. safety is reduced to E[AX] + F. F = (7.53) E[Br] + Hs By introducing Equation (7.50) into (7.52) the Az term for each slice can be calculated in three steps as follows: A;' = [cL +(PPJ+TTJ-ur)tan0i ]b- (7.54) 1 + (1/F)tan0't- tano^ NA = (7.551 + tan2at A = A '/NA (7.56) The A and B values depend on the interslice shear force which in turn depends on the factor of safety. Hence, the need for an iteration technique arises. The stresses on the shear surface may be calculated as follows: T; = — = (7.57) F F-{(1+tan2o:)bc] and V; = PP: + TTt - Tttanac . (7.58) in accordance with Equation (7.39). The interslice forces may be found as follows: Introduce Equations (7.51) and (7.52) into (7.40) to find AEi = FHj.+ B : - Ar/F (7.59) Summing the AE values for each slice gives rise to EC = E[AE:] (7.60The vertical shear force T-t, is given by (Equation 7.41) 167 Tt = -E-ttan6-t + Ah-t (dE^/dbj) - h- (dFH[/db\) (7.61) All of the preceding equations must be satisfied simultaneously by an iteration procedure. The average factor of safety on a slice interface is found from c'jh; + (E;-UHc)tan0'c = (7.62) Tt where UH-t is the water pressure force on the slice interface in question. For a theoretically correct solution, F'~ must be greater than F. Note that average soil properties are used here. 7.5 Modified Sarma Method Sarma's (1973) approach to deriving a slice method differs substantially from the other slice methods applicable to slip surfaces of arbitrary shape: those of Janbu (1973), and Morgenstern and Price (1965). These methods solve the slice equilibrium equations and find the factor of safety by changing the value of the forces on the shear surface until slice equilibrium is satisfied. Since these forces are expressed in terms of the factor of safety, which is initially unknown, an iteration procedure is required. Sarma (1973) also solves the slice equilibrium equations, but in a different way. A destabilizing force is introduced into the equilibrium equations for each slice. This force is equal to the product of the acceleration coefficient K and the slice weight - a pseudo-earthquake force. A factor of safety is assumed and the forces which depend on the degree of strength 168 mobilization in the soil are expressed in terms of the limit equilibrium parameters. The analysis is performed for several values of the factor of safety. Since the solution requires no numerical iterations for the factor of safety, only distinct calculations, there is no possibility of numerical instability. This factor can be invaluable at times when the other slice methods provide no reasonable answer. The solution for any particular shear surface is found when the acceleration coefficient is equal to zero for some value of the factor of safety. A curve of F vs. K may be drawn as shown in figure 7.3. The "static" factor of safety may then be picked off the curve. In the case of an earthquake stability analysis, the solution is found when the acceleration coefficient is equal to some specified value. A distribution of the destabilizing force within the soil mass may be assumed for this type of work. Sarma's (1973) approach has been adopted for offshore stability analyses at the University of British Columbia. Finn and Lee (1978) modified this method to analyze the stability^ of underwater slopes subjected to seismic loading. A pore water pressure generation model was included in the analysis and the number of cycles to failure was found for undrained loading. Further work in this area is presently (1982) being done. Sarma's (1973) method is derived in such a way that many modifications may easily be made. A pseudo-three-dimensional analysis may be made by including the forces acting on the side areas (see figure 6.2(b)) in the slice equilibrium equations. 169 * 0-7 8 •12 to * 0 Si*of/c factor of safety 2 Factor of safety , F Figure 7.3 - Curve used for evaluating the safety factor (After Finn and Lee, 1978) 170 7.5.1 Assumptions The fundamental assumptions on which this slice method is based are: A - Plane strain conditions apply. B - The point of application of the normal force acting on the base of each slice is assumed to be known. C - The relative magnitudes of the interslice shear forces are assumed to be known. Additionally, for a pseudo-three-dimensional analysis D - The magnitudes and positions of the side forces on each slice are assumed to be known. 7.5.2 Derivation of Equilibrium Equations The following equations include all the forces acting on the slice shown in figures 7.4 and 7.5. The geometrical variables and forces for any slice are defined in figure 7.4. The forces acting on the side areas are included in the slice equilibrium equations as mobilized forces acting parallel to the base. For any slice this force may be defined as SS-L = [2(bE)(ytl-yb:)(c + cr'tan0)]/Lo (7.63) Note that this force is expressed in terms of its equivalent magnitude per unit length. The effective normal stress may be computed from earth pressure theory. For vertical and horizontal equilibrium of a slice NjCOSG;: + StSinc-t = W; - AT; + FN-COS0; + FSjsinp; - SScsino; S;;COScc - Njisino; = KWC + AEX - FN;sin0r + FS;COSPj - SScCOSO; The limiting equilibrium condition may be defined by (7.64) (7.65) 171 Figure 7.5 - Typical (Sarma) slice showing side forces 172 Sz = (N'c tan0'- + c'-t b-tsecc; )/F (7.66) This equation may be written as Sc = N'£ tan0-t + Cibisecai (7.67) by using the mobilized strength parameters. The effective normal force can be calculated as N'-t = N- - US: (7.68) where US-L is the force due to pore water pressure on the base of any slice due to pore water pressure. This is simply USt = Uxbcseccj (7.69) Equations (7.64) and (7.65) can be used to solve for N-t and Tj, namely, N; = (Wt-AT:)coscc " (KW;+AEC)sinoj + FNfcos(«i-pt) - FS;sin(o;-»K ) (7.70) S-t = (W;-AT;;) sina; + (KW t + AE r ) c OS a : + FN-tsin(o-t-pj ) + FSrcos(o£-0r ) - SS: (7.71) These equations together with (7.68) can be substituted into (7.67) to obtain AT-tan(0-t-oj ) + AE- = BBC - KW- (7.72) where BB;; = (W£-USj )tan(0--a j) + [cr b£ secat-cos0i+FNcsin(0--a.+pI ) -FS;COs(0j-o;+*j )+SSjCOS0£ ]sec(0£-a-) (7.73) Equation (7.72) is simplified by summing both sides of the equation. Note that E[AEt] = E(n+1) - E(1) = 0 (7.74) since both E(n+1) and E(1) equal zero. Hence, [ (ATt)tan(0;-o-t) ] = E[BB-J - E[KWC] (7.75) Equation (7.75) is therefore one equation which satisfies both vertical and horizontal equilibrium for all slices. A second 173 equation is obtained if moment equilibrium of the whole sliding body about the origin is considered. [ (Nj;Coso;+Sjsincc)xbc ] - E[ (-N£ sinctj+S tcoso- )ybt- ] - E[(W:)xg:] + E[(KWt-)yg-J - E[FNE (xtccos*c +ytt- sin*. ) ] - E[FS; (xt: sinp£ -yt-cos$r) ] + E[SSj (xst sincc -ys-tcosa£) ] = 0 (7.76) This equation may be combined with Equations (7.64), (7.65), and (7.72) to eliminate N-t, Sj , and AEj, resulting in I [Wftxbj-xgj)] + E[KWrygT] - E[BB;ybt-] + E [ FNj sin*- (ybc -yt£ ) ] + EtSSiCOSo^yb.