Springer Series in Reliability Engineering Series Editor Hoang Pham For furthervolumes: http://www.springer.com/series/6917 Halil Karadeniz Stochastic Analysis of Offshore Steel Structures An Analytical Appraisal 123 Halil Karadeniz Vedat Togan Faculty ofCivil Engineering Department of CivilEngineering and Geosciences Karadeniz Technical University Delft Universityof Technology Trabzon Delft Turkey The Netherlands MehmetPolat Saka Department of EngineeringSciences Middle EastTechnical University Ankara Turkey Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com/ ISSN 1614-7839 ISBN 978-1-84996-189-9 ISBN 978-1-84996-190-5 (eBook) DOI 10.1007/978-1-84996-190-5 SpringerLondonHeidelbergNewYorkDordrecht LibraryofCongressControlNumber:2012940560 (cid:2)Springer-VerlagLondon2013 Content of Section 1.3 in Chapter one adapted from OMAE-2010, Paper No. OMAE2010-20971, Shanghai,ChinaforuseherewithkindpermissionfromASME Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyright ClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Increasing oil consumption in the world and scarcity of land-oil resources due to political and economical reasons has caused offshore oil exploration and pro- duction to become a growing investigation field in the past six decades. The analysis of structures to use energy deposits and other recourses, or for other purposes, in ocean environments requires a special consideration since environ- mental and loading conditions offshore are very complicated and contain large uncertainties. Offshore structures are continuously subjected to random ocean waves producing stochastic loads that cause mainly fatigue failure in structural components. In tectonic offshore environments, structures are also subjected to earthquakeandearthquake-inducedhydrodynamicloadings,whichareconsidered to be important as they can cause structural collapse in a short time. Since the ocean environment and random wave phenomenon are highly uncertain, a prob- abilistic structural analysis needs to be carried out essentially. This requires the knowledge of probability theory and applied probability models, which construct the basis of reliability analysis. For the fatigue damage and fatigue reliability analysis, theoretical knowledge is combined with experimental information to predict correct results. Under these complexities, offshore structures should be designedtogiveoptimalperformancewithinthesafemargin.Thiscanbeachieved by applying methods of the reliability-based design optimization. This book aims to cover difficult issues encountered in the analysis of offshore steel structures underrandomwaveandearthquakeloadings.Itprovidesbroadanalyticaltoolsfor advancedanalysisofoffshorestructures.Itservesasastand-alonereferencebook for design engineers, researchers, graduate and post graduate students, and for higher education in the field of offshore structural engineering. A corresponding computer program, SAPOS, is also attached to the book via Springer Extras. The program is run at the responsibility of the users. The publisher and the authors of this book are not responsible in any form that may arise from the use of this program.ThebookcontainssevenindependentchaptersofwhichChaps.1–6have been writtenby Prof. Dr. H.Karadeniz and Chap.7 has been writtenby Prof. Dr. M. P. Saka and Dr. V. Togan. Each chapter is devoted to handle the specific subject as briefly summarized in the following paragraphs. v vi Preface Chapter 1 explains the mechanics of space frame structures and presents necessary formulations for the finite element analysis of space frames since offshoresteelstructuresareconstitutedfromspaceframes.Inthischapter,the3D Timoshenkobeamtheoryispresentedwithemphasisonformulationofpartlyand eccentrically connected members, as well as formulation of a soil-beam interface element to account for foundation effects in the analysis. This chapter highlights alsotheeigenvalueproblemrelatedtothedynamicanalysisofoffshorestructures. Chapter 2 presents basic information and essential formulation of random vibration and stochastic analysis that are needed in offshore structural analysis. Having presented commonly used probability models, stochastic processes are summarized. Then, the spectral analysis, transfer functions, and the crossing analysis are highlighted. Chapter 3 is devoted to ocean wave mechanics and wave forces. Having summarized wave theories in general, the Airy wave theory is explained in