-ys^) ] - E[FSt-cos<»t (yb£-yt- ) ] = [ (ATj; {xbi-ybctan(0i-oi;)} ] (7.77) 7.5.3 Working Formulas The distribution of the interslice shear forces is assumed so that Tc = X-Qj or AT; = X.-DQj; (7.78) The Q-t values may be found as follows: Q. = [(Et-UHt;)tan0,£ +c'ch^]f(x) (7.79) where f(x) is a distribution parameter. This may be taken as unity since it gives acceptable results (Sarma, 1973). E^ may be expressed as Et = K;(0.5 ht2 + QSjh-) (7.80) where QS; is the average surcharge above any slice interface, is given by Sarma (1973) as 1 - sin(2ol-0'f )[ (1-2Rtt)sin0,'u/ h-)cos0V ] K-t = (7.81) 1 + sin(2o;-0'^ )sin0'; R* is the ratio of excess pore water pressure to vertical stress and is taken as an average value. The strength parameters may 174 also be taken as average values. Sarma (1973) discusses in detail how to obtain the Q- values. Upon substituting Equation (7.78) into (7.75) and (7.77) the following equations are obtained-S1 • X + S2-K = S3 (7 .82) S4- X - S5-K = S6 (7 .83) jre S1 thru S6 are defined as follows: SI = E[DQttan(0i-o'l)] (7 .84) S2 = E[W-J (7 .85) S3 = I[BB t] (7 .86) S4 = E[DQt{xbi;-yb:tan(0t-c {)} ] (7 .87) S5 = l[Wiybf] (7 .88) S6 = E[FN;(yb£-ytc )sin* ,] - E[FSt- (yb--ytc Jcos*;; ] + E[SS;(ybt-ys-t)cosac] - EtBB-.yg;;] (7 .89) The interslice forces and the forces acting on the base can be found using the equilibrium equations derived in the preceding section once K and > are known. The average factor of safety on a slice interface is found from c'c hj + (Ei-UHc )tan0V F\ = (7.90) CHAPTER 8 EXAMPLES AND APPPLICATION OF ANALYSES 175 8.1 Description of Computer Procedure A computer program GRAVSTAB was developed to perform the analyses described in the preceding chapter. A thorough description of the program may be found in the program documentation which is available through the Soil Dynamics Group at the University of British Columbia. A brief description of the program will be given here. The routine for the modified Sarma analysis was taken from the program STESL (Lee and Finn, 1978) and modified for a gravity structure stability analysis. The routine used to perform a modified Janbu analysis was written by the author; the equations derived in the preceding chapter were used. This routine (and the Sarma routine as well) was tested with numerous example problems, including those given in the papers by Janbu (1973) and Finn and Lee (1978) to eliminate any programming errors. The two slice methods agree very closely for slope analyses and for foundation analyses of cohesive deposits. This is not the case for foundation analyses of cohesionless deposits. It was found that the stresses computed at the base of each slice from Janbu's (1973) method did not form a smooth curve over the slip surface. There is some instability associated with this. In fact, Janbu's (1973) method often did not provide an answer for these problems. The solution was very unstable and divergent. 176 The loads used in an analysis are found from the equations in sections 7.2 and 7.3. These loads, which are expressed in terms of the mobilized strength, are found iteratively. As a first approximation the effective foundation width is assumed to be constant in any analysis. Although the effective width is initially unknown, it may be estimated by assuming a reasonable factor of safety which when used in the appropriate equations yields a numerical value for the effective width. The variation of the effective foundation width with the safety factor is minimal and in most cases may be ignored. The soil data may be input in one of two ways: layer by layer, or slice by slice. The former method requires less data input and lends itself well to investigating different potential slip surfaces in multi-layered deposits since the strength parameters will be automatically calculated for each slice every time a new slip surface is chosen. To aid in finding the critical shear surface, a simple rerun control option may be used (as many times as desired). The slice coordinates on the shear surface (between the two end coordinates "b" and "d" shown in figure 7.1) may be incremented by a given percentage of their current values to vary the position of the slip surface. Hence, the depth coordinates for any shape of shear surface need to be input only once since they may be moved up and down between the two end values to locate the critical position of the shear surface for that particular shape. The critical shear surface may thus be found approximately with a minimum amount of effort since only a few shapes (that is, shear surface coordinate sets) 177 need to be specified. Pore water pressures may be taken as being hydrostatic or they may be input individually for each slice. The latter method is used for an effective stress analysis; only the excess pore water pressures need to be input. For a three-dimensional analysis, the pore water pressures on the sides of the slices are also required. It is often adequate to use the hydrostatic pore water pressures here since this type of analysis is quite approximate. The stresses computed on the base of each slice are standard output for GRAVSTAB. They may be examined to see if overstressing occurs anywhere. These stresses may also be used in an effective stress analysis to estimate the instantaneous pore water pressures assuming that the pore water pressure distribution due to the short-term (i.e. undrained) wave loading may be found based on Equation (6.6). This is demonstrated for Example 2. 8.2 Example 1 - A Multi-layered Cohesive Deposit The first example problem to be analyzed is a CONDEEP type structure founded on a nonhomogeneous cohesive deposit. This problem was taken from Lauritzsen and Schjetne (1976) and represents, an • actual offshore platform in the North Sea. The required geometry and loading data is given in Table IX. The actual platform base is nearly circular with a diameter of approximately 100 meters. The shear strength profile is shown in figure 8.1. One profile is that given by Lauritzsen and Schjetne (1976). The other is an approximation of this profile Table IX Geometry and Loading Data for Example 4 Equivalent Platform Length, L0 68.3 m Equivalent Platform Width, B0 68.3 m Effective Foundation Depth, D0 3.5 m Total Vertical Load, Pv + APW 187,000 t Horizontal Wave Load, PH 49,100 t Moment at Seafloor, M 2,240,000 t-m Dynamic Wave Pressure, Ap, 3.5 t/mJ Dynamic Wave Pressure, Ap2 -3.5 t/m* Unit Weight of Soil, V 2.0 t/m3 Figure 8.1 - Shear strength profile for Example 1 179 using layers with different undrained strengths. The latter profile was used for the slice analyses discussed below. The procedure used to search for the most critical shear surface is as follows: A first estimate of the critical shear surface is made using the NGI slip surface method. A computer program SLIPSURF was written for this purpose. The critical slip surface found in this way is then rounded off at the sharp corner and the shape is altered several times. As described before, each shear surface shape which is input is moved up and down to find the minimum value of the factor of safety. By using just a few different geometries for the shear surface, the general shape of the critical shear surface may be estimated. More iterations based on information provided by the early runs may be done if greater accuracy is desired. The critical shear surfaces evaluated by a number of methods are shown in figure 8.2. The corresponding safety factors are given in Table X. The angle which the bearing capacity rupture surface makes with the horizontal may be determined approximately (Lauritzsen and Schjetne, 1976). For this example problem the safety factors determined by the NGI slip surface method and the procedures developed in this thesis agree quite well. The slice methods predict a slightly lower factor of safety. This is in part due to the position of the critical slip surface determined by the two methods; the critical slip surface determined by the method of slices passes through more of the weak zone. For most total stress analyses of cohesive foundations, the NGI slip method is adequate. However, if a thin layer of very weak material exists within the strata, then the NGI method is of 180 Table X Comparison of Computed Safety Factors for Example 1 CALCULATED SAFETY FACTOR METHOD OF ANALYSIS Plane Strain B/L = 0 Actual Case B/L = 1 Hansen's (1970) Formula, modified 2 . 1 5 1 2.351 Meyerhof's (1963) Formula, modified 2. 17 1 2.491 NGI Slip Surface Method 2.001 2 . 1 5 1 Modified Janbu Method of Slices 1 .92 -Modified Sarma Method of Slices 1 .93 2.06 'From Lauritzsen and Schjetne (1976) Method of SI Ices NGI Slip Surface Method Hansen Bearing Capacity Theory Figure 8.2 - Critical shear surfaces for Example 1 as evaluated by different stability methods 181 little use in assessing the platform stability. It is also important to note that the increase in the safety factor for the pseudo-three-dimensional analyses is nearly identical for both the NGI method and the procedure developed in this thesis based on Sarma's method. In the foregoing analyses, the undrained strength was assumed to be constant with horizontal position. Since the stress conditions will vary considerably along a potential failure surface, the use of laboratory shear test results which reflect the different states of stress may be appropriate. There are four distinct states of stress existing within the soil mass. These are shown in figure 8.3. The undrained shear strength may be found by using the stress path method (Lambe, 1967). Laboratory samples are subjected to the estimated in-situ and total stresses that they will be subjected to in the field. The resulting undrained shear strengths determined in this way may then be used directly in a stability analysis. Since there is often a lack or absence of good quality samples, all the tests shown in figure 8.3 may not always be performed. In these cases, the undrained strength estimated from in-situ tests may be related to the standard or triaxial compression value. The other shear strengths may be taken as various percentages of this value. Some possible values for these coefficients are given in Table XI. The shear strength profile must then be represented by both depth and horizontal variations. Assuming that a layered profile may be used, the strength variation with horizontal position may easily be incorporated into the numerical technique 182 Table XI Coefficients for Estimating Undrained Strength from Triaxial Compression Data SHEAR TEST x C,, Direct Shear Triaxial Extension Triaxial Compression 0.75 0.50 1 .00 1 \ 11 ACTIVE - ZONE -T PASSIVE | TRIAXIAL f -Figure 8.3 - Zones of shear on the potential failure surface and relevant laboratory tests (Adapted from Kjekstad and Lunne, 1979) 183 by using the triaxial compression values multiplied by the appropriate coefficients in the various shear zones. Another stability analysis was performed using these new values for the undrained strength. The critical shear surface found using this new profile was almost identical to the one found for the original profile. Results of this study are reported in Table XII. The use of the shear zone concept for specifying the strength along the failure surface is a matter which has not yet been resolved. It is important to note that reducing the safety factor by such a substantial amount has a tremendous effect on the cost of the platform since the base size must be increased to reduce the average bearing pressure. This cost increase may be on the order of many millions of dollars. The critical shear surface found for the aforementioned analyses is shown in figure 8.4. This type of plot is standard output of GRAVSTAB and is useful for examining the location of this surface with respect to different strata. (It is also useful for locating this surface.) Note the length of that part of the surface which runs through the weaker layers. 8.3 Example 2 - A Cohesionless Deposit; Ekofisk Tank The second example problem to be analyzed is the Ekofisk tank. The required geometry and loading data is given in Table XIII. The distribution of residual pore water pressures due to cyclic loading is given in figure 8.5. This distribution was developed based on observations reported by Clausen et al (1975); these pore water pressures were shown in figure 4.11. The contours drawn on the figure are based on Rahman et al's Table XII Effect of Shear Zone Representation on the Safety Factor STRENGTH PROFILE USED COMPUTED SAFETY FACTOR Regular Strengths Shear Zone Strengths 1 .93 1 .62 WflHPLf 1 - COUDfFP STRUCTURE 03.30 Q N. FEB. 02. 