detail andformulationfordeep-waterconditionsispresented.Thechapterfirstdescribes stochastic ocean waves, transfer functions, commonly used short-term sea spectra with directional distribution, wave-current interaction phenomenon, and probabi- listic description of sea states in the long term. Then, attention is paid to the calculation of wave and member consistent forces with emphasis on added mass and hydrodynamic damping concept. Chapter 4 presents spectral analysis of offshore structures under wave and earthquake actions. After describing the problem of spectral analysis and giving general information, formulation ofdynamic analysis ofoffshore structures in the frequencydomain,transferfunctionsofwaveandearthquakeforcesarepresented. Then, concentration is focused on response transfer functions of wave and earthquakeforces,followedbyformulationofthehydrodynamicandinertiaforces produced by earthquakes and their combination. This chapter also explains cal- culation of response spectra of offshore structures under stochastic wave and earthquake forces with earthquake ground motion and its spectral representation including non-uniform earthquake ground motions. Finally, the calculation of response statistical quantities is presented with illustrative examples. Chapter5isdevotedtothefatiguephenomenoninstructures.First,thefatigue process,sourceoffatigue,andmodelingoffatigueissummarizedingeneral.Then, the calculation of fatigue damages is explained by using the fracture mechanics andS–Ncurve approaches.Thecumulativedamage iscalculatedaccordingtothe Palmgren–Miner’s rule. For non-narrow banded stress processes, the fatigue damage is estimated using probability distribution of random stress ranges obtained fromthe rain-flow cycle counting algorithm. Finally, calculationof total spectral fatigue damage is presented for a given lifetime by using a multilinear S–N fatigue model. Chapter 6 presents reliability analysis of offshore structures. Having explained uncertaintiesingeneralandgiveninformationaboutthereliabilitymethods,basic definitions and structural reliability methods are presented in more detail. This is followed by the calculation of the reliability index b by the FORM and SORM methods for nonlinear failure functions and non-normal correlated design Preface vii variables.Thecalculationalgorithmsandflowdiagramsaregiven.Then,afterthe Level III reliability methods are outlined, the inverse reliability method and its calculation algorithm are presented. In the final sections, uncertainties in spectral stresses and fatigue damages of offshore structures are explained in a reduced uncertainty space which follows the fatigue reliability calculation and its algorithms. Chapter 7 is devoted to optimization techniques that are widely applied to determine the optimum solution of structural design problems. First, this chapter introduces the mathematical formulation of optimization problems and their solution techniques among which sequential programming technique and differ- ential evolution algorithm are briefly explained. Second, it presents the mathe- matical formulation of reliability-based design optimization problems in the uncertainty space regarding load, resistance, and structural response. Then, it summarizes the available solution techniques and explains methods of sensitivity analysis of the reliability-based design optimization of offshore structures. The first author of this book, Prof. Dr. H. Karadeniz, acknowledges Delft University of Technology (TUDelft), The Netherlands, for giving him the opportunitytocarryoutresearchworkandtousetheirfacilitiesforalongperiod, from 1978 until his retirement at the end of 2010. Especially, he acknowledges Prof. A. L. Bouma of the structural mechanics group in the Civil Engineering Department for accepting him into his research group in 1978. He thanks all the staff of the Structural Mechanics Group in the Civil Engineering Department at TUDelftforthememorablesocialatmospherethattheycreated.Hisspecialthanks and appreciation are for Prof. Ton Vrouwenvelder of the Civil Engineering Department, TUDelft, for his continuous, stimulating, and invaluable discussions during his stay at TUDelft. He appreciates and acknowledges all his colleagues, friends, and research collaborators from all over the world for their useful dis- cussions and inspiring suggestions. Finally, all the authors of this book thank Springer-Verlag for publishing this book and making it available for interested readers in worldwide. July 2012 H. Karadeniz M. P. Saka V. Togan Contents 1 Finite Element Analysis of Space Frame Structures . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Formulation of a 3D Timoshenko Beam Element. . . . . . . . . . . 2 1.2.1 Curvatures of 3D Beams Under Pure Bending. . . . . . . 3 1.2.2 Equilibrium Equations of 3D Beam Elements . . . . . . . 5 1.2.3 Contributions of Transverse Shear Forces on the Elastic Curve. . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4 Deformation of a Point on a Cross-Section of 3D Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.5 Rotation Matrix of a Point on a Cross-Section of 3D Beams and Deformation for Small Rotations. . . 10 1.2.6 Strains and Stresses at a Location on a Cross-Section of a 3D Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.7 Calculation of Forces and Moments of 3D Beams. . . . 14 1.2.8 Differential Equations of the 3D Beam Element . . . . . 16 1.2.9 Solution of Differential Equations of the Elastic Curve and Shape Functions of the 3D Beam Element . . . . . . 18 1.2.10 Total Potential Energy, Stiffness Matrix, and Static Equilibrium Equation . . . . . . . . . . . . . . . . 23 1.2.11 Total Kinetic Energy, Mass Matrix, Damping Matrix, and Dynamic Equilibrium Equation . . . . . . . . 30 1.2.12 Coordinate Systems and Transformations . . . . . . . . . . 37 1.2.13 Transformations of Element Stiffness Matrix, Consistent Load Vector, and Mass Matrix. . . . . . . . . . 41 1.3 Formulation of Member Releases and Partly Connected Members. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.3.1 Representation of a Partly Connected Beam Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ix x Contents 1.3.2 Formulation of Stiffness Matrix, Consistent Load Vector, and Mass Matrix of a Spring-Beam Element . . . . . . . . . . . . . . . . . . . . 46 1.3.3 Calculation of the Connectivity Matrix. . . . . . . . . . . . 53 1.3.4 Member Releases in a Different Coordinate System. . . 63 1.4 Formulation of Eccentrically Connected Members. . . . . . . . . . 65 1.5 An Interface Beam Element for the Soil–Structure Interaction and Deformation of Soil Under R-Wave Propagation. . . . . . . . 69 1.5.1 Modeling of Soil Medium and Calculation of Interface Loadings. . . . . . . . . . . . . . . . . . . . . . . . 71 1.5.2 Formulation of Interface Element for Soil–Beam Interactions . . . . . . . . . . . . . . . . . . . . 75 1.5.3 Ground Deformation Under R-Wave Propagation and Calculation of the Exerted Force Vector. . . . . . . . 79 1.6 Calculation of Natural Frequencies and Mode Shapes, Eigenvalue Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.6.1 Eigenvalue Solution. . . . . . . . . . . . . . . . . . . . . . . . . 89 1.6.2 Eigenvalue Solution of Deteriorated Structures . . . . . . 94 1.7 Dynamic Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 97 1.7.1 Time-Domain Solution. . . . . . . . . . . . . . . . . . . . . . . 98 1.7.2 Frequency Domain Solution . . . . . . . . . . . . . . . . . . . 105 1.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.8.1 Example of a Portal Frame. . . . . . . . . . . . . . . . . . . . 111 1.8.2 Example of 2D Offshore Jacket Structure. . . . . . . . . . 114 1.8.3 A Simple Beam for Exercise. . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 Introduction to Random Vibration and Stochastic Analysis. . . . . . 121 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.2 Probability, Random Variables, Stochastic Processes, Probability Distribution and Density Functions . . . . . . . . . . . . 122 2.2.1 Probability Measure. . . . . . . . . . . . . . . . . . . . . . . . . 123 2.2.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.2.3 Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . 126 2.2.4 Probability Distribution and Density Functions . . . . . . 127 2.3 Mean Values, Probability Moments, and Variances of Random Variables and Random Functions . . . . . . . . . . . . . 134 2.3.1 Functions of Random Variables. . . . . . . . . . . . . . . . . 136 2.3.2 Some Useful Probability Distributions . . . . . . . . . . . . 141 2.4 Random Processes, Ensemble Averages, Expected Values, Stationary and Ergodic Processes. . . . . . . . . . . . . . . . . . . . . . 147 2.4.1 Ensemble Averages and Expected Values. . . . . . . . . . 148 2.4.2 Stationary and Ergodic Processes. . . . . . . . . . . . . . . . 150 2.4.3 Differentiation of Stochastic Processes. . . . . . . . . . . . 153