1062 2-0 ANALYSIS »PLANE STRAIN) PLATFORN BRSE I 0 Figure 8.4 - Critical shear surface for Example 1 found from computer program GRAVSTAB Table XIII Geometry and Loading Data for Example 2 Equivalent Platform Length, L0 85.8 m Equivalent Platform Width, B0 85.8 m Effective Foundation Depth, D0 0.4 m Bouyant Platform Weight, Pv 190,000 t Vertical Wave Load, APW 10,000 t Horizontal Wave Load, PH 78,600 t Moment at Seafloor, M 2,800,000 t-m Dynamic Wave Pressure, Ap,« 3.0 t/m2 Dynamic Wave Pressure, Ap2 -3.0 t/m2 Unit Weight of Soil, K 2.0 t/m» Figure 8.5 - Distribution of pore water pressures in foundation soil used in Example 2 186 (1977) work. The most difficult part of this analysis is estimating the instantaneous pore water pressures due to the cycling loads. These were found based on Equation (6.6), which is AU = A0"3 + A (ACT, - ACT3) (9.1) The method of slices may be used to find the changes in the principal stresses. The procedure used to determine the principal stresses is as follows: A slip surface is analyzed and the factor of safety corresponding to Au=0 is found. The same slip surface is then analyzed without the wave loads, i.e. the vertical platform load only. This load is now assumed to act over the total foundation area. The "no load" factor of safety is then established. The stresses at the base of each slice are then determined from the equilibrium equations derived in the previous chapter. This procedure has been set up in GRAVSTAB. Only the A-parameter is required as input for any slip surface. The principal stresses and pore water pressure changes for the instantaneous wave loads are then computed and the new safety factor is determined. The principal stresses may be evaluated from cr, = cr+ [(1 - cos2*)/sin<*] (9.2) o3 = 6- [(1 + cos2*)/sin*] (9.3where * is defined by ot = 45°+ 0/2 (9.4) Here 0 is the mobilized friction angle. This will be different for each case. For the no-load analysis this value will be very low. The critical slip surface found for the tank is shown in 187 figure 8.6. Only the residual pore water pressures due to cyclic loading were used and pore water pressures associated with total stress increments were neglected (i.e. Au=0). This is felt to be conservative since the dense Ekofisk sand would probably dilate and create negative pore water pressures in much of the soil mass during loading. The value of the A-parameter has a marked effect on the computed factor of safety. For the slip surface shown in the figure, four values of the A-parameter were chosen between 0.0 and -0.33. The latter value corresponds to a dense sand at failure. The safety factors computed with these different values are shown in Table IX. For this problem, the increase in the safety factor due to added resistance from the side areas was minimal. For sand foundations with shallow failure mechanisms this will generally be the case. For deeper failure mechanisms which occur in cohesive foundations, the added resistance can increase the safety factor substantially. This was shown in the total stress analysis in Example 1 and would also show up in an effective stress analysis of the same foundation. 188 Table XIV Effect of A-parameter on the Safety Factor VALUE OF A-PARAMETER COMPUTED SAFETY FACTOR -0.33 1.77 -0.20 1.61 Au-0 1.51 -0.10 1.49 0.00 1.37 i DMU 1 - «C •na PA. m m « urn nr.. 'auJJJJJJJllV '1* i. i. Ah.k.^HcjM.^ iu JM iu Figure 8.6 - Critical shear surface for Example 2 found from computer program GRAVSTAB 189 CHAPTER 9 SUMMARY AND CONCLUSIONS Offshore gravity structures have a bright future in offshore development schemes. They offer the advantage of being nearly complete at tow-out, thereby minimizing installation time. This is particularly important in hostile environments such as the northern North Sea where conventional steel jacketed structures present a substantial risk in terms of short-term safety. It also allows for an earlier production start. By nature of their design, these structures incorporate storage and provide a large deck area which is necessary for production equipment. This is a significant advantage for marginal field recovery or for locations offshore where pipelines are not economically justified. Offshore engineering presents many challenges for the geotechnical specialist. Although offshore engineering has existed for many years, only recently has hydrocarbon exploration moved into deeper waters where the design of major structures has consistently required an advancement of the state of the art in not only geotechnical engineering but also in structural, hydrodynamic, and oceanographical engineering. The development of new techniques and design philosophies in all of these fields requires that the geotechnical engineer be familiar with them since he must play a key role in gravity structure design. Instrumentation has :' provided some useful data for 190 predicting the magnitudes of settlements expected for a large gravity structure. However, the availability of this data is limited and most engineers will have to design future structures based on what is "publically known". Published data for the Ekofisk tank demonstrates that platform settlement is difficult to assess. More recent data is available for other platforms which confirms this result. Since settlement due to cyclic effects is an important consideration, some assessment must be made. Presently, the evaluation of cyclic settlements for offshore gravity type structures is made using very simplistic models. Cyclic settlement analysis for offshore gravity type structures is an area which requires a considerable amount of research. Cooperation among those individuals and corporations involved in this research would greatly accelerate progress in this area. Stability is the other principal problem that geotechnical engineers must deal with. This is a complex problem which must be viewed in perspective. The procedures discussed and developed within this thesis may only be used if certain criteria are met. For example, full contact is usually assumed; this critical assumption must be verified by instrument data. Instrumentation is used to provide information on the stress distribution over the slab and also the distribution of pore water pressures within the soil mass. The evaluation of residual pore water pressures due to cyclic loading by theoretical methods is by no means reliable. In fact, substantial discrepancies exist between theoretical estimates and on-site observations. 191 Estimation of the stability of a gravity type structure is presently assessed by using bearing capacity theory, the finite element method, or the NGI slip surface method for clay foundations. Bearing capacity theory is of limited value for use in offshore analyses primarily because of the inability to (1) treat adequately complex loading, (2) analyze layered foundations, and (3) perform effective stress analyses. Extensions of classical theory have been proposed to deal with the latter problem, however, these theories are of limited value. The NGI slip surface method is an improvement over bearing capacity theory, however, this method is only applicable to total stress analyses of cohesive deposits. The finite element method may also be used to assess platform stability. The most serious problems associated with this method are that the soil stiffness parameters need to be known accurately and the assumed constitutive relations be reasonable for the soil analyzed. Both of these factors must be considered in the light of the reliability of offshore soil investigations. An alternative approach to the bearing capacity and finite element methods was presented. This procedure is based on the method of slices. Two methods were derived. Janbu's (1973) method, which is well known to most foundation engineers, was extended to treat gravity structure problems in two dimensions. A pseudo-three-dimensional analysis procedure was not developed within this framework. An alternative technique, based on Sarma's (1973) method of slices was developed. Three-dimensional effects were easily incorporated into Sarma's (1973) 192 method. This method is also numerically stable which makes a solution possible for any problem by using the method of slices. The analysis procedures developed in this thesis were applied to two example problems - one on a layered cohesive deposit and one on cohesionless soil. The first example illustrated how a total stress analysis could be performed. Results were compared with existing methods and shown to be good. The pseudo-three-dimensional analysis suggests that the safety factor can be increased by about six percent due to the added resistance from the side areas. This result agrees well with the NGI slip surface method. The NGI method provides reasonable results for cohesive foundations without thin weak seams. For the problem analyzed, the method of slices is shown to give a lower factor of safety, although not substantially lower for this profile. For other profiles the difference may be greater. The values of the undrained strength chosen along the slip surface have a pronounced effect on the computed safety factor. The use of the shear zone concept reduces the safety factor considerably. An effective stress analysis was also performed. The critical part of this analysis was the determination of the instantaneous pore water pressures due to the wave loads. The method of slices was used to predict these pore water pressures based on changes in the principal stresses. This procedure was incorporated' into the computer program GRAVSTAB. The value of the A-parameter has a significant effect on the pore water pressures developed and hence the factor of safety computed. For a dense sand, such as Ekofisk sand, where dilation will 193 occur, the stability will increase since the pore water pressures on the slip surface will be less. The procedures developed in this thesis were shown to provide reasonable answers to offshore gravity structure stability problems. The procedure based on Sarma's (1973) method of slices is preferred since it can provide information on three-dimensional effects and does not suffer from numerical instability. This method, which is easy to use and applicable to a wide range of problems, is suggested as a means of estimating offshore gravity structure stability under storm wave loading. $SIGNOFF 194 REFERENCES 0 1. Agostoni, Alberto; Di Telia, Vincenzo; Guone, Enzo; and Sebastiani, Gaetano. (1980): "TSG - Integrated Storage Platform for Early Production in the North Sea." Twelfth Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.245-260. 2. Airy, G. B. (1845): "On Tides and Waves." Encyclopedia  Metropolitana, London, pp.241-396. 3. American Petroleum Institute. (1978): API Recommended. Practice for Planning, Designing and Constructing Fixed  Offshore Structures, API/RP2A; 9th Ed., Dallas. 4. Andersen, K. H. (1972): "Bearing Capacity of Shallow Foundations on Cohesionless Soils." Norwegian Geotechnical Institute, Internal Report 51404-1. 5. Anderson, Knut H. (1976): "Behavior of Clay Subjected to Undrained Cyclic Loading." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.392-403. 6. Andersen, K. H.; Brown, S. F.; Foss, I.; Pool, J. H.; and Rosenbrand, W. F. (1976): "Effect of Cyclic Loading On Clay Behavior." Norwegian Geotechnical Institute, No.113, pp.1-6. 7. Andersen, K. H.; Seines, P. B.; Rowe, P. W.; and Craig, W. H. (1979): "Prediction and Observation of a Model Gravity Platform on Drammen Clay." Second International Conference on the Behavior of Offshore Structures, London, Proceedings, Vol.1, pp.427-446. 8. Augustine, F. E., Maxwell, F. D., and Lazanoff, S. M. (1978): "Extreme Wave Heights in the Gulf of Alaska." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.1551 -1562. 9. Bea, R. G., and Akky, M. R. (1979): "Seismic, Oceanogra-phic, and Reliability Considerations in Offshore Platform Design." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2251-2262. 10. Bea, R. G., and Lai, N. W. (1978): "Hydrodynamic Loading on Offshore Platforms." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.155-168. 11. Bell, W. E. (1974): "The Equipment Requirements for Oil and Gas in the North Sea." Offshore Europe, 2nd Ed., Bucks, England: Scientific Surveys (Offshore) Ltd., pp.27-36. 195 12. Bercha, F. G., and Stenning, D. G. (1979): "Arctic Offshore Deepwater Ice-Structure Interaction." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2377-2386. 13. Berman, M. Y., Blenkarn, K. A., and Dixon, D. A. (1978): "The Vertically Moored Platform, for Deepwater Drilling and Production." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.55-64. 14. Billington, C. J. (1979): "The Underwater Repair of Concrete Offshore Structures." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.927-938. 15. Bjerrum, L. (1973): "Geotechnical Problems Involved in Foundations of Structures in the North Sea." Geotechnique, Vol.23, No.3, pp.319-358. 16. Braun, Willi M. (1974): "Ekofisk Settlements and the Steady Sealab." Ground Engineering, Vol.7, No.4, pp.47-49. 17. Broughton, Peter. (1975): "Offshore Gravity Based Oil Production Platform Interaction with the Sea Bed." Seventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.387-398. 18. Brown, J. D., and Meyerhof, G. G. (1969): "Experimental Study of Bearing Capacity in Layered Clays." Seventh International Conference on Soil Mechanics and Foundation Engineering, Mexico City, Proceedings, Vol.2, pp.45-51. 19. Burkhardt, J. A., and Michie, T. W. (1979): "Submerged Production System--A Final Report." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.801-806. 20. Burns, G. E., and D'Amorim, G. D. (1977): "Bouyant Towers for Phase I Development of Garoupa Field." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.177-184. 21. Button, S. J. (1953): "The Bearing Capacity of Footings on a Two-layer Cohesive Subsoil." Third International Conference on Soil Mechanics and Foundation Engineering, Zurich, Proceedings, Vol.1, pp.332-335. 22. Callis, C; Knox, C; Sutton, D.; and Wiley, S. ( 1979): "An Assessment of Grouting Materials, Placement Methods, and Monitoring Equipment for Offshore Structures." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2755-2764. 196 23. Clausen, Carl J. Frimann. (1976): "The CONDEEP Story." Offshore Soil Mechanics, Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pp.256-270. 24. Clausen, C. J. F.; DiBagio, E.; Duncan, J. M.; and Andersen, K. H. (1975): "Observed Behavior of the Ekofisk Oil Storage Tank Foundation." Seventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.399-413. 25. Davis, E. H., and Booker, J. R. (1973): "The Effect of Increasing Strength with Depth on the Bearing Capacity of Clays." Geotechnique, Vol.23, No.4, pp.551-563. 26. Dean, R. G. (1965): "Stream Function Representation of Nonlinear Ocean Waves." Journal of the Geophysical Research, Vol.70, No.18. 27. Department of Energy. (1974): Guidance on the Design and  Construction of Off-shore Installations, London. 28. Department of the Interior, U.S.G.S. (1979): Approval  Procedure for Installation and Operation of Platforms,  Fixed and Mobile Structures, and Artificial Islands, esp.  PCS, Order 8, Washington. :29. Derrington, J. A. (1977): "Construction of the McAlpine/Sea Tank Gravity Platforms at Ardyne Point, Argyll." Design and Construction of Offshore Structures, London: Institute of Civil Engineers, Proceedings, pp.121-130. 30. Det Norske Veritas. (1977): Rules for the Design Construc  tion and Inspection of Fixed Offshore Structures, Oslo. 31. DiBagio, Elmo, Myrvoll, Frank, and Hansen, Svein Borg. (1976): "Instrumentation of Gravity Platforms for Performance Observations." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.516-527. 32. Duncan, J. M. (1972): "Foundation Study for North Sea Oil Tank." Norwegian Geotechnical Institute, No.92, pp.13-18. 33. Eide, Ove T. (1974): "Marine Soil Mechanics." Norwegian Geotechnical Institute, No.103, pp.1-20. 34. Falkner, C. B., and Franks, N. S. (1978): "Production Techniques from Tension Legged Platforms." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2079-2086. 197 35. Federation Internationale de la Precontrainte. (1977): Recommendations for the Design of Concrete Sea Structures, 3rd Ed., Slough. 36. Finn, L. D., Wardell, J. B., and Loftin, T. D. (1979): "The Guyed Tower as a Platform for Integrated Drilling and Production Operations." Journal of Petroleum Technology, Vol.31, No.12, pp.1531-1537. 37. Finn, W. D. Liam and Lee, Michael K. W. (1978): "Seafloor Stability Under Seismic and Wave Loading." ASCE Spring Conference, Boston, Specialty Session on Soil Mechanics in the Marine Environment. 38. Finn, W. D. Liam, Lee, Kwok W., and Martin, Geoffrey R. (1977): "An Effective Stress Model for Liquefaction." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.103, N0.GT6, pp.517-533. 39. Foss, Ivar. (1974): "Discussion—Settlement Observations of the Ekofisk Oil Storage Tank in the North Sea." British Geotechnical Society Settlement Conference, Cambridge University, United Kingdom, Proceedings, pp.674-676. 40. Franco, Alvaro. (1976): "Offshore Brazil due Concrete Platforms." Oil and Gas Journal, Vol.74, No.18, pp.153-159. 41. Furnes, 0. (1978): "Overview of Offshore Oil Industry with Emphasis on the North Sea." Lectures on Offshore Engineering, Combined Proceedings of a One-Day Conference Plus Eight Weekly Seminars, Aalborg University Center, Aalborg, Denmark. 42. Garrison, C. J. (1977): "Wave Loads on North Sea Gravity Platforms: A Comparison of Theory and Experiment." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.513-524. 43. Garrison, C. J. (1979): "Hydrodynamic Loading of Offshore Structures. Three Dimensional Source Distribution Methods." Numerical Methods in Offshore Engineering, Ed. 0. C. Zien-kiewicz, New York: John Wiley and Sons, Inc., pp.87-140. 44. Garrison, L. E., and Bea, R. G. (1977): "Bottom Stability as a Factor in Platform Siting and Design." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.127-134. 45. George, P. J. (1976): "Notes on Site Investigation with Respect to the Design of Offshore Structures." Offshore Soil Mechanics, Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pp.101-116. 198 46. Gerwick, Ben C, Jr. (1974): "Preparations of Foundations for Concrete Caisson Sea Structures." Sixth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.119-130. 47. Gerwick, B. C, and Hognstad, E. (1973): "Concrete Oil Storage Tank Placed on North Sea Floor." Civil Engineering, ASCE, Vol.43, No.8, pp.81-85. 48. Gumbel, Emil Julius. (1958): Statistics of Extremes, New York: Columbia University Press, 375 p. 49. Hansen, Bent. (1976): "Modes of Failure Under Inclined Eccentric Loads." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.488-500. 50. Hansen, F. J., and Ingerslev, L. C. F. (1977): "The Case for a Hybrid." Design and Construction of Offshore Structures, London: Institute of Civil Engineers, Proceedings, pp.135-141. 51. Hansen, J. B. (1961): "A General Theory for Bearing Capacity." The Danish Geotechnical Institute, Copenhagen, Bulletin No.11, pp.38-46. 52. Hansen, J. B. (1970): "A Revised and Extended Formula for Bearing Capacity." The Danish Geotechnical Institute, Copenhagen, Bulletin No.28, pp.5-11. 53. Haring, R. E., and Heideman, J.C. (1978): "Gulf of Mexico Rare Wave Return Periods." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.1537-1550. 54. Heijnen, W. J. (1981): "The Use of Physical Models in Solv ing Offshore Geotechnical Problems." Offshore Structures:  The Use of Physical Models in Their Design, Eds. G. S. T. Armer and F. K. Garas, Lancaster: The Construction Press, pp.263-272. 55. Henkel, D. J. (1970): "The Role of Waves in Causing Sub marine Landslides." Geotechnique, Vol.20, No.1, pp.75-80. 56. Hitchings, Gordon A., Bradshaw, Heath, and Labiosa, Thomas D. (1976): "The Planning and Execution of Offshore Site Investigations for a North Sea Gravity Platform." Eighth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.61-74. 57. H0eg, Kaare. (1976): "Foundation Engineering for Fixed Offshore Structures." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.39-69. 199 58. Hogben, N.; Miller, B. L.; Searle, J. W.; and Ward, G. (1977): "Estimation of Fluid Loading on Offshore Structures." Institute of Civil Engineers, London, Proceedings, Vol.63, Part 2, pp.515-562. 59. Hove, Knut and Foss, Ivar. (1974): "Quality Assurance for Offshore Concrete Gravity Structures." Sixth Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.829-842. 60. Huntemann, J. E., Anastasio, F. L., Jr., and Deshazar, W. A. (1979): "Concrete Gravity Platform in Shallow Offshore Louisiana Water." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.1003-1008. 61. Isaacson, M. de St. Q. (1980): "Wave Forces in the Diffraction Regime--A Review." Coastal/Ocean Engineering Report, Department of Civil Engineering, University of British Columbia. 62. Isaacson, M. de St. Q. (1981): Private Communication. 63. Janbu, Nilmar. (1973): "Slope Stability Computations." Embankment-Dam Engineering, Casagrande Volume, Eds. R. C. Hirschfield and S. J. Poulos. New York: John Wiley and Sons, Inc., pp.45-86. 64. Janbu, Nilmar, Grande, Lars, and Eggareide, Rare. (1976): "Effective Stress Stability Analysis for Gravity Structures." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.449-466. 65. de Jong, J. J. A., and Bruce, J. C. (1978): "Design and Construction of a Caisson Retained Island Drilling Platform for the Beaufort Sea." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2111-2120. 66. Kinsman, B. (1965): Wind Waves, their Generation and Propa  gation on the Ocean Surface, Englewood Cliffs, N. J.: Prentice-Hall, 676 p. 67. Kjekstad, O., and Lunne, T. (1979): "Soil Parameters Used for Design of Gravity Platforms in the North Sea." Second International Conference on the Behavior of Offshore Structures, London, Proceedings, Vol.1, pp.175-192. 68. Kliewer, Raymond M., and Forbes, Graeme S. (1980): "A Fixed Platform Providing an Integrated Deck on a Multiple Leg Ice Resistant Substructure." Twelfth Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.315-324. 200 69. Klitz, J. Kenneth. (1980): North Sea Oil: Resource Require  ments for Development of the U.K. Sector, Oxford: Pergammon Press, 260 p. 70. Korteweg, D. J., and De Vries, G. (1895): "On the Change of Form of Long Waves Advancing in a Rectangular Channel, and on a New Type of Long Stationary Wave." Philosophical Magazine, 5th Series, pp.422-443. 71. Lalli, D. (1975): "Discussion." Off-shore Structures, London: Institute of Civil Engineers, Proceedings, pp.92-93. 72. Lalli, D. (1977): "Design, Construction and Installation of the Loango Steel Gravity Platforms." Design and Construction of Offshore Structures, London: Institute of Civil Engineers, Proceedings, pp.31-38. 73. Lambe, T. W. (1967): "Stress Path Method." Journal of the Soil Mechanics and Foundation Division, ASCE, Proceedings, Vol.93, NO.SM6, pp.309-317. 74. Lauritzsen, Rolf, and Schjetne, Knut. (1976): "Stability Calculations for Offshore Gravity Structures." Eighth Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.76-82. 75. Lee, Kenneth L. (1976): "Predicted and Measured Pore Pressures in the Ekofisk Tank Foundation." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.2, pp.384-398. 76. Lee, K. L., and Albaisa, A. (1974): "Earthquake Induced Settlements in Saturated Sands." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.100, NO.GT4, pp.387-406. 77. Lee, Kenneth L., and Focht, John A., Jr. (1975a): "Lique faction Potential at Ekofisk Tank in North Sea." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.101, NO.GT1, pp.1-18. 78. Lee, Kenneth L., and Focht, John A., Jr. (1975b): "Cyclic Testing of Soil for Ocean Wave Loading Problems." Seventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.1, pp.343-354. 79. Lee, K. W., and Finn, W. D. Liam. (1978): "STESL - A Compu ter Program for Static and Earthquake Analyses of Underwater Slopes." Soil Dynamics Group, University of British Columbia. 201 80. Low, E. (1975): "Foundations for Gravity Type Off-shore Drilling/Production Platforms." Off-shore Structures, London: Institute of Civil Engineers, Proceedings, pp.27-35. 81. Lundgren, H., and Mortensen, K. (1953): "Determination by the Theory of Plasticity of the Bearing Capacity of Continuous Footings on Sand." Third International Conference on Soil Mechanics and Foundation Engineering, Zurich, Proceedings, Vol.1, pp.409-412. 82. MacCamy, R. C, and Fuchs, R. A. (1954): "Wave Forces on Piles: A Diffraction Theory." U. S. Army Corps of Engineers, Beach Erosion Board, Technical Memo No.69, Washington. 83. McClelland, B. (1977): "Geotechnical Problems in Ocean Engineering." Ninth International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Proceedings, Panel Discussion, Specialty Session 7, pp.513-523. 84. McCormick, M. E. (1973): Ocean Engineering Wave Mechanics, New York: John Wiley and Sons, Inc., 179 p. 85. McPhee, W. S., and Reeves, S. J. (1975): "Drilling and Production Platforms for the Oil Industry." Off-shore Structures, London: Institute of Civil Engineers, Proceedings, pp.189-196. 86. Maidl, B., and Schiller, W. (1979): "Testing and Exper iences of Different Scour Protection Technologies in the North Sea." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.981-988. 87. Marion, H. A. (1974): "Ekofisk Storage Tank." Symposium on Ocean Engineering, Teddington: The Royal Institute of Naval Architects, Proceedings, pp.83-90. 88. Martin, M. R., and Shaw, L. K. (1974): "A Decade of North Sea Platforms." Symposium on Ocean Engineering, Teddington: The Royal Institute of Naval Architects, Proceedings, pp.73-82. 89. Meyerhof, G. G. (1953): "The Bearing Capacity of Foun dations Under Eccentric and Inclined Loads." Third International Conference on Soil Mechanics and Foundation Engineering, Zurich, Proceedings, Vol.1, pp.440-445. 90. Meyerhof, G. G. (1963): "Some Recent Research on the Bearing Capacity of Foundations." Canadian Geotechnical Journal, Vol.1, No.1, pp.16-26. 202 91. Meyerhof, G. G. (1974): "Ultimate Bearing Capacity of Footings on Sand Layer Overlying Clay." Canadian Geotechnical Journal, Vol.1'1, No.2, pp.223-229. 92. Moinard, M. (1979): "Deep Sea Production Use of Articulated Columns." Symposium on New Technologies for Exploration and Exploitation of Oil and Gas Resources, London: Graham and Trotman Limited, Proceedings, Vol.2, pp.1010-1033. 93. Morgenstern, N. R., and Price, V. E. (1965): "The Analysis of the Stability of General Slip Surfaces." Geotechnique, Vol.15, No.1, pp.79-93. 94. Morison, J. R. (1950): "The Force Exerted by Surface Waves on Piles." Petroleum Transactions, TP 2846, pp.149-154. 95. Morrison, Allen. (1980a): "Cognac: World's Tallest Oil Platform." Civil Engineering, «Vol.50, No.6, pp.55-57. 96. Morrison, Allen. (1980b): "U.S. Offshore Future: New Techniques, Deeper Water, But Where's the Oil?." Civil Engineering, Vol.50, No.6, pp.58-59. 97. Murff, J. D., and Miller, T. W. (1977): "Stability of Off shore Gravity Structure Foundations by the Upper Bound Method." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.147-154. 98. Offshore Europe, 2nd Ed. (1974): Bucks, England: Scientific Surveys (Offshore) Ltd. 99. Offshore Soil Mechanic's. (1976): Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pg.431. 100. Penzien, Joseph. (1976): "Structural Dynamics of Fixed Off shore Structures." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.581-592. 101. Prandtl, L. (1921): "Uber die Eindrigung Festigkeit (Harte) Plastischer Baustoffe und die Festigkeit von Schneiden." Zietschrift Fur Angewandte Mathematic und Mechanic, Vol.1, pp.15-20. 102. Prev0st, Jean. H., Cuny, Bernard., and Scott, Ronald F. (1981a): "Offshore Gravity Structures: Centrifugal Modelling." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.107, No.GT2, pp.125-141. 103. Prev0st, Jean. H., Cuny, Bernard., Hughes, Thomas. J. R., and Scott, Ronald F. (1981b): "Offshore Gravity Structures: Analysis." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.107, No.GT2, pp.143-165. 203 104. Rahman, M. S., Seed, H. B., and Booker, J. R. (1977): "Pore Pressure Generation Under Offshore Gravity Structures." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.103, No.GT12, pp.1419-1436. 105. Ranney, William M. (1979): Offshore Oil Technology—Recent  Developments. Park Ridge, N. J.: Noyes Data Corporation, 399 p. 106. Reddy, A. S., and Srinivasan, R. J. (1967): "Bearing Capacity of Footings on Layered Clays." Journal of the Soil Mechanics and Foundation Division, ASCE, Proceedings, Vol.93, NO.SM2, pp.83-99. 107. R0ren, E. M. Q., and Fames, 0. (1976): "Behavior of Structures and Structural Design." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.70-112. 108. Rowe, P. W. (1975): "Displacement and Failure Modes of Model Offshore Gravity Platforms Founded on Clay." Offshore Europe Conference '75, Aberdeen, Scotland, pp.218.1-16. 109. Rowe, P. W., Craig, W. H., and Proctor, D. C. (1976): "Model Studies of Offshore Gravity Structures Founded on Clay." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.439-448. 110. de Ruiter, J. (1976): "North Sea Site Investigations—The Role of the Geotechnical Consultant." Offshore Soil Mechanics, Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pp.61-78. 111. Sangrey, D. A., Henkel, D. J., and Esrig, M. I. (1969): "The Effective Stress Response of Saturated Clay Soil to Repeated Loading." Canadian Geotechnical Journal, Vol.6, No.3, pp.241-252. 112. Sarma, S. K. (1973): "Stability Analysis of Embankments and Slopes." Geotechnique, Vol.23, No.3, pp.423-433. 113. Sarpkaya, T., and Isaacson, M. de St. Q. (1981): Mechanics  of Wave Forces on Offshore Structures, New York: Van Nos-trand Reinhold, 651 p~. 114. Schjetne, Knut. (1976): "Foundation Engineering for Gravity Structures in the- North Sea." Norwegian Geotechnical Institute, No.113, pp.23-33. 115. Seed, H. Bolton, Martin, Phillipe P., and Lysmer, John. (1976): "Pore Water Pressure Changes During Soil Liquefaction." Journal of the Geotechnical Engineering Division, ASCE, Proceedings, Vol.102, No.GT4, pp.323-346. 204 116. Seines, P. B. (1981): "Offshore Earthquake Technology— First Part." ASCE Fall Conference, St. Louis, Proceedings, pp.81 7-823. 117. Shore Protection Manual, 3rd Ed. (1977): 3 Vol., U. S. Army Coastal Engineering Research Center, Washington: United States Government Printing Office. 118. Sjoerdsma, G. W. (1975a): "General Appraisal of Off-shore Gravity Structures." Off-shore Structures, London: Institute of Civil Engineers, Proceedings, pp.61-66. 119. Sjoerdsma, G. W. (1975b): "Discussion - Session G." Off shore Structures, London: Institute of Civil Engineers, Proceedings, pg.l9B. 120. Skempton, A. W. (1954): "The Pore Pressure Coefficients A and B." Geotechnique, Vol.4, No.4, .pp.143-147. 121. Skempton, A. W., and Bjerrum, L. (1957): "A Contribution to the Settlement Analysis of Foundations on Clay." Geotechnique, Vol.7, pp.168-178. 122. Stenning, D. G., and Schumann, C. G. (1979): "Arctic Production Monocone." Eleventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.2357-2366. 123. Steven, Robert R. (1981a): "Some Offshore Feats Set World Standards, Others Set World Records." Offshore, Vol.41, No.7, pp.70-71. 124. Steven, Robert R. (1981b): "North Sea: Special Report, Future Bright for Norway Production." Offshore, Vol.41, No.2, pp.98-104. 125. Steven, Robert R. (1981c): "North Sea: Special Report, U.K. Output Slackens Due to Problems." Offshore, Vol.41, No.2, pp.89-96. 126. Stokes, G. C. (1880): "On the Theory of Oscillatory Waves." Mathematical and Physical Papers, Vol.1, Cambridge: Cambridge University Press. 127. Stubbs, S. B. (1975): "Seabed Foundation Considerations for Gravity Structures." Off-shore Structures, London: Institute of Civil Engineers, Proceedings, pp.67-74. 128. Taylor, K. L. (1976): "Practices Adopted in North Sea Investigations." Offshore Soil Mechanics, Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pp.1 17-130. 129. Terzaghi, Karl. (1943): Theoretical Soil Mechanics, New York: John Wiley and Sons, Inc., 510 p. 205 130. Terzaghi, Karl, and Peck, Ralph B. (1948): Soil Mechanics  in Engineering Practice, Second Edition 1966, New York: John Wiley and Sons, Inc., 729 p. 131. van Eekelen, H. A. M., and Potts, D. M. (1978): "The Beha vior of Drammen Clay Under Cyclic Loading." Geotechnique, Vol.28, No.2, pp.173-196. 132. Vaughan, P. R.; Davachi, M. M.; El Ghamrawy, M. K.; Hamza, M. M.; and Hight, D. W. (1976): "Stability Analysis of Large Gravity Structures." International Conference on the Behavior of Offshore Structures, Oslo, Proceedings, Vol.1, pp.467-487. 133. Vesic, Aleksandar S. (1975): "Bearing Capacity of Shallow Foundations." Foundation Engineering Handbook, Eds. Hans F. Winterkorn and Hsai-Yang Fang, New York: Van Nostrand Reinhold Company, pp.121-147. 134. Waagaard, Knut. (1977): "Fatigue of Offshore Concrete Structures—Design and Experimental Investigations." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.4, pp.341-350. 135. Ward, E. G., Evans, D. J., and Pompa, J. A. (1977): "Extreme Wave Heights Along the Atlantic Coast of the United States." Ninth Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.315-324. 136. Watt, B. J. (1976): "Gravity Structures—Installation and Other Problems." Offshore Soil Mechanics, Eds. Phillip George and David Wood, Cambridge: Cambridge University Engineering Department, pp.286-305. 137. Watt, B. J; Boaz, I. B.; Ruhl, J. A.; Shipley, S. A.; and Ghose, A. (1978): "Earthquake Survivability of Concrete Platforms." Tenth Annual Offshore Technology Conference, Houston, Proceedings, Vol.2, pp.957-975. 138. Watt, B. J. (1979): "Basic Structural Systems—A Review of Their Design and Analysis Requirements." Numerical Methods  in Offshore Engineering, Ed. 0. C. Zienkiewicz, New York: John Wiley and Sons, Inc., pp.1-42. 139. Werenskiold, K. (1977): "Maritime Operations Relative to Construction of Offshore Structures." Design and Construction of Offshore Structures, London: Institute of Civil Engineers, Proceedings, pp.97-105. 140. Yamaguchi, H., and Terashi, M. (1971): "Ultimate Bearing Capacity of the Multi-Layered Ground, Fourth Asian Regional Conference on Soil Mechanics and Foundation Engineering, Proceedings, Vol.1, pp.97-105. 206 141. Young, Alan G., Kraft, Leland M., Jr., and Focht, John A., Jr. (1975): "Geotechnical Considerations in Foundation Design of Offshore Gravity Structures." Seventh Annual Offshore Technology Conference, Houston, Proceedings, Vol.3, pp.367-386. 142. Zienkiewicz, 0. C; Norris, V. A.; Winnicki, L. A.; Naylor, D. J.; and Lewis, R. W. (1979): "A Unified Approach to the Soil Mechanics Problems of Offshore Foundations." Numerical  Methods in Offshore Engineering, Ed. 0. C. Zienkiewicz, New York: John Wiley and Sons, Inc., pp.361-411. 

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